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[ [ "3DFS: Deformable Dense Depth Fusion and Segmentation for Object\n Reconstruction from a Handheld Camera" ], [ "Abstract We propose an approach for 3D reconstruction and segmentation of a single object placed on a flat surface from an input video.", "Our approach is to perform dense depth map estimation for multiple views using a proposed objective function that preserves detail.", "The resulting depth maps are then fused using a proposed implicit surface function that is robust to estimation error, producing a smooth surface reconstruction of the entire scene.", "Finally, the object is segmented from the remaining scene using a proposed 2D-3D segmentation that incorporates image and depth cues with priors and regularization over the 3D volume and 2D segmentations.", "We evaluate 3D reconstructions qualitatively on our Object-Videos dataset, comparing to fusion, multiview stereo, and segmentation baselines.", "We also quantitatively evaluate the dense depth estimation using the RGBD Scenes V2 dataset [Henry et al.", "2013] and the segmentation using keyframe annotations of the Object-Videos dataset." ], [ "Introduction", "Our aim is to create a 3D model of a single object recorded by a handheld mobile phone camera.", "We assume only that the object is placed on a flat surface and that the object is approximately centered.", "The ability to easily and accurately create 3D models from handheld cameras has broad applications including virtual reality and 3D printing.", "But existing methods are unable to consistently produce 3D models of objects with specularities and irregular shapes without user interaction or controlled capture settings.", "3D reconstruction of scenes from image sequences is a well studied problem in computer vision.", "The underlying idea behind most existing work is that multiple pixel measurements of a point in the scene can be used to triangulate its 3D position.", "Therefore, reconstruction accuracy hinges on the ability to track a pixel with sub-pixel accuracy.", "Tracking and matching algorithms, however, assume brightness constancy which breaks down for specular surfaces.", "Tracking algorithms also falter in textureless regions where neighboring pixels have similar intensities.", "MonoFusion [17] and MobileFusion [16] demonstrate 3D reconstruction by explicitly modeling surfaces as depth maps and then performing volumetric fusion.", "However, to achieve real-time performance such methods compromise on accuracy of the depth maps by relying on stereo matching between the live frame and the last key frame.", "Similarly, volumetric fusion is performed by a weighted average of implicit surface representation of the depth surfaces, such as Truncated Signed Distance Fields (TSDF), computed in different views.", "Success of such volumetric fusion approaches depends on the accuracy of depth maps.", "While these techniques work well for depth maps acquired using active sensors like Kinect, they are not robust to localized but often large errors in depth maps that are common with multi-view stereo techniques.", "Another important aspect of single object 3D reconstructions is segmentation of the object from the scene.", "To be useful for 3D printing, the 3D volume occupied by the object needs to be identified.", "While TSDF gives an estimate of empty regions in space, the cues that distinguish the object from other surfaces come from scene priors and the images.", "Some approaches iteratively solve for 2D and 3D segmentations [19] or jointly segment pixels and sparse point clouds [21], but none, to the best of our knowledge, perform joint inference over pixels and a dense grid of voxels, which we find to be important for obtaining accurate 3D models.", "In this work, we propose to improve the accuracy and robustness of video based multi-view single object 3D reconstruction and segmentation systems by improving surface modeling, volumetric fusion, and segmentation.", "Our system computes accurate depth maps by posing dense per-pixel depth estimation as an optimization problem which incorporates multi-view stereo cues, sparse point cloud reconstruction from a VSLAM system, and a surface smoothness prior in the form of a rotation invariant bending energy.", "For 3D reconstruction, we reformulate volumetric fusion of depth maps by getting rid of the truncation in TSDF and using a soft-max based signed distance function (SDF).", "Our fusion approach has the benefit of being more robust to errors in depth maps and also produces smoother surfaces.", "This technique however leads to a shift in the zero-crossing of the function field, which we remedy by introducing a novel volume field deformation using a sparse surface point cloud such as that provided by patch-based multi-view stereo [7].", "For segmentation, we perform joint inference over pixel and voxel labels in a graph cut optimization framework.", "Pixels and voxels impose complementary constraints on the reconstruction.", "Pixels model the object color which helps distinguish the object surface from other background surfaces and provide cues to surface discontinuities through contour edges.", "Voxels impose constraints on empty regions, enforce continuity of objects in 3D space and incorporate our scene prior.", "In addition, surface voxels enforce consistency of pixel labeling across multiple images.", "We evaluate each of these components to demonstrate good performance even in the presence of specularities and textureless surfaces.", "To summarize our contributions: We propose a fully automated approach to produce a 3D mesh of a single object placed on a flat surface.", "We propose a method to robustly estimate depth surfaces from multi-view stereo cues and a sparse point cloud (optional) which is regularized by a rotation invariant second order bending energy.", "We demonstrate its performance on our Object-Videos dataset as well as RGBD Scenes v2 dataset and compare against other forms of regularization.", "We reformulate volumetric fusion by using a soft-max instead of the truncation and weighted averaging for computing the TSDF proposed in [14].", "To correct the bias in zero-crossing we perform a smooth function field deformation using a surface point cloud.", "Our approach is more robust to errors in depth maps and produces smooth surface meshes.", "We formulate joint 2D image segmentation and volumetric 3D reconstruction as the task of assigning discrete labels $\\lbrace object,background\\rbrace $ to every pixel and $\\lbrace object,background,empty\\rbrace $ to every voxel $v$ .The discrete optimization is solved using graph cut with $\\alpha $ -expansion procedure under constraints imposed by pixels, voxels and scene priors.", "We evaluate our entire system through pixel segmentation accuracy on Object-Videos dataset which is a video dataset of 12 objects recorded by a mobile phone camera.", "We provide ground-truth segmentation masks for selected frames to encourage advancement of state-of-the-art on this task." ], [ "Related Work", "We can group relevant literature into two broad areas: 3D reconstruction and multi-view segmentation.", "3D Reconstruction: Most of the existing work in this area focuses on reconstruction of an entire scene.", "Different approaches cater to different reconstruction resolution requirements and size of the scenes.", "Single object reconstruction mostly falls within the purview of literature that deals with small sized indoor scenes.", "MonoFusion [17] and MobileFusion [16] use stereo matching between the live frame and last selected key-frame to generate depth maps.", "The final surface reconstruction is computed by extracting zero-level set of the Signed Distance Field (SDF) obtained by weighted averaging of TSDFs for different views.", "This approach to volumetric fusion, popularized by KinectFusion [14], has demonstrated good performance for fusing accurate depth measurements from active sensors like the Kinect.", "However, this method has two drawbacks.", "First, due to weighted averaging, the TSDFs are forced to be asymmetric to avoid changing the zero-level set.", "This in turn hinders their ability to collect evidence of occupancy for voxels behind the visible surfaces.", "Second, this approach relies on accurate depth surface estimates which, while generally true for active sensors operating in indoor scenes, does not generalize to stereo based methods in presence of specular reflection and texture-less surfaces.", "To counter these problems we propose an alternative scheme that relies on maximum signed distance to the visible surfaces.", "For robustness we use a soft-max instead of a hard maximum.", "This is followed by zero-crossing correction using a smooth deformation field generated using a sparse point cloud provided by PMVS [7].", "Our joint 2D-3D segmentation also reduces dependence on the depth maps by utilizing multiple cues and hence improves the overall robustness of the system.", "DTAM [15] proposes a novel and robust alternative to pairwise stereo for computing dense per-pixel depth maps.", "They pose the problem of computing depth at evey pixel as a two step iterative optimization problem, where the first step ensures consistency of depth estimates with photometric evidence integrated over multiple frames (cost volume) and the second step provides a first order spatial regularization.", "Inspired by DTAM, we also pose depth surface computation as a two step iterative optimization problem but with some important improvements and simplifications.", "The major differences are: (1) relaxation of frontal-planar assumption imposed by the first order regularization; (2) removal of Huber loss on smoothness term and appropriate compensation through spatially varying weights that turns the optimization problem in the second step to linear least squares; (3) replacement of pixel based photometric error by patch based Zero-mean Normalized Cross Correlation (ZNCC) which is more robust to brightness changes, while computing the cost volume.", "Another approach to dense 3D reconstruction is to begin with a sparse point cloud, increase density by propagating measurements to nearby points using techniques like PMVS [7] and then fit a mesh to this semi-dense reconstruction [9].", "However to expand the point cloud, techniques like PMVS perform only local operations that may not produce a globally consistent result.", "It is also difficult to enforce surface regularization.", "Camera pose estimation is an integral component of any of the above mentioned methods.", "Direct monocular slam approaches like LSD-SLAM [6] and DTAM, which directly minimize photometric error to register live frame with the last selected key-frame have been shown to be more robust than feature tracking based approaches such as PTAM [10].", "LSD-SLAM also provides semi-dense depth estimates with inverse depth variance which expresses belief about the accuracy of the estimates.", "Figure: System overview: First, LSD-SLAM is used to estimate camera poses and to select key-frames from the input video.", "Then we compute dense depth maps which are volumetrically fused using a robust 2 stage process involving computation of a softmax based signed distance function followed by deformation using PMVS point cloud.", "Finally, a joint 2D-3D segmentation is performed to obtain a 3D reconstruction with consistent 2D segmentations.Multi-view Segmentation: The dominant approach for multi-view segmentation involves finding optimal labels for some subset of {pixels, superpixels, 3D points, voxels} that minimizes an energy function that encodes task specific priors and evidence from data.", "The work most closely related to ours is [19], where 3D reconstruction is posed as a volumetric Graph Cut [2] optimization problem defined over a 6-connected grid of voxels.", "Foreground and background color distributions were modeled from images and the unary term for a voxel was defined in terms of the average posterior probabilities of the pixels that the voxel projected to.", "The foreground voxels were then projected onto the images to get segmentation masks and the color model was updated.", "Similarly, [4] propose an approach for segmenting an object from a video by performing graph cut over superpixels and sparse 3D points.", "They have explicit edges between 3D points and superpixels to ensure multi-view coherence, and edges between superpixels across frames, which are related by optical flow, to model temporal consistency.", "Both the approaches generate image segmentation masks, but, while the first method generates a volumetric reconstruction by labeling voxels, the second approach only segments a sparse point cloud.", "Our formulation borrows the idea of jointly labeling superpixels and a voxel grid with edges between them to enforce consistency, but we incorporate rich information from volumetrically fused dense depth maps computed in early stages of our system and scene priors to get good quality reconstructions of objects with complicated shapes and varying material properties.", "Our system also differs from these approaches in that we use a richer set of labels that distinguish between the voxels that belong to the object of interest and those belonging to other objects in the scene, namely voxels behind the support surface.", "This distinction allows a more complete and accurate reasoning about the scene.", "Like us, [12] attempts to exploit the rich evidence available in dense depth maps, using stereo and color based appearance cues to first compute dense piecewise planar depth maps for every view.", "Then they assign a foreground or background label to polygons in every image independently.", "Finally, they fuse these independent segmentations using multi-view reasoning.", "Each of these steps is performed by $\\alpha $ expansion based energy minimization with carefully chosen terms.", "Unlike us, they only generate image segmentations.", "Also, the piecewise planar assumption also does not hold for many objects of interest.", "An alternative to graph cut optimization was proposed in [21], where joint segmentation is performed over 2D pixels and 3D scene points using affinity propagation.", "A graph is constructed over joint nodes each of which comprises of a 3D point and its image projections in different views.", "Similarity between nodes is defined using 3D features like spatial proximity and angle between normal directions, and 2D features like color differences and Kullback-Leibler divergence between patch histograms.", "This method, however, requires user initialization and includes only sparse 3D points in the optimization.", "All the above approaches perform 3D segmentation by converting the problem to a discrete optimization.", "[11] formulate volumetric reconstruction as that of minimizing a continuous convex functional by relaxing binary labels to lie in $[0,1]$ , and they show quantitative improvement over discrete graph cut based approaches." ], [ "Approach", "Our system takes as input an image sequence and automatically generates a volumetric reconstruction of the object, depth surfaces, and segmentation masks for selected key-frames, as shown in Fig.", "REF .", "There are four main stages: Pose Estimation: A state-of-the-art VSLAM system LSD-SLAM [6] is used to get camera poses, key-frame selection, and semi-dense depth maps which are used in later stages of the system.", "Surface Modeling: The visible surfaces from each view are modeled as dense depth maps.", "The procedure, inspired by DTAM [15], involves minimizing an objective function comprising of a cost volume based data term regularized by a spatially varying linearized bending energy.", "We experiment with second order bending energies and compare to the first order regularization used in [15].", "Volumetric Fusion: Depth maps from the previous stage are fused together volumetrically to generate a signed function field over a voxel grid.", "The function field indicates normalized and clipped signed distance of every voxel to its nearest surface.", "The fusion, however, introduces a bias in the function field and shifts the zero-crossing.", "PMVS [7] points which are known to lie on the surface are used to correct the bias by deforming the function field where the deformation is modeled by a radial basis expansion.", "Joint 2D-3D Segmentation: A joint segmentation of all key-frame images and a common voxel map is performed using graph cuts with $\\alpha $ -expansion over the set of all pixel and voxel nodes.", "The pixel unary models the color of the object and background regions.", "The voxel unary enforces constraints on the empty regions (using the SDF) and incorporates scene prior by encouraging voxels below the fitted plane to belong to the background.", "Pixel-pixel and voxel-voxel pairwise terms are used to impose smoothness in 2D (edge-aware) and 3D space respectively.", "Finally, edges connecting pixel nodes to sparse surface voxel nodes (identified by hashing sparse 3D points generated by LSD-SLAM into the voxel grid) implicitly enforce consistent labeling across views." ], [ "Pose Estimation", "Camera poses and an initial sparse point cloud with visibility information for each point are required in later stages.", "Camera focal length, optical center, and radial distortion parameters are precomputed using a checkerboard pattern.", "The video frames are also corrected for radial distortion.", "LSD-SLAM [6] with loop closure is then used to select a set of key-frames, compute camera poses for every frame, and generate semi-dense depth maps with corresponding estimates of inverse depth variance for all key-frames.", "Figure: Visualization of depth maps and surface normals generated for videos in the Object-Videos Dataset using our depth surface reconstruction method.", "Smoothness constraints in our optimization make the surface normals well behaved.", "Higher order bending energy helps preserve more fine details such as those on the dragon's head and keyboard's keys and wire." ], [ "Surface Modeling", "Let the set of all pixels in the current reference image $I_r$ be denoted by $\\mathcal {P}$ .", "The pixel coordinates will be referred to as $(u,v)$ or by a 2 dimensional vector $\\textbf {u}$ .", "The problem of estimating depth surface reduces to assigning a depth value $\\xi (\\textbf {u})$ to each pixel $\\textbf {u} \\in \\mathcal {P}$ such that the assignment results in low photometric error, agrees with the depth measurements $\\hat{\\xi }(\\textbf {u})$ wherever available and is mostly smooth barring depth discontinuities.", "In our system these depth measurements are provided by LSD-SLAM for a set of pixels $\\mathcal {T}$ .", "LSD-SLAM also provides an estimate of the inverse depth variance $\\sigma (\\textbf {u})$ for all $\\textbf {u}\\in \\mathcal {T}$ .", "Figure: Effect of different smoothness terms and number of measurement points available on depth map estimation for RGBD Scenes V2 dataset.", "Our method is robust in presence of specularity and textureless surfaces.", "The second order bending energies allow the reconstruction of long thin surfaces like the arm rest on the chair.Inspired by DTAM [15], we begin by constructing the cost volume $C(\\textbf {u},d)$ which specifies the average photometric error of pixel $\\textbf {u}$ with respect to neighboring frames for some discrete depth candidates $d\\in \\mathcal {D}$ .", "Instead of simply using the difference between the pixel intensities, we use a measure based on Zero-mean Normalized Cross-Correlation (ZNCC) between patches of size $3\\times 3$ .", "ZNCC has two advantages: (i) removal of brightness constancy assumption; and (ii) robustness to false matches between pixels with similar intensities but different neighborhoods.", "For a selected set of neighboring frames $N(m)$ , the cost volume is given by $C(\\textbf {u},\\xi (\\textbf {u})) = \\frac{1}{|N(m)|} \\sum _{I\\in N(m)} \\frac{1-Z(\\textbf {u},\\xi (\\textbf {u});I,I_r)}{2}$ where $Z(u,\\xi (\\textbf {u});I,I_r)$ is the ZNCC between a patch around $\\textbf {u}$ in $I_r$ and the patch around its reprojection in $I$ assuming depth $\\xi (\\textbf {u})$ .", "Depth estimation is now formulated as the following optimization problem $ \\min _{\\xi }\\sum _{\\textbf {u}\\in \\mathcal {P}} C(\\textbf {u},\\xi (\\textbf {u})) +\\\\\\sum _{\\textbf {u}\\in \\mathcal {T}}\\lambda _m(\\textbf {u})(\\xi (\\textbf {u})-\\hat{\\xi }(\\textbf {u}))^2+\\sum _{\\textbf {u}\\in \\mathcal {P}} \\lambda _s(\\textbf {u})\\mathcal {B}(\\textbf {u})$ where $\\mathcal {B}(\\textbf {u})$ is the surface bending energy at $\\textbf {u}$ and $\\lambda _s(\\textbf {u})$ is a spatially varying weight that prevents smoothing across edges.", "For ease of optimization, we restrict the choice of bending energy to those which can be approximated by finite differences.", "Ideally, bending energy must also be invariant to in-plane rotation of the camera.", "One such bending energy has the form $\\mathcal {B}(\\textbf {u}) = \\left(\\frac{\\partial ^2 \\xi (\\textbf {u})}{\\partial u^2}\\right)^2 + \\left(\\frac{\\partial ^2 \\xi (\\textbf {u})}{\\partial v^2}\\right)^2 + 2\\left(\\frac{\\partial ^2 \\xi (\\textbf {u})}{\\partial u \\partial v}\\right)^2$ Experiments show that dropping the cross term produces similar results but considerably speeds up the optimization procedure.", "A commonly used substitute, which is also used in [15], is squared norm of the gradient $\\Vert \\nabla \\xi (\\textbf {u})\\Vert ^2$ .", "However, this imposes an often misleading bias towards frontal-planar surfaces.", "The optimization problem in Equation REF is solved using a two-step iterative procedure, similar to [15], by introducing auxiliary variables $\\alpha $ and a coupling term whose role in the optimization is controlled by parameter $\\theta $ $\\min _{\\xi ,\\alpha }\\sum _{\\textbf {u}\\in \\mathcal {P}} \\lambda _c C(\\textbf {u},\\alpha (\\textbf {u})) + \\frac{1}{2\\theta }(\\alpha (\\textbf {u})-\\xi (\\textbf {u}))^2 +\\\\\\sum _{\\textbf {u}\\in \\mathcal {T}}\\lambda _m(\\textbf {u})(\\xi (\\textbf {u})-\\hat{\\xi }(\\textbf {u}))^2+\\sum _{\\textbf {u}\\in \\mathcal {P}} \\lambda _s(\\textbf {u})\\mathcal {B}(\\textbf {u})$ where $\\lambda _c$ is a constant that is empirically determined and $\\lambda _m(\\textbf {u})$ is a spatially varying weight that depends on the reliability of the measurement at $\\textbf {u}$ .", "The first step involves minimizing the data term by point-wise search; while the second step solves for the surface that is consistent with the estimates in the first step, agrees with the measurement, and has low bending energy.", "Step 1 involves a per-pixel point-wise search over the discrete depth candidates to obtain photo-consistent depth estimates (u) = dD Eaux(t)(u,d,(t-1)) Eaux(t)(u,d,(t-1)) = c C(u,d) + 12(t)(d-(t-1)(u))2 where $\\theta ^{(t)}$ is decreased after every iteration to increase the coupling between the discrete auxiliary variables and continuous depth estimates.", "This step is identical to [15] except that our final estimate in the current iteration $\\alpha ^{(t)}$ is obtained by median filtering the depth image $\\hat{\\alpha }$ to increase robustness to outliers and then performing a single Newton step using the numerical gradient of $E_{aux}^{(t)}$ (t)(u) (t)(u) - Eaux(t)(u,(t),(t-1))2 Eaux(t)(u,(t),(t-1)) where $\\eta $ is the learning rate.", "We use $\\eta =0.01$ for our experiments.", "$\\lambda _c=10$ was used for RGBD Scenes dataset and $\\lambda _c=1000$ was used for Object-Videos dataset.", "Step 2 enforces smoothness and consistency with depth measurements while being tied to photo-consistent depth estimates obtained in the previous step.", "$\\xi ^{(t)} = \\arg \\min _{\\xi }\\sum _{\\textbf {u}\\in \\mathcal {T}}\\lambda _m(\\textbf {u})(\\xi (\\textbf {u})-\\hat{\\xi }(\\textbf {u}))^2 + \\\\\\sum _{\\textbf {u}\\in \\mathcal {P}} \\frac{\\lambda _a(\\textbf {u})}{2\\theta ^{(t)}}(\\alpha ^{(t)}(\\textbf {u})-\\xi (\\textbf {u}))^2 + \\lambda _s^{(t)}(\\textbf {u})\\mathcal {B}(\\textbf {u})$ In contrast to [15], we remove the Huber loss on the smoothness term which turns our optimization problem into linear least squares thereby improving efficiency.", "We deal with outliers and large depth discontinuities by employing two techniques: 1) median filtering the optimal solution to equation (REF ); 2) appropriate choice of spatially varying constants $\\lambda _m, \\lambda _a$ and $\\lambda _s$ : Measurement weights: $\\lambda _m(\\textbf {u})$ is chosen to be inversely proportional to $\\sigma (\\textbf {u})$ , the inverse-depth variance estimate for each measurement provided by LSD-SLAM.", "We use $0.1$ as the constant of proportionality.", "Smoothness weights: Since we remove the Huber loss on smoothness term, we gradually increase $\\lambda _s^t(u)$ with every iteration so that by the time the smoothness term dominates, $\\xi $ is already very close to the true solution.", "The weight also depends upon edge strength to make smoothing edge aware.", "Specifically we use $\\lambda _s^{(t)}(\\textbf {u}) = \\gamma ^{t-1} \\lambda _s^{(0)}e^{-c_{\\text{edge}}\\; Edge(\\textbf {u})}$ where $Edge(\\textbf {u})$ is the edge strength obtained from Structured Edge Detector  [5] which was trained to detect object contours in images.", "For our experiments we used $\\lambda _s^{(0)}=0.01$ and $c_{\\text{edge}}=5$ .", "$\\gamma =1.6$ was used for RGBD Scenes dataset, whereas $\\gamma =1.8$ was used for Object-Videos dataset.", "Coupling weights: Similar to [15], we decrease $\\theta $ with every iteration to increase coupling $\\theta ^{(t)} = (1-\\beta t)\\theta ^{(t-1)}$ In addition, photometric error based spatially varying weights are used to propagate depth estimates from more confident regions to less confident ones such as textureless and specular surfaces.", "$\\lambda _a(\\textbf {u}) = e^{-c_{\\text{conf}}\\; \\rho (\\textbf {u})}$ where $\\rho (\\textbf {u})$ is the minimum photometric error stored in the cost volume at pixel $\\textbf {u}$ $\\rho (\\textbf {u}) = \\min _{d}C(\\textbf {u},d)$ In our experiments, we use $\\theta ^{(0)}=1$ , $\\beta =0.03$ , and $c_{\\text{conf}}=5$ .", "Note that with the exception of $\\lambda _c$ and $\\gamma $ , the same set of parameters generalized to two datasets with very different scales." ], [ "Volumetric Fusion", "The depth maps estimated for the key-frames are fused together to get a volumetric reconstruction of the scene represented by an implicit surface function field $S(\\textbf {v})$ defined on a voxel grid, $\\mathcal {V}$ .", "A desired property of such a function field is that $S(\\textbf {v})$ takes a positive value in empty regions and negative value otherwise.", "The surface mesh can then be extracted as the zero-level set of $S$ .", "Let $S_i(\\textbf {v})$ be the Signed Distance Function (SDF) computed over $\\mathcal {V}$ due to $i^{th}$ key-frame.", "Given $K$ key-frames, the actual choice of the functional mapping from $\\lbrace S_1(\\textbf {v}),\\cdots ,S_K(\\textbf {v})\\rbrace $ to $S(\\textbf {v})$ is closely related to the choice of the mapping from the $i^{th}$ key-frame's depth map $\\xi _i$ to $S_i(\\textbf {v})$ .", "Another important requirement from $S(\\textbf {v})$ is that its zero-crossing lies on the object's surface.", "If the depth maps were very accurate, then for any choice of $S_i(\\textbf {v})$ which assigns a negative value to voxels behind the depth surface and positive otherwise, $S(\\textbf {v})$ can simply be computed as follows $S(\\textbf {v}) = \\max _{i\\in \\lbrace 1,\\cdots ,K\\rbrace } S_i(\\textbf {v})$ The surface extracted from $S(\\textbf {v})$ will be guaranteed to be accurate up to the resolution of the voxel grid.", "Sub-voxel accuracy however still depends on the computation of $S_i$ .", "Naturally smooth and more accurate surfaces can be expected if $S_i(\\textbf {v})$ is proportional to the distance of $\\textbf {v}$ from the closest depth surface.", "However, since the estimated depth maps often have local errors in regions with textureless surfaces or specularities, the above computations need to be suitably modified for robustness.", "We propose to decompose volumetric fusion into two parts: i) robust fusion, and ii) zero-crossing or bias correction.", "The first step produces SDFs that are robust to errors in depth map or camera pose estimation at the cost of incurring a bias or a shift in the zero-crossing.", "The second step corrects for this bias by applying a smooth deformation field to force the function value close to zero at a set of points known to lie on the object surface.", "We compare our proposed approach to TSDF [14] in Fig.", "REF .", "Fig.", "REF shows the effect and importance of zero-crossing correction." ], [ "Robust Fusion", "Let $\\textbf {q}_i(\\textbf {v})$ be the point where the ray from the $i^{th}$ camera center to $\\textbf {v}$ intersects the depth surface due to $\\xi _i$ .", "$S_i(\\textbf {v})$ is simply given by $S_i(\\textbf {v}) = \\min \\left(1, \\frac{\\Vert \\mathbf {\\eta }_i(\\textbf {v})\\Vert _2}{\\mu }\\right) \\times \\text{sign}\\left(\\eta _{iz}(\\textbf {v})\\right)$ where $\\mathbf {\\eta }_i(\\textbf {v}) = \\textbf {T}_i(\\textbf {q}_i(\\textbf {v})) - \\textbf {T}_i(\\textbf {v})$ with $\\textbf {T}_i$ being the transformation from world coordinates to camera coordinates.", "Here $\\mu $ is a constant used to clip the SDF value further away from the surface.", "The SDFs from all key-frames are fused using a soft maximum operation defined as follows $S^{\\prime }(\\textbf {v}) = \\frac{\\sum _{i=1}^{K}S_i(\\textbf {v})e^{h S_i(\\textbf {v})}}{\\sum _{i=1}^{K}e^{h S_i(\\textbf {v})}}$ where $h$ is a hardness constant.", "We use $h=10$ for all our experiments.", "Soft maximum produces level surfaces that are naturally smooth and trades the property of preserving the zero-crossings for robustness to errors in depth and camera parameter estimates.", "Note that this is in contrast to [14] where $S_i$ were truncated to zero beyond a depth of $\\mu $ behind the depth surface in order to preserve zero-crossing as it relied on a weighted averaging of $S_i$ s. Views in which a voxel was more directly in the line of sight were given a higher weight while fusing the SDFs.", "Such a weighing is implicit in our soft-max formulation since voxels tend to have higher $S_i(\\textbf {v})$ in views where it is directly in the line of sight.", "Figure: Comparison of TSDF with the proposed robust fusion scheme involving deformation of Softmax SDF.", "The proposed SDF is more robust to errors in depth estimation and produces smoother more accurate surfaces." ], [ "Zero-crossing Correction", "To correct for the bias in $S^{\\prime }$ , we use a sparse set of 3D surface points, $\\mathcal {M}$ generated by PMVS [7].", "We pose this problem as that of finding a deformation field $\\Delta S(\\textbf {v})$ such that $S(\\textbf {p}) = S^{\\prime }(\\textbf {p}) + \\Delta S(\\textbf {p}) = 0 \\: \\forall \\: \\textbf {p} \\in \\mathcal {M}$ We choose to parametrize $\\Delta S$ by a radial basis expansion $\\Delta S(\\textbf {v}) = \\sum _{j=1}^{C} a_j \\phi (\\Vert \\textbf {v} - \\textbf {c}_j\\Vert _2)$ where $\\textbf {c}_j$ is a set of control points, $\\textbf {a}$ denotes the coefficients of expansion, and $\\phi $ is any radial basis kernel.", "For our experiments, we use $\\phi (x) = e^{-x^2/\\sigma ^2}$ with $\\sigma =0.1$ .", "For a voxel grid of maximum grid dimension 200, we use $C=4000$ randomly sampled control points out of which half are sampled in the region $\\lbrace \\textbf {v}\\in \\mathcal {V} | -0.5 < S^{\\prime }(\\textbf {v}) < 0.5\\rbrace $ and the other half outside this region.", "The coefficients are obtained by minimizing the following least squares objective function $\\sum _{\\textbf {p}\\in \\mathcal {M}^{\\prime }_1}\\left(S^{\\prime }(\\textbf {p}) +\\sum _{j=1}^{C}a_j\\phi (\\Vert \\textbf {p} - \\textbf {c}_j\\Vert _2)\\right)^2 + \\\\\\sum _{\\textbf {p}\\in \\mathcal {M}^{\\prime }_2}\\left(\\sum _{j=1}^{C}a_j\\phi (\\Vert \\textbf {p} - \\textbf {c}_j\\Vert _2)\\right)^2 +\\lambda \\frac{\\Vert \\textbf {a}\\Vert ^2}{2}$ The first term enforces $S(\\textbf {p})=0$ for $\\textbf {p}\\in \\mathcal {M}^{\\prime }_1$ which contains 500 points randomly sampled from $\\mathcal {M}$ .", "The second term constrains the deformation to be zero at points $\\textbf {p}\\in \\mathcal {M}^{\\prime }_2$ which contains 500 points each sampled from $0.9$ and $-0.9$ level sets of $S^{\\prime }$ .", "The third term is used to regularize the deformation." ], [ "Joint 2D-3D Segmentation", "Given a set of key-frame images $\\mathcal {I}$ , their camera pose estimates $\\mathcal {C}$ , fused and deformed SDF $S$ , and a sparse set of pixel-voxel correspondences, we want to label all 2D image pixels as $\\lbrace object,background\\rbrace $ and all voxels as $\\lbrace object,background,empty\\rbrace $ .", "Note that even though the objects are placed on simple planar surfaces, achieving good quality segmentations is challenging because of one or more of the following reasons: (i) specularities; (ii) significant color variations on the object surface; (iii) errors in camera pose estimation; (iv) local errors in depth map estimates, (v) noisy PMVS point cloud; and (vi) error in support surface estimation.", "Another challenge is the computational complexity that arises as a result of dealing with pixels in all images and a dense voxel grid.", "To keep computational complexity in check, we label superpixels instead of pixels.", "About 2000 superpixels are computed for each image $I_i \\in \\mathcal {I}$ using SLIC [1].", "We denote the label assigned to the $i^{th}$ superpixel and voxel by $s_i$ and $v_i$ , and the set of all pixels and voxels by $\\mathcal {S}$ and $\\mathcal {V}$ respectively .", "We formulate our objective as a joint 2D-3D segmentation and minimize it using graph cuts with $\\alpha $ -expansion.", "The objective function is given by $E(s,v) = \\sum _{i=1}^{|\\mathcal {S}|}\\psi _s(s_i) + \\sum _{i=1}^{|\\mathcal {V}|}\\psi _v(v_i) + \\sum _{(i,j)\\in \\mathcal {E}_{sv}}\\psi _{sv}(s_i,s_j) \\: + \\\\\\sum _{(i,j)\\in \\mathcal {E}_{ss}}\\psi _{ss}(s_i,s_j) + \\sum _{(i,j)\\in \\mathcal {E}_{vv}}\\psi _{vv}(s_i,s_j)$ The superpixel unary term $\\psi _s$ encodes object color, a scene prior that is informative about background superpixels, and consistency of image segmentations with volumetric reconstruction.", "The voxel unary term, $\\psi _v$ , encodes information about the empty regions using the SDF, location of background voxels using the scene prior pertaining to a flat support surface, and consistency of volumetric reconstruction with image segmentations.", "Super-pixel binary term, $\\psi _{ss}$ , encodes edge-aware smoothness between labeling of neighboring superpixels.", "Similarly, voxel binary term $\\psi _{vv}$ encodes smoothness in voxel labeling.", "Finally, superpixel - voxel pairwise term, $\\psi _{sv}$ , ensures consistency between surface voxels, and superpixels they project to in different views.", "Indirectly, $\\psi _{sv}$ ensures consistent superpixel labeling across views.", "Next we describe the graph structure, initialization, and each of the unary and pairwise energy terms in detail." ], [ "Graph Structure", "There is a node in the graph, $\\mathcal {G}$ , for every superpixel $\\textbf {s}_i$ in $\\mathcal {S}$ and every voxel $\\textbf {v}_i$ in $\\mathcal {V}$ .", "We connect the neighboring voxels using a 6-connected grid.", "We insert an edge between two superpixels if they both lie in the same image and share a boundary.", "We would ideally like to connect each superpixel with all voxels that project inside the superpixel, are in direct line of sight in the corresponding view, and lie on the surface of the object.", "However, this would require knowing the surface voxels and their visibility information.", "We can compute this information for a sparse set of points using semi-dense depth maps provided by LSD-SLAM.", "For each image we back-project the LSD-SLAM depth map and hash the points into the voxel grid $\\mathcal {V}$ .", "This gives us a sparse set of correspondences between superpixels and voxels.", "The region to be discretized by the voxel grid (our region of interest or ROI), is set to $median \\pm 2$ standard deviations computed from the LSD-SLAM point cloud along each dimension.", "The resolution of the grid is chosen such that the largest dimension is divided into 200 voxels." ], [ "Initialization", "We first extract the zero level set of the fused and deformed SDF and fit a plane to the points to identify the flat surface that supports the object.", "All voxels whose signed distance to the plane (negative denoting behind the plane) is less than a small positive theshold, $\\tau $ , are set to $background$ .", "All voxels above the surface which contain at least one point from LSD-SLAM point cloud are marked as $object$ and the remaining ones are maked $empty$ .", "We initialize the segmentation with a per-image trimap (see Fig.", "REF ), computed by projecting an over-estimate, $\\mathcal {U}$ , and an under-estimate, $\\mathcal {L}$ , of the 3D volume on to each image.", "$\\mathcal {U}$ is obtained by selecting the largest connected component from the set of voxels with $S(\\textbf {v}) < 0.5$ and distance to the plane greater than $\\tau $ .", "We project $\\mathcal {U}$ on to each image $I_i$ to get segmentation masks.", "Super-pixels that have more than $90\\%$ pixels in the background region of the segmentation mask are permanently set to $background$ .", "Next we create a set of voxels $\\mathcal {L}=\\lbrace \\textbf {v}\\in \\mathcal {U}|S(\\textbf {v}) < -0.5\\rbrace $ .", "$\\mathcal {L}$ is projected on to each image plane to get segmentation masks and each superpixel with more than $90\\%$ pixels in the foreground is permanently marked $object$ .", "The remaining superpixels are randomly initialized.", "Note that only the superpixels that lie in this narrow band around the object silhouette are being solved for by graph cuts.", "A good initialization must include as few voxels on or below the plane in $\\mathcal {U}$ as possible.", "For this $\\tau $ is set to $0.05$ times the mean of the standard deviations along each dimension of the points used to fit the plane." ], [ "Energy Terms", "Super-Pixel Unary $\\psi _s$ comprises 3 components: $\\psi _c$ encodes foreground and background colors; $\\psi _{roi}$ ensures that superpixels which correspond to surfaces farther away from the ROI are assigned to background; and iii) $\\psi _{sil}$ ensures consistency of silhouettes with volumetric reconstruction.", "For the color term, separate GMMs are used to model the color distribution of foreground and background in Lab color space across all key-frame images.", "We learn both GMMs with 10 components and learn a full covariance matrix.", "5000 superpixels are randomly sampled across images from the currently labeled foreground or background regions and from them 50000 pixels are sampled to learn each GMM model.", "Next, the posterior probabilities, denoted by $Prob(.", ")$ , are computed for each pixel using the learned GMMs.", "In order to limit the influence of the color term, the probabilities are scaled and truncated by using the following function $M(x) = \\min (5,-log(x))$ Let $\\mathcal {P}(i)$ denote the set of pixels in the $i^{th}$ superpixel.", "Then the color based superpixel unary term is given by $\\psi _c(s_i) = \\frac{1}{|\\mathcal {P}(i)|} \\sum _{p \\in \\mathcal {P}(i)} M(\\rm P_{gmm}(s_i|Color(p))$ To compute $\\psi _{roi}$ , we back-project pixels using the depth map and count how many lie inside the ROI.", "Let $f(i)$ denote the fraction of pixels in $\\mathcal {P}(i)$ which fall outside the ROI.", "Then the ROI based term is given by $\\psi _{roi}(s_i) ={\\left\\lbrace \\begin{array}{ll}f(i) & s_i=object \\\\0 & s_i=background\\end{array}\\right.", "}$ For the silhouette based term, the current voxels labeled as object are used to render segmentation masks.", "Let $sil(i)$ denote the fraction of pixels in $\\mathcal {P}(i)$ labeled as foreground.", "Then silhouette consistency term is given by $\\psi _{sil}(s_i) ={\\left\\lbrace \\begin{array}{ll}1-sil(i) & s_i=object \\\\sil(i) & s_i=background\\end{array}\\right.", "}$ Finally $\\psi _s(s_i) = \\psi _c(s_i) + \\lambda _{roi}\\psi _{roi}(s_i) + \\lambda _{sil}\\psi _{sil}(s_i)$ .", "For our experiments we used $\\lambda _{roi} = 10$ and $\\lambda _{sil} = 1$ .", "In our experiments, we found that using an average of $\\psi _{sil}$ over all past iterations of graph cut leads to more stable solutions.", "Figure: Comparison of reconstructions produced by our system “SDF + 2D-3D Voxel” with other baseline methods on videos in Object-Videos dataset.", "Our approach incorporates: dense depth maps with an improved cost function and improved signed distance function for fusion; sparse patch-based stereo 3D points (PMVS); and joint 2D-3D segmentation based on color, edge, and depth cues.", "By using a broad set of techniques, our system avoids many of the errors and artifacts inherent in any individual.", "Note that our method produces smoother and more detailed reconstructions as compared to the baseline methods while being more robust to errors in PMVS point cloud and depth estimation errors that show their effect in “PMVS + Poisson” and “Deformed SDF”.", "Also unlike “Volumetric GC”, our approach is less sensitive to the color model, as evident from the books example.Voxel Unary $\\psi _v$ has 3 components: $\\psi _{sdf}$ encodes the information contained in the SDF, $S$ ; $\\psi _{scene}$ encodes our scene prior; and $\\psi _{carve}$ ensure that voxel labeling is consistent with superpixel labeling.", "Note that SDF helps in identifying empty regions ($S(\\textbf {v})>0$ ) but cannot distinguish between object of interest and other background regions, such as those below the support surface or regions further away from the object, since SDF attains a positive value in both cases.", "This complementary information is provided by the other two components, $\\psi _{scene}$ and $\\psi _{carve}$ .", "First, for each voxel $\\textbf {v}$ , a normalized distance to the fitted plane is computed as follows $d_{n}(\\textbf {v}) = \\frac{1}{1+e^{-d(\\textbf {v})/\\sigma _{d}}}$ where $d(\\textbf {v})$ denotes the distance of $\\textbf {v}$ from the plane and $\\sigma _d$ is the standard deviation of the distance of all voxels from the plane.", "Next, all voxels with $d_n(\\textbf {v}) < 0.3$ are permanently set to $background$ .", "All voxels with $S(\\textbf {v}) > 0.9$ are permanently set to $empty$ .", "For the remaining voxels, $\\textbf {v}$ the $SDF$ based energy term is given by $\\psi _{sdf}(v_i) ={\\left\\lbrace \\begin{array}{ll}S(\\textbf {v}) & v_i=object \\\\S(\\textbf {v}) & v_i=background \\\\-S(\\textbf {v}) & v_i=empty\\end{array}\\right.", "}$ The scene prior energy term, $\\psi _{scene}$ , penalizes labeling of voxels above the support surface as background and those below the surface as either $object$ or $empty$ .", "It is defined as $\\psi _{scene}(v_i) ={\\left\\lbrace \\begin{array}{ll}\\mathbb {1}(d_{n}(\\textbf {v})<0.5) \\; (1-d_{n}(\\textbf {v})) & v_i=object \\\\\\mathbb {1}(d_{n}(\\textbf {v})>0.5) \\; d_{n}(\\textbf {v}) & v_i=background \\\\\\mathbb {1}(d_{n}(\\textbf {v})<0.5) \\; (1-d_{n}(\\textbf {v})) & v_i=empty\\end{array}\\right.", "}$ In order to compute $\\psi _{carve}$ , we first perform silhouette carving to get a set of voxel $\\mathcal {V}_{carve}$ which project on the superpixels labeled $object$ in all key-frame views.", "Let the complementary set be $\\mathcal {V}_{carve}^c = \\mathcal {V}\\setminus \\mathcal {V}_{carve}$ .", "$\\psi _{carve}$ is then given by $\\psi _{carve}(v_i) = \\mathbb {1}(\\textbf {v}_i\\in \\mathcal {V}_{carve}^c)\\mathbb {1}(v_i=object)$ The final voxel unary for the variable voxels is given by $\\psi _v(v_i)~=~\\lambda _{sdf}\\psi _{sdf}(v_i) + \\lambda _{scene}\\psi _{scene}(v_i) + \\lambda _{carve}\\psi _{carve}(v_i)$ .", "For all our experiments we have set $\\lambda _{sdf}=4$ , $\\lambda _{scene}=1$ , and $\\lambda _{carve}~=~1$ .", "Similar to $\\psi _{sil}$ , we also use the average $\\psi _{carve}$ over all past iterations.", "Super-pixel Binary term $\\psi _{ss}$ imposes edge-aware smoothness constraints on the superpixel labeling.", "For a pair of superpixels with indices $(i,j)$ that share boundary, with the set of boundary pixels denoted by $\\mathcal {B}(i,j)$ , the superpixel pairwise energy term is given by $\\psi _{ss}(s_i,s_j) = \\mathbb {1}(s_i\\ne s_j)\\exp \\left(-2\\sum _{\\textbf {p}\\in \\mathcal {B}(i,j)}Edge(\\textbf {p})\\right)$ where $Edge(\\textbf {p})$ denotes the contour edge strength of pixel $\\textbf {p}$ obtained using Structured Edge Detector [5].", "Voxel Binary term $\\psi _{vv}$ imposes smoothness constraints on the voxels.", "For a pair of voxels indexed by $(i,j)\\in \\mathcal {E}_{vv}$ , the voxel binary term is defined as $\\psi _{vv}(v_i,v_j) = \\mathbb {1}(v_i\\ne v_j)$ .", "Super-pixel - Voxel Pairwise term $\\psi _{sv}$ ensures that for every superpixel and voxel with an edge between them, with indices $(i,j)\\in \\mathcal {E}_{sv}$ , superpixel and voxel labels are consistent with each other.", "Note that $\\mathcal {E}_{sv}$ only connects surface voxels with superpixels.", "The surface voxels can never be $empty$ hence both $s_i$ and $v_j$ lie in $\\lbrace object,background\\rbrace $ .", "Given this restriction on the labels $\\psi _{sv}(s_i,v_j) = \\mathbb {1}(v_i\\ne v_j)$ ." ], [ "Details of $\\alpha $ -expansion", "We run 5 iterations of optimization with 3 $\\alpha $ -expansion steps per iteration or until the sum of the number of label swaps for all expansion moves in an iteration falls below 6000, whichever comes first.", "To make the set of labels the same for pixels and voxels, we assign a very high cost to $empty$ for any superpixel.", "The color models are updated in each iteration using the current superpixel labeling hence $\\psi _c$ needs to be recomputed in each iteration.", "Besides this only $\\psi _{sil}$ and $\\psi _{carve}$ are updated in each iteration since they depend on the current voxel map and segmentation masks." ], [ "Post-processing", "As an important post-processing step, we select the largest connected component from the voxels labeled as $object$ .", "While this could be used to generate a surface mesh of the object directly, we found that qualitatively better results are obtained by using it as a mask to set $S(\\textbf {v})$ to 1 for all $\\textbf {v}$ such that $v\\ne object$ , extract its zero-level set, and perform mesh smoothing using [22]." ], [ "Experiments", "We evaluate our system qualitatively (Fig.", "REF and Fig.", "REF ) with displays of 3D reconstructed objects from our Object-Videos dataset.", "We also quantitatively evaluate dense depth map estimation (Fig.", "REF ) with the RGBD Scenes V2 Dataset [8] and segmentation accuracy (Tab.", "REF ) on key frames from the Object-Videos dataset." ], [ "Object Reconstruction", "Our Object-Videos dataset consists of 12 videos of 10 objects captured using a commercial mobile phone camera Samsung Galaxy S4.", "Many of the objects have complex shapes, low-texture surfaces, and specular materials.", "Thus, while many graphics and vision papers use carefully designed experimental setups and/or objects with smooth Lambertian surfaces that satisfy model assumptions, we attempt to reconstruct common objects filmed with a typical camera in a casual process.", "In Fig.", "REF , we show results of our method “SDF + 2D-3D Voxel” with comparisons to baseline methods: (i) Poisson surface reconstruction [9] using PMVS [7] point cloud; (ii) Volumetric graph cut method of  [19]; and (iii) Zero level set of deformed SDF $S$ after selecting appropriate region using aggressive and conservative thresholds on $S$ ." ], [ "Dense Depth Estimation", "To evaluate our dense depth map estimation procedure we use the available camera poses to estimate depth maps for five selected key frames of each video in the RGBD Scenes V2 dataset.", "We compare three different smoothness priors: first order ($\\Vert \\nabla \\xi (u)\\Vert _2^2$ ); second order (eq.", "REF without the cross term); and second order rotation invariant bending energy (eq.", "REF ).", "We also evaluate the effect of increasing the number of measurements (known depth points) from 100 to 1000 (see Fig.", "REF for a qualitative comparison on RGBD Scenes V2 dataset).", "We compare to bilinear interpolation of known depth values as a baseline.", "For indoor scenes that consist of many planar surfaces, the baseline works quite well, but it provides poor estimates for objects that have curved surfaces.", "Fig.", "REF shows the qualitative comparison of depth estimation with different smoothness priors on 2 videos from our Object-Videos dataset.", "In Fig.", "REF , we plot root mean squared error as a function of ground truth depth.", "All the variants of our approach beat the baseline for small depths, which is most relevant for our application.", "For large depths, non-linear discretization used for the cost volume results in high quantization errors, causing the dense depth estimation to underperform the baseline.", "While inverse depth and log space discretization are common, in our experiments we found that scaling $\\lbrace (0.2 + 0.8\\times i/l)^5\\rbrace _{i=1}^l$ to the expected range of depth values, where $l$ is the number of discrete values, performed best.", "Among smoothness priors, the rotation invariant second order bending energy performs the best, beating the second order bending energy by a small but noticeable amount.", "However, the latter may be preferred because it requires less computation.", "Both the second order energies perform significantly better than the first order energy.", "Finally, as the number of measurements increase the error reduces, as expected, especially for regions further from the camera." ], [ "Object Segmentation", "We also compare variants of our algorithm on 2D pixel segmentation.", "These results give a sense of 3D segmentation and reconstruction quality, but multiple 3D segmentations are consistent with a single 2D segmentation (e.g., the visual hull and the true shape have the same silhouettes) and different kinds of 2D segmentation errors have different impact on 3D reconstruction.", "In Table REF , we compare segmentation performance, measured by intersection over union (IoU) of ground truth and estimated segmentation masks over annotated keyframes, for several variants of our technique: Fused SDF: Segmentations obtained by backprojecting the largest connected component of interior voxels above the fitted plane from the signed distance function (SDF) created from depth maps.", "Deformed SDF: Same as above, but after deforming the SDF so that its surface lies close to PMVS points.", "Volumetric GC: Our implementation of volumetric graph cut method of [3].", "2D GC: Image graph cuts co-segmentation using color terms computed from multiple images but without voxel segmentation, the pixel-voxel constraints, or the $\\psi _{sil}$ term.", "2D-3D GC Pixel: Pixel segmentations resulting from our joint 2D-3D graph cuts method.", "2D-3D GC Voxel: Backprojects segmented voxels using our joint 2D-3D method.", "SDF + 2D-3D Voxel: Backprojects the 3D volume obtained by slightly dilating the voxels obtained from 2D-3D segmentation, intersecting with Deformed SDF, and smoothing.", "This method provides the best qualitative results and is used for reconstruction results.", "The SDF deformation yields better results in most cases, but for teabox, books, helmet, hedgehog1, and godzilla1, the results are much worse due to large errors in camera pose estimation that decreased accuracy of the PMVS point cloud used for deformation.", "The PMVS results are included in the supplementary video.", "Our graph cuts methods are robust to these errors, outperforming “Fused SDF” and “Deformed SDF” by large margins.", "Our joint 2D-3D segmentation (“2D-3D GC Pixel”) performs equally or better than 2D-only segmentation (“2D GC”) in all but two cases, supporting the value of joint segmentation.", "The main case in which 2D-only outperforms is teabox in which errors in camera pose estimation harm the 2D-3D result.", "Although backprojecting voxels from the 2D-3D segmentation (“2D-3D GC Voxel”) slightly underperforms the 2D-3D pixel segmentation (partly due to coarser voxel discretization), the voxel-based 3D model is better than that obtained by shape carving from 2D segmentations because shape carving is sensitive to errors in individual images.", "Our most qualitatively pleasing 3D models are produced by combining meshes from “Deformed SDF” and “2D-3D GC Voxel”, but when backprojected to images, the resulting silhouettes are slightly less accurate than “2D-3D GC Voxel”.", "As an alternative to graph cut optimization, we tried Spectral Matting [13] for performing independent image segmentations using the trimap.", "Spectral Matting first identifies a basis set of matting components where each component is obtained as a linear combination of a Laplacian matrix.", "Then it uses the trimap initialization to assign each component to foreground or background and constructs an $\\alpha $ -matte.", "However, we found that in order to obtain consistent segmentations across views it is necessary to compute eigenvectors of a prohibitively large Laplacian matrix defined over pixels in all key-frames and all voxels." ], [ "Failure Modes and Limitations", "Based on qualitative (see Fig.", "REF ) and quantitative evaluation, we have identified a set of failure modes for our approach - A major source of error that affects all stages of our system is camera pose estimation.", "Camera pose estimates directly affect the accuracy of constructed cost volume and depth measurements for computing dense depth maps, accuracy of PMVS point cloud, alignment of depth maps during volumetric fusion, deformation of SDF using PMVS point cloud, and superpixel-voxel consistency constraints during joint 2D-3D segmentation.", "The pose estimation errors are largely due to severe occlusion and breaking of brightness constancy assumption due to textureless and specular surfaces.", "The videos in Object-Videos dataset are also collected in a casual fashion with blurry frames and large displacement between consecutive frames.", "Secondly, errors in estimated depth maps due to specular and textureless surfaces adversely affect quality of the fused SDF which is the driving force behind the 2D-3D segmentation mechanism.", "We have demonstrated some degree of robustness in reconstructing textureless and specular surfaces such as in stapler, paper-punch, teabox, and helmet, but explicit removal of specularities would further improve performance.", "As a byproduct of the above two, incorrect estimation of the support surface is a main source of error while reconstructing flat objects like the keyboard.", "Our approach also has some limitations.", "Our method was targeted towards reconstruction of small objects for 3D printing or augmented reality applications and hence applies to a scale of objects and scenes which can afford computation with a discrete grid of voxels.", "We are also limited by computation of the cost volume for depth estimation which grows linearly in the number of discrete depth values used.", "We limit our selves to planar support surfaces with limited background clutter.", "Finally, our method cannot recover fine geometric details such as the scale pattern on the dragon.", "Such detail recovery would require shape from shading and material analysis which are still open research problems." ], [ "Conclusion", "In this work, we proposed a system for 3D reconstruction of an object from a video taken with hand-held mobile phone camera.", "Our three major contributions are: (1) improved objective function for dense depth map computation; 2) robust estimation of an implicit surface using a softmax signed distance function with zero-crossing correction; and (3) a method for joint $2D$ -$3D$ segmentation.", "Through qualitative and quantitative results we demonstrate robustness to textureless surfaces, specularities, and errors in camera pose estimation.", "Potential directions for future work include extending the proposed approach for category specific reconstruction using data driven priors and recovering high frequency details in the reconstruction through shape-from-shading and material analysis." ], [ "Acknowledgements", "This research was supported in part by NSF Award 14-21521.", "We are also thankful to David Forsyth for helpful discussion on linearized bending energy and smoothness priors, and to Jason Rock for suggesting the region of interest based component of superpixel unary." ] ]
1606.05002
[ [ "GMC Evolution in a Barred Spiral Galaxy with Star Formation and Thermal\n Feedback" ], [ "Abstract We explore the impact of star formation and thermal stellar feedback on the giant molecular cloud (GMC) population forming in a M83-type barred spiral galaxy.", "We compare three high-resolution simulations (1.5 pc cell size) with different star formation/feedback models: one with no star formation, one with star formation but no feedback, and one with star formation and thermal energy injection.", "We analyze the resulting population of clouds, finding that we can identify the same population of massive, virialized clouds and transient, low-surface density clouds found in our previous work (that did not include star formation or feedback).", "Star formation and feedback can affect the mix of clouds we identify.", "In particular, star formation alone simply converts dense cloud gas into stars with only a small change to the cloud populations, principally resulting in a slight decrease in the transient population.", "Feedback, however, has a stronger impact: while it is not generally sufficient to entirely destroy the clouds, it does eject gas out of them, increasing the gas density in the inter-cloud region.", "This decreases the number of massive clouds, but substantially increases the transient cloud population.", "We also find that feedback tends to drive a net radial inflow of massive clouds, leading to an increase in the star formation rate in the bar region.", "We examine a number of possible reasons for this and conclude that it is possible that the drag force from the enhanced intercloud density could be responsible." ], [ "Introduction", "Understanding the `where' and `how' of gas conversion into stars underpins many areas of astrophysics.", "In particular, it links the large, galaxy-scale evolution with the collapsing cores embedded within the giant molecular clouds.", "Key to this process is the evolution of the clouds themselves.", "Their properties, interactions and motion within the galaxy determine the rate and efficiency of the local star formation and from there, the global distribution.", "This global distribution is far from uniform.", "Observationally, there is the Kennicutt-Schmidt empirical power-law relation between the gas surface density and the surface density of the star formation rate [30], [29], [4].", "Yet closer inspection shows environmental differences between galaxy types and the nucleus, bar, spiral and inter-arm regions within a single galaxy [10], [35], [41], [24], [42].", "These changes include systematic variations in the star formation efficiency ($\\rm {SFE}=\\Sigma _{\\rm {SFR}}/\\Sigma _{\\rm {gas}}$ ) that suggest galactic structure plays a larger role than simply gathering gas so that it can form stars.", "Other factors beyond simply the local gas density are controlling the star formation rate.", "[31] looked at GMCs in the spiral galaxy M51 and found that clouds assembled in the spiral arm to become giant molecular associations but broke apart into smaller clouds in the interarm region.", "This result was replicated in a simulation by [11].", "[39] and [9] also observed the same galaxy, finding that shearing flows and shocks from the spiral arms could stabilise the GMCs and prevent the formation of stars.", "This was supported by observations of the intermediate spiral galaxy, IC342 by [23], who found that the GMCs that were forming stars were downstream of the spiral arm and tended to be more massive.", "This points to differences in the GMC population that control the star formation on local scales in these different regions.", "The impact of the global environment on cloud properties was investigated theoretically by [14] (hereafter, Paper I).", "This work examined the GMC populations forming in the bar, spiral and quiescent outer disc regions of an M83-type galaxy with a grand design architecture.", "They found that while the typical GMC properties were uniform between the three environments, the cloud-cloud interaction rate was strongly dependent on the global structure.", "The elliptical gas motion in the bar and spiral potentials increased the interaction rate to create a larger number of both giant molecular associations from multiple collisions, and small, transient clouds found in tidal tails.", "The resultant three cloud types were clearly distinguished by their locations on the mass-radius scaling relation: Type A clouds were the most common cloud in all environments, with properties matching those typically observed in local galaxies with a median mass of $5\\times 10^5\\ M_{\\odot }$ , radius $15\\ \\rm pc$ and velocity dispersion $6\\ \\rm km/s$ [58], [22], [49], [43].", "Type B clouds were the giant molecular associations with radii above 30 pc, created during multiple mergers, while Type C were unbound, transient clouds forming in the tidal tails, predominantly around the gravitationally dominating Type Bs.", "The cloud interaction rate has previously been linked to the production of stars.", "In analytical calculations by [63], it was suggested that a frequent collision rate could drive the star formation to create the observed relation with gas surface density.", "The required rate was supported by simulations [64], while observational evidence that such collisions could trigger star formation has been seen by [18], [45], [17].", "With this in mind, Paper I estimated the star formation rate based on the cloud interactions.", "This resulted in the highest star formation efficiency being in the bar region, which is not seen in observations [41], [59].", "However, simulations of individual cloud collisions suggested that high velocity interactions might be less productive than lower velocity encounters [62].", "By incorporating this into the model, [15] found that while the interactions were frequent in the bar region, they were too fast to produce a high yield of stars, resulting in a lower star formation efficiency in that region.", "While cloud interactions are not the only controlling factor in star formation, their importance –and therefore the importance of the cloud environment– should not be underestimated.", "While Paper I explored the environmental influences on GMC evolution, it did not include the internal processes of active star formation or stellar feedback.", "Stars emit UV radiation and stellar winds during their lifetime, with massive stars larger than 8 M$_\\odot $ dying in supernovae explosions.", "This energy is injected into the star's surrounding gaseous cradle, having a strong or even disruptive influence on the GMC.", "It is therefore another major player in determining star formation in the galaxy.", "Exactly how much of an impact this feedback has on GMCs and future star formation is strongly debated in the literature.", "For example, In simulations of a Milky Way-type galaxy, [65] found that thermal energy feedback could effectively suppress the star formation, but the surrounding GMC could survive to form more stars.", "This result was also seen by [56], in two-dimensional simulations of the galactic disc, where the stars inject momentum into the gas.", "On the other hand, feedback was found to be a much more detrimental force for GMCs in simulations performed by [68], creating a younger and less massive population than when no feedback is included.", "[12] however, finds the profile slope of the GMCs is unaffected by thermal and momentum feedback for a star formation efficiency above 5%, although its normalization does vary.", "The effect of different feedback mechanisms in simulations was tested by [25], [26], who individually included the effects of supernovae, stellar winds, radiation pressure and HII photoionisation heating.", "They found that the star formation rates in the galaxy disc match the observed Kennicutt-Schmidt relation independent of the feedback used, providing it was effective at breaking up the densest gas.", "[1] investigated the star formation-feedback cycle in cosmological galaxy simulations, finding that, in order to reproduce Milky Way properties, they required early momentum feedback and a large efficiency of star formation per dynamical time.", "One way to understand these questions is to determine the source of ISM turbulence in the galactic disc.", "[13] found that turbulence could be driven by a combination of accretion, disk instabilities, and energetic feedback by young stars.", "[20], [21] explored the effects of gravitational instability and stellar feedback on turbulence in the disc, comparing a pure self-gravity model with one that included stellar feedback.", "They found that gravitational instabilities are likely to be the dominant source of turbulence, and that they can transport mass inwards, fueling star formation in the inner parts of galactic discs over cosmological time.", "The interplay between the environmental and internal impact factors on the GMC has been less well studied.", "[12] notes that the collection of gas in the galactic spiral arms to make giant molecular associations operates with and without feedback.", "However, there has not been a more detailed investigation as to whether internal feedback or external environment play the major role.", "In this paper, we explore these processes in a barred spiral galaxy simulation with resolutions down to 1.5 pc, comparing the clouds formed in the galaxy disc.", "The two new simulations presented here have star formation without stellar feedback and a run which also includes localized thermal energy injection.", "We compare the results with those in Paper I, where no active star formation was included.", "Our simulations are based on M83, a nearby face-on galaxy, a target for the Atacama Large Millimeter Array (ALMA) in cycles 0, 1 and 2.", "This will ultimately allow our GMC populations to be compared with the high spatial resolution achieved in the ALMA observations.", "The main analysis of our results is performed at $t = 200$  Myr.", "This is earlier than in Paper I, which primarily focused on cloud properties at $t = 240$  Myr.", "This change is due to the effectiveness of our feedback, which disperses the grand design spiral at later times, making the comparison with M83 difficult.", "In order to compare with future comparisons of M83 with instruments such as ALMA, we selected an earlier analysis time, when the grand design is still clearly present.", "This paper is organized as follows: In Section , we present our model of the barred galaxy, along with the star formation, stellar feedback and cloud identification models.", "In Section , we describe our main findings, showing the effect of the stellar feedback on the ISM, GMC properties and star formation.", "The simulations presented in this paper are of an isolated galaxy disc run using Enzo, a three-dimensional adaptive mesh refinement (AMR) hydrodynamics code [7].", "The box size is 50 kpc across, covered by a root grid of $128^3$ cells and eight levels of refinement, giving a limiting resolution (smallest cell size) of about 1.5 pc.", "The cell was refined by a factor of two whenever the mass within the cell exceeded $1000\\ \\mathrm {M_{\\odot }}$ .", "We confirmed that this refinement criteria, even at our temperature floor, resolves the Jeans length [66] up to a density of $10^4$ cm$^{-3}$ , at which point we reach our maximum refinement.", "Beyond this, we add an artificial pressure ($P \\propto \\rho ^2$ ) to prevent unresolved collapse at the finest cell level so that the Jeans length is resolved by at least four cells [38].", "The evolution of the gas was computed using a three-dimensional version of the ZEUS hydrodynamics algorithm [61].", "The gas was self-gravitating and allowed to cool radiatively down to 300 K. The cooling rates were taken from the analytical expression of [54] for solar metallicity down to $10^5$ K, and continued to 300 K with rates provided by [50].", "Our galaxy was modeled on the nearby barred spiral galaxy, M83.", "For a stellar potential, we used the model from [24], who analysed the 2MASS $K$ -band image of M83 to produce a description of the density in the axisymmetric bulge and disc and non-axisymmetric bar and spiral galactic regions.", "The non-axisymmetric stellar components consisted of $10^5$ fixed-motion star particles that rotate at the estimated pattern speed for M83, $54\\ \\rm km\\ s^{-1}\\ kpc^{-1}$ .", "The details of the potential are described fully in Paper I.", "While using individual star particles allows for a more complex potential than a fixed analytic expression, there is a risk of numerical issues from the discreteness of the potential particles.", "To minimize this, we smoothed the particle gravitational contribution by adding their mass to the grid at AMR level 4, with a cell size of 50 pc.", "The dark matter halo was included using a static NFW profile [44], with parameters given in Paper I.", "For the initial galaxy radial gas distribution, we assumed an exponential density profile based on the observations of M83 by [37].", "These are shown in equation (2) of Paper I, and the initial gas temperature was $10^4$ K. The gas was given a circular velocity calculated as $V_{\\rm cir}(r) = ({\\rm G} M_{\\rm tot}(r)/r)^{1/2}$ , where $M_{\\rm tot}(r)$ is the enclosed mass of stars, dark matter and gas within the radius, $r$ ." ], [ "Star formation and feedback", "Star formation and stellar feedback were included in two out of the three simulations, starting from $t = 120 \\rm \\ Myr$ .", "During this first period, the global gas structure of the galaxy was forming, as gas fell into the potential created by the non-axisymmetric bar and spiral star potential particles.", "120 Myr takes the galaxy through roughly one rotation of the pattern speed and allows the gas to fully fragment.", "We used a star formation and feedback algorithm based on [8].", "A star particle forms in a grid cell when the following five criteria are met: (1) The cell's gas density exceeds $n_{\\textrm {threshold}} = 1.3 \\times 10^4\\ \\textrm {cm}^{-3}$ ($n_{\\textrm {cell}} > n_{\\textrm {threshold}}$ ).", "This is consistent with the density at which star formation is observed to occur [33], [19], [47].", "(2) There must be a net gas inflow into the grid cell, $\\nabla \\cdot {\\bf v_{\\rm cell}} < 0$ .", "(3) The cooling time is less than the dynamical time ($t_{\\rm cool} < t_{\\rm dyn} \\equiv \\sqrt{3 \\pi / 32 {\\rm G} \\rho _{\\rm tot}}$ ) or the temperature is less than 11,000 K. (4) The star particle mass is greater than $m_{\\textrm {min}} = 500\\ \\textrm {M}_{\\odot }$ .", "This last condition is primarily a numerical restraint imposed to avoid creating an excessive number of star particles that would slow the computation.", "However, this choice also justifies the thermal feedback from supernovae, since a stellar cluster less than 200 $\\rm M_{\\odot }$ is unlikely to contain any Type II supernovae.", "(An assumption based on [52] stellar initial mass function for the frequency of massive stars $> 8\\ \\rm M_{\\odot }$ .)", "(5) Finally, the cell must be maximally refined.", "If all of these criteria are met, a star particle is created at the centre of the cell with a mass, $m_{\\rm star} = m_{\\mathrm {cell}} (\\Delta t/t_{\\mathrm {dyn}}) f_{\\rm SFE}$ , where $f_{\\rm SFE}$ is the star formation efficiency parameter (roughly star formation per dynamical time).", "The particle velocity matches that of the gas in its birth cell.", "The efficiency parameter was selected to be $f_{\\rm SFE} = 0.002$ .", "This value was selected to match the star formation rate of the simulation to that of M83 (see Figure REF later in this paper).", "This value is a factor of ten times lower than the observed GMC average star formation efficiency [32] (but note that we are applying it on a cell-by-cell basis rather than averaged over an entire GMC).", "If a cell does not have sufficient mass to form the particle, but otherwise fulfills the criteria for star formation, its mass is added to a global tally of unborn stellar mass.", "When this summation exceeds the minimum mass, a star particle is formed.", "This stochastic system is employed for all the stars formed in our simulation, since the low efficiency parameter prevents any one cell having sufficient mass immediately.", "While the star formation algorithm creates each star particle instantaneously, the stellar feedback takes place over an extended period of time to mimic the evolution of the cluster.", "The cluster is assumed to form its stars at a rate $ \\propto \\tau {\\rm e}^{-\\tau }$ , where $\\tau = (t - t_{\\rm form})/t_{\\rm dyn}$ and $t_{\\rm form}$ is the formation time of the star particle.", "The mass of stars formed at a time $t$ with time step $\\Delta t$ is therefore $\\Delta m_{\\rm sf} = M(t+\\Delta t) - M(t)= m_{\\rm star} [(1+\\tau _0){\\rm e}^{-\\tau _0} - (1+\\tau _1){\\rm e}^{-\\tau _1}]$ where $M(t)$ is the total stellar mass formed between $t_{\\rm form}$ and $t$ , $\\tau _0 = (t - t_{\\rm form})/t_{\\rm dyn}$ , and $\\tau _1 = (t + \\Delta t - t_{\\rm form})/t_{\\rm dyn}$ .", "This newly created stellar mass is then used in the feedback routine.", "Our feedback scheme adds thermal energy at each time step equivalent to $\\Delta E = f_{\\rm SN} (\\Delta m_{\\rm sf} {\\rm c}^2)$ , where $f_{\\rm SN}$ is the fraction of the rest-mass energy of the star particle that has been converted into heat.", "Our value of $f_{\\rm SN} = 3 \\times 10^{-6}$ is equivalent of about three supernovae for every $500\\ \\textrm {M}_{\\odot }$ star particle formed assuming one supernova ejects $10^{51} {\\rm erg}$ .", "The thermal energy is distributed over the 19 neighbouring cells.", "To account for winds and other ejecta, $m_{\\rm ej} = f_{\\rm ej} \\Delta m_{\\rm sf}$ , is subtracted from the star particle and returned as gas to the grid cell with momentum $m_{\\rm ej} \\bf {v_{\\rm *}}$ , where $\\bf {v_{\\rm *}}$ is the star particle velocity.", "The assumed fraction of mass ejected by all stars integrated over their life is taken to be $f_{\\rm ej} = 0.25$ .", "Table REF shows the simulations we performed.", "To compare the effects of the star formation and thermal stellar feedback, we performed three different runs: no star formation or feedback run (NoSF), only star formation run (SFOnly) and thermal feedback run (SNeHeat).", "NoSF was published in Paper I.", "Table: Summary of the simulations compared in this paper.", "Star formation and stellar feedback are included from t=t = 120 Myr." ], [ "Cloud analysis", "The GMCs in our simulation were identified as coherent structures contained within contours at a threshold density of $n_{\\rm gas} = 100$  cm$^{-3}$ , similar to the observed mean volume densities of typical galactic GMCs.", "Note that we do not include formation or destruction of molecules.", "Instead, we assume that the cloud would consist of both a molecular core and atomic envelope.", "Our clouds were assigned to an environment group (spiral, bar or disc) based on their physical location within the galaxy.", "If a cloud is found within galactic radii $2.5 < r < 7.0$  kpc, it is recognised as a spiral cloud.", "Bar clouds form in a box-like region at the galactic centre, with a length of 5.0 kpc and width 1.2 kpc that rotates with the bar potential.", "The nuclear region inside 600 pc is excluded from cloud analysis due to the difficulty in accurately tracking clouds in such a high density area.", "Outside $r = 7.0$  kpc, clouds are designated disc clouds.", "The outermost ring ($r > 8$ kpc) is formed in a Toomre instability during the fragmentation of the initial conditions and is also excluded from cloud analysis, since it is affected by our idealized initial conditions.", "These three environment regions are shown in Figure REF .", "The monchrome background image is the surface density of the gas in run SNeHeat at $t = 200$  Myr and is overlaid with the cloud positions, coloured to indicate their assigned environment.", "Blue shows clouds identified as being in the disc region, green for the spiral and red for the bar.", "Black clouds do not sit in any of the analysed regions and are not included in the analysis.", "Figure: The location of the three different galactic environments: bar, spiral and disc.", "Coloured markers show the positions of the identified clouds: red triangles are clouds in the bar region, green diamonds are spiral clouds and blue circles are in the disc.", "The black squares show clouds not included in the analysis.", "The background image shows the gas surface density of SNeHeat run at t=200t = 200 Myr." ], [ "The Interstellar Medium", "Figure REF shows the gas distribution in the global galactic disc in all three runs at $t = 200$  Myr.", "The non-axisymmetric bar and spiral pattern, rotates anticlockwise, taking 120 Myr for one complete rotation.", "In the runs without feedback, NoSF and SFOnly, the galactic disc settles into a quasi-equilibrium state with no large structural change after about $t = 150$  Myr.", "The grand design bar and spiral can be clearly seen in the left and centre panels, showing a gas distribution qualitatively similar to the lower resolution CO observations of M83 [37].", "While showing strong simularities, these two non-feedback runs are not identitical.", "The inclusion of star formation removes high density gas, converting it into star particles (not shown).", "This can be seen most clearly in the bar and spiral regions, where the gas is typically denser than in the disc.", "In these regions, the SFOnly run shows a clear reduction in surface density.", "The inclusion of thermal feedback in run SNeHeat, seen in the right-hand panel of Figure REF , produces a stronger global change.", "The galaxy disc remains structurally similar, but the green and yellow mid-density regions at $10^1 \\sim 10^2\\ \\rm M_{\\odot } pc^{-2}$ are more widely distributed in each of the bar, spiral and disc regions.", "There is also evidence of gas outflows, especially in the densest bar region.", "This is the effect of the thermal stellar feedback injecting energy into the dense cloud gas surrounding the newly formed star particles and causing it to expand.", "This gas is ejected from the dense clouds, increasing the density of the warm interstellar medium.", "In addition, comparing the structure of the bar and spiral features in SNeHeat with the other two runs, more gas appears to have been funneled towards the centre.", "We will return to this observation later in the cloud analysis.", "Figure: Close-up images of the bar-end region of the galactic disc at t=200t = 200 Myr in the three simulations.", "The left-hand column shows the disc for the NoSF simulation, middle is SFOnly and the right most column is for SNeHeat.", "Each image is a 5 kpc across.", "Top to bottom, the images show the gas surface density, the gas volume density in the disc midplane and the gas temperature in the midplane.", "The marked `x' is the location of the galactic centre.", "The three circles in the right-hand panels are clouds at various stages at evolution, as discussed in the text.Figure REF shows 5 kpc $\\times $ 5 kpc close-up images of the face-on view of the bar end at $t = 200$  Myr, for all three simulations.", "The top row is the gas surface density, middle is the gas volume density in the disc mid-plane, and the bottom row is the mid-plane temperature.", "In the left-hand column, the NoSF run shows the gas evolution without the influence of star formation or feedback.", "There are clearly defined dense knots of material corresponding to the clouds.", "These show up as red collapsed regions in the density panels and dark red cold gas in the temperature panels.", "These clouds are undergoing tidal interactions in the denser environments, pulling spiral filaments of structure around them.", "Closest to the galaxy centre, in the bar itself, more massive clouds are forming and dominating the local gravitational environment.", "This increases the density of the tidal tails to produce a transient population of clouds, as discussed in more detail in Paper I.", "In between the densest regions of cloud material at the beginning of the spiral arms is warm, low-density gas.", "The SFOnly simulation in the middle column shows a similar structure of dense knots, but noticeably fewer tidal filaments.", "This is due to the star formation eroding the reservoirs of dense gas to shrink the size of the largest clouds.", "The result is a less extended population of massive structures to pull gas away from neighbouring clouds.", "The effect of feedback in the SNeHeat images (right-hand column of Figure REF ) has made even more significant differences.", "The gas surface density images have lost their ordered clump and filament structure seen in NoSF and SFOnly, and instead show a higher density reservoir of gas surrounding the clouds.", "While the inclusion of star formation has reduced the number of massive clouds and tidal tails, adding stellar feedback has dispersed part of the gas in the clouds to form a new, more turbulent mix of filaments and clumps.", "In the mid-plane slice of density and temperature for run SNeHeat, three circles show the early (circle A), mid (circle B) and late (circle C) phase of a feedback site.", "In the centre of circle A, we see a high density knot of gas that has risen to high temperatures.", "This is where a star particle has just injected thermal energy into the surrounding gas.", "This thermal energy causes the gas to expand, leading to a hot cavity that can be seen inside circle B.", "The thermal energy is eventually radiatively lost, leading a high density rim to the expanding cavity, as can be seen in circle C. Figure: Left: Edge-on gas surface density of the galactic disc at t=t = 200 Myr for the three runs: NoSF, SFOnly, and SNeHeat.", "Each image is 10 kpc across.", "The `x' mark at the left side of the image shows the galactic centre.", "Right: 1D profile of the scale height for the galactic disc at t=200t = 200 Myr.", "The y-axis is the mass weighted average density as a function of the position, zz.", "The black solid line is NoSF, the blue dashed line is SFOnly, and the red dotted line is SNeHeat.The effect of the different stellar physics on the vertical profile of the disc is shown in Figure REF .", "The left-hand side of the figure shows the projected density along the disc edge for each of the three runs, while the right-hand side is the disc scale height.", "Without thermal feedback, the vertical height of the disc is primarily controlled by cloud interactions that can scatter denser material off the disc plane.", "This effect is more marked in the NoSF run images compared to the SFOnly for the same reasons that fewer filaments were seen in Figure REF ; the star formation results in fewer high mass clouds that are the most efficient at promoting cloud interactions and tidally stripping other clouds.", "This difference results in a small change in the scale height shown in the right-hand plot, where the disc height for SNOnly is marginally lower than for the NoSF run.", "The very densest gas in the SNeHeat run extends to a similar height as that in NoSF and SFOnly, with a scale height of about 400 pc.", "However, there is a large difference in the lower density gas above and below the disc.", "While the density drops sharply beyond 500 pc in NoSF and SFOnly, the density in the SNeHeat run is $10^4$ times higher.", "Visually, plumes of gas are being ejected from the disc by the thermal stellar feedback to form a galactic fountain that stretches up to several kiloparsecs above the mid-plane.", "The fact that the densest gas seems largely unaffected suggests that the thermal feedback is having the strongest affect on the medium density warm ISM, in keeping with the very extended filamentary structure seen in Figure REF .", "Since M83 is a face-on galaxy, its scale height cannot be measured.", "Our value of about 400 pc for the dense gas compares favourably to other galaxies, e.g.", "$100 \\sim 500$  pc in the Milky Way [36], [53], 180 pc in LMC [46], 200 pc in NGC 891 [55].", "Figure: The ISM mass distribution in temperature versus number density at 200 Myr.", "Left is NoSF, middle is SFOnly, and right is SNeHeat.The structural differences in the ISM can also be seen clearly in Figure REF , which shows the two-dimensional phase diagrams of temperature versus density in the three runs.", "All three simulations show a continuous distribution of densities and temperatures, demonstrating that the ISM phases are not distinct bodies of gas but part of a smoothly changing system.", "The NoSF and SFOnly simulations show almost the same ISM phase distributions: mass collects in the cold and dense clouds, sitting at the radiative cooling limit of 300 K. The surrounding warm ISM is in rough pressure equilibrium (but at multiple pressures) at temperatures around $10^4$  K and $0.1$  cm$^{-3}$ and there is a small hot phase around $10^6$  K. Below our cooling limit at 300 K, gas above and below the disc expands adiabatically expansion to cool to about 100 K. This is slightly more marked in the NoSF case, as the increased number of cloud interactions due to the formation of more massive clouds scatters more gas off the mid-plane.", "Likewise, the conversion of gas into star particles reduces the amount of very dense gas in the SFOnly run.", "Compared to the NoSF and SFOnly ISM, the addition of thermal feedback greatly changes the distribution of the warm and hot gas.", "Thermal stellar feedback is injected into the densest star forming regions, producing hotter gas at cloud densities above 100 cm$^{-3}$ .", "As with the disc images in Figures REF and REF , the main difference is seen at lower densities outside the clouds, between $10^{-3} \\sim 10^1\\ \\mathrm {cm^{-3}}$ .", "The warm ISM now contains more gas mass and spreads to higher densities, reflecting the heavy tangle of filaments visible in Figure REF .", "At densities below $< 10^{-1}\\ \\mathrm {cm^{-3}}$ , we see a much broader range of temperatures corresponding to the galactic fountain, which throws hot gas off the disc where it cools and returns to the midplane, as seen in the edge-on disc views of Figure REF .", "Figure: Radial profiles of the evolution of the three galactic discs at t=120,170,220t = 120, 170, 220 Myr.", "Bins are azimuthally averaged with a width of 333 pc.", "Top to bottom rows show (1) the gas surface density, Σ gas =∫ -1 kpc +1 kpc ρ(z)dz\\Sigma _{\\rm gas} = \\int _{-1\\mathrm {kpc}}^{+1\\mathrm {kpc}}\\rho (z)\\mathrm {d}z, (2) gas circular velocity (mass-weighted average over -1 kpc <z<1 kpc -1\\ \\mathrm {kpc} < z < 1\\ \\mathrm {kpc}) and (3) 1D gas velocity dispersion defined by σ gas =(𝐯-𝐯 cir ) 2 /3\\sigma _{\\rm gas} = \\sqrt{(\\mbox{$v$} - \\mbox{$v$}_{\\rm cir})^2/3} (also mass-weighted average over -1 kpc <z<1 kpc -1\\ \\mathrm {kpc} < z < 1\\ \\mathrm {kpc}).", "Star formation and feedback is included only after t=120 Myr t = 120 \\rm \\ Myr, so that the profiles of the three runs at 120 Myr are the same in each column.The evolution of the ISM is shown in the one dimensional radial profiles in Figure REF .", "At 120 Myr, all galaxy discs have the same profile, since this is just prior to the beginning of star formation and feedback, as described in Section REF .", "For the NoSF run, all three profiles in gas surface density, circular velocity and velocity dispersion remain steady over time.", "Once star formation in include in SFOnly, the gas surface density drops with time as gas is removed from the disc to form stars.", "This is most pronounced in the galactic bar region ($r < 3\\ \\mathrm {kpc}$ ) where the star formation timescale is shortest, due to the high gas surface density.", "Once feedback is included in SNeHeat, the same decrease in gas surface density is seen in the outer parts of the disc beyond $\\sim 4$  kpc, but the the central region of the galaxy remains gas rich.", "This suggests one of two possibilities are occurring.", "The first option is that star formation is being suppressed more strongly in the central region by the thermal feedback than in the outer parts of the disc.", "The second possibility is that gas is inflowing to the galactic centre, replenishing the material lost due to star formation.", "This latter option is supported by the images of the global galaxy disc in Figure REF , where the gas surface density appears to be raised in the central region compared to the SFOnly and NoSF runs.", "Our exploration of the star formation in the next section will confirm this view.", "The middle and bottom rows show the changes in circular velocity and velocity dispersion over time.", "The velocity dispersion rises in the central $< 2$  kpc of all discs due to the elongated elliptical motion of gas in the bar potential.", "The densely packed bar region is also the site of the highest number of cloud collisions, further raising the velocity dispersion in this region.", "The inclusion of star formation in SFOnly produces a slight rise in the velocity dispersion as the coldest gas is removed from the disc to form stars, leaving the average to be weighted by the faster moving warm sector.", "This effect increases far more significantly when thermal feedback is injected in SNeHeat, heating the dense gas and creating a more extensive warm and hot ISM phase.", "The average circular velocity also shows changes in the SNeHeat run, implying that the predominantly circular motion of the gas is being disrupted.", "This again points to an increase in radial gas motions, as suggested by the possible inflow of material into the galactic center just discussed." ], [ "Star formation", "The left panel of Figure REF shows the star formation history of the galaxy in SFOnly (blue solid line) and SNeHeat (red dashed line) from when star formation begins at t=120 Myr.", "In the first 10 Myr, both discs show a rising star formation rate, reaching over 25 M$_\\odot $ yr$^{-1}$ .", "This burst phase is a consequence of starting star formation in the pre-fragmented gas at $t = 120$  Myr.", "The large number of dense clumps all rapidly form stars.", "After this starburst, the star formation rate gradually decreases as the quantity of dense gas decreases due to conversion into stars.", "Both the SFOnly and SNeHeat simulations show similar histories over the next 100 Myr, decreasing to around 15 M$_\\odot $ yr$^{-1}$ by 200 Myr.", "This value roughly corresponds to the observed SFR of $5 \\sim 20\\ \\mathrm {M_{\\odot }yr^{-1}}$ in M83 [24].", "It should be noted that this match is partially due to our choice of star formation efficiency, as described in Section REF .", "Despite the similar trend, the star formation rate when feedback is included in SNeHeat is slightly higher than for SFOnly for all times after $\\sim 140$  yr.", "This suggests that the addition of thermal feedback may initially disrupt the star forming dense gas (between $t = 120 - 140$  Myr) but this effect is swiftly overwhelmed by a positive feedback loop.", "The result is a nearly constant SFR in the last 20 Myr of the SNeHeat simulation, while SFOnly continues to use up the available dense gas.", "This positive effect on the star formation can stem from multiple possible effects.", "Star formation can be triggered in shells of compressed gas in the wake of strong outflows following the thermal energy injection.", "Alternatively, gas may be more effectively recycled by the galactic fountain and other outflows, making the net efficiency of gas into stars higher.", "The ejection of gas into the warm ISM may also encourage the replenishment of gas in the galactic centre, where short dynamical times in the dense region encourage higher star formation rates.", "While triggering and recycling may play a role, the feeding of gas into the galaxy centre appears to be a strongly controlling factor.", "The maintenance of a high central gas density is seen in Figures REF and REF , and the star formation history in Figure REF confirms this is not due to a low star formation rate, but from a replenishment of fresh material.", "Feedback therefore has minimal effect on star-forming cloud gas, but boosts star formation by maneuvering gas to maintain dense areas.", "The right panel of Figure REF shows the relation between the SFR surface density ($\\Sigma _{\\rm SFR}$ ) and the gas surface density ($\\Sigma _{\\rm gas}$ ), averaged in cylindrical regions of radius 500 pc and height 5 kpc.", "The colours of each region marker indicate the location in the galaxy: bar (red crosses), spiral (green circles) and disc (blue squares).", "The resulting correlation is known as the Kennicutt-Schmidt relation: $\\Sigma _{\\rm SFR} \\propto {\\Sigma _{\\rm gas}^N}$ , where $N$ is typically observed to be between 1 and 2 for combined molecular and atomic hydrogen [29].", "Recent sub-kpc resolution observations also show an index around 1 for the pure molecular gas and slightly steeper for the atomic component [4], [34].", "Both the SFOnly and SNeHeat relations follow an index of $\\sim 1.7$ , in reasonable agreement with observations.", "This super-linear relation results in a trend in the star formation efficiency (SFE) in each region: bar $>$ spiral $>$ disc.", "Such ordering is present in both SFOnly and SNeHeat, although the thermal feedback does slightly reduce the SFE in all regions.", "Moreover, there are bar regions in the SNeHeat simulation with markedly higher gas surface density ($> 10^2\\ \\mathrm {M_{\\odot }pc^{-2}}$ ) and SFR ($> 10^0\\ \\mathrm {M_{\\odot }yr^{-1}kpc^{-2}}$ ), further adding to the picture that this region is fed by gas inflow.", "This high SFE in the bar is contrary to observations that show a lower efficiency in this region than in the spiral arms [41], [24].", "In [15], we showed that such a difference could be reproduced by assuming that star formation was triggered during cloud collisions using an adaption of the star formation model of [63] that allowed for a dependence on cloud collision velocity as suggested by [62].", "In this situation, cloud interactions in the bar region were too violent to result in productive star formation, due to the elliptical global gas motion in the stellar bar potential.", "This result is not replicated in this model as our star formation model is based on local gas density, rather than cloud interactions.", "This is presently an unfortunate necessity, since even at our resolution, we do not have the spatial accuracy to resolve a shockfront during a cloud interaction that would lead to star formation, nor is it possible to calculate the cloud interaction rate swiftly enough to be used during the hydrodynamical calculation.", "Our results are therefore consistent with the estimates in Paper I, which also primarily considers gas density, but some variation in the bar region is possible if an alternative cloud interaction-based scheme were to be employed." ], [ "Cloud properties", "Clouds in our simulations were defined as coherent structures of gas contained within contours at a threshold density of $n_{\\rm gas} = 100$  cm$^{-3}$ as described in Section REF .", "In this section, we discuss the effects of star formation and stellar feedback on these cloud properties.", "In Paper I, we introduced three different types of clouds that form in the simulation.", "The most common Type A clouds constitute the largest fraction of clouds in each galactic environment.", "Their properties are typical of those observed in observations; a peak mass of $5\\times 10^5 M_{\\odot }$ , radius 15 pc, and velocity dispersion of 6 km/s.", "The larger Type B clouds are giant molecular associations (GMAs) that formed through successful mergers between smaller clouds and had radii greater than 30 pc.", "The third, Type C, clouds are transient clouds that are gravitationally unbound with short lifetimes, typically below 1 Myr.", "They form in the dense filaments and tidal tails caused by cloud interactions, most commonly between the massive Type B clouds and Type A.", "Within the NoSF disc, Paper I found that the fractions of each cloud type depended on the frequency of interactions between clouds.", "This made their relative quantities environment dependent, with the bar having the highest fraction of the merger-induced Type B and the resultant Type C clouds." ], [ "Cloud scaling relations", "Figure REF shows three relationships between a cloud's radius and its mass (top row), one dimensional velocity dispersion (middle row) and virial parameter measuring gravitational binding (bottom row).", "The cloud mass, $M_{\\rm c}$ , is a sum of the mass in each cell within the cloud, while the cloud radius is defined as $R_{\\rm c} = \\sqrt{(A_{xy} + A_{yz} + A_{zx})/3\\pi }$ , where $A_{xy}$ is the projected area of the cloud in the $x$ -$y$ plane, $A_{yz}$ is that in the $y$ -$z$ plane, and $A_{zx}$ is in the $z$ -$x$ plane.", "The top row also shows the definitions for the three kinds of clouds.", "The clouds lie on two sequences of surface density above and below 230 M$_\\odot $ pc$^{-3}$ .", "The bottom trend is the transient Type C clouds, while the top trend consists of the typical Type A clouds and the giant associations, Type B, with sizes larger than 30 pc.", "These definitions are the same as in Paper I.", "All three runs show the same split into three cloud types, ensuring these are good tracers of environmental change in the presence of feedback.", "The distributions for the common Type A clouds do not change significantly with the addition of star formation, with the radius extending between $5 \\sim 30$  pc for all runs and peak values sitting at 13, 17 and 13 pc for runs NoSF, SFOnly and SNeHeat, respectively.", "The mass also shows little variation.", "The mass range runs between $10^4 \\sim 10^6 M_{\\odot }$ in all three simulations and the peak cloud mass is $5 \\times 10^5 M_{\\odot }$ for NoSF, increasing by a factor of two in the SFOnly and SNeHeat runs to give a mass of $1 \\times 10^6 M_{\\odot }$ .", "This small rise towards higher values is because a few of the giant Type B clouds that sit close to the 30 pc divide become large Type A clouds as star formation reduces their gas content, boosting the larger Type A end of the distribution.", "These values are slightly larger than the GMCs observed in the Milky Way and M33, which have peak masses of $5\\times 10^4$  M$_\\odot $ and $1\\times 10^5$  M$_\\odot $ respectively [49], [51].", "However, since we do not consider molecular gas separately from atomic gas in the simulations, the observed values should be doubled to include the atomic envelope [5], [16] and may even be higher by an order of 2 or 3 to allow for uncertainties in areas such as survey resolution [2], [27].", "The mass range for our clouds from $10^4 \\sim 10^7$  M$_\\odot $ , comfortably includes the observed peak for these two galaxies.", "At present, there is no observational survey of M83, but this may change with ALMA.", "The reduction in the size of the Type B clouds can be seen in the maximum size they reach in the three simulations.", "Without star formation, the maximum cloud size is $10^8 M_{\\odot }$ and 80 pc.", "Once star formation converts these gas-rich associations into stars, the maximum size becomes $2 \\times 10^7 M_{\\odot }$ and 60 pc in SFOnly.", "The Type B clouds are most sensitive to this effect, as their high mass results in a high star formation rate, as will be later shown.", "Once feedback is included, there is a similar stunting of the Type B clouds, which reach a maximum mass of $3 \\times 10^7 M_{\\odot }$ and 50 pc, with the exception of one outlier, which lies in a particularly crowded region in the global design.", "Compared to the SFOnly run, the Type B cloud masses show more scatter, indicative of outflows that may remove part of their diffuse outer envelope to leave a more compact object with a similar mass.", "The total number of Type B clouds is also the smallest here, suggesting that, at least in some cases, enough mass is lost to convert some of the Type B clouds into Type A.", "Figure: The number of each cloud type for all three simulations at t=t = 200 Myr.The most prominent difference across the three runs occurs for the transient Type C clouds.", "The cloud mass-radius distribution shows a similar trend in all three runs, but the number of Type C clouds drops markedly when star formation (but not feedback) is included.", "This is due primarily to the formation mechanism for this cloud type.", "As the number and mass of the giant GMA Type B clouds reduces with the addition of star formation, the amount of gas in tidal tails also declines, reducing the environment where Type C transient clouds predominantly form.", "This situation reverses once feedback is introduced.", "In the cloud-mass relation of the SNeHeat run, the number of Type C clouds blooms, forming a more extended trend that has both lower mass and higher mass tails.", "This can be seen most clearly in the bar chart in Figure REF .", "The red bar shows the number of Type C clouds in each run, with the number dropping to roughly half its value between NoSF and SFOnly, but increasing by a factor of 2.5 from NoSF to SNeHeat, becoming the most dominant cloud type, by number, in that simulation.", "The presence of a large number of Type C clouds without a strong Type B cloud population suggests an alternative formation mechanism in the SNeHeat simulation that does not involve cloud production within tidal tails.", "The addition of the thermal stellar feedback boosts not only the warm ISM as discussed in Section REF , but also produces a population of Type C clouds in this intercloud material.", "Since these Type C clouds are formed in a different environment to those in the tidal tails of NoSF and SFOnly, their properties are not completely identical, with a larger number having smaller and larger radii, as seen in the scaling relations in Figure REF .", "Figure REF also confirms the slight reduction in Type B clouds seen in the cloud-mass relation for SNeHeat.", "These clouds have lost their outer layers to become Type A clouds, while the smaller end of the Type A clouds have been dispersed.", "This keeps the total number of Type A clouds roughly constant between runs SFOnly and SNeHeat.", "The middle and bottom panels of Figure REF show the scaling relations for the 1D velocity dispersion and the virial parameter versus radius.", "The mass-weighted 1D velocity dispersion of a cloud is defined as: $\\sigma _{\\rm 1D} = \\sqrt{\\frac{(v_x-v_{{\\rm c},x})^2+(v_y-v_{{\\rm c},y})^2+(v_z-v_{{\\rm c},z})^2}{3}},$ where $(v_{x}, v_{y}, v_{z})$ is the velocity of the gas, and $(v_{{\\rm c},x}, v_{{\\rm c},y}, v_{{\\rm c},z})$ is the velocity of the cloud's centre of mass.", "The virial parameter is defined as $\\alpha _{\\mathrm {vir}} = 5\\frac{(\\sigma _{\\rm 1D}^2 + {c_{\\rm s}}^2)R_{\\rm c}}{\\mathrm {G}(M_{\\rm c} + M_{\\rm s})},$ where $c_{\\mathrm {s}}$ is the sound speed, $M_{\\rm c}$ is the total gas mass of the cloud, and $M_{\\rm s}$ is total mass of star particles that are in the cloud's boundary.", "The virial parameter, $\\alpha _{\\mathrm {vir}}$ , is a measure of gravitational binding, with a value less than 2 indicating that the cloud is gravitationally bound [3].", "The main features of the $\\sigma _{\\mathrm {1D}}$ -$R_{\\rm c}$ and $\\alpha _{\\mathrm {vir}}$ -$R_{\\rm c}$ relations are reproduced in all three runs for all cloud types: The Type A and Type B clouds lie on the same trend with increasing velocity dispersion and virial parameter value with radius.", "The massive GMA Type B clouds are borderline gravitationally unbound, in keeping with their higher velocity dispersion, while the smaller Type A clouds tend to sit just on the other side of the cut-off, and are borderline bound.", "The transient Type Cs follow a similar trend with velocity dispersion and radius, but with a lower velocity dispersion than the larger Type A and B clouds.", "They lie in a different part of the $\\alpha _{\\mathrm {vir}}$ -$R_{\\rm c}$ relation, being generally small and unbound, but becoming borderline bound as their masses increases.", "While the overall properties are similar, the addition of star formation and feedback does have some impact.", "The gradient of the $\\sigma _{\\mathrm {1D}}$ -$R_{\\rm c}$ correlation steepens when star formation is included in SFOnly and SNeHeat.", "This is due to the densest parts of the cloud now forming stars, leaving the remaining gas with a higher average velocity dispersion.", "Without stellar physics, the peak value for the velocity dispersion is $\\sigma _{\\mathrm {1D}} = 6$  km/s but rises to 10 km/s in SFOnly and SNeHeat.", "These numbers are all close to what is observed for GMCs, with M33 clouds having a characteristic velocity dispersion of 6 km/s and the Milky Way, a lower value of 1 km/s.", "This increase in slope for the velocity dispersion also affects the virial parameter, steepening the $\\alpha _{\\mathrm {vir}}$ -$R_{\\rm c}$ relation in SFOnly and SNeHeat compared to NoSF as well.", "The clouds remain borderline gravitationally bound, in keeping with the observations of clouds in the Milky Way.", "Figure: The percentages of each cloud type in each galactic region (=N type /N region ×100= N_{\\mathrm {type}}/N_{\\mathrm {region}}\\times 100) at t=200t =200 Myr.As for the $M_{\\rm c}$ -$R_{\\rm c}$ relation, the distribution of Type C clouds shows the largest difference between the simulations.", "Between NoSF and SFOnly, the trend for these transient clouds remains similar, with less clouds present when star formation has reduced the size of the Type Bs and thus reduced the filament environment.", "With the addition of feedback, the number of Type C clouds is substantially larger and the scatter in their relation also increases.", "The additional formation process in the denser warm ISM dispersed by stellar feedback produces a wider range of radii and velocity dispersions, leading to a more variable (yet still mostly unbound) virial parameter." ], [ "Cloud lifetime and merger rate", "The cloud lifetime and merger rate were discussed extensively in Paper I, so we include only a brief discussion here.", "The distributions of lifetimes for each cloud type are very similar between the three runs: all clouds have a typical lifetime of less than 10 Myr, which agrees well with estimates that suggest lifespans of 1-2 dynamical times, with ages in the range 5-30 Myr [6], [28], [40].", "The maximum Type A cloud lifetime ranges up to 30 Myr; by contrast, Type B clouds are longer lived, with a few per cent living longer than 40 Myr.", "The smallest lifetimes are for Type C clouds, the vast majority of which live only a few Myr.", "As discussed in Paper I, the cloud lifetime is linked to interactions between clouds.", "The massive Type B clouds have the highest merger rate, as high as 1 merger every 2-3 Myr.", "They undergo many mergers during their long lifetime, accounting for their large mass and size.", "On the other hand, the transient Type C clouds have the lowest merger rate, in keeping with their short lifetimes.", "They either merge or their high virial parameter causes them to dissipate shortly after birth.", "The lack of difference in the cloud lifetime and merger rate between the three runs shows that stellar feedback does not disperse the clouds (entirely), instead cloud interaction primarily controls their lifespan." ], [ "Cloud Properties By Environment", "The difference in cloud properties in the three galactic environments can be seen clearly in Figure REF .", "This bar chart shows the percentage of each cloud type in the bar, spiral and disc for the three simulations at 200 Myr, showing how the three types are divided.", "In both NoSF and SFOnly, the galaxy is dominated throughout by the Type A clouds.", "The other two types are most prevalent in the bar region, where the high cloud interaction rate due to the elliptical gas motions encourages the formation of GMAs and the tidal tails that give rise to Type C clouds.", "This is followed by the spiral region, where gas is gathered in the spiral-arm potential also boosts cloud interactions, while the quiescent disc region shows the smallest percentage of Type B and Type C clouds.", "(Paper I contains a more detailed discussion of these differences.)", "Run SFOnly does show a smaller fraction of Type C clouds in the bar region than NoSF, due to the smaller Type B population.", "With thermal feedback included, the percentage of Type A clouds remains high, but in the central bar and spiral regions of the disc, it is overtaken by the fraction of Type C clouds.", "Now forming in the dispersed gas from feedback, Type C clouds make up a larger percentage of the cloud population in every environment, but become markedly more prominent towards the galaxy centre.", "In contrast to this, the percentage of massive Type B clouds becomes lower and more uniform across the three environments, reinforcing our claim that they are not the primary cause of the boost in the Type C's.", "These large GMAs are kept smaller than in other runs by star formation and stripping of their outer layers through feedback.", "This stripped gas is funnelled towards the galaxy bar to keep the star formation highest in the central regions.", "This feedback results in an increase in the amount of feedback in the bar and spiral environments, boosting the dense filamentary structure of the warm ISM and forming more Type C clouds.", "Figure: Normalised distributions of the cloud properties in the bar region (triangle solid lines), spiral region (circle dotted lines) and disc region (square dashed line) at t=t = 200 Myr.", "From top to bottom, each row shows the cloud mass, the cloud surface density, Σ c =M c /(πR c 2 )\\Sigma _{\\rm c} = M_{\\rm c}/(\\pi R_{\\rm c}^2) and the virial parameter.", "From left to right are the cloud populations of the NoSF, SFOnly and SNeHeat runs.The distribution of properties within the cloud population itself is shown in Figure REF .", "The top row of three panels shows the normalised distribution of cloud mass in the three runs.", "The population of clouds in the bar region (red solid line with triangular markers) is most strongly affected by the addition of stellar physics, while the lower density, quiescent outer disc region (Blue dashed line with square markers) is more uniform across all three runs.", "As star formation erodes the highest density gas, the maximum cloud mass is reined back from above $10^8$  M$_\\odot $ to about $3\\times 10^7$  M$_\\odot $ in the bar region, where the highest fraction of Type B clouds form.", "This population of giant GMAs result in a distinctive bump in the bar profile that remains marked in all three simulations, peaking at around $10^7$  M$_\\odot $ .", "The split between these Type B and the smaller Type A clouds occurs at around $M_{\\mathrm {c}} = 5 \\times 10^6\\ \\mathrm {M_{\\odot }}$ , corresponding to $R_{\\mathrm {c}} = 30\\ \\mathrm {pc}$ , where the border is marked in the scaling relations of Figure REF .", "The relative number of Type B clouds in the bar region matches that already seen in the bar chart in Figure REF : the fraction rises in SFOnly and then drops in SNeHeat.", "This is not due to the absolute numbers of the GMAs increasing, but rather due to the fluctuations in the relative amount of the small Type C clouds.", "These can be seen at the low mass end of each distribution.", "In both the bar and spiral environments, the number of clouds below $M_c < 10^5$  M$_\\odot $ rises sharply when feedback is included in SNeHeat.", "By contrast, the disc region maintains a peak cloud value typical of that of the Type A clouds at $7\\times 10^5$  M$_\\odot $ , indicating that the outer parts of the disc are less influenced by feedback's effect on the surrounding ISM.", "The middle row of Figure REF shows the surface density of the cloud populations.", "With the clear near parallel relations in Figure REF seen between the transient Type C and larger Type A/B clouds, these profiles are expected to be bimodal, with the dip at $\\Sigma _{\\mathrm {c}} = 230\\ \\mathrm {M_{\\odot } pc^{-2}}$ that divides these two trends.", "This is seen in all simulations, but most prominently when feedback is included in SNeHeat, which bumps the Type C clouds to a population even larger than the Type A and B combined.", "A very clear trend can be seen going from disc to spiral to bar environments in the Type C low-mass mode, with the bar environment gaining the largest number of Type C clouds due to the influx of interstellar material from the outer parts of the disc.", "In contrast, the trend is weakest in the SFOnly run, where the formation of extremely massive clouds is throttled by star formation, reducing the filament environment where Type C clouds form.", "Without feedback to boost the filamentary structure in the warm ISM, the number of Type C clouds also declines.", "The bottom row in Figure REF shows the distribution of the cloud's virial parameters, with the vertical dotted line marking $\\alpha _{\\rm vir} = 2$ , the boundary between gravitationally bound and unbound clouds.", "In NoSF, the clouds are predominantly bound (although only just) in all regions, with a peak at $\\alpha _{\\mathrm {vir}} \\sim 1$ .", "The bar region has a high-end tail of unbound clouds due to the high fraction of Type C clouds.", "Once star formation is included, this high-end unbound tail retreats, as the lower numbers of massive Type B clouds reduce the filament environment for Type C formation.", "The width of the distributions in SFOnly does increase, due to the boost in velocity dispersion from to the densest gas being removed to form stars, as discussed earlier.", "In SNeHeat, the drastic increase in Type C clouds greatly broadens the distribution, especially in the galaxy's central region.", "The peak in the bar environment moves from 1 to 3 to reflect the dominant Type C distribution." ], [ "Cloud star formation", "The previous section focussed on the changes in the cloud gas under different assumptions about star formation and feedback.", "This section considers the result of these variations on the cloud star formation.", "Figure: Cloud SFR (averaged over the previous 1 Myr) versus cloud mass at t=t = 200 Myr.", "Type C clouds do not form stars and so do not show up in this plot.The SFR per cloud as a function of its gas mass is shown in Figure REF for the two simulations that include active star formation.", "The estimate of the SFR within the cloud is calculated from the number of newly formed star particles within the preceding 1 Myr that are born within the cloud boundary.", "The overall trend is similar for both SFOnly and SNeHeat, with a clear relationship between cloud gas mass and the resultant SFR.", "With their much larger mass, Type B clouds sit in the top-right of the relation, while the smaller Type A clouds are in the left-bottom.", "The transient Type C clouds cannot be seen on the plots at all, since their low surface densities and short lifetimes prevents them from reaching the densities needed to form star particles.", "Notably –unlike the mass-radius scaling relation in Figure REF – the Type A and Type B clouds do not quite form a continuation of the same trend.", "In the region around $3 \\times 10^6 \\sim 8 \\times 10^6 M_{\\odot }$ , both Type A and Type B clouds overlap, but the Type A clouds have a consistently higher SFR for the same cloud gas mass.", "This difference is due to the large radii of the Type B clouds.", "Formed through multiple mergers of smaller clouds and defined as having radii above 30 pc, the Type B population includes less compact objects that tend to feature an envelope of diffuse tidal interaction gas.", "By contrast, a Type A cloud of the same mass have a radius below 30 pc, resulting in a denser cloud.", "This higher density gives Type A clouds a higher SFR, producing the observed offset.", "When stellar feedback is included, the spread in the Type A clouds remains very similar, but the Type B clouds show variation at the low and high mass tails.", "In particular, at the high-mass end, the SNeHeat simulation contains a group of Type B clouds with significantly higher masses and SFRs than the other clouds in the simulation.", "These clouds were also seen in Figure REF and exist in the central bar region, which benefits most strongly from the gas inflow towards the galaxy centre.", "In this region, the gas density can get very high, with a corresponding high SFR." ], [ "The cloud drag force", "As we have seen, the addition of stellar feedback boosts star formation in the galaxy's central regions by gas inflow; however, the mechanism for moving the gas is less obvious.", "This section takes a closer look at how gas flows may be driven.", "First, we demonstrate that there is a net transfer of dense, star-forming clouds from larger to smaller radii.", "The left-hand panel of Figure REF shows the total mass in clouds of each cloud type within the three galactic regions for the SFOnly and SNeHeat runs.", "While there is little difference in the Type A clouds, the amount of mass in Type B and C shows substantial change.", "For the massive (and strongly star-forming) Type B clouds, the amount of mass in the spiral region reduces and (largely) appears in the bar region as feedback is included.", "Note, this appears contrary to what was seen in Figure REF , where the relative number of clouds dropped once feedback was included.", "That was due to the sheer number of forming transient Type C clouds, which contain very little mass.", "When mass is considered, the bar gets an obvious boost of Type B cloud material that moves inwards from the outer disc regions.", "Since cloud material ejected via thermal feedback would not result in a strong change in radii of cloud material, this boost in Type B cloud mass implies that the clouds themselves are moving inwards through the disc.", "The total amount of gas in Type B is slightly reduced compared to SFOnly, as outer layers are removed in feedback processes to convert some clouds into Type As.", "However, this reduction is more than compensated for in the inner galaxy regions due to the substantial influx of cloud mass.", "The right-hand panel in Figure REF shows the radial profile of the cloud-forming gas (densities above 100 cm$^{-3}$ ) for the SFOnly and SNeHeat runs.", "This shows the dense gas collecting in the central bar region ($R < 1\\ \\rm kpc$ ) when feedback is included, and a corresponding dip in the outer regions of the spiral and disc ($R > 3\\ \\rm kpc $ ).", "Cloud material is therefore moving from the outer regions to the inner regions, creating a boost in Type B cloud mass in the inner parts of the disc.", "Figure: Normalised histogram of dl z /dt\\mathrm {d}l_z/\\mathrm {d}t, where l z l_z is a zz component of the specific angular momentum of the clouds around the galactic centre.", "The variation is calculated between 199 and 201 Myr as dl z /dt={l z (201 Myr )-l z (199 Myr )}/2 Myr \\mathrm {d}l_z/\\mathrm {d}t = \\lbrace l_z(201\\ \\mathrm {Myr}) - l_z(199\\ \\mathrm {Myr})\\rbrace /2\\ \\mathrm {Myr}.", "We exclude the typeCtype\\ C clouds from this distribution due to their small size.To confirm that clouds are actually moving through the disc, we analysed the angular momentum loss of the Type A and Type B clouds.", "The result is shown in Figure REF , which plots the normalised histogram of the change in the $z$ -component of the specific angular momentum, $l_z$ , for each cloud around the galactic centre.", "To compute this change, we compared the specific angular momentum at 199 and 201 Myr for each cloud, $\\mathrm {d}l_z/\\mathrm {d}t = \\lbrace l_z(201\\ \\mathrm {Myr}) - l_z(199\\ \\mathrm {Myr})\\rbrace /2\\ \\mathrm {Myr}$ .", "Type C clouds are excluded in this distribution due to their small size and short lifetimes.", "For the two runs without feedback, the angular momentum distribution peaks at $\\mathrm {d} l_z / \\mathrm {d} t = 0$ , suggesting no net angular momentum change during the considered 2 Myr.", "On the other hand, clouds in the SNeHeat simulation peak to the left, indicating that the clouds tend to lose angular momentum during their lifetime when stellar feedback is included.", "Such a loss will cause clouds to spiral in towards the galactic centre, as seen in Figure REF .", "One possible mechanism for the movement of cloud material via angular momentum loss is the impact of drag.", "Drag opposes the cloud motion around the galaxy disc.", "The drag force, $\\mathbf {F}_{\\mathrm {drag}}$ , is defined as, $\\mathbf {F}_{\\mathrm {drag}} = \\frac{1}{2} D \\rho (\\mathbf {v} - \\mathbf {v}_{\\mathrm {c}})^2 A\\ \\frac{\\mathbf {v} - \\mathbf {v}_{\\mathrm {c}}}{|\\mathbf {v} - \\mathbf {v}_{\\mathrm {c}}|},$ where $D$ is a dimensionless drag coefficient of order unity, $\\rho $ is the density of the surrounding ISM around the cloud, $\\mathbf {v}$ is the velocity of the ISM, $\\mathbf {v}_{\\mathrm {c}}$ is the cloud velocity, and $A$ is a cross sectional area of the cloud.", "The surrounding density is calculated in the region the cloud is moving towards, within a sphere that is twice the average radius of the cloud.", "This drag equation means that the denser the surrounding ISM, the stronger the drag.", "Earlier, we say that stellar feedback resulted in an increase in the density of the intercloud medium.", "This then would explain an increased drag force on the largest Type B clouds, pulling those objects towards the galaxy centre.", "Their motion would both explain the increase in Type B mass seen in Figure REF and the replenished gas supply to keep the star formation high.", "Drag is therefore a major effect from the addition of stellar feedback.", "This view can be further confirmed by estimating the drag force on each cloud and comparing this with the angular momentum loss.", "To estimate the magnitude of the drag force, the cloud cross sectional area is calculated as $A = \\pi {R_{\\rm c}}^2$ .", "For rigid bodies, Newton's second law for rotation is $\\mathbf {r} \\times \\mathbf {F}_{\\mathrm {drag}} = \\frac{\\mathrm {d}}{\\mathrm {d}t} \\mathbf {L}= m \\frac{\\mathrm {d}\\mathbf {l}}{\\mathrm {d}t} + \\mathbf {l} \\frac{\\mathrm {d}m}{\\mathrm {d}t},$ where $\\mathbf {r}$ is a radial position of the cloud, $m$ is the cloud mass, $\\mathbf {L} = m \\mathbf {l}$ is the total cloud angular momentum, and $\\mathbf {l}= \\mathbf {r} \\times \\mathbf {v}$ is the specific angular momentum.", "Figure: Relation of -(𝐫×𝐅 drag ) z -(\\mathbf {r} \\times \\mathbf {F}_{\\mathrm {drag}})_z versus -m(dl z /dt)-m(\\mathrm {d}l_z/\\mathrm {d}t) for each cloud.", "The green circles are the Type A clouds, the blue squares are the Type B clouds, and the red x are the Type C clouds.The relationship between the drag force and the loss in angular momentum can be seen in Figure REF .", "The graph shows $-(\\mathbf {r} \\times \\mathbf {F}_{\\mathrm {drag}})_z$ and $-m(\\mathrm {d}l_z/\\mathrm {d}t)$ (calculated to have the same units) for each cloud, ignoring the mass loss term, $l_z(\\mathrm {d}m/\\mathrm {d}t)$ .", "Since both axes would contain negative values, the numbers are multiplied by $-1$ to make the scale positive.", "While the drag force term, $-(\\mathbf {r} \\times \\mathbf {F}_{\\mathrm {drag}})_z$ , is about one order of magnitude less than the angular momentum loss term, $-m(\\mathrm {d}l_z/\\mathrm {d}t)$ , there is a clear relation between them for the Type A and Type B clouds.", "The difference in magnitude means that factors in addition to drag must also be controlling the cloud motion.", "One of these is the mass loss from the cloud during the thermal energy injection.", "A normalised histogram (not shown for space) gave a value for the mass loss rate of the clouds, $dm/dt = \\lbrace M_{\\rm c}(201 \\ \\mathrm {Myr}) - M_{\\rm c}(199 \\ \\mathrm {Myr})\\rbrace /2 \\ \\mathrm {Myr} \\sim -10^4$  M$_\\odot $ Myr$^{-1}$ for SNeHeat run and $\\sim 0$ for the NoSF and SFOnly runs.", "This mass loss when stellar feedback is included means that the $\\mathbf {l} (\\mathrm {d}m/\\mathrm {d}t)$ must be calculated when estimating the drag force.", "The typical absolute value of the $l_z (\\mathrm {d}m/\\mathrm {d}t)$ is roughly the same order as that of $m (\\mathrm {d}l_z/\\mathrm {d}t)$ .", "A second factor is our simplified estimation of the drag force.", "The clouds are in reality poorly approximated as rigid bodies, since feedback will remove part of the gas and external forces distort the surface.", "Similarly, the cross-sectional area may be an underestimate, but this is hard to judge since clouds do not have a clear boundary [48].", "The exact value of the drag coefficient, $D$ is also unknown.", "All these uncertainties mean that the magnitude of the drag force is a difficult quantity to pin down.", "However, the trend shown in Figure REF suggests that drag is playing a role in the reduction of the angular momentum of the larger clouds.", "For the small Type C clouds, there is no obvious relation between drag and angular momentum loss due to their small size.", "That is consistent with Figure REF , which showed that the main clouds being pulled inwards were the massive Type B clouds.", "Although previous works have investigated the effect of stellar feedback on GMCs, the process of the expelled gas creating a strong drag force on the clouds has previously been unreported.", "This might be due to the focus on the high density, star-forming gas in previous works, rather than the lower density warm ISM that plays a role here.", "Clouds can lose their angular momentum through additional differing physical processes.", "[69] showed that clouds could drop towards the galactic central region through dynamical friction between clouds, but they stated that lifetime of the clouds must be greater than 100 Myr; a time period inconsistent with observational estimates (5 $\\sim $ 30 Myr.", "e.g.", "[28], [40]) and our own results (typically less than 10 Myr)." ], [ "Cooling and feedback", "How much damage stellar feedback inflicts on a GMC is hotly debated.", "In previous simulations, [65] found that the cloud is largely unaffected by thermal feedback, which exits the structure through the easiest route.", "Conversely, [68] finds that feedback plays a far more destructive role on the star formation nursery.", "Figure: Cooling time of a cell injected with thermal energy from stellar feedback, depending on the cell gas density.To assess the effectiveness of our feedback, we explored the time needed to radiatively cool away the energy injected during a supernovae event.", "The cooling time of the gas is given by the thermal energy divided by the cooling rate, $t_{\\rm cool} = E/ \\dot{E}$ .", "Due to the magnitude of the thermal feedback, the energy is approximately equal to the injected energy.", "The maximum value for this in a cell is $E_{\\rm cell} = f_{SN} m_{\\rm star}c^2/19$ , as described in Section REF , while the cooling rate is $\\dot{E} = n_{\\rm HI}^2\\Lambda \\Delta x^3$ , for number density $n_{\\rm HI}$ and cooling rate, $\\Lambda $ .", "Figure REF shows the cooling time as a function of the number density.", "At our star formation threshold density at $10^4\\ \\mathrm {cm^{-3}}$ , the cooling time is only a few years.", "This value is considerably shorter than the time step of the finest cell for the simulation ($dt \\sim 200\\ \\mathrm {year}$ ).", "Thermal feedback injected into this region will cool rapidly and have no opportunity to affect the ISM.", "However, our model additionally adds thermal energy to the cells surrounding the star-forming cell.", "This region will contain lower density gas consistent with the cloud identification threshold of $100\\ \\mathrm {cm^{-3}}$ .", "At these densities, the cooling time is about $10^5$ year, allowing time for the gas to respond.", "This is consistent with the discussion in [57].", "Overcooling is a well-known problem in galaxy simulations (e.g.", "[60]).", "Feedback energy injected into our densest cells will suffer from this issue, which makes the effects of our feedback a lower limit.", "Despite this, the result that the warm ISM can change the flow of star-forming gas in the simulation is unlikely to change." ], [ "Conclusions", "We performed a simulation of an M83-type barred spiral galaxy with resolution down to the pc-scale, including star formation and thermal stellar feedback.", "By comparing simulations with different added stellar physics, we explored the effect of feedback on the properties of dense clouds in the bar, spiral and outer disc regions of the galaxy.", "As initially described in Paper I, we divided our clouds into three types based on the mass-radius scaling relation: Type A clouds have properties typical to those of observed GMCs, Type B clouds were massive GMAs and Type C clouds were low density, transient objects.", "Our main results are as follows.", "We find that, overall, the inclusion of star formation (without feedback) has only a minor impact on our simulated galaxy, primarily through the conversion of dense gas in large, self-gravitating clumps into stars.", "In particular, we find that: The addition of a star formation prescription allows us to estimate star formation rates on a cloud-by-cloud basis.", "The Type B clouds have the highest star formation rate, although their diffuse outer layers result in a slightly lower SFR compared to the more compact Type A clouds of similar mass.", "The diffuse Type C clouds do not produce any stars, as they do not become dense enough to reach the star formation threshold of $10^4$  cm$^{-3}$ .", "The smaller number of massive clouds results in a decrease in the number of transient Type C clouds because these small clouds form out of the tidal debris produced from the large (Type A and Type B) clouds.", "The addition of feedback results in much more substantial changes due to outflows from the star-forming clumps and the resultant increase in the amount of gas in the interstellar medium.", "Our primary results from the simulation including feedback are: Stellar feedback does not usually destroy the GMCs, but it does disperse a substantial fraction of its gas mass.", "This gas flows into the inter-cloud region, raising the density of the warm ISM and reducing the mass in dense clouds.", "The denser inter-cloud ISM becomes a new site for the formation of Type C transient clouds.", "Rather than forming in the dense filaments associated with tidal interactions, these new Type C clouds are formed in filamentary material in the inter-cloud ISM.", "This strongly increases their number compared to both the no-star formation and star formation only runs.", "Massive clouds (especially Type B clouds) lose their specific angular momentum and move towards the inner galactic regions.", "The reason for this loss likely has multiple causes, but a trend between the drag induced by the higher density inter-cloud gas is one possible candidate.", "The inflow of the dense gas towards the galactic centre supplies a significant amount of gas to the central bar region, replenishing that which is used during star formation.", "While further study will benefit from including additional forms of feedback such as radiation and ionising winds, this work strongly points at the importance of lower density inter-cloud gas.", "It is unlikely that star formation can be considered an entity controlled solely by the surround GMC nursery, instead being both affected and effecting the gas on a much wider scale." ], [ "Acknowledgments", "We acknowledge extensive use of the yt package [67] in analyzing these results and the authors would like to thank the yt development team for their generous help.", "Numerical computations were carried out on the Cray XC30 at the Center for Computational Astrophysics (CfCA) of the National Astronomical Observatory of Japan.", "YF is financially supported as Research Fellow of Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 15J02294).", "YF is funded by Clark Memorial Foundation, and Strategic Young Researches Overseas Visits Program for Accelerating Brain Circulation commissioned by the Japan Society for the Promotion of Science (R2405).", "EJT is funded by the MEXT grant for the Tenure Track System.", "GB acknowledges support from NSF grant AST-1312888, and NASA grant NNX15AB20G, as well as computational resources from NSF XSEDE, and Columbia University's Yeti cluster.", "EJT and AH are supported by the Japanese Promotion of Science Sciety (KAKENHI Grant Number 15K05014).", "CMS acknowledges support from the European Research Council under ERC-StG grant EXAGAL-308037 and from the Klaus Tschira Foundation.", "Computations described in this work were performed using the publicly-available Enzo code (http://enzo-project.org), which is the product of a collaborative effort of many independent scientists from numerous institutions around the world.", "Their commitment to open science has helped make this work possible." ] ]
1606.04981
[ [ "On well-posedness of vector-valued fractional differential-difference\n equations" ], [ "Abstract We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \\begin{equation*} \\left\\{ \\begin{array}{rll} \\Delta^{\\alpha} u(n) &= Au(n+2) + f(n,u(n)), \\quad n \\in \\mathbb{N}_0, \\,\\, 1< \\alpha \\leq 2; u(0) &= u_0; u(1) &= u_1, \\end{array} \\right.", "\\end{equation*} where $A$ is an closed linear operator defined on a Banach space $X$.", "Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences.", "Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on strongly continuous sequences of bounded operators generated by $A,$ and natural restrictions on the nonlinearity $f$.", "Finally we present some original examples to illustrate our results." ], [ "Introduction", "In an interesting paper published in 1943, H. Bateman [13] studied the differential-difference equation $\\begin{array}{lcl}M u_{xx}(n,x)+ 2K u_x(n,x) + S u(n,x) & = & (na+a+b)[u(n+1,x)-u(n,x)] \\\\ & - & (na+c)[u(n,x)-u(n-1,x)],\\end{array}$ where $M,K,S, a,b,c $ are positive constants.", "This general formulation includes previous work of B. Taylor, J. Bernoulli and D. Bernoulli.", "In particular, it includes the temporal discretization of the diffusion equation $u(n+1)-u(n,x)= u_{xx}(n,x),$ and the wave equation $u(n+1,x)-2u(n,x)+u(n-1,x) = u_{xx}(n,x).$ Defining $v(n)(x):= u(n,x)$ and $A=\\displaystyle \\frac{\\partial ^2}{\\partial x^2}$ the diffusion and wave equations can be rewritten as $\\Delta v(n)= Av(n), \\qquad \\hbox{and}\\qquad \\Delta ^2 v(n)= Av(n+1),\\qquad n\\in \\mathbb {{N}},$ respectively, where $A$ is the generator of a $C_0$ -semigroup of operators given by $T(t)f(x)= \\frac{1}{\\sqrt{4\\pi t}} \\int _{-\\infty }^{\\infty } e^{\\frac{-(x-s)^2}{4t}}f(s)ds.$ The difference-differential equation $\\quad \\lambda p(n+1,x) - \\lambda p(n,x)= -p_x(n+1,x)$ is even covered by (REF ).", "With initial value $p(0,x)= e^{-\\lambda x}$ and border condition $p(n,0)=0, \\,\\, n=1,2,...$ it defines the probability distribution function of the Poisson process given by $p(n,x) = \\frac{\\lambda ^n x^n}{n!}", "e^{-\\lambda x}, \\quad n \\in \\mathbb {N}.$ Note that equation $(\\ref {eq-a})$ is likewise used in the so called car-following problems, see for example [16].", "Defining $u(n)(x):= \\lambda p(n,x)$ and $A=-\\frac{\\partial }{\\partial x}$ equation $(\\ref {eq-a})$ can be rewritten in abstract form as $\\Delta u(n) = Au(n+1), \\quad n \\in \\mathbb {N}_0.$ Note that $A$ is the generator of the translation semigroup $T(t)f(x)=f(x -t).$ The difference-differential equation $\\quad T(n+2,x)=2xT(n+1)-T(n,x), \\quad n \\in \\mathbb {N}_0,$ with appropriate initial values define Chebyshev polynomials of the first and second kind.", "Indeed, take $T(0,x)=1$ and $T(1,x)=x$ in the first case and $T(0,x)=1$ and $T(1,x)=2x$ in the second case.", "Defining $u(n)(x):= T(n,x),$ equation (REF ) can be rewritten in abstract form as $\\Delta ^2 u(n) = Au(n+1), \\quad n \\in \\mathbb {N}_0,$ where $Af(x)=2(x-1)f(x).$ We observe that $A$ is the generator of the multiplication semigroup: $T(t)f(x)= e^{2(x-1)}f(x).$ These abstract models, with unbounded operators $A$ defined on Banach spaces, are closely connected with numerical methods for partial differential equations, integro-differential equations [17], [28] and evolution equations [24], [26].", "See also the recent monograph [3].", "Recently, it has been shown that the extension of some of this models to fractional difference equations is a promising tool for several biological and physical applications where a memory effect appears [8], [12].", "In spite of the significant increase of research in this area, there are still many significative questions regarding fractional difference equations.", "In particular, the study of fractional difference equations with closed linear operators and their qualitative behaviour remains an open problem.", "In this paper, we propose a novel method to deal with this classes of abstract fractional difference problems.", "This method is inspired on sampling by means of the Poisson distribution.", "We will use it to develop a theory on well-posedness for the abstract fractional difference problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{\\alpha } u(n) &= Au(n+2)+f(n,u(n)), \\quad n \\in \\mathbb {N}_0, \\quad 1 < \\alpha \\le 2; \\\\u(0) &= u_0 \\in X; \\\\ u(1) & = u_1 \\in X,\\end{array}\\right.$ where $A:D(A)\\subset X \\rightarrow X$ is a closed linear operator defined on a Banach space $X$ and $f$ a suitable function.", "We notice that first studies on the model (REF ) when $A$ is a complex or real valued matrix, have only recently appeared [9], [19].", "However, the study of this equation when $A$ is a closed linear operator, not necessarily bounded, has not been considered in the literature.", "The approach followed here is purely operator-theoretic and has as main ingredient the use of the Poisson distribution $p_n(t)= e^{-t} \\frac{t^n}{n!", "}, \\quad n \\in \\mathbb {N}_0, \\quad t \\ge 0.$ Our method relies in to take advantage of the properties of this distribution when it is applied to discrete phenomena.", "More precisely, given a continuous evolution $(u(t))_{t \\in [0, \\infty )}$ we can discretize it by means of that we will call the Poisson transformation $(\\mathcal {P}u)(n):= \\int _0^{\\infty } p_n(t)u(t)dt, \\quad n \\in \\mathbb {N}_0.$ In this paper, we will show that when this procedure is applied to continuous fractional processes, these transformations are well behaved and fit perfectly in the discrete fractional concepts.", "A remarkable feature of this work will be to show that by this method of sampling we recover the concept of fractional nabla sum and difference operator in [7], which has been used recently and independently of the method used here by other authors in order to obtain qualitative properties of fractional difference equations, notably concerning stability properties [14], [15].", "The outline of this paper is as follows: In Section 2, we give some background on the definitions to be used.", "The remarkable fact here is that we highlight a particular choice of the definition introduced in [10] for the nabla operator; see the recent paper [23].", "This choice, that has been implicitly used by other authors [14], [15], is proved to be the right notion in the sense that $\\mathcal {P}(D^{\\alpha }_t u)(n+m) = \\Delta ^{\\alpha } \\mathcal {P}(u)(n),\\quad n \\in \\mathbb {N}_0,$ where $D^{\\alpha }_{t}$ denotes the Riemann-Liouville fractional derivative on $\\mathbb {R}_+$ of order $\\alpha >0, \\,$ $m-1<\\alpha \\le m$ and $\\Delta ^{\\alpha }v(n)= \\Delta ^m \\Bigg ( \\sum _{j=0}^n k^{m-\\alpha }(n-j)v(j) \\Bigg ),$ where, $k^{\\beta }(n):= \\frac{\\Gamma (\\beta +n)}{\\Gamma (\\beta )\\Gamma (n+1)} = \\int _0^{\\infty } p_n(t)\\frac{t^{\\beta -1}}{\\Gamma (\\beta )}dt, \\quad n \\in \\mathbb {N}_0, \\quad \\beta >0.$ See Definition REF and Theorem REF below.", "Then, we can show an interesting connection between the Delta operator of order $\\alpha $ (i.e.", "the Riemann-Liouville-like fractional difference) in the right hand side of (REF ) and the Caputo-like fractional difference by means of the identity (Theorem REF ): $_C\\Delta ^{\\alpha }u(n)= \\Delta ^{\\alpha }u(n) - k^{2-\\alpha }(n+1)[u(1)-2u(0)] - k^{2-\\alpha }(n+2)u(0), \\quad n \\in \\mathbb {N}_0.$ In Section 3, we use successfully the preceding definitions and properties to solve the problem (REF ), firstly, in the homogeneous linear case.", "In order to do that, we construct a distinguished sequence of bounded and linear operators $\\lbrace \\mathcal {S}_{\\alpha }(n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ that solves the homogeneous linear initial value problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{\\alpha } u(n) &= Au(n+2), \\quad n \\in \\mathbb {N}_0, \\quad 1 < \\alpha \\le 2; \\\\u(0) &= u_0 \\in D(A); \\\\ u(1) & = u_1 \\in D(A).\\end{array}\\right.$ See Theorem REF .", "In particular, when the operator $A$ is bounded, we derive an explicit representation of the solution (Proposition REF ).", "From a different point of view, this representation can be considered as the discrete counterpart of the Mittag-Leffler function $t^{\\alpha -1}E_{\\alpha ,\\alpha }(At^{\\alpha })$ (when $A$ is a complex number) which interpolates between the exponential and hyperbolic sine function for $1<\\alpha <2.$ In Section 4 we study the fully nonlinear problem (REF ).", "After to introduce the notion of solution, which is motivated by the representation of the solution in the non-homogeneous linear case (Corollary REF ), we consider a distinguished class of vector-valued spaces of weighted sequences, that behaves like $l^{\\infty }_w(\\mathbb {N}_0;X):= \\left\\lbrace \\xi :\\mathbb {N}_0\\rightarrow X \\quad / \\quad \\sup _{n\\in \\mathbb {N}_0}\\frac{\\Vert \\xi (n)\\Vert }{nn!", "}<\\infty \\right\\rbrace .$ This vector-valued Banach spaces of sequences will play a central role in the development of this section.", "The main ingredient for the success of our analysis is the observation that the special weight $w(n)=nn!,$ that represents the factorial representation of a positive integer, proves to be suitable to find existence of solutions for (REF ) in the above defined space $l^{\\infty }_w(\\mathbb {N}_0;X)$ under the hypothesis of only boundedness of the sequence of operators $\\mathcal {S}_{\\alpha }(n).$ We give two positive results in this direction.", "See Theorem REF and Theorem REF .", "In Section 5, we prove several relations between the continuous and discrete setting, including the notable identity (REF ).", "See Theorem REF below.", "This relations are obtained in the context of the Poisson transformation (REF ) whose main properties are established in Theorem REF .", "Note that the idea of discretization of the fractional derivative in time was employed in the paper [18] (see also [17] and references therein).", "Section 6 is devoted to the construction of sequences of operators $\\lbrace {\\mathcal {S}}_{\\alpha }(n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ via subordination by the Poisson transformation of $\\alpha $ -resolvent families generated by $A$ (Theorem REF ).", "A remarkable consequence is Theorem REF , which proves existence of solution for the nonlinear problem (REF ) in the space $l^{\\infty }_w(\\mathbb {N}_0;X)$ under the hypothesis that $A$ is the generator of a bounded sine family such that the resolvent operator $(\\lambda -A)^{-1}$ is a compact operator for some $\\lambda $ large enough.", "Finally, Section 7 provide us with several examples and applications of our general theorems, notably concerning the cases where either $A$ is a multiplication operator or the second order partial differential operator $\\frac{\\partial ^2}{\\partial x^2}.$ We also pay special attention to the case $\\alpha =2$ and to some related problems formatted in a slighty different way than (REF ).", "Notation We denote by $\\mathbb {N}_0:=\\lbrace 0, 1, 2, ...\\rbrace ,$ the set of non-negative integer numbers and $X$ a complex Banach space.", "We denote by $s( \\mathbb {N}_0; X)$ the vectorial space consisting of all vector-valued sequences $u:\\mathbb {N}_0 \\rightarrow X.$ We recall that the $Z$ -transform of a vector-valued sequence $f\\in s(\\mathbb {N}_0;X)$ , is defined by $\\widetilde{u}(z) := \\sum _{j=0}^{\\infty } z^{-j} u(j)$ where $z$ is a complex number.", "Note that convergence of the series is given for $|z|>R$ with $R$ sufficiently large.", "Recall that the finite convolution $*$ of two sequences $u\\in s( \\mathbb {N}_0; \\mathbb {C})$ and $v\\in s( \\mathbb {N}_0; X)$ is defined by $(u*v)(n) := \\sum _{j=0}^n u(n-j)v(j), \\quad n \\in \\mathbb {N}_0.$ It is well known that $\\widetilde{(u*v)}(z)=\\widetilde{u}(z)\\widetilde{v}(z),\\quad |z|>\\max \\lbrace R_1,R_2\\rbrace ,$ where $R_1$ and $R_2$ are the radius of convergence of the $Z$ -transforms of $u$ and $v$ respectively.", "The Banach space $\\ell ^1(X)$ is the subset of $s( \\mathbb {N}_0; X)$ such that $\\Vert u\\Vert _1:=\\sum _{n=0}^\\infty \\Vert u(n)\\Vert <\\infty ; $ and the Lebesgue space $L^1(\\mathbb {R}_+; X)$ is formed by measurable functions $f:\\mathbb {R}_+\\rightarrow X$ such that $\\Vert f\\Vert _1:= \\int _0^\\infty \\Vert f(t)\\Vert dt\\,<\\infty .$ The usual Laplace transform is given by $\\hat{f}(\\lambda ):= \\int _0^{\\infty } e^{-\\lambda t} f(t)dt,\\qquad \\Re \\lambda >0, \\quad f\\in L^1(\\mathbb {R}_+; X).$ In the case $X= \\mathbb {C}$ , the Banach space $L^1(\\mathbb {R}_+)$ is, in fact, a Banach algebra with the usual convolution product $\\ast $ given by $f\\ast g(t):=\\int _0^t f(t-s)g(s)ds, \\qquad t\\ge 0, \\quad f, g\\in L^1(\\mathbb {R}_+).$ The same holds in the case of $(\\ell ^1, \\ast )$ .", "The Banach space $ C^{(m)}(\\mathbb {R}_+;X)$ is formed for continuous functions which have $m$ -continuous derivatives defined on $\\mathbb {R}_+$ with $m\\in \\mathbb {N}_0$ .", "Let $S: \\mathbb {R}_+ \\rightarrow \\mathcal {B}(X)$ be strongly continuous, that is, for all $x \\in X$ the map $ t\\rightarrow S(t)x$ is continuous on $\\mathbb {R}_+.$ We say that a family of bounded and linear operators $\\lbrace S(t)\\rbrace _{t\\ge 0}$ is exponentially bounded if there exist real numbers $M >0$ and $\\omega \\in \\mathbb {R}$ such that $\\Vert S(t) \\Vert \\le Me^{\\omega t}, \\quad t \\ge 0.$ We say that $\\lbrace S(t)\\rbrace _{t\\ge 0}$ is bounded if $\\omega =0.$ Note that if $\\lbrace S(t)\\rbrace _{t\\ge 0}$ is exponentially bounded then the Laplace transform $\\hat{S}(\\lambda )x$ exists for all $\\Re (\\lambda ) >\\omega .$" ], [ "Fractional difference operators", "The forward Euler operator $\\Delta :s( \\mathbb {N}_0; X) \\rightarrow s( \\mathbb {N}_0; X)$ is defined by $\\Delta u(n):=u(n+1)-u(n), \\quad n \\in \\mathbb {N}_0.$ For $m \\in \\mathbb {N}$ , we define recursively $\\Delta ^m :s(\\mathbb {N}_0; X) \\rightarrow s( \\mathbb {N}_0; X)$ by $\\Delta ^1= \\Delta $ and $\\Delta ^m:=\\Delta ^{m-1}\\circ \\Delta .", "$ The operator $\\Delta ^m$ is called the $m$ -th order forward difference operator and $\\Delta ^m u(n)=\\sum _{j=0}^m\\binom{m}{j}(-1)^{m-j}u(n+ j), \\quad n\\in \\mathbb {N}_0,$ for $u \\in s(\\mathbb {N}_0;X)$ .", "We also denote by $\\Delta ^0 = I,$ where $I$ is the identity operator.", "We define $k^{\\alpha }(n) := \\frac{\\Gamma (\\alpha +n)}{\\Gamma (\\alpha )\\Gamma (n+1)}, \\quad n \\in \\mathbb {N}_0, \\quad \\alpha >0.$ This sequence has appeared in [1] and [23] in connection with fractional difference operators.", "The semigroup property $k^\\alpha \\ast k^\\beta = k^{\\alpha +\\beta }$ and the generating formula $\\sum _{n=0}^\\infty k^\\alpha (n)z^n={1\\over (1-z)^\\alpha }, \\qquad \\vert z\\vert <1,$ hold for $\\alpha , \\beta >0$ , see for example [32].", "The following definition of fractional sum (also called Cesàro sum in [32]) has appeared recently in some papers, see for example [1], [23].", "It has proven to be useful in the treatment of fractional difference equations.", "Note that this definition is implicitly included in e.g.", "[4], [7], [25].", "Definition 2.1 [23] Let $\\alpha >0.$ The $\\alpha $ -th fractional sum of a sequence $u: \\mathbb {N}_0 \\rightarrow X$ is defined as follows $\\Delta ^{-\\alpha }u(n): = \\sum _{j=0}^n k^{\\alpha }(n-j)u(j)=(k^\\alpha \\ast u)(n), \\quad n \\in \\mathbb {N}_0.$ One of the reasons to choose this operator in this paper is because their flexibility to be handled by means of $Z$ -transform methods.", "Moreover, it has a better behavior for mathematical analysis when we ask, for example, for definitions of fractional sums and differences on subspaces of $s(\\mathbb {N}_0;X)$ like e.g.", "$l_p$ spaces.", "We notice that, recently, this approach by means of the $Z$ -transform has been followed by other authors, see [14], [15].", "The next concept is analogous to the definition of a fractional derivative in the sense of Riemann-Liouville, see [6], [25].", "In other words, to a given vector-valued sequence, first fractional summation and then integer difference are applied.", "Definition 2.2 [23] Let $\\alpha \\in \\mathbb {R}^+\\backslash \\mathbb {N}_0$ .", "The fractional difference operator of order $\\alpha $ in the sense of Riemann-Liouville, $\\Delta ^{\\alpha }: s(\\mathbb {N}_0;X) \\rightarrow s(\\mathbb {N}_0;X),$ is defined by $\\Delta ^{\\alpha } u(n) := \\Delta ^m ( \\Delta ^{-(m-\\alpha )} u)(n), \\quad n \\in \\mathbb {N}_0,$ where $m-1 < \\alpha < m.$ Example 2.3 Let $1<\\beta $ .", "Then $\\Delta k^\\beta (n)=\\frac{\\Gamma (\\beta +n+1)}{\\Gamma (\\beta )\\Gamma (n+2)}-\\frac{\\Gamma (\\beta +n)}{\\Gamma (\\beta )\\Gamma (n+1)}= {(\\beta -1)\\Gamma (\\beta +n)\\over \\Gamma (\\beta )\\Gamma (n+2)}=k^{\\beta -1}(n+1), \\qquad n \\in \\mathbb {N}_0.$ We iterate $m$ -times with $m\\in \\mathbb {N}$ to get for $\\beta >m$ that $\\Delta ^m k^\\beta (n)=k^{\\beta -m}(n+m), \\qquad n \\in \\mathbb {N}_0.$ Let $0<\\alpha <\\beta $ and $m-1<\\alpha <m$ for $m\\in \\mathbb {N}$ .", "By Definition REF and (REF ), we get that $\\Delta ^{\\alpha } k^{\\beta }(n)= \\Delta ^m ( \\Delta ^{-(m-\\alpha )} k^{\\beta })(n)=\\Delta ^m ( k^{m-\\alpha } \\ast k^{\\beta })(n)=\\Delta ^m ( k^{m-\\alpha +\\beta })=k^{\\beta -\\alpha }(n+m) ,$ for $ n \\in \\mathbb {N}_0.$ This equality extends [22] given for $0<\\alpha <1.$ Interchanging the order of the operators in the definition of fractional difference in the sense of Riemann-Liouville, and in analogous way as above, we can introduce the notion of fractional difference in the sense of Caputo as follows.", "Definition 2.4 [23] Let $\\alpha \\in \\mathbb {R}^+\\backslash \\mathbb {N}_0$ .", "The fractional difference operator of order $\\alpha $ in the sense of Caputo, $_C\\Delta ^{\\alpha }: s(\\mathbb {N}_0;X) \\rightarrow s(\\mathbb {N}_0;X),$ is defined by $_C\\Delta ^{\\alpha } u(n) :=\\Delta ^{-(m-\\alpha )}(\\Delta ^m u)(n), \\quad n \\in \\mathbb {N}_{0},$ where $m-1< \\alpha < m$ .", "For further use, we note the following relation between the Caputo and Riemann-Liouville fractional differences of order $1<\\alpha <2.$ The connection between the Caputo and Riemann-Liouville fractional differences of order $0<\\alpha <1$ is given in [22].", "Theorem 2.5 For each $1< \\alpha < 2$ and $u \\in s(\\mathbb {N}_0;X)$ , we have $_C\\Delta ^{\\alpha }u(n)= \\Delta ^{\\alpha }u(n) - k^{2-\\alpha }(n+1)[u(1)-2u(0)] - k^{2-\\alpha }(n+2)u(0), \\quad n \\in \\mathbb {N}_0.$ By definition and (REF ) we have $\\begin{array}{lll}\\Delta ^{-(2-\\alpha )}(\\Delta ^2 u)(n) &=& \\displaystyle \\sum _{j=0}^n k^{2-\\alpha }(n-j)\\Delta ^2 u(j)= \\displaystyle \\sum _{j=0}^n k^{2-\\alpha }(n-j)u(j+2) \\\\ &\\quad &\\qquad - \\displaystyle 2\\sum _{j=0}^n k^{2-\\alpha }(n-j)u(j+1) +\\sum _{j=0}^n k^{2-\\alpha }(n-j)u(j)\\\\&=& \\displaystyle \\sum _{j=2}^{n+2} k^{2-\\alpha }(n+2-j)u(j)-2\\sum _{j=1}^{n+1} k^{2-\\alpha }(n+1-j)u(j)\\\\ & \\quad &\\qquad + \\displaystyle \\sum _{j=0}^n k^{2-\\alpha }(n-j)u(j) = \\displaystyle \\sum _{j=0}^{n+2} k^{2-\\alpha }(n+2-j)u(j) \\\\ &\\quad &\\qquad - \\displaystyle 2\\sum _{j=0}^{n+1} k^{2-\\alpha }(n+1-j)u(j)+\\sum _{j=0}^n k^{2-\\alpha }(n-j)u(j) \\\\ \\\\&\\quad & \\qquad - k^{2-\\alpha }(n+2)u(0)-k^{2-\\alpha }(n+1)u(1)+2k^{2-\\alpha }(n+1)u(0)\\\\ \\\\&=& \\displaystyle \\Delta ^2(\\Delta ^{-(2-\\alpha )} u)(n)- k^{2-\\alpha }(n+1)(u(1)-2u(0)) - k^{2-\\alpha }(n+2)u(0),\\end{array}$ and so we obtain the desired result.", "We also have the following property for the Riemann-Liouville fractional difference of the convolution.", "Theorem 2.6 Let $1< \\alpha \\le 2$ , $u \\in s(\\mathbb {N}_0;\\mathbb {C})$ and $v \\in s(\\mathbb {N}_0;X)$ .", "Then, for each $n\\in \\mathbb {N}_0$ the following identity holds $\\Delta ^\\alpha (u*v)(n)=(\\Delta ^\\alpha u*v)(n)+ (u(1)-\\alpha u(0))v(n+1)+ u(0)v(n+2).$ For each $n\\in \\mathbb {N}_0,$ $\\Delta ^\\alpha (u*v)(n)&=& \\Delta ^{-(2-\\alpha )}(u*v)(n+2) -2 \\Delta ^{-(2-\\alpha )}(u*v)(n+1) + \\Delta ^{-(2-\\alpha )}(u*v)(n)\\\\&=& \\sum _{j=0}^{n+2}(k^{2-\\alpha }*u)(n+2-j)v(j) - 2\\sum _{j=0}^{n+1}(k^{2-\\alpha }*u)(n+1-j)v(j) \\\\&\\,&+ \\sum _{j=0}^{n}(k^{2-\\alpha }*u)(n-j)v(j) \\\\&=& \\sum _{j=0}^{n}(k^{2-\\alpha }*u)(n+2-j)v(j)-2\\sum _{j=0}^{n}(k^{2-\\alpha }*u)(n+1-j)v(j) \\\\ &\\,&+ \\sum _{j=0}^{n}(k^{2-\\alpha }*u)(n-j)v(j)+ (k^{2-\\alpha }*u)(1)v(n+1) \\\\ &\\, & +(k^{2-\\alpha }*u)(0)v(n+2)-2(k^{2-\\alpha }*u)(0)v(n+1)\\\\&=& \\sum _{j=0}^{n}\\Delta ^2(k^{2-\\alpha }*u)(n-j)v(j) + (k^{2-\\alpha }(0)u(1)+k^{2-\\alpha }(1)u(0))v(n+1) \\\\&\\,&+ (k^{2-\\alpha }(0)u(0))v(n+2)-2(k^{2-\\alpha }(0)u(0))v(n+1)\\\\ &=& \\sum _{j=0}^{n}\\Delta ^\\alpha u(n-j)v(j)+ (u(1)+(2-\\alpha )u(0))v(n+1)+u(0)v(n+2) \\\\ &\\,&- 2u(0)v(n+1)\\\\&=& (\\Delta ^\\alpha u*v)(n) + (u(1)-\\alpha u(0))v(n+1)+ u(0)v(n+2),$ proving the claim.", "We notice that for $0<\\alpha \\le 1$ the above property reads $\\Delta ^\\alpha (u*v)(n)= (\\Delta ^{\\alpha }u*v)(n) + u(0)v(n+1), \\quad n \\in \\mathbb {N}_0.$ It has been proved only recently in [23]." ], [ "Linear fractional difference equations on Banach spaces", "Let $A$ be a closed linear operator defined on a Banach space $X$ .", "In this section we study the problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{\\alpha } u(n) &= Au(n+2) + f(n), \\quad n \\in \\mathbb {N}_0, \\quad 1<\\alpha \\le 2; \\\\u(0) &= u_0 ;\\\\ u(1) & =u_1.\\end{array}\\right.$ We say that a vector valued sequence $u\\in s(\\mathbb {N}_0;X)$ is a solution of (REF ) if $u(n) \\in D(A)$ for all $n \\in \\mathbb {N}_0$ and $u$ satisfies (REF ).", "We will use the notion of discrete $\\alpha $ -resolvent family introduced in [1] to obtain the solution of the problem (REF ).", "Note that the knowledge of the abstract properties of this family of bounded operators provide insights on the qualitative behavior of the solutions of fractional difference equations.", "Definition 3.1 Let $\\alpha >0$ and $A$ be a closed linear operator with domain $D(A)$ defined on a Banach space $X.$ An operator-valued sequence $\\lbrace {\\mathcal {S}}_{\\alpha }(n)\\rbrace _{n\\in \\mathbb {{N}}_0}\\subset \\mathcal {B}(X)$ is called a discrete $\\alpha $ -resolvent family generated by $A$ if it satisfies the following conditions (i) ${\\mathcal {S}}_{\\alpha }(n)Ax=A{\\mathcal {S}}_{\\alpha }(n)x$ for $n\\in \\mathbb {{N}}_0$ and $x\\in D(A);$ (ii) ${\\mathcal {S}}_{\\alpha }(n)x=k^{\\alpha }(n)x+A(k^{\\alpha }*{\\mathcal {S}}_{\\alpha })(n)x,$ for all $n\\in \\mathbb {{N}}_0$ and $x\\in X.$ The family $\\lbrace {\\mathcal {S}}_{\\alpha }(n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ is said bounded if $\\Vert {\\mathcal {S}}\\Vert _\\infty := \\sup _{n\\in \\mathbb {{N}}_0 }\\Vert {\\mathcal {S}}_{\\alpha }(n) \\Vert <\\infty $ .", "An explicit representation of discrete $\\alpha $ -resolvent family generated by bounded operators $A$ with $\\Vert A\\vert <1$ is given in the following proposition.", "Proposition 3.2 Let $\\alpha >0$ and $A\\in \\mathcal {B}(X)$ , with $\\Vert A\\Vert <1$ .", "Then the operator $A$ generates a discrete $\\alpha $ -resolvent family $\\lbrace {\\mathcal {S}}_{\\alpha }(n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ given by $\\mathcal {S}_\\alpha (n)=\\sum _{j=0}^\\infty k^{\\alpha (j+1)}(n)A^j, \\qquad n\\in \\mathbb {N}_0.$ Since $k^{\\alpha }(n)=\\frac{n^{\\alpha -1}}{\\Gamma (\\alpha )}(1+O({1\\over n})),$ for $ n\\in \\mathbb {{N}},$ (see for example [32]), then the series is convergent for $\\Vert A\\Vert <1$ .", "Take $x\\in X$ and $n \\in \\mathbb {N}_0$ .", "Then we get that $A(k^{\\alpha }*\\mathcal {S}_\\alpha )(n)x&=&A\\displaystyle \\sum _{j=0}^n k^{\\alpha }(n-j)\\mathcal {S}_\\alpha (j)x=A\\displaystyle \\sum _{j=0}^n k^{\\alpha }(n-j)\\sum _{i=0}^{\\infty }k^{\\alpha (i+1)}(j)A^ix \\\\&=&\\sum _{i=0}^{\\infty }A^{i+1}x\\displaystyle \\sum _{j=0}^n k^{\\alpha }(n-j)k^{\\alpha (i+1)}(j) =\\sum _{i=0}^{\\infty }k^{\\alpha (i+2)}(n)A^{i+1}x,$ where we have applied the semigroup property of the kernel $k^{\\alpha }.$ Then we obtain $k^{\\alpha }(n)x+A(k^{\\alpha }*\\mathcal {S}_\\alpha )(n)x=\\sum _{i=0}^{\\infty }k^{\\alpha (i+1)}(n)A^{i}x=\\mathcal {S}_\\alpha (n)x,\\qquad n\\in \\mathbb {N}_0,$ and we conclude the proof.", "For $\\alpha >0$ fixed and each $n \\in \\mathbb {N}$ the sequence $\\lbrace \\beta _{\\alpha ,n}(j)\\rbrace _{j=1,...,n}$ was introduced in [1] as follows: For $n=1$ , $\\beta _{\\alpha ,1}(1)=k^{\\alpha }(1)=\\alpha .$ For $n=2,$ $\\beta _{\\alpha ,2}(1)&=&k^{\\alpha }(2)-k^{\\alpha }(1)\\beta _{\\alpha ,1}(1)=k^{\\alpha }(2)-\\left(k^{\\alpha }(1)\\right)^2,\\cr \\beta _{\\alpha ,2}(2)&=&k^{\\alpha }(1)\\beta _{\\alpha ,1}(1)=\\left(k^{\\alpha }(1)\\right)^2=\\alpha ^2.$ For $n=3,$ $ \\beta _{\\alpha ,3}(1)&=&k^{\\alpha }(3)-k^{\\alpha }(2)\\beta _{\\alpha ,1}(1)-k^{\\alpha }(1)\\beta _{\\alpha ,2}(1)=k^{\\alpha }(3)-2k^{\\alpha }(2)k^{\\alpha }(1)+(k^\\alpha (1))^3,\\\\\\beta _{\\alpha ,3}(2)&=&k^{\\alpha }(2)\\beta _{\\alpha ,1}(1)+k^{\\alpha }(1)\\beta _{\\alpha ,2}(1)-k^{\\alpha }(1)\\beta _{\\alpha ,2}(2)=2 k^\\alpha (2)k^\\alpha (1)-2\\left(k^{\\alpha }(1)\\right)^3,\\\\\\beta _{\\alpha ,3}(3)&=&k^{\\alpha }(1)\\beta _{\\alpha ,2}(2)= \\left(k^{\\alpha }(1)\\right)^3=\\alpha ^3.$ For $n\\ge 4,$ $\\beta _{\\alpha ,n}(1)&=&k^{\\alpha }(n)-\\displaystyle \\sum _{j=1}^{n-1}k^{\\alpha }(n-j)\\beta _{\\alpha ,j}(1),\\\\\\beta _{\\alpha ,n}(l)&=&\\displaystyle \\sum _{j=l-1}^{n-1}k^{\\alpha }(n-j)\\beta _{\\alpha ,j}(l-1)-\\displaystyle \\sum _{j=l}^{n-1}k^{\\alpha }(n-j)\\beta _{\\alpha ,j}(l) \\quad \\hbox{for} \\quad 2\\le l\\le n-1,\\\\\\beta _{\\alpha ,n}(n)&=&k^{\\alpha }(1)\\beta _{\\alpha ,n-1}(n-1)= \\left(k^{\\alpha }(1)\\right)^n=\\alpha ^n$ In case that $A$ is closed, but not necessarily bounded, the authors in [1] proved that given $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}\\subset \\mathcal {B}(X)$ a discrete $\\alpha $ -resolvent family generated by $A,$ then $1\\in \\rho (A)$ and ${\\mathcal {S}}_\\alpha (0)=(I-A)^{-1}$ ; ${\\mathcal {S}}_\\alpha (0)x\\in D(A)$ and ${\\mathcal {S}}_\\alpha (n)x\\in D(A^2)$ for all $n\\in \\mathbb {{N}},$ and $x\\in X$ ; and ${\\mathcal {S}}_\\alpha (n)x=\\displaystyle \\sum _{j=1}^{n}\\beta _{\\alpha ,n}(j)(I-A)^{-(j+1)}x,\\quad n\\in \\mathbb {{N}}, \\quad x\\in X.$ The last equality provides an explicit representation of discrete $\\alpha $ -resolvent families in terms of a bounded linear operators which is, in fact, a characterization of this family of operators as the next theorem shows.", "Theorem 3.3 Let $\\lambda ,\\alpha >0$ , $(A, D(A))$ be a closed operator on the Banach space $X$ and $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}\\subset \\mathcal {B}(X)$ be a sequence of bounded operators.", "Then the following conditions are equivalent.", "(i) The family $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}\\subset \\mathcal {B}(X)$ is a discrete $\\alpha $ -resolvent family generated by $A.$ (ii) $1\\in \\rho (A)$ , the operator ${\\mathcal {S}}_\\alpha (0)=(I-A)^{-1}$ and ${\\mathcal {S}}_\\alpha (n)x=\\displaystyle \\sum _{j=1}^{n}\\beta _{\\alpha ,n}(j)(I-A)^{-(j+1)}x,\\quad n\\in \\mathbb {{N}}, \\quad x\\in X.$ If there exists $\\lambda _{0}>0$ such that $\\sup _{n\\in {\\mathbb {{N}}_0}}\\lambda _{0}^{-n}\\Vert {\\mathcal {S}}_\\alpha (n)\\Vert <\\infty ,$ both equations are equivalent to (iii) $\\displaystyle {\\left(\\lambda -1\\over \\lambda \\right)^\\alpha }\\in \\rho (A)$ and $\\left(\\left(\\lambda -1\\over \\lambda \\right)^\\alpha -A\\right)^{-1}x= \\sum _{n=0}^\\infty \\lambda ^{-n}{\\mathcal {S}}_\\alpha (n)x, \\qquad x\\in X, |\\lambda |>\\max \\lbrace \\lambda _0,1\\rbrace .$ The condition (i) implies the condition (ii) is given in [1].", "Now we suppose that the condition (ii) holds.", "Then ${\\mathcal {S}}_\\alpha (n)x\\in D(A)$ for any $x\\in X$ and $n\\in {\\mathbb {{N}}_0}.$ For $n\\in \\mathbb {{N}}$ and $x\\in X$ we have that $(I-A)\\mathcal {S}_{\\alpha }(n)x&=&\\displaystyle \\sum _{j=1}^{n}\\beta _{\\alpha ,n}(j)(I-A)^{-j}x=(k^{\\alpha }(n)-\\sum _{i=1}^{n-1}k^{\\alpha }(n-i)\\beta _{\\alpha ,i}(1))(I-A)^{-1}x \\\\&&+\\displaystyle \\sum _{j=2}^{n-1}(\\displaystyle \\sum _{i=j-1}^{n-1}k^{\\alpha }(n-i)\\beta _{\\alpha ,i}(j-1)-\\displaystyle \\sum _{i=j}^{n-1}k^{\\alpha }(n-i)\\beta _{\\alpha ,i}(j))(I-A)^{-j}x \\\\&&+k^{\\alpha }(1)\\beta _{\\alpha ,n-1}(n-1)(I-A)^{-n}x\\\\&=&k^{\\alpha }(n)(I-A)^{-1}x+\\displaystyle \\sum _{j=1}^{n-1}\\sum _{i=j}^{n-1} k^{\\alpha }(n-i) \\beta _{\\alpha ,i}(j)((I-A)^{-(j+1)}-(I-A)^{-j})x.$ Applying that $(I-A)^{-1}-I=A(I-A)^{-1}$ and $\\mathcal {S}_{\\alpha }(0)=(I-A)^{-1}$ we get that $(I-A)\\mathcal {S}_{\\alpha }(n)x&=&k^{\\alpha }(n)(I+A\\mathcal {S}_{\\alpha }(0))x +A\\displaystyle \\sum _{i=1}^{n-1} k^{\\alpha }(n-i) \\sum _{j=1}^{i} \\beta _{\\alpha ,i}(j)(I-A)^{-(j+1)}x \\\\&=&k^{\\alpha }(n)(I+A\\mathcal {S}_{\\alpha }(0))x + A\\displaystyle \\sum _{i=1}^{n-1} k^{\\alpha }(n-i) \\mathcal {S}_{\\alpha }(i)x,$ and clearly it follows that $\\mathcal {S}_{\\alpha }(n)x=k^{\\alpha }(n)x+A(k^{\\alpha }*{\\mathcal {S}}_{\\alpha })(n)x$ for $n\\in \\mathbb {{N}}.$ The case $n=0$ is a simple check.", "Finally we prove that if there exists $\\lambda _{0}>0$ such that $\\sup _{n\\in {\\mathbb {{N}}_0}}\\lambda _{0}^{-n}\\Vert {\\mathcal {S}}_\\alpha (n)\\Vert <\\infty ,$ (iii) is equivalent to (i).", "Assume that $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}\\subset \\mathcal {B}(X)$ is a discrete $\\alpha $ -resolvent family generated by $A,$ then applying $Z$ -transform we get for $|\\lambda |>\\max \\lbrace \\lambda _0,1\\rbrace $ that $\\widetilde{\\mathcal {S}_{\\alpha }} (\\lambda )x &=& \\sum _{j=0}^{\\infty } \\lambda ^{-j} \\mathcal {S}_{\\alpha }(j)x=\\widetilde{k^{\\alpha }}(\\lambda )x+A\\widetilde{k^{\\alpha }}(\\lambda )\\widetilde{\\mathcal {S}_{\\alpha }} (\\lambda )x\\\\&=&\\biggl (\\frac{\\lambda }{\\lambda -1}\\biggr )^{\\alpha }x+\\biggl (\\frac{\\lambda }{\\lambda -1}\\biggr )^{\\alpha }A\\widetilde{\\mathcal {S}_{\\alpha }} (\\lambda )x,\\quad x\\in X,$ and $\\widetilde{\\mathcal {S}_{\\alpha }} (\\lambda )x=\\biggl (\\frac{\\lambda }{\\lambda -1}\\biggr )^{\\alpha }x+\\biggl (\\frac{\\lambda }{\\lambda -1}\\biggr )^{\\alpha }\\widetilde{\\mathcal {S}_{\\alpha }} (\\lambda )Ax,\\quad x\\in D(A),$ where we have used that (REF ) and (REF ).", "Thus the operator $\\left(\\lambda -1\\over \\lambda \\right)^\\alpha -A$ is invertible, and we get (REF ).", "Conversely, let $|\\lambda |,|\\mu |>\\max \\lbrace \\lambda _0,1\\rbrace $ and $x\\in D(A),$ then there exists $y\\in X$ such that $x=\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)^{-1}y.$ Using that $\\left(\\left(\\lambda -1\\over \\lambda \\right)^\\alpha -A\\right)^{-1}$ and $\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)^{-1}$ are bounded operators and commute, and $A$ is closed we have that $\\widetilde{\\mathcal {S}_{\\alpha }} (\\lambda )x&=&\\widetilde{\\mathcal {S}_{\\alpha }}(\\lambda )\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)^{-1}y \\\\&=&\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)^{-1}\\left(\\left(\\lambda -1\\over \\lambda \\right)^\\alpha -A\\right)^{-1}y \\\\&=&\\sum _{n=0}^\\infty \\lambda ^{-n}\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)^{-1}\\mathcal {S}_{\\alpha }(n)\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)x.$ The uniqueness of $Z$ -transform proves that $\\mathcal {S}_{\\alpha }(n)x=\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)^{-1}\\mathcal {S}_{\\alpha }(n)\\left(\\left(\\mu -1\\over \\mu \\right)^\\alpha -A\\right)x.$ Then we have $\\mathcal {S}_{\\alpha }(n)x\\in D(A),$ and therefore $A\\mathcal {S}_{\\alpha }(n)x=\\mathcal {S}_{\\alpha }(n)Ax$ for all $x\\in X.$ Finally note that for $|\\lambda |>\\max \\lbrace \\lambda _0,1\\rbrace $ and $x\\in D(A)$ we have using (REF ) that $\\widetilde{k^{\\alpha }}(\\lambda )x&=&\\widetilde{k^{\\alpha }}(\\lambda )\\widetilde{\\mathcal {S}_{\\alpha }}(\\lambda )\\left(\\left(\\lambda -1\\over \\lambda \\right)^\\alpha -A\\right)x \\\\&=&\\widetilde{\\mathcal {S}_{\\alpha }}x-\\widetilde{(k^{\\alpha }*\\mathcal {S}_{\\alpha })}(\\lambda )Ax,$ and by the uniqueness of $Z$ -transform we get the result.", "A beautiful consequence of Theorem REF is the following result about sums of combinatorial numbers which seems to be new.", "Corollary 3.4 Take $\\alpha >0$ , $n\\in \\mathbb {N}$ and $\\lbrace \\beta _{\\alpha ,n}(j)\\rbrace _{j=1,...,n}$ defined as above.", "Then (i) $\\displaystyle \\sum _{j=1}^n {\\beta _{\\alpha , n}(j)\\over (1-\\lambda )^{j+1}}=\\sum _{l=0}^{\\infty }\\lambda ^l\\frac{\\Gamma (\\alpha (l+1)+n)}{\\Gamma (\\alpha (l+1))\\Gamma (n+1)},\\quad \\text{for }\\vert \\lambda \\vert <1.$ (ii) $\\displaystyle \\sum _{j=1}^n \\beta _{\\alpha , n}(j)\\frac{\\Gamma (l+1+j)}{\\Gamma (l+1)\\Gamma (j+1)}=\\frac{\\Gamma (\\alpha (l+1)+n)}{\\Gamma (\\alpha (l+1))\\Gamma (n+1)},\\quad \\text{for } l\\in \\mathbb {{N}}.$ (i) We take $|\\lambda |<1,$ then using Proposition REF and Theorem REF in the scalar case we have that $\\displaystyle \\sum _{j=1}^n {\\beta _{\\alpha , n}(j)\\over (1-\\lambda )^{j+1}}=\\sum _{l=0}^\\infty {\\lambda ^l}k^{\\alpha (l+1)}(n)=\\sum _{l=0}^\\infty {\\lambda ^l}\\frac{\\Gamma (\\alpha (l+1)+n)}{\\Gamma (\\alpha (l+1))\\Gamma (n+1)}, \\qquad n\\in \\mathbb {{N}}.$ (ii) Let $|\\lambda |<1,$ then $\\displaystyle \\sum _{j=1}^n {\\beta _{\\alpha , n}(j)\\over (1-\\lambda )^{j+1}}=\\sum _{l=0}^{\\infty }\\lambda ^l\\sum _{j=1}^n \\beta _{\\alpha , n}(j)\\frac{\\Gamma (l+1+j)}{\\Gamma (l+1)\\Gamma (j+1)},\\qquad n\\in \\mathbb {{N}},$ where we have applied that $\\displaystyle {\\frac{1}{(1-\\lambda )^{j+1}}=\\sum _{l=0}^{\\infty }\\lambda ^l \\frac{\\Gamma (l+1+j)}{\\Gamma (l+1)\\Gamma (j+1)}.", "}$ Then we apply (i) to get the result.", "Our main result in this section is the following theorem.", "Theorem 3.5 Suppose that $A$ is the generator of a discrete $\\alpha $ -resolvent family $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ on a Banach space $X.$ Then the fractional difference equation $\\Delta ^{\\alpha } u(n) = Au(n+2), \\quad n \\in \\mathbb {N}_0, \\,\\, 1 < \\alpha \\le 2$ with initial conditions $u(0)=u_0 \\in D(A)$ and $u(1)=u_1 \\in D(A) $ admits the unique solution $u(n) = \\mathcal {S}_{\\alpha }(n)(I-A)u(0) - \\alpha \\mathcal {S}_{\\alpha }(n-1)u(0) + \\mathcal {S}_{\\alpha }(n-1)(I-A)u(1), \\,\\, n \\in \\mathbb {N}_0.$ Convolving the identity given in Definition REF (ii) by $k^{2-\\alpha }$ , we obtain $(k^{2-\\alpha }*\\mathcal {S}_{\\alpha })(n)x = (k^{2-\\alpha }*k^{\\alpha })(n)x + A(k^{2-\\alpha }*k^{\\alpha }*\\mathcal {S}_{\\alpha })(n)x, \\quad n \\in \\mathbb {N}_0.$ Using the semigroup property for the kernels $k^{\\alpha }$ we have $(k^{2-\\alpha }*\\mathcal {S}_{\\alpha })(n)x = k^{2}(n)x + A(k^{2}*\\mathcal {S}_{\\alpha })(n)x, \\quad n \\in \\mathbb {N}_0.$ This is equivalent, by definition of fractional sum and convolution, to the following identity $\\Delta ^{-(2-\\alpha )} \\mathcal {S}_{\\alpha }(n)x = k^{2}(n)x + A \\sum _{j=0}^n k^2(n-j)\\mathcal {S}_{\\alpha }(j)x, \\quad n \\in \\mathbb {N}_0.$ Therefore, we get using $\\Delta ^2 k^2(j)=0$ for $j\\in \\mathbb {N}_0$ that $\\Delta ^2 \\circ \\Delta ^{-(2-\\alpha )}\\mathcal {S}_{\\alpha }(n)x & = \\Delta ^2 k^{2}(n)x + A \\Delta ^2 \\sum _{j=0}^{n}k^2(n-j)\\mathcal {S}_{\\alpha }(j)x \\\\ &= A \\Big [\\sum _{j=0}^{n+2}k^2(j)\\mathcal {S}_{\\alpha }(n+2-j)x -2\\sum _{j=0}^{n+1}k^2(j)\\mathcal {S}_{\\alpha }(n+1-j)x \\\\ & \\quad + \\sum _{j=0}^{n}k^2(j)\\mathcal {S}_{\\alpha }(n-j)x \\Big ] \\\\ &= A \\Big [\\sum _{j=2}^{n+2}k^2(j)\\mathcal {S}_{\\alpha }(n+2-j)x -2\\sum _{j=1}^{n+1}k^2(j)\\mathcal {S}_{\\alpha }(n+1-j)x \\\\ &\\quad + \\sum _{j=0}^{n}k^2(j)\\mathcal {S}_{\\alpha }(n-j)x +\\mathcal {S}_{\\alpha }(n+2)k^2(0)x \\\\ & \\quad + \\mathcal {S}_{\\alpha }(n+1)k^2(1)x -2\\mathcal {S}_{\\alpha }(n+1)k^2(0)x \\Big ] \\\\ &= A\\Big [\\sum _{j=0}^{n}k^2(j+2)\\mathcal {S}_{\\alpha }(n-j)x -2\\sum _{j=0}^{n}k^2(j+1)\\mathcal {S}_{\\alpha }(n-j)x \\\\ & \\quad + \\sum _{j=0}^{n}k^2(j)\\mathcal {S}_{\\alpha }(n-j)x + \\mathcal {S}_{\\alpha }(n+2)x \\Big ]$ for all $ n \\in \\mathbb {N}_0.$ We note that the left hand side in the above identity corresponds to the fractional difference of order $\\alpha \\in (0,2)$ in the sense of Riemann-Liouville.", "Therefore, we obtain $\\Delta ^{\\alpha }\\mathcal {S}_{\\alpha }(n)x = A\\mathcal {S}_{\\alpha }(n+2)x,$ for all $n \\in \\mathbb {N}_0$ and all $x \\in X.$ Define $u(n)$ as $u(n)= \\mathcal {S}_{\\alpha }(n)(I-A)u(0) - \\alpha \\mathcal {S}_{\\alpha }(n-1)u(0) + \\mathcal {S}_{\\alpha }(n-1)(I-A)u(1), \\quad n \\in \\mathbb {N}_0.$ It then follows from (REF ) that $u$ solves (REF ).", "Finally, from the identities $\\mathcal {S}_{\\alpha }(0)x = k^{\\alpha }(0) x + A(k^{\\alpha }* \\mathcal {S}_{\\alpha })(0)x = x + Ak^{\\alpha }(0) \\mathcal {S}_{\\alpha }(0)x= x + A \\mathcal {S}_{\\alpha }(0)x$ and $\\mathcal {S}_{\\alpha }(1)x= k^{\\alpha }(1)x + A(k^{\\alpha }* \\mathcal {S}_{\\alpha })(1)x = \\alpha \\mathcal {S}_{\\alpha }(0)x + A \\mathcal {S}_{\\alpha }(1)x$ which follow from Definition REF (ii), we obtain $u(0)= \\mathcal {S}_{\\alpha }(0)(I-A)u_0 =u_0$ and $u(1)= \\mathcal {S}_{\\alpha }(1)(I-A)u_0 - \\alpha \\mathcal {S}_{\\alpha }(0)u_0 + \\mathcal {S}_{\\alpha }(0)(I-A)u_1 = u_1,$ and we conclude the proof.", "In the non homogeneous case, we derive the following result.", "Corollary 3.6 Suppose that $A$ is the generator of a discrete $\\alpha $ -resolvent family $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ on a Banach space $X$ and $f$ be a vector-valued sequence.", "The fractional difference equation $\\Delta ^{\\alpha } u(n) = Au(n+2)+f(n), \\quad n \\in \\mathbb {N}_0, \\,\\, 1 < \\alpha \\le 2,$ with initial conditions $u(0)=u_0 \\in D(A)$ and $u(1)=u_1 \\in D(A)$ , admits the unique solution $u(n) = \\mathcal {S}_{\\alpha }(n)(I-A)u(0) - \\alpha \\mathcal {S}_{\\alpha }(n-1)u(0) + \\mathcal {S}_{\\alpha }(n-1)(I-A)u(1)+(\\mathcal {S}_{\\alpha }*f)(n-2), \\,\\, $ for all $ n \\ge 2.$ Indeed, by Theorem REF and Theorem REF we have $u(n) \\in D(A)$ for all $n \\ge 2$ and $\\Delta ^\\alpha u(n) &=& \\Delta ^\\alpha (\\mathcal {S}_{\\alpha }(n)(I-A)u(0) - \\alpha \\mathcal {S}_{\\alpha }(n-1)u(0) + \\mathcal {S}_{\\alpha }(n-1)(I-A)u(1)) \\\\&&+ \\Delta ^\\alpha (\\mathcal {S}_{\\alpha }*f)(n-2)\\\\&=& A (\\mathcal {S}_{\\alpha }(n+2)(I-A)u(0) - \\alpha \\mathcal {S}_{\\alpha }(n+1)u(0) + \\mathcal {S}_{\\alpha }(n)(I-A)u(1))\\\\&&+ (\\Delta ^\\alpha \\mathcal {S}_{\\alpha }*f)(n-2) + (\\mathcal {S}_{\\alpha }(1)-\\alpha \\mathcal {S}_{\\alpha }(0))f(n-1)+\\mathcal {S}_{\\alpha }(0)f(n)\\\\&=& Au(n+2)-(A\\mathcal {S}_{\\alpha }*f)(n) + (\\Delta ^{\\alpha }\\mathcal {S}_{\\alpha }*f)(n-2) \\\\&&+ (\\mathcal {S}_{\\alpha }(1)-\\alpha \\mathcal {S}_{\\alpha }(0))f(n-1)+\\mathcal {S}_{\\alpha }(0)f(n).$ From (REF ) it follows $(\\Delta ^{\\alpha }\\mathcal {S}_{\\alpha }*f)(n-2)=(A\\mathcal {S}_{\\alpha }*f)(n)-A\\mathcal {S}_{\\alpha }(1)(n-1)-A\\mathcal {S}_{\\alpha }(0)(n)$ , and hence we obtain $\\Delta ^\\alpha u(n) &=& Au(n+2) + (I-A)\\mathcal {S}_{\\alpha }(0)f(n) + ((I-A)\\mathcal {S}_{\\alpha }(1)-\\alpha \\mathcal {S}_{\\alpha }(0))f(n-1)\\\\&=& Au(n+2) + f(n),$ where we have used that $\\mathcal {S}_{\\alpha }(0)=(I-A)^{-1}$ and $(I+A)\\mathcal {S}_{\\alpha }(1)=\\alpha \\mathcal {S}_{\\alpha }(0)$ .", "For $n=0$ and $n=1$ it is a simple check, using the same above arguments, that $u$ is solution of (REF )." ], [ "Non-linear fractional difference equations on Banach spaces", "Let $A$ be a closed linear operator defined on a Banach space $X$ .", "In this section we study the non linear problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{\\alpha } u(n) &= Au(n+2) + f(n, u(n)), \\quad n \\in \\mathbb {N}_0, \\, \\, 1<\\alpha \\le 2; \\\\u(0) &=0 ;\\\\ u(1) & =0.\\end{array}\\right.$ The following definition is motivated by Corollary REF .", "In particular, it shows their consistence with the problem (REF ).", "Definition 4.1 Under the assumption that the operator $A$ is the generator of a discrete $\\alpha $ -resolvent family $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ on a Banach space $X$ , we say that $u:\\mathbb {N}_0 \\rightarrow X$ is a solution of the non-linear problem (REF ) if $u$ satisfies $u(n) = \\sum _{k=0}^{n-2}\\mathcal {S}_\\alpha (n-2-k)f(k,u(k)), \\,\\, n=2,3,4,...$ The next concept of admissibility is one of the keys ingredients for the estimates that we will use in the proofs of our main results on existence of solutions to (REF ).", "Definition 4.2 We say that a sequence $h:\\mathbb {N}_0\\rightarrow (0,\\infty )$ is an admissible weight if $\\displaystyle \\lim _{n\\rightarrow \\infty }\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k)=0.$ Example 4.3 The sequence $h(n)=nn!,$ that represents the factorial number system, is an admissible weight function, since by [27], we have $\\displaystyle \\sum _{k=1}^n kk!=(n+1)!-1.$ For each admissible weight sequence $h$ , we consider the vector-valued weighted space $ l_h^\\infty (\\mathbb {N}_0;X)=\\left\\lbrace \\xi :\\mathbb {N}_0\\rightarrow X\\,\\, \\vert \\,\\,\\Vert \\xi \\Vert _h<\\infty \\right\\rbrace , $ where the norm $\\Vert \\quad \\Vert _h$ is defined by $\\Vert \\xi \\Vert _h:=\\displaystyle \\sup _{n\\in \\mathbb {N}_0}\\frac{\\Vert \\xi (n)\\Vert }{h(n)}$ .", "The following is our first positive result on existence of solutions for the problem (REF ).", "It uses a Lipschitz type condition.", "Theorem 4.4 Let $h$ be an admissible weight and define $H:=\\displaystyle \\sup _{n\\in \\mathbb {N}_0}\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k).$ Let $A$ be the generator of a bounded discrete $\\alpha $ -resolvent family $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ on a Banach space $X$ for some $1<\\alpha \\le 2$ , and let $f:\\mathbb {N}_0\\times X\\rightarrow X$ be such that $f(k,0)=0$ for all $k\\in \\mathbb {N}_0$ , verifying the following hypothesis: The function $f$ satisfies a Lipschitz condition in $x\\in X$ uniformly in $k\\in \\mathbb {N}_0$ , that is, there exists a constant $L>0$ such that $\\Vert f(k,x)-f(k,y)\\Vert \\le L\\Vert x-y\\Vert $ , for all $x,y\\in X$ , $k\\in \\mathbb {N}_0$ , with $L<(\\Vert \\mathcal {S}_\\alpha \\Vert _\\infty H)^{-1}$ .", "Then the problem (REF ) has an unique solution in $l_h^\\infty (\\mathbb {N};X)$ .", "Let us define the operator $G:l_h^\\infty (\\mathbb {N}_0;X)\\rightarrow l_h^\\infty (\\mathbb {N}_0;X)$ given by $Gu(n)=\\sum _{k=0}^{n-2}\\mathcal {S}_\\alpha (n-2-k)f(k,u(k)), \\quad n\\ge 2.$ First, we show that $G$ is well defined: Let $u\\in l_h^\\infty (\\mathbb {N}_0;X)$ be given.", "By using the assumption (L) for $y=0$ and the boundedness of $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ we get that, $\\Vert Gu(n)\\Vert \\le \\sum _{k=0}^{n-2}\\Vert \\mathcal {S}_\\alpha (n-2-k)\\Vert \\Vert f(k,u(k))\\Vert \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty L\\sum _{k=0}^{n-2}\\Vert u(k)\\Vert \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty L\\Vert u\\Vert _h\\sum _{k=0}^{n-2}h(k),$ for each $n\\in \\mathbb {N}_0.$ Hence, $\\frac{\\Vert Gu(n)\\Vert }{h(n)} \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty L\\Vert u\\Vert _h\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k).$ It proves that $Gu\\in l_h^\\infty (\\mathbb {N}_0;X)$ .", "We next prove that $G$ is a contraction on $l_h^\\infty (\\mathbb {N}_0;X).$ Indeed, let $u,v\\in l_h^\\infty (\\mathbb {N}_0;X)$ be given.", "Then, for each $n\\in \\mathbb {N}_0$ , $\\Vert Gu(n)-Gv(n)\\Vert &\\le &\\sum _{k=0}^{n-2}\\Vert \\mathcal {S}_\\alpha (n-2-k)\\Vert \\Vert f(k,u(k))-f(k,v(k))\\Vert \\\\&\\le & \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty \\sum _{k=0}^{n-2}\\Vert f(k,u(k))-f(k,v(k))\\Vert \\\\&\\le & \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty \\sum _{k=0}^{n-2}L\\Vert u(k)-v(k)\\Vert \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty L\\Vert u-v\\Vert _h\\sum _{k=0}^{n-2}h(k),$ where we have used the assumption (L).", "Therefore $ \\frac{\\Vert Gu(n)-Gv(n)\\Vert }{h(n)} \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty L\\Vert u-v\\Vert _h\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k),$ and consequently $ \\Vert Gu-Gv\\Vert _h \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty HL\\Vert u-v\\Vert _h,$ with $\\Vert \\mathcal {S}_\\alpha \\Vert _\\infty HL<1$ .", "Then, $G$ has a unique fixed point in $l_h^\\infty (\\mathbb {N}_0;X)$ , by the Banach fixed point theorem.", "The next Lemma provide a necessary tool for the use of the Schauder's fixed point theorem, needed in the second main result on existence and uniqueness of solutions to (REF ).", "Lemma 4.5 Let $h$ be an admissible weight and $U\\subset l_h^\\infty (\\mathbb {N}_0;X)$ such that: The set $H_n(U)=\\left\\lbrace \\frac{u(n)}{h(n)}:u\\in U\\right\\rbrace $ is relatively compact in $X,$ for all $n\\in \\mathbb {N}_0$ .", "$\\displaystyle \\lim _{n\\rightarrow \\infty }\\frac{1}{h(n)}\\sup _{u\\in U}\\Vert u(n)\\Vert =0$ , that is, for each $\\varepsilon >0$ , there are $N>0$ such that $\\displaystyle {\\frac{\\Vert u(n)\\Vert }{h(n)}<\\varepsilon }$ , for each $n\\ge N$ and for all $u\\in U$ .", "Then $U$ is relatively compact in $l_h^\\infty (\\mathbb {N}_0;X)$ .", "Let $\\lbrace u_m\\rbrace _m$ be a sequence in $U$ , then by (a) for $n\\in \\mathbb {N}_0$ there is a convergent subsequence $\\lbrace u_{m_j}\\rbrace _j\\subset \\lbrace u_m\\rbrace _m$ such that $\\displaystyle \\lim _{j\\rightarrow \\infty }\\frac{u_{m_j}(n)}{h(n)}=a(n)$ , that is, for each $\\varepsilon >0$ there exists $N(n,\\varepsilon )>0$ such that $\\Vert \\frac{u_{m_j}(n)}{h(n)}-a(n)\\Vert <\\varepsilon $ for all $j\\ge N(n,\\varepsilon )$ .", "Let $\\varepsilon >0$ and $N$ the value of the assumption (b).", "If we consider $N^*:=\\displaystyle \\min _{0\\le n< N}N(n,\\varepsilon )$ , then for $j,k\\ge N^*$ we have $ \\sup _{0\\le n< N}\\frac{\\Vert u_{m_j}(n)-u_{m_k}(n)\\Vert }{h(n)} \\le \\sup _{0\\le n< N}\\Vert \\frac{u_{m_j}(n)}{h(n)}-a(n)\\Vert + \\sup _{0\\le n< N}\\Vert \\frac{u_{m_k}(n)}{h(n)}-a(n)\\Vert <\\varepsilon /2+\\varepsilon /2=\\varepsilon , $ and also $ \\sup _{n\\ge N} \\frac{\\Vert u_{m_j}(n)-u_{m_k}(n)\\Vert }{h(n)} \\le \\sup _{n\\ge N} \\frac{\\Vert u_{m_j}(n)\\Vert }{h(n)} + \\sup _{n\\ge N} \\frac{\\Vert u_{m_k}(n)\\Vert }{h(n)}<\\varepsilon /2+\\varepsilon /2=\\varepsilon .", "$ Consequently, $ \\Vert u_{m_j}-u_{m_k}\\Vert _h = \\sup _{n\\in \\mathbb {N}_0}\\frac{\\Vert u_{m_j}(n)-u_{m_k}(n)\\Vert }{h(n)}<\\varepsilon , $ therefore $\\lbrace u_{m_j}\\rbrace _j$ is a Cauchy subsequence in $l_h^\\infty (\\mathbb {N}_0;X)$ which finishes the proof.", "For $f:\\mathbb {N}_0 \\times X \\rightarrow X$ we recall that the Nemytskii operator $\\mathcal {N}_f:l_h^\\infty (\\mathbb {N}_0;X) \\rightarrow l_h^\\infty (\\mathbb {N}_0;X) $ is defined by $\\mathcal {N}_f(u)(n) := f(n,u(n)), \\quad n \\in \\mathbb {N}_0.$ The next theorem is the second main result for this section.", "It gives one useful criteria for the existence of solutions without use of Lipchitz type conditions.", "Theorem 4.6 Let $h$ be an admissible weight function.", "Let $A$ be the generator of a bounded discrete $\\alpha $ -resolvent family $\\lbrace {\\mathcal {S}}_\\alpha (n)\\rbrace _{n\\in \\mathbb {{N}}_0}$ on a Banach space $X$ for some $1<\\alpha \\le 2$ , and $f:\\mathbb {N}_0\\times X\\rightarrow X$ .", "Suppose that the following conditions are satisfied: There exist a sequence $M\\in l^\\infty (\\mathbb {N}_0)$ and a function $W:\\mathbb {R}^+\\rightarrow \\mathbb {R}^+$ , with $W(y)\\le Cy$ for $y\\in \\mathbb {R}^+$ , such that $\\Vert f(k,x)\\Vert \\le M(k)W(\\Vert x\\Vert )$ for all $k\\in \\mathbb {N}_0$ and $x\\in X$ .", "The Nemytskii operator is continuous in $l_h^\\infty (\\mathbb {N}_0;X)$ , that is, for each $\\varepsilon >0$ , there is $\\delta >0$ such that for all $u,v\\in l_h^\\infty (\\mathbb {N}_0;X)$ , $\\Vert u-v\\Vert _h <\\delta $ implies that $\\Vert \\mathcal {N}_f(u)-\\mathcal {N}_f(v)\\Vert _h < \\varepsilon $ .", "For all $a\\in \\mathbb {N}_0$ and $\\sigma >0$ , the set $\\lbrace \\mathcal {S}_\\alpha (n)f(k,x):0\\le k\\le a,\\Vert x\\Vert \\le \\sigma \\rbrace $ is relatively compact in $X$ for all $n\\in \\mathbb {N}_0$ .", "Then the problem (REF ) has an unique solution in $l_h^\\infty (\\mathbb {N};X)$ .", "Let us define the operator $G:l_h^\\infty (\\mathbb {N}_0;X)\\rightarrow l_h^\\infty (\\mathbb {N}_0;X)$ given by $Gu(n)=\\sum _{k=0}^{n-2}\\mathcal {S}_\\alpha (n-2-k)f(k,u(k)), \\quad n\\ge 2.$ To prove that $G$ has a fixed point in $l_h^\\infty (\\mathbb {N}_0)$ , we will use Leray-Schauder alternative theorem.", "We verify that the conditions of the theorem are satisfied: $G$ is well defined: Let $u\\in l_h^\\infty (\\mathbb {N}_0)$ and $M_\\infty :=\\displaystyle \\sup _{n\\in \\mathbb {N}_0}M(n)$ , then by the assumption (i) $\\Vert Gu(n)\\Vert &\\le & \\sum _{k=0}^{n-2}\\Vert \\mathcal {S}_\\alpha (n-2-k)\\Vert \\Vert f(k,u(k))\\Vert \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty \\sum _{k=0}^{n-2}M(k)W(\\Vert u(k)\\Vert )\\\\&\\le & \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty M_\\infty C \\sum _{k=0}^{n-2}\\Vert u(k)\\Vert \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty M_\\infty C\\Vert u\\Vert _h\\sum _{k=0}^{n-2}h(k).$ Therefore, for each $n\\in \\mathbb {N}_0$ , we have $\\frac{\\Vert Gu(n)\\Vert }{h(n)} \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty M_\\infty C\\Vert u\\Vert _h\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k).$ Since $h$ is admissible, the claim follows.", "$G$ is continuous: Let $\\varepsilon >0$ and $u,v\\in l_h^\\infty (\\mathbb {N}_0).$ Then, for each $n\\in \\mathbb {N}_0$ , $\\Vert Gu(n)-Gv(n)\\Vert &\\le &\\sum _{k=0}^{n-2}\\Vert \\mathcal {S}_\\alpha (n-2-k)\\Vert \\Vert f(k,u(k))-f(k,v(k))\\Vert \\\\&\\le & \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty \\sum _{k=0}^{n-2}\\Vert f(k,u(k))-f(k,v(k))\\Vert \\\\&\\le & \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty \\Vert \\mathcal {N}_f(u)-\\mathcal {N}_f(v)\\Vert _h\\sum _{k=0}^{n-2}h(k).$ Therefore $ \\frac{\\Vert Gu(n)-Gv(n)\\Vert }{h(n)} \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty \\Vert \\mathcal {N}_f(u)-\\mathcal {N}_f(v)\\Vert _h\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k).$ Hence, by the assumption (ii) and admissibility of $h$ we obtain $\\Vert Gu-Gv\\Vert _h <\\varepsilon .$ $G$ is compact: For $R>0$ given, let $B_R(l_h^\\infty (\\mathbb {N}_0;X)):=\\lbrace w\\in l_h^\\infty (\\mathbb {N}_0;X):\\Vert w\\Vert _h< R\\rbrace $ .", "To prove that $V:=G(B_R(l_h^\\infty (\\mathbb {N}_0;X)))$ is relatively compact, we will use Lemma REF .", "We check that the conditions in such Lemma are satisfied: (a) Let $u\\in B_R(l_h^\\infty (\\mathbb {N}_0;X))$ and $v=Gu$ .", "We have $ v(n)=Gu(n)=\\sum _{k=0}^{n-2} \\mathcal {S}_\\alpha (n-2-k)f(k,u(k))=\\sum _{k=0}^{n-2} \\mathcal {S}_\\alpha (k)f(n-2-k,u(n-2-k)), $ and then, $ \\frac{v(n)}{h(n)}=\\frac{n-1}{h(n)}\\left(\\frac{1}{n-1}\\sum _{k=0}^{n-2} \\mathcal {S}_\\alpha (k)f(n-2-k,u(n-2-k))\\right).", "$ Therefore $\\frac{v(n)}{h(n)}\\in \\frac{n-1}{h(n)}co(K_n)$ , where $co(K_n)$ denotes the convex hull of $K_n$ for the set $ K_n=\\bigcup _{k=0}^{n-2}\\lbrace \\mathcal {S}_\\alpha (k)f(\\xi ,x):\\xi \\in \\lbrace 0,1,2,\\dots , n-2\\rbrace ,\\Vert x\\Vert \\le R\\rbrace , \\quad n\\in \\mathbb {N}_0.", "$ Note that each set $K_n$ is relatively compact by the assumption (iii).", "From the inclusions $H_n(V)=\\left\\lbrace \\frac{v(n)}{h(n)}:v\\in V\\right\\rbrace \\subseteq \\frac{n-1}{h(n)}co(K_n)\\subseteq \\frac{n-1}{h(n)}co(\\overline{K_n})$ , we conclude that the set $H_n(V)$ is relatively compact in $X,$ for all $n\\in \\mathbb {N}_0$ .", "(b) Let $u\\in B_R(l_h^\\infty (\\mathbb {N}_0;X))$ and $v=Gu$ .", "For each $n\\in \\mathbb {N}_0$ , we have $\\frac{\\Vert v(n)\\Vert }{h(n)} &\\le & \\frac{1}{h(n)}\\sum _{k=0}^{n-2} \\Vert \\mathcal {S}_\\alpha (n-2-k)\\Vert \\Vert f(k,u(k))\\Vert \\\\&\\le & \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty M_\\infty C\\Vert u\\Vert _h\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k)\\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty M_\\infty CR\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k),$ then the admissibility of $h$ implies $\\displaystyle \\lim _{n\\rightarrow \\infty }\\frac{\\Vert v(n)\\Vert }{h(n)}=0$ independently of $u\\in B_{R}(l_h^\\infty (\\mathbb {N}_0;X))$ .", "Therefore, $V=G(B_R(l_h^\\infty (\\mathbb {N}_0;X)))$ is relatively compact in $l_h^\\infty (\\mathbb {N}_0;X)$ by Lemma REF and we conclude that $G$ is a compact operator.", "The set $U:=\\lbrace u\\in l_h^\\infty (\\mathbb {N}_0;X):u=\\gamma Gu,\\gamma \\in (0,1)\\rbrace $ is bounded: In fact, let us consider $u\\in ł_h^\\infty (\\mathbb {N}_0)$ such that $u=\\gamma Gu$ , $\\gamma \\in (0,1)$ .", "Again by (i), $\\Vert u(n)\\Vert =\\Vert \\gamma Gu(n)\\Vert \\le \\sum _{k=0}^{n-2} \\Vert \\mathcal {S}_\\alpha (n-2-k)\\Vert \\Vert f(k,u(k))\\Vert \\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty M_\\infty C\\Vert u\\Vert _h\\sum _{k=0}^{n-2}h(k).$ Then for each $n\\in \\mathbb {N}_0$ we have $\\frac{\\Vert u(n)\\Vert }{h(n)}\\le \\Vert \\mathcal {S}_\\alpha \\Vert _\\infty M_\\infty C\\Vert u\\Vert _h\\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k).$ We deduce that $U$ is a bounded set in $l_h^\\infty (\\mathbb {N}_0;X)$ .", "Finally, by using the Leray-Schauder alternative theorem, we conclude that $G$ has a fixed point $u\\in l_h^\\infty (\\mathbb {N}_0)$ ." ], [ "The Poisson transformation of fractional difference operators", "For each $n \\in \\mathbb {N}_0,$ the Poisson distribution is defined by $p_n(t):= e^{-t} \\frac{t^{n}}{n!", "}, \\quad t \\ge 0.$ The Poisson distribution arises in connection with classical Poisson processes and semigroups of functions; note that it is also called fractional integral semigroup in [29].", "In this section we study in detail this sequence of functions (Proposition REF ), the Poisson transformation (considered deeply in Theorem REF ) and give their connection with fractional difference and differential operators in Theorem REF .", "Proposition 5.1 Let $n \\in \\mathbb {N}_0$ and $(p_n)_{n\\ge 0}$ given by (REF ).", "Then (i) For $t\\ge 0$ , the inequality $p_n(t) \\ge 0$ holds, $\\int _0^{\\infty } p_n(t) dt =1,$ and $\\int _0^\\infty p_n(t)p_m(t)dt= {1\\over 2^{n+m+1}}{(n+m)!\\over n!m!", "}, \\qquad n,m \\in \\mathbb {N}_0.$ (ii) The semigroup property $p_n\\ast p_m=p_{n+m}$ holds for $n, m\\in \\mathbb {N}_0$ .", "(iii) Given $t\\ge 0$ , then $(p_{(\\cdot )})(t)\\ast p_{(\\cdot )}(t))(n)=2^ne^{-t}p_n(t), \\qquad n\\in \\mathbb {N}_0.$ (iv) For $m,n \\in \\mathbb {N}_0$ , we have $\\Delta ^m p_n=(-1)^mp^{(m)}_{n+m}$ .", "(v) The $Z$ -transform and the Laplace transform are given by $\\widetilde{p_{(\\cdot )}(t)}(z)&=&e^{-t(1-{1\\over z})}, \\qquad z\\ne 0,\\quad t>0;\\cr \\widehat{p_n}(\\lambda )&=&{1\\over (\\lambda +1)^{n+1}}, \\qquad \\Re \\lambda >-1,\\quad n\\in \\mathbb {N}_0.$ The proof of (i) and (ii) is straightforward, and also may be found in [29].", "To show (iii), note that $(p_{(\\cdot )})(t)\\ast p_{(\\cdot )}(t))(n)=e^{-2t}{t^n\\over n!", "}\\sum _{j=0}^n {n!\\over j!(n-j)!", "}=2^ne^{-t}p_n(t),$ for $n\\in \\mathbb {N}_0$ and $t\\ge 0$ .", "Now we get that $(p_{n+1})^{\\prime }(t)= -e^{-t}{t^{n+1}\\over (n+1)!", "}+e^{-t}{t^n\\over n!", "}=-\\Delta p_n(t),$ and we iterate to obtain the equality $\\Delta ^m p_n=(-1)^mp^{(m)}_{n+m}$ for $m,n \\in \\mathbb {N}_0$ .", "Finally the $Z$ -transform and the Laplace transform of $(p_n)_{n\\ge 0}$ are easily obtained.", "Now we introduce an integral transform using the Poisson distribution as integral kernel.", "Some of their properties are inspired in results included in [22] in particular a remarkable connection between the vector-valued $Z$ -transform and the vector-valued Laplace transform, Theorem REF (ii).", "Theorem 5.2 Let $\\psi \\in L^1(\\mathbb {R_+};X)$ and we define $(\\mathcal {P}\\psi )\\in s(\\mathbb {N}_0;X)$ by $(\\mathcal {P}\\psi )(n):= \\int _0^{\\infty } p_n(t)\\psi (t)dt, \\quad n \\in \\mathbb {N}_0.$ Then the following properties hold.", "(i) The map $\\mathcal {P}$ defines a bounded linear operator from $L^1(\\mathbb {R_+};X)$ to $\\ell ^1({\\mathbb {N}_0};X)$ and $\\Vert \\mathcal {P} \\Vert = 1.$ (ii) For $\\psi \\in L^1(\\mathbb {R_+};X)$ , we have that $\\mathcal {P}(\\psi )(n)={(-1)^n\\over n!", "}\\left[\\widehat{\\psi }(\\lambda )\\right]^{(n)}\\vert _{\\lambda =1}, \\qquad n \\in \\mathbb {N}_0.$ In particular the map $\\mathcal {P}$ is inyective.", "(iii) We have that $\\displaystyle {\\widetilde{(\\mathcal {P} \\psi )}(z) = \\widehat{\\psi }(1-1/z),} $ for $|z| >1.$ (iv) For $a \\in L^{1}(\\mathbb {R}_+)$ and $\\psi \\in L^{1}(\\mathbb {R_+} ;X)$ then $\\mathcal {P}(a*\\psi )=\\mathcal {P}(a)*\\mathcal {P}(\\psi ).$ (v) If there are constants $M>0$ and $\\omega \\ge 0$ such that $\\Vert \\psi (t) \\Vert \\le Me^{-\\omega t}$ for a.e.", "$t\\ge 0$ then $ \\Vert \\mathcal {P}(\\psi )(n) \\Vert \\le \\displaystyle \\frac{M}{(1+\\omega )^{n+1}}$ for all $n \\in \\mathbb {N}_0.$ In particular if $\\psi $ is bounded then $\\lbrace \\mathcal {P}(\\psi )(n)$ for $n \\in \\mathbb {N}_0$ is well-defined by (REF ) and $\\lbrace \\mathcal {P}(\\psi )(n)\\rbrace _{n\\in \\mathbb {N}_0}$ is bounded.", "(vi) Let $X$ be a Banach lattice and $\\psi (t) \\ge 0$ for all $x \\ge 0$ and a.e.", "$t \\ge 0$ then $\\mathcal {P}(\\psi )(n) \\ge 0 $ for $n \\in \\mathbb {N}_0.$ (vii) Suppose that $\\lbrace S(t)\\rbrace _{t\\ge 0}\\subset {\\mathcal {B}}(X)$ is a uniformly bounded family of operators.", "If $\\lbrace S(t)\\rbrace _{t\\ge 0}$ is compact then $\\lbrace \\mathcal {P}(S)(n)\\rbrace _{n\\in \\mathbb {N}_0}$ is compact.", "To prove (i) is enough to observe that $\\Vert \\mathcal {P} \\psi \\Vert _1\\le \\sum _{n=0}^{\\infty } \\int _0^{\\infty } p_n(t) \\Vert \\psi (t) \\Vert dt = \\int _0^{\\infty } \\sum _{n=0}^{\\infty } \\frac{t^n}{n!}", "e^{-t} \\Vert \\psi (t)\\Vert dt = \\int _0^{\\infty } \\Vert \\psi (t) \\Vert dt=\\Vert \\psi \\Vert _1,$ for $\\psi \\in L^1(\\mathbb {R_+};X)$ .", "Take $0\\ne x\\in X$ and define $(e_{\\lambda }\\otimes x)(t):=e^{-\\lambda t}x$ for $t,\\lambda >0$ .", "Note that $e_\\lambda \\otimes x\\in L^1(\\mathbb {R_+};X)$ and $\\Vert e_\\lambda \\otimes x\\Vert _1={1\\over \\lambda }\\Vert x\\Vert $ for $\\lambda >0$ .", "It is straightforward to check that $\\mathcal {P}(e_\\lambda \\otimes x)(n)={1\\over (1+\\lambda )^{n+1}}x, \\qquad \\lambda >0, \\quad n\\in \\mathbb {N}_0,$ and $\\Vert \\mathcal {P}(e_\\lambda \\otimes x)\\Vert _1={1\\over \\lambda }\\Vert x\\Vert $ for $\\lambda >0$ .", "We conclude that $\\Vert \\mathcal {P}\\Vert =1$ .", "By properties of Laplace transform, see for example [5], we have that $\\mathcal {P}(\\psi )(n)={(-1)^n\\over n!", "}\\left[\\widehat{\\psi }(\\lambda )\\right]^{(n)}\\vert _{\\lambda =1}, \\qquad \\psi \\in L^1(\\mathbb {R_+};X).$ Now take $\\psi \\in L^1(\\mathbb {R_+};X)$ such that $\\mathcal {P}(\\psi )(n)=0$ for all $n\\in \\mathbb {N}_0$ .", "Then we also get that $\\left[\\widehat{\\psi }(\\lambda )\\right]^{(n)}\\vert _{\\lambda =1}=0$ for $n\\in \\mathbb {N}_0$ .", "Since $\\widehat{\\psi }$ is an holomorphic function, we conclude that $\\widehat{\\psi }=0$ and then $\\psi =0$ where we apply that the Laplace transform is injective, see for example [5].", "Part (iii) is proved following similar ideas than in [22].", "For (iv) note that because $a \\in L^{1}(\\mathbb {R}_+)$ and $\\psi \\in L^{1}(\\mathbb {R_+} ;X)$ we have $a*\\psi \\in L^1(\\mathbb {R}_+;X)$ and $\\mathcal {P}(a*\\psi )(n)&=&\\int _0^\\infty {t^n\\over n!", "}e^{-t}\\int _0^ta(s)\\psi (t-s)dsdt=\\int _0^\\infty a(s)e^{-s}\\int _0^\\infty {(s+u)^n\\over n!", "}e^{-u}\\psi (u)duds\\\\&=&\\sum _{j=0}^\\infty \\int _0^\\infty a(s)e^{-s}{s^j\\over j!", "}ds\\int _0^\\infty {u^{n-j}\\over (n-j)!", "}e^{-u}\\psi (u)du=\\left(\\mathcal {P}(a)*\\mathcal {P}(\\psi )\\right)(n),$ for $n\\in {\\mathbb {N}_0}$ .", "Assertion (v) and (vi) are easily checked and assertion (vii) is obtained from [31].", "We check Poisson transforms of some known functions in the next example.", "Note that, in fact, the Poisson transform can be extended to other sets than $L^1(\\mathbb {R_+};X)$ , for example, $\\mathcal {P}(f)(n)$ is well-defined for measurable functions $f$ such that $\\hbox{ess sup}_{t\\ge 0}\\Vert e^{\\omega t}f(t)\\Vert <\\infty .$ Also the identity given in Theorem REF (iii) holds for the Dirac distribution $\\delta _t$ for $t>0$ , see Proposition REF (v).", "Definition 5.3 The map $\\mathcal {P}: L^1(\\mathbb {R_+};X) \\rightarrow \\ell ^1({\\mathbb {N}_0};X)$ defined by (REF ) is called the Poisson transformation.", "Example 5.4 (i) Note that $\\mathcal {P}(e_{\\lambda })(n)=\\displaystyle {1\\over (\\lambda +1)^{n+1}}$ for $n\\in \\mathbb {N}_0$ , where $e_\\lambda (t):=e^{-\\lambda t}$ for $t>0$ .", "(ii) By Proposition REF (i), $\\displaystyle {\\mathcal {P}(p_{m})(n)={1\\over 2^{n+m+1}}{(n+m)!\\over n!m!", "}}= {1\\over 2^{n+m+1}}k^{m+1}(n), \\qquad n,m \\in \\mathbb {N}_0,$ where the kernel $k^\\alpha $ is defined in (REF ).", "(iii) For $\\alpha >0$ , define $g_{\\alpha }(t) := \\left\\lbrace \\begin{array}{cr}\\displaystyle \\frac{t^{\\alpha -1}}{\\Gamma (\\alpha )}, \\, & \\quad t>0; \\\\0, \\, \\, & \\quad t= 0.\\\\\\end{array}\\right.$ Then the identity $\\mathcal {P}(g_{\\alpha })= k^{\\alpha }$ holds, see more details in [22].", "(iv) The Mittag-Leffler function is an entire function defined by $E_{\\alpha ,\\beta }(z):=\\sum _{k=0}^\\infty \\frac{z^k}{\\Gamma (\\alpha k+\\beta )}, \\quad \\alpha ,\\beta >0, \\qquad z\\in \\mathbb {C},$ see for example [11].", "Now take $\\lambda \\in \\mathbb {C}$ such that $\\vert \\lambda \\vert <1$ and $s_{\\alpha , \\beta }(t):= t^{\\beta -1}E_{\\alpha , \\beta }(\\lambda t^\\alpha )$ , for $t>0$ .", "Then $\\mathcal {P}(s_{\\alpha , \\beta })(n)=\\sum _{k=0}^\\infty \\frac{\\lambda ^k}{n!\\Gamma (\\alpha k+\\beta )}\\int _0^\\infty t^{n+\\alpha k+\\beta -1}e^{-t}dt=\\sum _{k=0}^\\infty \\frac{\\lambda ^k\\Gamma (n+\\alpha k+\\beta )}{n!\\Gamma (\\alpha k+\\beta )}, \\qquad n \\in \\mathbb {N}_0,$ which extends the result [22] proved for $\\beta =1$ .", "In the particular case $\\beta =\\alpha $ , we get that $\\mathcal {P}(s_{\\alpha , \\alpha })(n)=\\sum _{j=0}^\\infty \\frac{\\lambda ^j\\Gamma (n+\\alpha (j+1))}{n!\\Gamma (\\alpha (j+1))}=\\sum _{j=0}^\\infty {\\lambda ^j}k^{\\alpha (j+1)}(n), \\qquad n \\in \\mathbb {N}_0.$ Now we are interested to establish a notable relation between the discrete and continuous fractional concepts in the sense of Riemann-Liouville.", "In order to give our next result, we recall that the Riemann-Liouville fractional integral of order $\\alpha >0$ , of a locally integrable function $u:\\;[0,\\infty ) \\rightarrow X$ is given by: $I_t^{\\alpha }u(t): = (g_{\\alpha }*u)(t) := \\int _0^tg_{\\alpha }(t-s)u(s)ds, \\qquad t\\ge 0.$ The Riemann-Liouville fractional derivative of order $\\alpha ,$ for $m-1< \\alpha <m$ , $m\\in \\mathbb {N}$ , is defined by ${D}_t^\\alpha u(t) := \\frac{d^m}{dt^m}\\int _0^tg_{m-\\alpha }(t-s)u(s)ds = \\frac{d^m}{dt^m}(g_{m-\\alpha }*u)(t), \\qquad t\\ge 0,$ for $u\\in C^{(m)}(\\mathbb {R}_+;X)$ , see for example [11] and [12].", "Compare these definitions with Definitions REF and REF .", "Theorem 5.5 Let $m\\in \\mathbb {N}$ and $m-1<\\alpha \\le m$ .", "Take $u\\in C^{(m)}(\\mathbb {R}_+;X)$ such that $e_{-\\omega }u^{(m)}$ is integrable for some $0<\\omega <1$ .", "Then we have $\\mathcal {P}(D_t^\\alpha u)(n+m)=\\int _0^{\\infty } p_{n+m}(t)D^{\\alpha }_{t} u(t)dt = \\Delta ^{\\alpha }\\mathcal {P}(u)(n), \\quad n \\in \\mathbb {N}_0.$ Set $n\\in \\mathbb {N}_0$ , $m\\in \\mathbb {N}$ and $u\\in C^{(m)}(\\mathbb {R}_+;X)$ such that $e_{-\\omega }u^{(m)}$ is integrable for some $0<\\omega <1$ .", "We integrate by parts $m$ -times to get $\\mathcal {P}(D_t^m u)(n+m)&=&\\int _0^\\infty p_{n+m}(t)D_t^m u(t)dt =\\dots = (-1)^m\\int _0^\\infty p^{(m)}_{n+m}(t)u(t)dt\\\\&=& \\int _0^\\infty \\Delta ^m p_n(t)u(t)dt=\\Delta ^{m}\\mathcal {P}(u)(n)$ where we have applied Proposition REF (iv).", "Now consider $m-1<\\alpha <m$ .", "By the definition of Riemann-Liouville fractional derivative, see formula (REF ), we have that $\\mathcal {P}(D_t^\\alpha u)(n+m)&=&\\int _0^\\infty p_{n+m}(t)D_t^\\alpha u(t)dt = \\int _0^\\infty p_{n+m}(t)\\frac{d^m}{dt^m}(g_{m-\\alpha }*u)(t)dt \\\\&=& \\Delta ^{m}\\mathcal {P}(g_{m-\\alpha }*u)(n) = \\Delta ^{m}\\left(k^{m-\\alpha }\\ast \\mathcal {P}(u)\\right)(n)\\\\&=&\\Delta ^m\\left(\\Delta ^{-(m-\\alpha )}\\mathcal {P}(u)\\right)(n) = \\Delta ^\\alpha \\mathcal {P}(u)(n),$ where we have applied Theorem REF (iv), Example REF (iii) and Definition REF ." ], [ "Discrete $\\alpha $ -resolvent families via Poisson subordination", "We recall the following concept (see [2], [21] and references therein).", "Definition 6.1 Let $A$ be a closed and linear operator with domain $D(A)$ defined on a Banach space $X$ and $ \\alpha >0.$ We call $A$ the generator of an $\\alpha $ -resolvent family if there exists $\\omega \\ge 0$ and a strongly continuous function $S_{\\alpha }:[0,\\infty ) \\rightarrow \\mathcal {B}(X)$ (respectively $S_{\\alpha }:(0,\\infty ) \\rightarrow \\mathcal {B}(X)$ in case $0<\\alpha <1$ ) such that   $\\lbrace \\lambda ^{\\alpha } \\, : \\, \\mbox{Re}(\\lambda )> \\omega \\rbrace \\subset \\rho (A),$ the resolvent set of $A$ , and $(\\lambda ^{\\alpha } -A)^{-1}x = \\int _0^{\\infty }e^{-\\lambda t} S_{\\alpha }(t)x dt, \\quad \\mbox{Re}(\\lambda ) > \\omega ,\\quad x \\in X.$ In this case, $S_{\\alpha }(t)$ is called the $\\alpha $ -resolvent family generated by $A$ .", "By the uniqueness theorem for the Laplace transform, a 1-resolvent family is the same as a $C_0$ -semigroup, while a 2-resolvent family corresponds to a strongly continuous sine family.", "See for example [5] and the references therein for an overview on these concepts.", "Some properties of $(S_{\\alpha }(t))_{t>0}$ are included in the following Lemma.", "For a proof, see for example [21].", "Lemma 6.2 Let $\\alpha >0$ .", "The following properties hold: $S_{\\alpha }(0)=g_{\\alpha }(0)I$ (respectively $\\displaystyle \\lim _{t\\rightarrow 0^+}\\frac{S_{\\alpha }(t)x}{g_\\alpha (t)}=x$ for all $x\\in X$ in case $0<\\alpha <1$ ).", "$S_{\\alpha }(t)D(A) \\subset D(A)$ and $AS_{\\alpha }(t)x =S_{\\alpha }(t)Ax $ for all $x \\in D(A), \\, t \\ge 0.$ For all $x \\in D(A):$ $S_{\\alpha }(t)x = g_{\\alpha }(t) x + \\displaystyle \\int _0^t g_{\\alpha }(t-s)AS_{\\alpha }(s)xds, \\, t \\ge 0.$ For all $x \\in X:$ $(g_{\\alpha }*S_{\\alpha })(t)x \\in D(A)$ and $S_{\\alpha }(t)x = g_{\\alpha }(t) x + \\displaystyle A \\int _0^tg_{\\alpha }(t-s)S_{\\alpha }(s)xds, \\,\\,\\, t \\ge 0.$ The next theorem is the main result of this section.", "Theorem 6.3 Suppose that $A$ is the generator of an $\\alpha $ -resolvent family $(S_{\\alpha }(t))_{t>0}$ on a Banach space $X,$ of exponential bound less than 1.", "Then $A$ is the generator of a discrete $\\alpha $ -resolvent family $(\\mathcal {S}_{\\alpha }(n))_{ n \\in \\mathbb {N}_0}$ defined by $\\mathcal {S}_\\alpha (n):=\\mathcal {P}(S_\\alpha )(n), \\qquad n \\in \\mathbb {N}_0.$ Take $x\\in D(A)$ .", "Since $(A, D(A))$ is a closed operator and the condition in Lemma REF (ii) we have that $\\mathcal {S}_\\alpha (n)A(x)= \\int _0^\\infty p_n(t) S(t)A(x)dt= \\int _0^\\infty p_n(t) AS(t)(x)dt=A\\mathcal {S}_\\alpha (n)(x).$ From the identity $S_{\\alpha }(t)x = g_{\\alpha }(t)x + A \\int _0^t g_{\\alpha }(t-s)S_{\\alpha }(s)x ds, \\quad t \\ge 0,$ valid for all $x \\in X,$ we obtain $\\begin{array}{lll}\\mathcal {S}_\\alpha (n)x=\\mathcal {P}(S_{\\alpha })(n)x &=& \\mathcal {P}(g_{\\alpha })(n) + A\\mathcal {P} (g_{\\alpha }*S_{\\alpha })(n)x =k^\\alpha (n)+A\\left(k^\\alpha \\ast \\mathcal {S}_\\alpha \\right)(n)x,\\end{array}$ where we have applied Example REF (iii) and Theorem REF (iv) and the second condition in Definition REF .", "The theorem is proved.", "Example 6.4 Consider the Mittag-Leffler function $E_{\\alpha ,\\beta }$ studied in Example REF (iv).", "Suppose that $A$ is a bounded operator on the Banach space $X$ .", "It then follows from Definition REF that $S_\\alpha (t)=t^{\\alpha -1}E_{\\alpha ,\\alpha }(A t^\\alpha ), \\quad t\\ge 0,\\,\\alpha >0$ is the $\\alpha $ -resolvent family generated by $A$ .", "If $\\Vert A \\Vert <1,$ then $\\mathcal {S}_\\alpha (n)x:=\\int _0^\\infty e^{-t}\\frac{t^n}{n!", "}t^{\\alpha -1}E_{\\alpha ,\\alpha }(A t^\\alpha )xdt= \\sum _{k=0}^\\infty \\frac{\\Gamma (\\alpha (k+1)+n)}{\\Gamma (\\alpha (k+1))\\Gamma (n+1)}A^k x, \\quad n \\in \\mathbb {N}_0,$ for $x\\in X$ .", "Compare with Proposition REF .", "Example 6.5 Let us recall the definition of $\\omega $ -sectorial operator.", "A closed and densely defined operator $A$ is said to be $\\omega $ -sectorial of angle $\\theta $ if there exist $0<\\theta <\\pi /2$ , $M>0$ and $\\omega \\in \\mathbb {R}$ such that its resolvent exists outside the sector $\\omega +\\Sigma _\\theta :=\\lbrace \\omega +\\lambda :\\lambda \\in \\mathbb {C},\\left|\\arg (-\\lambda )\\right|<\\theta \\rbrace $ and $\\Vert (\\lambda -A)^{-1}\\Vert \\le \\frac{M}{\\left|\\lambda -\\omega \\right|}, \\quad \\lambda \\notin \\omega +\\Sigma _\\theta .$ Suppose that $A$ is a $\\omega $ -sectorial operator of angle $\\theta < \\alpha \\pi /2$ and $\\omega <0.$ Then $A$ is the generator of a bounded $\\alpha $ -resolvent family $(S_{\\alpha }(t))_{t>0}$ on $X$ for $1<\\alpha < 2$ given by $S_\\alpha (t)x=\\frac{1}{2\\pi i}\\int _{\\Gamma }e^{\\lambda t}(\\lambda ^\\alpha -A)^{-1}xd\\lambda ,\\qquad t>0,\\quad x\\in X,$ where $\\Gamma $ is a suitable path where the resolvent operator is well defined.", "By Theorem REF and REF , its Poisson transformation $\\mathcal {S}_\\alpha (n) :=\\mathcal {P}(S_\\alpha )(n)$ defines a bounded discrete $\\alpha $ -resolvent family $\\lbrace \\mathcal {S}_\\alpha (n)\\rbrace _{n\\in \\mathbb {N}_0} \\subset \\mathcal {B}(X).$ Example 6.6 Suppose that $A$ is the generator of a bounded sine family $(S(t))_{t>0}$ on $X.$ Then $A$ is the generator of a bounded $\\alpha $ -resolvent family $(S_{\\alpha }(t))_{t>0}$ on $X$ for $1<\\alpha <2$ given by $S_\\alpha (t)x=\\int _0^\\infty \\psi _{\\alpha /2,0}(t,s)S(s)xds, \\qquad t>0,\\quad x\\in X,$ where $\\psi _{\\alpha /2,0}(t,s)$ is the stable Lévy process, see [2].", "Then, by [2] $\\Vert S_\\alpha (t)\\Vert \\le M \\int _0^\\infty \\psi _{\\alpha /2,0}(t,s)ds= M g_{\\alpha /2}(t),\\quad t>0,$ and since $S_{\\alpha }(0)=0,$ $\\frac{1}{2}<\\frac{\\alpha }{2}<1$ and $(S_{\\alpha }(t))_{t>0}$ is strongly continuous we conclude that $(S_{\\alpha }(t))_{t>0}$ is bounded.", "Hence, again by Theorem REF and REF , we obtain a bounded discrete $\\alpha $ -resolvent family $\\lbrace \\mathcal {S}_\\alpha (n)\\rbrace _{n\\in \\mathbb {N}_0} \\subset \\mathcal {B}(X).$ Our next corollary imposes a natural and useful condition of compactness on a given family of operators in order to obtain existence and uniqueness of solutions.", "Theorem 6.7 Suppose that $A$ is the generator of a bounded sine family $(S(t))_{t>0}$ on $X$ such that $(\\lambda -A)^{-1}$ is a compact operator for some $\\lambda $ large enough.", "Let $h$ be an admissible weight and $f:\\mathbb {N}_0 \\times X \\rightarrow X$ satisfying the following conditions: There exist a function $M\\in l^\\infty (\\mathbb {N}_0)$ and a function $W:\\mathbb {R}^+\\rightarrow \\mathbb {R}^+$ , with $W(x)\\le Cx$ for $x\\in \\mathbb {R}^+$ , such that $\\Vert f(k,x)\\Vert \\le M(k)W(\\Vert x\\Vert )$ for all $k\\in \\mathbb {N}_0$ and $x\\in X$ .", "The Nemytskii operator $\\mathcal {N}_f$ is continuous in $l_h^\\infty (\\mathbb {N}_0;X).$ Then, for each $1<\\alpha \\le 2,$ the problem (REF ) has an unique solution in $l_h^\\infty (\\mathbb {N}_0;X).$ To prove this result we only have to check that the assumption (iii) in Theorem REF is satisfied.", "Indeed, by hypothesis we have that $(\\lambda ^\\alpha -A)^{-1}$ is compact for all $\\lambda ^{\\alpha } \\in \\rho (A) $ and all $1<\\alpha \\le 2.$ By Example REF we obtain that $A$ is the generator of a bounded $\\alpha $ -resolvent family $(S_\\alpha (t))_{t>0},$ which is moreover compact by [31].", "From Theorem REF (vii) it follows that $\\lbrace \\mathcal {S}_\\alpha (n)\\rbrace _{n\\in \\mathbb {N}_0}$ is compact.", "Also, for all $a\\in \\mathbb {N}_0$ and $\\sigma >0$ , the set $\\lbrace f(k,x):0\\le k\\le a,\\Vert x\\Vert \\le \\sigma \\rbrace $ is bounded because $\\Vert f(k,x)\\Vert \\le M(k)C\\Vert x\\Vert \\le MC\\sigma $ for all $0\\le k\\le a$ and $\\Vert x\\Vert \\le \\sigma $ .", "Consequently, the set $\\lbrace \\mathcal {S}_\\alpha (n)f(k,x):0\\le k\\le a,\\Vert x\\Vert \\le \\sigma \\rbrace $ is relatively compact in $X$ for all $n\\in \\mathbb {N}_0$ ." ], [ "Examples, applications and final comments", "In this section, we provide several concrete examples and applications of the abstract results developed in the previous sections.", "Finally we present some related problems with problem (REF ) for $\\alpha =2$ .", "Example 7.1 Let $m:[a,b] \\rightarrow (0,1)$ be a continuous function.", "Let $A$ be the multiplication operator given by $Af(x)=m(x)f(x)$ defined on $L^2(a,b).$ We known that $A$ is a bounded operator [20].", "Since $(\\lambda ^2-A)^{-1} = \\frac{1}{\\lambda ^2 -m(x)}$ for $\\lambda $ sufficiently large, we have by Definition REF that $A$ generates a sine family $(S(t))_{t>0}$ on $L^2(a,b)$ , given by $ S(t)f(x)=\\frac{1}{2\\sqrt{m(x)}}\\left(e^{\\sqrt{m(x)}t}-e^{-\\sqrt{m(x)}t}\\right)f(x), \\qquad t>0.", "$ Since $0<m(x)<1$ we obtain by subordination $\\mathcal {S}(n)f(x) &=& \\int _0^\\infty p_n(t)S(t)f(x)dt\\\\&=& \\int _0^\\infty e^{-t}\\frac{t^n}{n!", "}\\frac{1}{2\\sqrt{m(x)}}\\left(e^{\\sqrt{m(x)}t}-e^{-\\sqrt{m(x)}t}\\right)f(x)dt\\\\&=& \\frac{1}{ n!", "2\\sqrt{m(x)}}\\left(\\int _0^\\infty t^ne^{-(1-\\sqrt{m(x)})t}f(x)dt-\\int _0^\\infty t^ne^{-(1+\\sqrt{m(x)})t}f(x)dt\\right)\\\\&=& \\frac{1}{2\\sqrt{m(x)}}\\left(\\frac{1}{(1-\\sqrt{m(x)})^{n+1}}-\\frac{1}{(1+\\sqrt{m(x)})^{n+1}}\\right)f(x),$ for $n\\in {n\\in \\mathbb {N}_0}$ .", "By Theorem REF and Theorem REF , we conclude that the fractional difference equation $ \\Delta ^{2} u(n) = Au(n+2), \\quad n \\in \\mathbb {N}_0, $ with initial conditions $u(0)=u_0$ and $u(1)=u_1$ , admits the explicit solution $u(n) &=& (\\mathcal {S}(n)(I-A)-2\\mathcal {S}(n-1))u_0 + \\mathcal {S}(n-1)(I-A)u_1\\\\&=& \\sqrt{A^{-1}} \\left(\\frac{I-A}{2}(1-\\sqrt{A})^{-(n+1)}-\\frac{I-A}{2}(1+\\sqrt{A})^{-(n+1)}\\right)u_0 \\\\ && - \\sqrt{A^{-1}} \\left((1-\\sqrt{A})^{-n} -(1+\\sqrt{A})^{-n}\\right)u_0 \\\\&&+ \\frac{1}{2}\\sqrt{A^{-1}} \\left((1-\\sqrt{A})^{-n}- (1+\\sqrt{A})^{-n}\\right)(I-A)u_1 ,\\quad n\\in \\mathbb {N}_0.$ Example 7.2 We study the existence of solutions for the problem $\\left\\lbrace \\begin{array}{rll}&\\Delta ^{\\alpha } u(n,x) = u_{xx}(n+2,x) + \\displaystyle \\frac{\\sin {n}}{1+n^3}\\frac{u(n,x)}{1+ \\Big (\\displaystyle \\int _0^{\\pi }|u(n,s)|^2ds \\Big )^{1/2}}, \\quad n \\in \\mathbb {N}_0, \\quad 0<x<\\pi ; \\\\&u(0,x) =0 ;\\quad u(1,x) =0;\\\\ &u(n,0) =0 ;\\quad u(n,\\pi ) =0;\\end{array}\\right.$ for $1<\\alpha <2$ .", "We will use Corollary REF .", "Let $X=L^2[0,\\pi ]$ and let us define the operator $A=\\displaystyle \\frac{\\partial ^2}{\\partial x^2}$ , on the domain $D(A)=\\lbrace v\\in L^2[0,\\pi ] / v,v^{\\prime } \\textrm { absolutely continuous, } v^{\\prime \\prime }\\in L^2[0,\\pi ], v(0)=v(\\pi )=0\\rbrace .$ Observe that the operator $A$ can be written as $ Av=-\\sum _{n=1}^\\infty n^2(v,z_n)z_n, \\quad v\\in D(A), $ where $z_n(s)=\\sqrt{2/\\pi }\\sin {ns}$ , $n=1,2,\\dots $ , is an orthonormal set of eigenvectors of $A$ .", "Note that $A$ is the infinitesimal generator of a sine family $S(t)$ , $t\\in \\mathbb {R}$ , in $L^2[0,\\pi ]$ , given by $ S(t)v=\\sum _{n=1}^\\infty \\frac{\\sin {nt}}{n}(v,z_n)z_n, \\quad v\\in L^2[0,\\pi ].", "$ The resolvent of $A$ is given by $ R(\\lambda ; A)v=\\sum _{n=1}^\\infty \\frac{1}{\\lambda +n^2}(v,z_n)z_n, \\quad v\\in L^2[0,\\pi ], -\\lambda \\ne k^2, k\\in \\mathbb {N}.", "$ The compactness of $R(\\lambda ; A)$ follows from the fact that eigenvalues of $R(\\lambda ; A)$ are $\\lambda _n=\\frac{1}{\\lambda +n^2}$ , $n=1,2,\\dots $ , and thus $\\displaystyle \\lim _{n\\rightarrow \\infty }\\lambda _n=0$ , see for example [30].", "Let us consider the weighted space $ l_h^\\infty (\\mathbb {N}_0; L^2[0,\\pi ])=\\left\\lbrace \\xi :\\mathbb {N}_0\\rightarrow L^2[0,\\pi ]/\\sup _{n\\in \\mathbb {N}}\\frac{\\Vert \\xi (n)\\Vert _{L^2}}{nn!", "}<\\infty \\right\\rbrace , $ where the function $h(n)=nn!$ is an admissible weight function (see Example REF ).", "For the function $f:\\mathbb {N}_0\\times L^2[0,\\pi ]\\rightarrow L^2[0,\\pi ]$ , defined by $f(n,v):=\\displaystyle \\frac{\\sin {n}}{1+n^3}\\frac{v}{1+\\Vert v\\Vert }$ , we consider the Nemystkii operator $\\mathcal {N}_f$ associated to $f.$ That is, $N_f(u):\\mathbb {N}_0\\rightarrow L^2[0,\\pi ]$ is such that $N_f(u)(n):=f(n,u(n))$ for $u:\\mathbb {N}_0\\rightarrow L^2[0,\\pi ].$ Then: There exists $M(n)=\\frac{1}{1+n^3}$ in $l^{\\infty }(\\mathbb {N}_0)$ and $W(t):= \\frac{t}{1+t}$ such that $\\Vert f(n,v)\\Vert \\le M(n) W(\\Vert v\\Vert ) $ , for all $n\\in \\mathbb {N}_0$ and $v\\in L^2[0,\\pi ]$ .", "For $u_1,u_2\\in l_h^\\infty (\\mathbb {N}_0; L^2[0,\\pi ])$ and each $n\\in \\mathbb {N}_0$ , we have $\\Vert N_f(u_1)(n)-N_f(u_2)(n)\\Vert &\\le & \\Vert \\frac{u_1(n)}{1+\\Vert u_1(n)\\Vert }-\\frac{u_2(n)}{1+\\Vert u_2(n)\\Vert }\\Vert \\\\&=& \\Vert \\frac{(u_1(n)-u_2(n))(1+\\Vert u_2(n)\\Vert )+u_2(n)(\\Vert u_2(n)\\Vert -\\Vert u_1(n)\\Vert )\\Vert }{(1+\\Vert u_1(n)\\Vert )(1+\\Vert u_2(n)\\Vert )}\\Vert \\\\&\\le & \\frac{\\Vert u_1(n)-u_2(n)\\Vert }{1+\\Vert u_1(n)\\Vert }+\\frac{\\Vert u_2(n)\\Vert \\Vert u_1(n)-u_2(n)\\Vert }{(1+\\Vert u_1(n)\\Vert )(1+\\Vert u_2(n)\\Vert )}\\\\&\\le & \\frac{2}{1+\\Vert u_1(n)\\Vert }\\Vert u_1(n)-u_2(n)\\Vert \\le 2\\Vert u_1(n)-u_2(n)\\Vert .$ Consequently, by Corollary REF , we conclude that the problem (REF ) has an unique solution $u\\in l_h^\\infty (\\mathbb {N}_0)$ , that is, $u$ satisfies $\\sup _{n\\in \\mathbb {N}_0} \\frac{\\Vert u(n)\\Vert _{L^2}}{nn!", "}=\\sup _{n\\in \\mathbb {N}_0} \\frac{1}{nn!", "}\\left(\\int _0^{\\pi }|u(n,x)|^2dx\\right)^{1/2}<\\infty ,$ Therefore, there exist a constant $K>0$ such that $\\int _0^{\\pi }|u(n,x)|^2dx<K(nn!", ")^2, \\quad n\\in \\mathbb {N}.$" ], [ "Final comments", "In some circumstances, the equation (REF ) for $\\alpha =2$ may have a different format on the right hand side.", "For instance, the problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{2} u(n) &= Bu(n+1)+g(n,u(n)), \\quad n \\in \\mathbb {N}_0; \\\\u(0) &=u_0 ;\\\\ u(1) & =u_1.\\end{array}\\right.$ where $B$ is a linear operator defined on a Banach space $X.$ In such cases, and under mild conditions, we can still handle this problem with our theory.", "That is the content of the following two results.", "Proposition 7.3 Let $B$ be a linear operator defined on a Banach space $X$ , such that $-2\\in \\rho (B)$ .", "Then, (REF ) is equivalent to the problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{2} u(n) &= Tu(n+2)+Tu(n)+(I-T)g(n,u(n)), \\quad n \\in \\mathbb {N}_0; \\\\u(0) &=u_0 ;\\\\ u(1) & =u_1.\\end{array}\\right.$ where $T=I-2(2+B)^{-1}.$ From the definition $\\Delta ^2u(n)=u(n+2)-2u(n+1)+u(n),$ we obtain $u(n+1)=\\frac{1}{2}(u(n+2)-\\Delta ^2u(n)+u(n)).$ On the other hand, by (REF ) we have $u(n+2)-2u(n+1)+u(n)=Bu(n+1)+g(n,u(n))$ that is, for $-2\\in \\rho (B)$ we have $u(n+1)=(2+B)^{-1}u(n+2)+(2+B)^{-1}u(n)-(2+B)^{-1}g(n,u(n)).$ By identifying both expressions for $u(n+1)$ , we achieve $(2+B)^{-1}u(n+2)+(2+B)^{-1}u(n)-(2+B)^{-1}g(n,u(n))=\\frac{1}{2}(u(n+2)-\\Delta ^2u(n)+u(n)),$ and therefore $\\Delta ^2u(n)=(I-2(2+B)^{-1})u(n+2)+(I-2(2+B)^{-1})u(n)+2(2+B)^{-1}g(n,u(n)).$ So, assuming $-2\\in \\rho (B)$ , the original problem (REF ) is equivalent to the problem (REF ), with $T=I-2(2+B)^{-1}.$ Observe that the operator $T$ in the above proposition is bounded whenever $B$ is a closed linear operator and $-2\\in \\rho (B).$ A second case of interest is the following.", "Proposition 7.4 Let $B$ be a linear operator defined on a Banach space $X$ , such that $1\\in \\rho (B)$ .", "Then, the problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{2} u(n) &= Bu(n)+g(n+1,u(n+1)), \\quad n \\in \\mathbb {N}_0; \\\\u(0) &=u_0 ;\\\\ u(1) & =u_1.\\end{array}\\right.$ is equivalent to the problem $\\left\\lbrace \\begin{array}{rll}\\Delta ^{2} u(n) &= Tu(n+2)-2Tu(n+1)+(I-T)g(n+1,u(n+1)), \\quad n \\in \\mathbb {N}_0; \\\\u(0) &=u_0 ;\\\\ u(1) & =u_1.\\end{array}\\right.$ where $T=I-(I-B)^{-1}.$ From the definition $\\Delta ^2u(n)=u(n+2)-2u(n+1)+u(n),$ we obtain $u(n)=\\Delta ^2u(n)-u(n+2)+2u(n+1).$ On the other hand, by (REF ) we have $u(n+2)-2u(n+1)+u(n)=Bu(n)+g(n+1,u(n+1))$ that is, for $1\\in \\rho (B)$ we have $u(n)=-(I-B)^{-1}u(n+2)+2(I-B)^{-1}u(n+1)+(I-B)^{-1}g(n+1,u(n+1)).$ By identifying both expressions for $u(n)$ , we achieve $(I-B)^{-1}\\left(-u(n+2)+2u(n+1)+g(n+1,u(n+1))\\right)=\\Delta ^2u(n)-u(n+2)+2u(n+1),$ and therefore $\\Delta ^2u(n)=(I-(I-B)^{-1})u(n+2)-2(I-(I-B)^{-1})u(n+1)+(I-B)^{-1}g(n+1,u(n+1)).$ So, assuming $1\\in \\rho (B)$ , the problem (REF ) is equivalent to the problem (REF ), with $T=I-(I-B)^{-1}.$ For instance, let $B$ be a linear operator defined on a Banach space $X$ , and $\\gamma $ a positive constant.", "We study the existence of solutions of the problem $\\left\\lbrace \\begin{array}{rll}&&\\Delta ^{2} u(n,x) = (B+2\\gamma )u(n+1,x), \\quad n \\in \\mathbb {N}_0, x\\in [a,b]; \\\\&&u(0,x) =0 ;\\quad u(1,x) =0,\\\\&&u(n,a) =0 ;\\quad u(n,b) =0.\\end{array}\\right.$ By Proposition REF the solution of (REF ) coincides with the solution of the problem $\\left\\lbrace \\begin{array}{rll}&&\\Delta ^{2} u(n,x) = Tu(n+2,x)+Tu(n,x), \\quad n \\in \\mathbb {N}_0, x\\in [a,b]; \\\\&&u(0,x) =0 ;\\quad u(1,x) =0,\\\\&&u(n,a) =0 ;\\quad u(n,b) =0.\\end{array}\\right.$ where $T=I-2(2(1+\\gamma )+B)^{-1},$ provided that $2+2\\gamma \\in \\rho (-B).$ As an example of application to Theorem REF with $\\alpha =2$ , let us consider $X=L^2(\\pi ,2\\pi )$ and define $Bf(x)= 2\\Big ( \\frac{1}{1+x}- (1+\\gamma ) \\Big )f(x), \\quad x \\in [\\pi , 2\\pi ].$ Note that $B$ is bounded.", "A computation shows that $Tf(x)= -xf(x)$ and therefore generates the sine family $S(t)f(x)= \\frac{\\sin (\\sqrt{x}t)}{\\sqrt{x}}f(x), \\quad x \\in [\\pi , 2\\pi ]$ It follows that $ \\Vert T \\Vert \\le 2\\pi $ and $\\Vert S\\Vert _\\infty \\le \\sqrt{\\pi }.", "$ Let $h$ the admissible weight function defined by $h(n)=nn!$ , for which we have $\\sup _{n \\in \\mathbb {N}_0} \\frac{1}{h(n)} \\sum _{k=0}^{n-2}h(k)=\\frac{1}{18}$ since $\\displaystyle \\frac{1}{h(n)}\\sum _{k=0}^{n-2}h(k)$ is a decreasing sequence for $n\\ge 3.$ Let us consider the function $f:\\mathbb {N}_0\\times X\\rightarrow X$ defined by $f(n,\\xi )=T\\xi $ .", "Then $f(n,0)=0$ for all $n\\in \\mathbb {N}_0$ and the function $f$ verifies: There exists $L:= \\Vert T \\Vert $ such that $\\Vert f(n,x)-f(n,y)\\Vert \\le \\Vert T\\Vert \\Vert x-y\\Vert ,$ for all $x,y \\in X.$ Moreover, $\\displaystyle \\Vert T\\Vert \\Vert S\\Vert _\\infty \\frac{1}{18} < \\frac{2 \\pi \\sqrt{\\pi }}{18} <1.$ Therefore, by Theorem REF we conclude that the problem (REF ) has an unique solution or, equivalently, the problem (REF ) has an unique solution $u\\in l_h^\\infty (\\mathbb {N}_0;X)$ ." ] ]
1606.05237
[ [ "Localized Reduced Basis Approximation of a Nonlinear Finite Volume\n Battery Model with Resolved Electrode Geometry" ], [ "Abstract In this contribution we present first results towards localized model order reduction for spatially resolved, three-dimensional lithium-ionbattery models.", "We introduce a localized reduced basis scheme based on non-conforming local approximation spaces stemming from a finite volume discretizationof the analytical model and localized empirical operator interpolation for the approximation of the model's nonlinearities.", "Numerical examples are provided indicating the feasibility of our approach." ], [ "Introduction", "Over the recent years, three dimensional lithium (Li) ion battery models that fully resolve the microscopic geometry of the battery electrodes have become a subject of active research in electrochemistry [10].", "These models are also studied in the collaborative research project Multibat, where the influence of the microscopic electrode geometry plays in important role in understanding the degradation process of Li-plating [9].", "Due to the strongly nonlinear character of these models and the large number of degrees of freedom of their discretization, numerical simulations are time consuming and parameter studies quickly turn prohibitively expensive.", "Our work in context of the Multibat project has shown that model reduction techniques such as reduced basis (RB) methods are able to vastly reduce the computational complexity of parametrized microscale battery models while retaining the full microscale features of their solutions [13], [14].", "Still, such methods depend on the solution of the full high-dimensional model for selected parameters during the so-called offline phase.", "When only relatively few simulations of the model are required – as it is typically the case for electrochemistry simulations where one is mainly interested in the qualitative behaviour of the battery cell – the offline phase can quickly take nearly as much time as the simulation of the full model for all parameters of interest would have required.", "It is, therefore, paramount to reduce the number of full model solves as much possible.", "Localized RB methods construct spatially localized approximation spaces from few global model solves or even by only solving adequate local problems (see also [5], [6], [12] and the references therein).", "Thus, these methods are a natural choice for accelerating the offline phase of RB schemes, in particular for problems with a strong microscale character such as geometrically resolved electrochemistry simulations.", "While localized RB methods have been studied extensively for linear problems and while there are first results for instationary problems [12], [13], we are not aware of any previous work treating nonlinear models.", "In this contribution, we introduce a localized RB scheme for nonlinear finite volume battery models, which builds local approximation and interpolation spaces by decomposition of global solution snapshots w.r.t.", "a given coarse triangulation of the computational domain (sec:lrb).", "As a preparation, we will first briefly review the microscale battery model under consideration (sec:model), its discretization (sec:discretization) and finally its RB approximation (sec:rb).", "We will close with first numerical experiments that investigate the applicability of localized RB techniques to the problem at hand (sec:experiment)." ], [ "Analytical model", "As in [13], [14], we consider the microscale battery model introduced in [10] (without taking thermal effects into account and assuming constant $t_+$ ).", "In this model, the battery cell is described via coupled systems of partial differential equations for the concentration of Li$^+$ -ions and the electrical potential $\\phi $ for each part of the cell: the electrolyte, positive and negative electrode, as well as positive and negative current collector.", "In the electrolyte, the change of the concentration $c$ is governed by a diffusion process, whereas $\\phi $ is determined by a stationary potential equation with source term depending non-linearly on $c$ : $\\frac{\\partial c}{\\partial t} - \\nabla \\cdot (D_e \\nabla c) &= 0,\\\\-\\nabla \\cdot \\Bigl (\\kappa \\frac{1 - t_+}{F}RT \\frac{1}{c} \\nabla c - \\kappa \\nabla \\phi \\Bigr ) &= 0.$ In the electrodes, the evolution of $c$ , i.e.", "the intercalation of Li-ions into the active particles, is again driven by diffusion.", "The potential $\\phi $ no longer depends on the Li-ion distribution: $\\frac{\\partial c}{\\partial t} - \\nabla \\cdot (D_s \\nabla c) &= 0,\\\\-\\nabla \\cdot (\\sigma \\nabla \\phi ) &= 0.$ No Li-ions enter the metallic current collectors, so $c \\equiv 0$ on this part of the domain, whereas $\\phi $ is again given as: $-\\nabla \\cdot (\\sigma \\nabla \\phi ) = 0.$ The reaction at the interface between active electrode particles and the electrolyte is governed by the so-called Butler-Volmer kinetics which determine the electric current $j = \\nabla \\phi \\cdot n$ from the active particle into the electrolyte as $j = 2k\\sqrt{c_ec_s(c_{max}- c_s)} \\sinh \\left(\\frac{\\phi _s - \\phi _e -U_0(\\frac{c_s}{c_{max}})}{2RT} \\cdot F \\right),$ where $c_e$ , $\\phi _e$ ($c_s$ , $\\phi _s$ ) are the concentration and potential on the electrolyte (solid particle) side of the interface.", "The Li-ion flux $N$ over the interface proportionally depends on $j$ and is given by $N = j / F$ .", "Note that the Butler-Volmer relations ensure the coupling between both considered variables and, through the exponential dependence on the overpotential $\\phi _s - \\phi _e - U_0(c_s/c_{max})$ , lead to a highly nonlinear behaviour of the system.", "Finally, continuity conditions for $\\phi $ are imposed between electrode and current collector, whereas there is no coupling between electrolyte and current collector.", "The following boundary conditions are imposed: $\\phi = U_0(c(0) / c_{max})$ at the negative current collector boundary, Neumann boundary conditions at the positive current collector (applied fixed charge/discharge rate) and periodic boundary conditions for $c$ and $\\phi $ at the remaining domain boundaries.", "We denote the initial concentration at time $t=0$ by $c_0 = c(0)$ , the final time is denoted as $T$ .", "All appearing natural/material constants as well as the initial data is summarised in tab:constants." ], [ "Finite volume discretization", "Following [16], the continuous model is discretized using a basic cell centered finite volume scheme on a voxel grid.", "Each voxel is assigned a unique subdomain and the Butler-Volmer conditions are chosen as numerical flux on grid faces separating an electrolyte from an electrode voxel.", "We obtain a single nonlinear finite volume operator $A_\\mu : V_h \\oplus V_h \\rightarrow V_h \\oplus V_h$ for the whole computational domain, where $V_h$ denotes the space of piecewise constant grid functions and $\\mu $ indicates a parameter we want to vary.", "In the following, we will consider the applied charge current as parameter of interest.", "Implicit Euler time stepping with constant time step size $\\Delta t$ leads to the $N:=T/\\Delta t$ nonlinear equation systems $\\begin{bmatrix}\\frac{1}{\\Delta t}(c_{\\mu }^{(n+1)} - c_{\\mu }^{(n)}) \\\\0\\end{bmatrix}+ A_\\mu \\left(\\begin{bmatrix}c_{\\mu }^{(n+1)} \\\\\\phi _{\\mu }^{(n+1)}\\end{bmatrix}\\right)= 0, \\qquad (c_{\\mu }^{(n)}, \\phi _{\\mu }^{(n)}) \\in V_h \\oplus V_h.$ The equation systems are solved using a standard Newton iteration scheme.", "Table: Constants used in the battery model.", "The open circuit potential U 0 U_0for a state of charge ss is give as U 0 (s)=(-0.132+1.41·e -3.52s )·VU_0(s) = (-0.132+1.41\\cdot e^{-3.52 s})\\cdot Vfor the negative electrode and as U 0 (s)=[0.0677504·tanh(-21.8502·s+12.8268)-0.105734·(1.00167-s) -0.379571 - 1 .", "576-0.045·e -71.69·s 8 +0.01·e -200·(s-0.19) +4.06279]·VU_0(s) = [0.0677504 \\cdot \\tanh (-21.8502\\cdot s+ 12.8268)- 0.105734 \\cdot \\bigl ( (1.00167 - s)^{-0.379571} - 1.576\\bigr ) - 0.045\\cdot e^{-71.69 \\cdot s^8}+ 0.01\\cdot e^{-200\\cdot (s-0.19)} + 4.06279]\\cdot V for the positive electrode.Note that we can decompose $A_\\mu $ as $A_\\mu = A_\\mu ^{(aff)} + A^{(bv)} + A^{(1\\!/\\!c)}$ where $A^{(bv)}, A^{(1\\!/\\!c)}: V_h \\oplus V_h \\rightarrow V_h \\oplus V_h$ accumulate the numerical fluxes corresponding to eq:butlervolmer and $\\kappa \\frac{1 - t_+}{F}RT \\frac{1}{c} \\nabla c$ .", "Thus, the operator $A_\\mu ^{(aff)}$ collecting the remaining numerical fluxes is affine linear and the only operator in the decomposition depending on the charge rate.", "$A_\\mu ^{(aff)}$ can be further decomposed as $A_\\mu ^{aff} = A^{const} + \\mu \\cdot A^{bnd} + A^{lin},$ with constant, non-parametric operators $A^{const}, A^{bnd}$ corresponding to the boundary conditions and a non-parametric linear operator $A^{(lin)}$ ." ], [ "Reduced basis approximation", "As reduced model we consider the Galerkin projection of eq:detailed onto an appropriate reduced basis space $\\tilde{V} \\subseteq V_h \\oplus V_h$ , i.e.", "we solve $P_{\\tilde{V}} \\left\\lbrace \\begin{bmatrix}\\frac{1}{\\Delta t}(\\tilde{c}_{\\mu }^{(n+1)} - \\tilde{c}_{\\mu }^{(n)}) \\\\0\\end{bmatrix}+A_\\mu \\left(\\begin{bmatrix}\\tilde{c}_{\\mu }^{(n+1)} \\\\\\tilde{\\phi }_{\\mu }^{(n+1)}\\end{bmatrix}\\right) \\right\\rbrace = 0, \\quad (\\tilde{c}_{\\mu }^{(n)}, \\tilde{\\phi }_{\\mu }^{(n)}) \\in \\tilde{V},$ where $P_{\\tilde{V}}$ denotes the orthogonal projection onto $\\tilde{V}$ .", "In order to obtain at an online efficient scheme, the projected operator $P_{\\tilde{V}} \\circ A_\\mu $ has to be approximated by an efficiently computable approximation.", "Considering the decompositions eq:operatordecomposition,eq:affineoperatordecomposition, only the nonlinear operators $A_\\mu ^{(bv)}$ , $A_\\mu ^{(1\\!/\\!c)}$ require special treatment for which we employ empirical operator interpolation [8] based on the empirical interpolation method [2].", "Denoting the discrete time differential operator by $B$ , the fully reduced scheme is then given as $\\begin{aligned}\\Bigl \\lbrace P_{\\tilde{V}} \\circ B & +P_{\\tilde{V}} \\circ A^{(const)} + \\mu \\cdot P_{\\tilde{V}}\\circ A^{(bnd)}+ P_{\\tilde{V}} \\circ A^{(lin)} \\\\& + \\lbrace P_{\\tilde{V}} \\circ I_{M^{(1\\!/\\!c)}}^{(1\\!/\\!c)}\\rbrace \\circ \\tilde{A}_{M^{(1\\!/\\!c)}}^{(1\\!/\\!c)} \\circ R_{M^{\\prime (1\\!/\\!c)}}^{(1\\!/\\!c)} \\\\& + \\lbrace P_{\\tilde{V}} \\circ I_{M^{(bv)}}^{(bv)}\\rbrace \\circ \\tilde{A}_{M^{(bv)}}^{(bv)} \\circ R_{M^{\\prime (bv)}}^{(bv)}\\ \\ \\ \\Bigr \\rbrace \\left(\\begin{bmatrix}\\tilde{c}_{\\mu }^{(t+1)} \\\\\\tilde{\\phi }_{\\mu }^{(t+1)}\\end{bmatrix}\\right)= 0,\\end{aligned}$ where $\\tilde{A}^{(*)}_{M^{(*)}}: \\mathbb {R}^{M^{\\prime (*)}} \\rightarrow \\mathbb {R}^{M^{(*)}}$ ($* = bv,1\\!/\\!c$ ) denotes the restriction of $A^{(*)}$ to certain $M^{(*)}$ image degrees of freedom given the required $M^{\\prime (*)}$ source degrees of freedom, $R^{(*)}_{M^{\\prime (*)}}: V_h \\oplus V_h \\rightarrow \\mathbb {R}^{M^{\\prime (*)}}$ is the linear operator restricting finite volume functions to these $M^{\\prime (*)}$ source degrees of freedom, and $I^{(*)}_{M^{(*)}}: \\mathbb {R}^{M^{(*)}} \\rightarrow V_h \\oplus V_h$ is the linear interpolation operator to the $M^{(*)}$ evaluation points and an appropriately selected interpolation basis.", "Note that for the considered finite volume scheme we have $M^{\\prime (*)} \\le 14\\cdot M^{(*)}$ , thus $\\tilde{A}^{(*)}_{M^{(*)}}$ can be computed quickly for sufficiently small $M^{(*)}$ .", "The remaining terms in eq:eireduced are linear (or constant) and can be pre-computed for a given reduced basis of $\\tilde{V}$ .", "Figure: Small battery geometry used in numerical experiment.Domain: 104μm×40μm×40μm104\\mu m \\times 40\\mu m \\times 40\\mu m, 4.6004.600 degrees of freedom.Coloring: Li + ^+-concentration at final simulation time T=2000sT=2000s, electrolyte not depicted.In [13] we have considered the solution of eq:eireduced where $\\tilde{V}$ and the interpolation data for $A^{(bv)}$ , $A^{(1\\!/\\!c)}$ have been generated using standard model order reduction techniques.", "The reduced space $\\tilde{V}$ was determined by computing a proper orthogonal decomposition (POD) [17] of solution trajectories of eq:detailed for an equidistant training set of charge rate parameters.", "Since the $c$ and $\\phi $ variables are defined on different scales, the POD had to be applied separately for both variables, yielding a reduced space of the form $\\tilde{V} = \\tilde{V}_c \\oplus \\tilde{V}_\\phi $ , in order to obtain a stable scheme.", "Moreover, the intermediate stages of the Newton algorithms used for solving eq:detailed were included in the snapshot data to ensure the convergence of the Newton algorithms when solving the reduced scheme.", "The interpolation bases and interpolation points have been obtained by evaluating $A^{(*)}$ on the computed solution trajectories and then performing the EI-Greedy algorithm [7] on these evaluations.", "Note that for solution trajectories of eq:detailed, $A_\\mu $ vanishes identically in the $\\phi $ -component.", "Thus, applying the EI-Greedy algorithm directly to evaluations of $A_\\mu $ would not have yielded usable interpolation spaces." ], [ "Localized basis generation", "Localized RB methods can be seen as RB schemes where the reduced space $\\tilde{V}$ has a certain direct sum decomposition $\\tilde{V} = \\tilde{V}_1 \\oplus \\ldots \\oplus \\tilde{V}_K$ with subspaces $\\tilde{V}_i$ associated to some partition $\\overline{\\Omega } =\\overline{\\Omega _1} \\cup \\ldots \\cup \\overline{\\Omega _K}$ of the computational domain $\\Omega $ .", "Since this imposes an additional constraint on the possible choices of reduced spaces $\\tilde{V}$ , it is not to be expected that such methods yield better approximation spaces for the same (total) dimension of $\\tilde{V}$ than classical RB methods.", "However, these methods can yield enormous saving in computation time during basis generation.", "In particular, we are interested in the following aspects: When the parametrization of the problem mainly affects the global solution behaviour, only few global solution snapshots may be required to observe all relevant local behaviour.", "This can be exploited by computing local approximation spaces from global solutions which have been decomposed according to the partition $\\Omega _1 \\cup \\ldots \\cup \\Omega _K$ (e.g. [1]).", "The local approximation spaces $\\tilde{V}_i$ may be enriched by solving appropriate local problems on a neighbourhood of $\\Omega _i$ .", "The solution of the local problems can be trivially parallelized, and each local problem will be solvable much faster than the global problem, which might even be unsolvable with the available computational resources (e.g. [15]).", "When the problem undergoes local changes (e.g.", "geometry change due to Li-plating), the spaces $\\tilde{V}_i$ which are unaffected by the change can be reused and only few new local problems have to be solved (e.g. [5]).", "For many applications, the time for basis generation must be taken into account when considering the overall efficiency of the reduction scheme.", "Hence, such localization approaches can be an essential tool for making model order reduction profitable for these applications.", "This is also the case for battery simulations, where typically only relatively few parameter samples are required to gain an appropriate idea of the behaviour of the model and these same computational resources are available for all required simulations.", "Also note that while reduced system matrices/Jacobians are dense matrices for standard RB approaches, one typically obtains block sparse matrices for localized RB approaches, so the increased global system dimension can be largely compensated by appropriate choices of linear solvers.", "In this contribution we investigate if spatially resolved electrochemistry simulations are in principle amenable to such localization techniques.", "For this we partition the computational domain with a cuboid macro grid with elements $\\Omega _1, \\ldots \\Omega _K$ that are aligned with the microscale voxel grid of the given finite volume discretization (cf.", "fig:rbdims).", "This partition induces a direct sum decomposition of $V_h$ : $V_h = V_{h,1} \\oplus \\ldots \\oplus V_{h,K}, \\qquad V_{h_i} = \\lbrace f \\in V_h\\,|\\, \\operatorname{supp}(f) \\subseteq \\overline{\\Omega _i}\\rbrace .$ We now compute local reduced spaces $\\tilde{V}_{c,i}$ , $\\tilde{V}_{\\phi ,i}$ by first computing global solution snapshots $c^{(n)}_{\\mu _s}$ , $\\phi ^{(n)}_{\\mu _s}$ for preselected parameters $\\mu _1, \\ldots , \\mu _S$ and then performing local PODs of the $L^2$ -orthogonal projections of these snapshots onto the local finite volume spaces $V_{h,i}$ .", "Hence, $\\tilde{V}_{c,i} \\subseteq \\operatorname{span}\\lbrace P_{V_{h,i}}(c^{(n)}_{\\mu _s})\\ |\\ 1 \\le s\\le S,\\ 1 \\le n \\le N\\rbrace ,\\\\\\tilde{V}_{\\phi ,i} \\subseteq \\operatorname{span}\\lbrace P_{V_{h,i}}(\\phi ^{(n)}_{\\mu _s})\\ |\\ 1 \\le s\\le S,\\ 1 \\le n \\le N\\rbrace .$ Since our high-dimensional model is already given as a non-conforming discretization, we con now simply obtain a reduced model by solving eq:reduced with the reduced space $\\tilde{V} = (\\tilde{V}_{c,1} \\oplus \\ldots \\oplus \\tilde{V}_{c,K}) \\oplus (\\tilde{V}_{\\phi ,1} \\oplus \\ldots \\oplus \\tilde{V}_{\\phi , K}).$ In order to obtain a fully localized model, localized treatment of the nonlinearities $A^{(bv)}$ , $A^{(1\\!/\\!c)}$ is required as well.", "Not only will most of the speedup during the offline phase be lost when the interpolation data is computed without localization.", "Global interpolation basis vectors will also induce a coupling between all local approximation spaces $\\tilde{V}_{c,i}$ , $\\tilde{V}_{\\phi ,i}$ .", "Thus the block sparsity structure of the Jacobians appearing in the Newton update problems for solving eq:eireduced is lost, strongly deteriorating reduced solution times.", "Moreover, the additional reduced degrees of freedom due to localization can exhibit a destabilizing effect when not accounted for while generating the interpolation spaces: in the limit when each subdomain $\\Omega _i$ corresponds to a single voxel, we have $\\tilde{V} = V_h \\oplus V_h$ whereas the images of the interpolated operators are only $M^{(bv)}/M^{(1\\!/\\!c)}$ -dimensional with $M^{(bv)}/M^{(1\\!/\\!c)} \\ll \\dim (V_h \\oplus V_h)$ .", "As a first approach to localized treatment of the nonlinear operators, we proceed similar to the reduced basis generation.", "We first construct local empirically interpolated operators $I^{(*)}_{i,M^{(*)}_i}\\circ \\tilde{A}^{(*)}_{i,M^{(*)}_i}\\circ R^{(*)}_{i,M^{\\prime (*)}_i}$ ($* = bv,1\\!/\\!c$ ) by applying the EI-Greedy algorithm to the projected evaluations $\\lbrace P_{V_{h,i}}(A^{(*)}([c^{(n)}_{\\mu _s}, \\phi ^{(n)}_{\\mu _s}]^T))\\ |\\ 1\\le s\\le S,\\ 1\\le n \\le N\\rbrace .$ We then obtain the localized interpolated operators $A^{(*)} \\approx I^{(*)} \\circ \\tilde{A}^{(*)} \\circ R^{(*)},$ where $I^{(*)} = \\Bigl [I^{(*)}_{1,M^{(*)}_1}, \\ldots , I^{(*)}_{K,M^{(*)}_K}\\Bigr ], \\quad \\tilde{A}^{(*)} =\\operatorname{diag}\\Bigl (\\tilde{A}^{(*)}_{1,M^{(*)}_1}, \\ldots , \\tilde{A}^{(*)}_{K,M^{(*)}_K}\\Bigr ), \\\\R^{(*)} = \\Bigl [R^{(*)}_{1,M^{\\prime (*)}_1}, \\ldots , R^{(*)}_{K,M^{\\prime (*)}_K}\\Bigr ]^T.", "$ Using these operators in eq:eireduced leads to a basic, fully localized and fully reduced approximation scheme for eq:detailed.", "In order to obtain a stable reduced scheme, accurate approximation of the Butler-Volmer fluxes is crucial.", "However, each localized interpolated operator only takes interface fluxes into its associated domain $\\Omega _i$ into account: Let $T_1$ be a finite volume cell at the boundary of $\\Omega _i$ and $T_2$ an adjacent cell in a different subdomain $\\Omega _j$ , $i \\ne j$ .", "Unless both cells are selected as interpolation points for the respective operators, local mass conservation will be violated at the $T_1$ /$T_2$ interface due to the errors introduced by empirical interpolation.", "To investigate whether these jumps in the interface fluxes of the interpolated operators have a destabilizing effect, we consider the following modified scheme: We denote by $A^{\\prime (*)}_i: V_h \\oplus V_h \\rightarrow V_h \\oplus V_h$ , ($* = bv,1\\!/\\!c$ ) the operator which accumulates all numerical fluxes associated with $A^{(*)}$ which correspond to grid faces contained in $\\overline{\\Omega _i}$ .", "Fluxes corresponding to faces which are also contained in some $\\overline{\\Omega _j}$ , $i \\ne j$ , are scaled by $1/2$ .", "This scaling ensures that we have $A^{(*)} = \\sum _{i=1}^K A^{\\prime (*)}_i.$ Each operator $A^{\\prime (*)}_i$ is interpolated separately yielding approximations $I^{\\prime (*)}_{i,M^{(*)}_i}\\circ \\tilde{A}^{\\prime (*)}_{i,M^{(*)}_i}\\circ R^{\\prime (*)}_{i,M^{\\prime (*)}_i}$ , where the interpolation data is again obtained via EI-Greedy algorithms for the evaluations $\\bigl \\lbrace A^{\\prime (*)}([c^{(n)}_{\\mu _s}, \\phi ^{(n)}_{\\mu _s}]^T)\\ |\\ 1\\le s\\le S,\\ 1\\le n \\le N\\bigr \\rbrace .$ We then proceed as before by defining $I^{\\prime (*)}$ , $\\tilde{A}^{\\prime (*)}$ , $R^{\\prime (*)}$ as in eq:defIK,eq:defR, obtaining the localized approximation $A^{(*)} \\approx I^{\\prime (*)} \\circ \\tilde{A}^{\\prime (*)} \\circ R^{\\prime (*)}$ .", "Figure: Relative model order reduction errors for the concentration variable cc.The error is measured in the L 2 L^2-in space, L ∞ L^\\infty -in time,L ∞ L^\\infty -in μ\\mu norm for 10 randomly sampled parameters μ∈𝒫:=[0.00012,0.0012]A/cm 2 ≈[0.1,1]C\\mu \\in \\mathcal {P} := [0.00012, 0.0012]\\, A/cm^2 \\approx [0.1, 1]\\, C. Top left: errors for the fully localized scheme, V ˜ i :=V ˜ c,i ⊕V ˜ φ,i \\tilde{V}_i:= \\tilde{V}_{c,i} \\oplus \\tilde{V}_{\\phi ,i}, M loc :=max i (max(M i (bv) ,M i (1/c) ))M_{loc}:= \\max _i (\\max (M^{(bv)}_i, M^{(1\\!/\\!c)}_i)).Top right: errors for the fully localized scheme with additional special treatment of theinterface fluxes, M loc ' :=max i (max(M i '(bv) ,M i '(1/c) ))M^\\prime _{loc}:= \\max _i (\\max (M^{\\prime (bv)}_i, M^{\\prime (1\\!/\\!c)}_i)).Bottom left: errors for reduced basis approximation without localization, M:=max(M (bv) ,M (1/c) )M:=\\max (M^{(bv)},M^{(1\\!/\\!c)}).", "Bottom right: errors for reduced basis approximation without localization with sameaxis scaling as in top row.Figure: Maximum local reduced basis dimensions dim(V ˜ c,i )\\dim (\\tilde{V}_{c,i}) (left) and dim(V ˜ φ,i )\\dim (\\tilde{V}_{\\phi ,i}) (right)obtained in the numerical experiment." ], [ "Numerical experiment", "As a first numerical experiment we consider the small battery geometry depicted in fig:geometry.", "For this geometry we compare the performance of the two localized RB approximation schemes introduced in sec:lrb with the scheme without localization described in sec:rb.", "The model was simulated for 2000 seconds with equidistant time steps of size $\\Delta t:=10s$ .", "In order to preclude any effects from possibly insufficient sampling of the solution manifold, the reduced models were constructed using the relatively large amount of $S=20$ equidistant parameters in the parameter domain $\\mathcal {P}:= [0.00012, 0.0012]\\, A/cm^2 \\approx [0.1, 1]\\, C$ .", "All reduced approximation and interpolation spaces were computed with relative POD/EI-Greedy error tolerances of $10^{-7}$ .", "The resulting local reduced basis dimensions for the concentration and potential variables are depicted in fig:rbdims.", "The maximum model reduction errors were estimated by computing the reduction errors for a test set of 10 random parameters and are shown for the concentration variable in fig:errors (the errors in the potential variable show similar behaviour).", "All simulations of the high-dimensional finite volume battery model have been performed within the DUNE software framework [3], [4], which has been integrated with our model order reduction library pyMOR [11].", "We observe (fig:errors, top row) that both localized schemes yield stable reduced order models with good error decay, provided a sufficiently large number of interpolation points is chosen.", "The localized scheme with special treatment of the boundary fluxes (top right) is indeed overall more stable than the localized scheme without boundary treatment (top left) and yields slightly smaller reduction errors.", "In comparison to the global RB approximation (bottom left), less reduced basis vectors/interpolation points are required per subdomain to obtain a good approximation for the localized schemes.", "As expected for localized RB schemes, the total number of basis vectors/interpolation points is larger (cf.", "bottom right) than for the global scheme, however.", "Given the small size of the full order model, we cannot expect any speedup for the localized reduced models.", "Nevertheless, based on our experience with global RB approximation of this model [13], we expect only a small increase in the number of required basis vectors/interpolation points to approximate larger, finely resolved geometries.", "Thus, good speedups can be expected for large-scale applications.", "Verifying this hypothesis, as well as developing algorithms for efficient localized construction and enrichment of the local approximation spaces, will be subject to future work." ], [ "Conclusion", "In this contribution we demonstrated the applicability of the Localized Reduced Basis Method for an instationary nonlinear finite volume Li-ion battery model with resolved pore scale electrode geometry.", "To this end, we have extended the Localized Reduced Basis Method to parabolic systems of equations, while simultaneously employing the localized empirical operator interpolation in order to deal with the strong nonlinearities of the underlying electrochemical reaction processes.", "Numerical experiments were given to demonstrate the model order reduction potential of this approach." ] ]
1606.05070
[ [ "Nonequilibrium Tuning of the Thermal Casimir Effect" ], [ "Abstract In net-neutral systems correlations between charge fluctuations generate strong attractive thermal Casimir forces and engineering these forces to optimize nanodevice performance is an important challenge.", "We show how the normal and lateral thermal Casimir forces between two plates containing Brownian charges can be modulated by decorrelating the system through the application of an electric field, which generates a nonequilibrium steady state with a constant current in one or both plates, reducing the ensuing fluctuation-generated normal force while at the same time generating a lateral drag force.", "This hypothesis is confirmed by detailed numerical simulations as well as an analytical approach based on stochastic density functional theory." ], [ "Numerical simulations", "We used the standard three-dimensional formulation of the Ewald summation introducing a real-space range $\\alpha $ .", "The complementary long-range part of the interaction is calculated in Fourier space, requiring modes out to wave-vectors $q_m\\sim 1/\\alpha $ .", "The interaction energy as well as the forces are independent of $\\alpha $ which can then be used to optimize the speed of the calculation.", "Balancing the effort in real and Fourier space leads to a value of $\\alpha $ which decreases with the number of simulated particles so that the computer effort to calculate the force on all of the particles scales as $O(N^{8/5})$ , with $N$ the number of particles.", "In practice, for each value of $N$ , we made a series of test runs with variable $\\alpha $ chosen to produce the most rapid code.", "We used a relative precision of $10^{-11}$ in the Ewald evaluation.", "We integrated the Langevin equation using the Euler method with a time step $dt$ .", "We chose this step as follows: First we performed a series of short simulations with very large time steps to find the stability limit of the integrator.", "We then divided this value of $dt$ by 100 to find a stable regime where integration errors are not too large.", "We performed high statistics simulations for different time steps about our first estimate for the production time step.", "We measured the force between plates with statistical errors better than 2%, estimated using the method of [39], and found the value of $dt$ which also gives systematic errors of order of 2%.", "In practice it is immediately clear when the step size has been chosen too large when comparing several different simulations.", "We note that the effect of a finite step size can be studied analytically using the concept of inverse error analysis.", "A set of variables $i$ with energy function $\\mathcal {H}$ when simulated by a Euler integrator generates an equilibrium measure characterised by an effective energy [40] $\\bar{\\mathcal {H}} = \\mathcal {H} + \\frac{D dt}{4} \\sum _i \\left( 2 \\nabla _i^2 \\mathcal {H} -\\beta (\\nabla _i \\mathcal {H})^2 \\right),$ where $D$ is the particle diffusion constant.", "When applied to a pair of Coulomb particles this gives an extra effective potential in $dt/r^4$ .", "Remarkably this correction is very similar to the Wigner expansion [41] in quantum mechanics.", "The integration error looks very much like the quantum Casimir interaction in the high temperature limit, with $dt$ playing a role comparable to $\\hbar ^2$ .", "With the choice of the time step set as above, we performed preliminary simulations in order to choose the temperature and the number of particles for the main simulations.", "In these initial studies we found that finite size corrections are rather strong, requiring a large number of particles to observe the correct asymptotic form of the Casimir interaction at zero driving field.", "We interpret this as a consequence of the unusual form of screening in quasi-two-dimensional plasmas, coupled through the three dimensional Coulomb interaction, which leads to a charge structure factor, in the Debye-Hückel limit (for a single plate), of the form [42] $S({\\bf Q}) = \\frac{|{\\bf Q}|}{(|{\\bf Q}| +m/2)}$ with $m = q^2 \\bar{n} \\beta /\\epsilon $ and $\\bar{n} = N/H^2$ .", "This singular form in $|{\\bf Q}|$ leads to a power-law decay of correlations in real space [42], in contrast to the exponential decay of correlations in three-dimensional plasmas.", "We chose to run our simulations at a temperature such the Bjerrum length is comparable to the particle separation; specifically we took the values $T =1.03\\, q^2 \\sqrt{\\bar{n}}/(4 \\pi \\epsilon )$ .", "This corresponds to a region intermediate between the high and low temperature limits.", "Simulations at lower temperatures showed strong deviations from the Debye-Hückel structure factor, in particular $S({\\bf Q})$ develops a weak peak.", "At large temperatures the screening length $m^{-1}$ becomes large, physically thermal fluctuations destroy screening, and the far field limit is pushed to distances $L$ comparable to the system size and thus into a region where finite size corrections become very important.", "We finally chose a series of systems with three different values of $N$ : $N=1000, 2000, 4000$ .", "With these numbers of particles the un-driven systems shows a Casimir interaction, with the correct amplitude, over a range of separations from $0.02H$ to $0.12H$ .", "A simulation for $N=500$ showed strong modification of the Casimir amplitude for the separation of $0.02H$ .", "Proper scaling of the data as the number of particles increases is a strong test of being in the continuum limit.", "For the values of $N$ that we study and separations below $0.02H$ we cross-over into a single particle regime, rather than a collective Casimir regime.", "For separations large than $0.12H$ there is an exponential decay of interactions, coming from the discrete nature of Fourier modes within a box.", "Within this range of distances Fig.", "(2) in the main text shows that we control systematic errors at zero driving force to within about 2%.", "The largest systems that we study require one month of simulation per point, to reduce statistical errors to within 2%.", "The residual vertical spread in the points of Fig.", "(2) is dominated by finite size corrections, which would require larger numbers of particles to reduce further.", "With a series of systems calibrated at zero driving force we performed the final series of simulations to measure interactions as a function of driving field.", "We note that at the largest driving field it is necessary to re-calibrate the criterion for the choice of $dt$ : Relative motion of driven/un-driven particles is the fastest process in the simulation, which must be well resolved to generate accurate results.", "The data presented is evaluated for an electric field which is skewed compared to the simulation cell.", "Other simulations (data not shown) show that the curves of Fig.", "2 become anisotropic for the largest fields that we studied.", "This anisotropy can be understood by the fact that the driving reduces the effective number of interacting modes contributing to the Casimir interactions.", "This small number of modes in the summation then leads to angular modulation of the curves.", "This anisotropy can be treated analytically within the stochastic density functional theory by replacing the Fourier integrals in Eqs.", "(9, 10) by a discrete Fourier sum." ], [ "Analytical results for the force", "The compact form of Eqs.", "(9, 10), as derived from Eq.", "(8), are deduced from the expressions $\\langle F_\\perp (L)\\rangle = -T S \\int \\!\\!", "\\frac{d^2{\\bf Q}}{{(2\\pi )}^2} \\frac{Q \\, m_1 m_2 \\exp (-2QL)}{(m_1+2Q)(m_2+2Q)\\left( 1+\\frac{4 {({\\bf Q}\\cdot ({\\bf v}_1-{\\bf v}_2))}^2}{Q^2{(D_1m_1+D_2m_2+2(D_1+D_2)Q)}^2} \\right) - m_1 m_2 \\exp (-2QL)},$ $\\langle {\\bf F}_{||}(L) \\rangle &=& - T S \\int \\frac{d^2{\\bf Q}}{ {(2\\pi )}^2}\\frac{2 m_1 m_2 \\, e^{-2 Q L} }{(m_1 + 2Q) (m_2 + 2Q) \\left(1 + \\frac{4{({\\bf Q}\\cdot ({\\bf v}_1 - {\\bf v}_2))}^2}{ Q^2 {(D_1 m_1 + D_2 m_2 + 2(D_1 + D_2) Q)}^2}\\right) - m_1 m_2 \\, e^{-2 Q L} } \\nonumber \\\\ && \\times \\frac{{({\\bf Q}\\!\\cdot \\!", "({\\bf v}_1 - {\\bf v}_2))} {\\bf Q}}{Q(D_1 m_1 + D_2 m_2 + 2(D_1 + D_2) Q)},$ which are derived from Eqs.", "(6) and (7) under averaging and using the density fluctuation correlation functions derived from Eq.", "(4)." ] ]
1606.04911
[ [ "Functional Horseshoe Priors for Subspace Shrinkage" ], [ "Abstract We introduce a new shrinkage prior on function spaces, called the functional horseshoe prior (fHS), that encourages shrinkage towards parametric classes of functions.", "Unlike other shrinkage priors for parametric models, the fHS shrinkage acts on the shape of the function rather than inducing sparsity on model parameters.", "We study the efficacy of the proposed approach by showing an adaptive posterior concentration property on the function.", "We also demonstrate consistency of the model selection procedure that thresholds the shrinkage parameter of the functional horseshoe prior.", "We apply the fHS prior to nonparametric additive models and compare its performance with procedures based on the standard horseshoe prior and several penalized likelihood approaches.", "We find that the new procedure achieves smaller estimation error and more accurate model selection than other procedures in several simulated and real examples.", "The supplementary material for this article, which contains additional simulated and real data examples, MCMC diagnostics, and proofs of the theoretical results, is available online." ], [ "Introduction", "Since the seminal work of [23], shrinkage estimation has been immensely successful in various statistical disciplines and continues to enjoy widespread attention.", "Many shrinkage estimators have a natural Bayesian flavor.", "For example, one obtains the ridge regression estimator as the posterior mean arising from an isotropic Gaussian prior on the vector of regression coefficients [24], [21].", "Along similar lines, an empirical Bayes interpretation of the (positive part) James–Stein estimator can be obtained [11].", "Such connections have been extended to the semiparametric regression context, with applications to smoothing splines and penalized splines [40], [38].", "Over the past decade and a half, a number of second-generation shrinkage priors have appeared in the literature in relation to high-dimensional sparse estimation.", "Such priors can be almost exclusively expressed as global-local scale mixtures of Gaussians [33]; examples include the relevance vector machine [39], normal/Jeffrey's prior [2], the Bayesian lasso [32], [17], the horseshoe [8], normal/gamma and normal/inverse-Gaussian priors [7], [16], generalized double Pareto priors [1] and Dirichlet–Laplace priors [3].", "These priors typically have a large spike near zero with heavy tails, thereby providing an approximation to the operating characteristics of sparsity inducing discrete mixture priors [12], [25].", "For more on connections between Bayesian model averaging and shrinkage, refer to [33].", "A key distinction between ridge-type shrinkage priors and the global-local priors is that while ridge-type priors typically shrink towards a fixed point, most commonly the origin, the global-local priors provide shrinkage towards the union of subspaces consisting of sparse vectors, with the amount of sparsity controlled by certain hyperparameters [3].", "In this article, we further enlarge the scope of shrinkage to present a class of functional shrinkage priors, namely the functional horseshoe priors (fHS), that facilitate shrinkage towards pre-specified subspaces.", "The shrinkage factor (defined in Section 3) is assigned a $\\mbox{Beta}(a, b)$ prior with $a, b < 1$ , which has the shape of a horseshoe [8].", "However, while the horseshoe prior of [8] shrinks towards sparse vectors, the proposed functional horseshoe prior shrinks functions towards arbitrary subspaces.", "As a preliminary example, consider a nonparametric regression model with unknown regression function $f : \\mathcal {X} \\rightarrow \\mathbb {R}$ given by $Y = F + \\varepsilon , \\quad \\varepsilon \\sim { \\mathrm {N} }(0, \\sigma ^2 \\mathrm {I}_n),$ where $Y = (y_1, \\ldots , y_n)^{{ \\mathrm {\\scriptscriptstyle T} }}$ , and $F = (f(x_1), \\ldots , f(x_n))^{{ \\mathrm {\\scriptscriptstyle T} }} = \\mathbb {E}(Y \\mid {\\bf x})$ , with the covariates $x_i \\in \\mathcal {X} \\subset \\mathbb {R}$ .", "In (REF ), we can either make parametric assumptions (e.g., linear or quadratic dependence on $x$ ) regarding the shape of $f$ , or model it nonparametrically using splines, wavelets, Gaussian processes, etc.", "Although one can examine a scatter plot or perform a goodness of fit test to ascertain the validity of a linear or quadratic model in (REF ), such an exercise is only feasible in relatively simple settings.", "In relatively complex and/or high dimensional problems, there is clearly a need for an automatic data-driven procedure to adapt between models of varying complexity.", "With this motivation, we propose the functional horseshoe prior that encourages shrinkage towards a parametric class of models embedded inside a larger semiparametric model, as long as a suitable projection operator can be defined.", "For example, in (REF ), $f$ will be shrunk towards a linear or quadratic function if such parametric assumptions are supported by the data, and will remain unshrunk otherwise.", "As noted already, our approach is not limited to the univariate regression context and can be extended to the varying coefficient model [20], density estimation via log-spline models [26], and additive models [19], among others; further details are provided in Section .", "In the additive regression context, the proposed approach is highly competitive to state-of-the-art procedures such as the Sparse Additive Model (SpAM) of [37] and the High-dimensional Generalized Additive Model (HGAM) by [29].", "We provide theoretical support to the method by showing an adaptive property of the approach in the context of (REF ).", "Specifically, we show that the posterior contracts [13] at the parametric rate if the true function belongs to the pre-designated subspace, and contracts at the optimal rate for $\\alpha $ -smooth functions otherwise.", "In other words, our approach adapts to the parametric shape of the unknown function while allowing deviations from the parametric shape in a nonparametric fashion." ], [ "Preliminaries", "We begin by introducing some notation.", "For $\\alpha > 0$ , let $\\lfloor \\alpha \\rfloor $ denote the largest integer smaller than or equal to $\\alpha $ and $\\left\\lceil \\alpha \\right\\rceil $ denote the smallest integer larger than or equal to $\\alpha $ .", "Let $C^{\\alpha }[0,1]$ denote the Hölder class of $\\alpha $ smooth functions on $[0,1]$ that have continuously differentiable derivatives up to order $\\lfloor \\alpha \\rfloor $ , with the $\\lfloor \\alpha \\rfloor $ th order derivative being Lipschitz continuous of order $\\alpha - \\lfloor \\alpha \\rfloor $ .", "For a vector $x \\in \\mathbb {R}^d$ , let $\\big \\Vert x\\big \\Vert $ denote its Euclidean norm.", "For a function $g:[0,1] \\rightarrow \\mathbb {R}$ and points $x_1, \\ldots , x_n \\in [0,1]$ , let $\\big \\Vert g\\big \\Vert _{2,n}^2 = n^{-1} \\sum _{i=1}^n g^2(x_i)$ ; we shall refer to $\\big \\Vert \\cdot \\big \\Vert _{2,n}$ as the empirical $L_2$ norm.", "For an $m \\times d$ matrix $A$ with $m > d$ and $\\mbox{rk}(A) = d$ , let $\\mathfrak {L}(A)=\\lbrace A\\beta : \\beta \\in \\mathbb {R}^d \\rbrace $ denote the column space of $A$ , which is a $d$ -dimensional subspace of $\\mathbb {R}^m$ .", "Let $\\mathrm {Q}_A = A(A^{{ \\mathrm {\\scriptscriptstyle T} }} A)^{-1} A^{{ \\mathrm {\\scriptscriptstyle T} }}$ denote the projection matrix on $\\mathfrak {L}(A)$ ." ], [ "The functional horseshoe prior", "In the nonparametric regression model in (REF ), we model the unknown function $f$ as spanned by a set of pre-specified basis functions $\\lbrace \\phi _j\\rbrace _{1\\le j\\le k_n}$ as follows: $f(x) = \\sum _{j=1}^{k_n} \\beta _j \\phi _j(x).$ We shall work with the B-spline basis in the sequel, though the methodology generalizes to a larger class of basis functions.", "The B-splines basis functions can be constructed in a recursive way.", "Let the positive integer $q$ denote the degree of the B-spline basis functions satisfying $k_n>q+1$ .", "Define a sequence of knots $0=t_0<t_1<\\cdots <t_{k_n-q}=1$ In addition, define $q$ knots $t_{-q}=\\cdots =t_{-1} = t_0$ and another set of $q$ knots $t_{k_n-q}=\\cdots =t_{k_n}$ .", "As in [10], the B-spline basis functions are defined as $\\phi _{j,1}(x) &=& {\\left\\lbrace \\begin{array}{ll}1,\\:\\: t_j\\le x<t_{j+1},\\\\0,\\:\\:\\mbox{otherwise},\\end{array}\\right.", "}\\\\\\phi _{j,q+1}(x) &=& \\frac{x-t_j}{t_{j+q}-t_j}\\phi _{j,q}(x) + \\frac{t_{j+q+1}-x}{t_{j+q+1}-t_{j+1}}\\phi _{j+1,q}(x),$ for $j=-q,\\dots ,k_n-q-1$ .", "We reindex $j=-q,\\dots ,k_n-q-1$ to $j=1,\\dots ,k_n$ and the number of basis functions is $k_n$ .", "Letting $\\beta = (\\beta _1, \\ldots , \\beta _{k_n})^{{ \\mathrm {\\scriptscriptstyle T} }}$ denote the vector of basis coefficients and $\\Phi = \\lbrace \\phi _j(X_i)\\rbrace _{1 \\le i \\le n, 1 \\le j \\le k_n}$ denote the $n \\times k_n$ matrix of basis functions evaluated at the observed covariates, the model (REF ) can be expressed as $Y \\mid \\beta \\sim \\mbox{N}(\\Phi \\beta , \\sigma ^2 \\mathrm {I}_n).$ A standard choice for a prior on $\\beta $ is a $g$ -prior $\\beta \\sim \\mbox{N}(0, g (\\Phi ^{{ \\mathrm {\\scriptscriptstyle T} }} \\Phi )^{-1})$[43].", "$g$ -priors have been commonly used in linear models, since they incorporate the correlation structure of the covariates inside the prior variance.", "The posterior mean of $\\beta $ with a $g$ -prior can be expressed as $\\lbrace 1 - 1/(1+g)\\rbrace \\widehat{\\beta }$ , where $\\widehat{\\beta } = \\mathrm {Q}_{\\Phi } Y$ is the maximum likelihood estimate of $\\beta $ .", "Thus, the posterior mean shrinks the maximum likelihood estimator towards zero, with the amount of shrinkage controlled by the parameter $g$ .", "[4] studied asymptotic properties of the resulting posterior by providing bounds on the total variation distance between the posterior distribution and a Gaussian distribution centered at the maximum likelihood estimator with the inverse Fisher information matrix as covariance.", "In his work, the $g$ parameter was fixed a priori depending on the sample size $n$ and the error variance $\\sigma ^2$ .", "His results in particular imply minimax optimal posterior convergence for $\\alpha $ -smooth functions.", "Among related work, [15] established minimax optimality with isotropic Gaussian priors on $\\beta $ .", "Our goal is to define a broader class of shrinkage priors on $\\beta $ that facilitate shrinkage towards a null subspace that is fixed in advance, rather than shrinkage towards the origin or any other fixed a priori guess $\\beta _0$ .", "For example, if we have a priori belief that the function is likely to attain a linear shape, then we would like to impose shrinkage towards the class of linear functions.", "In general, our methodology allows shrinkage towards any null subspace spanned by the columns of a null regressor matrix $\\Phi _0$ , with $d_0 = \\mbox{rank}(\\Phi _0)$ the dimension of the null space.", "For example in the linear case, we define the null space as $\\mathfrak {L}(\\Phi _0)$ with $\\Phi _0 = \\lbrace {\\bf 1},{\\bf x}\\rbrace \\in \\mathbb {R}^{n \\times 2}$ , where ${\\bf 1}$ is a $n \\times 1$ vector of ones and $d_0 = 2$ .", "Shrinkage towards quadratic, or more generally polynomial, regression are achieved similarly.", "With the above ingredients, we propose the functional horseshoe prior through the following hierarchical specification: $\\pi (\\beta \\mid \\tau ) &\\propto & (\\tau ^2)^{-(k_n-d_0)/2}\\exp \\left\\lbrace -\\frac{1}{2\\sigma ^2\\tau ^2} \\beta ^{ \\mathrm {\\scriptscriptstyle T} }\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\beta \\right\\rbrace , \\\\\\pi (\\tau ) &\\propto & \\frac{(\\tau ^2)^{b-1/2 }}{(1+\\tau ^2)^{(a+b)}} \\mathbb {1}_{(0, \\infty )}(\\tau ), $ where $a, b > 0$ and recall that $\\mathrm {Q}_0 = \\Phi _0(\\Phi _0^{ \\mathrm {\\scriptscriptstyle T} }\\Phi _0)^{-1}\\Phi _0^{ \\mathrm {\\scriptscriptstyle T} }$ denotes the projection matrix of $\\Phi _0$ .", "When $\\Phi _0 = 0$ , (REF ) is equivalent to a $g$ -prior with $g = \\tau ^2$ .", "The key additional feature in our proposed prior is to introduce the quantity $(\\mathrm {I}-\\mathrm {Q}_0)$ in the exponent, which enables shrinkage towards subspaces rather than single points.", "Although the proposed prior may be singular, it follows from the subsequent results that the joint posterior of $(\\beta , \\tau ^2)$ is proper.", "Note that the prior on the scale parameter $\\tau $ follows a half-Cauchy distribution when $a = b = 1/2$ .", "Half-Cauchy priors have been recommended as a default prior choice for global scale parameters in the linear regression framework [34].", "Using the reparameterization $\\omega = 1/(1 + \\tau ^2)$ , the prior () can be interpreted as the prior induced on $\\tau $ via a $\\mbox{Beta}(a, b)$ prior on $\\omega $ .", "We shall work in the $\\omega $ parameterization subsequently for reasons to be evident shortly.", "Exploiting the conditional Gaussian specification, the conditional posterior of $\\beta $ is Gaussian, $\\beta \\mid Y, \\omega \\sim \\mbox{N}( { \\widetilde{\\beta } }_{\\omega }, { \\widetilde{\\Sigma } }_{\\omega }),$ where ${ \\widetilde{\\beta } }_{\\omega } = \\left(\\Phi ^{{ \\mathrm {\\scriptscriptstyle T} }} \\Phi + \\frac{\\omega }{1-\\omega }\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\right)^{-1} \\Phi ^{{ \\mathrm {\\scriptscriptstyle T} }} Y, \\quad { \\widetilde{\\Sigma } }_{\\omega } = \\sigma ^2 \\left(\\Phi ^{{ \\mathrm {\\scriptscriptstyle T} }} \\Phi +\\frac{\\omega }{1-\\omega }\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\right)^{-1}.$ We now state a lemma which delineates the role of $\\omega $ as the parameter controlling the shrinkage.", "Lemma 3.1 Suppose that $\\mathfrak {L}(\\Phi _0) \\subsetneq \\mathfrak {L}(\\Phi )$ .", "Then, $\\mathbb {E}\\left[\\Phi \\beta \\mid Y,\\omega \\right] = \\Phi { \\widetilde{\\beta } }_\\omega = (1-\\omega )\\mathrm {Q}_{\\Phi } Y + \\omega \\mathrm {Q}_0 Y,$ where $\\mathrm {Q}_\\Phi $ is the projection matrix of $\\Phi $ .", "The above lemma suggests that the conditional posterior mean of the regression function given $\\omega $ is a convex combination of the classical B-spline estimator $\\mathrm {Q}_{\\Phi } Y$ and the parametric estimator $\\mathrm {Q}_0 Y$ .", "The parameter $\\omega \\in (0,1)$ controls the shrinkage effect; the closer $\\omega $ is to 1, the greater the shrinkage towards the parametric estimator.", "We learn the parameter $\\omega $ from the data with a $\\mbox{Beta}(a, b)$ prior on $\\omega $ .", "The hyperparameter $b < 1$ controls the amount of prior mass near one.", "Figure REF illustrates the connection between the choice of the hyperparameters $a$ and $b$ and the shrinkage behavior of the prior.", "The first and the second column in Figure REF , with $a$ fixed at $1/2$ shows that the prior density of $\\omega $ increasingly concentrates near 1 as $b$ decreases from $1/2$ to $10^{-1}$ .", "The third column in Figure REF depicts the prior probability that $\\omega >0.95$ and $\\omega <0.05$ .", "Clearly, as $b$ decreases, the amount of prior mass around one increases, which results in stronger shrinkage towards the parametric estimator.", "In particular, when $a= b = 1/2$ , the resulting “horseshoe\" prior density derives its name from the shape of the prior on $\\omega $ [8].", "Figure: The first two columns illustrate the prior density function of ω\\omega with different hyperparameters (a,b)(a,b): (1/2,1/2)(1/2,1/2) for the first column and (1/2,10 -1 )(1/2,10^{-1}) for the second column.", "The third column shows the prior probability that ω>0.95\\omega >0.95 (solid line) and ω<0.05\\omega <0.05 (dotted line) for varying bb and a fixed a=1/2a=1/2.When $\\mathfrak {L}(\\Phi _0) \\subsetneq \\mathfrak {L}(\\Phi )$ , we can orthogonally decompose $\\mathrm {Q}_\\Phi = \\mathrm {Q}_1 + \\mathrm {Q}_0$ , where the columns of $\\mathrm {Q}_1$ are orthogonal to $\\mathrm {Q}_0$ , i.e., $\\mathrm {Q}_1^{{ \\mathrm {\\scriptscriptstyle T} }} \\mathrm {Q}_0 = 0$ .", "To see this, since $\\mathfrak {L}(\\Phi _0) \\subsetneq \\mathfrak {L}(\\Phi )$ , we can use Gram-Schmidt orthogonalization to create ${ \\widetilde{\\Phi } } = [\\Phi _0; \\Phi _1]$ of the same dimension as $\\Phi $ such that $\\Phi _1^{{ \\mathrm {\\scriptscriptstyle T} }} \\Phi _0 = 0$ and $\\mathfrak {L}(\\Phi ) = \\mathfrak {L}({ \\widetilde{\\Phi } })$ .", "Then, we let $\\mathrm {Q}_{1}$ denote the projection matrix on $\\mathfrak {L}(\\Phi _1)$ .", "Simple algebra shows that $& \\pi (\\omega \\mid Y) = \\int \\pi (\\omega , \\beta \\mid Y) d \\beta = \\frac{\\pi (\\omega )}{m(Y)} \\int f(Y \\mid \\beta , \\omega ) \\pi (\\beta \\mid \\omega ) d \\beta \\\\& = \\omega ^{a+(k_n-d_0)/2-1}(1-\\omega )^{b-1}\\exp \\lbrace -H_n \\omega \\rbrace /m(Y), $ where $H_n = Y^{{ \\mathrm {\\scriptscriptstyle T} }} \\mathrm {Q}_{1} Y/(2\\sigma ^2)$ and $m(Y) = \\int ^1_0 \\omega ^{a+(k_n-d_0)/2-1}(1-\\omega )^{b-1}\\exp \\left\\lbrace -H_n\\omega \\right\\rbrace d\\omega $ .", "To investigate the asymptotic behavior of the implied posterior, it is crucial to find tight two-sided bounds on $m(Y)$ , which is stated in Lemma REF .", "Lemma 3.2 (Bounds on the normalizing constant) Let $A_n$ and $B_n$ be arbitrary sequences satisfying $A_n\\rightarrow \\infty $ as $n \\rightarrow \\infty $ and $B_n=O(1)$ .", "Define $t_n = \\int _0^1 \\omega ^{A_n-1}(1-\\omega )^{B_n-1}\\exp \\lbrace -H_n\\omega \\rbrace d\\omega $ .", "Then, $\\frac{\\Gamma (A_n)\\Gamma (B_n)}{\\Gamma (A_n+B_n)}\\exp \\lbrace -H_n\\rbrace (1+Q_n^L) \\le t_n \\le \\frac{\\Gamma (A_n)\\Gamma (B_n)}{\\Gamma (A_n+B_n)}\\exp \\lbrace -H_n\\rbrace (1+Q_n^U),$ where, $Q_n^U &=& \\frac{B_n}{A_n+B_n}\\exp ( H_n),\\\\Q_n^L &=& \\frac{B_nH_n}{A_n+B_n} + \\frac{DB_n(B_n+T_n)^{-A_n}}{(A_n+B_n)^{3/2}}\\left(\\exp \\lbrace H_n\\rbrace - 1 -H_n - (T_n+2)^{-1/2}\\right)_+,$ where $T_n=\\max \\lbrace A_n^2, 3\\left\\lceil H_n \\right\\rceil \\rbrace $ and $D$ is some positive constant.", "By setting $A_n=a+k_n/2$ and $B_n = b$ , Lemma REF suggests that the magnitude of the normalizing constant $m(Y)$ in (REF ) is determined by an interplay between the relative sizes of $b$ and $\\exp (H_n)$ .", "When $b$ is small enough to dominate $\\exp (H_n)$ , $m(Y) \\approx \\mbox{Be}(a+k_n/2,b) \\exp (-H_n)$ , where $\\mbox{Be}(\\cdot ,\\cdot )$ denotes the beta function.", "Otherwise, $m(Y) \\approx Be(a+k_n/2,b)b$ ignoring polynomial terms.", "This asymptotic behavior of $m(Y)$ is the key ingredient to identify the posterior contraction rate of the functional horseshoe prior.", "We also note that the magnitude of $a$ asymptotically does not affect the strength of shrinkage as long as $a$ is a fixed constant, since the prior contribution $\\omega ^{a-1}$ is dominated by the likelihood contribution $\\omega ^{k_n/2}$ ." ], [ " ", "We state a set of assumptions ([44], [9]) that have been used in the literature to prove minimax optimality of B-spline estimators.", "Assume that the following conditions hold: (A1).", "Let $u=\\max _{1\\le j \\le (k_n-1)}(t_{j+1}-t_j)$ .", "There exists a constant $C>0$ , such that $u/\\min _{1\\le j\\le (k_n-1)}(t_{j+1}-t_j) \\le C$ and $u=o(k_n^{-1})$ .", "(A2).", "There exists some distribution function $G$ with a positive continuous density such that $\\sup _{x\\in [0,1]}|G_n(x)-G(x)|= o(k_n^{-1}),$ where $G_n$ is the empirical distribution of the covariates $\\lbrace x_i\\rbrace _{1\\le i \\le n}$ , which are fixed by design.", "Under (A1) and (A2), [44] showed that the mean square error of the B-spline estimator $\\mathrm {Q}_\\Phi Y$ achieves the minimax optimal rate.", "If the true function $f_0 \\in C^{\\alpha }[0, 1]$ is $\\alpha $ -smooth and the number of basis functions $k_n \\asymp n^{1/(2 \\alpha + 1)}$ , then [44] shows that $\\mathbb {E}_0\\left[ \\big \\Vert \\mathrm {Q}_\\Phi Y - F_0\\big \\Vert _{2,n}^2\\right] = O\\left(n^{-2\\alpha /(1+2\\alpha )}\\right),$ where $\\mathbb {E}_0(\\cdot )$ represents an expectation with respect to the true data generating distribution of $Y$ .", "We now state our main result on the posterior contraction rate of the functional horseshoe prior.", "Theorem 3.3 Consider the model (REF ) equipped with the functional horseshoe prior (REF )-().", "Assume (A1) and (A2) hold and $\\mathfrak {L}(\\Phi _0) \\subsetneq \\mathfrak {L}(\\Phi )$ .", "Further, assume that for some integer $\\alpha \\ge 1$ , the true regression function $f_0\\in C^\\alpha [0,1]$ and the B-spline basis functions $\\Phi $ are constructed with $k_n-\\lfloor \\alpha \\rfloor $ knots and $\\lfloor \\alpha \\rfloor -1$ degree, where $k_n \\asymp n^{1/(1+2\\alpha )}$ .", "Suppose that the prior hyperparameters $a$ and $b$ in () satisfy $a \\in (\\delta , 1 - \\delta )$ for some constant $\\delta \\in (0, 1/2)$ , and $k_n\\log k_n\\prec -\\log b \\prec (n k_n)^{1/2}$ .", "Then, $\\mathbb {E}_0\\left[ P\\left\\lbrace \\big \\Vert \\Phi \\beta - F_0\\big \\Vert _{2,n}>M_n(f_0)^{1/2} \\mid Y \\right\\rbrace \\right] = o(1),$ where $M_n(f_0) ={\\left\\lbrace \\begin{array}{ll} \\zeta _n n^{-1},\\:\\:\\mbox{if}\\:\\:F_0 \\in \\mathfrak {L}(\\Phi _0), \\\\\\zeta _n n^{-2\\alpha /(1+2\\alpha )}\\log n,\\:\\:\\mbox{if}\\:\\: F_0^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F_0 \\asymp n,\\end{array}\\right.", "}$ where $\\zeta _n$ can be any arbitrary sequence that diverges to infinity as $n$ tends to $\\infty $ .", "Theorem REF exhibits an adaptive property of the functional horseshoe prior.", "If the true function is $\\alpha $ -smooth, then the posterior contracts around the true function at the near minimax rate of $n^{-\\alpha /(2\\alpha + 1)} \\log n$ .", "However, if the true function $f_0$ belongs to the finite dimensional subspace $\\mathfrak {L}(\\Phi _0)$ , then the posterior contracts around $f_0$ in the empirical $L_2$ norm at the parametric $1/\\sqrt{n}$ rate.", "We note that the bound $k_n \\log k_n \\prec -\\log b \\prec (n k_n)^{1/2}$ is key to the adaptivity of the posterior, since the strength of the shrinkage towards $\\mathfrak {L}(\\Phi _0)$ is controlled by $b$ .", "If $-\\log b \\prec k_n \\log k_n$ , then the shrinkage towards $\\mathfrak {L}(\\Phi _0)$ is too weak to achieve the parametric rate when $F_0\\in \\mathfrak {L}(\\Phi _0)$ .", "On the other hand, if $-\\log b \\succ (nk_n)^{1/2}$ , the resulting posterior distribution would strongly concentrate around $\\mathfrak {L}(\\Phi _0)$ , and it would fail to attain the optimal nonparametric rate of posterior contraction when $F_0\\notin \\mathfrak {L}(\\Phi _0)$ .", "We ignore the subspace of functions such that $\\lbrace F \\in \\mathbb {R}^n: F^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F = o(n),\\:\\:F\\notin \\mathfrak {L}(\\Phi _0)\\rbrace $ .", "We only focus on the function space that can be strictly separated from the null space $\\mathfrak {L}(\\Phi _0)$ , although it would be meaningful to illustrate the shrinkage behavior when the regression function $f$ approaches the null space in a sense that $F^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F/n \\rightarrow 0$ as $n\\rightarrow \\infty $ ." ], [ "Extensions to Gaussian process priors", "Even though the procedure based on the functional horseshoe prior can be interpreted as a partial linear model, its scope of applicability extends to a more general class of nonparametric models.", "We outline such an extension to Gaussian process (GP) priors below [30], [36]: $F \\mid \\Sigma ,\\tau &\\sim & { \\mathrm {N} }\\left({\\bf 0},\\lbrace \\Sigma ({\\bf x})^{-1}+(\\mathrm {I}-\\mathrm {Q}_0)/\\tau ^2\\rbrace ^{-1} \\right)\\\\\\pi (\\tau ) &\\propto & \\frac{(\\tau ^2)^{b-1/2 }}{(1+\\tau ^2)^{(a+b)}} \\mathbb {1}_{(0, \\infty )}(\\tau ),\\nonumber $ where $\\Sigma (\\cdot , \\cdot )$ is a positive definite covariance kernel with $\\Sigma ({\\bf x}) = (\\Sigma (x_i, x_j))$ , and $\\mathrm {Q}_0$ , $a$ and $b$ are defined in (REF ) and ().", "We note that the proposed prior does not define a stochastic process.", "However, it can be used as a prior on $F$ given the set of locations ${\\bf x}$ .", "To investigate the shrinkage effect of the modified GP prior, we considered two examples of GP priors with the functional shrinkage idea: shrinking towards a class of linear functions and a class of piece-wise linear functions.", "For shrinking towards linearity, it is straightforward to choose $\\Phi _0$ , which is defined in Section , as being equivalent to $\\lbrace {\\bf 1}, {\\bf x}\\rbrace $ .", "In the same sense, for the shrinkage towards a class of piece-wise linear functions with the knots $-1$ and 1, we can consider $\\Phi _0=\\lbrace {\\bf 1}, ({\\bf x}+{\\bf 1})_+,(-{\\bf x}-{\\bf 1})_+,({\\bf x}-{\\bf 1})_+ \\rbrace $ , where $(t)_+ = t$ , if $t>0$ and zero otherwise.", "Figure: Samples from the classical GP prior (the first column), the GP prior with shrinkage towards linear functions (the second column), and the GP prior with shrinkage towards piecewise linear functions with knots -1-1 and 1 (the third column).Figure REF illustrates a comparison between the classical GP prior and the shrinkage version of the GP prior.", "The covariates were independently generated from a uniform distribution between $-\\pi $ and $\\pi $ .", "The exponential covariance function, i.e., $\\Sigma ({\\bf x})_{i,k} = \\exp \\lbrace -|x_i-x_k| \\rbrace $ for $1\\le i, k \\le n$ , was considered, and we set $a=1/2$ and $b=n^{-2}$ with sample size $n=100$ .", "The first plot shows five sample curves generated from the classical GP prior, i.e., ${ \\mathrm {N} }({\\bf 0}, \\Sigma ({\\bf x}))$ .", "The second and the third plots display five sample curves from the modified GP prior in (REF ) with shrinkage towards linear and piece-wise linear functions, respectively.", "The near parametric forms of the sample paths from the modified GP prior suggest a promising way to shrink GP regression towards simpler parametric classes." ], [ "Univariate examples", "In this section, we examine the performance of the functional horseshoe prior on various simulated data sets.", "We consider three models as follows: $\\mbox{i) simple regression model:}&& Y_i = f(x_i) + \\epsilon _i \\\\\\mbox{ii) varying coefficient model:}&\\:& Y_i = w_if(x_i) +\\epsilon _i\\\\\\mbox{iii) density function estimation:}&\\:& p(Y_i) = \\frac{\\exp \\lbrace f(Y_i) \\rbrace }{\\int \\exp \\lbrace f(t) \\rbrace dt }, $ with $\\epsilon _i\\overset{i.i.d}{\\sim }{ \\mathrm {N} }(0,\\sigma ^2)$ for $i=1,\\dots ,n$ in (i) and (ii), and $p(\\cdot )$ the density function of $Y$ in (iii).", "The varying coefficient model [20] in () reduces to a linear model when the coefficient function $f$ is constant, and the density function $p$ is Gaussian when the log-density function $f$ is quadratic in the log-spline model [26] in (), motivating the usage of the functional horseshoe prior in these examples to shrink towards the respective parametric alternatives.", "For each setting, we considered the case corresponding to the relevant parametric model, as well as the parametric model was not adequate.", "For (i) and (ii), we generated the covariates independently from a uniform distribution between $-\\pi $ and $\\pi $ and set the error variance $\\sigma ^2 = 1$ .", "For each case (i) - (iii), we considered three parametric choices for $f$ .", "For case (i), we considered $f$ to be linear, quadratic, and sinusoidal.", "For case (ii), we considered constant, quadratic and sinusoidal functions.", "For (iii), we considered normal, log-normal and mixture of normal distributions.", "For the first two cases, we standardized the true function so as to obtain a signal-to-noise ratio of $1.0$ .", "We used the B-spline basis with $k_n=8$ in (REF ) to model the function $f$ in each setting.", "To shrink the regression function in (REF ) towards linear subspaces, we set $\\Phi _0= \\lbrace {\\bf 1}, {\\bf x}\\rbrace $ in the fHS prior (REF ).", "For the varying coefficient model (), we set $\\Phi _0= \\lbrace {\\bf 1} \\rbrace $ to shrink $f$ towards constant functions, whence the resulting model reduces to a linear regression model.", "Finally, we set $\\Phi _0 = \\lbrace {\\bf 1}, Y, Y^2\\rbrace $ to shrink $f$ towards the space of quadratic functions in (), which results in the density $p$ being shrunk towards the class of Gaussian distributions.", "We note that the prior for $p$ in () is data-dependent.", "An inverse-gamma prior with parameters $(1/100,1/100)$ was imposed on $\\sigma ^2$ for the fHS prior in (i) and (ii).", "In all three examples, we set $b = \\exp \\lbrace -k_n\\log n/2 \\rbrace $ to satisfy the conditions of Theorem REF and arbitrarily set $a=1/2$ .", "Although Theorem REF only applies to the regression model (REF ), the empirical results for these hyperparameter choices are promising for the varying coefficient model and the log-density model as well.", "Table: Results of univariate examplesWe considered the Jeffrey's prior, $\\pi (\\beta ,\\sigma ^2)\\propto 1/\\sigma ^2$ , on the B-spline coefficients for the simple regression model and the varying coefficient model as a competitor to the functional horseshoe prior.", "Following [14], we assigned independent $U(-\\pi , \\pi )$ priors on the B-spline coefficients, which are known to guarantee the minimax rate of posterior convergence rate for the log-density model.", "For each prior, we used the posterior mean $\\hat{f}$ as a point estimate for $f$ , and report the empirical Mean Square Error (MSE), i.e.", "$\\big \\Vert \\widehat{f} - f\\big \\Vert _{n,2}^2$ .", "Figure: Examples when the underlying true functions are parametric.", "Posterior mean of each procedure (red solid), its 95% pointwise credible bands (red dashed), and the true function (black solid) from a single example with n=200n=200 for each model.", "The top row is for the simple regression model; the second row is for the varying coefficient model; the last row is for the density estimation.", "The Bayesian B-spline procedure, the Bayesian parametric model procedure, and functional horseshoe priors are illustrated in the first, second, and third columns, respectively.Figure: Examples when the underlying true functions are nonparametric.", "Posterior mean of each procedure (red solid), its 95% pointwise credible bands (red dashed), and the true function (black solid) from a single example with n=200n=200 for each model.", "The top row is for the simple regression model; the second row is for the varying coefficient model; the last row is for the density estimation.", "The Bayesian B-spline procedure, the Bayesian parametric model procedure, and functional horseshoe priors are illustrated in the first, second, and third columns, respectively.In Table REF , we report 100 times MSE of the posterior mean estimator and its standard deviation over 100 replicates in estimating the unknown function $f$ for all three models, for sample sizes $n = 200$ , 500, and 1000.", "The first top three rows are for the simple regression model; the second three rows for the varying coefficient model; the last three rows for the density estimation.", "“Mixture\" in the last row indicates a mixture of Gaussian densities as $0.3{ \\mathrm {N} }(2,1)+0.7{ \\mathrm {N} }(-1,0.5)$ .", "In all three settings, when the true function $f$ belongs to the nominal parametric class, the posterior mean function resulting from the functional horseshoe prior clearly outperforms the B-spline prior.", "When the true function does not belong to the parametric model, the functional horseshoe prior performs comparably to the B-spline prior.", "Figure REF depicts the point estimate (posterior mean) and pointwise 95% credible bands for the unknown function $f$ for a single data set for each of the three examples when the true function belongs to the parametric class, that is a linear function in (REF ), a constant function in (), and a quadratic function in ().", "Figure REF depicts the corresponding estimates when the data generating function does not fall in the assumed parametric class.", "It is evident from Figure REF that when the parametric assumptions are met, the fHS prior performs similarly to the parametric model, which empirically corroborates our findings in Theorem REF that the posterior contracts at a near parametric rate when the parametric assumptions are met.", "It is also evident that the fHS procedure automatically adapts to deviations from the parametric assumptions in Figure REF , again confirming the conclusion of Theorem REF that when the true function is well-separated from the parametric class, the posterior concentrates at a near optimal minimax rate.", "We reiterate that the same hyperparameters $a=1/2$ and $b=\\exp \\lbrace -k_n\\log n /2\\rbrace $ for the fHS prior were used in the examples in Figure REF and Figure REF ." ], [ "Comparisons to additive models", "Our regression examples in the previous subsection involved one predictor variable.", "In the case of multiple predictors, a popular modeling framework is the class of additive models ([19], [19]), where the unknown function relating $p$ candidate predictors to a univariate response is modeled as the sum of $p$ univariate functions, with the $j$ th function only dependent on the $j$ th predictor.", "In this section, we apply the fHS prior to additive models and compare results obtained under this prior to several alternative methods.", "To be consistent with our previous notation, we express additive models as $Y = \\sum _{j=1}^p F_j + \\epsilon ,$ where $F_j = (f_j(x_{1j}),\\dots ,f_j(x_{nj}) )^{ \\mathrm {\\scriptscriptstyle T} }$ for $j=1,\\dots ,p$ and $\\epsilon \\sim { \\mathrm {N} }(0,\\sigma ^2\\mathrm {I}_n)$ .", "In the specific case where each $f_j$ is linear, we obtain a linear regression model.", "In general, each component function can be modeled nonparametrically, for example, using the B-spline basis functions as described in the previous section; $f_j(x) = \\sum _{l=1}^{k_n}\\beta _{jl}\\phi _{jl}(x)$ for $j=1,\\dots ,p$ .", "However, if there are many candidate predictors, then nonparametrically estimating $p$ functions may be statistically difficult, and in addition, may result in a loss of precision if only a small subset of the variables are significant.", "With this motivation, we extend the fHS framework to additive models, where we assign independent fHS priors to the $f_j$ 's with $Q_0 = 0$ in (REF ) to facilitate shrinkage of each of these functions towards the null function.", "We use the resulting posterior mean as a point estimate and compare its performance with a host of penalized likelihood estimators.", "For the additive model, [37] proposed penalized likelihood procedures called Sparse Additive Models (SpAM) that combine ideas from model selection and additive nonparametric regression.", "The penalty term of SpAM can be described as a weighted group Lasso penalty [42], and the coefficients for each component function $f_j$ for $j=1,\\dots ,p$ are forced to simultaneously shrink towards zero, so that the resulting procedure selects the variables that are associated with the response.", "[29] proposed the High-dimensional Generalized Additive Model (HGAM) that differs from SpAM in the sense that its penalty term not only imposes shrinkage towards zero, but also regularizes the smoothness of the function.", "[22] introduced the two step procedure of adaptive group Lasso (AdapGL) for the additive model, which first estimates the weight of the group penalty, then applies it to the adaptive group lasso penalty.", "Since the performance of penalized likelihood methods is sensitive to the choice of the tuning parameter, in the simulation studies that follow we considered two criterion for tuning parameter selection: AIC and BIC.", "R packages SAM, hgam, and grplasso were used to implement SpAM, HGAM, and AdapGL, respectively.", "We denote the signal-to-noise ratio as $\\mbox{SNR} =\\mathop {\\rm Var}(f(X))/Var(\\epsilon )$ , where f is the true underlying function, and we examine the same settings that were considered in [29] as follows: Setting 1: ($p=200$ , $\\mbox{SNR}\\approx 15$ ).", "This is same with Example 1 in [29], and a similar setting was also considered in [18] and [37].", "The model is $Y_i = f_1(x_{i1}) + f_2(x_{i2}) + f_3(x_{i3}) + f_4(x_{i4}) + \\epsilon _i,$ where $\\epsilon _i\\overset{i.i.d}{\\sim }{ \\mathrm {N} }(0,1)$ for $i=1,\\dots ,n$ , with $f_1(x) &=& -\\sin (2x), \\:\\: f_2(x) = x^2-25/12,\\:\\: f_3(x) = x,\\\\f_4(x) &=& \\exp \\lbrace -x\\rbrace -2/5\\cdot \\sinh (5/2).$ The covariates are independently generated from a uniform distribution between $-2.5$ to $2.5$ .", "Setting 2: ($p=80$ , $\\mbox{SNR}\\approx 7.9$ ).", "This is equivalent to Example 3 in [29] and similar with an example in [27].", "The model is $Y_i = 5f_1(x_{i1}) + 3f_2(x_{i2}) + 4f_3(x_{i3}) + 6f_4(x_{i4}) + \\epsilon _i,$ where $\\epsilon _i\\overset{i.i.d}{\\sim }{ \\mathrm {N} }(0,1.74)$ for $i=1,\\dots ,n$ , with $f_1(x) &=& x, \\:\\: f_2(x) = (2x-1)^2,\\:\\: f_3(x) = \\frac{\\sin (2\\pi x)}{2-\\sin (2\\pi x)},\\\\f_4(x) &=& 0.1\\sin (2\\pi x) + 0.2* \\cos (2\\pi x) + 0.3 \\sin ^2(2\\pi x)\\\\&&+0.4 \\cos ^3(2\\pi x) + 0.5 \\sin ^3(2\\pi x).$ The covariate ${\\bf x}_j=(x_{1j},\\dots ,x_{nj})^{ \\mathrm {\\scriptscriptstyle T} }$ for $j=1,\\dots ,p$ is generated by ${\\bf x}_j = (W_j + U)/2$ , where $W_1,\\dots ,W_p$ and $U$ are independently simulated from U$(0, 1)$ distributions.", "Setting 3 ($p=60$ , $\\mbox{SNR}\\approx 11.25$ ).", "This is equivalent to Example 4 in [29], and a similar example was also considered in [27].", "The same functions are used and the same process to generate the covariates is considered as in Setting 2.", "The model is $Y_i &=& f_1(x_{i1}) + f_2(x_{i2}) + f_3(x_{i3}) + f_4(x_{i4}) \\\\&& +1.5 f_1(x_{i5}) + 1.5 f_2(x_{i6}) + 1.5 f_3(x_{i7}) + 1.5 f_4(x_{i8}) \\\\&& +2.5 f_1(x_{i9}) + 2.5 f_2(x_{i10}) + 2.5 f_3(x_{i11}) + 2.5 f_4(x_{i12}) + \\epsilon _i,$ where $\\epsilon _i\\overset{i.i.d}{\\sim }{ \\mathrm {N} }(0,0.5184)$ for $i=1,\\dots ,n$ .", "To evaluate the estimation performance of the functional horseshoe prior, we report the MSE for each method.", "To measure the performance of model selection, we considered the proportion of times the true model was selected, as well as the Matthews correlation coefficient (MCC; [28]), defined as, $\\mbox{MCC} = \\frac{\\mbox{TP}\\cdot \\mbox{TN} - \\mbox{FP}\\cdot \\mbox{FN}}{(\\mbox{TP}+\\mbox{FP})(\\mbox{TP}+\\mbox{FN})(\\mbox{TN}+\\mbox{FP})(\\mbox{TN}+\\mbox{FN})},$ where TP, TN, FP, and FN denote the number of true positive, true negatives, false positives, false negatives, respectively.", "MCC is generally regarded as a balanced measure of the performance of classification methods, which simultaneously takes into account TP, TN, FP, and FN.", "We note that MCC is bounded by 1, and the closer MCC is to 1, the better the model selection performance is.", "Figure: The first column illustrates the logarithm of the MSE of each method; the second column displays the MCC; the third column is the proportion of times the each procedure selected the true model.", "The top row, the middle row, and the bottom row represent the Setting 1, Setting 2, and Setting 3, respectively.", "For penalized likelihood methods, AIC (black) and BIC (grey) were used to choose the tuning parameter.For model selection using the fHS prior, we used 95% pointwise credible bands for each component function to exclude component functions whose credible bands uniformly contained the zero function on the entire support of the corresponding covariate.", "To investigate the performance achieved by the proposed method, we compared it with the partial oracle estimator that refers to the B-spline least squares estimator under the situation where the variables in the true model are given, but the true component functions in the additive model are not provided.", "Results from a simulation study to compare these methods are depicted in Figure REF .", "In all three settings it is clear that the procedure based on the functional horseshoe prior outperforms the penalized likelihood estimators in terms of MSE.", "In terms of model selection performance, the proposed procedure is also better or at least comparable to that of the competitors.", "We note that the SpAM procedure with tuning parameter selected by BIC provides comparable model selection performance to the fHS prior in Setting 1, yet its MSE is at least 8 times bigger than that of the procedure based on the functional horseshoe prior (note that the reported scale is logarithmic).", "The results suggest that the fHS prior provides improvement over the penalized likelihood methods in terms of both MSE and model selection performance combined, at least under the considered settings." ], [ "Real data analysis", "In this section, we apply the functional horseshoe prior to two well known data sets: the first concerns ozone levels and the second considers housing prices in Boston.", "Both data sets are available in the R package mlbench.", "These two data sets have been previously analyzed in various places, including [6], [5], [27] and [41].", "Following the pre-processing step in [41], we standardized both the response and independent variables prior to our analyses.", "We first consider the Boston housing data set that contains the median value of 506 owner-occupied homes in the Boston area, together with several variables that might be associated with the median value.", "To examine the performance of our method in eliminating extraneous predictors, we add 40 spurious variables generated as i.i.d.", "standard Gaussian deviates.", "Using the standard notation for the variable in this data set, we then assumed a model of the following form: $\\texttt {medv} &=& \\beta _0 +f_1(\\texttt {crim}) + f_2(\\texttt {indus}) + f_3(\\texttt {nox})+f_4(\\texttt {rm})+f_5(\\texttt {age})+f_6(\\texttt {dis})+f_7(\\texttt {tax})\\\\&&+f_8(\\texttt {ptratio})+f_9(\\texttt {b}) + f_{10}(\\texttt {lstat}) +\\epsilon ,$ where $\\epsilon \\sim { \\mathrm {N} }(0,\\sigma ^2\\mathrm {I}_n)$ .", "Each component function is modeled by the B-spline bases with $k_n=8$ , and 50 test data points were randomly selected to estimate the out-of-sample prediction error.", "Five hundreds simulations of each procedure were used to generate the plots in Table REF .", "We also modeled the ozone data set using each of the procedures that were applied the housing data.", "The ozone data consists of the daily maximum one-hour-average ozone readings and nine meteorological variables for 330 days in the Los Angeles basin in 1976.", "The model applied to these data can be expressed as follows: $\\texttt {ozone} &=& \\beta _0 +f_1(\\texttt {height}) + f_2(\\texttt {wind}) + f_3(\\texttt {humidity})+f_4(\\texttt {temp1})+f_5(\\texttt {temp2})\\\\&&+f_6(\\texttt {inv height})+f_7(\\texttt {gradient})+f_8(\\texttt {inv temp})+f_9(\\texttt {visibility}) +\\epsilon .$ Like the Boston Housing data case, we added 40 spurious variables generated as i.i.d.", "standard Gaussian deviates.", "We used B-spline bases with $k_n=5$ were considered to model the component functions.", "We performed a cross-validation experiment to assess the predictive performance of the competing methods.", "In each of 500 simulated data sets, we held out 30 data values as the test set and used the remaining observations to estimate the model.", "The parameter settings described in Section REF were again used for the functional horseshoe prior.", "Also, for each training data set we generated $30,000$ posterior samples by following the MCMC algorithm described in the Appendix, and only the last $20,000$ samples were used in the analysis.", "We compared the performance of the procedure based on the proposed priors with that of SpAM, HGAM, AdapGL and the classical B-spline estimator was fit without the spurious noise variables.", "For the penalized likelihood methods, AIC and BIC were used to choose tuning parameters.", "Table REF displays the average of test set errors, the average number of selected noise variables, and the most frequently selected model for each method.", "Table: Results of real data examplesIn Table 2, “Test Error\" refers to the average of empirical $L_2$ test errors, and “NN\" represents the averaged number of selected spurious variables, and “Full\" indicates the B-spline least square estimator from the full model without spurious variables.", "Table 2 shows that for both data sets the procedure based on the functional horseshoe prior achieved the smallest test errors, and it also selected the minimum number of spurious variables.", "Moreover, even though 40 spurious variables are added to the proposed procedure, its test error was smaller than that of the full estimator that was estimated without the spurious variables.", "For both data sets, the model selected by the fHS prior was similar to that chosen by SpAM with BIC.", "However, the test error of the SpAM procedure was roughly twice that of fHS.", "More generally, the fHS procedure outperformed all of the other procedures in these examples." ], [ "Conclusion", "We have proposed a class of shrinkage priors which we call the functional horseshoe priors.", "When appropriate, these priors imposes strong shrinkage towards a pre-specified class of functions.", "The shrinkage term in the prior is new, as it directly allows the nonparametric function shrink towards parametric functions, so it preserves the minimax optimal parametric rate of posterior convergence $n^{-1/2}$ when the true underlying function is parametric, and it also comes within $O(\\log n)$ of achieving the minimax nonparametric rate when the true function is strictly separated from the class of parametric functions.", "The novel shrinkage term contained in the proposed prior, $F^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F$ (i.e., (REF )), can be naturally applied to a new class of penalized likelihood methods having a general form expressible as $- l(Y\\mid F) + p_{\\lambda }\\big ( F^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F \\big ),$ where $l(Y\\mid F)$ is the logarithm of a nonparametric likelihood function and $p_\\lambda $ is the penalty term.", "In constrast to other penalized likelihood, this form of penalty allows shrinkage towards the space spanned by a projection matrix $Q_0$ , rather than simply a zero function." ], [ "Proofs of Theorems", "Lemma A.1 For arbitrary positive sequences $u_n$ and $w_n$ , $\\left( 1-\\frac{u_n}{u_n+w_n} \\right)^{u_n+w_n} \\ge \\exp \\left\\lbrace - u_n +\\frac{u_n^2}{2(u_n+w_n)} \\right\\rbrace .$ By Talyor's theorem, there exists $q_n^*\\in (0,u_n/(u_n+w_n))$ such that $\\left( 1 - \\frac{u_n}{u_n+w_n} \\right)^{u_n+w_n} &=& \\exp \\left\\lbrace (u_n+w_n)\\log \\left( 1 - \\frac{u_n}{u_n+w_n} \\right) \\right\\rbrace \\\\&=& \\exp \\left\\lbrace (u_n+w_n)\\left( - \\frac{u_n}{u_n+w_n} + \\frac{1}{(1-q_n^*)^2}\\frac{u_n^2}{2(u_n+w_n)^2} \\right) \\right\\rbrace \\\\&\\ge &\\exp \\left\\lbrace -u_n+\\frac{u_n^2}{2(u_n+w_n)} \\right\\rbrace .$ $\\Box $ Lemma A.2 Suppose $W$ follows a non-central chi-square distribution with $m_n$ degrees of freedom and non-centrality parameter $\\lambda _n\\ge 0$ , i.e, $W \\sim \\chi ^2_{m_n}(\\lambda _n)$ .", "Also, let $w_n\\rightarrow 0$ and $t_n\\rightarrow \\infty $ as $n\\rightarrow \\infty $ and assume that $m_n \\prec t_n$ .", "Then, $P(W\\le \\lambda _n w_n) \\le c_1 \\lambda _n^{-1}\\exp \\lbrace - \\lambda _n (1-w_n)^2 \\rbrace ,$ and $P(W > \\lambda _n + t_n) \\le c_2\\left(\\frac{t_n}{2m_n}\\right)^{m_n/2}\\exp \\left\\lbrace m_n/2-t_n/2 \\right\\rbrace +c_3\\lambda _n^{1/2}t_n^{-1}\\exp \\left\\lbrace -\\frac{t_n^2}{32\\lambda _n} \\right\\rbrace ,$ where $c_1$ , $c_2$ , and $c_3$ are some positive constants.", "$W$ can be expressed as $W = \\sum _{i=1}^{m_n}\\lbrace Z_i+(\\lambda _n/m_n)^{1/2}\\rbrace ^2$ , where $Z_i \\stackrel{i.i.d}{\\sim } N(0,1)$ for $i = 1, \\ldots , m$ .", "Then, by the fact that $P(Z>a) \\le (2\\pi )^{-1/2}a^{-1}\\exp \\lbrace -a^2/2\\rbrace $ for any $a>0$ , we can show that there exist some positive constants $c_1$ such that $P\\left( W \\le \\lambda _n w_n \\right)&=& P\\big \\lbrace \\sum _{i=1}^{m_n} Z_i^2 + 2(\\lambda _n/m_n)^{1/2}\\sum _{i=1}^{m_n} Z_i + \\lambda _n \\le \\lambda _n w_n \\big \\rbrace \\\\&\\le & P\\big \\lbrace m_n^{-1/2}\\sum _{i=1}^{m_n} Z_i \\le - \\lambda _n^{1/2} (1-w_n)/2 \\big \\rbrace \\\\&=& P\\big \\lbrace |Z_1| \\ge \\lambda _n^{1/2} (1-w_n)/2 \\big \\rbrace /2\\\\&\\le & c_1\\lambda _n^{-1}\\exp \\lbrace - \\lambda _n (1-w_n)^2/2\\rbrace ,$ since $Z_1$ follows a standard normal distribution.", "By using Chernoffs's bound and the fact that $P(Z>a) \\le (2\\pi )^{-1/2}a^{-1}\\exp \\lbrace -a^2/2\\rbrace $ for any $a>0$ , one can show that $&& P(W > \\lambda _n + t_n) =P\\left\\lbrace \\sum _{i=1}^{m_n} Z_i^2 + 2 (\\lambda _n/m_n)^{1/2}\\sum _{i=1}^{m_n} Z_i > t_n\\right\\rbrace \\\\&\\le & P\\left(\\sum _{i=1}^{m_n} Z_i^2 > t_n/2\\right) + P\\left\\lbrace m_n^{-1/2}\\sum _{i=1}^{m_n} Z_i > \\lambda _n^{-1/2}t_n/4\\right\\rbrace \\\\&\\le & c_2\\left(\\frac{t_n}{2m_n}\\right)^{m_n/2} \\exp \\left\\lbrace m_n/2-t_n/2\\right\\rbrace +c_3\\lambda _n^{1/2}t_n^{-1}\\exp \\left\\lbrace -\\frac{t_n^2}{32\\lambda _n} \\right\\rbrace ,$ where $c_2$ and $c_3$ are some positive constants.", "$\\Box $ Proof of Lemma REF .", "As discussed in the paragraphs following Lemma REF when $\\mathfrak {L}(\\Phi _0) \\subsetneq \\mathfrak {L}(\\Phi )$ , we can generate a new basis ${ \\widetilde{\\Phi } }= [\\Phi _0, \\Phi _1]$ such that $\\Phi _0^{ \\mathrm {\\scriptscriptstyle T} }\\Phi _1 = {\\bf 0}$ and $\\mathfrak {L}(\\Phi )=\\mathfrak {L}({ \\widetilde{\\Phi } })$ , which implies $Q_{{ \\widetilde{\\Phi } }} = Q_\\Phi $ .", "Then, $&& \\Phi \\left(\\Phi ^T \\Phi + \\frac{\\omega }{1-\\omega } \\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\right)^{-1}\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }\\\\&=& { \\widetilde{\\Phi } }\\left( { \\widetilde{\\Phi } }^T { \\widetilde{\\Phi } }+ \\frac{\\omega }{1-\\omega } { \\widetilde{\\Phi } }^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0){ \\widetilde{\\Phi } }\\right)^{-1}{ \\widetilde{\\Phi } }^{ \\mathrm {\\scriptscriptstyle T} }\\\\&=& [\\Phi _0, \\Phi _1]\\begin{bmatrix}(\\Phi _0^{ \\mathrm {\\scriptscriptstyle T} }\\Phi _0)^{-1} & {\\bf 0} \\\\{\\bf 0} & (1-\\omega )(\\Phi _1^{ \\mathrm {\\scriptscriptstyle T} }\\Phi _1)^{-1}\\end{bmatrix}\\begin{bmatrix}\\Phi _0^{ \\mathrm {\\scriptscriptstyle T} }\\\\\\Phi _1^{ \\mathrm {\\scriptscriptstyle T} }\\end{bmatrix}\\\\&=& (1-\\omega )Q_{{ \\widetilde{\\Phi } }} + \\omega Q_0 \\\\&=& (1-\\omega )Q_{\\Phi } + \\omega Q_0.$ $\\Box $ Lemma A.3 $n\\big \\Vert Q_0\\Phi \\beta -Q_0Y\\big \\Vert ^2_{n,2}/\\sigma ^2 \\mid Y, \\omega \\sim \\chi ^2_{d_0},$ and $n\\big \\Vert Q_1\\Phi \\beta - (1-\\omega )Q_1Y\\big \\Vert ^2_{n,2}/\\lbrace (1-\\omega )\\sigma ^2\\rbrace \\mid Y, \\omega \\sim \\chi ^2_{k_n-d_0}.$ Recall that $\\beta \\mid Y, \\omega \\sim \\mbox{N}( { \\widetilde{\\beta } }_{\\omega }, { \\widetilde{\\Sigma } }_{\\omega }),$ where ${ \\widetilde{\\beta } }_{\\omega } = \\left(\\Phi ^{{ \\mathrm {\\scriptscriptstyle T} }} \\Phi + \\frac{\\omega }{1-\\omega }\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\right)^{-1} \\Phi ^{{ \\mathrm {\\scriptscriptstyle T} }} Y, \\quad { \\widetilde{\\Sigma } }_{\\omega } = \\sigma ^2 \\left(\\Phi ^{{ \\mathrm {\\scriptscriptstyle T} }} \\Phi +\\frac{\\omega }{1-\\omega }\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\right)^{-1}.$ As shown in the proof of Lemma REF , $ \\Phi \\left(\\Phi ^T \\Phi + \\frac{\\omega }{1-\\omega } \\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\right)^{-1}\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }=(1-\\omega ) Q_\\Phi + \\omega Q_0$ , so $\\mathbb {E}\\left[ Q_0\\Phi \\beta \\mid Y, \\omega \\right] = Q_0Y \\\\\\mathop {\\rm Var}\\left[ Q_0\\Phi \\beta \\mid Y, \\omega \\right] = \\sigma ^2 Q_0,$ which shows that $n\\big \\Vert Q_0\\Phi \\beta -Q_0Y\\big \\Vert ^2_{n,2}/\\sigma ^2 \\mid Y, \\omega \\sim \\chi ^2_{d_0}$ .", "Similarly, $\\mathbb {E}\\left[ Q_1\\Phi \\beta \\mid Y, \\omega \\right] = (1-\\omega )Q_1Y \\\\\\mathop {\\rm Var}\\left[ Q_1\\Phi \\beta \\mid Y, \\omega \\right] = \\sigma ^2 (1-\\omega ) Q_1,$ which proves that $n\\big \\Vert Q_1\\Phi \\beta - (1-\\omega )Q_1Y\\big \\Vert ^2_{n,2}/\\lbrace (1-\\omega )\\sigma ^2\\rbrace \\mid Y, \\omega \\sim \\chi ^2_{k_n-d_0}$ .", "$\\Box $ Proof of Lemma REF .", "From [34] it follows that $\\int _0^1\\omega ^{A_n-1}(1-\\omega )^{B_n-1}\\exp \\lbrace -H_n\\omega \\rbrace d\\omega = \\frac{\\Gamma (A_n)\\Gamma (B_n)}{\\Gamma (A_n+B_n)}\\exp \\lbrace -H_n\\rbrace \\sum _{m=0}^{\\infty }\\frac{(A_n)_{(m)}}{(A_n + B_n)_{(m)}}\\frac{{H_n}^m}{m !", "},$ where $(a)_{(m)}=a(a+1)\\dots (a+m-1)$ .", "We shall show that $\\sum _{m=0}^{\\infty }\\left\\lbrace \\frac{(B_n)_{(m)}}{(A_n + B_n)_{(m)}}\\frac{{H_n}^m}{m !", "}\\right\\rbrace \\ge 1 + Q_n^L$ .", "By using Lemma REF and Stirling's approximation, i.e., $m!", "\\asymp m^{m+1/2}\\exp \\lbrace -m\\rbrace $ , it follows that $&&\\sum _{m=0}^{\\infty }\\left\\lbrace \\frac{(B_n)_{(m)}}{(A_n + B_n)_{(m)}}\\frac{{H_n}^m}{m !", "}\\right\\rbrace \\nonumber \\\\&=& 1+\\frac{B_n}{A_n+B_n}\\left\\lbrace H_n +\\sum _{m=1}^\\infty \\left[\\frac{(B_n+1)_{(m)}}{(A_n + B_n+1)_{(m)}}\\frac{{H_n}^{m+1}}{(m+1) !", "}\\right] \\right\\rbrace \\nonumber \\\\&\\ge &1+\\frac{B_n}{A_n+B_n}\\left\\lbrace H_n +\\sum _{m=1}^\\infty \\left[\\frac{(B_n+m)!", "}{(A_n + B_n+m)!", "}\\frac{{H_n}^{m+1}}{( m+1 ) !", "}\\right] \\right\\rbrace \\nonumber \\\\&\\ge & 1+\\frac{B_n}{A_n+B_n}\\left\\lbrace H_n + D\\sum _{m=1}^\\infty \\left[\\left( \\frac{B_n+m}{A_n+B_n+m} \\right)^{A_n+B_n+m+1/2}(B_n+m)^{-A_n}e^{A_n} \\frac{{H_n}^{m+1}}{ (m+1) !", "}\\right] \\right\\rbrace \\nonumber \\\\&\\ge & 1+\\frac{B_n}{A_n+B_n}\\left\\lbrace H_n + D\\sum _{m=1}^{T_n} \\left[\\left( \\frac{B_n+1}{A_n+B_n+1} \\right)^{1/2}(B_n+m)^{-A_n}\\left(\\frac{B_n+m}{A_n+B_n+m}\\right)^{A_n+B_n+m}e^{A_n} \\frac{{H_n}^{m+1}}{ (m+1) !", "}\\right] \\right\\rbrace \\nonumber \\\\&\\ge & 1+\\frac{B_n}{A_n+B_n}\\left\\lbrace H_n + D\\left( \\frac{B_n+1}{A_n+B_n+1} \\right)^{1/2}(B_n+T_n)^{-A_n}\\exp \\left\\lbrace \\frac{A_n^2}{2(A_n+B_n+T_n)} \\right\\rbrace \\sum _{m=2}^{T_n+1} \\frac{{H_n}^{m}}{ m !}", "\\right\\rbrace ,$ where $T_n= \\max \\lbrace A_n^2,3\\left\\lceil H_n \\right\\rceil ]\\rbrace $ , and $D$ is some positive constant.", "Since $H_n<(T_n+2)\\exp \\lbrace 1\\rbrace $ , by using the Stirling's approximation, the term $\\sum _{m=2}^{T_n+1} H_n/m!$ in (REF ) can be expressed as follows: $\\sum _{m=2}^{T_n+1} \\frac{H_n^m}{m!}", "&=&\\exp \\lbrace H_n\\rbrace - 1 -H_n- \\sum _{m=T_n+2}^{\\infty } \\frac{H_n^m}{m!", "}\\\\&\\preceq &\\exp \\lbrace H_n\\rbrace - 1 -H_n- (T_n+2)^{-1/2}\\sum _{m=T_n+2}^{\\infty }\\left(\\frac{\\exp \\lbrace 1\\rbrace H_n}{T_n+2}\\right)^m\\\\&\\le & \\exp \\lbrace H_n\\rbrace - 1 -H_n - (T_n+2)^{-1/2}$ Therefore, (REF ) can be bounded by $&&1+\\frac{B_n}{A_n+B_n}\\left\\lbrace H_n + D\\left( \\frac{B_n+1}{A_n+B_n+1} \\right)^{1/2}(B_n+T_n)^{-A_n} \\left(\\exp \\lbrace H_n\\rbrace - 1 -H_n - (T_n+2)^{-1/2}\\right)_+ \\right\\rbrace \\\\&\\ge &1+\\frac{B_nH_n}{A_n+B_n} + \\frac{DB_n}{(A_n+B_n)^{3/2}}(B_n+T_n)^{-A_n}\\left(\\exp \\lbrace H_n\\rbrace - 1 -H_n - (T_n+2)^{-1/2}\\right)_+,$ where $(\\cdot )_+$ denotes the positive hinge function (i.e., for any $t\\in \\mathbb {R}$ , $(t)_+=t$ , if $t>0$ , and $(t)_+=0$ , otherwise).", "Also, since $(B_n+m)!/(A_n+B_n+m)!", "<1 $ for any positive integer $m$ , it follows that $H_n +\\sum _{m=1}^\\infty \\left[\\frac{(B_n+m)!", "}{(A_n + B_n+m)!", "}\\frac{{H_n}^{m+1}}{( m+1 ) !", "}\\right] \\le \\exp \\lbrace H_n\\rbrace ,$ which completes the proof.", "$\\Box $ Proof of Theorem REF.", "Let $\\beta ^*$ denote the projection of the true $F_0$ on the basis $\\lbrace \\phi _j\\rbrace _{1\\le j\\le k_n}$ , i.e., $\\beta ^* = \\mbox{argmin}_{\\beta \\in \\mathbb {R}^{k_n} } \\big \\Vert F_0 - \\Phi \\beta \\big \\Vert _{2, n}.$ We shall treat $\\beta ^*$ as the pseudo-true parameter and study the posterior concentration of $\\Phi \\beta $ in the posterior around $\\Phi \\beta ^*$ .", "To prove Theorem REF , it is sufficient to show that the posterior probability in the equation (REF ) converges in probability to zero.", "The quantity in (REF ) can be decomposed as follows: $&&P\\left[ \\big \\Vert \\Phi \\beta - F_0\\big \\Vert _{n,2} > M_n^{1/2}\\mid Y \\right]\\\\&\\le &P\\left[ \\big \\Vert \\Phi \\beta - \\Phi \\beta ^*\\big \\Vert _{n,2} > M_n^{1/2}/2 \\mid Y\\right] + \\mathbb {1}\\left[ \\big \\Vert \\Phi \\beta ^* - F_0\\big \\Vert _{n,2} > M_n^{1/2}/2 \\right],$ where $\\beta ^*$ is defined in (REF ) and $\\mathbb {1}(\\cdot )$ is the indicator function.", "The second term on the right-hand side of this expression is always zero when $F_0 \\in \\mathfrak {L}(\\Phi _0)$ , since we assume that the column space of $\\Phi _0$ is contained in the column space of $\\Phi $ , and its expectation with respect to the true density is asymptotically zero when $F_0^{{ \\mathrm {\\scriptscriptstyle T} }}(\\mathrm {I}-\\mathrm {Q}_0)F_0 \\asymp n$ from (REF ).", "Therefore, we focus on the first term on the right-hand side.", "Since $\\Phi \\beta = Q_1 \\Phi \\beta + Q_0 \\Phi \\beta $ , by Lemma REF .", "the first term can be decomposed as $&&P\\left[ \\big \\Vert \\Phi \\beta - \\Phi \\beta ^*\\big \\Vert _{n,2} > M_n^{1/2}/2 \\mid Y\\right] = E_{\\omega \\mid Y}\\left[ P\\left( \\big \\Vert \\Phi \\beta -\\Phi \\beta ^* \\big \\Vert _{n,2} > M_n^{1/2}/2\\mid Y, \\omega \\right) \\right]\\\\&\\le & \\mathbb {E}_{\\omega \\mid Y}\\left[ P\\left( \\big \\Vert \\Phi \\beta - \\Phi {\\widetilde{\\beta }}_\\omega \\big \\Vert _{n,2} > M_n^{1/2}/4 \\mid Y, \\omega \\right) \\right] + \\mathbb {E}_{\\omega \\mid Y}\\left[ P\\left( \\big \\Vert \\Phi {\\widetilde{\\beta }}_\\omega - \\Phi \\beta ^*\\big \\Vert _{n,2} > M_n^{1/2}/4 \\mid Y, \\omega \\right) \\right]\\\\&\\le & \\mathbb {E}_{\\omega \\mid Y}\\left[ P\\left( \\big \\Vert \\mathrm {Q}_1\\Phi \\beta - (1-\\omega )\\mathrm {Q}_1 Y\\big \\Vert _{n,2} > M_n^{1/2}/8 \\mid Y, \\omega \\right) \\right] \\\\&&+ \\mathbb {E}_{\\omega \\mid Y}\\left[ P\\left( \\big \\Vert Q_1\\Phi \\beta ^*-(1-\\omega )Q_1 Y \\big \\Vert _{n,2} > M_n^{1/2}/8 \\mid Y, \\omega \\right) \\right] \\\\&&+\\mathbb {E}_{\\omega \\mid Y}\\left[ P\\left( \\big \\Vert Q_0 \\Phi \\beta - Q_0 Y\\big \\Vert _{n,2} > M_n^{1/2}/8 \\mid Y , \\omega \\right) \\right] \\\\&&+ \\mathbb {1}\\left[ \\big \\Vert Q_0\\Phi \\beta ^* - Q_0 Y \\big \\Vert _{n,2} > M_n^{1/2}/8 \\right],$ where $\\Phi {\\widetilde{\\beta }}_\\omega = (1-\\omega )\\mathrm {Q}_\\Phi Y+\\omega \\mathrm {Q}_0 Y = (1-\\omega )\\mathrm {Q}_1 Y+\\mathrm {Q}_0 Y$ .", "We denote $W_1 &=& P\\left( \\big \\Vert \\mathrm {Q}_1\\Phi \\beta - (1-\\omega )\\mathrm {Q}_1 Y\\big \\Vert _{n,2} > M_n^{1/2}/8 \\mid Y, \\omega \\right),\\\\W_2 &=& P\\left( \\big \\Vert Q_1\\Phi \\beta ^* - (1-\\omega )Q_1 Y \\big \\Vert _{n,2} > M_n^{1/2}/8 \\mid Y, \\omega \\right),\\\\W_3 &=& P\\left( \\big \\Vert Q_0 \\Phi \\beta - Q_0 Y\\big \\Vert _{n,2} > M_n^{1/2}/8 \\mid Y , \\omega \\right).$ The indicator function in the fourth term converges to zero in probability, since $\\big \\Vert Q_0Y - Q_0 \\Phi \\beta ^*\\big \\Vert ^2_{2,n}$ achieves the parametric optimal rate.", "To complete the proof we show that the expectations of $W_1$ , $W_2$ , and $W_3$ with respect to the marginal posterior distribution of $\\omega $ converge to zero in probability.", "First consider $W_3$ .", "Since $n\\big \\Vert \\mathrm {Q}_0\\Phi \\beta - \\mathrm {Q}_0Y\\big \\Vert ^2_{2,n}/\\sigma ^2 \\mid Y, \\omega \\sim \\chi ^2_{d_0}$ by Lemma REF , by using Lemma REF it follows that $&&E_{\\omega \\mid Y} \\left[ W_3 \\right] = E_{\\omega \\mid Y} \\left[ P\\left\\lbrace \\big \\Vert \\mathrm {Q}_0\\Phi \\beta - \\mathrm {Q}_0 Y \\big \\Vert _{2,n} >M_n^{1/2}/8 \\mid Y,\\omega \\right\\rbrace \\right]\\\\\\ &\\le &C \\left( \\frac{nM_n}{64\\sigma d_0} \\right)^{d_0/2}\\exp \\lbrace -nM_n/(128\\sigma ^2) \\rbrace ,$ for some constant $C$ .", "The last quantity converges to zero as $n$ tends to $\\infty $ , which implies that $\\mathbb {E}_{\\omega \\mid Y}[W_3]=o_p(1)$ .", "Now we obtain the bounds on $W_1$ .", "By Lemma REF $n\\big \\Vert \\mathrm {Q}_1\\Phi \\beta - (1-\\omega )\\mathrm {Q}_1 Y\\big \\Vert ^2_{2,n}/\\lbrace (1-\\omega )\\sigma ^2\\rbrace \\mid Y \\sim \\chi ^2_{k_n-d_0}$ .", "By using Lemma REF , it follows that $W_1 &\\le & \\left[ \\frac{nM_n}{64\\sigma ^2(k_n-d_0)}(1-\\omega )^{-1} \\right]^{\\frac{k_n-d_0}{2}} \\exp \\left\\lbrace \\frac{k_n-d_0}{2} - \\frac{nM_n}{128\\sigma ^2}(1-\\omega )^{-1} \\right\\rbrace \\mathbb {1}\\left[ \\frac{nM_n}{64\\sigma ^2}(1-\\omega )^{-1}> k_n-d_0\\right]\\\\&&+ \\mathbb {1}\\left[ \\frac{nM_n}{64\\sigma ^2}(1-\\omega )^{-1}\\le k_n-d_0\\right].$ We denote the two terms in this expression as $W_{1,1}$ and $W_{1,2}$ .", "By using Lemma REF and defining $\\widehat{\\omega }= (k_n-d_0)/\\lbrace nM_n/(64\\sigma ^2)+k_n-d_0\\rbrace $ , it follows that $&&\\mathbb {E}_{\\omega \\mid Y} \\left[ W_{1,1} \\right]\\nonumber \\\\&=& \\frac{1}{m(Y)} \\left[ \\frac{nM_n\\exp \\lbrace 1\\rbrace }{64\\sigma ^2(k_n-d_0)} \\right]^{\\frac{k_n-d_0}{2}} \\int ^1_{m_n} \\omega ^{a+\\frac{k_n-d_0}{2}-1}(1-\\omega )^{b-\\frac{k_n-d_0}{2}-1}\\exp \\left\\lbrace - \\frac{nM_n}{128\\sigma ^2}(1-\\omega )^{-1} -H_n\\omega \\right\\rbrace d\\omega \\nonumber \\\\&\\le &\\frac{1}{m(Y)} \\left[ \\frac{nM_n\\exp \\lbrace 1\\rbrace }{64\\sigma ^2(k_n-d_0)} \\right]^{\\frac{k_n-d_0}{2}} \\int ^1_{m_n} \\omega ^{a-1}(1-\\omega )^{b-1}\\exp \\left\\lbrace -H_n\\omega \\right\\rbrace d\\omega \\nonumber \\\\&&\\times \\widehat{\\omega }^{\\frac{k_n-d_0}{2}}(1- \\widehat{\\omega })^{-\\frac{k_n-d_0}{2}}\\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2}(1- \\widehat{\\omega })^{-1} \\right\\rbrace \\nonumber \\\\&=& \\frac{1}{m(Y)} \\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} \\right\\rbrace \\int ^1_{m_n} \\omega ^{a-1}(1-\\omega )^{b-1}\\exp \\left\\lbrace -H_n\\omega \\right\\rbrace d\\omega ,$ where $m_n = \\max [0,1-nM_n/\\lbrace 16\\sigma ^2(k_n-d_0)\\rbrace ]$ .", "Also, $&& \\mathbb {E}_{\\omega \\mid Y}[W_{1,2}]=P_{\\omega \\mid Y}\\left[ \\omega <1-\\frac{nM_n}{64\\sigma ^2(k_n-d_0)} \\right]\\nonumber \\\\&=& \\frac{1}{m(Y)}\\int _0^{1-\\frac{nM_n}{64\\sigma ^2(k_n-d_0)}} \\omega ^{a+(k_n-d_0)/2-1} (1-\\omega )^{b-1} \\exp \\lbrace -H_n\\omega \\rbrace d\\omega \\nonumber \\\\&\\le &\\frac{1}{m(Y)}\\left( \\frac{nM_n}{64\\sigma ^2(k_n-d_0)} \\right)^{b-1}\\int _0^1 \\omega ^{a+(k_n-d_0)/2-1}\\exp \\lbrace -H_n\\omega \\rbrace d\\omega \\nonumber \\\\&\\le & \\left( \\frac{nM_n}{64\\sigma ^2(k_n-d_0)} \\right)^{b-1}\\frac{\\Gamma (a+b+(k_n-d_0)/2)}{\\Gamma (a+(k_n-d_0)/2)\\Gamma (b)}H_n^{-1}\\mathbb {1}\\left( 1-\\frac{nM_n}{64\\sigma ^2(k_n-d_0)} \\ge 0 \\right) \\exp \\lbrace H_n\\rbrace \\nonumber \\\\&&\\times \\left[1 +\\frac{bH_n}{a+b+(k_n-d_0)/2}+D \\frac{b(b+T_n)^{-a-(k_n-d_0)/2}}{(a+b+(k_n-d_0)/2)^{3/2}} \\left( \\exp \\lbrace H_n\\rbrace -1 -H_n - (T_n+2)^{-1/2} \\right)_+ \\right]^{-1},\\nonumber \\\\$ where $T_n = \\max \\lbrace (a+(k_n-d_0)/2)^2,3\\left\\lceil H_n \\right\\rceil \\rbrace $ and $D$ is some constant.", "We now consider two cases: (i) when $F_0\\in \\mathfrak {L}(\\Phi _0)$ and (ii) when $F_0^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F_0 \\asymp n$ .", "Case (i) $F_0\\in \\mathfrak {L}(\\Phi _0)$: Recall that in this case $M_n = \\zeta _n n^{-1}$ for any arbitrary diverging sequence $\\zeta _n$ .", "First, we show that $\\mathbb {E}_{\\omega \\mid Y}[W_1]\\overset{p}{\\rightarrow }0$ by proving that $\\mathbb {E}_{\\omega \\mid Y}[W_{1,1}]\\overset{p}{\\rightarrow }0$ and $\\mathbb {E}_{\\omega \\mid Y}[W_{1,2}]\\overset{p}{\\rightarrow }0$ .", "Applying Lemma REF , it follows that (REF ) is bounded above by $&&\\mathbb {E}_{\\omega \\mid Y}\\left[W_{1,1}\\right] \\le C \\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} \\right\\rbrace \\frac{1+\\frac{b}{a+b}\\exp \\lbrace H_n \\rbrace }{1 +\\delta _n+u_n \\frac{Db}{a+b} \\left( \\exp \\lbrace H_n\\rbrace -1 -H_n - (T_n+2)^{-1/2} \\right)_+}\\nonumber \\\\&\\le &C \\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} \\right\\rbrace \\left(1+\\frac{b}{a+b}\\exp \\lbrace H_n \\rbrace \\right),$ where $\\delta _n= bH_n/(a+b+(k_n-d_0)/2)$ and $u_n= (a+b)(b+T_n)^{-a_n-(k_n-d_0)/2}/(a+b+(k_n-d_0)/2)^{3/2}$ with $T_n=\\max \\lbrace (a+(k_n-d_0)/2)^2,3\\left\\lceil H_n \\right\\rceil \\rbrace $ , and $C$ and $D$ are some constants.", "Since $2H_n\\sim \\chi _{k_n-d_0}^2$ , by Lemma REF and defining $q_n = k_n^{-1/2}(\\log k_n)^{1/2}(-\\log b)^{1/2}$ , it follows that $P\\left[H_n > k_n q_n/2\\right]\\le \\exp \\lbrace -ck_nq_n\\rbrace ,$ for some constant $c$ .", "Hence, by the condition that $k_n\\log k_n \\prec -\\log b$ , it is clear that $b\\exp \\lbrace H_n\\rbrace = o_p(1)$ , which shows that $\\mathbb {E}_{\\omega \\mid Y}[W_{1,1}]=o_p(1)$ .", "Similarly, since $\\Gamma (b)^{-1}\\asymp b$ , (REF ) is bounded by $C^{\\prime }b\\exp \\lbrace H_n\\rbrace \\left( \\frac{nM_n}{64\\sigma ^2(k_n-d_0)} \\right)^{b-1},$ for some constant $C^{\\prime }$ .", "By (REF ), $b\\exp \\lbrace H_n\\rbrace = o_p(1)$ , which implies $\\mathbb {E}_{\\omega \\mid Y}[W_{1,2}]=o_p(1)$ .", "We next show that $E_{\\omega \\mid Y}[W_2]$ converges in probability to zero.", "Applying Lemma REF , it follows that $&&\\mathbb {E}_{\\omega \\mid Y}[W_2] =\\mathbb {E}_{\\omega \\mid Y}\\left[ P\\big [ \\big \\Vert (1-\\omega )Q_1Y-Q_1\\Phi \\beta ^*\\big \\Vert _{n,2}> M_n^{1/2}/8 \\mid Y, \\omega \\big ] \\right]= P_{\\omega \\mid Y}\\left[ \\omega < 1-\\left(\\frac{nM_n}{64\\sigma ^2H_n}\\right)^{1/2} \\right] \\\\&=& \\frac{1}{m(Y)}\\int _0^{1-\\left(\\frac{nM_n}{128\\sigma ^2H_n} \\right)^{1/2} } \\omega ^{a+(k_n-d_0)/2-1}(1-\\omega )^{b-1}\\exp \\lbrace -H_n\\omega \\rbrace d\\omega \\\\&\\le & \\mathbb {1}\\left\\lbrace 1-\\left(\\frac{nM_n}{128\\sigma ^2H_n} \\right)^{1/2}\\ge 0 \\right\\rbrace \\frac{1}{m(Y)}\\left(\\frac{nM_n}{64\\sigma ^2H_n} \\right)^{(b-1)/2}\\int _0^{1} \\omega ^{a+(k_n-d_0)/2-1}\\exp \\lbrace -H_n\\omega \\rbrace d\\omega \\\\&\\le & \\mathbb {1}\\left\\lbrace 1-\\left(\\frac{nM_n}{128\\sigma ^2H_n} \\right)^{1/2}\\ge 0 \\right\\rbrace \\frac{\\Gamma (a+b+(k_n-d_0)/2)}{\\Gamma (b)\\Gamma (a+(k_n-d_0)/2)} \\left(\\frac{nM_n}{128\\sigma ^2H_n} \\right)^{(b-1)/2} \\\\&&\\times \\exp \\lbrace H_n\\rbrace \\left\\lbrace 1 +\\delta _n+u_n \\frac{Db}{a+b} \\left( \\exp \\lbrace H_n\\rbrace -1 -H_n - (T_n+2)^{-1/2} \\right)_+\\right\\rbrace ^{-1}\\\\&\\le &C b\\left(\\frac{nM_n}{128\\sigma ^2} \\right)^{(b-1)/2}H_n^{1/2}\\exp \\lbrace H_n\\rbrace ,$ where $C$ is some constant, and $\\delta _n$ and $u_n$ are defined following (REF ).", "From (REF ), it follows that $b\\lbrace nM_n/(128\\sigma ^2)\\rbrace ^{(b-1)/2}H_n^{1/2}\\exp \\lbrace H_n\\rbrace $ is bounded by $b\\lbrace nM_n/(128\\sigma ^2)\\rbrace ^{(b-1)/2}(k_n q_n/2)^{1/2}\\exp \\lbrace k_nq_n/2\\rbrace $ with probability greater than $1-\\exp \\lbrace -ck_nq_n\\rbrace $ from which it follows that $\\mathbb {E}_{\\omega \\mid Y}[W_2]=o_p(1)$ .", "Case (ii) $F_0^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F_0 \\asymp n$: Recall that in this case $M_n = \\zeta _n n^{-2\\alpha /(1+2\\alpha )}\\log n$ for any arbitrary diverging sequence $\\zeta _n$ , and $\\delta _n$ and $u_n$ are defined following (REF ).", "From (REF ) it follows that $&&\\mathbb {E}_{\\omega \\mid Y}\\left[ W_{1,1} \\right] \\le \\frac{1}{m(Y)} \\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} \\right\\rbrace \\int ^1_{m_n} \\omega ^{a-1}(1-\\omega )^{b-1}\\exp \\left\\lbrace -H_n\\omega \\right\\rbrace d\\omega \\\\&\\le &C \\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} \\right\\rbrace \\frac{1+\\frac{b}{a+b}\\exp \\lbrace H_n \\rbrace }{1 +\\delta _n+u_n \\frac{Db}{a+b} \\left( \\exp \\lbrace H_n\\rbrace -1 -H_n - (T_n+2)^{-1/2} \\right)_+},$ for some constant $C$ .", "By Lemma REF , for any sequence $w_n\\rightarrow 0$ , $H_n$ is larger than $w_nF_0^{ \\mathrm {\\scriptscriptstyle T} }Q_1 F_0/\\sigma ^2$ with probability greater than $1-\\exp \\lbrace -cF_0^{ \\mathrm {\\scriptscriptstyle T} }Q_1 F_0(1-w_n)^2/\\sigma ^2\\rbrace $ for some constant $c$ .", "Since $F_0^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)F_0 \\asymp n$ implies $F_0^{ \\mathrm {\\scriptscriptstyle T} }Q_1 F_0 \\asymp n$ , the last line in the above display can be expressed as $C^{\\prime }\\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} (k_n-d_0)^{3/2}(b+T_n)^{(k_n-d_0)/2} \\right\\rbrace + o_p(1),$ where $T_n=\\max \\lbrace (a+(k_n-d_0)/2)^2,3H_n\\rbrace $ and $C^{\\prime }$ is some positive constant.", "Therefore, to show $\\mathbb {E}_{\\omega \\mid Y}[W_{1,1}]\\overset{p}{\\rightarrow }0$ , it is sufficient to prove that $T_n^{(k_n-d_0)/2}\\exp \\lbrace -nM_n/(128\\sigma ^2)\\rbrace =o_p(1)$ .", "For any $\\epsilon >0$ , $&& P\\left[ T_n^{(k_n-d_0)/2}\\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} \\right\\rbrace >\\epsilon \\right]\\\\& \\le & P\\left[ (3H_n)^{(k_n-d_0)/2}\\exp \\left\\lbrace -\\frac{nM_n}{128\\sigma ^2} \\right\\rbrace > \\epsilon \\right] + P\\left[ 3H_n < (a+(k_n-d_0)/2)^2 \\right]\\\\&\\le & P\\left[ \\log H_n > \\zeta _n \\log n \\right] + P\\left[ 3H_n < (a+(k_n-d_0)/2)^2 \\right].$ Since $\\zeta _n \\rightarrow \\infty $ as $n$ tends to $\\infty $ , from (REF ) in Lemma REF , it follows that the first term in the above display can be bounded above by $\\exp \\lbrace -c^{\\prime }(n^\\zeta _n-F_0^{ \\mathrm {\\scriptscriptstyle T} }Q_1 F_0/\\sigma ^2)\\rbrace $ for some constant $c^{\\prime }$ .", "Similarly, from (REF ) in Lemma REF , the second term is bounded by $\\exp \\lbrace -c^{\\prime \\prime }F_0^{ \\mathrm {\\scriptscriptstyle T} }Q_1 F_0/\\sigma ^2\\rbrace $ with some constant $c^{\\prime \\prime }$ , which proves that $\\mathbb {E}_{\\omega \\mid Y}[W_{1,1}]\\overset{p}{\\rightarrow }0$ .", "Since $nM_n \\succ k_n$ , the indicator function $\\mathbb {1}( 1-nM_n/(64\\sigma ^2(k_n-d_0)) \\ge 0 )$ in (REF ) is zero when $n$ is large enough, which results in $\\mathbb {E}_{\\omega \\mid Y}[W_{1,2}]\\overset{p}{\\rightarrow }0$ .", "The marginal posterior mean of $W_2$ can be decomposed as $\\mathbb {E}_{\\omega \\mid Y}[W_2]&\\le &P_{\\omega \\mid Y}\\left[ \\big \\Vert (1-\\omega )\\mathrm {Q}_1 Y - \\mathrm {Q}_1 Y\\big \\Vert _{n,2}>\\frac{1}{16}M_n^{1/2} \\right]+ \\mathbb {1}\\left[\\big \\Vert \\mathrm {Q}_1 Y-\\mathrm {Q}_1\\Phi \\beta ^*\\big \\Vert _{n,2}>\\frac{1}{16}M_n^{1/2} \\right].$ Results provided by [44] (see equation (REF ) on page REF ) show that the second term in the previous expression is $o_p(1)$ .", "The first term can be expressed as $&&P_{\\omega \\mid Y}\\left[ \\omega >\\left( \\frac{nM_n}{256\\sigma ^2 H_n} \\right)^{1/2} \\right] = \\frac{1}{m(Y)}\\int ^1_{\\left( \\frac{nM_n}{256\\sigma ^2 H_n} \\right)^{1/2}}\\omega ^{a+(k_n-d_0)/2-1}(1-\\omega )^{b-1}\\exp \\lbrace -H_n\\omega \\rbrace d\\omega \\\\&\\le & \\frac{1}{m(Y)} \\exp \\left\\lbrace -H_n^{1/2}\\left( nM_n/(256\\sigma ^2) \\right)^{1/2} \\right\\rbrace \\int ^1_0 \\omega ^{a+(k_n-d_0)/2-1}(1-\\omega )^{b-1} d\\omega \\\\&\\le & \\left[ u_n\\exp \\lbrace -H_n \\rbrace \\frac{Db}{a+b} \\left( \\exp \\lbrace H_n\\rbrace -1 -H_n - (T_n+2)^{-1/2} \\right)_+ \\right]^{-1} \\exp \\left\\lbrace -H_n^{1/2}\\left( nM_n/(256\\sigma ^2) \\right)^{1/2} \\right\\rbrace ,$ for some positive constant $D$ .", "Since $H_n/n = O_p(1)$ and $ -\\log b \\prec n^{1/2}k_n^{1/2}$ , the above quantity converges in probability to zero, which completes the proof.$\\Box $" ], [ "Computation Strategy: Slice Sampling", "In model (REF ), the conditional posterior distribution of $\\tau $ based on the functional horseshoe prior can be expressed as $\\pi (\\tau \\mid Y,\\beta ) \\propto (\\tau ^2)^{-(k_n-d_0)/2 + b - 1/2}(1+\\tau ^2)^{-a-b}\\exp \\lbrace - \\beta ^{ \\mathrm {\\scriptscriptstyle T} }\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\beta /(2\\sigma ^2)\\rbrace .$ By reparameterizing $\\eta =1/\\tau ^2$ , the resulting conditional posterior distribution of $\\eta $ can be expressed as $\\pi (\\eta \\mid Y,\\beta ) \\propto \\eta ^{a+(k_n-d_0)/2-1}\\exp \\lbrace -\\beta ^{ \\mathrm {\\scriptscriptstyle T} }\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\beta /(2\\sigma ^2)\\rbrace \\frac{1}{(1+\\eta )^{a+b}}.$ As in [35], a slice sampling method [31] can be used to sample $\\eta $ from its conditional posterior distribution.", "The resulting MCMC algorithm is described in Algorithm .", "[!h] MCMC algorithm for simple nonparametric regression models Choose an initial value $\\beta ^{(0)}$ and $\\tau ^{(0)}$ .", "For $l$ in $0:(L-1)$    Sample $\\beta ^{(l+1)}$ from ${ \\mathrm {N} }({ \\widetilde{\\beta } }_{\\omega ^{(l)}},\\sigma ^2{ \\widetilde{\\Sigma } }_{\\omega ^{(l)}})$ , where ${ \\widetilde{\\beta } }_{\\omega }$ and ${ \\widetilde{\\Sigma } }_{\\omega }$ are defined in (REF ).", "(Slice sampling step) Set $\\eta =1/\\tau ^{2(l)}$ and $t = (\\eta +1)^{-a-b}$ .", "Sample $u \\sim Unif(0,t)$ and set $t^*=u^{-(a+b)^{-1}}-1$ .", "Sample $\\eta ^* \\sim truncated\\:\\:Gamma(a+(k_n-d_0)/2,\\beta ^{(l+1){ \\mathrm {\\scriptscriptstyle T} }}\\Phi ^{ \\mathrm {\\scriptscriptstyle T} }(\\mathrm {I}-\\mathrm {Q}_0)\\Phi \\beta ^{(l+1)}/(2\\sigma ^2))$ on $(0,t^*)$ ,    Update $\\tau ^{(l+1)}$ by $\\eta ^{*-1/2}$ .", "End.", "In the additive model in (REF ) with a product of the functional horseshoe priors, the conditional posterior distribution of $\\beta _j$ given $\\omega _j$ and the other coefficients $\\beta _{(-j)}$ , for $j=1,\\dots ,p$ , can be expressed as $\\beta _j \\mid \\omega _j, \\beta _{(-j)}, Y \\sim { \\mathrm {N} }\\left({ \\widetilde{\\beta } }_{j,\\omega },\\sigma ^2{ \\widetilde{\\Sigma } }_{j,\\omega }\\right),$ where ${ \\widetilde{\\beta } }_{j,\\omega } = { \\widetilde{\\Sigma } }_{j,\\omega }\\Phi _j^{ \\mathrm {\\scriptscriptstyle T} }r_j,\\quad { \\widetilde{\\Sigma } }_{j,\\omega } = (1-\\omega _j)\\left(\\Phi _j^{ \\mathrm {\\scriptscriptstyle T} }\\Phi _j\\right)^{-1},\\quad r_j = Y - \\sum _{l\\ne j}\\Phi _l\\beta _l.$ It follows that sampling Algorithm can be extended to additive regression models to obtain Algorithm below.", "[!h] MCMC algorithm for additive regression models Choose an initial value $\\beta _j^{(0)}$ and $\\tau _j^{(0)}$ for $j=1,\\cdots ,p$ .", "For $l$ in $0:(L-1)$    For $j$ in $1:p$       Sample $\\beta _j^{(l+1)}$ from ${ \\mathrm {N} }({ \\widetilde{\\beta } }_{j,\\omega ^{(l)}},\\sigma ^2{ \\widetilde{\\Sigma } }_{j,\\omega ^{(l)}})$ , where ${ \\widetilde{\\beta } }_{j,\\omega }$ and ${ \\widetilde{\\Sigma } }_{j,\\omega }$ are defined in (REF ).", "End.", "For $j$ in $1:p$    (Slice sampling step)       Set $\\eta =1/\\tau _j^{2(l)}$ and $t = (\\eta +1)^{-a-b}$ .", "Sample $u \\sim Unif(0,t)$ and set $t^*=u^{-(a+b)^{-1}}-1$ .", "Sample $\\eta ^* \\sim truncated\\:\\:Gamma(a+k_n/2,\\beta _j^{(l+1){ \\mathrm {\\scriptscriptstyle T} }}\\Phi _j^{ \\mathrm {\\scriptscriptstyle T} }\\Phi _j\\beta _j^{(l+1)}/(2\\sigma ^2))$ on $(0,t^*)$ ,       Update $\\tau _{j}^{(l+1)}$ by $\\eta ^{*-1/2}$ .", "End.", "End." ] ]
1606.05021
[ [ "Decoupling of the re-parametrization degree of freedom and a generalized\n probability in quantum cosmology" ], [ "Abstract The high degree of symmetry renders the dynamics of cosmological as well as some black hole spacetimes describable by a system of finite degrees of freedom.", "These systems are generally known as minisuperspace models.", "One of their important key features is the invariance of the corresponding reduced actions under reparametrizations of the independent variable, a fact that can be seen as the remnant of the general covariance of the full theory.", "In the case of a system of $n$ degrees of freedom, described by a Lagrangian quadratic in velocities, one can use the lapse by either gauge fixing it or letting it be defined by the constraint and subsequently substitute into the rest of the equations.", "In the first case, the system is solvable for $n$ accelerations and the constraint becomes a restriction among constants.", "In the second case, the system can only be solved for $n-1$ accelerations and the \"gauge\" freedom is transferred to the choice of one of the scalar degrees of freedom.", "In this paper, we take the second path and express all $n-1$ scalar degrees of freedom in terms of the remaining one, say $q$.", "By considering these $n-1$ degrees of freedom as arbitrary but given functions of $q$, we manage to extract a two dimensional pure gauge system consisting of the lapse $N$ and the arbitrary $q$: in a way, we decouple the reparametrization invariance from the rest of the equations of motion.", "The solution of the corresponding quantum two dimensional system is used for the definition of a generalized probability for every configuration $f^i (q)$, be it classical or not.", "The main result is that, interestingly enough, this probability attains its extrema on the classical solution of the initial $n$-dimensional system." ], [ "Introduction", "One of the key elements of the theory of General Relativity is the diffeomorphism invariance of the action principle; one can map different line elements to each other by means of any (sufficiently smooth) invertible coordinate transformation without altering the geometry.", "This is in conjunction with the existence of (constraint) equations of motion involving only first derivatives of the metric and corresponds to the “gauge” freedom of choosing a coordinate system.", "In the Hamiltonian analysis of the full theory, where the configuration space becomes the space of all three-metrics (superspace), generators of this freedom are the four constraints: the quadratic, together with the three linear in the momenta [1].", "The presence of first-class constraints demands that symmetry considerations be taken into account.", "Symmetries play an important role in theoretical physics.", "In the case of cosmological models, one assumes a particular class of symmetries and proceeds with the reduction of the action from a field theory to a mechanical system.", "This procedure is called mini-superspace approximation and is an elegant way to truncate the theory to a system of finite degrees of freedom; in this context, symmetries of the configuration space have been considered.", "In particular, variational symmetries have been applied in the past (see e.g.", "[2]).", "Recently, the projective group was used in [3], [4], [5] while in [6] the correspondence of the conditional symmetries [7] with the variational symmetries in a certain lapse parametrization was proven and consequently used for the study of minisuperspace models in [8], [9], [10], [11], [12].", "If one considers a general line element of a spatially homogeneous Bianchi type $ds^2 = - N(t)^2 dt^2 + \\gamma _{\\mu \\nu }(t)\\sigma ^\\mu _i(x) \\sigma ^\\nu _j(x) dx^i dx^j,$ with the invariant basis one-form satisfying the relations $\\sigma ^\\alpha _{i,j} -\\sigma ^\\alpha _{j,i} = C^\\alpha _{\\beta \\gamma } \\sigma ^\\beta _j \\sigma ^\\gamma _i$ , then the Lagrangian function of the theory assumes the general form $L= \\frac{1}{2 N} G_{\\alpha \\beta }(q)\\dot{q}^\\alpha \\dot{q}^\\beta - N V(q), \\quad \\quad \\alpha ,\\,\\beta = 0,\\ldots , n-1,$ where $N$ is the lapse function and $q^\\alpha $ are the degrees of freedom that correspond to the components of the scale factor matrix $\\gamma _{\\mu \\nu }$ and possibly any additional fields that one might add in the theory.", "One of course has to check whether the Euler - Lagrange equations corresponding to (REF ) generate the same dynamics as the field theory equations of motion evaluated for the above line element.", "When this is the case, (REF ) is considered valid.", "This Lagrangian has certain characteristic properties: Let us write the corresponding action as $I = \\int dt N \\left( \\frac{1}{2} G_{\\alpha \\beta } \\frac{d q^\\alpha }{N dt} \\frac{d q^\\beta }{N dt} -V\\right) $ In this form, it can be easily seen that the action retains its form under reparametrizations of time and the corresponding changes of the dependent variables $&t = f (\\bar{t}), \\\\&q^\\alpha (t) = q^\\alpha (f(\\bar{t})) =: \\bar{q}^\\alpha (\\bar{t}), \\\\&N(t) dt = N (f(\\bar{t})) \\frac{d f (\\bar{t})}{d\\bar{t}} d\\bar{t} =: \\bar{N} (\\bar{t}) d\\bar{t}.$ The last transformation rule is evident from the line element (REF ) and designates $N$ as a density degree of freedom in contradistinction to $q^\\alpha $ 's which are scalars; when they refer to $\\gamma _{\\alpha \\beta }$ 's their transformation rule (REF ) is also evident from (REF ).", "Alternatively, one could guess these transformations by just looking at the action and demanding that it remains form invariant.", "The above invariance is also present at the level of the equations of motion.", "Indeed, one can check that the Euler-Lagrange equations $&E^0:= \\frac{1}{2 N^2} G_{\\alpha \\beta } \\dot{q}^\\alpha \\dot{q}^\\beta + V =0, \\\\&E^\\mu := \\ddot{q}^\\mu + \\Gamma ^\\mu _{\\nu \\lambda } \\dot{q}^\\nu \\dot{q}^\\lambda -\\frac{\\dot{N}}{N} \\dot{q}^\\mu + N^2 G^{\\mu \\kappa } V_{,\\kappa }$ do indeed respect this symmetry: by effecting the change $t= f(\\bar{t})$ , (REF ) is invariant, while () is multiplied by $(df/d\\bar{t})^{-2}$ since the only troublesome term arising from the second derivative of time is counterbalanced by the third term containing $\\frac{\\dot{N}}{N}$ .", "A second important consequence that this invariance under arbitrary changes of $t = f(\\bar{t})$ induces is that of the freedom to scale the lapse $N$ by an arbitrary function of the $q^\\alpha $ 's.", "In order to understand this property of the system it is useful to isolate the scalar degrees of freedom $q^\\alpha $ 's by solving (REF ) algebraically for $N^2$ , $N^2 = - \\frac{G_{\\alpha \\beta } \\dot{q}^\\alpha \\dot{q}^\\beta }{2 V} =: -\\frac{K}{2V} $ and then substituting it in () thereby obtaining $\\ddot{q}^\\mu + \\Gamma ^\\mu _{\\nu \\lambda } \\dot{q}^\\nu \\dot{q}^\\lambda -\\frac{1}{2}\\frac{\\dot{K}}{K} \\dot{q}^\\mu + \\frac{1}{2} \\frac{V_{,\\kappa }}{V} \\dot{q}^\\kappa \\dot{q}^\\mu - \\frac{K}{2V} G^{\\mu \\kappa } V_{,\\kappa }=0 $ after replacing $\\dot{V}= V_{,\\kappa } \\dot{q}^\\kappa $ .", "In this form, the equations are still covariant under arbitrary changes of time but they can only be solved for $n-1$ accelerations, as it can be seen by contracting with $2 G_{\\mu \\rho } \\dot{q^\\rho }$ , thereby obtaining an identity $0=0$ .", "At this stage, one can say that the freedom in defining the $t$ -variable is translated into the freedom of arbitrarily choosing one of the $q^\\alpha $ 's.", "In this context, the constraint equation (REF ) simply specifies the lapse, once all the equations have been solved (after, of course, the arbitrary function has been specified).", "Let us now take the Lagrangian (REF ) and consider a new independent variable $M= N e^{2 \\omega }$ , thereby obtaining the transformed Lagrangian $\\overline{L} =\\frac{1}{2M} \\overline{G}_{\\alpha \\beta } \\dot{q}^\\alpha \\dot{q}^\\beta - M\\overline{V}$ where $\\overline{G}_{\\alpha \\beta } =e^{2 \\omega } G_{\\alpha \\beta }, \\qquad \\overline{V}=e^{-2 \\omega } V. $ By applying the same process on the equations of motion emanating from $\\overline{L}$ corresponding to $M$ , $q^\\alpha $ 's we obtain the analogue of (REF ) $\\ddot{q}^\\mu + \\overline{\\Gamma }^\\mu _{\\nu \\lambda } \\dot{q}^\\nu \\dot{q}^\\lambda -\\frac{1}{2}\\frac{\\dot{\\overline{K}}}{\\overline{K}} \\dot{q}^\\mu + \\frac{1}{2} \\frac{\\overline{V}_{,\\kappa }}{\\overline{V}} \\dot{q}^\\kappa \\dot{q}^\\mu - \\frac{\\overline{K}}{2\\overline{V}}\\, \\overline{G}^{\\mu \\kappa }\\, \\overline{V}_{,\\kappa }=0.", "$ This equation becomes identical to (REF ) by using the transformations (REF ) and the well known relation $\\overline{\\Gamma }^\\mu _{\\nu \\lambda } = \\Gamma ^\\mu _{\\mu \\lambda } + \\delta ^\\mu _\\nu \\omega _{,\\lambda } + \\delta ^\\mu _\\lambda \\omega _{,\\nu } -G_{\\nu \\lambda } G^{\\mu \\rho } \\omega _{,\\rho }$ This proves that one can scale $N$ arbitrarily and still solve the same equations containing accelerations.", "A particularly convenient, and useful, choice is to use this property in order to transform the potential $V$ into an effective constant potential $\\overline{V}$ .", "Then, the dynamics is in a sense geometrized, since the Euler-Lagrange equations for the $q^\\mu $ 's become free geodesics in the scaled manifold's configuration metric (where of course the constraint selects the type of the geodesic vector).", "Therefore, one is allowed to assume the transformation $N\\mapsto \\overline{N} = N\\, V$ first introduced in [13] with the Lagrangian acquiring a constant potential $\\overline{L}= \\frac{1}{2\\overline{N}} \\overline{G}_{\\alpha \\beta }(q)\\dot{q}^\\alpha \\dot{q}^\\beta - \\overline{N}, \\quad \\quad \\alpha ,\\,\\beta = 0,\\ldots , n-1,$ with $\\overline{G}_{\\alpha \\beta }= V G_{\\alpha \\beta }$ being the scaled mini-supermetric.", "Of course, as explained above, the dynamics before and after the scaling remains the same.", "For more information on the usefulness of this particular parametrization, especially at the quantum level, see [8], [9], [10], [14].", "From this point on we choose to work exclusively in the constant potential parameterization, thus we omit the bar symbolism.", "As already mentioned, the time reparametrization invariance of the system makes it possible to solve the Euler - Lagrange equations of motion by leaving an unspecified degree of freedom, say $q^0$ .", "If one wishes, it is possible to select this degree as representing the independent variable and describe the other dependent variables in terms of this, e.g.", "$q^i=f^i(q^0)$ , where $i=1,\\ldots n-1$ .", "This description is independent of any time reparametrization since all $q^\\mu $ 's transform, unlike the lapse function, as scalars (REF ), thus leaving the relations $q^i=f^i(q^0)$ unaltered.", "The resulting solution of the Euler–Lagrange equations leads to a specific geometry for the base manifold which also satisfies the corresponding Einstein's equations and is designated by a particular set of $f^i$ 's.", "One can then proceed to the Hamiltonian description and successively to the quantization of the system by enforcing the quantum analogues of the constraints as conditions on the wave function; likewise, any other observables that may be obtained due to the existence of symmetries ([8], [9], [15], [16], [17], [18]).", "In the best case, a wave function can be completely determined by solving any existing eigen-equations together with the Wheeler-DeWitt constraint.", "However, this constitutes a quantization scheme that engulfs only the specific geometry under consideration.", "In this work, we try to expand the quantization procedure given previously in the literature, in order to encompass all possible line elements inside a specific geometrical class.", "We use the information contained in the time reparametrization invariance - associated with the fact that not all degrees of freedom are independent - at the Lagrangian level.", "We thus perform a reduction from the initial system to another consisting of $1+1$ degrees of freedom; its dynamics is governed by two equations, one of which is of first order, i.e.", "a constraint, while the second order is sattisfied by virtue of the constraint.", "In this sense, we get a pure gauge system in which the previous degrees associated with scale factors of the base manifold have been reduced to arbitrary functions which can take any value.", "Consequently, the quantization of this latter system gives weight to all geometries (in a certain class) and surprisingly enough, the extrema of a generalized probability we define are located on the classical geometry corresponding to the initial system prior to the reduction.", "The whole procedure we follow can be visualized by FIG.", "REF .", "Figure: The class of geometries G i G_i, define the configuration space (CS) through the minisuperspace Einstein–Hilbert action; the decoupling of the reparametrization invariance (R–invariance) leads to the reduced (1+1)(1+1) configuration space (RCS).", "The wave functional Ψ h \\Psi _h which describes the class of geometries G i G_i is obtained from the Dirac quantization of this system which finally gives a “weight\" h i h_i to every geometry of the space time through the generalized probability ().The paper is organized as follows: In section 2, we reduce the initial system by singling out one degree of freedom and in section 3 we proceed with the Hamiltonian formulation for the reduced system.", "In section 4, we perform its canonical quantization and we adopt a measure on the configuration space.", "This is necessary for the definition of a generalized probability function.", "In section 5, we study the conditions for the extrema of the generalized probability to exist, while section 6 is devoted to their characterization as minima or maxima.", "We conclude with two examples and some discussion of our results." ], [ "Decoupling of the initial system and its reduction", "We consider a system described by the Lagrangian (REF ), $L= \\frac{1}{2\\, N}\\, G_{\\alpha \\beta }\\dot{q}^\\alpha \\dot{q}^\\beta - N, \\quad \\alpha ,\\beta = 0,\\ldots n-1.$ which corresponds to a mini-superspace analysis that was initiated by a line element of the form (REF ) in the constant potential parametrization.", "The associated equations of motion are $ 2\\, N^2 + G_{\\alpha \\beta }\\, \\dot{q}^\\alpha \\, \\dot{q}^\\beta & = 0\\\\ \\ddot{q}^\\mu + \\Gamma ^\\mu _{\\alpha \\beta }\\, \\dot{q}^\\alpha \\, \\dot{q}^\\beta - \\frac{\\dot{N}}{N}\\, \\dot{q}^\\mu &= 0.$ Solving the constraint (REF ) for $N^2$ and replacing in (), we obtain $\\ddot{q}^\\mu + \\Gamma ^\\mu _{\\nu \\lambda } \\dot{q}^\\nu \\dot{q}^\\lambda -\\frac{1}{2} \\frac{\\dot{K}}{K} \\dot{q}^\\mu = 0 $ where $K$ is given by $K=G_{\\alpha \\beta } \\dot{q}^\\alpha \\dot{q}^\\beta .$ At this stage, the system is described by the equations (REF ) and (REF ).", "We wish to disentangle the true dynamics from the gauge freedom $t = f (\\bar{t})$ .", "To this end, we choose one of the dependent scalar variables, say $q^0= q(t)$ , and select the function $f$ as the inverse of $q$ , $f:=q^{-1}$ .", "Then, all of the $q^\\alpha $ 's can be expressed as functions of the selected $q$ and we are led to the definitions $& q^0=: q(t) = q(f (\\bar{t})) = q (q^{-1} (\\bar{t})) =\\bar{t}, \\\\& q^i := q^i (t) = q^{i} (q^{-1}(\\bar{t})) = q^i \\circ q^{-1} (\\bar{t}) =f^i(q), \\quad i = 1,\\ldots ,n-1 .", "$ This means that we are effectively selecting $q(t)$ as time.", "As a result, the expression for $K$ , using the obvious reletions $\\dot{q}^i (t) =\\acute{f}^i (q(t)) \\dot{q}(t)$ where $^{\\prime } = \\frac{d}{dq}$ , reads $ K =\\left[ G_{00}+2 G_{0i} \\acute{f}^i+G_{ij} \\acute{f}^i\\acute{f}^j \\right] \\dot{q}^2 =: h[q, f^i (q), \\acute{f}^i (q)] \\dot{q}^2.$ Equation (REF ) defines the functional $h[q, f^i (q) \\acute{f}^i (q)]$ , which, in the interest of concisness, we hereafter abbreviate as $h[q]$ .", "In order to simplify the forthcoming expressions, we adopt, without loss of generality, a Gauss-normal coordinate system in the configuration space, i.e.", "we assume coordinates in which the $(q, q^i), i=1,...,n-1$ are such that the metric elements are $G_{qq}=\\varepsilon =\\pm 1, G_{qi}=0$ .", "Note that this singling out of a particular variable is not a feature of the selected coordinate system, but a general feature of the constrained systems as it has been discussed in detail in the introduction.", "Now we can express equation (REF ) and decompose the equations (REF ) in $\\mu =0$ and $\\mu =i$ components, with $\\mu =0$ corresponding to the $q$ variable.", "After the replacement of the non-vanishing Christoffel symbols, where $\\Gamma ^q_{jk} =-\\frac{1}{2 G_{qq}} G_{jk,q}, \\,\\Gamma ^i_{qk} = \\frac{1}{2} G^{ij} G_{jk,q}$ and with $\\Gamma ^i_{jk}$ aquiring the regular form, as well as of the obvious relations $q^{\\prime }=1, \\ q^{\\prime \\prime } =0$ , the quantity $K$ and the constraint assume the form $K =\\left[ \\varepsilon +G_{ij} (q, f^i) \\acute{f}^i (q)\\acute{f}^j (q) \\right]$ $N^2 = -\\frac{h[q]}{2} ,$ while the $\\mu =0$ component of () takes the form $\\frac{h^{\\prime }}{h} = - \\frac{1}{\\varepsilon } G_{ij,q} \\acute{f}^i \\acute{f}^j $ and the $\\mu =i$ component becomes $\\acute{\\acute{{f}}}^i + G^{ik} G_{jk,q}\\acute{f}^j + \\Gamma ^i_{kl} \\acute{f}^k \\acute{f}^l -\\frac{1}{2} \\frac{h^{\\prime }}{h} \\acute{f}^i =0 .", "$ Replacing (REF ) in (REF ), we get $\\acute{\\acute{{f}}}^i +G^{ik} G_{jk,q}\\acute{f}^j + \\Gamma ^i_{kl} \\acute{f}^k \\acute{f}^l +\\frac{1}{2\\varepsilon } G_{jk,q} \\acute{f}^j \\acute{f}^k \\acute{f}^i =0, $ a set of equations that are explicitly solved for the accelerations and describe the dynamics of the $f^i$ 's degrees of freedom.", "These equations constitute a regular system with no other invariance under further arbitrary changes of the $q$ variable.", "An important property is that (REF ) already contains the information of (REF ); one can see this by contracting (REF ) with $2 G_{ir} \\acute{f}^r$ and using the derivative of the definition of $h$ to replace the term $2 G_{ir} \\acute{f}^r \\acute{\\acute{{f}}}^i$ .", "Thus the initial system of equations (), in which the dependent variables $N, q^\\alpha $ were coupled, has now been transformed into equations (REF ) and (REF ).", "In other words, the system is decoupled in the variables $N, q$ and $ f^i (q)$ .", "We now pose the question: what is the dynamics of the $N,q$ degrees of freedom when $f^i(q)$ are specified albeit arbitrary functions, not necessarily obeying (REF ), so that we can describe the entire family of geometries of a given base manifold's line element.", "To unravel the answer, it is sensible to turn to the Lagrangian (REF ) and replace the $q^i$ 's from () and their time derivatives by $\\dot{q}^i (t) =\\acute{f}^i (q(t)) \\dot{q}(t)$ .", "This action defines a new Lagrangian which - among other properties - could, upon quantization, be capable of associating a probability to each and every geometry in a given class defined by $f^i (q)$ 's.", "The reduced Lagrangian takes the form $L_{red} = \\frac{1}{2\\, N(t)}\\, h[q(t)]\\, \\dot{q}(t)^2- N(t),$ and describes a constrained system with two degrees of freedom, $N$ and $q$ .", "The reduced Lagrangian (REF ) is radically different from the original Lagrangian (REF ), since the former contains the arbitrary functions $f^i(q(t))$ (not to be considered as degrees of freedom), whilst the latter is a “usual” Lagrangian with the $q^\\alpha (t)$ playing the role of the ordinary degrees of freedom.", "Thus the equations of motion that result from the reduced Lagrangian are ordinary differential equations with arbitrary functions in them; therefore these equations describe a class of geometries.", "This fact is made more clear in the treament of the example of the static and spherically symmetric spacetime in section .", "The equations of motion for the above Lagrangian are: $ E^0_{red} &:= 2\\, N^2 + h[q]\\, \\dot{q}^2 = 0 \\\\ E_{red} &:= \\ddot{q} + \\frac{h^{\\prime }}{2\\, h} \\, \\dot{q}^2 - \\frac{\\dot{N}}{N}\\, \\dot{q}=0,$ It can be easily verified that equation () is a consequence of (REF ): by solving the former for $N$ and substituting in the later we get that its left hand side becomes identically zero.", "This is not just a property of the system under consideration.", "Every constrained system of $1+1$ degrees of freedom has its dynamics determined by only the first order constraint equation.", "Therefore, one can either regard the constraint as a first order differential equation for $q$ (when the lapse is to be prescribed), or as an algebraic equation for the lapse.", "The solution of the constraint leaves the other degree of freedom arbitrary.", "Thus, the equation of motion for $q(t)$ is satisfied by virtue of the constraint, without $q(t)$ itself being explicitly determined and, more importantly, without any conditions being imposed on the $f^i(q)$ 's.", "Any properties of the reduced system defined by (REF ) hold irrespectively to the particular form of the $n-1$ functions $f^i(q)$ .", "Additionally, we keep in mind that some or all of the $f^i(q)$ 's are components of the base manifold metric (REF ).", "Thus, the properties of the reduced system hold for any particular geometry contained in the general form of (REF ) (for fixed $\\sigma ^\\alpha _i(x)$ ); a key reason for considering this reduced Lagrangian." ], [ "Phase space description", "The conjugate momenta for the reduced system are: $p_N & := \\frac{\\partial L_{red}}{\\partial \\dot{N}}=0, \\\\p &:= \\frac{\\partial L_{red}}{\\partial \\dot{q}}= \\frac{h}{N}\\, \\dot{q}.$ By following Dirac's prescription for singular systems ([1], [19], [20], [21]), we recognize $p_N\\approx 0$ as the primary constraint, thus the Hamiltonian is written as $ \\nonumber H_T & = \\dot{q}\\, p - L + u^N\\, p_N \\\\ \\nonumber & = N\\left(\\frac{p^2}{2\\, h}+1\\right)+ u^N\\, p_N \\\\& = N\\, \\mathcal {H} + u^N\\, p_N,$ where we have set $\\mathcal {H}= \\frac{p^2}{2\\, h}+1 .$ The primary constraint must be preserved in time, i.e.", "$\\dot{p}_N = \\lbrace p_N, H_T\\rbrace \\approx -\\mathcal {H} \\approx 0.$ Consequently, we are led to the secondary constraint $\\mathcal {H}$ which produces no tertiary constraints.", "Obviously, the two constraints are first class, $\\lbrace p_N,\\mathcal {H}\\rbrace \\approx 0$ , meaning that their coefficients in the Hamiltonian are not fixed by the theory, since the Dirac-Bergmann algorithm for finding all the constraints is being terminated.", "From the secondary constraint, we can easily deduce that $\\mathcal {H} \\approx 0 \\Rightarrow \\frac{p^2}{-h}\\approx 2 \\Rightarrow \\frac{p}{\\sqrt{-h}} = \\pm \\sqrt{2},$ which means that $Q=\\frac{p}{\\sqrt{-h}}$ is a linear in the momenta integral of motion.", "Note here that the same result can be obtained formally: if we set $Q=A(q)p+B(q)$ , where $A(q)$ , $B(q)$ are arbitrary functions, and demand this quantity to be a conditional symmetry [7] (integral of motion by virtue of the constraints, i.e.", "$\\approx 0 \\Rightarrow \\; = F(q)\\mathcal {H}$ , with $F(q)$ an arbitrary function), we get $\\dot{Q} \\approx 0\\Rightarrow \\lbrace Q,H_T\\rbrace &= N\\, F(q)\\, \\mathcal {H} \\Rightarrow \\\\\\frac{N}{h}\\left(2\\, A^{\\prime }+ A\\, \\frac{h^{\\prime }}{h}\\right)\\, p^2 + \\frac{2\\, N\\, B^{\\prime }}{h}\\, p & = \\frac{F\\, N}{h}\\, p^2+F\\, N.$ and by gathering the coefficients of different powers of the momentum $p$ we deduce $ \\nonumber B(q) &= \\text{constant}\\\\ \\nonumber F(q) & = 0 \\\\2\\, A^{\\prime } + A\\, \\frac{h^{\\prime }}{h} &=0,$ with the last equation having as a solution $A=\\frac{c_1}{\\sqrt{-h}}$ .", "The general form of the linear integral of motion is thus $Q=\\frac{c_1\\, p}{\\sqrt{-h}}+ c_2$ .", "As it can be seen by the form of (REF ), the presence of $c_1$ corresponds to the multiplication of the Hamiltonian with an overall constant, while that of $c_2$ presents the trivial addition of any constant to an integral of motion; thus we restrict ourselves to the particular form $Q=\\frac{p}{\\sqrt{-h}}.$ It is important to note once more that the functions $f^i(q)$ have not suffered the imposition of any conditions.", "The quantity (REF ) is constant whenever the Euler-Lagrange equations () are satisfied and this happens for arbitrary $f^i(q)$ .", "Consequently, $Q$ is an integral of motion of (REF ) irrespective to the particular geometry that the non reduced system (REF ) might have as a solution." ], [ "Quantization of the reduced system", "We proceed with the quantization of the system by adopting the usual canonical commutation relations.", "The momenta operators are taken to be $\\widehat{p}_N &= - {i}\\frac{\\partial }{\\partial N} \\\\\\widehat{p} &= - {i}\\frac{\\partial }{\\partial q}$ and the Poisson brackets are mapped to the commutator as $\\lbrace \\;,\\;\\rbrace \\rightarrow -{i}[\\;,\\;],$ where for brevity we have adopted the units $\\hbar =1$ .", "Imposition of the primary constraint on the wave function, $\\widehat{p}_N\\Psi =0$ , leads to $\\Psi =\\Psi (q)$ .", "Thus, $\\Psi $ becomes a function of a single variable.", "For the operator counterpart of (REF ), we choose the most general form of an operator Hermitian under an arbitrary measure $\\mu (q)$ [22], $\\widehat{Q} = - \\frac{{i}}{2\\, \\mu }\\left(\\frac{\\mu }{\\sqrt{-h}}\\,\\frac{d}{d q}+\\frac{d}{d q}\\left(\\frac{\\mu }{\\sqrt{-h}}\\right)\\right).$ Since $Q$ corresponds to a classical constant of motion, we infer that at the quantum regime the eigenvalue equation $\\widehat{Q}\\Psi = \\kappa \\Psi $ where $\\kappa = \\pm \\sqrt{2}$ must be enforced.", "For the quadratic constraint, we use the operator $\\widehat{\\mathcal {H}} = -\\frac{1}{2\\, \\mu }\\, \\frac{d}{d q} \\left(\\frac{\\mu }{h}\\,\\frac{d}{d q}\\right)+1,$ whose kinetic part becomes the one dimensional Laplacian if $\\mu =\\sqrt{-h}$ (natural measure) and which, in general, has the property that when $\\mu $ transforms like a scalar density, $\\widehat{\\mathcal {H}}$ transforms as a scalar [22].", "For the time being we consider the measure arbitrary.", "Given the previous definitions, the Wheeler-DeWitt equation reads $\\widehat{\\mathcal {H}}\\Psi =0 \\Rightarrow \\frac{\\Psi ^{\\prime \\prime }}{2\\, h}+ \\frac{h\\, \\mu ^{\\prime } -\\mu \\, h^{\\prime }}{2\\, h^2\\, \\mu }\\, \\Psi ^{\\prime } - \\Psi =0.$ Equations (REF ) and (REF ) are to be satisfied by the wave function we seek.", "The eigenvalue equation (REF ) can, for an arbitrary measure, be easily integrated yielding $\\Psi _h [q]= \\frac{(-h[q])^{\\frac{1}{4}}}{\\sqrt{\\mu (q)}}\\, e^{{i}\\, \\kappa \\, \\int \\!\\!", "\\sqrt{-h[q]}\\,d q}.$ Of course, it is left for us to check what conditions (REF ) imposes on this solution.", "To this end, we insert the result (REF ) into the Wheeler-DeWitt equation and, for the sake of simplicity, we set $\\mu [q] = \\phi [q]^2\\, \\sqrt{-h[q]}$ ; the outcome, under the classical restriction $\\kappa = \\pm \\sqrt{2}$ , is the following expression for $\\phi [q]$ $h^{\\prime }\\, \\phi ^{\\prime } - 2\\, h\\, \\phi ^{\\prime \\prime } = 0 \\Rightarrow \\phi [q] = c_1 + c_2 \\int \\!\\!", "\\sqrt{-h[q]}\\, d q ,$ which sets a condition on the measure, $\\mu [q]= \\sqrt{-h[q]}\\left(c_1+c_2 \\int \\sqrt{-h[q]}\\, d q\\right)^2.$ As one can see, the natural measure is contained in this solution ($c_1=1$ , $c_2=0$ ).", "Had we originally adopted the physical measure, the Wheeler-DeWitt equation would have become an identity after the imposition of the eigenvalue equation.", "Additionally, one can easily check that, under the general measure (REF ), the relation $[\\widehat{Q},\\widehat{\\mathcal {H}}]=0$ holds in general, not only on the wave function $\\Psi $ The relation (REF ) is also satisfied for solutions of (REF ) that are not necessarily on-mass shell, i.e.", "for which $\\kappa \\ne \\pm \\sqrt{2}$ and the measure is not given by the form (REF )..", "However, the particular form of $\\mu (q)$ is not going to concern us any more, because the corresponding probability that one can define as $\\mu [q]\\, \\Psi [q]^* \\, \\Psi [q] dq = \\sqrt{-h[q]} dq$ is independent of it.", "Due to the form of (REF ) and since we know from the theory of quantum mechanics that the different states are distinguised by generically different (not up to a constant) phases, we can consider a generalized probability amplitude for a state of the system $h$ to be given by $P_{h} = \\int \\!\\!", "\\sqrt{-h[q]}\\, d q.$ The above generalized probability is a step towards the probabilistic description of the quantum system (spacetime geometries) described by the wave functional $\\Psi _h[q]$ .", "The basic difference with the standard probability is that (REF ) is not normalized to take values in $[0,1]$ ; in fact, the presence of the unspecified $f^i $ 's prohibits any attempt to check normalizability (for unormalized or relative probability see e.g.", "p.73 of [24]).", "However, this disadvantage is counterbalanced by the benefits of its introduction: The use of the generalized probability leads naturally to the connection of the quantum and classical treatment of a spacetime geometry.", "To the best of our knowledge this is the first time that someone can give a physical meaning to the wave functional of the space time $\\Psi _h[q]$ .", "Until now every information gained from the solution of the Wheller–DeWitt equation was restricted in the context of the semiclassical analysis, e.g.", "[25], [9].", "Now the choice of a particular configuration $q^i(t)$ in the base manifold connects directly with $\\Psi _h[q]$ through the functional $h[q]$ .", "As a consequence of the above, every configuration $q^i(t)$ acquires a weight through the probability (REF ).", "The dominant argument for announcing $P_h$ as a probability is that it attains its extreme value when $q^i(t)$ correspond to the classical solution, as it will be shown in the next section." ], [ "Candidates for extrema of the generalized probability $P_h$", "In order to check for the extrema of the functional (REF ), we are obliged to consider the functions $f^i (q)$ implicit in the functional $h[q]$ as eligible for variation, $\\delta P_{h} = \\delta \\int \\!\\!\\sqrt{-h[q]}\\, d q = 0,$ with $\\sqrt{-h[q,f,\\acute{f}]}= \\left(-G_{ij}(q)\\,\\acute{f}^i(q)\\, \\acute{f}^j(q)- \\varepsilon \\right)^{\\frac{1}{2}}.$ For simplicity, from now on we omit the minus sign inside the root (since $P_{-h}$ exhibits the same extrema as $P_{h}$ ).", "The demand that $\\delta P_{h}=0$ leads to the set of equations $\\acute{\\acute{f}}^m + G^{mk} G_{rk,q} \\acute{f}^r +\\Gamma ^m_{rn} \\acute{f}^r \\acute{f}^n -\\frac{\\acute{h}}{2 h} \\acute{f}^m =0 $ This proves that the extrema of the generalized probability lie on the classical solution of the initial Lagrangian, since (REF ) is identical to (REF ) and therefore equivalent to (REF ) which describes the classical dynamics.", "Hence, we can state the following: Theorem Given a cosmological mini-superspace model with a valid Lagrangian of the form $L = \\frac{1}{2\\, N}\\, G_{\\alpha \\beta }\\, \\dot{q}^\\alpha \\, \\dot{q}^\\beta - N\\, V, \\quad \\quad \\alpha , \\beta =0,\\ldots n-1,$ the possible existing extrema for the generalized probability $P_{h} = \\int \\!\\!", "\\sqrt{-h}\\, d q ,$ with $h:=\\left(V\\, G_{\\alpha \\beta }\\dot{q}^\\alpha \\dot{q}^\\beta \\right)\\big |_{q=t}$ and $t$ the dynamical independent variable of the system, rest at the classical solutions of Einstein's equations.", "This is the main result of this work.", "In order to make the application of the above theorem more concrete, we analyze the examples of a static, spherically symmetric metric and a spatially homogeneous one in section .", "Quantization of the reduced system, whose $q^i=f^i(q)$ do not acquire any particular values, encompasses all possible geometries spanning the base manifold.", "The generalized probability amplitude obtained herein reveals as its possible extrema the classical solutions of the initial system.", "By this method the reduced system gives weight to all the geometries contained in a certain class, but distinguishes the classical configurations.", "However, it must be noted that different functions $h(q,f,\\acute{f})$ may have their extrema on the same base manifold geometry, since the $f^i(q)$ can be re-parameterized with respect to each other or because a different gauge fixing condition might be taken into consideration.", "In other words, what we call here as state $h[q]$ corresponds to a specific parametrization of the initial configuration and it is thus possible for different states to exhibit their extrema on the same classical geometry.", "Finally we would like to point out that the quantity $\\sqrt{-V G_{\\alpha \\beta }q^{\\prime \\alpha }q^{\\prime \\beta }}$ with $\\alpha ,\\beta $ taking values from 0 to $n-1$ and $G_{\\alpha \\beta }$ being the unscaled mini-supermetric can be seen as a reduced form of the Baierlein-Sharp-Wheeler (BSW) action [26]: $S_{BSW} = \\int \\!\\!", "\\sqrt{g}\\,\\sqrt{R}\\,\\sqrt{k^{ij}k_{ij}-\\mathrm {tr}k^2}\\,d^4 x$ where $g=\\det g_{ij}$ is the determinant of the spatial metric and $k_{ij}=-2\\,N\\,K_{ij}=\\frac{\\partial g_{ij}}{\\partial t} - N_{i;j} - N_{j;i}$ , with $K_{ij}$ being the extrinsic curvature.", "It is known that (REF ) can be used to produce the same dynamics as the Einstein-Hilbert action (for more details see [27], [28]).", "The fact that $\\delta \\int {\\sqrt{-h}} dq=0$ leads to the equations of motion for the system described by (REF ) is in line with the property of the BSW action, even though the BSW action enjoys the reparametrization invariance, while the generalized probability does not.", "It is interesting to note that, in the semiclassical analysis, the wave function is taken to be $\\Psi \\propto e^{{i}S}$ , where $S$ is the action that describes the classical system integrating over all possible metrics [25].", "In the quantum description of the reduced system that we present here, the wave function for a given state becomes $\\Psi \\propto e^{{i}S_{BSW}}$ in a natural way, i.e.", "as the combined solution of the system consisting of the eigenfunction equation (REF ) and the quadratic constraint (REF ) [29]." ], [ "Characterization of the candidates for extrema", "The characterization of the candidates for extrema as either maxima or minima for a given functional is a rather tedious and complicated task that depends on many factors (among them even the range of integration).", "We refer the interested reader to various textbooks on the subject (for example [30], [31]).", "We briefly state some basic facts of the theory; one has to proceed with the second variation of the functional and check whether, on the extremizing configurations, has positive (minimum) or negative (maximum) value.", "In our case, this becomes $\\delta ^2 \\sqrt{-h}= \\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial \\acute{f}^j} \\, \\delta \\acute{f}^i \\, \\delta \\acute{f}^j+2\\, \\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial f^j} \\, \\delta \\acute{f}^i \\, \\delta f^j + \\frac{\\partial ^2 \\sqrt{-h}}{\\partial f^i \\partial f^j} \\, \\delta f^i \\, \\delta f^j .$ Then, by writing the second term as $2\\, \\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial f^j}\\, \\delta \\acute{f}^i \\, \\delta f^j = \\frac{d}{dq} \\left(\\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial f^j}\\, \\delta f^i \\, \\delta f^j\\right) - \\frac{d}{dq} \\left(\\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial f^j}\\right)\\, \\delta f^i \\, \\delta f^j ,$ and using the requirement that the variation $\\delta f^i$ should be zero on the boundary (together with the assumption of a well behaved term $\\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial f^j}$ ), (REF ) becomes $\\delta ^2 P_{h}= \\int _a^b\\!\\!", "\\left[\\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial \\acute{f}^j}\\, \\delta \\acute{f}^i \\, \\delta \\acute{f}^j + \\left(\\frac{\\partial ^2 \\sqrt{-h}}{\\partial f^i \\partial f^j}- \\frac{d}{dq}\\left(\\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial f^j}\\right)\\right)\\, \\delta f^i \\, \\delta f^j\\right]\\, dq.$ In the test for the characterization of the candidates for extrema, the following two symmetric matrices are important $W_{ij} & = \\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial \\acute{f}^j} \\\\A_{ij} & = \\frac{\\partial ^2 \\sqrt{-h}}{\\partial f^i \\partial f^j}- \\frac{d}{dq}\\left(\\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^{(i} \\partial f^{j)}}\\right).$ According to the theory, one has first to secure a matrix $U$ that is a valid solution of the following Riccati equation between matrices $A + \\acute{U} = U\\, W^{-1}\\, U$ in the region of integration $[a,b]$ , i.e.", "it is well behaved in this region.", "Given the existence of such a $U$ , an adequate total derivative term can be added to the right-hand side of (REF ) (on account of the boundary condition), thus making the sign of $\\delta ^2 P_{h}$ dependent on whether the matrix $W$ is positive or negative definite.", "In the first case, the solution of $\\delta P_{h}=0$ is a minimum of the functional, while in the latter it is a maximum.", "There is also the possibility that $W$ is neither positive nor negative definite, in this instance the test is considered inconclusive.", "In our case the coefficients of the matrix $W$ read $ \\nonumber W_{ij} & = \\frac{\\partial ^2 \\sqrt{-h}}{\\partial \\acute{f}^i \\partial \\acute{f}^j} = \\frac{\\partial }{\\partial \\acute{f}^i} \\left(-\\frac{1}{2\\, \\sqrt{-h}}\\frac{\\partial }{\\partial \\acute{f}^j}\\left( \\varepsilon + G_{kl}\\, \\acute{f}^k\\, \\acute{f}^l\\right)\\right) \\\\ \\nonumber & = \\frac{\\partial }{\\partial \\acute{f}^i} \\left(-\\frac{1}{\\sqrt{-h}}\\, G_{kj}\\, \\acute{f}^k\\right) \\\\ \\nonumber & =- \\frac{G_{ik}\\,G_{jl}\\acute{f}^k\\, \\acute{f}^l}{(-h)^{\\frac{3}{2}}}+ \\frac{G_{ij}}{\\sqrt{-h}} \\\\ & = - \\frac{1}{\\sqrt{-h}} \\left(\\frac{1}{-h}G_{ik}\\,G_{jl}\\acute{f}^k\\, \\acute{f}^l+G_{ij}\\right),$ where the $f^i$ 's are to be substituted from the solution of (REF )." ], [ "The Schwarzschild and the Bianchi Type III states", "The adoption of a static, spherically symmetric line element of the form $ds^2 = - a(r)^2 dt^2 + \\left(\\frac{N(r)}{2a(r)}\\right)^2 dr^2 + b(r)^2 (d\\theta ^2+ \\sin ^2\\theta \\,d\\phi ^2 )$ where the radial coordinate $r$ is taken as the evolution parameter of the system, leads to the following mini-superspace Lagrangian in the case of vacuum $L = -\\frac{1}{2\\, N}\\left(16\\, a\\, b\\, \\dot{a}\\, \\dot{b} + 8\\, a^2\\, \\dot{b}^2 \\right)-N$ where now $\\dot{}=\\frac{d}{dr}$ .", "Line element (REF ) is of the form (REF ) with $r$ being the dynamical parameter instead of $t$ .", "By adopting this $r$ -evolution, a valid mini-superpace can be given (a procedure first introduced in [32] and [33] and later used in [15], [16], [8], [9]).", "Note that in (REF ) we already parametrized the “lapse\" function in the line element as $\\frac{N(r)}{2a(r)}$ , so that the ensuing potential would be constant.", "Equivalently, we could set the coefficient of $dr^2$ as $N(r)^2$ and then re-scale the lapse inside the Lagrangian $N\\mapsto \\frac{\\overline{N}}{2a}$ .", "One can easily check that the Euler - Lagrange equations of (REF ) lead to the well known Schwarzschild solution $ds^2 = -c^2\\left(1-\\frac{2M}{b(r)}\\right)dt^2 + \\left(1-\\frac{2M}{b(r)}\\right)^{-1} \\dot{b}(r)^2 dr^2 +b(r)^2 \\left(d\\theta ^2+ \\sin ^2\\theta \\,d\\phi ^2\\right).$ Note that this particular form of the line-element has been acquired without any gauge fixing, hence the existence of an arbitrary function $b(r)$ .", "In order to keep in touch with the classical solution (REF ), we choose to construct the reduced system by choosing the scale factor $b(r)$ as our dynamical independent variable.", "Then, according to $h[q]$ and given that the mini-supermetric induced from (REF ) reads $\\overline{G}_{\\alpha \\beta } =\\begin{pmatrix}0 & -8\\, a\\, b \\\\ \\\\-8\\, a\\, b & -8 \\, a^2\\end{pmatrix},$ we deduce that the $h$ function in this case is $h_S[b] = - 8\\, \\left(2\\, b\\, a(b)\\, a^{\\prime }(b) + a(b)^2 \\right).$ As expected, the classical reduced system does not choose any particular form for the $a(b)$ .", "The Euler-Lagrange equation of $L=\\frac{1}{2N}h_S[b] \\dot{b}^2 -N$ for $b$ is satisfied whenever the corresponding equation for $N$ is satisfied.", "The Euler–Lagrange equation for $N(r)$ reads $b^{\\prime }(r)^2\\left(b a(b) \\right)^{\\prime }=\\frac{1}{4}N(r)^2,$ thus adopting the gauge $N(r)dr=2d\\tau $ , i.e $N(r)\\mapsto 2$ , we have the general solution of (REF ) in parametric form $\\tau &=\\rho f^{\\prime \\prime }(\\rho )-f^{\\prime }(\\rho ) \\nonumber \\\\b &=f^{\\prime \\prime }(\\rho ) \\nonumber \\\\a^2 &=\\frac{1}{f^{\\prime \\prime }(\\rho )}\\left( \\rho ^2 f^{\\prime \\prime }(\\rho )-2\\rho f^{\\prime }(\\rho )+2f(\\rho ) \\right)$ where $f(\\rho )$ is an arbitrary function.", "The above solution has no contact with the solution of the original Lagrangian (REF ), since the original degree of freedom $a(r)$ has transformed into the arbitrary function $a(b)$ .", "Eventually the arbitrariness of the function $a$ is transferred to the arbitrariness of the function $f$ .", "The generalized probability in our case is defined as $P_{h_S}= \\int \\!\\!", "\\sqrt{-h_S[b]}\\, d b$ and the value of $a(b)$ for which it may exhibit an extrema is given by the solution of $\\frac{d}{d b}\\left(\\frac{\\partial \\sqrt{-h}}{\\partial a^{\\prime }}\\right)- \\frac{\\partial \\sqrt{-h}}{\\partial a} = 0 \\Rightarrow (b\\, a^{\\prime \\prime } + 2\\, a^{\\prime })\\, a + b\\, a^2 =0$ The latter can be easily found to be $a_S(b) = c\\, \\sqrt{1-\\frac{2\\, M}{b}},$ which is identical to the classical solution for the scale factor, see line element (REF ) In order to proceed with the characterization of $a_S$ , we need to calculate the matrices $W$ and $A$ ().", "Here, due to the fact that there is only one $f^i(q)$ , that is $a(b)$ , we obtain the functions $W(b) = & \\frac{\\partial ^2 \\sqrt{-h}}{\\partial a^{\\prime } \\partial a^{\\prime }} = - \\frac{2\\, \\sqrt{2}\\, b^2\\, a^2}{\\left(a\\, (a+2\\, b\\, a^{\\prime })\\right)^{\\frac{3}{2}}} \\\\ \\nonumber A(b) = & \\frac{\\partial ^2 \\sqrt{-h}}{\\partial a \\partial a} - \\frac{d}{d b} \\left(\\frac{\\partial ^2 \\sqrt{-h}}{\\partial a \\partial a^{\\prime }}\\right) \\\\= & - \\frac{2\\,\\sqrt{2}\\, b\\, a \\left(b^2\\, a^{\\prime 3}-b^2\\, a\\, a^{\\prime }\\,a^{\\prime \\prime } + a^2\\, (2\\, a^{\\prime } + b\\, a^{\\prime \\prime })\\right)}{\\left(a\\, (a+2\\, b\\, a^{\\prime })\\right)^{\\frac{5}{2}}} .$ These, on the solution $a=a_S$ , assume the following values: $ W\\big |_{a =a_S} (b) = & -\\frac{2\\, \\sqrt{2}\\, b^2\\, (1-\\frac{2\\, M}{b})}{|c|}, \\\\A\\big |_{a =a_S} (b) = & -\\frac{2\\, \\sqrt{2}\\, M^2}{|c|\\, b\\, (b - 2\\, M)}.$ As we also mentioned in the previous section the Riccati equation $ \\nonumber & A\\big |_{a =a_S}(b) + U^{\\prime }(b) = \\frac{1}{W\\big |_{a =a_S} (b)}\\, U(b)^2 \\Rightarrow \\\\ & 2\\,\\sqrt{2}\\, c \\, (b-2\\, M)\\, U^{\\prime }(b) + c^2\\, U(b) -8\\, M^2 =0$ must have a well behaved solution $U(b)$ over the region of integration of the functional $P_{h_S}$ .", "Indeed one can integrate (REF ) to find $U(b) = \\frac{2\\, \\sqrt{2}\\, M}{c}\\, \\tanh \\left(\\frac{1}{2}\\left(\\mathrm {ln}|b-2\\, M|-\\mathrm {ln} b-4\\, \\sqrt{2}\\, c\\, M\\, c_1 \\right)\\right),$ with $c_1$ being the constant of integration.", "We observe that (REF ) is well behaved in $\\mathbb {R}$ , even at $b=0$ and $b=\\pm 2\\, M$ which at the classical level correspond to a curvature and a coordinate singularity respectively.", "Due to the fact that $\\tanh (x)$ is bounded, (REF ) is well behaved even when $b\\rightarrow \\pm \\infty $ .", "Thus, we can continue with the characterization of $a_S(b)$ .", "If we assume that we may, at the quantum level, correlate $b$ with the radial parameter of the classical regime, we can see from (REF ) that the sign of $W\\big |_{a =a_S} (b)$ does not remain constant in the region $(0,\\infty )$ .", "Outside the Schwarzschild horizon $b>2\\, M$ , the generalized probability assumes its maximum value when $a=a_S$ .", "On the contrary, inside the horizon $b<2\\, M$ the situation is exactly the opposite and $P_{h_s}$ takes its least value.", "Consequently, we may somewhat heuristically observe that the measurement outside the horizon is most likely to show that the space-time is characterized by the Schwarzschild metric, if we suppose that one can “measure\" geometry.", "On the other hand, inside the horizon there is no particular geometry that emerges as more probable.", "We can only say that the least probable is the one given by line element (REF ).", "Similar results are obtained in the case of the LRS Bianchi Type III cosmological model with spacetime metric $ds^2 = -\\left( \\frac{N(t)}{2\\, a(t)}\\right)^2 dt^2 + b(t)^2 dx^2 + e^{-2x} b(t)^2 dy^2 + a(t)^2 dz^2$ i.e.", "As a result, the generalized probability (REF ) results in a similar form of its extremum $a(b) = c_2 \\sqrt{1-\\frac{c_1}{b}}$ where $c_1$ is an essential constant.", "When $b>c_1$ , we have the maximum and the time parameter is $t$ , while for $b<c_1$ we obtain the minimum value and $z$ becomes the time parameter with the appropriate redefinitions of the constants.", "In this case, the space fails to be spatially homogeneous any more.", "A rather interesting fact that can be pointed out is that the minisupermetrics of these two examples are also flat; therefore, there exists a transformation of the configuration variables that maps one to the other.", "These two examples profoundly demonstrate the case in which the function $h$ is the same for two spacetimes not belonging to the same geometrical class.", "However, if the two line elements are given, the difference in the solutions can be inferred by their invariant relations $I_{Schw} &= q_2 - \\frac{2^{2/3} \\, 3^{5/3} q_1}{m^{2/3}} -6 \\sqrt{3} q_1^{5/2}, \\\\I_{III} &= q_2 - (6 \\sqrt{3} q_1^{5/2} -3 \\times 6^{2/3} c_1^{2/3} q_1^{7/3})$ where $q_1 $ is the Kretchmann scalar and $q_2 = g^{\\mu \\nu } q_{1,\\mu } q_{1,\\nu }$ for the corresponding geometry.", "$I_{Schw}, \\, I_{III}$ are zero only for the geometry for which they have been calculated, thus distinguishing them.", "By taking a step further besides considering single states, one could try and define a generalized wave function containing all possible geometries within a certain geometrical class (GC) as a sum of all different states $h(q)$ , $\\Psi _{GC} = \\int \\!\\frac{(-h[q])^{\\frac{1}{4}}}{\\sqrt{\\mu [q]}}\\, e^{{i}\\, \\kappa \\int \\!\\!\\sqrt{-h[q]}\\, d q}\\, \\mathcal {D}h(q)\\mathcal {D}\\delta (I(q)),$ where $I(q)$ are the appropriate invariant relations corresponding to a specific base manifold geometry.", "It should be noted that this expression is symbolic, in the sense that it must be considered as a sum over continuous functions not as an actual integration.", "As far as the state $h$ is concerned, it is meant to describe different configurations of the base manifold through the particular form of all $f^i (q)$ .", "Each set of the latter (for a given $h$ ) corresponds to a different geometry of the base manifold, since - in the context of the reduced system - the $f^i (q)$ need not satisfy the original classical equations." ], [ "Discussion", "The quantization of pure gravity has turned out to pose difficult enough obstacles, the most notable being the problem of time [34], [35].", "This problem appears explicitly in the canonical procedure for the quantization and has many manifestations, some of which are the absence of a unique choice of an evolution parameter, resulting in the non existence of a natural Hilbert space etc.", "Usually, there are two approaches of canonical quantization to follow: (i) one first classically restricts the dynamics on the physical phase space, by selecting gauge conditions, thus making the first class constraints second class; then performs the quantization, leading to a Schrödinger type equation.", "(ii) One performs the quantization on the full phase space.", "The latter case leads to the Wheeler-DeWitt equation, the solution of which still remains the holy grail of quantum gravity.", "An important issue to consider is the consistency and preservation of the spacetime covariance of the theory after canonical quantization.", "The problem is somewhat easier to be addressed in the context of the minisuperspace models, where the diffeomorphism invariance of the theory has been shrunk to the reparametrization invariance of the independent variable.", "In [36], an answer is given by imposing on the wave function, apart from the quadratic constraint, the quantum analogues of the spacetime invariant relations; at the classical level, these relations are weakly vanishing quantities and thus their quantum analogues also annihilate the wave function (if correctly factor-ordered).", "In [37], the demand for consistency of the Dirac quantization, after a procedure that takes account of the physical wave function defined on the reduced phase space, results in the Hermiticity of the momentum operator.", "Finally, in [38] the authors reconsider Kuchar's proposal of the milti-fingered time, in order to regain the diffeomorphism invariance of the Hamiltonian formalism through the construction of a diffeomorphism covariant Hamilton-Jacobi equation.", "Upon quantization, it turns out that the Wheeler-DeWitt equation is equivalent to an intrinsic Schrödinger equation that results from a particular choice of an intrinsic evolution parameter.", "The main characteristic of the current work is the exhibition of a general methodology concerning the classical and quantum treatment of spacetimes affording description in terms of a minisuperspace action principle.", "The aim is to include the entire family of geometries contained in a given base manifold's line element, not just the configurations satisfying the classical Einstein equations of motion.", "On the way towards this goal, the invariance under arbitrary reparametrizations of the independent evolution variable emerges as a key instrument in a twofold way: In the introduction, it is clearly demonstrated how the invariance of the action under the above mentioned change of independent dynamical variable generates the freedom to arbitrarily rescale the lapse (density) by any function of the scalar degrees of freedom.", "Subsequently, this leads to the possibility of selecting a lapse parametrization in which the potential term does not depend on any scalar degrees of freedom; thus, a particular minisuperspace metric is selected which is “natural” in the sense that the Euler-Lagrange equations become pure geodesics (not driven) of this metric.", "The desire to describe the family of geometries in a way that is independent of the particular choice of the evolution parameter leads to the expression of the $(n-1)$ scalar degrees of freedom in terms of one, as $f^i (q)$ ().", "Further use of the constraint equation by substitution into the rest of the Euler-Lagrange equations leads to reduction of the system into the $(n-1)$ second order equations (REF ).", "Subsequently, if one considers the $f^i (q)$ 's as arbitrary, albeit given, functions of $q$ , one is able to reduce the initial system (REF ) to a configuration of only two non-dynamical degrees of freedom (the lapse and one more arbitrary scalar degree of freedom).", "By performing the Hamiltonian analysis, we find that the reduced system still remains constrained and in addition it has an integral of motion.", "During the quantization procedure, we impose the quantum version of this conserved quantity together with the Hamiltonian constraint on the wave function, thus obtaining an additional eigenvalue equation.", "The outcome is a solution for the wave function and an expression for the measure in which the quantum operators are Hermitians.", "Therefore this line of thinking, leads us to the following important results: The first, of physical value, is an expression of a generalized probability corresponding to any configuration in the designated class of geometries.", "The second is that this probability has its extrema on the classical solution of the initial Lagrangian.", "Consequently, this generalized probability, interpreted as a regular action, can also describe the ensuing true dynamics of the innitial, non-reduced system; and this is so despite the fact that it has originated from quantizing the gauge part of the initial system.", "We thus find, at the quantum level, a justification of the adjective dynamical atributed to the classical quadratic constraint.", "Further information is obtained by examining the nature of these extrema.", "More specifically, the characterization of the candidates for extrema is possible whenever there exists a matrix $U$ satisfying the Riccati equation (REF ) and whether the extremum is maximum or minimum depends on the positive or negative definite nature of the matrix $W$ .", "A simple demonstration of the method was given in section 7 by some concrete examples.", "N. D. acknowledges financial support by FONDECYT postdoctoral grant no.", "3150016." ] ]
1606.05116
[ [ "Determination of Low Loss in Isotopically Pure Single Crystal $^{28}$Si\n at Low Temperatures and Single Microwave Photon Energy" ], [ "Abstract The low dielectric losses of an isotropically pure single crystal $^{28}$Si sample were determined at a temperature of 20 mK and at powers equivalent to that of a single photon.", "Whispering Gallery Mode (WGM) analysis revealed large Quality Factors of order $2\\times10^6$ (dielectric loss $\\sim 5\\times10^{-7}$) at high powers, degrading to $7\\times10^5$ (dielectric loss $\\sim 1.4\\times10^{-6}$) at single photon energy.", "A very low-loss narrow line width paramagnetic spin flip transition was detected with extreme sensitivity in $^{28}$Si, with very small concentration below $10^{11}$~cm$^{-3}$ (less than 10 parts per trillion) and g-factor of $1.995\\pm0.008$.", "Such determination was only possible due to the low dielectric photonic losses combined with the long lifetime of the spin transition (low magnetic loss), which enhances the magnetic AC susceptibility.", "Such low photonic loss at single photon energy combined with the narrow line width of the spin ensemble, indicate that single crystal $^{28}$Si could be an important crystal for future cavity QED experiments." ], [ "Determination of Low Loss in Isotopically Pure Single Crystal $^{28}$ Si at Low Temperatures and Single Microwave Photon Energy Nikita Kostylev ARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia Maxim Goryachev ARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia Andrey D. Bulanov G.G.", "Devyatykh Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences, 49 Tropinin Str., Nizhny Novgorod 603950, Russia N.I.", "Lobachevski State University, 23 Gagarin prosp., Nizhny Novgorod 603950, Russia Vladimir A. Gavva G.G.", "Devyatykh Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences, 49 Tropinin Str., Nizhny Novgorod 603950, Russia Michael E. Tobar [email protected] ARC Centre of Excellence for Engineered Quantum Systems, School of Physics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia The low dielectric losses of an isotropically pure single crystal $^{28}$ Si sample were determined at a temperature of 20 mK and at powers equivalent to that of a single photon.", "Whispering Gallery Mode (WGM) analysis revealed large Quality Factors of order $2\\times 10^6$ (dielectric loss $\\sim 5\\times 10^{-7}$ ) at high powers, degrading to $7\\times 10^5$ (dielectric loss $\\sim 1.4\\times 10^{-6}$ ) at single photon energy.", "A very low-loss narrow line width paramagnetic spin flip transition was detected with extreme sensitivity in $^{28}$ Si, with very small concentration below $10^{11}$  cm$^{-3}$ (less than 10 parts per trillion) and g-factor of $1.995\\pm 0.008$ .", "Such determination was only possible due to the low dielectric photonic losses combined with the long lifetime of the spin transition (low magnetic loss), which enhances the magnetic AC susceptibility.", "Such low photonic loss at single photon energy combined with the narrow line width of the spin ensemble, indicate that single crystal $^{28}$ Si could be an important crystal for future cavity QED experiments.", "Low loss crystals find a broad range of applications in many different areas of physics and engineering.", "One major purpose is the realization of high finesse and $Q$ -factor photonic and phononic systems allowing resonances with extremely narrow frequency linewidths.", "For example, such utilisation is of fundamental importance for the best stability frequency sources[1], [2], fundamental physics tests [3], [4], gravitational wave detection [5], [6] and opto-mechanical systems [7], [8].", "Another example is quantum electrodynamics involving dielectric crystals where quantum signal coherence, one of the most important parameters, is related to various loss mechanisms and thus on crystal properties[9], [10], [11], [12], [13], [14], [15], for these applications the properties at the energy level of a single photon is of paramount importance.", "In all these fields, special attention has been devoted to Silicon due to its excellent optical and mechanical properties and abundance of associated technologies for growth, treatment, implantation, etc.", "Furthermore, there is a constant need for new approaches and new materials due to emerging applications.", "In particular, a recent development in ion implementation with applications for quantum signal processing[16], [17], [18], [19] induced significant interest in isotopically purified silicon crystals due to its unique properties.", "Among three stable isotopes of Silicon with 28, 29 and 30 nucleons (approximately 0.92%, 0.04% and 0.03% abundance in nature), $^{28}$ Si and $^{30}$ Si have zero nuclear spin.", "This particular feature makes associated crystals act as a virtual vacuum where signals or implemented ions are not coupled to a bath of lattice magnetic Two Level Systems (TLS) leaving the medium inert [20].", "Thus, absence of nuclear spin in the lattice surpasses one of the most important channels of decoherence leaving only phonon dissipation that can be greatly reduced by cooling down the crystals.", "In this work, we investigate low temperature microwave properties of $^{28}$ Si single crystal down to the single photon level and undertake Electron Spin Resonance Spectroscopy.", "Figure: (A) Photo of the 28 ^{28}Si crystal sample.", "(B) Simulated field density for a WGM with of azimuthal mode number m=4m = 4 and ω 0 /2π\\omega _0/2\\pi = 7.01 GHz.Two samples of Silicon were investigated using the Whispering Gallery Mode (WGM) method.", "The first sample is a single crystal of isotopically purified $^{28}$ Si cylinder of $H = 55$  mm and $D=15$  mm, see Fig.", "REF , (A).", "This sample was grown at G.G.", "Devyatykh Institute of Chemistry of High-Purity Substances using the float zone melting method in Argon atmosphere[21].", "The specified concentration of $^{28}$ Si is 99.993%, with concentration of Oxygen and Carbon estimated to be less than $1.1\\times 10^{16}$ cm$^{-3}$ (on the order of a ppm) and Boron and Phosphorous to be less than $1.1\\times 10^{13}$ cm$^{-3}$ (on the order of a ppb).", "The second sample is float-zone high-resistivity single crystal silicon with a natural abundance of its isotopes.", "It is cut in a form of a cylinder with height $H = 6$  mm and diameter $D=15$  mm and crystallographic direction (100).", "The crystal has been mechanically polished to reduce surface resistance but the finish is below optical quality.", "Microwave properties of this sample have already been characterized for a temperature range of 10-400 K [22].", "It has been found that below 35 K the dielectric losses in the sample were decreasing, with Quality Factors reaching $10^{4}$ , suggesting existence of hopping or surface charge carrier conductivities.", "From our measurements, our work has shown good consistency with the losses at 10 K, with the best confined WGMs achieving a similar $Q$ -factor.", "Furthermore the permittivity of the sample was determined to be 11.45 at 10 K [22], from our simulations, both our samples (natural and isotopically pure) were found to be in good agreement with this value at 20 mK.", "Both crystals were thoroughly cleaned in an acid bath and using purified methanol before being enclosed in their respective Oxygen Free Copper Cavities.", "The natural Silicon sample was fixed through its middle point via a Copper and dielectric posts.", "This type of clamping removes dissipation via the conducting surfaces due to the fact that the energy density of a WGM are concentrated on the outer edges of the dielectric sample but not inside.", "The WGMs were excited and detected using two straight antenna probes located on the top of the cavity.", "Figure: Schematic diagram of a simple (A) and Bragg reflector-type (B) WGM cavities used with the 28 ^{28}Si sample.", "All dimensions are given in millimeters.The other sample was mounted on a teflon dielectric.", "Because of the absence of the middle hole, two cavity designs were implemented and are shown in Fig.", "REF .", "The first design (Fig.", "REF , (A)) provided a large contact area between one of the sample surfaces and copper walls to insure a good thermal connection at the expense of adding metallic loss, which degrade the $Q$ -factor.", "However, the long aspect ratio due to the large length of the sample relative to its diameter does reduce the metallic losses somewhat for WGMs compared to a smaller length sample.", "The second design (see Fig.", "REF , (B)) implements Bragg reflectors to decouple modes of quasi-Transverse Electric (TE) polarization from the metallic end walls[23], [24], [25], [26], [27], simultaneously improving the filling factor and reducing the metallic losses at the expense of a poor thermal contact.", "For both these configurations, the WGM waves were excited using loop probes located in the middle of the cavity side walls.", "Both samples were placed one at a time inside a bore of 7 T superconducting magnet and were cooled to 20 mK while in vacuum with a dilution refrigerator.", "The experimental setup was similar to one used for WGM spectroscopy of other materials[28], [29], [30].", "In this setup, the incident signal was attenuated to the level of few photons by a chain of room temperature and cold attenuators, and the output signal was amplified by a series of cold high-electron-mobility transistors (HEMT) and room temperature amplifiers.", "Using the described setups, a number of microwave modes of both crystals were measured in the frequency range 7-25 GHz.", "The detected resonances are higher order modes of different order, with the highest $Q$ -factor modes being the WGMs (confirmed with Finite Element Analysis).", "An example of a WGM is provided in Fig.", "REF , (B).", "Note that the loop electrodes made for the $^{28}$ Si sample may optimally couple to either $H_\\phi $ or $H_z$ , or a combination of fields, allowing coupling to a range of hybrid modes and WGMs of quasi TM and TE polarization respectively.", "Figure: Quality Factor as a function of frequency for all measured electromagnetic modes in the natural silicon sample and the isotopically purified 28 ^{28}Si crystal.", "(A) Blue diamonds show modes in the natural silicon cavity and red squares for the cavity in Fig.", "(A).", "The solid curve shows the upper limit associated with cavity metallic losses.", "(B) Quality Factors measured with a Bragg reflector-type cavity of Fig.", "(B).", "The measurements are for incident power of P in P_{in}=-60 dBm.A large number of microwave modes in both X and K$_\\text{u}$ frequency band allows us to establish microwave properties of the cavities as a function of frequency.", "In particular, the Quality Factor of the microwave modes shown in Fig.", "REF demonstrates photon losses in the material if the metallic losses can be made negligible.", "The measurements in Fig.", "REF , (A) are limited by the cavity wall losses due to the contact of the rod with the lid, with the WGMs exhibiting the highest $Q$ -factors.", "Nevertheless, we come close to measuring the intrinsic material loss for a high-$Q$ mode at 16.4 GHz, at high enough frequency where the coupling to the cavity lid is significantly reduced.", "The results in Fig.", "REF , (B) provide a good indication of the intrinsic losses of the crystal due to the decoupling from the cavity walls via Bragg reflection ($Q$ -factor $\\sim 2 \\times 10^6$ for quasi TE WGMs between 10 to 15 GHz)).", "The Quality Factors in natural Silicon reach $10^4$ independent of frequency, which is in good agreement with earlier experiments by Krupka et.", "al.[22].", "Figure: (A) Dependence of the crystal frequency response on the external magnetic field near the 11.8711.87 and 16.4016.40 GHz modes.", "(B) The detected points of interaction between the magnetic spin ensemble and the WG modes, assuming a spin flip transition that begins at the origin, an estimated Landé g-factor of 1.995±0.0081.995\\pm 0.008 is obtained from a linear fit with 2 sigma error.Information about crystal magnetic ion impurities may be obtained by monitoring the system microwave response while sweeping the external DC magnetic field $B_0$ .", "Since none of the samples were intentionally doped, in this work, we only deal with residual impurities introduced during crystal growth.", "In both cases, $B_0$ was applied along the cylinder axis of symmetry.", "When the energy splitting of residual impurities swept by the external field reaches resonance frequency $f_0$ of a crystal electromagnetic mode, the shape of this mode is altered, as it is coupled to another harmonic oscillator with a close enough frequency.", "One interaction of such kind for the $^{28}$ Si crystal is shown in Fig.", "REF , (A).", "The plot does not display an Avoided Level Crossing between the ion ensemble and the WGM since the cavity photon linewidth $\\delta $ is larger than the coupling, $g$ , between the ion ensemble and the WGMs.", "However, only the highest $Q$ -factor WGMs are sensitive enough to measure the transition, this is when the value of $g$ approaches the linewidth of the WGM.", "In any case, each of such interactions can be mapped as a point onto a two dimensional $B(f)$ plane giving a larger scale picture of magnetic response of residual ions[28].", "Such a plot for the isotopically purified crystal is shown in Fig.", "REF , (B), illustrating the existence of a spin-flip transition with an estimated Landé g-factor $g_\\text{DC}$ of $1.995\\pm 0.008$ .", "Note that no magnetic response have been detected for the natural Silicon due to significantly degraded photon Quality factors (observed experimentally) and potentially larger spin line widths arising because of nuclear spin of the crystal lattice.", "The sensitivity of the method depends on the average ion concentration and spin transition lifetime $T_2$ .", "The concentration can be determined from, $n=\\frac{4\\hbar }{\\omega _0\\mu _0\\xi }\\Big (\\frac{g}{g_{DC}\\beta _B}\\Big )^2$ , where $\\omega _0$ is a cavity mode angular frequency, $\\xi $ is the transverse magnetic filling factor, $\\beta _B$ is the Bohr magnetron.", "Since $\\delta >g$ (but of similar order for sensitive modes), $n$ cannot be accurately determined.", "However, the stated relation can give a upper bound for $n$ of the correct order of magnitude using the cavity half linewidth as the upper limit for the coupling $g$ .", "Given that the filling factor of the mode, $\\xi $ in Fig.", "REF , (B) is usually between 0.2 and 0.9 for the modes in the $^{28}$ Si sample (as modelled using Finite Element analysis), an estimated concentration of $10^{11}$ cm$^{-3}$ (on the order of ten parts per trillion) or less may be determined.", "Interestingly, the $Q$ -factor of the WGM was found to remain constant as a function of the magnetic field throughout the photon-spin interaction.", "In most other crystals in which ESR resonances are broadened to more than a MHz the $Q$ -factor will be significantly degraded.", "Thus the photon half linewidth also provides an upper bound of the spin transition linewidth, which is about 7 KHz.", "The actual spin linewidth is likely to be much lower than this, as very narrow spin linewidths have been measured previously, of order 10 Hz in $^{28}$ Si [20].", "It is this narrow spin linewidth, which enhances the AC susceptibility over the DC susceptibility by $\\sim \\omega _0 T_2$ , which allows such a low concentration to be detected.", "Such narrow spin impurity line width is an important consequence of the absence of nuclear spins in the crystal lattice specific to the isotopically pure Silicon-28.", "Figure: Power dependence of the 16.416.4GHz mode.", "(A) Cavity transmission profile near resonance for selected incident powers, showing significant change in loss with little frequency shift.", "(B) Quality factor of the 16.4 GHz mode in isotopically-purified 28 ^{28}Si, compared to the QQ-factor of a non-purified single crystal silicon capacitor lumped resonator at 5 GHz .As it has been specified above, the measurements were made down to very low levels of incident power (attenuated down to -132 dBm), required for a small number of excitations, with the average number of photons estimated to be on the order of $N=1$ to 2 [32] .", "The results indicate that the isotopically-purified silicon has an advantage in terms of losses in the single-photon regime when compared to natural silicon.", "A $Q$ -factor of $\\approx 7\\times 10^5$ (corresponding to a $Q\\times f_0$ product $\\sim 10^{16}$ ) was observed in our sample (Fig.", "REF , (C)), a factor of $3.5$ times higher than that reported in natural Si at 5 GHz[31].", "In terms of the $Q\\times f_0$ product, the mode in the $^{28}$ Si sample is a factor of 11.5 larger than that reported in natural Si.", "In general, the crystal microwave response shows nontrivial dependence on the incident power (see Fig.", "REF , (A, B) ), where transmission and mode Quality Factor grow with the increasing power.", "The dependence of $Q$ on incident power is different from the typical Kerr nonlinearity observed in many crystals, and is similar to that observed for rutile WGM resonators at 4K[33], where the $Q$ -factor increases with power with no corresponding frequency shift.", "For this case, such behavior was explained by WGM nonlinear losses originating from coupling to low frequency optical phonon modes.", "One can also attempt to explain the enhancement of $Q$ with increasing power by WGM coupling to a spin ensemble of TLS with considerable amount of dissipation.", "According to this model, a TLS ensemble becomes saturated at high incident power making the corresponding dissipation smaller compared to the overall circulating power[34].", "However, this mechanism induced by magnetic TLS has a narrow frequency range associated with the bandwidth of the ensemble.", "Moreover, we have shown in this work that the magnetic losses due to spins are extremely small.", "The current measurements are made at zero external magnetic field, where all existing magnetic impurities are detuned to zero frequency as shown in Fig.", "REF , (B), ruling out the described mechanism in favour of a dielectric loss mechanism.", "In summary, we have investigated the low temperature properties of isotopically purified $^{28}$ Si single crystals.", "The induced WGMs demonstrate higher Quality Factors than those observed in natural Silicon crystals.", "The magnetic field spectroscopy reveals small quantities of magnetic impurities with spin flip transitions with concentrations not exceeding $10^{11}$  cm$^{-3}$ .", "The crystal also demonstrates some power dependence in the form of increasing Quality Factor.", "We are grateful to Jerzy Krupka for supplying us with the natural silicon sample.", "This work was supported by Australian Research Council grant CE110001013." ] ]
1606.05083
[ [ "Radiative neutrino models in light of diphoton signals" ], [ "Abstract Viable explanations of a hinted 750 GeV scalar resonance may be sought within the extensions of the SM Higgs sector aimed at generating neutrino masses at the loop level.", "We confront a compatibility with the 750 GeV diphoton excess for two recent models which do not need to impose ad hoc symmetry to forbid the tree-level masses: a one-loop mass model providing the H(750) candidate within its real triplet scalar representation and a three-loop mass model providing it within its two Higgs doublets.", "Besides accounting for the 750 GeV resonance, we demonstrate that these complementary neutrino-mass scenarios have different testable predictions for the LHC which should show up soon as more data is accumulated during the ongoing 13 TeV run." ], [ "Introduction", "After discovery of the 125 GeV Standard Model (SM) Higgs boson [1], [2], there are alluring hints of a new scalar resonance responsible for the diphoton excess at 750 GeV in the ongoing run of the LHC [3], [4].", "Most of the existing studies which interpret the hinted resonance as an indication of a second Higgs boson, consider it in framework with an additional scalar singlet or with a second scalar doublet.", "In both cases one maintains the value of the electroweak precision parameter $\\rho = 1$ at the tree level; while for the scalar singlet this is obvious this issue has been studied in detail for two Higgs doublet models (2HDM) in [5].", "Thereby it was found that 2HDM cannot accommodate recent diphoton excess without introducing additional massive particles [6], [7], [8], [9], [10].", "If we employ instead the scalar field in a weak triplet representation, it is still possible to keep the $\\rho $ -parameter protected by using both real and complex triplet scalar fields, like in the custodial triplet model known as the Georgi-Machacek model [11].", "It has been introduced as another benchmark model for a diphoton study in [12] and [13].", "We study a possible appearance of the hinted resonance in the context of beyond-SM (BSM) fields which appear in models of radiative neutrino masses.", "Specifically, we confront the capacity to fit the 750 GeV excess of two different radiative neutrino mass scenarios displayed in Table REF : The one-loop neutrino mass model [14] with minimal BSM representations providing the neutral component of a real scalar field $\\Delta $ in the adjoint representation of the $SU(2)_L$ as the 750 GeV resonance candidate.", "The three-loop neutrino mass model [15] with exotic BSM representations where the 750 GeV candidate emerges in the form of the heavy CP-even neutral scalar field in the framework of the 2HDM.", "Table: Neutrino mass models.", "Scalar fields are in (light) yellow and fermion fields in three generations are in (dark) red.", "The fields containing the 750 GeV candidate are in (light grey) green.", "For the one-loop model (left) the SM Higgs doublet manifests itself only via its VEV vv in the neutrino mass diagram.The paper is structured as follows.", "In Sec.", "we briefly review these radiative neutrino mass models and study their implications for the diphoton signal in Sec. .", "We discuss the stability of the scalar potential as well as Landau poles of relevant couplings in Sec.", "and present our conclusions in Sec.", "." ], [ "The one-loop model", "The first mass model [14] in our focus is based on the one-loop diagram displayed on the LHS in Table REF .", "It has an appeal to invoke low non-singlet weak representations and to be free of imposing an additional ad hoc $Z_2$ symmetry to eliminate the tree-level contribution.", "Still, a Dark Matter (DM) stabilizing $Z_2$ symmetry is needed in the proposed attempts to account for the DM in “inert triplet” variants: the one realized with a $Z_2$ odd real triplet in [14], [16], [17] or another with a $Z_2$ odd complex scalar triplet [18].", "However, we will not consider here such cases where the new scalar field doesn't mix with the SM Higgs field.", "Our model may be viewed as a substitute for the original one-loop neutrino-mass model by Zee [19] which, in meantime, has been ruled out by data: a charged scalar singlet $h^+ \\sim (1,2)$ in Zee loop-diagram has been kept, while its second Higgs doublet has been replaced by hypercharge zero triplet scalar field $\\Delta =\\frac{1}{\\sqrt{2}}\\sum _{j}\\sigma _{j}\\Delta ^{j}=\\left( \\begin{array}{ccc}\\frac{1}{\\sqrt{2}} \\Delta ^0 & \\Delta ^+\\\\\\Delta ^- & -\\frac{1}{\\sqrt{2}} \\Delta ^0\\end{array} \\right) \\sim (3,0)\\ ,$ in conjunction with the vector-like lepton $E_{R,L} \\sim (2,-1)$ in three generations.", "Such modification of Zee model may be interesting in light of some findings that possible explanation of 750 GeV resonance requires both scalar and fermion BSM fields.", "The gauge invariant scalar potential of this model reads $&&V(H,\\Delta ,h^+)= -\\mu _H^2 H^\\dag H + \\lambda _1(H^\\dag H)^2 + \\mu _h^2 h^- h^+ + \\lambda _2 (h^- h^+)^2\\nonumber \\\\&&+ \\mu _\\Delta ^2 \\mathrm {Tr}[{\\Delta }^2] + \\lambda _3 (\\mathrm {Tr}[\\Delta ^2])^2 + \\lambda _4 H^\\dag H h^-h^+ + \\lambda _5 H^\\dag H \\mathrm {Tr}[\\Delta ^2] \\nonumber \\\\&&+ \\lambda _6 h^-h^+ \\mathrm {Tr}[\\Delta ^2] + (\\lambda _7 H^\\dag \\Delta \\tilde{H}h^+ + \\mathrm {H.c.}) + \\mu H^\\dag \\Delta H \\ ,$ where the vacuum expectation value (VEV) $v=246$ GeV of the neutral component of the Higgs doublet $H=(\\phi ^+,\\phi ^0)^T$ leads to electroweak symmetry breaking (EWSB).", "Without imposing $Z_2$ symmetry the trilinear $\\mu $ term in (REF ) leads to an induced VEV $\\langle \\Delta ^0\\rangle $ for the neutral triplet component, which is constrained by electroweak measurements to be smaller than a few GeV.", "Neutrino mass: The effective neutrino mass operator is generated via the $\\lambda _7$ coupling in (REF ) and appropriate Yukawa interactions from a gauge invariant Lagrangian $\\nonumber \\mathcal {L}&=& M \\overline{E_L} E_R + y \\overline{E_L} H l_R + g_1 \\overline{(L_L)^c} E_L h^+\\\\&+& g_2 \\overline{L_L} \\Delta E_R + g_3 \\overline{E_L} \\Delta E_R + g_4 \\overline{(L_L)^c} L_L h^+ + \\mathrm {h.c.} \\ .$ Here $y$ and $g_{1,2,3,4}$ are the Yukawa-coupling matrices and for simplicity we drop the flavour indices altogether.", "The resulting neutrino mass reads [14] $\\mathcal {M}_{ij}&=&\\sum _{k=1}^3\\frac{[(g_1)_{ik} (g_2)_{jk} + (g_2)_{ik}(g_1)_{jk}]}{16\\pi ^{2}} \\ \\lambda _7 \\; v^2 \\; M_{E_k} \\\\\\hspace{51.21504pt}&&\\frac{M_{E_k}^{2}m_{h^+}^{2}\\ln {\\frac{M_{E_k}^{2}}{m_{h^+}^{2}}}+M_{E_k}^{2}m_{\\Delta ^+}^{2}\\ln {\\frac{m_{\\Delta ^+}^{2}}{M_{E_k}^{2}}}+m_{h^+}^{2}m_{\\Delta ^+}^{2}\\ln {\\frac{m_{h^+}^{2}}{m_{\\Delta ^+}^{2}}}}{({m_{h^+}^{2}-m_{\\Delta ^+}^{2}})(M_{E_k}^{2}-m_{h^+}^{2})(M_{E_k}^{2}-m_{\\Delta ^+}^{2})} \\nonumber \\; .$ Assuming the mass values in the diphoton-preferred range, as we will use later, $M_E \\sim m_{\\Delta ^+} \\sim m_{h^+} \\sim 400$ GeV, (REF ) leads to $m_\\nu \\sim 0.1$ eV for the couplings $g_{1,2}$ and $\\lambda _7$ of the order of $10^{-4}$ ." ], [ "The three-loop model", "The second mass model [15] in our focus is based on the three-loop diagram displayed on the RHS in Table REF .", "Notably, this model contains 2HDM sector augmented by exotic scalar multiplets needed to close the three-loop mass diagram and motivated by the minimal dark matter (MDM) setup [20]: the complex scalar pentuplet $\\Phi $ and a real scalar field $\\chi $ in the septuplet representation.", "Since $\\Phi $ and $\\chi $ fields do not form gauge invariant couplings with the SM particles, there is again no need for an additional symmetry to eliminate the tree-level neutrino mass contributions.", "This model is ideally suited for producing small neutrino masses with non-suppressed couplings and the multiply-charged components in similar setup have already been claimed to explain the 750 GeV diphoton excess [21], [22].", "The three-loop model at hand is in addition fortuitously scotogenic [15]: a standard discrete $\\tilde{Z}_2$ symmetry imposed to produce a natural flavour conservation in 2HDM results in accidental $Z_2$ odd parity of its BSM sector shown in Table REF .", "On account of it, the lightest among the three generations ($\\alpha =1,2,3$ ) of exotic real fermions $\\Sigma _{\\alpha } \\sim (5,0)$ turns out to be a viable DM candidate.", "Out of four different ways the Higgs doublets are conventionally assigned charges under a $\\tilde{Z}_2$ symmetry [23], we adopt the “lepton-specific\" (Type X or Type IV) 2HDM implemented originally in [24], [25] and shown in Table REF .", "Table: Charge assignment under an automatic Z 2 Z_2 symmetry which is induced by the imposed Z ˜ 2 \\tilde{Z}_2 symmetryin the lepton-specific 2HDM.In terms of physical fields, the two Higgs doublet fields $H_{\\bf {1,2}}\\sim (2,1)$ are written as $H_{\\bf {1}}=\\left(\\begin{array}{c}\\displaystyle G^+\\cos \\beta -H^+\\sin \\beta \\\\\\displaystyle \\frac{1}{\\sqrt{2}}\\left(v_1-h\\sin \\alpha +H\\cos \\alpha +\\mathrm {i}\\left( G\\cos \\beta -A\\sin \\beta \\right)\\right)\\end{array}\\right),$ $H_{\\bf {2}}=\\left(\\begin{array}{c}\\displaystyle G^+\\sin \\beta +H^+\\cos \\beta \\\\\\displaystyle \\frac{1}{\\sqrt{2}}\\left(v_2+h\\cos \\alpha +H\\sin \\alpha +\\mathrm {i}\\left( G\\sin \\beta +A\\cos \\beta \\right)\\right)\\end{array}\\right),$ and their electroweak VEVs define $\\tan \\beta \\equiv v_2 /v_1$ .", "The physical charged scalars are $H^\\pm $ , and, besides the three Goldstone bosons ($G, G^\\pm $ ) eaten by $Z$ and $W^\\pm $ , there is a CP-odd physical neutral scalar $A$ .", "The two CP-even neutral states $h$ and $H$ (mixing with the angle $\\alpha $ ) are proposed to be the physical Higgs fields $h(125)$ and $H(750)$ .", "Conventionally, the VEVs $v_1=v\\cos \\beta $ and $v_2=v\\sin \\beta $ (which are related to the SM VEV $v=$ 246 GeV by $v^2 = v_1^2 + v_2^2$ ) originate from $m_{11}^2$ and $m_{22}^2$ terms through the minimization conditions of the most general CP-conserving 2HDM potential $\\begin{split}V(H_{\\bf {1}},H_{\\bf {2}}) &= m_{11}^2 H_{\\bf {1}}^\\dagger H_{\\bf {1}}+ m_{22}^2 H_{\\bf {2}}^\\dagger H_{\\bf {2}}-[m_{12}^2 H_{\\bf {1}}^\\dagger H_{\\bf {2}}+ \\, \\text{h.c.} ] \\\\& +\\frac{1}{2}\\lambda _1(H_{\\bf {1}}^\\dagger H_{\\bf {1}})^2+\\frac{1}{2}\\lambda _2(H_{\\bf {2}}^\\dagger H_{\\bf {2}})^2\\\\& +\\lambda _3(H_{\\bf {1}}^\\dagger H_{\\bf {1}})(H_{\\bf {2}}^\\dagger H_{\\bf {2}})+\\lambda _4(H_{\\bf {1}}^\\dagger H_{\\bf {2}})(H_{\\bf {2}}^\\dagger H_{\\bf {1}}) \\\\& +\\left\\lbrace \\frac{1}{2}\\lambda _5(H_{\\bf {1}}^\\dagger H_{\\bf {2}})^2+\\big [\\lambda _6(H_{\\bf {1}}^\\dagger H_{\\bf {1}})+\\lambda _7(H_{\\bf {2}}^\\dagger H_{\\bf {2}})\\big ]H_{\\bf {1}}^\\dagger H_{\\bf {2}}+\\, \\text{h.c.}\\right\\rbrace \\,.\\end{split}$ It is possible to trade the five quartic couplings $\\lambda _1$ to $\\lambda _5$ for the four physical Higgs boson masses (as free input parameters) and the mixing parameter $\\sin (\\beta -\\alpha )$ .", "The additional exotic scalar fields $\\Phi \\sim (5,-2)$ and $\\chi \\sim (7,0)$ are totally symmetric tensors $\\Phi _{abcd}$ and $\\chi _{abcdef}$ providing a number of multiply-charged component states $\\begin{matrix} \\Phi _{1111} = \\phi ^+ \\\\ \\Phi _{1112} = \\frac{1}{\\sqrt{4}}\\phi ^{0} \\\\ \\Phi _{1122} = \\frac{1}{\\sqrt{6}}\\phi ^{-} \\\\\\Phi _{1222} = \\frac{1}{\\sqrt{4}}\\phi ^{--} \\\\ \\Phi _{2222} = \\phi ^{---} \\end{matrix} \\ \\ ,&& \\begin{matrix} \\chi _{111111} = \\chi ^{+++} \\\\ \\chi _{211111} =\\frac{1}{\\sqrt{6}}\\chi ^{++} \\\\ \\chi _{221111} = \\frac{1}{\\sqrt{15}}\\chi ^{+}\\\\\\chi _{222111} = \\frac{1}{2\\sqrt{5}}\\chi ^{0} \\\\ \\chi _{222211} =\\frac{1}{\\sqrt{15}}\\chi ^- \\\\ \\chi _{222221} =\\frac{1}{\\sqrt{6}}\\chi ^{--} \\\\ \\chi _{222222} = \\chi ^{---} \\end{matrix} \\ \\ ,$ where we distinguish $\\phi ^{-}$ and $(\\phi ^{+})^*$ for the complex scalar.", "The full scalar potential contains gauge invariant pieces $\\nonumber V(H_{\\bf {1}},H_{\\bf {2}},\\Phi , \\chi ) &=& V(H_{\\bf {1}},H_{\\bf {2}}) + V(\\Phi ) + V(\\chi ) + V_m(\\Phi ,\\chi ) \\\\\\nonumber &+& V_m(H_{\\bf {1}},H_{\\bf {2}},\\Phi ) + V_m(H_{\\bf {1}},H_{\\bf {2}},\\chi ) \\\\&+& V_m(H_{\\bf {1}},H_{\\bf {2}},\\Phi , \\chi ) \\ ,$ where the first term $V(H_{\\bf {1}},H_{\\bf {2}})$ is explicated in (REF ) and we will not need explicit form of the terms $V(\\Phi )$ , $V(\\chi )$ and $V_m(\\Phi ,\\chi )$ in this paper.", "The terms $V_m(H_{\\bf {1}},H_{\\bf {2}},\\Phi )$ and $V_m(H_{\\bf {1}},H_{\\bf {2}},\\chi )$ are important for the diphoton signal and therefore will be introduced later in (REF ) and (REF ).", "Finally, the last term is relevant for neutrino mass and will be discussed next.", "Neutrino mass: This last term represents the $\\tilde{Z}_2$ -symmetric mixing potential $V_m(H_{\\bf {1}},H_{\\bf {2}},\\Phi , \\chi ) &=& \\kappa H_{\\bf {1}} H_{\\bf {2}} \\Phi \\chi + \\mathrm {h.c.} \\ ,$ which provides the couplings needed to close the three-loop neutrino mass diagram.", "After EWSB, the relevant 2HDM piece undergoes the substitution: $\\kappa (H^{+}_{\\bf {1}} H^{0}_{\\bf {2}} + H^{+}_{\\bf {2}} H^{0}_{\\bf {1}}) \\, \\rightarrow \\, v\\, \\kappa \\, \\cos 2\\beta \\, H^{+}\\ .$ so that the resulting quartic vertices together with the appropriate Yukawa couplings ${\\cal L}_Y= - y_{e_i} \\overline{L}_{i L} H_{\\bf {1}} e_{i R}- g_{i \\alpha } \\overline{(e_{i R})^c} {\\Phi }^* \\Sigma _{\\alpha R} + \\mathrm {h.c.} \\ .$ complete the neutrino mass diagram.", "In our lepton-specific 2HDM, only the Higgs doublet $H_{\\bf {1}}$ couples to the SM leptons, so that the SM lepton mass $m_e$ corresponds to the Yukawa strength $y_{e_i}^{SM}=y_{e_i} v_1/v=\\sqrt{2}m_{e_i}/v$ .", "Collecting all the pieces, we finally arrive at the resulting three-loop-generated lepton-number-breaking Majorana neutrino mass matrix $M^\\nu _{ij}$ for active neutrinos, which keeps the form of [25] and reads $M_{ij}^\\nu &=& \\sum _{\\alpha =1}^3C_{ij}^\\alpha \\, F(m_{H^\\pm }^{},m_\\Phi ^{}, m_\\chi , m_{\\Sigma _{\\alpha }}) \\ .$ Here the coefficient $C_{ij}^\\alpha $ comprises the vertex coupling strengths $C_{ij}^\\alpha &=&\\frac{7}{3} \\kappa ^2 \\tan ^2\\beta \\cos ^2 2\\beta \\,y_{e_i}^{\\rm SM} g_i^\\alpha y_{e_j}^{\\rm SM} g_j^\\alpha ,$ and the loop integral is represented by the function $F$ , expressed in terms of the Passarino-Veltman function for one-loop integrals [26].", "In the wide range of the parameter space, the magnitude of $F$ is of the order $10^{2}$ eV so that () reproduces the neutrino masses with the coefficient $C_{ij}^\\alpha \\le 10^{-4}$ that is easily achieved with natural values of $\\mathcal {O}(1)$ for the couplings of the model." ], [ "The one-loop model", "After EWSB, the neutral components of the SM Higgs doublet $\\phi ^0$ and the triplet $\\Delta ^0$ mix with an angle $\\theta _0$ , yielding $h(125)$ and $H(750)$ candidates.", "As discussed above, the VEV for the neutral triplet component is constrained by electroweak measurements to be $\\langle \\Delta ^0\\rangle <\\mathcal {O}$ (1) GeV so that we neglect effects of $\\mathcal {O}(\\langle \\Delta ^0\\rangle /v)$ .", "We also take the quartic coupling $\\lambda _7\\simeq 10^{-4}$ as deduced from the neutrino masses in Sec. 2.1.", "There are also charged components of the triplet $\\Delta ^\\pm $ and the charged scalar $h^+$ which enter into quantum loops relevant for production and decays of the light SM-like Higgs $h(125)$ and its heavy relative $H(750)$ .", "The 125 GeV Higgs: For a sole hypercharge-zero scalar triplet extension of the SM, studied previously in detail in [27], the LHC diphoton signal has been studied in [28].", "For the one-loop model at hand, containing additional charged singlet scalar $h^+$ , we extend for completeness the previous study of the diphoton signal [14] to new mass region of charged BSM scalars in the loop, as motivated by the recently hinted 750 GeV resonance.", "As in [14], the scalar $h(125)\\simeq \\phi ^0\\cos \\theta _0$ is predominantly given by the neutral component of the SM Higgs doublet $\\phi ^0$ , which couples via $c_S v \\phi ^0 S^\\dagger S$ to BSM charged scalars $S(h^+,\\Delta ^+)$ , and they in loop contribute to diphoton decay amplitude.", "Thereby, the $c_S$ couplings are linked to the couplings $\\lambda _4$ and $\\lambda _5$ in (REF ).", "In the conventions and notations from [29], [30], the enhancement factor with respect to the SM decay width is displayed in the left panel of Fig.", "REF .", "The horizontal lines in this figure highlight the current bound $R_{\\gamma \\gamma } = 1.17\\pm 0.27$  [31].", "Since the contribution of the lighter among the two charged scalars $S$ dominates, this figure sets a bound on the respective coupling $c_S$ .", "Only the charged scalars which are sufficiently light may produce significant effects in the LHC diphoton Higgs signals, so that there is poor constraint on $c_S$ couplings of the charged scalars which exceed a half of mass of the $H(750)$ scalar particle.", "Figure: Enhancement factor R γγ R_{\\gamma \\gamma } for the h→γγh \\rightarrow \\gamma \\gamma decay width in dependence on scalar coupling c S c_S and themass m S m_S of the lighter charged scalar (left).", "Region of parameter space whereone-loop model explains 750 GeV diphoton resonance (light/green) allowed (dark/grey) by theLHC 8 TeV constraints (right).The 750 GeV scalar: Here we attempt to fit the heavy state $H\\simeq \\Delta ^0\\cos \\theta _0 $ , which is predominantly $\\Delta ^0$ in this model, to the hinted $H(750)$ scalar particle.", "Let us first discuss the productions mechanisms for $H(750)$ .", "For $\\langle \\Delta ^0\\rangle =0$ there is no tree-level coupling of $H$ to the SM fermions and vector bosonsIn general $g_{Hff}$ and $g_{HVV}$ are $\\sim \\sin \\theta _0\\sim \\frac{\\langle \\Delta ^0\\rangle }{v}$ which is small.", "However, if $2M_{\\Delta ^+}^2=M^2_{H(750)}+M^2_{h(125)}$ the mixing can become sizeable [27].", "We assume this does not happen here.", "and therefore the gluon fusion production is negligible.", "We are thus led to consider the EW vector boson fusion (VBF) mechanisms.", "For resonance much heavier than electroweak scale, photon fusion production mechanism dominates and we neglect the contributions from fusion of weak bosons (see discussion in [32]).", "The diphoton signal strength at $\\sqrt{s}=13$ TeV from the photon fusion is given by [12]: $\\sigma _{\\gamma \\gamma }\\equiv \\sigma (pp \\rightarrow H\\rightarrow \\gamma \\gamma )=10.8\\, {\\rm pb}\\times \\frac{\\Gamma _H}{45 \\, {\\rm GeV}}\\times {\\rm Br}(H\\rightarrow \\gamma \\gamma )^2 \\;,$ where we account for the photoproduction that includes both elastic and inelastic contributions [33].", "To estimate the contributions of charged scalars to the one-loop generated $H\\gamma \\gamma $ coupling, we use Lagrangian (REF ).", "Here we notice that the leading trilinear couplings $\\lambda _{H \\Delta ^+ \\Delta ^-}\\sim \\lambda _3 \\langle \\Delta ^0\\rangle \\cos \\theta _0$ and $\\lambda _{H h^+ h^-}\\sim \\lambda _6 \\langle \\Delta ^0\\rangle \\cos \\theta _0$ relevant for the charged scalar loop vanish in the limit $\\langle \\Delta _0\\rangle =0$ and the remaining quartic couplings are negligible.", "We therefore need to consider the charged fermion loops and the leading contribution from Yukawa couplings in (REF ) is represented by $g_3 \\bar{E}_L \\Delta E_R+h.c.$ term.", "The vector-like fermion loop-generated couplings of 750 GeV candidate to different channels with SM gauge bosons for the degenerate coupling $\\lambda $ read (e.g.", "[34]): $g_{H\\gamma \\gamma } &= \\lambda \\alpha \\sum _F \\left\\lbrace Q_F^2 \\right\\rbrace \\frac{S_{1/2}(\\tau _F)}{m_F} \\;, \\nonumber \\\\g_{HZ\\gamma } &= \\lambda \\alpha \\sum _F \\left\\lbrace \\sqrt{2} Q_F\\frac{(T_{3F}-s_W^2Q_F)}{s_W c_W} \\right\\rbrace \\ \\frac{S_{1/2}(\\tau _F)}{m_F}\\;, \\nonumber \\\\g_{HZZ} &= \\lambda \\alpha \\sum _F \\left\\lbrace \\frac{(T_{3F}-s_W^2 Q_F)^2}{s_W^2 c_W^2} \\right\\rbrace \\frac{S_{1/2}(\\tau _F)}{m_F} \\;, \\nonumber \\\\g_{HWW} &= \\lambda \\alpha \\sum _F \\left\\lbrace \\sqrt{2}\\frac{(T_F-T_{3F})(T_F+T_{3F}+1)}{2 s_W^2} \\right\\rbrace \\frac{S_{1/2}(\\tau _F)}{m_F} \\;.$ Here, $T_F$ is the weak isospin of the loop-fermion $F$ , the triangle loop function is given by $S_{1/2}(\\tau _F)= 2\\tau _F (1+ (1-\\tau _F )\\arcsin ^2(1/\\sqrt{\\tau _F}))$ , and the respective variable is $\\tau _F=4 m_F^2/M_H^2$ .", "The couplings include symmetrization factors for identical particles in the final state, and are normalized so that, neglecting masses of the $W$ and $Z$ bosons give: $\\Gamma (H \\rightarrow VV)=\\frac{M_H}{64\\pi ^3}\\bigg |\\frac{M_H\\, g_{HVV}}{2}\\bigg |^2 \\ .$ For degenerate loop masses, the couplings can be compactly expressed in terms of quadratic Dynkin indices $I_1$ and $I_2$ of the loop-fermion SM group representations: $g_{H\\gamma \\gamma } = \\lambda \\alpha (I_1+I_2) \\frac{S_{1/2}(\\tau _F)}{m_F} \\,, \\qquad g_{HZ\\gamma } = \\sqrt{2} \\lambda \\left(\\frac{c_W}{s_W}I_2 - \\frac{s_W}{c_W}I_1\\right)\\frac{S_{1/2}(\\tau _F)}{m_F}\\,, \\nonumber \\\\g_{HZZ} = \\lambda \\alpha \\left(\\frac{c_{W}^2}{s_{W}^2}I_2 + \\frac{s_{W}^2}{c_{W}^2}I_1\\right)\\frac{S_{1/2}(\\tau _F)}{m_F} \\,, \\quad g_{HWW} = \\sqrt{2} \\lambda \\alpha \\frac{I_2}{s_W^2} \\frac{S_{1/2}(\\tau _F)}{m_F} \\,.$ For the vector-like fermion $E_{L,R}$ at hand with multiplicity $N_E=3$ , we have $\\lambda =g_3 \\cos \\theta _0 N_E$ , $I_1=1/2$ , $I_2=1/2$ .", "For the resulting ratio of the decay widths $R_{VV} \\equiv \\frac{\\Gamma (H\\rightarrow VV)}{\\Gamma (H\\rightarrow \\gamma \\gamma )},$ we obtain $R_{WW} \\approx 9.1 \\,, \\quad R_{ZZ} \\approx 3.2 \\,, \\quad R_{Z\\gamma } \\approx 0.8 \\,.$ This results in a branching ratio $Br(H\\rightarrow \\gamma \\gamma ) \\approx 7\\,\\% $ .", "Comparing to the diphoton signal strength (REF ), one can explain the diphoton resonance with cross-section of 3-9 fb by using narrow width of the resonance $\\Gamma _H \\sim 2.5-7.5\\,{\\rm GeV}$ .", "In this narrow width scenario, leading to $\\Gamma (H\\rightarrow \\gamma \\gamma ) = 0.18-0.53\\,{\\rm GeV}$ , we can now investigate the influence of the constraints coming from the searches for resonances decaying to gauge boson pairs at the LHC 8 TeV run.", "Constraints on the cross sections $\\sigma _{VV}^{8\\,{\\rm TeV}}\\equiv \\sigma (pp \\rightarrow H \\rightarrow VV)$ are [35], [36], [37], [38], [39], [40] $\\sigma _{\\rm WW}^{8 \\rm TeV}< 40 \\,{\\rm fb}\\,, \\quad \\sigma _{\\rm ZZ}^{8 \\rm TeV}< 12 \\,{\\rm fb}\\,, \\quad \\sigma _{\\rm Z\\gamma }^{8 \\rm TeV}< 11 \\,{\\rm fb}\\,, \\quad \\sigma _{\\rm \\gamma \\gamma }^{8 \\rm TeV}< 1.5 \\,{\\rm fb} \\,.$ To make a comparison between 8 TeV data (always explicitly indicated) and 13 TeV data, we need the value for the gain ratio $r_{\\gamma \\gamma }$ of the photon fusion production cross-sections at 13 TeV and at 8 TeV, so that $\\sigma _{VV}^{8\\,{\\rm TeV}} = \\frac{\\sigma _{\\gamma \\gamma }}{r_{\\gamma \\gamma }} R_{VV} \\,.$ This gain ratio is often taken to be $r_{\\gamma \\gamma }\\approx 2$ [41], which would create strong tension with non-observation of the diphoton resonance in the 8 TeV LHC data.", "However, more elaborate analyses [32], [42], taking into account also elastic emission of the photon as well as finite proton size effects lead to increased ratios up to 3.9, alleviating this tension.", "We take average value of $r\\approx 3$ and obtain for the $\\sigma _{\\gamma \\gamma } = 3-9\\,{\\rm fb}$ range $\\sigma _{\\rm WW}^{8 \\rm TeV} = 9-27\\,{\\rm fb}\\,, \\;\\sigma _{\\rm ZZ}^{8 \\rm TeV} = 3-10 \\,{\\rm fb}\\,, \\;\\sigma _{\\rm Z\\gamma }^{8 \\rm TeV} = 0.8-2.4 \\,{\\rm fb}\\,, \\;\\sigma _{\\rm \\gamma \\gamma }^{8 \\rm TeV} = 1-3 \\,{\\rm fb}\\,.$ We see that the LHC 8 TeV run constraint on $\\sigma _{\\rm \\gamma \\gamma }^{8 \\rm TeV}$ is violated for parameters corresponding to larger values of $\\sigma _{\\gamma \\gamma }$ , and that $\\sigma _{\\gamma \\gamma } \\sim 3-4.5\\,{\\rm fb}$ is preferred.", "Even in this case, one expects that additional gauge boson pairs from hinted 750 GeV resonance should show up soon as more data are gathered in the LHC 13 TeV run.", "Results above are summarized in Fig.REF (right), showing that, most importantly, for the dominant portion of the parameter space this model requires either non-perturbative value of the coupling $g_3 > 4\\pi $ or larger multiplicities $N_E>3$ .", "For $N_E=3$ , the value $g_3\\approx 4\\pi $ is achieved only for $m_E\\approx 375$ GeV.", "We might improve the capacity of our model to account for a diphoton excess by introducing appropriate coloured degrees of freedom [8].", "Numerous models employed a vector-like singlet quark to enhance the production cross section.", "In the present case it amounts to extending the radiative model [14] to the quark-lepton symmetric version containing the vector-like top-partner.", "Comparing to relatively weak bounds for charged and neutral leptons, typically around 100 GeV [43], the corresponding limits for new heavy charge-2/3 quarks are 720-920 GeV [44] and 715-950 GeV [45].", "Instead of trying to reproduce the diphoton excess with beyond SM fermions we can try to employ higher electroweak scalar multiplets containing a plethora of charged states.", "Such scenario is offered in a recent three-loop neutrino mass model [15], which we consider in the next section." ], [ "The three-loop model", "In this model the hinted $H(750)$ scalar particle is the heavy CP-even neutral scalar emerging from the 2HDM.", "The $H(750)$ state does not couple to exotic quintuplet fermion $\\Sigma $ in gauge invariant way.", "We therefore consider the contributions to diphoton signal from the exotic charged scalar particles contained in fields $\\Phi $ and $\\chi $ defined in (REF ).", "Let us start with quartic vertices which generate triangle loops with exotic charged scalars for diphoton decays.", "These couplings can be read from the scalar potentials contained in (REF ): $V_m(H_{\\bf {1}},H_{\\bf {2}},\\chi ) &\\supset & (\\tau _1H_{\\bf {1}}^\\dagger H_{\\bf {1}} + \\tau _2 H_{\\bf {2}}^\\dagger H_{\\bf {2}}) \\chi ^\\dagger \\chi \\ ,$ and $ V_m(H_{\\bf {1}},H_{\\bf {2}},\\Phi ) \\supset (\\sigma _1H_{\\bf {1}}^\\dagger H_{\\bf {1}}+\\sigma _2H_{\\bf {2}}^\\dagger H_{\\bf {2}}) \\Phi ^\\dagger \\Phi + (\\sigma ^{\\prime }_1H_{\\bf {1}}^* H_{\\bf {1}}+ \\sigma ^{\\prime }_2H_{\\bf {2}}^* H_{\\bf {2}}) \\Phi ^* \\Phi .", "$ We start with (REF ) where the trilinear couplings strengths $h(125)\\chi ^\\dagger \\chi $ and $H(750)\\chi ^\\dagger \\chi $ are extracted after using the VEVs $v_1=v\\ \\mathrm {cos}\\beta $ and $ v_2=v \\ \\mathrm {sin}\\beta $ in one of the doublets.", "This substitution leads to the universal coupling for all charged components of real scalar septuplet $\\chi $ to $h(125)$ and $H(750)$ $&V_{\\chi }&= (\\tau _1 H^{0}_{\\bf {1}} H^{0}_{\\bf {1}} + \\tau _2 H^{0}_{\\bf {2}} H^{0}_{\\bf {2}})\\chi ^\\dagger \\chi = \\\\\\nonumber &=&v\\chi ^\\dagger \\chi \\bigg [H (\\tau _1 \\mathrm {cos}\\alpha \\ \\mathrm {cos}\\beta + \\tau _2 \\mathrm {sin}\\alpha \\ \\mathrm {sin}\\beta )+h (- \\tau _1 \\mathrm {sin}\\alpha \\ \\mathrm {cos}\\beta + \\tau _2 \\mathrm {cos}\\alpha \\ \\mathrm {sin}\\beta ) \\bigg ]\\ .$ By working in the following in the “alignment limit\" of the 2HDM [22] $\\mathrm {tan}\\beta = 1 \\ \\ \\ , \\ \\ \\mathrm {sin}(\\beta -\\alpha ) = 1 \\ ,$ and assuming that couplings satisfy the relation $\\tau _1 = -\\tau _2\\equiv \\tau $ will lead us to $V_{\\chi }= \\ v \\ \\tau \\ \\big [\\mathrm {cos}(\\beta +\\alpha ) \\ H -\\mathrm {sin}(\\beta +\\alpha ) \\ h\\big ]\\ \\chi ^\\dagger \\chi =v \\ \\tau \\ H\\chi ^\\dagger \\chi .$ This alignment limit identifies the light state as SM-like $h(125)$ , such that its diphoton decay acquires no contribution from (REF ).", "Explicitly, the couplings of the charged components of the septuplet to $H(750)$ are : $V_\\chi = \\tau \\ v \\ (\\chi ^{+} \\chi ^{-} + \\chi ^{++} \\chi ^{--}+ \\chi ^{+++} \\chi ^{---}) \\ H \\ .$ The septuplet scalar components are degenerate at the tree-level $m^2_\\chi &=& \\mu _\\chi ^2 + \\frac{\\tau }{2} v^2(\\mathrm {cos}^2\\beta - \\mathrm {sin}^2\\beta )\\ ,$ where the EWSB correction vanishes for $\\tan \\beta =1$ .", "Similarly, for the quintuplet $\\Phi $ , we impose equivalent conditions on the couplings in (REF ), $\\sigma _1 = - \\sigma _2\\equiv \\sigma \\ \\ \\ , \\ \\ \\sigma ^{\\prime }_1 = - \\sigma ^{\\prime }_2\\equiv \\sigma ^{\\prime } \\ ,$ so that the trilinear couplings of $h(125)$ to the charged components of the quintuplet vanish.", "The $H(750)$ couplings to these charged components of the quintuplet, relevant for the $H \\rightarrow \\gamma \\gamma $ decay, are $ V_\\Phi =v H (c_{\\Phi ^{+}} \\Phi ^{+*} \\Phi ^{+} + c_{\\Phi ^{-}} \\Phi ^{-*} \\Phi ^{-}+ c_{\\Phi ^{--}} \\Phi ^{--*} \\Phi ^{--} + c_{\\Phi ^{---}} \\Phi ^{---*} \\Phi ^{---}) \\,$ where the newly introduced couplings simplify according to (REF ) as $c_{\\Phi ^{+}}= \\sigma , \\;c_{\\Phi ^{0}}= \\sigma + \\frac{\\sigma ^{\\prime }}{4} ,\\;c_{\\Phi ^{-}}= \\sigma + \\frac{\\sigma ^{\\prime }}{2} ,\\;c_{\\Phi ^{--}}= \\sigma + \\frac{3\\sigma ^{\\prime }}{4} ,\\;c_{\\Phi ^{---}}= \\sigma + \\sigma ^{\\prime } .$ In contrast to septuplet case, the EWSB contributions to the mass of different components of the complex quintuplet $\\Phi $ are not the same and are given as $m^2_{\\Phi ^{(Q)}} = \\mu _\\Phi ^2 + \\frac{1}{2} v^2(\\mathrm {cos}^2\\beta - \\mathrm {sin}^2\\beta )c_{\\Phi ^{(Q)}} \\ .$ Figure: Cross section for pp→H(750)→γγp p \\rightarrow H(750) \\rightarrow \\gamma \\gamma (dashed lines)in the three-loop neutrino mass model for ranges of values forcoupling (left) and mass parameters (right).", "Grey area is excluded by LHC search for pp→H→γγp p \\rightarrow H \\rightarrow \\gamma \\gamma at 8 TeV.Again for $\\tan \\beta =1$ the EWSB contributions vanish.", "In the three-loop model, the diphoton excess may be explained by the gluon-fusion production process of $H$ and $A$ .", "In the lepton-specific 2HDM at hand, only $H_2$ couples to the SM quarks and the relevant couplings of $H(750)$ in the alignment limit are given by ($V=W^\\pm , Z)$ [8]: $\\frac{g_{Htt}}{g_{Htt}^{SM}} &=& \\cos (\\beta -\\alpha )-\\frac{\\sin (\\beta -\\alpha )}{\\tan \\beta }=-1 \\qquad \\quad \\frac{g_{Att}}{g_{Htt}^{SM}} = \\frac{1}{\\tan \\beta } =1 \\nonumber \\\\g_{HVV}&=& 2 \\cos (\\beta -\\alpha )\\frac{m_V^2}{v}=0 \\ .$ The loop of the quintuplet and septuplet charged scalar states contributes only to the decay of the CP-even $H$ boson, so that the decay rate of $A$ into diphoton is not enhanced by these charged scalar loops.", "The dominant coupling of $A$ is to the top quark and for $A t\\bar{t}$ taking the SM $Ht\\bar{t}$ value, this coupling mediates the $\\sigma (pp \\rightarrow A\\rightarrow \\gamma \\gamma )\\sim 0.01$ fb which is about 1000 times smaller than required to explain the diphoton excess.", "For this reason we do not consider the contribution of the $A$ state to the diphoton signal further.", "For the same reason we neglect the decay of $H$ through the top-quark loop in the estimate of $\\sigma (pp \\rightarrow H\\rightarrow \\gamma \\gamma )$ and keep only the decay through the new charged states.", "In the scenario where $H$ is produced dominantly through gluon-gluon fusion, diphoton cross section is $\\sigma _{\\gamma \\gamma } = \\sigma _{ggF} Br(H\\rightarrow \\gamma \\gamma )\\;,$ where cross section for $pp\\rightarrow gg X \\rightarrow H X$ is $\\sigma _{ggF} = 737\\,{\\rm fb}$ at $\\sqrt{s} = 13\\,{\\rm TeV}$ , and $\\sigma _{ggF}^{8\\,{\\rm TeV}} =157\\,{\\rm fb}$ at $\\sqrt{s} = 8\\,{\\rm TeV}$ [46].", "Figure: The total decay width of H(750)H(750) particlein the three-loop neutrino mass model for the generic choice of the parameters.The decay width of the $H(750)$ to the $\\bar{t}t$ pair is: $\\Gamma (H\\rightarrow t\\bar{t})= N_c \\frac{\\alpha M_H}{8 \\sin ^2\\theta _W} \\frac{m_t^2}{m_W^2} \\bigg (1-\\frac{4 m_t^2}{m_H^2}\\bigg )^{3/2} \\approx 30 \\ \\rm {GeV}$ which is roughly what is observed by ATLAS [3].", "We therefore take the masses of the new charged scalar states to be $\\mu _{\\chi ,\\Phi }>$ 375 GeV as otherwise the decay channel of $H(750)$ to these states opens up and the resonance quickly becomes very wide.", "Additional subleading contributions to the $H(750)$ width are provided by the decays into SM vectors.", "In the alignment limit, the tree-level couplings $H(750) VV$ , from (REF ), are absent so that these decay modes are generated only at one-loop level.", "Again, it is convenient to introduce the effective couplings $g_{HVV}$ of $H(750)$ to the SM gauge bosons.", "They can be obtained from those in (REF ) with substitutions of the corresponding terms in curly braces $\\lambda \\sum _F \\left\\lbrace \\cdots \\right\\rbrace \\frac{A_{1/2}(\\tau _F)}{m_F} \\longrightarrow \\tau \\sum _S \\left\\lbrace \\cdots \\right\\rbrace \\frac{v A_{0}(\\tau _S)}{2 m_{S}^2} \\;,$ for the real septuplet contribution, and $\\lambda \\sum _F\\left\\lbrace \\cdots \\right\\rbrace \\frac{A_{1/2}(\\tau _F)}{m_F} \\longrightarrow \\sum _S \\left(\\sigma + \\sigma ^{\\prime }\\Big (\\frac{2-T_{3S}}{4}\\Big )\\right) \\left\\lbrace \\cdots \\right\\rbrace \\frac{v A_{0}(\\tau _S)}{m_{S}^2} \\;,$ for the complex quintuplet.", "Here the factor $(2-T_{3S})/4$ accounts for the non-universality of coupling to $H$ (REF ), and should be changed to $(3-T_{3S})/8$ in the sole case of $g_{HWW}$ .", "These constants are normalized so that, neglecting masses of the $W$ and $Z$ bosons, $\\Gamma (H\\rightarrow VV)=\\frac{M_H}{256\\pi ^3}\\bigg |\\frac{M_H\\, g_{HVV}}{2}\\bigg |^2 \\ .", "$ The variable $\\tau _S\\equiv 4 m_S^2/m_H^2$ and the loop function is given by $A_0(\\tau _S)\\equiv -\\tau _S (1-\\tau _S \\arcsin ^2 (1/\\sqrt{\\tau _S}))$ .", "For the degenerate couplings $\\tau =\\sigma =\\sigma ^{\\prime }$ , this leads to the ratios of diboson to diphoton decay widths (REF ) $R_{WW} \\approx 17.8 \\,, \\quad R_{ZZ} \\approx 4.9 \\,, \\quad R_{Z\\gamma } \\approx 3.1 \\, .$ The domination of the $WW$ channel above can be understood as the quintuplet contributes to both $H\\rightarrow W^+ W^-$ and $ZZ$ channels while septuplet, as a real multiplet, contributes only to $H\\rightarrow W^+ W^-$ .", "The LHC 8 TeV run data constraints (REF ) are shown by grey area in Fig.", "REF where, like in the one-loop model case, $\\gamma \\gamma $ channel provides the most stringent boundWe have also checked the 8 TeV data constraints from the remaining channels such as $t\\bar{t}$ and di-jets..", "The total width of $H(750)$ for the generic choice of the parameters is shown in Fig.", "REF .", "It is dominated by the $\\bar{t}t$ channel, so that even in the extreme case when $\\tau =\\sigma =\\sigma ^{\\prime }=8$ , $m_\\chi = m_\\Phi = 375\\,{\\rm GeV}$ , the branching ratio for diphoton channel is only $Br(H\\rightarrow \\gamma \\gamma ) = 0.013 \\,.$ Intriguingly, in the range of parameter space where the model can accommodate diphoton cross-section, it also robustly predicts the large total width of 30-50 GeV.", "In Fig.", "REF we show the diphoton cross section as a function of the parameters of the model compared to combined range of 3-9 fb for ATLAS and CMS diphoton anomaly.", "Figure: Cross section for pp→H(750)→γγp p \\rightarrow H(750) \\rightarrow \\gamma \\gamma in the three-loop neutrino mass model as a function of model parameters comparedto combined range of 3-9 fb for ATLAS and CMS diphoton anomaly (green)." ], [ "Vacuum stability and perturbativity", "Minimal scenarios relying on extra singlet scalars and vector-like BSM fermions correspond to the particle content used in widely studied class of “simplified models for the Higgs physics\" (e.g.", "[47] and Refs.", "[21–29] therein).", "By employing here a scalar field in the adjoint representation in the one-loop neutrino-mass scenario, we can only achieve the required diphoton signal strength for non-perturbative values of the couplings [49] or for many copies of vector-like fermions.", "A summary of the detailed outcome of this model is presented in the first row in Table REF .", "We can contrast it to a recent claim [48] that already one family of vector-like quarks and leptons with SM charges may be enough to explain the 750 GeV diphoton excess.", "Table: Comparison between the neutrino mass models.", "In the three-loop model the branching ratios are calculated for the benchmark point in () leading to the total width Γ 750 ≈\\Gamma _{750}\\approx 45 GeV.In the three-loop neutrino-mass scenario considered here, the charged components of exotic multicomponent scalar fields in a loop contribute to the diphoton decay of the neutral scalar in the 2HDM context, as presented in the second row in Table REF .", "We can contrast this to a recent three-loop radiative neutrino model with a local hidden U(1) symmetry [50] with another set of multiply charged particles introduced to explain the 750 GeV diphoton excess.", "Figure: Scale where the weak isospin coupling Landau pole appears in the three-loop neutrinomass model in dependence of masses of new particles.The three-loop model at hand is under a well known threat that invoking large multiplets [20] leads to Landau poles (LP) considerably below the Planck scale [51], potentially sensitive to two-loop RGE [52] effects.", "For the $SU(2)_L$ gauge coupling $g_2$ , this threat has been addressed in [53] for the particle content of two scotogenic three-loop neutrino mass models [15], [54] aiming at accidental DM protecting $Z_2$ symmetry.", "Thereby the three-loop model at hand [15] is less affected by this threat, and its exposure to additional scrutiny presented in Fig.", "REF shows that the LP appears around $10^6$ GeV.", "As for the quartic couplings, the large values of the “mixed\" scalar couplings $\\tau _{1,2}$ and $\\sigma _{1,2}^{(\\prime )}$ required to explain the di-photon excess and negative values for some of them from (REF ) and (REF ), put the stability of the scalar potential and perturbative control over the model in danger.", "Here, we highlight the possible ways out of these difficulties.", "First of all, we may depart from the limit of degenerate couplings, $\\tau =\\sigma =\\sigma ^\\prime $ , chosen for simplicity of the presentation in the previous section on the di-photon signal.", "In particular, we may choose initial value $\\tau =0$ or $\\sigma =\\sigma ^\\prime =0$ at the particle threshold to turn off contributions to the di-photon signal from the septuplet $\\chi $ or quintuplet $\\Phi $ , respectivelyOf course, this will require even bigger contribution of the remaining “mixed\" quartic to the di-photon signal to compensate for the absence of the other multiplet.. Related to this choice, we now discuss the different remedies that can be envisioned in the septuplet $\\chi $ and the quintuplet $\\Phi $ quartic sectors by activating them one at a time.", "In the quartic sectors at hand, there are three additional quartic self-couplings of the $\\Phi ^4$ -type and two additional quartic self-couplings of the $\\chi ^4$ -type [55] which we are still free to choose.", "There are additional quartics of the $\\chi ^2 \\Phi ^2$ -type which we choose to be zero in order to decouple the septuplet and the quintuplet quartic sectors.", "Now, the stability of the potential will be endangered only due to those active “mixed\" quartics which are negative by the virtue of (REF ) or (REF ), which may lead to an unbounded potential.", "Such quartics have to be balanced in the stability condition by appropriately chosen positive values of the corresponding quartic self-couplings (the stability condition for the septuplet sector has been explicated in [56]).", "For the other inactive “mixed\" quartics we may choose the “self\" quartics to be zero at the threshold as well.", "As for the perturbative control of the model, it was shown in [55] that for the inactive sector the LP will appear at: $\\Lambda _{\\Phi }\\sim 10^9 \\bigg (\\frac{m_\\Phi }{100 \\ \\hbox{GeV}} \\bigg )^{1.28}\\,{\\rm GeV}\\;, \\qquad \\Lambda _{\\chi }\\sim 10^6 \\bigg (\\frac{m_\\chi }{100\\ \\hbox{GeV}}\\bigg )^{1.13} \\,{\\rm GeV}\\;,$ for the quintuplet and the septuplet sectors, respectively.", "These values are not lower than $10^6$ GeV LP of the mentioned $SU(2)_L$ gauge coupling, so that we have a control over the inactive sector.", "As for the active scalar, we need to consider the possible Yukawa couplings of this scalar which provide a negative contribution to the one-loop beta function of the quartic self-couplings and may help to push the LP up.", "Unfortunately, for symmetry reasons, for the septuplet $\\chi $ the obvious $\\chi \\Sigma \\Sigma $ choice for the Yukawa term vanishes.", "Following [56], one may introduce the additional $SU(2)_L$ -triplet fermion $\\zeta =(3,0)$ to have a Yukawa coupling $\\chi \\Sigma \\zeta $ which may be fine-tuned to delay the appearance of the LP.", "For the quintuplet $\\Phi $ , the needed Yukawa coupling $g_{i\\alpha }$ already exists in our model in (REF ) and can be fine-tuned similarly.", "Finally, the dominant contribution to the 1-loop beta functions of the “mixed\" quartics $\\tau _{1,2}$ and $\\sigma _{1,2}$ is given in [55]: $\\beta _{x} \\sim 4 x^2 - \\frac{153}{2} x g_2^2 + 36 g_2^4 \\;, \\qquad \\beta _y \\sim 4 y^2 - \\frac{81}{2} y g_2^2 + 18 g_2^4 \\;.$ Here, these couplings are denoted by $x=\\tau _{1,2}$ and $y=\\sigma _{1,2}$ and obey the conditions $\\tau _1=-\\tau _2$ and $\\sigma _1=-\\sigma _2$ from (REF ) and (REF ).", "Due to large negative coefficients of $x g_2^2$ and $y g_2^2$ terms, it is easy to check that for $x<7.9$ and $y<4.1$ the sign of the beta function is such that by the running of the “mixed\" quartic coupling its initial value will be driven towards decreasing its absolute valueWe took the SM value of the $SU(2)_L$ gauge coupling $g_2 (100 \\ \\hbox{GeV})\\approx 0.65$ .. As seen in Fig.REF , this parameter space overlaps with the values needed to explain the di-photon signal.", "As we increase further the energy, the $SU(2)_L$ gauge coupling $g_2$ increases towards its LP and the $g_2^4$ -term will eventually start to dominate the evolution, driving these “mixed\" quartics to the LP as well.", "We therefore expect that the dangerously-large initial values of the “mixed\" quartics needed to explain the di-photon signal will develop LP $\\sim 10^6$ GeV together with the $g_2$ coupling." ], [ "Discussion and conclusions", "The very establishment of the SM is a successful bottom-up story: the Nature has been kind to us in revealing the SM degrees of freedom, providing the answers to emerging questions gradually, one at a time.", "Additional BSM degrees of freedom seem to be most tangible when addressing the contemporary riddle of the of neutrino-mass origin in the bottom-up way, since the BSM fields which produce neutrino masses radiatively may be accessible at the LHC.", "In the present account we take under scrutiny two radiative neutrino mass scenarios protected from tree-level contributions.", "An automatic $Z_2$ symmetry in the first (one-loop mass model) case forbids a tree-level mass contribution, and an accidental $Z_2$ symmetry in the second (three-loop mass model) case protects the stability of exotic BSM fields needed to close the three-loop mass diagram.", "Additional arguments exposed in the previous section justify a hope that the three-loop mass model at hand may provide an appealing UV completion, in the same way as it is expected that the TeV-extensions of the SM would preserve the accidental baryon number of the SM to sufficient accuracy.", "Let us stress that the underlying $\\tilde{Z}_2$ symmetry imposed on the 2HDM potential (REF ) is exact as long as $m_{12}^ 2$ , $\\lambda _6$ and $\\lambda _7$ terms vanish.", "A detailed study within the 2HDM scenario [57], [58] shows that in the absence of the soft breaking $m_{12}^ 2$ term the exact $\\tilde{Z}_2$ symmetry does not require intervention of new physics below $\\sim $ 10 TeV scale.", "Indeed, at this scale the exotic states of three-loop scotogenic model [15] already enter into the play.", "Despite the existence of the fortuitous DM-protecting symmetry $Z_2$ , induced by $\\tilde{Z}_2$ symmetry, the portion of the parameter space for the three-loop mass model which could reproduce the 750 GeV diphoton resonance seems to account only for a sub-dominant portion of the dark matter.", "The hinted diphoton signal constrains the value of particular “mixed” quartic couplings of the model as a welcome observable.", "On the other hand, a large value of this coupling leads to well known Landau pole threat.", "Interestingly enough, there is another virtue of the aligned 2HD sector completed with extra scalars in the context of the three-loop model.", "The mixture of the 2HD and exotic scalar sector provides a fortuitous remedy for the too early Landau pole for relevant couplings, due to signs and sizes of the coefficients in the relevant beta functions.", "To conclude, the existing hints of the diphoton resonance opened a hope that the history of “prediscoveries” of new particles through the loop amplitudes may be repeated in the scenarios taken under scrutiny.", "A verification of the diphoton signals at the LHC may enable us to discriminate between scenarios offering different BSM fields.", "On the other hand, if these hints disappear with a larger integrated luminosity, they will still constrain the parameter space of proposed extensions of the SM with new charged states affecting considered loop amplitudes.", "Note.", "In the interest of open and reproducible research, computer code used in production of final plots for this paper is made available at https://github.com/openhep/ackp16." ], [ "Acknowledgment", "This work is supported by the Croatian Science Foundation under the project number 8799, and by the QuantiXLie Center of Excellence." ] ]
1606.05163
[ [ "The Bootstrapped Robustness Assessment for Qualitative Comparative\n Analysis" ], [ "Abstract Qualitative Comparative Analysis (QCA) has been increasingly used in recent years due to its purported construction of a middle path between case-oriented and variable-oriented methods.", "Despite its popularity, a key element of the method has been criticized for possibly not distinguishing random from real patterns in data, rendering its usefulness questionable.", "Critics of the method suggest a straightforward technique to test whether QCA will return a configuration when given random data.", "We adapt this technique to determine the probability that a given QCA application would return a random result.", "This assessment can be used as a hypothesis test for QCA, with an interpretation similar to a p-value.", "Using repeated applications of QCA to randomly-generated data, we first show that generally, the tendency for QCA to return spurious results is attenuated by using reasonable consistency score and configurational N thresholds; however, this varies considerably according to the basic structure of the data.", "Second, we suggest an application-specific assessment of QCA results, illustrated using the case of Tea Party rallies in Florida.", "This method, which we coin the Bootstrapped Robustness Assessment for QCA (baQCA), can provide researchers with recommendations for consistency score and configurational N thresholds." ], [ "Introduction", "Qualitative Comparative Analysis (QCA), introduced in Charles Ragin's 1987 book The Comparative Method, is a set of techniques designed to be a “middle-path” between quantitative and qualitative analysis.", "Ragin and others have since published many works on practical applications and extensions of the method, including the use of `fuzzy-sets,' the development of software, and many substantive applications (, , , , , ).", "Though originally conceived has a method to analyze middle-sized samples, some herald its interpretive qualities above regression analysis in certain contexts (, , ).", "At base, QCA allows for a set-theoretic approach to social science grounded in Boolean algebra.", "Within this framework, QCA can identify necessary conditions and multiple explanatory combinations of conditions (“recipes”) for the value of an outcome.", "Using the logic of sets, QCA can provide a useful alternative for analyzing complex causation, broadening the reach of current research strategies to integrate a combinatorial, logic-of-sets framework versus the mean-based approaches in regression analysis.", "QCA can also be applied to middle-N sets, where regression may fail to provide robust results.", "This multi-method technique that enables the researcher to be “in dialogue” with the results through by way of `truth table' analysis.", "Researchers are encouraged to bring in their own case-oriented knowledge to establish causal conditions, examine the results of the truth table, investigate the placement of cases, and redefine conditions as needed.", "Despite recent popularity, QCA is not without its detractors.", "QCA is most often criticized for the use of the truth table and subsequent algorithm to determine a minimum configuration that derives a given outcome.", "The truth table summarizes agreement on an outcome value for cases with the same configuration of causal conditions; the algorithm combines many configurations in a reduced configuration that results in an outcome.", "Specifically, many are skeptical of its ability to successfully identify randomly-drawn data as being patternless, an important benchmark of any method.", "Stanley suggested that an application of QCA using random data would often lead to spurious configurations returned.", "Similarly, Simon questioned its usefulness on the grounds that it does not allow for error in hypothesis testing.", "Skaaning (2013) assessed the sensitivity of QCA when varying current robustness specifications.", "Though embarked on a permutation test to determine which configurations returned via QCA are due to randomness, a principled test of QCA against totally random data while varying configurational N, consistency score, and the structure of data has not been undertaken.", "A principled test of the assessment described is useful in two ways.", "First, to generally assess the ability of QCA, with reasonable applications of existing robustness parameters (i.e.", "configurational N and consistency score thresholds), to filter random patterns.", "Second, it can be used for directly calculating the probability of returning random results according to unique features of any data set.", "This would be useful as an application-specific diagnostic assessment for QCA's “truth table” procedure, and would help protect researchers against wrongfully spotting patterns via stochastic properties of measurable phenomena.", "The interpretation of such an assessment would be the probability of returning a random configuration using the data and robustness thresholds selected by the user.", "Systematically applying Lieberson's suggested technique could also provide specific recommendations to inform researchers of an ideal consistency score and configurational N threshold to reach a desired level of confidence against a random result.", "Here, we determine whether QCA is robust to randomness, generally.", "We systematically apply QCA to thousands of random data sets, incrementally changing elements of the data structure – sample size and the distribution of variables in the data set – as well as elements under the control of the researcher – consistency threshold, configurational N threshold and `complex' versus `parsimonious' solutions.", "We then use logistic regression to determine which of these elements affects the probability of returning a `random configuration,' or a result returned from random data.", "We also describe a related method, which is the primary purpose of this article.", "The Bootstrapped Robustness Assessment is to be used to evaluate any QCA result.", "This operates by generating many random data sets of the same data structure used in an application of QCA (i.e.", "based upon the sample size and variable distributions) and applying QCA repeatedly at the parameters under control of the researcher (i.e consistency score thresholds, configurational N, and parsimonious vs. complex solutions).", "The result is the probability that a given QCA application would return a random result, based upon random data of similar size and distribution.", "We hope that this method will provide straightforward, easily-interpreted recommendations for researchers who desire unarbitrarily-drawn parameters of choice." ], [ "`Probabilistic Processes' and QCA", "A recent Sociological Methodology symposium on the methodological merits of QCA hosted a number of papers evaluating the reliability of QCA under both statistical and epistemological conditions.", "One paper saw problems with QCA under optimal conditions () which subsequently could not be independently replicated by two other papers (, ).", "Seawright (2014) discusses the reliability of QCA when certain configurations are unobserved.", "One recent piece () points out several studies that verify whether QCA's algorithm returns robust configurations (e.g , , ).", "This paper will attempt to remedy what can be described as QCA's “randomness problem” – the extent to which QCA's `truth table' analysis and algorithm are able to filter randomness in data given the robustness checks currently available.", "QCA's “randomness problem” is best described by .", "Lieberson's chief criticism is that “QCA is less prepared to allow for chance and probabilistic processes” than other methods and that “procedures do not rule out the possibility that the observations are all a random matter” ().", "Although QCA is a broad set of techniques to analyze small-to-medium-n data, the primary criticism of QCA has been its analysis of the `truth table.'", "The QCA `truth table' is a decomposition of data that analyzes each combination of causal conditions found in the data, the number of cases within each combination, and the extent to which cases that share these causal conditions agree on an outcome.", "An algorithm is applied to the truth table, combining the information into one or more “causal recipes” or solutions that result in an outcome.", "Lieberson imagines a test of his assertion that truth table analysis returns random results: apply QCA to a collected data set versus data where values arre randomly reassigned, keeping the marginal distributions intact.", "If QCA returns a configuration in both cases, it has a serious problem with being able to distinguish real patterns from random ones.", "In a rebuttal, argue that such a test would show that random patterns would be filtered out by probabilistic procedures used in any application of QCA.", "One such procedure is the use of a high consistency threshold: the proportion of cases that are explained by a given a configuration or solution.", "This threshold is designed to prevent configurations that have high probability of being random from being included in the QCA algorithm and is the proportion of cases with the same combination of causal condition values and have the same value for the outcome.", "A recommended threshold is .85, meaning that 85% of all cases with a specific configuration of causal conditions all agree in the value of the outcome.", "However, depending upon the marginal distributions of the conditions used in the QCA, a consistency score threshold could have a differential effect for filtering random configurations.", "For example, imagine an application of QCA had an outcome whose cases have a 90% probability of being 1.", "Any combination of categories now has a .9 probability of having an outcome of value 1, with some rate of error.", "However, if attempting to explain the negation of the outcome, each combination of variables has a .1 probability of having an outcome with value 0.", "In these situations, the consistency score needed to protect the researcher against observing a random pattern differs quite a bit – at base, a .9 is needed in the first case, while the lowest value is needed for predicting the negation.", "A more direct estimation of randomness, one that takes into account the marginal distribution of the outcome used in the analysis, would thus be helpful for providing an application-specific recommendation for the consistency score.", "A second probabilistic procedure to prevent spurious results is the configurational N threshold: the minimum number of cases that have a certain combination of causal conditions, which allow the solution to be considered in the final result.", "This prevents the researcher from making conclusions about a small number of cases, especially about being overconfident about just one case with a unique set of conditions.", "To prevent such a scenario, the researcher can set a certain configurational N threshold (usually 2 or 3) to throw out those combinations of conditions that do not have a sufficient number of cases to make conclusions.", "Ostensibly, a high consistency score should be sufficient to account for error in causal conditions.", "As we have argued, these thresholds should have varying success rates according to the marginal probability of conditions present in the analysis.", "The utility of these procedures to distinguish a random data set from a collected one is an empirical question.", "Our first goal is thus to determine if the current researcher-set thresholds are enough to ensure that the results returned are not due to random chance.", "This paper addresses two areas of research inquiry.", "First, to what extent does a consistency score and configurational N threshold actually reduce the chance of returning a spurious result?", "Relatedly, how does this effectiveness differ according to the structure of the data?", "For example, is a consistency score of .9 effective at all sample sizes?", "How does the distribution of the outcome affect the usefulness of a high configurational N threshold?", "Our first set of results demonstrates the general effects of researcher choice on spuriousness in QCA's `truth table' analysis, with a special attention given to its variation with data structure.", "Our second set of questions refers to a practical application of these results.", "If results are highly dependent upon the structure of data analyzed, an application-specific robustness check would be useful for providing specific recommendations for researcher choice.", "The second set of results is a practical application of our procedure that uses a given QCA model, simulates many random data sets from this model, and 1) gives specific recommendations for ensuring against random results and 2) gives a specific value for the probability that a given QCA application would return a random result.", "The latter application has an analogous interpretation as a p-value for a QCA result." ], [ "Is QCA Robust to Randomness?", "There are researcher-set thresholds available prior to an analysis to reduce random configurations from being returned.", "The consistency threshold restricts the analysis to only consider configurational categories that have a certain proportion of cases that all agree in the direction of the dependent variable.", "The configurational N threshold restricts the analysis to only consider those configurational categories that have a certain number of cases within them.", "For example, we can choose to only include those combinations of causal conditions that have four or more cases that share all of the same causal condition states.", "Though these are attempts to introduce probabilistic checks for QCA configurations, their use is often flexible, and general recommendations for which thresholds are hard to determine without a principled test of their usefulness.", "This section assesses the relative importance of each probabilistic check for filtering out random configurations from being returned by QCA." ], [ "Assessing the Robustness of QCA", "We employ a straightforward assessment of QCA using simulations.", "First, we first simulate a random data set.", "Next, we apply QCA to this random data set, and record whether QCA returned a result at all from random data.", "If we discover that a result is returned, we know that QCA is returning a spurious result.", "We systematically vary several variables to determine which elements of data structure (marginal distribution of variables, number of causal conditions included in the model, and sample size) and features of researcher choice (consistency threshold, configurational N threshold, and complex versus parsimonious solutions) affect the probability of a spurious result.", "In these simulations, each causal condition is a dummy variable with a marginal distribution randomly varying between .1 to .9 probability of being “1.” Though variables vary between iterations, all variables have the same marginal distribution within each iteration.", "The number of causal conditions vary from one to six.", "In accordance with QCA's focus for small- to medium-n samples, the sample size of the random data set varies from 10 to 60.", "Between iterations, we systematically vary the configurational N threshold from one to six.", "We vary the consistency threshold from .5 to 1.", "The resulting data set is 2.5 million cases, each case an iteration of this procedure.", "We employ logistic regression on the results, with the dependent variable being a 0-1 outcome of returning a configuration from random data.", "The independent variables are the elements of data structure and researcher choice listed above.", "The primary question here is, which factors, when altered, independently decrease the probability of returning a random configuration in QCA?", "Secondly, we assess whether researcher choice differentially affects the probability of a result given the structure of the data.", "For example, does a high consistency score threshold filter out spuriousness across all sample sizes?", "Does a high configurational N threshold filter spuriousness given all marginal probabilities of causal conditions?", "This is tested using interaction effects between variables measuring researcher choice and variables measuring data structure.", "If interaction effects are substantial, it suggests that additional assessments that take into account data structure need to applied to QCA to ensure robustness." ], [ "Results", "Generally, the choice of a high consistency score and high configurational N in QCA do reduce the probability of returning a spurious result.", "Their effectiveness, however, is dependent upon the basic structure of the data: the distribution of the dependent variable, the number of variables used in the analysis, and the sample size affect how probable a result is spurious.", "First, we describe the main effects of each of these in turn.", "Then, we discuss how the structure of the data interacts strongly with checks." ], [ "The Effects of Data Structure and Researcher Choice on Spuriousness", "Model 1 in Table 1 shows the results of the logistic regression model predicting whether QCA returned a spurious result from simulated random data.", "Figure 1 plots these effects graphically.", "The largest Model 1 effect is for the distribution of the outcome.", "An increase in the distribution of 1s in the dependent variable from probability 0 to probability 1 increased the logged odds of a spurious result by 2.29 (about 10 times the odds).", "This is an expected result – in the case of an outcome with 90% `1's', each case, and thus each configuration, has a 90% of being 1 in the outcome.", "In Model 1, we see that increasing the number of variables decreases the probability of a spurious result by about 13% for each additional variable.", "Interestingly, increasing the sample size also increases the odds of a spurious result by about 3% for each additional case.", "This is in contrast to other mean-based approaches which typically become more robust as sample size increases.", "Table: Logistic Regression of Elements of Researcher Choice and Data Structure Predicting a Spurious QCA ResultModel 1 also shows the effect of researcher choice on returning a random configuration.", "Increasing the consistency score from and 0 to 1 in this case decreases the odds of returning a spurious result by 58%.", "Using a “complex” versus “parsimonious” solution reduces the odds of a spurious result by 56%.", "The most substantial effect, however, is the configurational N threshold, with a reduction in the odds of spuriousness by 51% at each additional increase in the configurational N threshold.", "With the exception of two interactions (between complex solution and configurational N threshold, and between complex solution and the dependent variable distribution), Model 2 shows that researcher choice significantly moderates the effects of the structure of the data in the application of QCA.", "Because there are many strong interaction effects, we encourage the reader to focus upon the interpretation of Model 2.", "These variegated effects are described in the section and figures below." ], [ "The Highly Contingent Effects of Researcher Choice on Spuriousness", "How effective is researcher choice in reducing the spuriousness of QCA under different conditions of data structure?", "One example question here is, To what extent does increasing the consistency score threshold effectively reduce the chance of a spurious result when the sample size is high?", "Figure: Predicted Interactions between Elements of Researcher Choice and Outcome DistributionFigure 2 shows the predicted interaction between researcher choice and the distribution of the dependent variable on the probability of spuriousness.", "On the y-axis is the expected probability predicted by the logistic regression model; the x-axis shows the outcome distribution in the data used to fit the QCA model.", "The top-left plot shows the predicted spuriousness across outcome distributions, with all other variables held at their means.", "The top-right shows that the consistency score threshold assists in reducing spuriousness by a great deal at lower distributions; when all cases have a .1 probability of being “1” on the outcome, increasing the consistency score from .5 to 1 decreases the probability of a spurious result from .5 to .2, a substantial reduction.", "The effectiveness of consistency score threshold on spuriousness decreases as the outcome distribution changes, however.", "When all cases have a .9 probability of being `1,' the predicted effect of increasing the consistency score is slightly negative.", "In data sets where cases almost all agree on an outcome, the consistency score may not be the most effective tool to prevent spuriousness.", "The higher likelihood that a lower consistency score improves robustness is likely due to lower consistency scores including all possible combinations into the analysis, which returns a non-result.", "Higher consistency scores in this circumstance will filter out configurations that randomly vary from the .9 baseline probability in the data set, returning a “random” configuration that made the cut while filtering out “random” configurations that predictably varied below the threshold in their incidence of the outcome.", "The configurational N threshold substantially impacts the probability of robustness at all levels of the outcome distribution.", "Unlike the consistency score threshold, there is no change in the direction of the effect; increasing the configurational N threshold decreases the probability of spuriousness at all levels of the outcome distribution.", "The extent to the effect, however, decreases with an increase in proportion being `1' in the outcome.", "The probability of spuriousness at a configurational N threshold of 6 increases from .1 to .25 when the outcome distribution of `1s' increases from .1 to .9.", "As shown in the bottom-right plot in Figure 2, the “complex” solution is always more robust than the parsimonious solution, regardless of the outcome distribution.", "As shown in the Table 1, the effect is not significant.", "Figure: Predicted Interactions between Elements of Researcher Choice and Number of Causal ConditionsFigure 3 shows the interactions between elements of researcher choice and the number of causal conditions used in the analysis.", "Generally, a larger number of causal conditions is predicted to have more spurious results.", "As the number of causal conditions increases, the effect of consistency score on spuriousness weakens.", "The opposite trend occurs when interacting configurational N threshold with the number of causal conditions – the effect is much greater when using more conditions.", "Ostensibly, this is due to fewer configurations being included in the truth table analysis when the number of possible configurations increases.", "When the number of causal conditions is seven, there is a 60% reduction in spuriousness when using a complex versus a parsimonious solution.", "At two causal conditions, however, the effect of using a complex versus parsimonious solution is negligible.", "Figure: Predicted Interactions between Elements of Researcher Choice and Sample SizeAs Figure 4 shows, the effect of researcher choice varies the least when interacted with sample size.", "Table 1 shows significant effects for these interactions, the effect size is small compared to the interactions with outcome distribution and number of causal conditions." ], [ "Discussion: QCA Robustness", "The results here show that the probabilistic checks set by the researcher in a QCA truth table analysis are effective in reducing the probability of a spurious result.", "However, they also show that the effectiveness of researcher-set parameters to ensure robust results vary according to the structure of the data.", "In some cases, consistency score alone will not effectively filter out random patterns, especially when the outcome is distributed such that all configurations have a high probability of having the same outcome value.", "In some cases, however, a high consistency score is not needed to ensure a robust result.", "The interaction effects above show that elements of researcher choice differed in their effectiveness at differing levels of number of causal conditions used.", "A high consistency score was more effective when using fewer causal conditions, while a high configurational N threshold was more effective at higher numbers of causal conditions.", "The difference between complex and parsimonious solutions was negligible at lower number of causal conditions, but represented a nearly four-fold increase in spuriousness – from 25% chance to 85% chance – when using seven causal conditions.", "These results show that the choices researchers make when conducting a truth table analysis of QCA data have a differential effect upon spuriousness according to variations of basic features of the data.", "The large variation of these effects, as well their complexity, justifies the need for a more straightforward approach to probabilistic assessments for QCA's truth table analysis.", "While it would be helpful to advise researchers on general practices for choosing a consistency score, a configurational N, and whether to use complex or parsimonious solution, the large variations in their effects according to features of the data used prevent the authors from doing so.", "In the section below, we present a method for providing a model-by-model estimate of spuriousness.", "This method has two functions: 1) to estimate the 'confidence level' of an existing QCA model; and 2) to provide a reasonable recommendation for setting the consistency score and configurational N thresholds to achieve a desired `confidence level.'", "We first generally describe the method; we then give an example of its application" ], [ "The Bootstrapped Robustness Assessment for QCA", "The Bootstrapped Robustness Assessment for QCA (baQCA) is a procedural check of a QCA result that takes into account data structure (e.g.", "marginal distribution of variables) and researcher choice (e.g.", "consistency score threshold) to provide an estimate of the probability of spuriousness for a given QCA result.", "Above, we show that the elements of researcher choice involved with ensuring robust QCA analysis of a truth table require vastly different thresholds according to the data structure.", "Identifying robust configurations would thus require taking into account the data structure.", "To build a robustness assessment while taking into account data structure, we first draw a random data set using the same data structure as a QCA result.", "This includes using 1) the same number of causal conditions as the observed QCA data set, 2) the marginal distributions of the causal conditions and dependent variable present in the QCA data, and 3) the sample size.", "We then run a QCA model matching 4) the consistency score threshold set by the researcher and 5) the configurational N threshold set by the researcher.", "After thousands of repetitions, we take the simple probability that QCA returned a configuration given those parameters.", "The inverse of this proportion can be interpreted as the confidence that the configuration returned in the QCA analysis is due to random chance.", "This interpretation is similar to the p-value used in regression analysis to determine the “significance” of a result.", "Software to run this assessment is available using the R software package baQCA.", "Although there is current work being done to test the robustness of QCA to randomness (, ), this method randomly samples the outcome, keeping the causal conditions fixed, and permutes configurations rather than individual variables.", "Though QCA is at base a “configurational method,” we think it makes more sense to permute values of variables independent of other variables, rather than treating combinations of values for causal conditions as inextricably linked at predefined rates.", "Specifically, we argue that Braumoeller's focus on consistency score threshold as the primary check to prevent spurious results is inadequate, given our finding that the configurational N threshold is the much more powerful check.", "Though his method implicitly takes into account configurational N thresholds by including counts of configurations in his permutation model, these counts can vary considerably when variables are treated as independently sampled.", "Importantly, baQCA allows us to vary configurational N thresholds via independent random sampling of variables, and allows us to calculate how changes in configurational N threshold affect spuriousness.", "The idea behind this strategy is to protect the researcher from spurious results by estimating the probability of spuriousness given any random data.", "By repeatedly sampling random data over thousands of iterations, we are in effect directly observing this probability.", "When a 95% confidence interval of the mean is calculated on a “random” variable, we conclude with 95% confidence that the interval covers the true value of the mean.", "Our strategy identifies the probability that an application of QCA, with the exact data structure and parameters of researcher choice, would, with some level of confidence, return a result given completely random data.", "If this confidence level is low, the researcher should be cautious; when this confidence level is high, the researcher can conclude with confidence that the result is unlikely due to random chance, and is robust to a direct comparison with random data." ], [ "baQCA in Practice", "This section outlines the step-by-step procedure of two methods for determining the probability that a given QCA application returns a spurious result.", "First, the Bootstrapped Assessment for QCA (baQCA) can be applied with a few steps: `Fit' a QCA model with $v$ causal conditions and $n$ number of cases.", "Simulate 2000 random data sets, each with $v$ causal conditions and size $n$ .", "Each causal condition and the outcome has the same distributions as the observed data, respectively.", "Apply QCA to all the generated data sets, matching the elements of researcher choice specified in the observed model (configurational N, consistency score, etc.).", "Record whether each QCA returned a spurious result.", "We chose 2000 to be the number of simulations using convergence diagnostics from .", "Take the simple proportion of times the QCA returned a configuration: $\\frac{R}{2000}$ Where $R$ = the number of times a QCA model returned a result from a randomly generated data set.", "We use bootstrapped standard errors for a measure of uncertainty ().", "We resample, with replacement, the vector of counts that sum to $R$ .", "We chose 1000 resamples using convergence diagnostics from .", "We then take 95% quantiles of $\\frac{R}{2000}$ for our confidence interval.", "The resulting scalar, and its interval, is the 95% confidence interval of the probability that the QCA application would return a random result.", "A useful, related tool is the recommendation of a consistency score/configurational N threshold given the data, without having fit a QCA model at all.", "To achieve this recommendation, we follow a similar trajectory: Simulate 2000 random data sets, each with $v$ causal conditions and size $n$ .", "Each causal condition and the outcome has the same distributions as the observed data, respectively.", "Apply QCA to all the generated data sets, matching systematically varying parameters of researcher choice (configurational N, consistency score, etc.).", "Record whether each QCA returned a spurious result.", "Again, we chose 2000 simulations based on convergence diagnostics.", "We apply a logistic regression model to the results, using the configurational N and consistency score thresholds as predictors.", "We use this model to provide fitted values, calculating the minimum consistency score needed at every configurational N threshold to achieve a desired “level of confidence.” We use the standard errors provided in the regression model to provide a confidence interval around each recommendation.", "Below, we provide a case study where reasonable thresholds were not quite enough to ensure good results, and where baQCA could be of service to suggest specific thresholds to ensure robustness." ], [ "Qualitative Comparative Analysis of Tea Party Rallies in Florida", "In an application of this technique, we use a subset of data constructed by as part of their project on the emergence of Tea Party organizations in U.S. counties.", "The data set includes several county-level measures, including demographic measures from the American Community Survey (ACS) 2005-2009 (), measures of religious adherence from the Association of Religion Data Archives (ARDA) 2001, 2008 Presidential election measures from Congressional Quarterly's America Votes, and the number of Tea Party organizations between 2009 and 2010 from the Institute for Research & Education on Human Rights (()).", "We extend the data set to include the a new outcome variable, the number of rallies in each county between 2009 and 2010, also from IREHR.", "We restrict the data set to counties in Florida for two reasons based on our own case knowledge ().", "First, we choose Florida counties because, with the exception of California, all other states had fewer numbers of organizations.", "Second, and substantively important for the choice of causal conditions, we choose Florida due to the perceived impact the Tea Party movement had in the 2010 midterm election ().", "Restricting the data to counties in Florida leaves us with 67 cases for analysis (see Figure 5).", "Our analysis addresses the multiple causal pathways that lead to the occurrence of one or more Tea Party rallies in a Florida county.", "Figure: Tea Party Rallies in Florida Counties.", "Note: Grey indicates at least one rally in county.Qualitative Comparative Analysis arguments are combinational and often overlapping, and a researcher's causal conditions require a deep knowledge of cases in the data set for adequate placement within particular sets of causal conditions ().", "For example, to fully belong to a crisp outcome set, a Florida county must meet or exceed a minimum criterion.", "Therefore, it is necessary to both establish adequate causal conditions and an inclusionary criteria for each condition.", "Some have argued that because these criteria are based on researcher selection, criteria are biased (), which can lead some to believe that a researcher has cherry-picked their analyses.", "To combat this assumption, and for the sake of clarity, we employ a simple inclusionary-exclusionary criterion for membership in a causal condition (outlined below) for the analysis.", "It is important to note that for many causal conditions, we dichotomize on the mean.", "Because QCA is designed to allow the researcher to be in dialogue between results and the cases and to recalibrate, it is generally considered bad practice to create inclusionary-exclusionary criteria in this way.", "We do so as a pedagogical exercise for the baQCA method and not as a manner with which to employ QCA.", "Research on the Tea Party movement finds that while most of their organizations were concentrated in conservative partisan environments (, ), much of their on-the-ground rally activity took place in heavily populated, left-leaning locales (, ).", "Research on Tea Party organizations has demonstrated the importance of educational background on support for the Tea Party (, ), finding that supporters of the Tea Party movement are highly educated, and that Tea Party organizations were more likely to be established in U.S. counties characterized by a predominance of college graduates.", "Although supporters of the Tea Party movement tended to be relatively impervious to the economic recession of 2008 (, ), many of the movement's grievances consisted of dismay about unemployment and the expanding reach of federal government, through a series of redistributive policies designed to remedy the economic situation (, , ).", "Scholars show that while much support for the Tea Party movement came from Protestants, these supporters were not of the Evangelical bent as was depicted by media (, , ).", "Finally, argue that support for the movement derived from racial backlash against the nation's first black President, as well as fears that Barack Obama would initiate policies that would favor blacks.", "The size of the black population has two possible effects on support for the Tea Party.", "First, blacks pose a pose a potential threat in places where the black population is large because their predominance might encourage redistributive action by the Obama administration.", "Secondly, because rallies were much more likely to take place in left-leaning, densely-populated areas, these locales are also much more likely to have larger black populations.", "The brief summary of extant research on the Tea Party provides insight into creating causal arguments about the presence of Tea Party rallies as a test of QCA and the baQCA method.", "Importantly, because the analysis here employs crisp-set QCA, each causal condition is coded as either one or zero.", "As previously mentioned, the outcome variable is Tea Party RALLIES.", "Full placement in the outcome set (1) requires that a county has at least one rally.", "In sum, there are 19 cases that have the outcome.", "Given the differential relationship between Republican partisan contexts and the presence organizations and the occurrence of rallies, we include the measure REPUBLICAN.", "This is coded as one if, during the 2008 Presidential Election, the Republican candidate received a majority of the votes in the Florida county.", "Based on the above literature, we expect the absence of (negation of) Republican context as an important component of a causal pathway to Tea Party rallies.", "We also include four causal conditions in which full inclusion is defined in a straightforward manner: full membership in the causal set (1) is determined by whether a value for a particular case falls at or above the mean for that variable.", "First, we include a measure of COLLEGE educated, the percentage of people in the county (aged 25 or above) who hold a Bachelor's degree.", "Second, we include a measure of UNEMPLOYMENT, measured as the percent of the county population which is unemployed.", "With regard to membership in the college educated set, we expect that the presence of a college educated population is an important component of the pathway to rallies.", "However, given that many Tea Party supporters were not actually unemployed (although the movement's rhetoric says otherwise), we expect that the absence of a high unemployed population is an important part of explaining Tea Party rallies.", "Third, we include a measure for the size of the BLACK population, measured as the percentage of African-Americans in the county.", "Fourth, we include a measure for the size of the EVANGELICAL population, measure as the percentage of Evangelical adherents in the Florida county.", "In QCA, the logical representation of the presence of a causal condition is indicated by the variable name in all upper-case letters whereas negation is represented by all lower-case letters.", "Combinations of conditions (e.g.", "complex combinations of variables) in a pathway or recipe to an outcome are expressed as a string of variable names delineated by an asterisk, representing the logical operator “AND.” If multiple pathways exist, each pathway is delineated by a plus symbol, the logical operator for “OR.” Therefore, our main expectation, expressed in QCA notation, is: republican * COLLEGE * unemployment * BLACK * evangelical In this analysis, there are a total of 32 possible pathways to the outcome, based on the five ($K$ ) causal conditions ($2^K = 2^5 = 32$ ).", "Before applying the baQCA method, we calculate Tea Party rally solutions based on plausible researcher preferences.", "In this test case, we use a default sufficiency inclusion or consistency score of .85 (the minimum proportion of cases explained by a causal configuration) and a default configurational N threshold of 1.0 (the minimum number of cases allowed for a particular configuration).", "The results in Table 2 indicate that only three configurations or combinations explain the presence of Tea Party rallies in Florida counties.", "Overall coverage for the three configurations is 47.4%, with only BLACK appearing in each.", "The presence or absence of REPUBLICAN, COLLEGE, UNEMPLOYMENT, and EVANGELICAL appear in various combinations, known as “insufficient but necessary components of causal combinations that are unnecessary but sufficient for the outcome” (, ).", "Table: Conclusion and General Recommendations" ] ]
1606.05000
[ [ "The number and size of subhalo-induced gaps in stellar streams" ], [ "Abstract Ample observational capabilities exist today to detect the small density perturbations that low-mass dark matter subhaloes impart on stellar streams from disrupting Galactic satellites.", "In anticipation of these observations, we investigate the expected number and size of gaps by combining an analytic prescription for gap evolution on circular orbits with the flux of subhaloes near the stream.", "We explore the distribution of gap sizes and depths for a typical cold stream around the Milky Way and find that for a given stream age and gap depth, each subhalo mass produces a characteristic gap size.", "For a stream with an age of a few Gyr, orbiting at a distance of 10-20 kpc from the Galactic center, even modest subhaloes with a mass of $10^6-10^7 M_\\odot$ produce gaps with sizes that are on the order of several degrees.", "We consider the number and distribution of gap sizes created by subhaloes with masses $10^5-10^9 M_\\odot$, accounting for the expected depletion of subhaloes by the Milky Way disk, and present predictions for six cold streams around the Milky Way.", "For Pal 5, we forecast 0.7 gaps with a density depletion of at least 25\\% and a typical gap size of $8^\\circ$.", "Thus, there appears to be {\\it no tension} between the recent non-detection of density depletions in the Pal 5 tidal tails and $\\Lambda$CDM expectations.", "These predictions can be used to guide the scale of future gap searches." ], [ "Introduction", "To date, only two promising techniques have been put forward to detect individual low-mass dark matter (DM) clumps devoid of stars.", "The presence of these so-called DM subhaloes may be betrayed by small perturbations in the images of cosmological gravitational lenses [49], [22], [39], or, alternatively, be revealed by gaps in the stellar streams around the Milky Way (MW) galaxy [42], [44], [63], [17].", "In principle, through gravitational lensing it might be possible to detect subhaloes with masses as low as $10^7 M_{\\odot }$ , and, encouragingly, measurements have already been reported of DM subhaloes with $M \\lesssim 10^9 M_{\\odot }$ [69], [70], [40].", "This is reassuring, as many dwarf galaxies have been shown to exist around the Milky Way with masses similar to that or lower [8].", "Moving forward, gaps in stellar streams offer the possibility to pin down the DM mass spectrum below the dwarf galaxy threshold, in other words in the completely dark regime [29].", "The importance of such a direct observational tool is difficult to overestimate, as $\\Lambda $ CDM expects any MW-size galaxy to be bathed in a plethora of DM subhaloes, with an overwhelming prevalence of low-mass objects [65].", "Over the last decade, detection of DM subhaloes via gaps in stellar streams has evolved from a plausible idea into an imminent measurement.", "This is because the haul of cold stellar streams (the prime contender to carry marks of an interaction with low-mass DM subhalos) has risen to at least a dozen structures [36] thanks to high-quality data from all-sky imaging surveys like SDSS [2], VST ATLAS [62] and DES [67].", "While the original survey data is clearly deep enough to identify the streams, it is likely too shallow to warrant an unambiguous detection of low-amplitude density fluctuations caused by DM subhalo flybys.", "Nonetheless, candidate gaps have been reported in Pal 5 [54] and GD-1 [38] streams using the SDSS photometry alone [21], [20].", "Curiously, the majority of these gaps are smaller than $2^{\\circ }$ in size, seemingly in agreement with $\\Lambda $ CDM-inspired predictions [71], [18], [53].", "The consensus in the literature is that the cosmological structure-formation predictions can be tested by measuring the shape and the normalisation of the gap size spectrum.", "Thus, the intuition is that the incidence of gaps encodes the sub-halo volume density in the vicinity of the stream, and the gap size is linked to the mass of the dark perturber.", "Recently, [43] measured the stellar density along the Pal 5 stream to a significantly fainter magnitude limit.", "Interestingly, they found no evidence for gaps on small scales.", "It appears, therefore, that the preliminary studies of the density fluctuations in the SDSS stellar streams might have erred on the side of risk, when considering the possible false positives.", "This is perhaps unsurprising as the interplay between the survey systematics and the stellar halo density field at faint magnitudes has not been studied in detail.", "The number of stars entering a particular color-magnitude box (used to trace the stream) will depend on the weather conditions at the epoch of observation.", "More precisely, sky brightness and seeing will determine the object detectability and the efficiency of star-galaxy classification.", "Unfortunately, only global estimates of the SDSS completeness exist [66], [1].", "Temporal changes in completeness and star-galaxy separation efficiency may therefore be reflected in spatial variations of the faint star counts.", "For example, in Figure 1 of [46], a stripy “patchwork” appearance of the SDSS stellar density distribution can be observed, which remains visible - albeit slightly subsided - even after application of the “uber-calibration” procedure to the SDSS photometry [55].", "Coupled with weather conditions, SDSS survey geometry can potentially induce spurious variations in the density field of faint stars on a variety of angular scales.", "The SDSS footprint consists of 25-wide stripes, each comprising of 125-wide strips.", "Thus, spurious power may be added on scales of 25 degrees and down to a small fraction of a degree, the latter due to the fact that individual stripes can overlap by different amounts depending on the distance from the survey poles.", "Worse still, because only a portion of a strip can be completed during one night (an SDSS “run”), bogus density fluctuations may exist on scales of several to tens of degrees due to changes in epoch (and hence weather conditions) along the individual stripe.", "Furthermore, various other sources of spurious density fluctuations are expected to exist, such as those associated with large scale structure and saturated stars.", "Naturally, many authors attempt to mitigate against the above problems by limiting their stellar samples to brighter magnitudes, e.g.", "$r<22$ .", "However, some of the issues discussed may unfortunately be exacerbated by the breakdown of the star-galaxy separation even at brighter magnitudes.", "While, globally, the SDSS completeness is 95$\\%$ at $r=22.2$Note, however, an increasingly erratic behaviour of the rms scatter in completeness estimate at $r=21$ in Figure 8 of [5], the star-galaxy separation is 95$\\%$ correct at only $r=21$ .", "This deteriorates to $90\\%$ at $r=21.6$ [5].", "This erroneous morphological classification can lead to spurious clumping in stellar density maps induced by the leakage of power from galaxy distributions as illustrated, for example, in Figure 4 of [45].", "Deeper follow-up imaging with better seeing and under a darker sky, additionally conforming to a different mosaic geometry, would naturally do away with most of the artifacts discussed above.", "But if the number and size of gaps reported earlier were in agreement with $\\Lambda $ CDM, does it mean that the null detection reported by [43] is in tension with the predictions of cosmological structure formation theory?", "Motivated by this conundrum, we re-visit the expectations for the frequency and the scale of gap creation.", "The first comprehensive attempt to describe the spectrum of stream gaps due to interactions with DM subhaloes can be found in [71] who lay out a simple framework to count the number of subhalo flybys near a stream.", "For example, for the Pal 5 stream, they estimated $\\sim 5$ close flybys for subhaloes with masses in the range of $10^7-10^8 M_\\odot $ .", "Exactly how large and deep a gap these flybys would create, and hence how detectable they would be, depends on the flyby geometry and the flyby velocity.", "[18] made the first attempt to answer this question by combining a similar flyby counting technique as [71] with fits to the properties of gaps created by subhalo flybys.", "In this work, we will build on the approaches of [71] and [18] by using a similar estimate for the number of subhalo encounters, while determining the effect of each flyby based on the results of [28].", "Theirs is an analytic model of the gap properties for density perturbations induced in streams on circular orbit.", "The advantage of this approach is that the analytic model works for any flyby geometry.", "This allows us to sample a wide range of encounters and determine the distribution of gap properties expected for a given stream and subhalo distribution.", "The assumptions of this method, both the flyby rates and their properties, as well as the gap properties, are tested against numerical simulations.", "Our predictions also account for the expected depletion of subhaloes by the Milky Way disk in the inner regions of the Milky Way [27].", "With this approach we find that dramatically fewer gaps are expected than the results of [71] and [18] suggested.", "In addition, we present predictions for the distribution of gap sizes expected from a $\\Lambda $ CDM spectrum of subhaloes and find that the characteristic size is rarely lower than several degrees, i.e.", "typically, an order of magnitude larger than that searched for by [43].", "Thus this paper will demonstrate that the lack of small-scale gaps in Pal 5 is not in tension with $\\Lambda $ CDM.", "The paper is organized as follows.", "In Section  the rate and properties of the subhalo flybys are derived.", "Next, in Section  we review and test the model of gap growth which translates the flybys into gap properties.", "In Section  we use this formalism to examine how the distribution of gap sizes and depths changes for various subhalo masses and stream ages.", "We also give the distribution of gap sizes expected from a $\\Lambda $ CDM population of subhaloes.", "In Section  we predict the number of gaps in six cold streams around the Milky Way and find significantly fewer gaps than was previously expected.", "The model of the rate and properties of the flybys, as well as the gaps they produce, is tested with N-body simulations in Section .", "We discuss implications of this work for gap searches, possible contamination from giant molecular clouds, and limitations of the method in Section .", "Finally, we conclude in Section ." ], [ "Setting up subhalo-stream encounters", "In order to compute the expected number of stream gaps, we must first compute the expected number of subhalo flybys, as well as their velocity distribution relative to the stream.", "The expected number of flybys is controlled by the velocity distribution of the subhaloes as well as their number density.", "Our approach is similar to that in [71] with several amendments.", "First, we will present a modified version of their derivation which correctly accounts for the velocity distribution of subhaloes.", "Second, we will use a lower number density of subhaloes since the presence of a baryonic disk in the Milky Way will deplete substructure by a factor of 2-3 [27].", "Finally, we will use a higher subhalo velocity dispersion motivated by models of the Milky Way and cosmological simulations.", "Below, we will assume that the subhaloes are uniformly distributed and that each component of their velocity follows a normal distribution with a mean of zero and a dispersion of $\\sigma $ .", "This is sometimes referred to as an isotropic Maxwellian distribution.", "We will also neglect the size of the stream, treating it as a line, and assume that each star in the stream is just moving along this line, thus neglecting the velocity dispersion in the stream." ], [ "Expected number of flybys", "The effect of each subhalo flyby depends on the impact parameter to the stream.", "Both [71] and [18] set up a straightforward scheme to count the number of subhaloes which pass within a given distance, $b_{\\rm max}$ , of the stream.", "We will now present a slightly modified version of their calculation and explain the difference with their result.", "We consider a cylinder of radius $b_{\\rm max}$ around the stream, as shown in Figure REF .", "The number of subhaloes piercing this cylinder in some time interval, $dt$ , is given by $ dN_{\\rm enc} &= (2 \\pi b_{\\rm max} l) \\times (| v_r | dt)\\times n_{\\rm sub} \\times P(v_r) dv_r , $ where $l$ is the length of the stream and $v_r$ is the cylindrical radial velocity in the stream coordinates, i.e.", "perpendicular to the motion of the stream.", "If we only consider the side of the cylinder and not the end caps, the flyby rate only depends on $v_r$ .", "Since the radial velocity is just a projection of the cartesian velocities, $P(v_r)$ is also a Gaussian with a mean of zero and a dispersion of $\\sigma $ .", "Figure: Imaginary cylinder around stream used to count how many subhaloes pass near a stream.", "The yellow line represents the stream.", "The region between the solid cylinder and the dashed cylinder shows the volume from which subhaloes with radial velocity v r v_r will enter within b max b_{\\rm max} of the stream in time dtdt.", "We restrict v r v_r to be negative since we only want to count the subhaloes entering this cylinder and not those leaving.", "v s v_s is the velocity of the stream.Integrating over the negative radial velocities, i.e.", "those that are about to enter the cylinder, we get $ \\frac{dN_{\\rm enc}}{dt} = \\sqrt{2\\pi } \\sigma b_{\\rm max} l n_{\\rm sub} .", "$ We can also compute the total number of encounters the stream will have by accounting for the growth of the stream in time.", "Assuming that the stream growth is linear in time, $l \\propto t$ , if we integrate the encounter rate until the present time, when the stream has a length of $l_{\\rm obs}$ , we get $ N_{\\rm enc} = \\sqrt{\\frac{\\pi }{2}} l_{\\rm obs} b_{\\rm max} n_{\\rm sub} \\sigma t .$ A similar result is presented in [71], who used $l_{\\rm obs}=4 (t/T_\\psi ) R_{\\rm circ} \\Delta \\Psi $ where $\\Delta \\Psi $ is the angular growth per orbit, $T_\\psi $ is the angular period, and $R_{\\rm circ}$ is the orbital radius of the stream.", "Plugging this value of $l_{\\rm obs}$ into Equation (REF ), we get $ N_{\\rm enc} = 2\\sqrt{2 \\pi } R_{\\rm circ} b_{\\rm max} \\sigma t n_{\\rm sub} \\Delta \\Psi \\Big ( \\frac{t}{T_\\psi } \\Big ) .$ This can be now compared with equation 15 of [71] in the limit that their encounter velocity, $v_{\\rm enc}$ , is taken to infinity, where we find that the number of encounters in this work is $2\\sqrt{2}$ smaller.", "The difference is due to what is assumed about the radial velocity distribution, $P(v_r)$ , which is used in Equation (REF ).", "While we have argued that this distribution should be a Gaussian, [71] instead used the relative speed distribution between two particles drawn from an isotropic Maxwellian distribution.", "This overestimates the radial velocity and hence the flux into the cylinder.", "We also note that our rate of flybys agrees with the rate per length derived in equation 3 of [18].", "To confirm that this rate is correct, we perform a simple numerical test.", "We take a cloud of particles with positions drawn from a uniform distribution.", "Each particle is assigned a velocity drawn from a normal distribution with $\\sigma = 100$ km/s in each component.", "The particles are stepped forwards in time and we count the number of particles entering a cylinder, representing the region near a stream, which had a height of 20 and a radius of 1 in arbitrary units.", "We show the comparison in Figure REF as a function of the stream velocity, $v_s$ .", "We see that our simple model in Equation (REF ) captures the bulk of the numerical encounter rate.", "We also see that the numerical rate has a slight dependence on the stream velocity which is due to subhaloes passing through the end caps of the cylinder.", "In Appendix  we derive the rate of subhaloes entering through the end caps which is presented in Equation (REF ).", "We show this model in Figure REF as the dashed red line and we find that it matches the numerical result.", "Figure: Rate of particles entering a cylinder around the stream as a function of the stream's velocity through the cloud of particles.", "The light grey broad line shows the result of our simple numerical experiment.", "The red solid line shows our prediction from Equation ().", "The red dashed line shows our fiducial model plus the contribution of subhaloes entering through the end caps of the cylinder which matches the numerical rate.", "We see that the fiducial model captures the bulk of the rate.", "Finally, the dot-dashed black curve is the rate from ." ], [ "Velocity Distribution of Flybys", "The distribution of flyby velocities is critical for understanding the gaps which subhaloes create since the encounter geometry and velocity affects the gap properties [28].", "While the velocity distribution of subhaloes in the galaxy is given by a Gaussian in each direction, the distribution of the subhaloes that interact with the stream, i.e.", "those that pass within $b_{\\rm max}$ , is not.", "First, the stream is moving in a given direction.", "The relative velocity in this direction, $w_{\\parallel }$ , is given by $ P(w_\\parallel |b_{\\rm max}) = \\frac{1}{\\sqrt{2\\pi \\sigma ^2}} \\exp \\left(-\\frac{(w_\\parallel +v_s)^2}{2\\sigma ^2} \\right) , $ where $v_s$ is the velocity of the stream.", "The distribution of the velocity perpendicular to the stream's motion, $w_\\perp $ , can be derived by considering the distribution of the radial and tangential velocity relative to the stream.", "As subhaloes enter a cylinder with radius $b_{\\rm max}$ around the stream, they have a radial velocity, $v_r$ , and a tangential velocity, $v_\\theta $ .", "As we saw in Section REF , the rate at which particles enter this cylinder is proportional to their radial velocity, e.g.", "Equation (REF ).", "Thus, the radial velocity distribution of subhaloes is given by $ P(v_r|b_{\\rm max}) = \\frac{|v_r|}{\\sigma ^2} \\exp \\left( -\\frac{v_r^2}{2\\sigma ^2} \\right) .", "$ Note that this distribution is only nonzero for negative $v_r$ , i.e.", "for the subhaloes heading towards the stream.", "The velocity distribution in the tangential direction, $P(v_\\theta |b_{\\rm max})$ , is Gaussian with a mean of zero and a dispersion of $\\sigma $ .", "By combining these two velocities into $w_\\perp = \\sqrt{v_r^2 + v_\\theta ^2}$ , we find $ P(w_\\perp |b_{\\rm max}) = \\sqrt{\\frac{2}{\\pi }} \\frac{w_\\perp ^2}{\\sigma ^3} \\exp \\left( - \\frac{w_\\perp ^2}{2\\sigma ^2} \\right) .", "$ Equation (REF ) and Equation (REF ) give us the velocity distribution of the subhaloes which have passed near the stream.", "The dispersion of the flyby speed, $w=\\sqrt{w_\\parallel ^2+w_\\perp ^2}$ , is given by $ \\langle w^2 \\rangle = v_s^2 + 4 \\sigma ^2 .$ From Equation (REF ) and Equation (REF ) we see that the distribution of relative speeds is not simply the relative speed distribution of two particles drawn from an isotropic Maxwellian distribution.", "This is because the radial velocity distribution is biased since subhaloes with higher radial velocities towards the stream are more likely to enter a region near the stream, i.e.", "Equation (REF ).", "As in Section REF , we can check these velocity distributions against a numerical example of subhaloes distributed uniformly in position with an isotropic Maxwellian velocity distribution and look at the properties of subhaloes which enter a cylinder around the stream.", "In Figure REF we compare the numerically derived velocity distribution against our model and find excellent agreement.", "For contrast, we also show the velocity distribution of particles inside the cylinder and the relative speed distribution assumed in [71].", "We also show the velocity distribution of the particles which were initially in the cylinder to emphasize that it is different from those entering the cylinder.", "For this example we used a velocity dispersion of $\\sigma = 100$ km/s, a cylinder with an aspect ratio of 1:10, and a stream velocity of $v_s = 200$ km/s.", "Figure: Velocity distribution relative to stream in numerical example.", "The solid red curve shows the velocity distribution of particles which entered the cylinder in the time interval.", "The dashed blue curve shows the velocity distribution of the particles which were initially inside the cylinder, highlighting that it is different from the distribution of those which enter.", "The solid green curve shows the distribution of our model using Equation () and Equation ().", "The dot-dashed black curve shows the relative speed distribution of two particles drawn from an isotropic Maxwellian distribution used in .", "In this example we have used a stream velocity of v s =200v_s = 200 km/s and a velocity dispersion of σ=100\\sigma = 100 km/s." ], [ "Impact parameter distribution", "Now that we have the velocity distribution of particles which enter a cylinder with radius $b_{\\rm max}$ around the stream, we can compute the distribution of impact parameters.", "A particle which enters the cylinder with a radial velocity of $v_r$ and a tangential velocity of $v_\\theta $ will have an impact parameter of $ b = b_{\\rm max} \\frac{ | v_\\theta |}{\\sqrt{v_\\theta ^2+v_r^2} } .", "$ We can then determine $P(b|b_{\\rm max})$ , the distribution of the impact parameters for the flybys within $b_{\\rm max}$ , by integrating the velocity distributions over all $v_r, v_\\theta $ which have an impact parameter of $b$ , i.e.", "$ P(b|b_{\\rm max}) &= \\int P(v_r|b_{\\rm max}) P(v_\\theta |b_{\\rm max}) \\delta (b - b_{\\rm max} \\frac{ |v_\\theta |}{\\sqrt{v_\\theta ^2+v_r^2} } ) dv_r dv_\\theta , \\nonumber \\\\&= \\frac{1}{b_{\\rm max}} .", "$ Thus we see that the impact parameters are distributed uniformly from 0 to $b_{\\rm max}$ .", "At first sight, this result may seem counterintuitive since if we looked at the subhaloes near the stream at any particular time, the distribution of their distances would increase linearly with distance from the stream.", "However, the impact parameter is the minimum distance between the subhalo's path and the stream track.", "As such, the impact parameter distribution is really the distribution of distances between two lines in three dimensions, which is independent of distance.", "Both [71] and [18] also used a uniform distribution of impact parameters." ], [ "Number density of subhaloes", "Next, we need an estimate of the number density of subhaloes.", "[65] studied the number density profile of subhaloes around a Milky Way-like analogue and found that it is well described by an Einasto profile: $ n_{\\rm sub} \\propto \\exp \\bigg ( - \\frac{2}{\\alpha } \\Big ( (\\frac{r}{r_{-2}})^\\alpha - 1 \\Big ) \\bigg ) , $ with $\\alpha = 0.678$ and $r_{-2} = 0.81 r_{\\rm 200} = 199$ kpc.", "They found that this fit worked for all mass ranges of subhaloes they explored, i.e.", "those from $10^5 M_\\odot $ to $10^{10} M_\\odot $ .", "The host halo they simulated had a mass of $M_{\\rm 200} = 1.839 \\times 10^{12} M_\\odot $ .", "We will scale this down to a mass of $M_{200} = 10^{12} M_\\odot $ , reducing $r_{200}$ by $M^\\frac{1}{3}$ and assume that the scaled down halo has the same concentration and same $\\alpha $ .", "Thus we would expect the same fit with $r_{-2} = 162.4$ kpc.", "Next we have the spectrum and normalization of the subhaloes.", "As in [65], we express the subhalo mass function as $ \\frac{dN_{\\rm sub}}{dM} = a_0 \\Big ( \\frac{M}{m_0} \\Big )^n , $ with $a_0 = 3.26 \\times 10^{-5} M_\\odot ^{-1}$ , $m_0 = 2.52 \\times 10^7 M_\\odot $ , and $n=-1.9$ .", "Note that this was for the total number within $r_{\\rm 50} = 433$ kpc.", "Scaling down to a MW mass of $10^{12} M_\\odot $ (i.e.", "scaling the virial radius by $M^{1/3}$ and the number of subhaloes within the virial radius by $M$ ), we would get $a_0 = 1.77 \\times 10^{-5} M_\\odot ^{-1}$ within 353 kpc.", "This can now be combined with the density profile to get the correctly normalized subhalo profile $ \\frac{dn_{\\rm sub}}{dM} = c_0 \\Big ( \\frac{M}{m_0} \\Big )^n \\exp \\bigg ( - \\frac{2}{\\alpha } \\Big ( (\\frac{r}{r_{-2}})^\\alpha - 1 \\Big ) \\bigg ) ,$ with $c_0 = 2.02\\times 10^{-13} M_\\odot ^{-1} {\\rm kpc}^{-3}$ .", "Using Equation (REF ), the number density of subhaloes in any mass range and location can be computed.", "We note that we found broadly similar results in the public catalogues of Via Lactea II [25] when we looked at the number of subhaloes within 50 kpc.", "This estimate of the number density of subhaloes from the Aquarius simulations [65] is based on a collisionless $N$ -body simulation which neglects baryonic effects.", "[27] found that the presence of a disk with a mass of 10% of that of the host galaxy decreases the number of subhaloes at $10^7 M_\\odot $ by a factor of 3.", "Similar reductions are found over a wide range of masses so we will assume that the disk decreases the abundance of all subhaloes in the inner region by a factor of 3.", "Furthermore, we assume that the presence of the disk only changes the normalization of the number density of subhaloes but not its shape.", "As such, we account for the disk's presence by simply dividing the right-hand side of Equation (REF ) by 3.", "For the properties of individual subhaloes, we make fits to the $M_{\\rm tidal}$ -$v_{\\rm max}$ relation to the subhaloes in VLII [25].", "If we model the subhaloes as Plummer spheres, where we take the Plummer sphere mass to be $M_{\\rm tidal}$ , this gives a scale radius of $ r_s = 1.62 {\\rm kpc}~\\Big ( \\frac{M_{\\rm sub}}{10^8 M_\\odot } \\Big )^{0.5} .", "$ If the fit is instead made for a Hernquist profile, the relation would be $ r_s = 1.05 {\\rm kpc}~\\Big ( \\frac{M_{\\rm sub}}{10^8 M_\\odot } \\Big )^{0.5} .", "$" ], [ "Velocity distribution of subhaloes", "In addition to the number density, we must also specify the velocity distribution of subhaloes.", "As described above, we assume that the velocity distribution of each component is a Gaussian with a mean of zero and a dispersion of $\\sigma $ .", "This simplification neglects the velocity anisotropy of subhaloes seen in simulations, as well as the fact that the velocity distributions in simulations are not Gaussian [26].", "In order to make a prediction for the gap properties in streams around our Galaxy, we need an estimate of this dispersion for the inner region, $r<30$ kpc, of the Milky Way.", "Observationally, the radial velocity dispersion has been measured for a collection of stars, globular clusters, and satellite galaxies and a value of $\\sigma \\sim 120$ km/s within 30 kpc was found [7].", "This value was also used as the fiducial subhalo velocity dispersion by both [71] and [18].", "However, the velocity dispersion of subhaloes appears to be substantially higher than this.", "[56] constructed self-consistent equilibrium models for the Milky Way and found that the dark matter had velocity dispersions of 150-205 km/s near the location of the Sun.", "[26] compared the velocity dispersions of dark matter particles and subhaloes in cosmological simulations and found that the subhaloes had a velocity dispersion which is $\\sim 10\\%$ higher.", "The velocity dispersion can also be computed from cosmological simulations of Milky Way-like galaxies.", "Using the public catalogues of VLII [25], we find velocity dispersions of 160-200 km/s in the three cartesian velocity components for subhaloes within 30 kpc of the Milky Way-analogue, although we note that those simulations do not include the effect of the disk and the halo is more massive than the Milky Way.", "With these results in mind we take $\\sigma = 180$ km/s as our velocity dispersion for each velocity component.", "It is not immediately clear what this increased velocity dispersion means for the number of gaps since while it will result in a larger number of flybys, i.e.", "Equation (REF ), it will also increase the relative speed of the flybys, i.e.", "Equation (REF ,REF ), which results in smaller perturbations to the stream and less pronounced gaps.", "However, the effect of the velocity dispersion was investigated in [15] where they found that the main effect is from the number of flybys and the change in the gap properties is subdominant." ], [ "Evolution of the number density of subhaloes", "Streams are sensitive to the number density of subhaloes they encounter starting from the epoch of the onset of the progenitor's disruption to present day.", "However, they only interact with subhaloes in the radial range which the stream explores.", "[24] studied the number of subhaloes within a fixed mass aperture in a Milky Way analogue.", "Their Figure 6 demonstrates that the number of subhaloes within a shell containing the mass fraction $M/M_{200} < 1/6$ decreases by a factor of almost 3 from $z=1$ to $z=0$ .", "They show that this mass shell has stabilized between $z=3$ and $z=2$ so this estimate of the subhalo disruption can also be thought of as for a given radial range.", "Many of the streams in the Milky Way have been disrupting for a period of time similar to this, and, thus, the change of the subhalo number density with redshift should be taken in account.", "Although not shown in this work, our numerical experiments indicated that the effect on a stream like Pal 5 is not very significant, likely due to the relatively young age of 3.4 Gyr we assume [48], and thus currently we choose to ignore it.", "However the older streams like GD-1 might be more affected.", "We will come back to the importance of the subhalo number density evolution with redshift in future work." ], [ "Stream Gap Fabrication", "Now that we have computed the rate of flybys, we need a model for the effect of each encounter to determine the detectability of the gap it produces, and hence the number of gaps expected.", "[28] presented an analytic model for the evolution of a stream gap after a flyby with an arbitrary geometry, and provided analytic expressions for the width and depth of the gap.", "As in that work, we define the gap depth as $f\\equiv \\rho /\\rho _0$ where $\\rho $ and $\\rho _0$ are the minimum perturbed density and the unperturbed density respectively.", "By combining this analytic model with the rate and properties of the flybys described in the previous Section, we can make a prediction for the properties of the gaps.", "We will provide a forecast for the expected number of gaps deeper than some density threshold, $f < f_{\\rm cut}$ .", "A similar approach was taken in [18] where a rate calculation was combined with fits to the gaps created by N-body simulations of flybys.", "We note that the gap widths and depths we consider are for an observer at the center of the Galaxy." ], [ "Review of gap evolution", "[28] found that the gap growth proceeds in three phases.", "The first phase, the compression stage, is short-lived and leads to a minor increase in the density near the point of closest approach.", "We ignore this short-lived phase and instead focus on the second and third phases, the expansion and the caustic phase respectively, where the gap is created and then becomes wider and deeper.", "During both the expansion and the caustic phase, the gap depth, is given by $ \\frac{\\rho }{\\rho _0} = \\left( 1 + \\frac{4-\\gamma ^2}{\\gamma ^2} \\frac{w_\\perp ^2}{w^3} \\frac{2 G M}{b^2+r_s^2} t \\right)^{-1} , $ where $\\gamma $ is related to the host's gravitational potential, $\\psi (r)$ : $ \\gamma ^2 = 3 + \\frac{r \\partial _r^2 \\psi (r)}{ \\partial _r \\psi (r) } .", "$ During the expansion phase, the size of the gap is given by $ \\Delta \\psi _{\\rm gap} = 2 \\frac{w}{w_\\perp } \\frac{\\sqrt{r_s^2+b^2}}{r_0} + \\frac{2 G M w_\\perp }{w^2 r_0 \\sqrt{r_s^2+b^2} }\\frac{4-\\gamma ^2}{\\gamma ^2} t .", "$ The expansion phase continues until the caustic timescale, $ t_{\\rm caustic} = \\frac{4 \\gamma ^2}{4-\\gamma ^2} \\frac{w^3}{w_\\perp ^2} \\frac{b^2+r_s^2}{GM} , $ after which the caustic phase begins and the gap size is given by $ \\Delta \\psi _{\\rm gap} = 4 \\left( \\frac{4-\\gamma ^2}{\\gamma ^2} \\frac{2 G M}{w r_0^2} t \\right)^{\\frac{1}{2}} .", "$ These expressions describe the average time evolution of the gap density and size.", "Thus, given any impact, we can rapidly compute the density and size of the gap.", "By integrating over the impact parameter, subhalo velocity, and impact time, we can determine the distribution of gaps that are created.", "In this work we will assume that the rotation curve is locally flat and hence that $\\gamma ^2 = 2$ .", "These expressions were derived assuming that the unperturbed stream is on a circular orbit and neglect both the eccentricity of the stream's orbit, and the energy and angular momentum dispersion in the stream.", "These effects were studied in [61] which found that while the picture in [28] is mostly correct, the dispersion in the stream can cause the density in gaps to plateau and that gaps can grow at slightly different rates depending on their location along the stream due to energy sorting of debris." ], [ "Effective N-body Simulation", "In order to test the limits of the analytic picture above, we need to compare it against simulations.", "Since an N-body disruption of a globular cluster progenitor can take several tens of CPU hours to run, it is not feasible to investigate a large number of flybys.", "Instead, we have developed an effective N-body simulation where we first run a disruption of a progenitor to produce a stream.", "We then take an earlier snapshot of the simulation and compute the velocity kicks from a single subhalo flyby using the impulse approximation, accounting for the stream curvature as in [61].", "The particles whose fractional energy change due to the kick is larger than some threshold are then evolved to the final time as tracers in the host potential.", "The particles within the progenitor do not receive a kick since they will be affected by the gravitational field of the progenitor and cannot be treated as tracers.", "At the final time, the perturbed particles are combined with the unperturbed particles from the final snapshot to give all particles in the stream.", "This method allows us to rapidly evaluate the gap profile from a variety of impacts.", "For the N-body simulation, we simulate a Pal 5-like stream whose progenitor matches the measured line-of-sight velocity and proper motions in [48] at the present time.", "These simulations are performed with the N-body part of gadget-3 which is similar to gadget-2 [64].", "The progenitor is modelled as a King profile with a mass of $2\\times 10^4 M_\\odot $ , a scale radius of 15 pc, $w=2$ , and is modelled with $10^5$ equal mass particles and a softening of 1 pc.", "The progenitor is evolved in the MWPotential2014 potential given in [12] with the bulge replaced with a Hernquist profile with $M=5 \\times 10^9 M_\\odot $ and a scale radius of $0.5$ kpc.", "The simulation is run for 5 Gyr and snapshots are stored every 10 Myr.", "We considered a wide range of impactors from $10^5$ to $10^8 M_\\odot $ and found that evolving the particles whose change in energy exceeded a threshold of $\\Delta E/E \\ge 0.1$ was sufficient to reproduce the effect of the flyby.", "These simulations allow us to gauge when our simple analytic model begins to break down, i.e.", "when the velocity kicks become sufficiently small that the velocity dispersion in the stream becomes important, as well as when particles begin to fill in the gap.", "Specifically, we consider a $1.6 \\times 10^6 M_\\odot $ subhalo and sample a grid of subhalo velocities in each direction, $-500$ to 400 km/s in steps of 100 km/s, a grid of impact parameters, from 0 to 1 kpc in steps of $0.1$ kpc, and four different impact times of 1,2,2.08, and 3 Gyr ago.", "We find that the flybys which have a maximum velocity kick of $\\Delta v \\sim 0.1$ km/s can still produce an appreciable gap depth as shown in Figure REF where we compare the maximum velocity kicks for gaps of different ages.", "Somewhat surprisingly, this is substantially smaller than the velocity dispersion in the stream, $\\sigma \\sim 1$ km/s.", "We note that this cutoff to produce appreciable gaps will depend on the mass of the progenitor.", "Figure: Comparison of density from effective N-body simulations with maximum velocity kick for flybys of subhaloes with a mass of 1.6×10 6 M ⊙ 1.6 \\times 10^6 M_\\odot evolved for varying amounts of time.", "We see that low velocity kicks do not produce a substantial gap and as the velocity increases, the gaps get deeper.", "We chose 0.1 km/s (the vertical black dashed line) as the cutoff since for these velocities we reliably get a significant depletion.", "The velocity kick where appreciable gaps are produced is similar for a range of masses so we use the same cutoff for all masses.", "The large difference between the gaps produced 2 Gyr ago and 2.08 Gyr is due to the stream being at a different orbital phase.The same simulations can be used to test the density formula given in Equation (REF ).", "We use the same grid as described above for the impacts 1 Gyr ago.", "For each flyby, the expected density using Equation (REF ) is also computed.", "These are compared in Figure REF which demonstrates that while the gaps do get filled in (the numerical density is slightly higher than the analytic approximation), the agreement is rather good.", "We note that the level of agreement depends on the phase at which the gap is observed and that the numerical gap depth oscillates between being deeper and shallower than our analytic prediction depending on this phase.", "Thus our analytic model should be thought of as giving the average evolution of the gap depth.", "Figure: Comparison of analytic density result versus the densities computed with the effective N-body for subhaloes with a mass of 10 6 M ⊙ 10^6 M_\\odot 1 Gyr after impact.", "The analytic result matches the numerically computed density, justifying the method.", "The match is similar for more massive subhaloes." ], [ "Expected Properties of stream gaps", "With the method now in place, the properties of gaps created by a distribution of subhaloes can be explored.", "First, we consider a single subhalo and find the properties of the gaps it creates.", "It turns out that the stream gaps produced by a single DM clump have a characteristic size which depends on the age of the stream.", "Then, we consider subhaloes with masses drawn from the $\\Lambda $ CDM distribution and examine the properties of the gaps thus created.", "In this section, the properties of the fiducial stream model, i.e.", "its age, velocity dispersion, and orbital properties, are chosen to be similar to those of the Pal 5 stream given in [48], namely we assume the age of the stream is 3.4 Gyr and that it is on a circular orbit with a radius of 13 kpc and a velocity of 220 km/s [14]." ], [ "Gap density threshold and minimum gap size", "When discussing observable gaps, we must introduce a density threshold below which the gap can be detected.", "As we saw in Section , subhalo flybys can produce arbitrarily shallow gaps which evidently will not be observable.", "However, since the gap size and gap depth both grow with time, in order for a gap to be deeper than some threshold, the gap must have grown to a certain extent.", "Thus, gap density and gap width are closely linked, and a gap density threshold gives a corresponding minimum gap size.", "Expressions for the gap size are given in Equation (REF ) and Equation (REF ).", "If the gap has a depth of $f=\\rho /\\rho _0$ , this gives a gap size of $ \\Delta \\psi _{\\rm gap} = B\\left(1 + f^{-1}\\right) , $ in the expansion phase and $ \\Delta \\psi _{\\rm gap} = 4 B \\sqrt{f^{-1}-1} , $ in the caustic phase, where $ B = \\frac{\\sqrt{b^2+r_s^2}}{r_0} \\frac{w}{w_\\perp } .$ Thus we see that a deep gap with $f \\sim 0.5$ will have a gap size of several $B$ as seen from the center of the galaxy.", "The value of $B$ is the smallest for a direct impact where the relative velocity is perpendicular to the stream's motion.", "In this case, $B = \\frac{r_s}{r_0}$ .", "For example, if the stream is located at a typical Galactocentric distance we assume for Pal 5, $r_0 \\sim 13$ kpc, a modest impactor with a mass of $10^7 M_\\odot $ with a scale radius of $r_s \\sim 500$ pc will have $B = 2.2^\\circ $ .", "Likewise, a smaller subhalo with a mass of $10^6 M_\\odot $ and a scale radius of $r_s = 100$ pc will have $B = 0.4^\\circ $ .", "Therefore, it is clear that the smallest gap size will be at least $1^\\circ $ and likely larger since most of the impacts will not be direct and will have velocity components which are aligned with the stream's motion.", "As a result, it is required that deep gaps should be fairly large, at least several degrees, or, in other words, deep gaps smaller than this size are not expected.", "Figure REF shows how the minimum gap size depends on the gap density threshold, $f_{\\rm cut}$ .", "As evidenced by the Figure, even for a $10^6 M_\\odot $ subhalo, the minimum gap size is larger than $1^\\circ $ , while an object with a mass of $10^7 M_\\odot $ will produce gaps in excess of $4^\\circ $ .", "Figure: Minimum gap size as a function of density in gap center for various mass subhaloes.", "These curves come from Equation () for the growth phase with f cut >1/9f_{\\rm cut} > 1/9 and Equation () for the caustic phase with f cut <1/9f_{\\rm cut} < 1/9.", "The discontinuity at f cut =1/9f_{\\rm cut} = 1/9 is due to the fact that the leading order contribution to the gap size is not continuous as the gap progresses from the expansion to the caustic phase." ], [ "Gap size distribution for single subhalo population", "Let us now explore the gap size distribution created by a homogenous population of subhaloes with a single mass and scale radius.", "This can be achieved by sampling the distribution of flyby velocities and impact parameters.", "In practice, we randomly draw the parallel and perpendicular flyby velocities from Equation (REF ) and Equation (REF ) respectively, and the impact parameter from a uniform distribution between 0 and $b_{\\rm max}=5 r_s$ .", "Finally, we draw the impact epoch from a linear distribution since the stream grows roughly linearly in time [15].", "Figure REF shows the gap size distribution for three different density thresholds, $f_c$ .", "The dashed vertical lines show the minimum gap size given by Equation (REF ).", "We use a Plummer sphere subhalo with $M=10^7 M_\\odot $ and a size of $r_s = 512$ pc.", "We see that as we decrease the density threshold, i.e.", "as we require deeper gaps, the gap size increases since in the time it takes the gap to achieve such a depth, it will also have grown to a substantial size.", "Figure: Distribution of gap sizes for a subhalo with M=10 7 M ⊙ M=10^7 M_\\odot and r s =512r_s = 512 pc with varying gap density thresholds.", "The vertical dashed black lines come from Equation ().", "We see that as the threshold decreases, the minimum gap size increases.", "This is expected since gaps which have had time to grow sufficiently deep will also have grown sufficiently large.In Figure REF we examine how the gap size varies as we change the subhalo properties.", "We consider three different subhaloes with masses of $10^6 M_\\odot , 10^7 M_\\odot ,$ and $10^8 M_\\odot $ with scale radii of 162 pc, 512 pc, and 1.62 kpc respectively.", "As the Figure demonstrates, each subhalo creates gaps with a characteristic scale and this scale increases with the mass of the perturber.", "Thus, rather intuitively, the gap size is related to the mass of the subhalo which created the gap as was first pointed out in [71].", "We also see that even for low mass subhaloes with $M=10^6 M_\\odot $ , the characteristic gap size is $\\sim 3^\\circ $ , while for those with $M=10^7 M_\\odot $ , the typical width is of order of $\\sim 10^\\circ $ .", "Figure: Distribution of gap sizes created by subhaloes with varying mass and scale radii which create a gap deeper than 50%.", "The vertical dashed black lines come from Equation ().", "We see that as we increase the mass of the subhalo, we get a corresponding increase in the gap size.", "We also see that each mass subhalo creates gaps of a characteristic size.", "We emphasize that these gaps are quite large: even the 10 6 M ⊙ 10^6 M_\\odot subhalo creates gaps with a characteristic size of ∼3 ∘ \\sim 3^\\circ .Next, we consider the effect of changing the age of the stream.", "In Figure REF , we show the distribution of gap sizes created by subhaloes of the same mass, but with three different stream ages.", "We see that as the stream grows in age, so too do the sizes of the gaps in the stream.", "This is simply because the gaps in these streams had more time to expand.", "Figure: Distribution of gap sizes created by a M=10 7 M ⊙ M=10^7 M_\\odot and r s =512r_s = 512 pc subhalo in streams of various ages which create gaps deeper than 50%.", "The vertical dashed black line comes from Equation ().", "Unsurprisingly, we see that for older streams, the gaps are larger since they have had more time to grow.Finally, we explore how the gap size depends on the velocity cutoff we use.", "This cutoff specifies what the maximum velocity of a subhalo kick must be in order to be included.", "As we discussed in Section REF , our effective N-body simulations suggest that detectable density depletions exist for kicks with $\\Delta v > 0.1$ km/s.", "In Figure REF we show the gap size distributions for various $\\Delta v$ thresholds.", "We see that requiring $\\Delta v > 0.1$ km/s captures almost all of the visible kicks so our model is not missing very much in this example.", "We note that the gaps used in this figure were required to have $f<0.5$ and that for shallower gaps, a larger fraction of the kicks would be below the velocity kick threshold.", "Figure: Distribution of gap sizes for a subhalo with M=10 7 M ⊙ M=10^7 M_\\odot and r s =512r_s = 512 pc for gaps deeper than 50%, i.e.", "f<0.5f<0.5.", "The colored lines correspond to different thresholds for the maximum velocity kick during the flyby.", "The vertical dashed black line comes from Equation ().", "We note that the curve with no cut and the curve with Δv>0.1\\Delta v > 0.1 km/s are indistinguishable.As we will discuss in Section REF , our approach is based on circular orbits and does not account for the eccentricity of the stream.", "This eccentricity causes the gap size to oscillate with the galactocentric distance as $r^{-2}$ , e.g.", "Fig.", "13 of [61].", "As a result, the distribution of gap sizes will depend on the exact phase at which the stream is measured and our results should be thought of as giving the average behavior of the gap size.", "These effects are beyond the scope of this work but are included in the recent work of [15] who find broadly similar results by studying the power spectrum of the perturbed stream density and find the the majority of the power is on scales larger than 10$^\\circ $ ." ], [ "Gap depth for single subhalo population", "The detectability of the stream gap depends not only on its size, but also on the density contrast between the center of the gap and the unperturbed stream.", "This Section therefore looks at the gap depth distribution.", "We repeat the same procedure as in Section REF and sample the appropriate distributions for velocity components, impact parameter, and impact time.", "Figure REF shows the distribution of gap depths imparted by three different mass subhaloes.", "As the mass of the perturber is decreased, the number of shallow gaps decreases but the number of deep gaps is unchanged.", "The decrease at the shallow end is due to the imposition of a minimal velocity kick, which affects the lower mass subhaloes more.", "The convergence for deep gaps is due to the fact that for direct impacts, and hence the deepest gaps, the gap depth itself (i.e.", "Equation (REF )) does not depend on mass for the scaling between mass and scale radius used here.", "Figure: Distribution of gap depths for a subhalo with varying mass and scale radius which create gaps larger than 1 ∘ 1^\\circ .", "As the mass is decreased, the number of shallow gaps decreases while the behavior for deep gaps remains unchanged.", "This decrease is due to the requirement of a minimum velocity kick of Δv>0.1\\Delta v > 0.1 km/s.In Figure REF we show the effect of changing the age of the stream on the distribution of gap depths.", "As expected, younger streams in which gaps have less time to grow have shallower gaps and older streams have deeper gaps.", "Figure: Distribution of gap depths for a subhalo with a mass of M=10 7 M ⊙ M=10^7 M_\\odot and r s =512r_s = 512 pc in streams of various ages which create gaps larger than 1 ∘ 1^\\circ .", "As expected the younger streams have shallower gaps and the older streams have deeper gaps." ], [ "Gap size distribution from $\\Lambda $ CDM background", "Having looked at the individual facets of the stream gap behaviour, we combine the intuition gained in the previous sub-sections to analyse the gap properties expected from a population of $\\Lambda $ CDM subhaloes.", "We consider subhaloes with masses in the range $10^5-10^9M_\\odot $ .", "Subhaloes below this mass range create too small of a velocity kick to cause a significant density depletion.", "We repeat the same procedure as in Section REF , marginalizing over the flyby velocities, impact parameter, and impact time, as well as marginalizing over the subhalo mass assuming cosmologically motivated halo mass function described above.", "Figure REF gives the expected distribution of gap sizes these subhaloes would create for various gap density thresholds.", "We see that the peak of the distribution depends on what density threshold is used but for a feasible depth of $f<0.75$ , the characteristic scale is on the order of several degrees.", "Figure: Distribution of gap sizes for subhaloes with 10 5 M ⊙ <M<10 9 M ⊙ 10^5 M_\\odot < M < 10^9 M_\\odot for the Pal 5 stream.", "This distribution is not sensitive to masses below 10 5 M ⊙ 10^5 M_\\odot since most of the velocity kicks from those subhaloes are too small to create a visible gap.In Figure REF we show the two dimensional distribution of gap sizes and depths created from a $\\Lambda $ CDM spectrum of subhaloes with masses between $10^5-10^9 M_\\odot $ .", "For shallow gaps, there are a wide range of gap sizes with the larger gap sizes dominating.", "However, as we proceed to deeper gaps, the lower mass subhaloes become more important which leads to smaller gaps.", "This somewhat counterintuitive result is due to the requirement of a minimum velocity kick.", "For deep gaps, this is satisfied for both low and high mass subhaloes and the gap size is dominated by the low mass subhaloes since they are more numerous.", "However, for shallow gaps, many of the low mass flybys will produce a negligible kick (e.g.", "Fig.", "REF ) so the shallow gaps have a larger contribution from larger subhaloes and hence larger gaps.", "Figure: Two dimensional distribution of gap depths and sizes for subhaloes with 10 5 M ⊙ <M<10 9 M ⊙ 10^5 M_\\odot < M < 10^9 M_\\odot for the Pal 5 stream with velocity kicks above Δv>0.1\\Delta v > 0.1 km/s.", "For very shallow gaps, there is a larger range of gap sizes as seen in Fig.", "which shows various slices of this figure.", "As we move towards deeper gaps, the typical gap size becomes smaller.", "This is because of the increasing influence of the lower mass haloes which create smaller gaps." ], [ "Number of gaps expected in observed MW streams", "Now that we have explored the properties of gaps from a realistic population of subhaloes, we can compute these quantities for the stellar streams observed in the Milky Way.", "Here, we will focus on globular cluster streams since these have the smallest velocity dispersion and hence should be sensitive to the widest range of subhalo masses [29].", "There are at least 13 claimed globular cluster streams to date [36].", "In Table REF we give the properties of 6 of these streams.", "These stream characteristics can be used to make a prediction for the expected number of gaps in each case.", "To proceed, the estimate of the progenitor mass is obtained from the model of the stream width described in [30].", "Finally, the stream's age can be gleaned using a simple model for the stream growth rate which we will present below.", "Note that the fiducial stream used in Section  was based on Pal 5 so Figure REF shows the distribution of gap sizes expected in Pal 5.", "Table: Observed properties for six cold streams around the Milky Way.", "Pal 5 data: a ^a , b ^b, c ^c .", "GD-1 data: d ^d , e. Acheron and Sytx data: f ^f.", "Tri/Psc data: g ^g, h ^h, i ^i.", "ATLAS data: j ^j.", "Phoenix data: k ^k.", "Note that the ages for Styx, Tri/Psc, ATLAS, and Phoenix come from the stream length results in Sec. .", "For the same four streams, the progenitor masses are estimated using their respective streams widths and the results of .", "We note that the inferred age of the Styx stream is greater than the age of the universe so we use an age of 13 Gyr when computing its gap properties.", "Note that we only have information on the pericenters and apocenters for Pal 5 and GD-1.", "For the other four streams these fields are left blank." ], [ "Estimating the age of a stream", "As we saw in Section REF and Section REF , the properties of gaps depend on the age of the stream.", "This is because gaps will have had more time to grow in older streams.", "In addition to changing the properties of the gaps, the age of the stream also determines how many interactions there will have been, as prescribed in Equation (REF ).", "As such, in order to estimate the number of gaps in a stream we need to know the age of the stream.", "For some of the streams we consider here, dynamical modelling has already been performed using Lagrange point stripping methods, e.g.", "[48] for Pal 5 and [16] for GD-1.", "This modelling gives an estimate of the age for both of these streams.", "However, for the other four streams there are no such estimates yet.", "Let us use the observed width of the stream to estimate the mass of the progenitor.", "The evolution of the width of the stream perpendicular to the orbital plane was studied in [30] for both axisymmetric and triaxial potentials.", "Here, we will just use the result for spherical potentials that the stream width is given by $ w = \\frac{1}{\\sqrt{2}} \\frac{\\sqrt{\\frac{Gm}{3 r_{\\rm tidal}} }}{v_{\\rm peri}} , $ where $m$ is the mass of the progenitor, $r_{\\rm tidal}$ is the tidal radius, and $v_{\\rm peri}$ is the velocity of the progenitor at pericenter.", "For the streams whose orbits we do not know, we will assume they are on circular orbits and the orbital velocity is given by $ v = \\sqrt{\\frac{GM(<r)}{r}} ,$ where $M(<r)$ is the mass of the host potential enclosed within a radius of $r$ .", "For a progenitor on a circular orbit, the tidal radius is given by $ r_{\\rm tidal} = r \\Big ( \\frac{m}{(4-\\gamma ^2) M(<r)} \\Big )^{\\frac{1}{3}} , $ where $\\gamma $ is given by Equation (REF ).", "For a host galaxy with a flat rotation curve, i.e.", "a logarithmic potential, we get $\\gamma ^2 = 2$ .", "Plugging these into the formula for the width we see $ w = \\frac{1}{\\sqrt{6}} \\Big ( \\frac{m}{2 M(<r)} \\Big )^{\\frac{1}{3}} .", "$ Finally, this can be arranged to give $ m = 2^{\\frac{1}{2}} 3^{\\frac{3}{2}} w^3 M(<r) .", "$ Using the widths of these streams reported in [30] we give the estimates of their progenitor masses in Table REF .", "For Pal 5, we assume a circular velocity of 220 km/s at a radius of 13 kpc, as measured in [14], and get a progenitor mass of $2.3 \\times 10^4 M_\\odot $ , very similar to that reported in [48].", "For GD-1, we assume a circular velocity of 220 km/s at a radius of 19 kpc and get a mass of $1.2 \\times 10^4 M_\\odot $ , which is below the range of masses considered in [16] although they have broad broad posteriors.", "Instead of this estimate, we use the central value from [16] of $10^5 M_\\odot $ but we will also discuss the gap predictions if GD-1 is as low mass as our method suggests.", "For Styx, we assume a circular velocity of 190 km/s at a radius of 45 kpc, as measured by [23], and get a mass of $2.6 \\times 10^5 M_\\odot $ .", "For Tri/Psc, we assume a circular velocity of 190 km/s at a radius of 40 kpc and get a mass of $2.4\\times 10^4 M_\\odot $ .", "For ATLAS, we assume a circular velocity of 220 km/s at a radius of 22 kpc and get a mass of $1.3 \\times 10^4 M_\\odot $ .", "Finally, for Phoenix we assume a circular velocity of 220 km/s a radius of 19 kpc and get a mass of $3.8 \\times 10^4 M_\\odot $ ." ], [ "Estimating the stream age", "With the progenitor mass in hand, we can now estimate the age of a stream.", "Once again, the stream is assumed to follow a circular orbit.", "The length of a stream can be estimated by considering small perturbations to the circular orbit of the progenitor and determining how quickly these stars move away from the release point.", "This is similar to the analysis presented in [28], where the effect of changing the velocity is considered.", "However, we must also include the fact that stripped material will be offset from the progenitor in radius, i.e.", "it is released from the Lagrange points.", "Thus, we find that the stream length in radians grows as $ l \\sim 2 \\frac{(4-\\gamma ^2)^{\\frac{2}{3}}}{\\gamma ^2} \\Big ( \\frac{m}{M} \\Big )^{\\frac{1}{3}} \\frac{v t}{r} , $ where $v$ is the circular velocity at the radius $r$ .", "We note that similar expressions have been derived in [3] (see their equation 20).", "Equation (REF ) can then be re-arranged to give an estimate for the age of the stream in terms of its length and mass, as well as the mass of the host enclosed within that radius.", "As above, we assume that the potential has a flat rotation curve so $\\gamma ^2 = 2$ , and turn Equation (REF ) into $ t = \\frac{2^{-\\frac{2}{3}} l }{ \\Big (\\frac{m}{M}\\Big )^{\\frac{1}{3}} \\sqrt{\\frac{GM}{r^3}} } .", "$ For each of the streams, we give an estimate of their age in Table REF .", "For Pal 5 we get an age of 3.4 Gyr which matches [48].", "For GD-1, if we use a mass of $10^5 M_\\odot $ we get an age of 3.8 Gyr, in the center of the range found in [16].", "We will use a value of 4 Gyr but we note that if we instead use our inferred mass, we would get an age of 7.7 Gyr, making GD-1 substantially older than previous fits have suggested.", "We note that the estimates of the progenitor mass and stream age in this section relied on results from circular orbits, however stream modelling for more realistic streams on eccentric orbits can also be used to estimate these quantities more robustly [32], [11], [60], [48], [31], [16].", "With these ages, we can now estimate the number and properties of the gaps in these streams.", "We can now make tailor-made predictions for each of the six streams reported in Table REF , starting with Pal 5.", "Based on Table REF , we will model Pal 5 as being on a circular orbit with a radius of 13 kpc.", "The number density of subhaloes between $10^6-10^7 M_\\odot $ at this radius is $1.01 \\times 10^{-3} {\\rm kpc}^{-3}$ .", "Assuming a flat rotation curve with a circular velocity of 220 km/s found in [14], we get an orbital period of 360 Myr.", "Figure REF shows the distribution of gap sizes in Pal 5 from subhaloes with a mass in the range $10^5 M_\\odot - 10^9 M_\\odot $ .", "It predicts that Pal 5 should have a characteristic gap size of $\\Delta \\psi _{\\rm gap} \\sim 4-7^\\circ $ for deep gaps with $f<0.5$ but the distribution is quite broad.", "This also guides the scales on which one should be looking for gaps in Pal 5.", "In addition to the distribution of gap sizes, we can also compute the expected number of gaps.", "We do this by performing the same marginalization over the flyby velocities, impact parameter, impact time, and subhalo mass described in Section REF .", "We include all flybys within 5 kpc of the stream and only find a 1-2% change in the number of gaps if we extend this to 10 kpc, suggesting that our results have converged.", "In Figure REF we show the expected number of gaps in Pal 5 deeper than a given threshold.", "We see that over a range of gap density thresholds, the subhaloes from $10^6 M_\\odot -10^8 M_\\odot $ produce the dominant contribution.", "Above a gap density threshold of $f_{\\rm cut} \\sim 0.75$ , gaps from the subhaloes with a mass of $10^7 M_\\odot - 10^8 M_\\odot $ outnumber those from the $10^6-10^7 M_\\odot $ range.", "We see that if we take $f_{\\rm cut} = 0.5$ we expect to find 0.3 gaps in Pal 5.", "However, if we take $f_{\\rm cut} = 0.75$ , we would expect almost 0.7 gaps.", "Thus, the null detection reported in [43] is not be very surprising although as noted above, the search was performed on scales significantly smaller than described here.", "The prediction for the number of gaps in Pal 5, and the other five streams, is given in Table REF .", "Figure: Expected number of gaps deeper than f cut f_{\\rm cut} with gap sizes bigger than 1 ∘ ^\\circ in Pal 5.", "The different colored curves show the expected number of gaps in different mass ranges.", "We see that gaps from subhaloes with masses between 10 6 -10 7 M ⊙ 10^6-10^7 M_\\odot dominate for most density thresholds.", "Combining all the mass ranges together, Pal 5 should have ∼\\sim 0.7 gaps with densities less than 0.75.Table: Expected number of gaps and their characteristic size for six observed cold streams ranked by the number of gaps.", "The second and fourth column respectively give the number of gaps deeper than 50% and 75% of the unperturbed density.", "The third and fifth column give the most common gap size for gaps deeper than 50% and 75% of the unperturbed density.", "Styx has by far the most expected gaps and if it is a cold stream with the reported length, it will be the best candidate for finding subhaloes.", "Tri/Psc, Pal 5, and GD-1 all have a relatively similar number of gaps we can expect at least one deep gap with f<0.5f<0.5 amongst them.", "These predictions include the gaps created from subhaloes with masses in the range 10 5 M ⊙ <M<10 9 M ⊙ 10^5 M_\\odot < M < 10^9 M_\\odot and account for the factor of 3 depletion expected from the presence of the MW disk ." ], [ "GD-1", "GD-1 is modeled as being on a circular orbit with a radius of 19 kpc.", "The number density of subhaloes between $10^6-10^7 M_\\odot $ at this radius is $8.66 \\times 10^{-4} {\\rm kpc}^{-3}$ .", "Assuming a circular velocity of 220 km/s, we get an orbital period of 530 Myr.", "Performing the same sampling as for Pal 5, we find 0.3 and 0.6 gaps in GD-1 for $f<0.5$ and $f<0.75$ respectively.", "This is roughly the same as the prediction for Pal 5.", "Since the age and length of GD-1 and Pal 5 are similar, the distribution of gap depths will look very similar to Figure REF .", "However, if we instead use the age of 7.7 Gyr we found in Section REF , we would expect 0.9 and 1.7 gaps for $f<0.5$ and $f<0.75$ respectively." ], [ "Tri/Psc", "Next we model the Tri/Psc stream.", "We take it to be on a circular orbit at 40 kpc.", "The number density of subhaloes between $10^6-10^7 M_\\odot $ at this radius is $5.51 \\times 10^{-4} {\\rm kpc}^{-3}$ .", "Taking a circular velocity of 190 km/s [23], we get an orbital period of 1.3 Gyr.", "The number of gaps in Tri/Psc is 0.9 and 1.6 for $f<0.5$ and $f<0.75$ respectively.", "Thus, Tri/Psc is a better candidate in the search for gaps than GD-1 or Pal 5.", "Furthermore, Tri/Psc is sufficiently far away from the disk that the factor of 3 depletion may be an overestimate.", "[27] found that the depletion is the strongest near the disk and decreases as we move away from the disk.", "Figure REF presents the cumulative distribution of gap depths for Tri/Psc.", "This can be compared against Figure REF where we show the distribution for Pal 5.", "Since Tri/Psc is both longer and older than Pal 5, there are more gaps.", "In addition, the increased age of Tri/Psc gives the gaps more time to grow and hence it has a larger fraction of deep gaps.", "This can also be seen in Figure REF where the effect of the stream age on the distribution of gap depths was examined and the same result was found.", "Figure: Expected number of gaps deeper than f cut f_{\\rm cut} with gap sizes bigger than 1 ∘ ^\\circ in Tri/Psc.", "The different colored curves show the number of gaps in various mass ranges.", "As with Pal 5, we see that subhaloes in the range 10 6 -10 8 M ⊙ 10^6-10^8 M_\\odot produce the most subhaloes." ], [ "ATLAS", "We model the ATLAS stream as being on a circular orbit with a radius of 22 kpc.", "The number density of subhaloes between $10^6 - 10^7 M_\\odot $ at this radius is $8.06 \\times 10^{-4} {\\rm kpc}^{-3}$ .", "Taking a circular velocity of 220 km/s, we get an orbital period of 610 Myr.", "The expected number of gaps in the ATLAS stream is 0.02 and 0.1 for $f<0.5$ and $f<0.75$ respectively.", "This small number of gaps is due to the young age of ATLAS and the fact that the currently observed stream is quite short.", "Due to its young age, ATLAS will have shallower distribution of gaps than Pal 5." ], [ "Phoenix", "Phoenix is modelled as being on a circular orbit with a radius of 19 kpc.", "The number density of subhaloes between $10^6-10^7 M_\\odot $ at this radius is $8.66 \\times 10^{-4} {\\rm kpc}^{-3}$ .", "Assuming a circular velocity of 220 km/s, we get an orbital period of 530 Myr.", "The number of gaps in Phoenix is 0.01 and 0.06 for $f<0.5$ and $f<0.75$ respectively.", "As with ATLAS, this is due to the young age of Phoenix and its short length.", "As a result, neither Phoenix nor ATLAS appear to be good candidates for detecting gaps.", "However, we note that [35] have recently suggested that the Phoenix stream may be part of a significantly longer stream which includes the Hermus stream [34].", "The purported length of 76 kpc would make it the longest cold stream in the Milky Way and hence an ideal candidate for studying gaps.", "Since the increased length of a stream also increases its estimated age, the expected number of gaps increases roughly quadratically with the stream's length.", "Thus, if the estimated length is correct, the combined stream would have approximately 20 gaps deeper than $f<0.75$ ." ], [ "Styx", "The Styx stream [33] is substantially longer than the other streams.", "If the stream is indeed a cold stream, then it is one of the best candidates for detecting subhaloes.", "We model Styx as being on a circular orbit with a radius of 45 kpc and take its age to be 13 Gyr, younger that the estimate in Table REF but consistent with the age of the universe.", "The number density of subhaloes between $10^6-10^7 M_\\odot $ at this radius is $5.01 \\times 10^{-4} {\\rm kpc}^{-3}$ .", "Assuming a circular velocity of 190 km/s, we get an orbital period of 1.46 Gyr.", "Sampling over the flybys we find 6 and 9 gaps expected with $f<0.5$ and $f<0.75$ respectively.", "Thus, Styx could have an order of magnitude more gaps than the second best stream, Tri/Psc.", "However, [33] argued that Styx is a dwarf galaxy stream.", "If this is correct, the number of gaps will decrease since the stream is younger and the stream from a dwarf galaxy is substantially hotter which will mask out many of the expected gaps." ], [ "Numerical Testing", "The framework we have discussed here has mostly been based on analytic methods, using the gap size formulae from [28] and the number of properties of the flybys as derived in Section .", "In this section we will test these assumptions and see how well they work." ], [ "Flyby properties", "First we compare the properties of the subhalo flybys.", "We take the N-body simulation of Pal 5 described in Section REF and include three times the expected subhalo population between $10^5-10^6 M_\\odot $ using the fits from the number density profiles in Aquarius [65] as described in Section REF .", "We include three times the expected population to improve the statistics since the total number of flybys is not that large.", "Note that we also have not decreased the number density of subhaloes by a factor of 3 due to the effect of the MW disk so in reality, this example has roughly 9 times the expected subhaloes.", "These subhaloes are included as tracer particles which are sourcing the force expected if they were Hernquist profiles with their given mass and scale radius.", "The simulation is run for 3.4 Gyr which is the best fit age from [48].", "Despite including significantly more substructure than expected, we see only a minor effect in the density from these low mass subhaloes.", "While the simulation is running, at each timestep we record whenever a subhalo passes within 2 kpc of a stream particle and record the position and velocity of both the subhalo and stream particle.", "During the simulation 2668 subhaloes passed within 2 kpc.", "We repeat our analysis above based on the stream length in the simulation at the present time and estimate that 2320 subhaloes should pass within 2 kpc of the stream in this time.", "Thus even though the model was based on circular orbits, it gives a good estimate for the number of flybys for a realistic stream on an eccentric orbit.", "In Figure REF we examine the distribution of relative flyby velocities for subhaloes which pass within 2 kpc.", "For each flyby in the simulation, we find the stream particle which has the smallest impact parameter and use the relative velocity to the subhalo.", "The figure also shows the prediction of our model and the relative velocity distribution from [71].", "We see that our model is a better match than using the relative velocity in a Maxwellian distribution.", "However, our model is not a perfect match because it does not account for the eccentricity of the stream.", "Figure: Distribution of relative velocities between the stream and the subhalo flyby.", "In solid blue is the result of an N-body simulation of a Pal 5-like stream described in the text.", "In solid green is the model in this work from Equation () and Equation ().", "In dashed black is the relative velocity of particles in a Maxwellian distribution which was used in .", "While the agreement is not as good as in Fig.", ", our model roughly matches the distribution in the simulation." ], [ "Distribution of gap depths and sizes", "In addition to testing our assumptions about the flyby properties, we can also test how well our formalism works for determining the gap depths.", "We take the stream described in Section REF and use the effective N-body formalism described in Section REF to simulate a large number of impacts.", "The properties of these impacts were chosen to cover a large range in the parameter space as follows: each component of the subhalo velocity was chosen uniformly from -500 to 400 km/s in steps of 100 km/s relative to the host potential, the impact parameter was chosen uniformly from 0 to 1 kpc in steps of 100 pc, five different impact times are chosen as 0.5, 0.85, 1, 1.13, 2 Gyr, and finally the subhalo mass is chosen uniformly in log space from $10^5 M_\\odot $ to $1.024 \\times 10^8 M_\\odot $ in steps of $\\log _{10}(2)$ .", "The position of the impact along the stream was chosen to be halfway between the progenitor and the end of the stream.", "Each subhalo is modelled as a Plummer sphere with a scale radius given by Equation (REF ).", "For each of these samples the effective N-body simulation is run, resulting in 500,000 simulated flybys.", "For each flyby, the particles are binned into $0.1^\\circ $ bins and the density is saved for the region within 20$^\\circ $ of the gap center.", "This density is then divided by the unperturbed density to get the gap depth and size.", "In addition, we use our model to make a prediction of the gap depth.", "For the model, we assume the stream is on a circular orbit at 15 kpc and compute the gap properties described in Section .", "In Figure REF we compare the distribution of gap depths between the effective N-body simulation with that of our analytic model and find a good match.", "There is a slight discrepancy in that the model predicts deeper gaps and in Section REF we will discuss the limitations of our model which are likely responsible for these differences.", "However, the match in Figure REF suggests that for a wide range of subhalo flybys, the analytic model used in this work produces a reasonable estimate of the actual gap depth.", "Figure: Normalised distribution of gap depths in effective N-body simulations compared against the model.", "The blue curve shows the distribution of gap depths for a wide range of flybys simulated using the effective N-body method.", "The green curve shows the distribution of gap depths from our model.", "The match is quite good suggesting that the analytic model provides a reasonable estimate of the gap depth.", "The model predicts slightly deeper gaps than in the simulations which is likely due to our neglecting the dispersion of the stream in energy and angular momentum and the eccentricity of Pal 5's orbit.Similarly, we can compute the distribution of gap sizes using the effective N-body simulations.", "Using the same distribution of flybys above, we show the distribution of gap sizes in Figure REF .", "As with the gap depths, we get a similar distribution suggesting that our model is reproducing the gap size for a larger range of parameters.", "We also see that in the effective N-body simulation, the characteristic gap size is on the order of a few degrees.", "For this comparison we only considered flybys in the effective N-body simulation which created gaps less than 10$^\\circ $ in size.", "This is because we only record the density profile within 20$^\\circ $ of the gap center and the tails of the unperturbed stream only have lengths of roughly 20$^\\circ $ .", "Since we define the gap size by dividing the perturbed density by the unperturbed density and finding the size of the underdense region, we cannot find gaps which are longer than the length of the stream.", "Figure: Distribution of gap sizes in effective N-body simulations compared against the model.", "The blue curve shows the distribution of gap sizes for a wide range of flybys simulated using the effective N-body method.", "The green curve shows the distribution of gap sizes from our model.", "The match is quite good suggesting that the analytic model provides a reasonable estimate of the gap size as well.", "As noted in the text, we have restricted this comparison to gaps whose length is less than 10 ∘ ^\\circ in the effective N-body simulation since we cannot measure gaps which extend outside of the range of the unperturbed stream." ], [ "Searching for gaps", "The results of this work can be used to guide targeted searches for stream gaps.", "In Section , the gap size distribution was explored for various mass subhaloes, and in Figure REF the distribution of gap sizes was shown for a $\\Lambda $ CDM population.", "The characteristic size of the gaps in known streams will depend on the age of the stream, but as we can see from Figure REF , it peaks between $4-8^\\circ $ for a wide range of density thresholds, with a large number of gaps with sizes of order $10^\\circ $ and above.", "In addition, the distribution drops off as we proceed to smaller gaps so gap searches should be performed on a scale of at least $1-2^\\circ $ .", "In recent work, [43] searched for gaps in Pal 5 on the scale of $0.2-1^\\circ $ and found none.", "The results of this work suggest that future searches will be more fruitful if they are made on larger scales.", "However, we stress that the gap sizes we found are as viewed from the center of the Galaxy.", "A heliocentric observer may see smaller or larger gaps depending on their distance to the stream.", "In addition, the gap may appear foreshortened if it is not oriented perpendicular to the line of sight.", "Thus while we have identified a characteristic size of gaps, these observational effects, as well as the stretching and compressing of gaps due to eccentric orbits discussed in Section REF , can modify the distribution of sizes depending on the exact orientation of the stream and its orbital phase.", "In Section  we explored the number of gaps expected in known streams around the Milky Way.", "The majority of these gaps are quite shallow, with $\\rho /\\rho _0 > 0.5$ .", "Thus, searches for gaps should be looking for shallow gaps over the scales of several degrees.", "This will require an accurate measurement of the stream density on fairly large scales.", "The detection of a wide and shallow gap will also require a careful modeling of the unperturbed stream density since changes in the stripping rate can also create features in the stream density profile.", "In Table REF we give the expected number of gaps for six cold streams around the Milky Way.", "We find that if Styx is a genuine cold stream (i.e.", "originating from a globular cluster), it is by far the best candidate with 6 gaps expected with gaps deeper than $f<0.5$ .", "After this, Tri/Psc is the best candidate with $\\sim 0.9$ gaps expected at this depth.", "Next, Pal 5 and GD-1 have a similar number of gaps with $~0.3$ expected.", "Finally, Phoenix and ATLAS both appear to be poor candidates for detecting subhaloes with $0.02$ and $0.01$ gaps at this depth.", "We note that these predictions rely on the assumed properties in Table REF .", "As a result, these predictions represent a lower bound on the number of gaps since we have assumed that we have observed the full stream length.", "If these streams are found to be longer than currently observed, the number of gaps would naively increase quadratically with the length of the stream since the number of flybys is proportional to the age of the stream times its length (i.e.", "Equation (REF )) and the age itself is proportional to the length.", "However, the increased age of the stream would also give the gaps more time to grow, potentially making the scaling even stronger.", "Thus, deep observations of the streams to determine their full length and characterise their densities are critical.", "Lastly, we emphasize that this work has only focused on the basic properties of the gaps, such as depth and width.", "As discussed in [29], [15], a flyby will also create wiggles in the track of the stream which can be seen in the debris centroid on the sky, the distance to the stream, and the velocities in the stream.", "An underdensity alone is not sufficient to show a gap is present since other mechanisms like a variable stripping rate can create density variations in the stream.", "Thus, searches for gaps should also aim to identify these oscillations which will be crucial for constraining the properties of the perturbing subhalo." ], [ "Extension to dwarf galaxy streams", "We note that the results of this work cannot be immediately extended to dwarf galaxy streams like Sagittarius [41] or Orphan [9] which are substantially longer than the cold streams discussed here.", "While the flyby rates are still applicable, we would need to determine which flybys can create appreciable gaps.", "Since streams from disrupting dwarf galaxies have substantially higher velocity dispersions, a larger velocity kick will be needed to form a gap and these streams will not be sensitive to low mass substructure.", "We expect that the minimum velocity kick needed will likely scale as $m_{\\rm prog}^{1/3}$ , where $m_{\\rm prog}$ is the progenitor mass, just like the stream width and length (e.g.", "Equation (REF ) and Equation (REF )).", "Since the fraction of velocity kicks above some threshold drops rapidly as the threshold is increased, e.g.", "Figure REF , it appears unlikely that dwarf galaxy streams will have more gaps than cold streams." ], [ "Comparison with other works", "Let us compare the results of this work against those already reported in the literature.", "[71] and [18] both made similar assumptions as in this work to compute the number of flybys, as well as their relative velocity distribution.", "As discussed in Section REF , our derivation for the number of flybys differs slightly from [71] but matches [18].", "[18] also made predictions for the number of observable gaps which gives significantly more gaps than our model expects.", "Part of this difference is due to our decreasing the number of subhaloes due to the presence of the Milky Way disk [27].", "[53] consider the distribution of gap sizes in N-body simulations and find that the $\\Lambda $ CDM spectrum of subhaloes primarily create gaps larger than $1^\\circ $ [53], qualitatively agreeing with the results of this work.", "[19] simulate a stream on an orbit similar to GD-1 in the presence of subhaloes and find substantially more gaps than expected here.", "However, their stream age is 10.67 Gyr, significantly older than what we assume for GD-1, and they also perform the gap search over a stream which is $\\sim $ 2 radians long, significantly longer than what is observed for GD-1.", "We note that if we naively scale up our predictions to the age and length of their GD-1, we would get $\\sim 5$ gaps deeper than $f<0.9$ , in agreement with Figure 7 of [19]." ], [ "Baryonic effects", "In this work we have characterized the frequency of gaps due to dark matter subhaloes.", "However, as was recently shown in [4], clumps of baryons such as giant molecular clouds (GMCs) can also create gaps in streams.", "The mass spectrum of GMCs in the Milky Way [59], [57] shows that within the solar circle, the mass function is a power-law with an index of $\\gamma =-1.6$ (not too different from that of the subhaloes) and there are no GMCs above $10^{7} M_\\odot $ .", "Outside the solar circle, the mass spectrum is steeper, with $\\gamma = -2.1$ , and the upper mass drops to $\\sim 10^6 M_\\odot $ .", "Thus, streams whose pericenters are outside the solar circle should not be strongly affected by GMCs since, as we have shown, the effect from flybys of objects with masses below $10^6 M_\\odot $ will not produce a noticeable gap in the stream.", "However, for streams which pass within the solar circle (i.e.", "the Pal 5 stream), the most massive GMCs could impart noticeable density fluctuations in the stream.", "[4] evolved N-body realizations of Pal 5 and GD-1 like streams in the presence of the expected distribution of GMCs and reached a similar conclusion with GMCs producing notable gaps in Pal 5 but very few gaps in GD-1.", "Their analysis also accounted for whether the streams were on a prograde or retrograde with respect to the GMCs and found that prograde orbits produce more substantial gaps since the smaller relative velocities increases the size of the kick from the GMC.", "Now we can attempt to estimate the importance of GMCs within the framework developed for subhaloes in this work.", "In order to determine the relative importance of these GMCs in the inner part of the Milky Way, we can compare their number density with that of the subhaloes, taking into account what fraction of the stream's orbit is within the disk.", "Figure 21 of [57] shows that their survey found $~40$ GMCs with masses between $10^6-10^7 M_\\odot $ within the solar circle.", "They estimate their completeness by comparing the total mass of the GMCs they found to the total molecular hydrogen mass in the Milky Way and find they are 28% complete.", "Thus, we can estimate that there are $\\sim 140$ GMCs in this mass range within the solar circle.", "If we further assume that the number density of GMCs is constant within the solar circle and that the GMCs are confined to the region within a scale-height of the Milky Way disk, $\\sim 250$ pc, we find an average number density of $2.6$ kpc$^{-3}$ .", "In order to compare the effect of the GMCs against the subhaloes, we must estimate the relative number of interactions for a segment of the stream.", "If a segment travels a length $l$ through a region where perturbers have a number density of $n$ and affect the stream if they pass within a distance $b$ , the number of impacts is proportional to $n l b$ .", "If we only consider the effect of the subhaloes within the same mass range as the GMCs considered here, i.e.", "$10^6-10^7 M_\\odot $ , then we can assume that $b$ is the same.", "Thus, we only need to compare the quantity $nl$ .", "The stream is sensitive to subhaloes for its entire orbit, however it is only sensitive to the GMCs when it passes through the disk.", "For a Pal 5-like stream, if we take the average radius to be 13 kpc and use $n_{\\rm sub} = 1.01\\times 10^{-3}$ kpc$^{-3}$ , we get $n_{\\rm sub}l_{\\rm sub} = 0.08{\\rm kpc}^{-2}$ where $l_{\\rm sub}$ is the length of the orbit.", "For the GMCs, if we assume that the orbit passes straight through the disk we get $n_{\\rm GMC}l_{\\rm GMC} = 1.3{\\rm kpc}^{-2}$ where $l_{\\rm GMC}$ is twice the disk scale height.", "Therefore, even though the stream only spends a fraction of its orbit within the disk, the GMCs will have an order of magnitude more interactions and we must consider the effect of GMCs as suggested by [4].", "Increasing the mass range of the subhaloes we consider does not change this conclusion since the number of gaps created by subhaloes with masses between $10^6-10^7 M_\\odot $ is similar to those created by subhaloes between $10^7-10^8 M_\\odot $ .", "This simple analysis has many caveats.", "First, the number density of GMCs is not constant within the solar circle.", "In [58] the surface density of molecular gas in the Milky Way is shown to peak around 4 kpc and then drop-off: by 7 kpc, the surface density has dropped by an order of magnitude.", "Thus the relative importance of GMCs depends sensitively on the stream's pericenter.", "The pericenter of Pal 5 is between $\\sim 6-8$ kpc (see Table REF ) so if the number density of GMCs were 10 times lower, the effect of GMCs and subhaloes would be comparable.", "Indeed, [4] who assumed a pericenter of 8 kpc found that GMCs should produce 0.5 gaps with $\\rho /\\rho _0<0.71$ in the observed section of Pal 5, similar to our prediction in Table REF .", "Second, this analysis only considered the number of flybys and not the gaps they created.", "This will be controlled by the relative velocity of the GMCs as compared to the subhaloes and warrants further study [4].", "Third, we did not account for the increased path length of a stream through the disk if it is not on a polar orbit, however Pal 5 is relatively close to polar with an orbital inclination of $\\sim 65^\\circ $ relative to the disk [30].", "Fourth, this analysis does not account for the time evolution of GMCs.", "If the star-formation was stronger in the past, this could increase the number density of GMCs.", "Finally, this analysis does not account for the finite lifetime of GMCs which are expected to only survive a few free-fall times [52].", "For the GMCs we consider here that only corresponds to a $\\sim 10-20$ Myr and at the typical speeds within the disk, the GMCs will only move a few kpc before dispersing.", "This can be compared to the timescales over which the GMC would deliver a substantial kick: the region where the accelerations are the largest is on the order of the scale radius for a direct impact so the timescale where the kick is important is on the order of a Myr.", "Thus, it appears that the disruption of the GMC can safely be neglected.", "Finally we note that in [29] it was shown that given measurements of the density profile of the stream, and two other observables such as the stream track on the sky and the radial velocity along the stream, it is possible to recover the mass and scale radius of the perturber, as well as the time since impact.", "Thus, in principle, it should be possible to distinguish an impact from a globular cluster from that of a subhalo by the gap properties and by the time since impact.", "If the interaction occurred within the disk plane, this will lend additional credence to a GMC while if the impact can be convincingly be shown to have occurred far from the disk plane, a subhalo impact will be preferred." ], [ "Limitations of the method", "The method used in this work is based on the perturbation of streams on circular orbits [28].", "In this approximation, the stream is treated as being arbitrarily thin and having no velocity dispersion.", "This neglects the energy and angular momentum dispersion in the stream which can cause the gap depth to plateau as described in [61].", "[61] also found that the evolution of the gap size depends on where along the stream the impact occurs, with flybys far from the progenitor giving rise to more rapidly growing gaps due to the stretching of the stream itself.", "Thus, the analysis in this work may be slightly overestimating the depth of the gaps and understimating their size.", "We have attempted to test this in Figure REF where we compared the flyby of a $10^6 M_\\odot $ in this formalism with the flyby in an effective N-body simulation.", "This showed a fairly good match indicating that our method is robust.", "However, if anything our method will over-estimate the depth of gaps and so the number of gaps should be even less than reported in Table REF .", "In addition, our method does not account for the change in gap size along the orbit.", "We have treated the streams as being on circular orbits but naturally a large fraction of them are on orbits with substantial eccentricity.", "This eccentricity causes the gap size and depth to oscillate as seen in Figures 4,5 of [28] and Figure 13 of [61].", "If we neglect the growth of the gap during an orbit, conservation of angular momentum tells us that the gap sizes goes as $r^{-2}$ , e.g.", "Fig.", "13 of [61].", "The gap depth relative to the unperturbed stream exhibits a weaker oscillation but the gaps are deepest at pericenter and shallowest at apocenter.", "Thus, the predictions in Table REF should be seen as an average of the number of gaps expected.", "At pericenter these gaps will be easier to detect and at apocenter they will be more difficult to spot.", "We note that both of these limitations are addressed in [15] where they find broadly similar conclusions.", "Finally, this method does not account for the ongoing disruption of subhaloes.", "In the regions of the potential where a globular cluster can be tidally stripped, the subhaloes should be disrupting much more vigorously due to their lower density, resulting in dark matter streams as discussed in [13].", "Including the effect of these partially disrupted subhaloes will create shallower gaps, further lowering the expected number of gaps." ], [ "Conclusion", "In this work we have made a prediction for the expected number of stream gaps created by subhaloes and found far fewer gaps than previously expected.", "This prediction is based on counting the number of subhalo flybys near the stream, similar to the approaches of [71] and [18], and a model for the growth of the resulting gap created by each flyby from [28].", "The model for the rate and properties of the flybys in Section  is broadly similar to that in [71] and [18] but we expect significantly fewer flybys with a hotter relative velocity distribution.", "This is partially due to an updated derivation, and partially due to accounting for the depletion of subhaloes by the Milky Way disk [27].", "The rate and properties of the flybys are then combined with the analytic model for gap growth described in Section .", "While this analytic prescription is based on perturbations of streams on circular orbits and neglects the dispersion in a real stream, the tests performed in Section  indicate that it is relatively robust.", "Using the Pal 5 stream as an example, the distribution of gap sizes and depths is examined in Section  and we find several interesting results.", "First, the gap sizes are larger than previously expected with the majority of the gaps in Pal 5 having typical size of $\\sim 5^\\circ $ and many as large as $\\sim 10^\\circ $ and above (see Fig.", "REF ).", "As a result, any searches for gaps in Pal 5 should focus on sizes larger than $\\sim 1^\\circ $ .", "Second, for a given age of the stream, each perturber mass gives rise to a characteristic gap size (see Fig.", "REF ).", "This can be used to roughly estimate a perturber mass from the size of a gap.", "Third, we find that the typical gap size is larger for older streams since these gaps have had more time to grow (see Fig.", "REF ).", "This formalism was also used to make predictions for the number of gaps in six cold streams around the Milky Way.", "These predictions are summarised in Table REF where it is clear that most streams will have very few gaps.", "Pal 5 is expected to have 0.3 and 0.7 gaps deeper than $f<0.5$ and $f<0.75$ respectively.", "As a result, the null detection reported in [43] is not surprising.", "This should be contrasted with the 6 gaps detected in [21].", "Indeed, [68] argue that the claimed detections are due to a combination of variation in the Milky Way background with a smooth stream density.", "The six streams are ranked by the expected number of gaps and the Tri/Psc stream appears to be the most promising candidate with 0.9 and 1.6 gaps deeper than $f<0.5$ and $f<0.75$ respectively.", "GD-1 is also a promising candidate with a similar number of gaps to Pal 5.", "Substantially fewer gaps are expected in the ATLAS and Phoenix stream due to their short length and young age.", "Finally, if the Styx stream is a cold stream with the reported length then it would have the most gaps.", "In addition to the total number of gaps, we also investigate the contribution from each mass decade of subhaloes for Pal 5 and Tri/Psc in Figure REF ,REF respectively.", "This shows that the vast majority of gaps are due to subhaloes with masses in the range $10^6 M_\\odot < M < 10^8 M_\\odot $ .", "This is a previously unexplored mass range and the detection of even a single confirmed subhalo in this range would be an important test of $\\Lambda $ CDM and would improve constraints on the mass of a warm dark matter particle.", "In Section REF we estimate the number of gaps created by GMCs and find that for streams which enter the solar circle, they will be comparable to the number from subhaloes, in agreement with [4].", "While these predictions may appear to dampen the prospects of using cold streams to detect subhaloes, they should instead be thought of as setting realistic expectations for the number of gaps and their properties.", "With exquisite observations of streams now possible as demonstrated in [43], these predictions show that a lack of gaps in Pal 5 is unsurprising but also imply that the search for gaps should be performed on larger scales.", "Our results suggest that in the near future, deep observations of GD-1 and Tri/Psc, combined with the existing observations of Pal 5, should allow us to begin to uncover the presence of dark subhaloes expected in $\\Lambda $ CDM." ], [ "Acknowledgements", "We thank the anonymous referee for their helpful comments.", "We thank the Streams club at Cambridge for valuable discussions and in particular Sergey Koposov and Thomas de Boer.", "The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no.", "308024.", "J.B. received financial support from the Natural Sciences and Engineering Research Council of Canada.", "JLS acknowledges the support of the Science and Technology Facilities Council (STFC)." ], [ "Number of subhaloes entering cylinder", "It is also possible to come up with a simple expression for the number of subhaloes entering the caps of the stream.", "From the end caps on the left and right side of the cylinder in Figure REF , we would expect $ dN^{\\rm L,R}_{\\rm enc} = \\pi b_{\\rm max}^2 \\times (|v_s-v_z|dt) \\times n_{\\rm sub} \\times P(v_z) dv_z ,$ subhaloes to enter the region within $b$ of the stream in time $dt$ .", "Both of these must be integrated over the subhaloes which enter the stream, i.e.", "on the left side we consider $v_z < v_s$ and on the right side we consider $v_z > v_s$ .", "Performing these integrals over $v_z$ for both endcaps and summing the result, we get $ \\frac{d N^L_{\\rm enc}}{dt}+\\frac{d N^R_{\\rm enc}}{dt} = \\pi b_{\\rm max}^2 n \\sigma \\left( \\sqrt{\\frac{2}{\\pi }} \\exp (-\\frac{v_s^2}{2\\sigma ^2}) + \\frac{v_s }{\\sigma } {\\rm erf}(\\frac{v_s}{\\sigma \\sqrt{2}}) \\right) $" ] ]
1606.04946
[ [ "Asynchronous simulation of Boolean networks by monotone Boolean networks" ], [ "Abstract We prove that the fully asynchronous dynamics of a Boolean network $f:\\{0,1\\}^n\\to\\{0,1\\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with $2n$ components.", "We then use this result to prove that, for every even $n$, there exists a monotone Boolean network $f:\\{0,1\\}^n\\to\\{0,1\\}^n$, an initial configuration $x$ and a fixed point $y$ of $f$ such that: (i) $y$ can be reached from $x$ with a fully asynchronous updating strategy, and (ii) all such strategies contains at least $2^{\\frac{n}{2}}$ updates.", "This contrasts with the following known property: if $f:\\{0,1\\}^n\\to\\{0,1\\}^n$ is monotone, then, for every initial configuration $x$, there exists a fixed point $y$ such that $y$ can be reached from $x$ with a fully asynchronous strategy that contains at most $n$ updates." ], [ "Introduction", "A Boolean network with $n$ components is a discrete dynamical system usually defined by a global transition function $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n,\\qquad x=(x_1,\\dots ,x_n)\\mapsto f(x)=(f_1(x),\\dots ,f_n(x)).$ Boolean networks have many applications.", "In particular, since the seminal papers of McCulloch and Pitts [13], Hopfield [7], Kauffman [9], [10] and Thomas [19], [20], they are omnipresent in the modeling of neural and gene networks (see [3], [12] for reviews).", "They are also essential tools in Information Theory, for the network coding problem [1], [5].", "The structure of a Boolean network $f$ is usually represented via its interaction graph, which is the signed digraph $G(f)$ defined as follows: the vertex set is $[n]:=\\lbrace 1,\\dots ,n\\rbrace $ and, for all $i,j\\in [n]$ , there exists a positive (resp.", "negative) arc from $j$ to $i$ is there exists $x\\in \\lbrace 0,1\\rbrace ^n$ such that $f_i(x_1,\\dots ,x_{j-1},1,x_{j+1},\\dots ,x_n)-f_i(x_1,\\dots ,x_{j-1},0,x_{j+1},\\dots ,x_n)$ is positive (resp.", "negative).", "Note that $G(f)$ may have both a positive and a negative arc from one vertex to another.", "Note also that $G(f)$ may have loops, that is, arcs from a vertex to itself.", "The sign of a cycle of $G(f)$ is, as usual, the product of the signs of its arcs (cycles are always directed and without “repeated” vertices).", "From a dynamical point of view, there are several ways to derive a dynamics from $f$ , depending on the chosen updating strategy.", "With the so-called synchronous or parallel strategy, each component is updated at each step: if $x^t$ is the configuration of the system at time $t$ , then $f(x^t)$ is the configuration of the system at time $t+1$ .", "Hence, the dynamics is just given by the successive iterations of $f$ .", "On the opposite way, with the so-called (fully) asynchronous strategy, exactly one component is updated at each time.", "This strategy is very often used in practice, in particular in the context of gene networks [20].", "More formally, given an infinite sequence $i_0i_1i_2\\dots $ of indices taken in $[n]$ , the dynamics of $f$ resulting from an initial configuration $x^0$ and the asynchronous strategy $i_0i_1i_2\\dots $ is given by the following recurrence: for all $t\\in \\mathbb {N}$ and $i\\in [n]$ , $x^{t+1}_i=f_i(x^t)$ if $i=i_t$ and $x^{t+1}_i=x^t$ otherwise.", "All the possible asynchronous dynamics can be represented in a compact way by the so-called asynchronous graph $\\Gamma (f)$, defined as follows: the vertex set is $\\lbrace 0,1\\rbrace ^n$ and, for all $x,y\\in \\lbrace 0,1\\rbrace ^n$ , there is an arc from $x$ to $y$ , called transition, if there exists $i\\in [n]$ such that $f_i(x)=y_i\\ne x_i$ and $y_j=x_j$ for all $j\\ne i$ .", "Note that $f$ and $\\Gamma (f)$ share the same information.", "The distance between two configurations $x$ and $y$ in $\\Gamma (f)$ , denoted $d_{\\Gamma (f)}(x,y)$ , is the minimal length of a path of $\\Gamma (f)$ from $x$ to $y$ , with the convention that the distance is $\\infty $ if no such paths exist.", "Note that $d_{\\Gamma (f)}(x,y)$ is at least the Hamming distance $d_H(x,y)$ between $x$ and $y$ .", "A path from $x$ to $y$ in $\\Gamma (f)$ is then called a geodesic if its length is exactly $d_H(x,y)$ .", "In other words, a geodesic is a path along which each component is updated at most one.", "The diameter of $\\Gamma (f)$ is $\\mathrm {diam}(\\Gamma (f)):=\\max \\lbrace d_{\\Gamma (f)}(x,y):x,y\\in \\lbrace 0,1\\rbrace ^n,d_{\\Gamma (f)}(x,y)<\\infty \\rbrace .$ In many contexts, as in molecular biology, the first reliable information are represented under the form of an interaction graph, while the actual dynamics are very difficult to observe [12], [21].", "A natural question is then the following: What can be said about $\\Gamma (f)$ according to $G(f)$ only?", "Robert proved the following partial answer [17], [18].", "If $G(f)$ is acyclic then $f$ has a unique fixed point $y$ .", "Furthermore, $\\Gamma (f)$ is acyclic and, for every configuration $x$ , $\\Gamma (f)$ has a geodesic from $x$ to $y$ .", "In other words, $d_{\\Gamma (f)}(x,y)=d_H(x,y)$ for every $x\\in \\lbrace 0,1\\rbrace ^n$ .", "However, the acyclicity of $G(f)$ is not sufficient for $\\Gamma (f)$ to have a short diameter.", "Indeed, in a rather different setting, Domshlak [4] proved (a slightly stronger version of) the following result.", "For every $n\\ge 8$ there exists $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ such that $G(f)$ is acyclic and $\\mathrm {diam}(\\Gamma (f))\\ge 1.5^{\\frac{n}{2}}$ .", "Now, what can be said if $G(f)$ contains cycles ?", "Thomas highlighted the fact that the distinction between positive and negative cycles is highly relevant (see [20], [21] for instance).", "The subtlety and versatility of the influences of interactions between positive and negative cycles lead researchers to first focus on networks with only positive cycles or only negative cycles.", "In particular, the following basic properties was proved in [2], [15], [16]: If $G(f)$ has no positive (resp.", "negative) cycles, then $f$ has at most (resp.", "at least) one fixed point.", "This gives a nice proof by dichotomy of the first assertion in Theorem .", "In [14], the authors showed that the absence of negative cycles essentially corresponds to the study of monotone networks, that is, Boolean networks $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ such that $x\\le y~\\Rightarrow ~f(x)\\le f(y)$ where $\\le $ is the usual partial order ($x\\le y$ if and only if $x_i\\le y_i$ for all $i\\in [n]$ ).", "More precisely, they proved the following: If $G(f)$ is strongly connected and without negative cycles, then there exists a monotone network $f^{\\prime }:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ such that: $G(f)$ and $G(f^{\\prime })$ have the same underlying unsigned digraph, and $\\Gamma (f)$ and $\\Gamma (f^{\\prime })$ are isomorphic.", "Furthermore, they proved the following reachability result, that shares some similarities with Theorem .", "If $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ is monotone, then, for every configuration $x$ , $\\Gamma (f)$ has a geodesic from $x$ to a fixed point $y$ of $f$ .", "Here, we prove the following theorem, that shows that there may exist, under the same hypothesis, a configuration $x$ and a fixed point $y$ such that $y$ is reachable from $x$ with paths of exponential length only.", "This result contrasts with the previous one, and may be seen as an adaptation of Theorem  for monotone networks.", "For every even $n$ , there exists a monotone network $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ , two configurations $x$ and $y$ such that $y$ is a fixed point of $f$ and $\\mathrm {diam}(\\Gamma (f))\\ge d_{\\Gamma (f)}(x,y)\\ge 2^{\\frac{n}{2}}.$ The proof is by construction, and the idea for the construction is rather simple.", "Let $A$ , $B$ and $C$ be the sets of configurations that contains $n/2-1$ , $n/2$ and $n/2+1$ ones.", "Clearly, $A$ , $B$ and $C$ are antichains of exponential size, and, in these antichains, obviously, the monotonicity of $f$ doesn't apply.", "This leaves enough freedom to defined $f$ on $A\\cup B\\cup C$ in such a way that subgraph $\\Gamma (f)$ induced by $A\\cup B\\cup C$ contains a configuration $x$ and fixed point $y$ reachable from $x$ with paths of exponential length only.", "To obtain a network as in the theorem, it is then sufficient to extend $f$ on the whole space $\\lbrace 0,1\\rbrace ^n$ by keeping $f$ monotone and without creating shortcuts from $x$ to $y$ in the asynchronous graph.", "This idea, that consists in using large antichains to construct special monotone functions, is also present in [6] and [8] for instance.", "Let $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ be any Boolean network such that $G(f)$ has no negative loops.", "With the technic described above, we can go further and prove that $\\Gamma (f)$ can be embedded in the asynchronous graph $\\Gamma (f^{\\prime })$ of a monotone network $f^{\\prime }:\\lbrace 0,1\\rbrace ^{2n}\\rightarrow \\lbrace 0,1\\rbrace ^{2n}$ in such a way that fixed points and distances between configurations are preserved.", "The formal statement follows.", "If $x,y\\in \\lbrace 0,1\\rbrace ^n$ , then the concatenation $(x,y)$ is seen as a configuration of $\\lbrace 0,1\\rbrace ^{2n}$ and, conversely, each configuration in $\\lbrace 0,1\\rbrace ^{2n}$ is seen as the concatenation of two configurations in $\\lbrace 0,1\\rbrace ^n$ .", "As usual, we denote by $\\overline{x}$ the configuration obtained from $x$ by switching every component.", "[Main results] Let $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ .", "If $G(f)$ has no negative loops, then there exists a monotone network $f^{\\prime }:\\lbrace 0,1\\rbrace ^{2n}\\rightarrow \\lbrace 0,1\\rbrace ^{2n}$ such that the following two properties holds.", "First, $x$ is a fixed point of $f$ if and only if $(x,\\overline{x})$ is a fixed point of $f^{\\prime }$ .", "Second, for all $x,y\\in \\lbrace 0,1\\rbrace ^n$ , $\\Gamma (f)$ has a path from $x$ to $y$ of length $\\ell $ if and only if $\\Gamma (f^{\\prime })$ has a path from $(x,\\bar{x})$ to $(y,\\bar{y})$ of length $2\\ell $ .", "Theorem  is now an easy corollary of Theorem .", "[of Theorem  assuming Theorem ] Let $r=2^n$ , and let $x^1,x^2,\\dots ,x^r$ be any enumeration of the elements of $\\lbrace 0,1\\rbrace ^n$ such that $d_H(x^k,x^{k+1})=1$ for all $1\\le k<r$ (take the Gray code for instance).", "Let $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ be defined by $f(x^k)=x^{k+1}$ for all $1\\le k<r$ and $f(x^r)=x^r$ .", "Let $x=x^0$ and $y=x^r$ .", "Then $y$ is the unique fixed point of $f$ .", "Furthermore, since the set of transitions of $\\Gamma (f)$ is $\\lbrace x^k\\rightarrow x^{k+1}:1\\le k<r\\rbrace $ , we deduce that $d_{\\Gamma (f)}(x,y)=2^n-1$ .", "We also deduce that $G(f)$ has no negative loops (this is an easy exercise to prove that $G(f)$ has a negative loop if and only if $\\Gamma (f)$ has a cycle of length two).", "Hence, by Theorem , there exists a monotone network $f^{\\prime }:\\lbrace 0,1\\rbrace ^{2n}\\rightarrow \\lbrace 0,1\\rbrace ^{2n}$ such that $(y,\\overline{y})$ is a fixed point and $d_{\\Gamma (f^{\\prime })}((x,\\overline{x}),(y,\\overline{y}))=2d_{\\Gamma (f)}(x,y)=2^{n+1}-2\\ge 2^n.$ $\\Box $ The paper is organized as follows.", "The proof of Theorem  is given in Section .", "A conclusion and some open questions are then given in Section ." ], [ "Proof of Theorem ", "We first fix some notations: $\\overline{x}^i&:=(x_1,\\dots ,\\overline{x_i},\\dots ,x_n)&(\\textrm {x\\in \\lbrace 0,1\\rbrace ^n and i\\in [n]}),\\\\w(x)&:=|\\lbrace i\\in [n]:x_i=1\\rbrace |&(\\textrm {x\\in \\lbrace 0,1\\rbrace ^n}),\\\\w(x,y)&:=w(x)+w(y)&(\\textrm {x,y\\in \\lbrace 0,1\\rbrace ^n}),\\\\\\Omega &:=\\lbrace (x,\\overline{x}):x\\in \\lbrace 0,1\\rbrace ^n\\rbrace .$ The function $f^{\\prime }$ in Theorem  is defined as follows from $f$ .", "Given $f:\\lbrace 0,1\\rbrace ^n\\rightarrow \\lbrace 0,1\\rbrace ^n$ , we define $f^{\\prime }:\\lbrace 0,1\\rbrace ^{2n}\\rightarrow \\lbrace 0,1\\rbrace ^{2n}$ by: for all $i\\in [n]$ and $x,y\\in \\lbrace 0,1\\rbrace ^n$ , $f^{\\prime }_i(x,y)=\\left\\lbrace \\begin{array}{ll}f_i(x)&\\text{ if $y=\\overline{x}$ or $\\overline{y}^i=\\overline{x}$}\\\\[2mm]\\overline{x_i}&\\text{ if $w(x,y)=n$ and $y\\ne \\overline{x}$}\\\\[2mm]1&\\text{ if $w(x,y)=n+1$ and $\\overline{y}^i\\ne \\overline{x}$}\\\\[2mm]0&\\text{ if $w(x,y)=n-1$ and $\\overline{y}^i\\ne \\overline{x}$}\\\\[2mm]1&\\text{ if $w(x,y)\\ge n+2$}\\\\[2mm]0&\\text{ if $w(x,y)\\le n-2$}\\end{array}\\right.\\quad \\text{and}\\quad f^{\\prime }_{n+i}(x,y)=\\overline{f^{\\prime }_i(\\overline{y},\\overline{x})}.$ $f^{\\prime }_i(x,y)=\\overline{f^{\\prime }_{n+i}(\\overline{y},\\overline{x})}$ .", "Let $A$ , $B$ and $C$ be sets of configurations $(x,y)\\in \\lbrace 0,1\\rbrace ^{2n}$ such that $w(x,y)$ is $n-1$ , $n$ and $n+1$ , respectively (these are the three sets discussed in the introduction) (we have $\\Omega \\subseteq B$ ).", "One can see that $f^{\\prime }_i$ behave as $f_i$ when $x$ and $y$ are mirroring each other ($y=\\overline{x}$ ) or almost mirroring each other ($\\overline{y}^i=\\overline{x}$ ); and in both cases, $(x,y)$ lies in $A\\cup B\\cup C$ .", "One can also see that $f^{\\prime }_i$ equals 0 below the layer $A$ and equals 1 above the layer $C$ .", "The same remarks apply on $f^{\\prime }_{n+i}$ , excepted that $f^{\\prime }_{n+1}$ behaves as the negation $\\overline{f_i}$ in $A\\cup B\\cup C$ .", "Hence, roughly speaking, $f$ behaves as $(f,\\overline{f})$ in the middle layer $A\\cup B\\cup C$ , and it converges toward the all-zeroes or all-ones configuration outside this layer.", "If $G(f)$ has no negative loops, then $f^{\\prime }$ is monotone.", "Suppose, for a contradiction, that there exists $a,b,c,d\\in \\lbrace 0,1\\rbrace ^n$ and $i\\in [n]$ such that $(a,b)<(c,d)\\text{ and }f^{\\prime }_i(a,b)>f^{\\prime }_i(c,d).$ Then we have $n-1\\le w(a,b)<w(c,d)\\le n+1.$ This leaves three possibilities.", "Case 1: $w(a,b)=n-1$ and $w(c,d)=n+1$.", "Since $f^{\\prime }_i(a,b)=1$ , we fall in the first case of the definition of $f^{\\prime }_i$ , that is, $f^{\\prime }_i(a,b)=f_i(a)=1\\text{ and }\\overline{b}^i=\\overline{a}.$ Similarly $f^{\\prime }_i(c,d)=f_i(c)=0\\text{ and }\\overline{d}^i=\\overline{c}.$ Thus $(a,b)=(a,\\overline{\\overline{a}}^i)<(c,d)=(c,\\overline{\\overline{c}}^i).$ So for all $j\\ne i$ , we have $a_j\\le c_j$ and $\\overline{a_j}=(\\overline{\\overline{a}}^i)_j\\le (\\overline{\\overline{c}}^i)_j=\\overline{c_j}$ thus $c_j\\le a_j$ .", "So $a_j=c_j$ for all $j\\ne i$ , that is, $c\\in \\lbrace a,\\overline{a}^i\\rbrace $ .", "Since $f_i(a)<f_i(c)$ we have $c=\\overline{a}^i$ , and since $a\\le c$ we deduce that $a_i=0$ .", "Thus $G(f)$ has a negative arc from $i$ to $i$ , a contradiction.", "Case 2: $w(a,b)=n-1$ and $w(c,d)=n$.", "As in Case 1, we have $f^{\\prime }_i(a,b)=f_i(a)=1\\text{ and }\\overline{b}^i=\\overline{a}.$ For $f^{\\prime }_i(c,d)$ we have two cases.", "Suppose first that $f^{\\prime }_i(c,d)=f_i(c)=0\\text{ and }d=\\overline{c}.$ Then $(a,b)=(a,\\overline{\\overline{a}}^i)<(c,d)=(c,\\overline{c}).$ So for all $j\\ne i$ , we have $a_j\\le c_j$ and $\\overline{a_j}=(\\overline{\\overline{a}}^i)_j\\le \\overline{c_j}$ thus $c_j\\le a_j$ .", "So $a_j=c_j$ for all $j\\ne i$ , that is, $c\\in \\lbrace a,\\overline{a}^i\\rbrace $ .", "Since $f_i(a)<f_i(c)$ we have $c=\\overline{a}^i$ , and since $a\\le c$ we deduce that $a_i=0$ .", "Thus $G(f)$ has a negative arc from $i$ to $i$ , a contradiction.", "The other case is $f^{\\prime }_i(c,d)=\\overline{c_i}=0\\text{ and }d\\ne \\overline{c}.$ First, observe that for all $j\\ne i$ , if $c_j=0$ then $a_j=0$ thus $1=(\\overline{\\overline{a}}^i)_j\\le d_j$ .", "Since $c_i=1$ we deduce that $\\overline{c}\\le d$ .", "Now, suppose that $c_j=d_j=1$ for some $j\\in [n]$ .", "Since $w(c,d)=n$ , we deduce that there exists $k\\ne j$ such that $c_k=d_k=0$ , and this contradicts $\\overline{c}\\le d$ .", "Thus, for all $j\\in [n]$ , either $d_j=0$ or $d_j>c_j$ , that is, $d\\le \\overline{c}$ .", "Thus $c=\\overline{d}$ , a contradiction.", "Case 3: $w(a,b)=n$ and $w(c,d)=n+1$.", "We obtain a contradiction as in Case 2.", "So we have proven that $f^{\\prime }_i$ is monotone for all $i\\in [n]$ .", "It remains to prove that $f^{\\prime }_{n+i}$ is monotone.", "Using the monotony of $f^{\\prime }_i$ for the implication we get: $\\begin{array}{rcl}(a,b)\\le (c,d)&\\iff & (\\overline{c},\\overline{d})\\le (\\overline{a},\\overline{b})\\\\[2mm]&\\iff & (\\overline{d},\\overline{c})\\le (\\overline{b},\\overline{a})\\\\[2mm]&\\Longrightarrow & f^{\\prime }_i(\\overline{d},\\overline{c})\\le f^{\\prime }_i(\\overline{b},\\overline{a})\\\\[2mm]&\\iff & \\overline{f^{\\prime }_i(\\overline{b},\\overline{a})}\\le \\overline{f^{\\prime }_i(\\overline{d},\\overline{c})}\\\\[2mm]&\\iff & f^{\\prime }_{i+n}(a,b)\\le f^{\\prime }_{i+n}(c,d).\\end{array}$ $\\Box $ For all $x\\in \\lbrace 0,1\\rbrace ^n$ we have $f(x)=x$ if and only if $f^{\\prime }(x,\\overline{x})=(x,\\overline{x})$ .", "By definition we have $f(x)=x\\quad \\iff \\quad f^{\\prime }_i(x,\\overline{x})=x_i~\\forall i\\in [n].$ So it is sufficient to prove that $f^{\\prime }(x,\\overline{x})=(x,\\overline{x})\\quad \\iff \\quad f^{\\prime }_i(x,\\overline{x})=x_i~\\forall i\\in [n].$ The direction $\\Rightarrow $ is obvious, and $\\Leftarrow $ is a consequence of the following equivalences: $f^{\\prime }_i(x,\\overline{x})=x_i&\\iff \\overline{f^{\\prime }_{n+i}(\\overline{\\overline{x}},\\overline{x})}=x_i\\\\&\\iff f^{\\prime }_{n+i}(x,\\overline{x})=\\overline{x_i}\\\\&\\iff f^{\\prime }_{n+i}(x,\\overline{x})=(x,\\overline{x})_{n+i}.$ $\\Box $ For all $x,y\\in \\lbrace 0,1\\rbrace ^n$ , if $\\Gamma (f^{\\prime })$ has a path from $(x,y)$ to $\\Omega $ then $n-1\\le w(x,y)\\le n+1$ .", "It is sufficient to prove that, $w(x,y)\\le n-2~\\Rightarrow f^{\\prime }(x,y)=0\\quad \\text{and}\\quad w(x,y)\\ge n+2~\\Rightarrow f^{\\prime }(x,y)=1.$ Let $i\\in [n]$ .", "If $w(x,y)\\le n-2$ (resp.", "$w(x,y)\\ge n+2$ ) then $f^{\\prime }_i(x,y)=0$ (resp.", "$f^{\\prime }_i(x,y)=1$ ) by definition.", "Now, if $w(x,y)\\le n-2$ then $w(\\bar{y},\\bar{x})\\ge n+2$ thus $f^{\\prime }_{n+i}(x,y)=\\overline{f^{\\prime }_i(\\bar{y},\\bar{x})}=\\overline{1}=0,$ and if $w(x,y)\\ge n+2$ then $w(\\bar{y},\\bar{x})\\le n-2$ thus $f^{\\prime }_{n+i}(x,y)=\\overline{f^{\\prime }_i(\\bar{y},\\bar{x})}=\\overline{0}=1.$ $\\Box $ If $G(f)$ has no negative loops, then, for all $x,y\\in \\lbrace 0,1\\rbrace ^n$ , the following assertions are equivalent: $x\\rightarrow y$ is a transition of $\\Gamma (f)$ .", "$(x,\\overline{x})\\rightarrow (y,\\overline{x})\\rightarrow (y,\\overline{y})$ is a path of $\\Gamma (f^{\\prime })$ .", "$(x,\\overline{x})\\rightarrow (x,\\overline{y})\\rightarrow (y,\\overline{y})$ is a path of $\\Gamma (f^{\\prime })$ .", "$\\Gamma (f^{\\prime })$ has a path from $(x,\\overline{x})$ to $(y,\\overline{y})$ without internal vertex in $\\Omega $ .", "Furthermore, the only possible paths of $\\Gamma (f^{\\prime })$ from $(x,\\overline{x})$ to $(y,\\overline{y})$ without internal vertex in $\\Omega $ are precisely the ones in (2) and (3).", "Suppose that $\\Gamma (f)$ has a transition $x\\rightarrow y$ , and let $i\\in [n]$ be such that $y=\\overline{x}^i$ .", "We have $f^{\\prime }_i(x,\\overline{x})=f_i(x)\\ne x_i$ thus $\\Gamma (f^{\\prime })$ has a transition from $(x,\\overline{x})$ to $(\\overline{x}^i,\\overline{x})=(y,\\overline{x})$ .", "Since $f^{\\prime }_{n+i}(\\overline{x}^i,\\overline{x})=\\overline{f_i(\\overline{\\overline{x}},\\overline{\\overline{x}^i})}=\\overline{f_i(x,\\overline{\\overline{x}^i})}=\\overline{f_i(x)}=x_i\\ne (\\overline{x}^i,\\overline{x})_{n+i},$ $\\Gamma (f^{\\prime })$ has a transition from $(\\overline{x}^i,\\overline{x})$ to $\\overline{(\\overline{x}^i,\\overline{x})}^{n+i}=(\\overline{x}^i,\\overline{\\overline{x}}^i)=(y,\\overline{y}).$ This proves the implication $(1)\\Rightarrow (2)$ .", "Now, if $\\Gamma (f^{\\prime })$ contains the transition $(x,\\overline{x})\\rightarrow (y,\\overline{x})$ then there exists $i\\in [n]$ such that $y=\\overline{x}^i$ and $y_i=f^{\\prime }_i(x,\\overline{x})=f_i(x)$ .", "Thus $x\\rightarrow y$ is a transition of $\\Gamma (f)$ .", "So we have $(1)\\iff (2)$ and we prove similarly that $(1)\\iff (3)$ .", "Since $[(2)\\text{ or }(3)]\\Rightarrow (4)$ is obvious, to complete the proof it is sufficient to prove that if $\\Gamma (f^{\\prime })$ has a path $P$ from $(x,\\overline{x})$ to $(y,\\overline{y})$ without internal vertex in $\\Omega $ then either $P=(x,\\overline{x})\\rightarrow (y,\\overline{x})\\rightarrow (y,\\overline{y})$ or $P=(x,\\overline{x})\\rightarrow (x,\\overline{y})\\rightarrow (y,\\overline{y})$ .", "Let $a$ be the configuration following $(x,\\overline{x})$ in $P$ , and let $b$ be the configuration following $a$ in $P$ .", "We will prove that $b=(y,\\overline{y})$ and $a=(x,\\overline{y})$ or $a=(y,\\overline{x})$ .", "We have $w(a)=n\\pm 1$ and thus $w(b)\\in \\lbrace n-2,n,n+2\\rbrace $ , but if $w(b)=n\\pm 2$ then we deduce from Lemma  that $\\Gamma (f^{\\prime })$ has no paths from $b$ to a configuration in $\\Omega $ , a contradiction.", "Thus $w(b)=n$ .", "Let $i\\in [n]$ be such that $a=(\\overline{x}^i,\\overline{x})$ or $a=(x,\\overline{\\overline{x}}^i)$ .", "We have four cases.", "Case 1: $a=(\\overline{x}^i,\\overline{x})$ and $w(a)=n-1$.", "Since $w(a)=n-1$ we have $x_i=1$ , and thus $f^{\\prime }_i(x,\\overline{x})=f_i(x)=0$ .", "Also $f^{\\prime }_i(a)=f^{\\prime }_i(\\overline{x}^i,\\overline{x})=f_i(\\overline{x}^i)=0$ since otherwise $G(f)$ has a negative loop on vertex $i$ .", "Let $1\\le j\\le 2n$ be such that $b=\\overline{a}^j$ .", "Since $w(a)<w(b)=n$ , we have $a_j=0$ and $f^{\\prime }_j(a)=1$ .", "If $1\\le j\\le n$ then $j\\ne i$ (since $f^{\\prime }_i(a)=0$ ) so $\\overline{\\overline{x}}^j\\ne \\overline{\\overline{x}^i}$ and since $w(\\overline{x}^i,\\overline{x})=n-1$ , we deduce from the definition of $f^{\\prime }$ that $f^{\\prime }_j(a)=f^{\\prime }_j(\\overline{x}^i,\\overline{x})=0$ , a contradiction.", "So $n< j\\le 2n$ .", "Let $k=j-n$ .", "We have $f^{\\prime }_j(a)=f^{\\prime }_{n+k}(a)=f^{\\prime }_{n+k}(\\overline{x}^i,\\overline{x})=\\overline{f^{\\prime }_k(\\overline{\\overline{x}},\\overline{\\overline{x}^i})}=\\overline{f^{\\prime }_k(x,\\overline{\\overline{x}^i})}.$ Since $w(\\overline{x}^i,\\overline{x})=n-1$ we have $w(x,\\overline{\\overline{x}}^i)=n+1$ .", "So if $k\\ne i$ we have $\\overline{\\overline{\\overline{x}}^i}^k\\ne \\overline{x}$ .", "Thus by the definition of $f^{\\prime }$ we have $f^{\\prime }_k(x,\\overline{\\overline{x}^i})=1$ thus $f^{\\prime }_j(a)=0$ , a contradiction.", "We deduce that $k=i$ , that is, $j=n+i$ .", "Thus $b=\\overline{a}^{n+i}=(\\overline{x}^i,\\overline{\\overline{x}}^i)\\in \\Omega $ , and we deduce that $P=(x,\\overline{x})\\rightarrow (y,\\overline{x})\\rightarrow (y,\\overline{y}).$ Case 2: $a=(\\overline{x}^i,\\overline{x})$ and $w(a)=n+1$.", "We prove with similar arguments that $P=(x,\\overline{x})\\rightarrow (y,\\overline{x})\\rightarrow (y,\\overline{y}).$ Case 3: $a=(x,\\overline{\\overline{x}}^i)$ and $w(a)=n-1$.", "We prove with similar arguments that $P=(x,\\overline{x})\\rightarrow (x,\\overline{y})\\rightarrow (y,\\overline{y}).$ Case 4: $a=(x,\\overline{\\overline{x}}^i)$ and $w(a)=n+1$.", "We prove with similar arguments that $P=(x,\\overline{x})\\rightarrow (x,\\overline{y})\\rightarrow (y,\\overline{y}).$ $\\Box $ If $G(f)$ has no negative loops, then for all $x,y\\in \\lbrace 0,1\\rbrace ^n$ , the following two assertions are equivalent: $\\Gamma (f)$ has a path from $x$ to $y$ of length $\\ell $ .", "$\\Gamma (f^{\\prime })$ has a path from $(x,\\bar{x})$ to $(y,\\bar{y})$ of length $2\\ell $ .", "According to Lemma , $x^0\\rightarrow x^1\\rightarrow x^2\\rightarrow \\cdots \\rightarrow x^\\ell $ is a path of $\\Gamma (f)$ if and only if $(x^0,\\overline{x^0})\\rightarrow (x^0,\\overline{x^1})\\rightarrow (x^1,\\overline{x^1})\\rightarrow (x^1,\\overline{x^2})\\rightarrow (x^2,\\overline{x^2}) \\cdots \\rightarrow (x^\\ell ,\\overline{x^\\ell })$ is a path of $\\Gamma (f^{\\prime })$ .", "This proves $(1)\\Rightarrow (2)$ .", "To prove $(2)\\Rightarrow (1)$ suppose that $\\Gamma (f^{\\prime })$ has a path $P$ from $(x,\\bar{x})$ to $(y,\\bar{y})$ of length $2\\ell $ .", "Let $(a^0,\\overline{a^0}),(a^1\\overline{a^1}),\\dots ,(a^p,\\overline{a^p})$ be the configurations of $P$ that belongs to $\\Omega $ , given in the order (so $a^0=x$ and $a^p=y$ ).", "According to Lemma , there exists $b^1,b^2,\\dots ,b^p$ with $b^q\\in \\lbrace (a^{q-1},\\overline{a^q}),(a^q,\\overline{a^{q-1}})\\rbrace $ for all $1\\le q\\le p$ such that $P=(a^0,\\overline{a^0})\\rightarrow b^1 \\rightarrow (a^1,\\overline{a^1})\\rightarrow b^2\\rightarrow \\cdots \\rightarrow b^p\\rightarrow (a^p,\\overline{a^p}).$ Thus $p=\\ell $ , and again by Lemma , $x^0\\rightarrow x^1\\rightarrow \\cdots \\rightarrow x^\\ell $ is a path of $\\Gamma (f)$ .$\\Box $ Theorem  result from Lemmas , and ." ], [ "Conclusion and open questions", "In this paper we have proved that the asynchronous graph of every $n$ -component Boolean network without negative loop can be embedded in the asynchronous graph of a $2n$ -component monotone Boolean network, in such a way that fixed points and distances between configurations are preserved.", "A consequence of this result, which was our initial goal, is that the asynchronous graph of a monotone network may have an exponential diameter.", "More precisely, it may exist a configuration $x$ and a fixed point $y$ reachable from $x$ such that the distance between $x$ and $y$ is at least $2^{\\frac{n}{2}}$ .", "This contrasts with the fact that for every configuration $x$ there exists a fixed point $y$ such that the distance between $x$ and $y$ is at most $n$ .", "These results raise several questions.", "Could it be possible to embed, in a similar way, a $n$ -component network with negative loops into a $m$ -component monotone network?", "Maybe this would require $m$ to be even larger than $2n$ .", "Besides, the embedding we propose is based on the injection $x\\mapsto (x,\\overline{x})$ from $\\lbrace 0,1\\rbrace ^n$ to the balanced words of length $2n$ .", "The well-known Knuth's balanced coding scheme [11] provides a rather simple injection from $\\lbrace 0,1\\rbrace ^n$ to the balanced words of length $n+2\\log _2n$ only.", "Could this technique be used to decrease the number of components in the host monotone network from $2n$ to $n+2\\log _2n$ ?", "Finally, it could be interesting to study the interaction graph of monotone networks with large diameter.", "Does it necessarily contain long cycles, or many disjoint cycles?" ], [ "Acknowledgment", "This work has been partially supported by the project PACA APEX FRI. We wish also to thank Pierre-Etienne Meunier, Maximilien Gadouleau and an anonymous reviewer for stimulating discussions and interesting remarks." ] ]
1606.05172
[ [ "Inversion Copulas from Nonlinear State Space Models" ], [ "Abstract We propose to construct copulas from the inversion of nonlinear state space models.", "These allow for new time series models that have the same serial dependence structure of a state space model, but with an arbitrary marginal distribution, and flexible density forecasts.", "We examine the time series properties of the copulas, outline serial dependence measures, and estimate the models using likelihood-based methods.", "Copulas constructed from three example state space models are considered: a stochastic volatility model with an unobserved component, a Markov switching autoregression, and a Gaussian linear unobserved component model.", "We show that all three inversion copulas with flexible margins improve the fit and density forecasts of quarterly U.S. broad inflation and electricity inflation." ], [ "A: Simulation Study", "We simulate data from two time series models.", "The first (Sim1) is the UCAR1 model outlined in Equation (3.4), but with $Z_t$ replaced by $Y_t$ .", "The second (Sim2) is a copula model with a Gamma(2,2) marginal distribution $G$ , and copula function InvCop3 constructed from the same latent UCAR1 model.", "The parameter values are $\\bar{\\mu }=0$ , $\\rho _1=0.7$ , $\\sigma ^2_\\mu = 0.25$ , and $\\sigma ^2=0.5$ , which satisfy the constraints in Section 3.3 Both Sim1 and Sim2 exhibit the same simple first order serial dependence structure, but Sim2 has a highly positively skewed margin with skew coefficient 1.41.", "One hundred time series datasets, each of length $T=1000$ , are generated from the data generating processes Sim1 and Sim2.", "To each dataset we fit two models.", "The first is the UCAR1 state space model fit directly the data using Bayesian methods, and labeled `M1'.", "The second is a copula model with copula function InvCop3 (with $p=1$ ) and a nonparametric margin, and labeled `M2'.", "In the copula literature it is popular to use the empirical distribution function as a nonparametric estimator of $G$ (Shih & Louis 1995; Tsukahara 2005).", "However, this can give inaccurate estimates of the tails of $G$ , as illustrated by Smith & Vahey (2015) for macroeconomic series.", "Therefore, a kernel density estimator (KDE) is fit using the locally adaptive bandwidth method of Shimazaki & Shinomoto (2010).", "The copula parameters are estimated using the Bayesian method outlined in Section 3.3.", "There are four combinations of data generating process and fitted model: Sim1/M1, Sim1/M2, Sim2/M1 and Sim2/M2.", "For each of these we construct one-step-ahead predictive distributions.", "For the UCAR1 model (ie.", "M1) these are Gaussian distributions with moments computed using the Kalman Filter, while for the copula model (ie.", "M2) they are non-Gaussian as outlined in Section 2.5.", "Predictive distributions are computed for time points $t=2,\\ldots ,1000$ and for each of the 100 simulated datasets.", "Point forecast accuracy is measured using the root mean squared error (RMSE).", "Density forecast accuracy is measured using the (negative) logarithm of the predictive score (LP), the cumulative rank probability score (CRPS) and the tail-weighted CRPS (TW-CRPS).", "The latter two measures are discussed in Gneiting & Raftery (2007) and Gneiting & Ranjan (2011), and computed directly from the quantile score.", "For each of the four cases, Table REF reports the mean metric values computed over all time points and datasets.", "More accurate forecasts correspond to higher values for LP, and lower values for all other metrics.", "Model M1 is the correct parametric model for Sim1, yet there is almost no loss of accuracy when fitting the copula model M2.", "However, for Sim2 where the margin is truly asymmetric, ignoring this fact and fitting M1 leads to poor forecasts.", "In contrast, fitting model M2 increases forecast accuracy by all measures because it allows for flexibility in the margin.", "The differences in mean metric values between fitting models M1 and M2 are statistically significant between the 100 replicated datasets at the 1% level for every metric.", "In summary, employing time series models with inaccurate margins can substantially decrease forecast accuracy, compared to employing a copula time series model with flexible margins.", "Table: Summary of the accuracy of the one-step-ahead predictivedistributions in the simulation study.", "The rows report results for the four combinations ofsimulation (Sim1 and Sim2) and fitted model (M1 and M2).", "The columnsreport results for the four metrics considered, whichare the mean (negative) logarithm predictive score (LP),the mean cumulative rank probability score (CRPS), the mean tail-weightedCRPS (TW-CRPS), and theroot mean squared error (RMSE).The meansare computed over the 999 time points and 100 simulated datasets.", "Note that testsat the 1% levelindicate the values in the bottom row are all statistically significantlydifferentthan those in the row immediately above." ] ]
1606.05022
[ [ "Wellposedness and regularity of steady-state two-sided\n variable-coefficient conservative space-fractional diffusion equations" ], [ "Abstract We study the Dirichlet boundary-value problem of steady-state two-sided variable-coefficient conservative space-fractional diffusion equations.", "We show that the Galerkin weak formulation, which was proved to be coercive and continuous for a constant-coefficient analogue of the problem, loses its coercivity.", "We characterize the solution to the variable-coefficient problem in terms of the solutions of second-order diffusion equations along with a two-sided fractional integral equation.", "We then derive a Petrov-Galerkin formulation for this problem and prove that the weak formulation is weakly coercive and so the problem is well posed.", "We then prove high-order regularity estimates of the true solution in a properly chosen norm of Riemann-Liouville derivatives." ], [ "Introduction", "In recent years nonlocal models are emerging as powerful tools for modeling challenging phenomena including overlapping microscopic and macroscopic scales, anomalous transport, and long-range time memory or spatial interactions in nature, science, social science, and engineering [24], [25], [26], [43].", "Data-driven fractional-order differential operators can be constructed to model a specific phenomenon instead of the current practice of tweaking the coefficients that multiply pre-set integer-order differential operators.", "It was shown that the misspecification of physical models using an integer-order partial differential equation often leads to a variable coefficient fit (struggling to fit the data at each location, for example) whereas a physical model using a fractional-order partial differential equation can fit all the data with a constant coefficient [5].", "In short, nonlocal models open up great opportunities and flexibility for modeling and simulation of multiphysical phenomena, e.g.", "from local to nonlocal dynamics [43].", "Because of their significantly improved modeling capabilities, various related but different nonlocal models, including fractional Laplacian, nonlocal diffusion and peridynamics, and fractional partial differential equations, have been developed to describe diverse nonlocal phenomena.", "The fractional Laplacian operator $(-\\Delta u)^s$ of order $0 < s < 1$ has been used to model nonlocal behavior in many physical problems [2], [4], [12], [21] and has appeared as the infinitesimal generator of a stable Lévy process [2], [17], [18], [38].", "$(-\\Delta )^s$ can be defined as a pseudodifferential operator of symbol $|\\mbox{$\\xi $}|^{2s}$ on the entire space $\\mathbb {R}^d$ [2] $(-\\Delta )^{s} u = \\mathcal {F}^{-1} \\big ( | \\mbox{$\\xi $}|^{2s} \\mathcal {F}{u}(\\mbox{$\\xi $}) \\big ), \\qquad \\forall u \\in \\mathcal {S}$ where $\\mathcal {S}$ refers to the Schwartz space and $\\mathcal {F}$ denotes the Fourier transform [1].", "It can equivalently be defined by the prescription [27] $(-\\Delta )^{s} u(\\mbox{$x$}) = C(d,s) ~\\mathrm {P.V.}", "\\int _{\\mathbb {R}^d} {|\\mbox{$x$}-\\mbox{$y$}|^{d+2s}}d\\mbox{$y$},$ where the parameter $C(d,s)$ depends on the space dimension $d$ and the order $s$ of the fractional Laplacian.", "The $(-\\Delta )^{s}$ can be extended to an integer-order partial differential equation on the half space $\\mathbb {R}_+^{d+1}$ via a Dirichlet-to-Neumann mapping [7].", "However, subtlety occurs in the corresponding “boundary value\" problem of the fractional Laplacian when the domain $\\Omega $ under consideration is bounded, as there are more than one defitions of $(-\\Delta )^{s}$ in the literature which are not necessarily equivalent.", "A feasible definition is to restrict the function $u$ in (REF ) to those supported in $\\Omega $ .", "And the corresponding boundary value problem is formulated as $\\begin{array}{rcll}(-\\Delta )^s u(\\mbox{$x$}) &=& f(\\mbox{$x$}), \\quad & \\mbox{$x$}\\in \\Omega , \\\\[0.025in]u(\\mbox{$x$}) &=& 0, & \\mbox{$x$}\\in \\Omega ^c = \\mathbb {R}^d\\backslash \\Omega .\\end{array}$ By the Feynman-Kac formula [2], [29]: $u(x)$ can be obtained by an ensemble of the boundary data at the feet of the sample paths of a stochastic Lévy process that start from $\\mbox{$x$}$ and just jump out of domain $\\Omega $ .", "As the sample paths of a Lévy process admit jumps of arbitrary lengths, the boundary data must be imposed on the entire complement $\\Omega ^c$ of the domain $\\Omega $ .", "On the other hand, for the Laplacian equation ((REF ) with $s=1$ ), the underlying stochastic process is a Brownian motion that has continuous sample paths that intersect the boundary $\\partial \\Omega $ of the domain $\\Omega $ almost surely.", "Hence, the boundary condition needs only be specified at the boundary $\\partial \\Omega $ .", "Alternatively, let $\\lbrace \\lambda _n,\\psi _n\\rbrace _{n=1}^\\infty $ be the set of eigenvalues and ($L^2$ orthogonal and) normalized eigenfunctions of the Laplace operator in $\\Omega $ with the homogeneous Dirichlet boundary condition on $\\partial \\Omega $ .", "$(-\\Delta )^s u = \\sum _{n=1}^\\infty (u,\\psi _n) \\lambda _n^s \\psi _n, \\quad \\forall u \\in C^\\infty _0(\\Omega ).$ In [36] $(-\\Delta )^s$ defined in (REF ) was extended to a integer-order partial differential equation posed on $\\Omega \\times (0,\\infty )$ by generalizing the result in [7].", "This result was then utilized in [28] in the numerical approximation of the fractional Laplacian $(-\\Delta )^s$ defined in (REF ), by solving the integer-order equation on $\\Omega \\times (0,\\infty )$ via graded meshes in the extended variable.", "An alternative numerical discretization of the fractional Laplacian defined by (REF ) was presented in [45] via a discrete version of the spectral decomposition of (REF ).", "The constitutive models in peridynamics depend on finite deformation vectors, instead of deformation gradients in classical constitutive models [34], [35].", "Consequently, peridynamic models yield nonlocal mathematical formulations that are based on long-range interactions and present more appropriate representation of discontinuities in displacement fields and the description of cracks and their evolution in materials than classical continuum solid mechanics that are based on local interactions.", "For instance, a bond-based linear peridynamic model takes the form [9], [34], [35] $\\begin{array}{rlll}\\displaystyle C(d,\\delta ,k) \\int _{B_{\\delta }(\\mbox{$x$})} \\frac{(\\mbox{$x$}- \\mbox{$y$}) \\otimes (\\mbox{$x$}- \\mbox{$y$})}{|\\mbox{$x$}- \\mbox{$y$}|^3} \\big [ \\mbox{$u$}(\\mbox{$x$}) - \\mbox{$u$}(\\mbox{$y$}) \\big ] d \\mbox{$y$}& = \\mbox{$f$}(\\mbox{$x$}), &\\mbox{$x$}\\in \\Omega ,\\\\\\mbox{$u$}(\\mbox{$x$}) &= \\mbox{$g$}(\\mbox{$x$}), ~& \\mbox{$x$}\\in \\Omega _\\delta .\\end{array}$ Here $\\mbox{$u$}$ is the displacement vector, $\\mbox{$f$}$ is the prescribed body force density field, $\\Omega _\\delta $ denotes a boundary zone surrounding $\\Omega $ with width $\\delta > 0$ , and $\\mbox{$g$}$ is the prescribed displacement imposed on the domain $\\Omega _\\delta $ .", "The constant $C$ depends on the space dimension $d$ , the radius $\\delta $ , and the bulk modulus $k$ .", "The material horizon $B_{\\delta }(\\mbox{$x$})$ is a closed ball centered at $\\mbox{$x$}$ with the radius $\\delta $ .", "In other words, all the interactions in peridynamic models are allowed to be nonlocal, they are indeed assumed to be short ranged so the particle at $\\mbox{$x$}$ does not have any interaction with particles outside of $B_\\delta (\\mbox{$x$})$ .", "Moreover, the “boundary condition\" is imposed neither on the classical boundary $\\partial \\Omega $ of the domain $\\Omega $ nor the entire complement $\\Omega ^c$ of $\\Omega $ as in (REF ), but rather on the “collar\" $\\Omega _\\delta $ of the domain $\\Omega $ .", "In the context of one space dimension $d=1$ , the peridynamic model (REF ) reduces to the nonlocal diffusion model [10], [11] which corresponds to (REF ) with $d=1$ and $s=0$ and $\\mathbb {R}^d$ being replaced by $B_\\delta (x)$ .", "A variable-coefficient peridynamic model was derived in [23] in which an extra coefficient $K(\\mbox{$x$}) + K(\\mbox{$y$})$ appears in the integrand in (REF ).", "A variable-coefficient analogue $(-\\nabla (K(\\mbox{$x$}) \\nabla ))^s$ of (REF ) is defined by the right-hand side of (REF ) except that the $\\lbrace \\lambda _n,\\psi _n\\rbrace _{n=1}^\\infty $ are now the set of eigenvalues and eigenfunctions of the operator $-\\nabla (K(\\mbox{$x$}) \\nabla )$ in $\\Omega $ with the homogeneous Dirichlet boundary condition on $\\partial \\Omega $ .", "Finally, we turn to fractional differential equations (FDEs).", "Classical Fickian diffusion equation was derived under the assumptions of (i) the existence of a mean free path and (ii) the existence of a mean waiting time in the underlying identical and independently distributed random particle jumps [13], [16], [30].", "Under these assumptions, long walks in the same direction are rare so the variance of a particle excursion distance is finite.", "The central limit theorem concludes that the probability density function of finding a particle somewhere in space satisfies a canonical Fickian diffusion equation and thus gives rise to a probabilistic description of a normal diffusion precess [22].", "However, the random particle movements in heterogeneous media often undergo long jumps and so violate the assumptions (i) and (ii).", "These processes may have arbitrarily long jumps and so have large deviations from the stochastic process of Brownian motion.", "This is the reason why these processes cannot be described appropriately by the second-order diffusion equation.", "Consequently, the probability density function of finding a particle somewhere in space satisfies a Lévy distribution, which satisfies a space-fractional diffusion equation and thus gives rises a probabilistic description of an anomalous diffusion process.", "This explains why FDEs provide a more appropriate description of anomalous diffusion processes than classical Fickian diffusion equation and why FDE models have been used in many applications [22], [25], [26], [31], [43].", "We take the two-sided variable-coefficient conservative Caputo space-fractional diffusion equation as an example to demonstrate the idea $\\begin{array}{c}\\displaystyle - D \\bigl [ K(x) \\bigl (\\theta ~{}^{l}_aI_x^{\\beta } D + (1-\\theta ) ~{}^{r}_xI_b^{\\beta }D \\bigr )u \\big ] = f(x), \\quad x \\in (a,b), \\\\[0.05in]u(a) = u(b) = 0.\\end{array}$ Here $D$ is the first-order differential operator, $2-\\beta $ with $0 < \\beta < 1$ represents the order of anomalous diffusion of the problem, $0 < K_m \\le K(x) \\le K_M <+\\infty $ is the diffusivity coefficient, and $0 \\le \\theta \\le 1$ indicates the relative weight of forward versus backward transition probability, and $f$ is the source term [6], [8], [22], [44], [46].", "The left and right fractional integrals of order $\\sigma > 0$ are defined for any $w \\in L^1(a,b)$ by [1], [31], [32] ${}^{l}_aI_x^{\\sigma }w(x) := {\\Gamma (\\sigma )} \\int _a^x {(x-s)^{1-\\sigma }}ds, \\quad {}^{r}_xI^{\\sigma }_bw(x) := {\\Gamma (\\sigma )} \\int _x^b {(s-x)^{1-\\sigma }}ds$ where $\\Gamma (\\cdot )$ is the Gamma function.", "(REF ) is derived by combining a conventional mass balance law in terms of the flux $J$ $D J = f$ with the fractional Fick's law that accounts for the contributions of the particles that jumps to $x$ from any point in the domain $(a,b)$ [33] $J = - K(x) \\bigl (\\theta ~{}^{l}_aI_x^{\\beta } D + (1-\\theta ) ~{}^{r}_xI_b^{\\beta }D \\bigr )u.$ Although FDEs share some common mathematical properties with fractional Laplacian, peridynamics and nonlocal diffusion models due to their common nonlocality, the wellposedness of the boundary-value problems of the FDEs is more subtle to analyze than those of the other nonlocal models partly due to the following reason: The Fractional Laplacian operators $(-\\Delta )^s$ defined in (REF ) or (REF ) and the peridynamic (and nonlocal diffusion) models (REF ) as well as their variable-coefficient analogues can be formulated as a minimization of an quadratic energy functional and are symmetric and coercive with respect to appropriate (possibly weighted) fractional Sobolev spaces [1], [10], [11], [37].", "Hence, the Lax-Milgram theorem concludes that the corresponding boundary-value problems are wellposed exactly like in the context of integer-order PDEs [15].", "However, in the FDE in (REF ) the external operator is a first-order differential operator $D$ and the internal operator is a two-sided fractional differential operator of order $1-\\beta $ .", "Hence, the FDE problem (REF ) cannot be formulated as the minimization of an energy functional and the inherent trial and test spaces are fundamentally different especially in the presence of a variable diffusivity coefficient $K$ .", "This makes the analysis of the FDE problem (REF ) difficult to analyze.", "In a pioneer and foundational work on the wellposedness of problem (REF ) with a constant diffusivity coefficient $K$ , Ervin and Roop [14] derived a Galerkin weak formulation and proved that its bilinear form is coercive and bounded on the product space $H^{1-\\beta /2}_0(a,b) \\times H^{1-\\beta /2}_0(a,b)$ even though the problem cannot be formulated as the minimization of an energy functional.", "Thus, the Lax-Milgram theorem concludes that the problem is well posed [14], [15].", "In this paper we show that the bilinear form of the Galerkin formulation may lose its coercivity for problem (REF ) with a variable diffusivity coefficient.", "Numerical results show that the Galerkin finite element method does not necessarily converge in this case [40]!", "We then characterize the solution to problem (REF ) in terms of the solutions of second-order diffusion equations along with an integral equation.", "We then accordingly derive a Petrov-Galerkin formulation for problem (REF ) and prove that the bilinear form of the Petrov-Galerkin weak formulation is weakly coercive and so problem (REF ) is well posed." ], [ "Preliminaries", "Let $C[a,b]$ denote the space of continuous functions on $[a,b]$ , and $C^\\infty _0(a,b)$ denote the space of infinitely many times differentiable functions on $(a,b)$ that are compactly supported in $(a,b)$ .", "Let $L^p(a,b)$ , with $1 \\le p \\le +\\infty $ , be the Banach spaces of $p$ -th power Lebesgue integrable functions on $(a,b)$ and $W^{m,p}(a,b)$ be the Sobolev spaces of functions whose weak derivatives up to order $m$ are in $L^p(a,b)$ [1], [32].", "For any $\\mu > 0$ , define the (semi) norms [1], [14] $ | w |_{H^{\\mu }(\\mathbb {R})} := \\big \\Vert |\\omega |^\\mu \\mathcal {F}(w) \\big \\Vert _{L^2(\\mathbb {R})},\\qquad \\Vert w \\Vert _{H^{\\mu }(\\mathbb {R})} := \\big (\\Vert w\\Vert ^2_{L^2(\\mathbb {R})} + |w|_{H^{\\mu }(\\mathbb {R})}^2 \\bigr )^{1/2},$ and the fractional Sobolev space $H^{\\mu }(\\mathbb {R})$ is the completion of $C^\\infty _0(\\mathbb {R})$ with respect to the norm $\\Vert \\cdot \\Vert _{H^{\\mu }(\\mathbb {R})}$ .", "Let $0 < \\sigma < 1$ , $m$ be a postive integer, and $\\mu := m - \\sigma $ .", "Then the left and right (Caputo) fractional derivatives of order $\\mu $ are defined to be ${}_a^l D_x^{\\mu }w := {}_a^lI_x^{\\sigma } D^m w, \\quad {}_x^r D_b^{\\mu }w := {}_x^r I_b^{\\sigma }~ (-D)^m w.$ We then introduce the corresponding (semi) norms for the left, right, and weighted two-sided fractional derivatives, respectively for $w \\in C^\\infty _0(\\mathbb {R})$ [14], [32]: $\\begin{array}{ll}|w|_{J_l^{\\mu }(\\mathbb {R})} &:= \\big \\Vert {}_{-\\infty }^{l}D_x^{\\mu }w \\big \\Vert _{L^2(\\mathbb {R})}, \\qquad \\Vert w\\Vert _{J_l^{\\mu }(\\mathbb {R})} := \\big (\\Vert w\\Vert ^2_{L^2(\\mathbb {R})} + |w|_{J_l^{\\mu }(\\mathbb {R})}^2 \\bigr )^{1/2}, \\\\[0.075in]|w|_{J_r^{\\mu }(\\mathbb {R})} &:= \\big \\Vert {}_x^r D_\\infty ^{\\mu }w \\big \\Vert _{L^2(\\mathbb {R})}, ~~ \\qquad \\Vert w\\Vert _{J_r^{\\mu }(\\mathbb {R})} := \\big (\\Vert w\\Vert ^2_{L^2(\\mathbb {R})} + |w|_{J_r^{\\mu }(\\mathbb {R})}^2 \\bigr )^{1/2}, \\\\[0.075in]|w|_{J^{\\mu ,\\theta }(\\mathbb {R})} &:= \\big (\\theta ^2 | w |^2_{J_l^{\\mu }(\\mathbb {R})}+(1-\\theta )^2 | w |^2_{J_r^{\\mu }(\\mathbb {R})} \\bigr )^{1/2},\\\\[0.075in]\\Vert w\\Vert _{J^{\\mu ,\\theta }(\\mathbb {R})} &:= \\big (\\Vert w\\Vert ^2_{L^2(\\mathbb {R})} + |w|_{J^{\\mu ,\\theta }(\\mathbb {R})}^2 \\bigr )^{1/2}.\\end{array}$ Let $J_l^{\\mu }(\\mathbb {R})$ , $J_r^{\\mu }(\\mathbb {R})$ , and $J^{\\mu ,\\theta }(\\mathbb {R})$ denote the spaces that are completion of $C^\\infty _0(\\mathbb {R})$ with respect to the norms $\\Vert \\cdot \\Vert _{J_l^{\\mu }(\\mathbb {R})}$ , $\\Vert \\cdot \\Vert _{J_r^{\\mu }(\\mathbb {R})}$ , and $\\Vert \\cdot \\Vert _{J^{\\mu ,\\theta }(\\mathbb {R})}$ , respectively.", "In addition, we let $J^{\\mu }_{l,0}(a,b)$ , $J^{\\mu }_{r,0}(a,b)$ , $J^{\\mu ,\\theta }_0(a,b)$ , and $H^\\mu _0(a,b)$ denote the function spaces that are the completion of $C^\\infty _0(a,b)$ with respect to the norms $\\Vert \\cdot \\Vert _{J^{\\mu }_l(\\mathbb {R})}$ , $\\Vert \\cdot \\Vert _{J^{\\mu }_r(\\mathbb {R})}$ , $\\Vert \\cdot \\Vert _{J^{\\mu ,\\theta }(\\mathbb {R})}$ , and $\\Vert \\cdot \\Vert _{H^{\\mu }(\\mathbb {R})}$ , respectively.", "Finally, we let $H^{\\mu }(a,b)$ be the fractional Sobolev space of order $\\mu $ which can be defined to be the restriction of functions in $H^\\mu (\\mathbb {R})$ to the interval $(a,b)$ , and $H^{-\\mu }(a,b)$ be the dual space of $H^\\mu _0(a,b)$ [1].", "Below we cite some known results in the literature and prove some others that relate different fractional derivatives, spaces and (semi-)norms.", "We use $C$ to denote a generic constant that may assume different values at difference occurrences.", "We use $C_i$ to denote fixed constants.", "The following lemmas were proved in [14], [31], [32].", "(Fractional Poincaré inequality) Let $\\mu > 1/2 $ .", "Then there exists a positive constant $C_0 = C_0(\\mu )$ such that the following inequality holds $\\Vert w \\Vert _{L^2(a,b)} \\le C_0 | w |_{H^\\mu (a,b)}, \\qquad \\forall \\ w \\in H^\\mu _0(a,b).$ Let $\\mu > 0$ and $\\mu \\ne m - 1/2$ with $m \\in \\mathbb {N}$ .", "The spaces $J^{\\mu }_{l,0}(a,b)$ , $J^{\\mu }_{r,0}(a,b)$ , and $H^\\mu _0(a,b)$ are equal with equivalent semi-norms and norms, i.e., there exist positive constants $C_1 = C_1(\\mu )$ and $C_2 = C_2(\\mu )$ such that $\\begin{array}{l}C_1 | w |_{H^\\mu (a,b)} \\le |w|_{J_l^{\\mu }(a,b)} = |w|_{J_r^{\\mu }(a,b)} \\le C_2 | w |_{H^\\mu (a,b)}, \\\\[0.1in]C_1 \\Vert w \\Vert _{H^\\mu (a,b)} \\le \\Vert w\\Vert _{J_l^{\\mu }(a,b)} = \\Vert w\\Vert _{J_r^{\\mu }(a,b)} \\le C_2 \\Vert w \\Vert _{H^\\mu (a,b)}.\\end{array}$ Under the condition of Lemma and $0 \\le \\theta \\le 1$ , we have $\\begin{aligned}& (C_1/2) | w |_{H^\\mu (a,b)} \\le |w |_{J^{\\mu ,\\theta }(a,b)} \\le C_2 | w |_{H^\\mu (a,b)}, \\\\& (C_1/2) \\Vert w \\Vert _{H^\\mu (a,b)} \\le \\Vert w \\Vert _{J^{\\mu ,\\theta }(a,b)} \\le C_2 \\Vert w \\Vert _{H^\\mu (a,b)}.\\end{aligned}$ The left and right fractional integral operators are adjoint in the $L^2$ -sense, i.e., for all $\\mu > 0$ $\\big ({}_a^lI^{\\mu }_x ~w, v \\big )_{L^2(a,b)} = \\big (w,{}_x^rI^{\\mu }_b~ v \\big )_{L^2(a,b)}, \\quad \\forall ~ w, v \\in L^2(a,b).$ The left and right fractional integral operators follow the properties of a semigroup, i.e., for any $w \\in L^p(a,b)$ with $p \\ge 1$ , $\\begin{array}{rcll}{}_a^lI^{\\mu }_x ~ {}_a^lI^{\\sigma }_x w &=& {}_a^lI^{\\mu +\\sigma }_x w \\qquad \\forall x \\in [a,b], ~~\\forall \\mu , \\sigma > 0, \\\\[0.05in]{}_x^rI^{\\mu }_b ~{}_x^rI^{\\sigma }_b w &=& {}_x^rI^{\\mu +\\sigma }_b w \\qquad \\forall x \\in [a,b], ~~\\forall \\mu , \\sigma > 0.\\end{array}$ For $\\mu > 0$ , the following relations hold for any $w \\in H^{\\mu }_0(a,b)$ $\\bigl ({}_a^lD^{\\mu }_xw, {}_x^rD^{\\mu }_bw \\bigr )_{L^2(a,b)} = \\cos (\\pi \\mu ) \\Vert {}_a^l{D}^{\\mu }_xw\\Vert ^2_{L^2(a,b)}= \\cos (\\pi \\mu ) \\Vert {}_x^rD^{\\mu }_bw\\Vert ^2_{L^2(a,b)}.$ Let $0 < \\sigma <1$ .", "Then for any $w \\in W^{1,1}(a,b)$ with $w(a)= 0$ $D~{}^{l}_aI_x^\\sigma w = {}^l_a I_x^\\sigma Dw, \\quad x \\in (a,b),$ and for any function $w \\in W^{1,1}(a,b)$ with $w(b)= 0$ $D~{}^{r}_{x}I^\\sigma _bw = {}^r_xI^\\sigma _b Dw, \\quad x \\in (a,b).$ By symmetry, we only prove the first equation.", "$\\begin{aligned}D ~{}^l_a I_x^{\\sigma }w & = {\\Gamma (\\sigma )} d̑{dx} \\int _a^x {(x-s)^{1-\\sigma }} ds\\\\& \\displaystyle = {\\Gamma (1+\\sigma )} d̑{dx} \\big [ -(x-s)^{\\sigma } w(s) \\Big |_{s=a}^{s=x}+ \\int _a^x (x-s)^{\\sigma } w^{\\prime }(s) ds \\big ] \\\\& \\displaystyle = {\\Gamma (\\sigma )} \\int _a^x {(x-s)^{1-\\sigma }} ds = {}^{l}_{a}I_x^{\\sigma } Dw.\\end{aligned}$ This concludes the proof of the lemma." ], [ "Previous results for constant-coefficient FDEs", "In their pioneer work [14] Ervin and Roop studied the wellposedness of problem (REF ) with a constant diffusivity coefficient $K$ .", "They introduced a Galerkin weak formulation: For $f \\in H^{-(1-\\beta /2)}(a,b)$ , find $u \\in H^{1-\\beta /2}_0(a,b)$ such that for any $v \\in H^{1-\\beta /2}_0(a,b)$ $B(u,v) := \\theta K \\bigl \\langle {}^l_aI_x^{\\beta } D u, Dv \\bigr \\rangle + (1-\\theta ) K \\bigl \\langle {}^{r}_xI_b^{\\beta }D u, Dv \\bigr \\rangle = \\bigl \\langle f, v \\bigr \\rangle .$ They proved the following theorems for the wellposedness of the Galerkin weak formulation (REF ) and its corresponding Galerkin finite element approximations [14].", "The bilinear form $B(\\cdot ,\\cdot )$ is coercive and bounded on $H^{1-\\beta /2}_0(a,b) \\times H^{1-\\beta /2}_0(a,b)$ , so problem (REF ) is well posed.", "Let $S_h^m(a,b) \\subset H^{1-\\beta /2}_0(a,b)$ consist of piecewise polynomials of degree up to $(m-1)$ with respect to a quasiuniform partition of diameter $h$ and $u_h \\in S_h^m(a,b)$ satisfy $B(u_h,v_h) = \\bigl \\langle f, v_h \\bigr \\rangle \\qquad \\forall v_h \\in S_h^m(a,b).$ Assume that the weak solution $u$ to problem (REF ) is in $H^m(a,b) \\cap H^{1-\\beta /2}_0(a,b)$ .", "Then an optimal-order error estimate in the energy norm holds $\\Vert u_h - u \\Vert _{H^{1-\\beta /2}} \\le C h^{m-1+\\beta /2} \\Vert u\\Vert _{H^m}.$ Furthermore, if the true solution $w_g$ to the dual problem of (REF ) is in $H^{2-\\beta }(a,b) \\cap H^1_0(a,b)$ for each $g \\in L^2(a,b)$ such that $\\Vert w_g \\Vert _{H^{2-\\beta }} \\le C \\Vert g\\Vert _{L^2},$ then an optimal-order error estimate in the $L^2$ norm holds $\\Vert u_h - u\\Vert _{L^2} \\le C h^m \\Vert u\\Vert _{H^m}.$ It was shown in [19], [40], [41], [42] that the true solution to the homogeneous Dirichlet boundary-value problem of one-dimensional steady-state FDEs (REF ) of constant coefficients and right-hand side is not even in $W^{1,1/\\beta }(0,1)$ for any $0 < \\beta < 1$ .", "In particular, $u \\notin H^1(0,1)$ for any $1/2 \\le \\beta < 1$ !", "This is in sharp contrast to the case of integer-order elliptic PDEs.", "To date there are no verifiable conditions on the coefficients and source terms of FDEs in the literature that can ensure the existence of smooth true solutions to FDEs.", "Consequently, there are no verifiable conditions to guarantee the high-order convergence rates of the numerical discretizations to FDEs.", "Moreover, the lack of full regularity (REF ) of the solution to the dual FDE also implies that any Nitsche-lifting based proof of the optimal-order $L^2$ error estimates of the form (REF ) in the literature [14] is invalid!", "Another natural and fundamental question is as follows: Whether a variable-coefficient analogue of the bilinear form $B(\\cdot ,\\cdot )$ in (REF ) is coercive, which in turn ensures the wellposedness of problem (REF )?", "However, the following lemma gives rise to a negative answer to the question.", "For any $ 0 < \\beta <1$ and $ 0 \\le \\theta \\le 1$ , there exists a variable diffusivity coefficient $K = K(x,\\beta ,\\theta )$ with positive lower and upper bounds and a function $w \\in H^{1-\\beta /2}_0(0,1)$ such that $B(w,w) < 0$ .", "In fact, $K$ can be chosen as piecewise constant with just three pieces or its smooth modification.", "We prove the lemma by construction.", "Choose $w$ to be the following continuous and piecewise-linear function $w(x) :=\\left\\lbrace \\begin{array}{ll}4x, & \\displaystyle x \\in \\Big [0,{4} \\Big ],\\\\[0.1in]\\displaystyle 4\\Big ({2} - x \\Big ), & \\displaystyle x \\in \\Big [{4},{4} \\Big ],\\\\[0.1in]-4(1-x), \\quad & \\displaystyle x \\in \\Big [ {4},1 \\Big ].\\end{array}\\right.$ Apparently, $w \\in H^1_0(0,1)$ .", "Direct calculation yields ${}_0^l I_x^{\\beta } D w(x) ={\\Gamma (\\beta +1)} \\left\\lbrace \\begin{array}{ll}\\displaystyle x^\\beta , & \\displaystyle x \\in \\Bigl [0,{4}\\Bigr ],\\\\[6pt]\\displaystyle x^\\beta -2\\Bigl (x-{4}\\Bigr )^{\\beta }, & \\displaystyle x \\in \\Bigl [{4},{4}\\Bigr ],\\\\[6pt]\\displaystyle x^\\beta -2\\Bigl (x-{4}\\Bigr )^{\\beta }+2\\Bigl (x-{4}\\Bigr )^{\\beta }, ~~& \\displaystyle x \\in \\Bigl [{4},1\\Bigr ]\\end{array}\\right.$ and $\\begin{array}{l}\\displaystyle {}_x^rI_1^{\\beta } Dw(x) = {\\Gamma (\\beta +1)}\\left\\lbrace \\begin{array}{ll}\\displaystyle (1-x)^\\beta -2\\Bigl ({4}-x\\Bigr )^\\beta +2\\Bigl ({4}-x\\Bigr )^\\beta , & \\displaystyle x \\in \\Bigl [0,{4}\\Bigr ],\\\\[6pt]\\displaystyle (1-x)^\\beta -2\\Bigl ({4}-x\\Bigr )^{\\beta }, & \\displaystyle x \\in \\Bigl [{4},{4}\\Bigr ],\\\\[6pt]\\displaystyle (1-x)^\\beta , ~~ &\\displaystyle x \\in \\Bigl [{4},1\\Bigr ].\\end{array}\\right.\\end{array}$ We now prove that there exists a variable diffusivity coefficient such that $B(w,w)<0$ .", "In fact, direct calculation shows $\\begin{aligned}&\\big (\\theta {}_0^l I_x^{\\beta } + (1-\\theta ) {}_x^r I_1^{\\beta }\\big ) D w(x) \\bigl |_{x={4}} \\\\&\\quad = \\frac{4^{1-\\beta }}{\\Gamma (\\beta +1)} \\big [(2\\theta -1) + (1-\\theta ) (1+3^\\beta -2^{1+\\beta })\\bigr ]\\end{aligned}$ and $\\begin{aligned}&\\big (\\theta {}_0^lI_x^{\\beta } + (1-\\theta ){}_x^rI_1^{\\beta }\\big ) D w(x) \\bigl |_{x={4}} \\\\& \\quad = \\frac{4^{1-\\beta }}{\\Gamma (\\beta +1)} \\bigl [(1-2\\theta ) + \\theta (1+3^\\beta -2^{1+\\beta })\\bigr ].\\end{aligned}$ It is easy to check that $1+3^\\beta -2^{1+\\beta } \\le 0$ for $ 0 \\le \\beta \\le 1$ .", "Moreover, $1+3^\\beta -2^{1+\\beta }=0$ if and only if $\\beta =0$ or $\\beta =1$ .", "Hence, in the current context of $0 < \\beta < 1$ , $\\lambda = \\lambda (\\beta ) = 2^{1+\\beta }-1-3^\\beta >0$ .", "We now consider the case $0 \\le \\theta \\le 1/2$ .", "We observe from (REF that $\\begin{aligned}-\\frac{4^{1-\\beta }(1+\\lambda )}{\\Gamma (\\beta +1)} &\\le \\big (\\theta {}_0^l I_x^{\\beta } + (1-\\theta ) {}_x^r I_1^{\\beta }\\big ) D w(x) \\bigl |_{x={4}} \\\\&= \\frac{4^{1-\\beta }}{\\Gamma (\\beta +1)} \\big [(2\\theta -1) -(1-\\theta )\\lambda \\bigr ] \\le -\\frac{4^{1-\\beta }\\lambda }{2\\Gamma (\\beta +1)}.", "\\end{aligned}$ By the continuity of ${}_0^lI_x^{\\beta } D w(x)$ and ${}_x^r I_1^{\\beta }D w(x)$ there exists a $0 < \\delta \\le 1/4$ such that $-\\frac{2\\cdot 4^{1-\\beta }(1+\\lambda )}{\\Gamma (\\beta +1)} \\le \\bigl (\\theta {}_0^l{D}_x^{-\\beta } + (1-\\theta ) {}_x^r D_1^{1-\\beta } \\big ) D w\\le -\\frac{4^{1-\\beta }\\lambda }{4\\Gamma (\\beta +1)}, \\quad x \\in \\Bigl [{4}-\\delta ,{4}\\Bigr ].$ We accordingly define a diffusivity coefficient $K(x)$ as follows $K(x) :=\\left\\lbrace \\begin{array}{ll}K_l, \\quad & \\displaystyle x \\in \\Bigl (0,{4}-\\delta \\Bigr ),\\\\[0.05in]1, \\quad & \\displaystyle x \\in \\Bigl ({4}-\\delta ,{4}\\Bigr ),\\\\[0.1in]K_r, & \\displaystyle x \\in \\Bigl ({4},1\\Bigr ),\\end{array}\\right.$ where $K_l$ and $K_r$ are positive constants to be determined.", "Then we have $\\begin{aligned}B(w,w)= & 4 K_l\\int ^{{4}-\\delta }_0(\\theta {}_0^lI_x^{\\beta } + (1-\\theta ) {}_x^r I_1^{\\beta })D wdx + 4 \\int ^{{4}}_{{4}-\\delta }(\\theta {}_0^lI_x^{\\beta } + (1-\\theta ){}_x^rI_1^{\\beta })D wdx \\\\& -4 K_r\\int ^{{4}}_{{4}}(\\theta {}_0^lI_x^{\\beta } + (1-\\theta ) {}_x^r I_1^{\\beta }) D wdx + 4 K_r\\int ^1_{{4}}(\\theta {}_0^lI_x^{\\beta } + (1-\\theta ){}_x^rI_1^{\\beta })D wdx \\\\\\le & -\\frac{4^{1-\\beta }\\lambda \\delta }{\\Gamma (\\beta +1)} + 4 K_l \\int ^{{4}-\\delta }_0 \\bigl | (\\theta {}_0^lI_x^{\\beta }+ (1-\\theta ){}_x^rI_1^{\\beta })D w \\bigr |dx \\\\& + 4 K_r\\int ^1_{{4}} \\bigl |(\\theta {}_0^lI_x^{\\beta } +(1-\\theta ){}_x^rI_1^{\\beta })D w \\bigr |dx.\\end{aligned}$ We note that the integrands in the two integrals on the right-hand side are uniformly bounded from above with respect to $\\theta \\in [0,1/2]$ .", "Hence, by choosing the positive constants $K_l$ and $K_r$ sufficiently small we can enforce $B(w,w) < 0$ .", "We can similarly prove the conclusion for the case of $1/2 \\le \\theta \\le 1$ by using (REF ).", "We also observe from the proof that we can connect the piecewise constant diffusivity coefficient $K(x)$ as smooth as one desired such that the bilinear form $B(w,w)$ still loses its coercivity at least for some $w \\in H^1_0(0,1)$ .", "Remark 3.1 We observe from the proof of Lemma that the fundamental reason for the bilinear form $B(w,w)$ to lose its coercivity is that $Dw$ and $(\\theta {}_0^l{I}_x^{\\beta } + (1-\\theta ) {}_x^rI_1^{\\beta }) Dw$ do not always retain the same sign for all the functions $w \\in H^1_0(0,1)$ .", "As long as there exists one function $w$ and a subinterval on which $(\\theta {}_0^l{I}_x^{\\beta } +(1-\\theta ){}_x^rI_1^{\\beta }) Dw) ~Dw < 0$ , one can always enforce $B(w,w) < 0$ by choosing a specific diffusivity coefficient $K$ appropriately.", "Finally, a careful examination of the counterexample shows that the bilinear form $B(w,w)$ with a variable diffusivity coefficient having large variations might lose its coercivity." ], [ "A two-sided fractional integral operator $I^{\\beta }_\\theta $ and its properties", "Besides the pioneer work [14] of Ervin and Roop on the well-posedness of the two-sided FDE (REF ) with a constant diffusivity coefficient $K$ , virtually almost all the rest of the well-posedness results were proved for the FDE (REF ) are only for a one-sided simplification of problem (REF ) with either a constant diffusivity coefficient $K$ [19] or a variable diffusivity coefficient $K$ [39], [40].", "To the best of our knowledge, there is no well-posedness result on problem (REF ) with a variable diffusivity coefficient $K$ .", "Moreover, there is no regularity result in the literature for the two-sided problem (REF ) even for a constant diffusivity coefficient $K$ .", "To study the well-posedness and some regularity result of the two-sided problem (REF ) we introduce the following two-sided fractional integral operator $I^{\\beta }_\\theta $ for $ 0 < \\beta <1$ and $ 0 \\le \\theta \\le 1$ : $ I^{\\beta }_\\theta w = \\theta ~{}^{l}_aI^{\\beta }_xw+(1-\\theta )~{}^{r}_xI^{\\beta }_bw.$ We note that in the case of $\\theta = $ 0 or 1 the two-sided integral operator $I^{\\beta }_\\theta $ reduces to the well known Volterra integral operators, which have been well studied [20], [32].", "However, $I^{\\beta }_\\theta $ is a convex combination of two Volterra integral operators for $0 < \\theta < 1$ , for which there seems to be little study in the literature.", "We study its properties in the following theorem.", "Without loss of generality from now on we assume $a = 0$ and $b = 1$ for simplicity of presentation.", "We apply Lemma to express the boundary-value problem (REF ) in terms of the fractional integral operator $I^{\\beta }_\\theta $ as follows $\\begin{array}{rcl}\\displaystyle - D \\bigl ( KD I^{\\beta }_\\theta u \\bigr ) &=& f(x), \\quad x \\in (0,1), \\\\[0.075in]u(0) = u(1) &=& 0.\\end{array}$ In other word, the two-sided variable-coefficient FDE in problem (REF ) can naturally be rewritten as a canonical second-order diffusion equation in terms of $I^{\\beta }_\\theta u$ .", "However, the homogeneous Dirichlet boundary condition in terms of $u$ cannot be naturally expressed in terms of $I^{\\beta }_\\theta u$ , and in fact becomes one of the fundamental difficulties to overcome in the study of problem (REF ).", "Let $ 0 < \\beta <1/2$ and $0 \\le \\theta \\le 1$ .", "Then for any $w \\in H^{1-\\beta }_0(0,1)$ we have $\\begin{array}{rl}\\displaystyle 2 |w|_{J^{1-\\beta ,\\theta }(0,1)}^2 & \\displaystyle \\le \\big (1-\\cos (\\pi \\beta )\\big ) \\big (\\theta ^2 + (1-\\theta )^2 \\big ) |w|_{J^{1-\\beta ,\\theta }(0,1)}^2 \\\\[0.05in]& \\displaystyle \\le \\big \\Vert D I^{\\beta }_\\theta w \\big \\Vert _{L^2(0,1)}^2 = \\big \\Vert I^{\\beta }_\\theta D w \\big \\Vert _{L^2(0,1)}^2 \\\\ [0.1in]& \\displaystyle \\le \\big (1 + \\cos (\\pi \\beta )\\big ) \\big (\\theta ^2 + (1-\\theta )^2 \\big ) |w|_{J^{1-\\beta ,\\theta }(0,1)}^2\\\\[0.1in]& \\displaystyle \\le \\big (1 + \\cos (\\pi \\beta )\\big ) |w|_{J^{1-\\beta ,\\theta }(0,1)}^2\\end{array}$ where $| \\cdot |_{J^{1-\\beta ,\\theta }(0,1)}$ is defined below (REF ).", "We use the definition of $I^{\\beta }_\\theta w$ to obtain $\\begin{aligned}& \\big \\Vert D I^{\\beta }_\\theta w \\big \\Vert _{L^2(0,1)}^2 \\\\& \\displaystyle = \\bigl ( \\theta ~{}_0^lD^{1-\\beta }_x w - (1-\\theta )~{}_x^rD^{1-\\beta }_1 w,\\theta ~{}_0^lD^{1-\\beta }_x w - (1-\\theta )~{}_x^rD^{1-\\beta }_1 w \\bigr )_{L^2(0,1)}\\\\& \\quad = \\theta ^2 \\Vert {}_0^lD^{1-\\beta }_x w \\Vert _{L^2(0,1)}^2 + (1-\\theta )^2\\Vert {}_x^rD^{1-\\beta }_1 w \\Vert _{L^2(0,1)}^2\\\\[0.025in]&\\displaystyle \\qquad -2\\theta (1-\\theta ) \\bigl ({}_0^lD^{1-\\beta }_xw, {}_x^rD^{1-\\beta }_1w \\bigr )_{L^2(0,1)}\\\\[0.025in]&\\quad = \\theta ^2 \\Vert {}_0^lD^{1-\\beta }_x w \\Vert _{L^2(0,1)}^2 + (1-\\theta )^2\\Vert {}_0^l D^{1-\\beta }_x w \\Vert _{L^2(0,1)}^2\\\\[0.025in]&\\displaystyle \\qquad +2\\theta (1-\\theta ) \\cos (\\pi \\beta ) \\Vert {}_0^lD^{1-\\beta }_x w \\Vert _{L^2(0,1)}^2 \\\\[0.025in]&\\quad \\ge \\theta ^2 \\Vert {}_0^lD^{1-\\beta }_x w \\Vert _{L^2(0,1)}^2 + (1-\\theta )^2\\Vert {}_0^l D^{1-\\beta }_x w \\Vert _{L^2(0,1)}^2\\\\[0.025in]&\\displaystyle \\qquad -(\\theta ^2 + (1-\\theta )^2) \\cos (\\pi \\beta ) \\Vert {}_0^lD^{1-\\beta }_x w \\Vert _{L^2(0,1)}^2 \\\\[0.025in]&\\quad = (1 - \\cos (\\pi \\beta )) |w|_{J^{1-\\beta ,\\theta }(0,1)}^2.\\end{aligned}$ The other inequality can be proved similarly.", "We are now in the position to study the properties of $I^{\\beta }_\\theta w$ .", "Let $0 < \\beta < 1/2$ and $0 \\le \\theta \\le 1$ .", "The fractional integral operator $I^{\\beta }_\\theta $ is a bounded linear bijection from $H^{1-\\beta }_0(0,1)$ to its closed range $\\mathcal {R}(I^{\\beta }_\\theta ) \\subset H^1(0,1)$ with $\\mathcal {R}(I^{\\beta }_\\theta ) :=\\big \\lbrace w \\in H^1(0,1): \\exists \\phi \\in H^{1-\\beta }_0(0,1),~~ s.t.", "~~ w = I^{\\beta }_\\theta \\phi \\big \\rbrace .$ Moreover, the inverse operator $(I^{\\beta }_\\theta )^{-1}$ of $I^{\\beta }_\\theta $ is also bounded.", "We prove the theorem in two steps.", "At step 1 we prove that $I^{\\beta }_\\theta $ is a bounded linear operator from $H^{1-\\beta }_0(0,1)$ to $H^1(0,1)$ .", "It is clear that the integral $I^{\\beta }_\\theta \\phi $ is well defined for any $\\phi \\in H^{1-\\beta }(0,1) \\subset C[0,1]$ .", "Then we apply Lemma to obtain $\\begin{aligned}\\big \\Vert DI^{\\beta }_\\theta \\phi \\big \\Vert _{L^2(0,1)} &\\le \\theta \\Vert {}_0^lD^{1-\\beta }_x \\phi \\Vert _{L^2(0,1)}+(1-\\theta )\\Vert {}_x^rD^{1-\\beta }_1 \\phi \\Vert _{L^2(0,1)}\\\\&\\le C| \\phi |_{H^{1-\\beta }(0,1)}, \\quad \\forall \\ \\phi \\in H^{1-\\beta }_0(0,1).\\end{aligned}$ Furthermore, we have $\\begin{aligned}\\big | {}_0^lI^{\\beta }_x \\phi \\big | & = \\Bigl |\\int ^x_0D({}_0^lI^{\\beta }_s \\phi ) ds \\Bigr | \\le \\sqrt{x} \\big \\Vert {}_0^lD^{1-\\beta }_x \\phi \\big \\Vert _{L^2(0,1)},\\\\\\big | {}_x^rI_1^{\\beta } \\phi \\big | & = \\Bigl |\\int ^1_x D({}_s^rI_1^{\\beta } \\phi ) ds \\Bigr | \\le \\sqrt{1-x} \\big \\Vert {}_x^r D^{1-\\beta }_1\\phi \\big \\Vert ^2_{L^2(0,1)}.\\end{aligned}$ Hence, $\\big \\Vert I^{\\beta }_\\theta \\phi \\big \\Vert _{L^2(0,1)} \\le C | \\phi |_{H^{1-\\beta }(0,1)}, \\quad \\forall \\ \\phi \\in H^{1-\\beta }_0(0,1).$ We combine the preceding estimates to finish the proof of step 1.", "At step 2 we prove that $I^{\\beta }_\\theta $ has a bounded inverse operator $(I^{\\beta }_\\theta )^{-1}$ from $\\mathcal {R}(I^{\\beta }_\\theta )$ onto $H^{1-\\beta }_0(0,1)$ and that $\\mathcal {R}(I^{\\beta }_\\theta )$ is a closed subspace of $H^1(0,1)$ .", "In fact, we apply Lemmas and to conclude that for any $\\phi \\in H^{1-\\beta }_0(0,1)$ $\\big \\Vert I^{\\beta }_\\theta \\phi \\big \\Vert ^2_{H^1(0,1)} \\ge \\big \\Vert D I^{\\beta }_\\theta \\phi \\big \\Vert ^2_{L^2(0,1)}\\ge 2 | \\phi |_{J^{1-\\beta ,\\theta }(0,1)}^2 \\ge \\eta \\Vert \\phi \\Vert ^2_{H^{1-\\beta }(0,1)} $ for some $\\eta = \\eta (\\beta ) >0$ .", "Hence the operator $I^{\\beta }_\\theta $ is invertible and its inverse operator is bounded by $1/\\eta $ .", "Further, since both $I^{\\beta }_\\theta $ and its inverse operator are bounded linear operators, $\\mathcal {R}(I^{\\beta }_\\theta )$ is a closed subspace of $H^1(0,1)$ .", "To further study the properties of the range $\\mathcal {R}(I^{\\beta }_\\theta )$ , let $P$ be a projection operator from $H^1(0,1)$ onto $H^1_0(0,1)$ defined as follows: for any $w \\in H^1(0,1)$ , seek $Pw \\in H^1_0(0,1)$ such that $(D Pw, D v)_{L^2(0,1)} = (D w, D v)_{L^2(0,1)}, \\quad \\forall \\ v \\in H^1_0(0,1).$ Let $\\mathcal {N}(P)$ be the null space of the operator $P$ $\\begin{array}{rl}\\mathcal {N}(P) := & \\bigl \\lbrace w \\in H^1(0,1): \\ Pw=0 \\bigr \\rbrace \\\\[0.05in]=& \\bigl \\lbrace w \\in H^1(0,1): \\ (D w, D v)_{L^2(0,1)} =0, \\ \\ \\forall \\ v \\in H^1_0(0,1) \\bigr \\rbrace .\\end{array}$ Choosing any $w \\in H^1_0(0,1)$ concludes immediately that $H^1_0(0,1) = P(H^1(0,1))$ .", "Hence, the following decomposition $H^1(0,1) = H^1_0(0,1) \\oplus \\mathcal {N}(P)$ holds.", "Since $\\mathcal {R}(I^{\\beta }_\\theta ) \\subset H^1(0,1)$ , $P(\\mathcal {R}(I^{\\beta }_\\theta )) \\subset H^1_0(0,1)$ .", "In the next theorem we prove that the equality actually holds.", "Let $0 < \\beta < 1/2$ and $0 \\le \\theta \\le 1$ .", "The following equalities hold $ \\mathcal {N}(P) = span\\lbrace w_l^c, w_r^c\\rbrace , \\quad P\\bigl (\\mathcal {R}\\bigl (I^{\\beta }_\\theta \\bigr )\\bigr ) = H^1_0(0,1), \\quad H^1(0,1) =\\mathcal {R}\\bigl (I^{\\beta }_\\theta \\bigr ) \\oplus \\mathcal {N}(P)$ where $w_l^c := 1-x$ and $w_r^c: = x$ .", "We prove the theorem in four steps.", "As the first step we prove the first equality in (REF ).", "Note that $w_l^c$ and $w_r^c$ in $H^1(0,1)$ satisfy $\\begin{aligned}& (Dw_l^c,Dv)_{L^2(0,1)} = 0, \\ \\ \\forall \\ v \\in H^1_0(0,1); \\quad w_l^c(0)=1, ~w_l^c(1)=0;\\\\[0.025in]& (Dw_r^c,Dv)_{L^2(0,1)} = 0, \\ \\ \\forall \\ v \\in H^1_0(0,1); \\quad w_r^c(0)=0, ~w_r^c(1)=1.\\end{aligned}$ Hence, $w_l^c, w_r^c \\in \\mathcal {N}(P)$ .", "Conversely, for any $w \\in \\mathcal {N}(P)$ , it is clear that $w - w(0)w_l^c - w(1)w_r^c \\in H^1_0(0,1) \\cap \\mathcal {N}(P)$ as $\\bigl (D(w - w(0)w_l^c - w(1)w_r^c), Dv \\bigr )_{L^2(0,1)} = 0, \\ \\ \\forall \\ v \\in H^1_0(0,1).$ Choosing $v = w - w(0)w_l^c - w(1)w_r^c$ in this equation concludes that $w = w(0)w_l^c + w(1)w_r^c$ .", "Thus, the first equality in (REF ) holds.", "As the next step, we prove that the space $\\mathcal {U} := \\bigl \\lbrace w \\in H^1_0(0,1): \\exists \\ \\phi \\in H^{1-\\beta }_0(0,1)~\\mathrm {and}~c, c^{\\prime } \\in \\mathbb {R} ~\\mathrm {s.t.}", "~I^{\\beta }_\\theta \\phi = w + c w_l^c + c^{\\prime } w_r^c \\bigr \\rbrace $ is a closed subspace of $H^1_0(0,1)$ .", "First, it is clear that $\\mathcal {U}$ is a subspace of $H^1_0(0,1)$ .", "Let $w_1, w_2 \\in \\mathcal {U}$ and $a_1, a_2 \\in \\mathbb {R}$ , then there exist $\\phi _1, \\phi _2 \\in H^{1-\\beta }_0(0,1)$ such that $I^{\\beta }_\\theta (a_1 \\phi _1 + a_2 \\phi _2) = (a_1w_1 + a_2w_2) + (a_1 c_1 + a_2 c_2) w_l^c + (a_1 c^{\\prime }_1 + a_2 c^{\\prime }_2) w_r^c.$ Hence, $a_1 w_1 + a_2 w_2 \\in \\mathcal {U}$ .", "We now prove that $\\mathcal {U}$ is closed.", "To do so, let $\\lbrace w_n\\rbrace ^\\infty _{n=1} \\subset \\mathcal {U}$ be a sequence that converges to $w \\in H^1_0(0,1)$ .", "By definition of $\\mathcal {U}$ , there exist sequences $\\lbrace \\phi _n\\rbrace _{n=1}^\\infty \\subset H^{1-\\beta }_0(0,1)$ and $\\lbrace c_n\\rbrace _{n=1}^\\infty $ , $\\lbrace c^{\\prime }_n\\rbrace _{n=1}^\\infty \\subset \\mathbb {R}$ such that $I^{\\beta }_\\theta \\phi _n = w_n + c_n w_l^c + c^{\\prime }_n w_r^c \\in \\mathcal {R}(I^{\\beta }_\\theta ), \\quad n \\ge 1.$ As $\\lbrace \\Vert w_n\\Vert _{H^1(0,1)}\\rbrace ^\\infty _{n=1}$ is bounded, we claim that both sequences $\\lbrace c_n\\rbrace _{n=1}^\\infty $ and $\\lbrace c^{\\prime }_n\\rbrace _{n=1}^\\infty $ are bounded.", "Otherwise, there exists a subsequence $\\lbrace n_k\\rbrace _{k=1}^\\infty $ such that $\\lim _{k \\rightarrow \\infty } \\max \\lbrace |c_{n_k}|, |c^{\\prime }_{n_k}|\\rbrace =+\\infty $ .", "Without loss of generality, we assume that $|c_{n_k}| = \\max \\lbrace |c_{n_k}|, |c^{\\prime }_{n_k}|\\rbrace $ so $\\lim _{n_k \\rightarrow \\infty } |c_{n_k}| = \\infty $ .", "Since $|c^{\\prime }_{n_k}|/|c_{n_k}| \\le 1$ , there exists a subsequence which we still denote by $\\lbrace c^{\\prime }_{n_k}/c_{n_k}\\rbrace $ such that $\\lim _{k \\rightarrow \\infty }c^{\\prime }_{n_k}/c_{n_k} = c^{\\prime }$ .", "Let $\\psi _{n_k} = \\phi _{n_k}/c_{n_k}$ .", "Then $\\psi _{n_k} \\in H^{1-\\beta }_0(0,1)$ .", "We have $I^{\\beta }_\\theta \\psi _{n_k}= \\frac{1}{c_{n_k}}\\big (w_{n_k}+c_{n_k}w_l^c+c^{\\prime }_{n_k}w_r^c \\big ) \\rightarrow w_l^c + c^{\\prime } w_r^c, \\ \\ \\ as \\ \\ n_k \\rightarrow \\infty .$ As $\\mathcal {R}(I^{\\beta }_\\theta )$ is closed, $w_l^c + c^{\\prime } w_r^c \\in \\mathcal {R}(I^{\\beta }_\\theta )$ .", "By Theorem , there exists a $\\psi \\in H^{1-\\beta }_0(0,1)$ such that $I^{\\beta }_\\theta \\psi = w_l^c + c^{\\prime } w_r^c$ .", "Consequently, $\\bigl (D(I^{\\beta }_\\theta \\psi ),Dv \\bigr )_{L^2(0,1)} = \\bigl (D (w_l^c + c^{\\prime } w_r^c),Dv \\bigr )_{L^2(0,1)} =0, \\ \\ \\forall \\ v \\in H^1_0(0,1).$ Since this problem apparently has a trivial solution, the uniqueness of the solution of the problem ensured by Lemma and Theorem concludes that $\\psi \\equiv 0$ .", "That is, $0 = w_l^c + c^{\\prime } w_r^c$ .", "This contradicts to the linear independence of $w_l^c$ and $w_r^c$ .", "We thus have proved that $\\max \\lbrace |c_n|,|c^{\\prime }_n|\\rbrace $ is bounded.", "Consequently, there exist convergent subsequences $(c_{n_k},c^{\\prime }_{n_k})$ that converge to $(c,c^{\\prime })$ as $k \\rightarrow \\infty $ .", "We pass the limit in (REF ) to the subsequence to deduce that there exists a $\\phi \\in H^{1-\\beta }_0(0,1)$ such that $\\lim _{k \\rightarrow \\infty }\\phi _n = \\lim _{k \\rightarrow \\infty }(I^{\\beta }_\\theta )^{-1}(w_{n_k}+c_{n_k} w_l^c + c^{\\prime }_{n_k} w_r^c) = (I^{\\beta }_\\theta )^{-1}(w +c w_l^c + c^{\\prime } w_r^c) = \\phi .$ That is, $I^{\\beta }_\\theta \\phi = w + c w_l^c + c^{\\prime } w_r^c$ which implies that $w \\in \\mathcal {U}$ .", "We have thus proved that $\\mathcal {U}$ is closed.", "At the third step, we prove that $\\mathcal {U} = H^1_0(0,1)$ .", "In fact, for any $g \\in H^1_0(0,1) \\bigcap H^2(0,1)$ , Lemma and Theorem ensure that the problem $ \\big (D I^{\\beta } \\phi ,Dv \\big )_{L^2(0,1)} = - \\big (D^2 g, v \\big )_{L^2(0,1)}, \\ \\ \\forall \\ v \\in H^{1-\\frac{\\beta }{2}}_0(0,1)$ has a unique solution $\\phi \\in H^{1-\\frac{\\beta }{2}}_0(0,1)$ .", "This equation can then be rewritten as $\\bigl (D(I^{\\beta }\\phi - g),Dv \\bigr )_{L^2(0,1)} = 0, \\quad \\forall \\ v \\in H^1_0(0,1).$ This implies that $I^{\\beta }\\phi -g \\in \\mathcal {N}(P)$ .", "Hence, there exist constants $c$ and $c^{\\prime }$ such that $I^{\\beta }\\phi = g + c w_l^c + c^{\\prime } w_r^c$ .", "This shows that $g \\in \\mathcal {U}$ for any $g \\in H^1_0(0,1) \\cap H^2(0,1)$ .", "That is, $H^1_0(0,1) \\cap H^2(0,1) \\subset \\mathcal {U} \\subset H^1_0(0,1)$ .", "Since $H^1_0(0,1) \\cap H^2(0,1)$ is dense in $H^1_0(0,1)$ and $\\mathcal {U}$ is closed, we conclude that $\\mathcal {U} = H^1_0(0,1)$ .", "Finally, at step 4 we prove the last equality in (REF ).", "For any $w \\in H^1(0,1)$ , it is clear that $v = w - w(0)w_l^c - w(1)w_r^c \\in H^1_0(0,1) = \\mathcal {U}$ .", "By (REF ), there exist $\\phi \\in H^{1-\\beta }_0(0,1)$ and $c, c^{\\prime } \\in \\mathbb {R}$ such that $I^{\\beta }_\\theta \\phi = v + c w_l^c + c^{\\prime } w_r^c$ .", "That is, $w = I^{\\beta }_\\theta \\phi + (w(0)-c)w_l^c + (w(1)-c^{\\prime }) w_r^c.$ We thus prove $H^1(0,1) = \\mathcal {R}(I^{\\beta }_\\theta ) + \\mathcal {N}(P)$ .", "We now prove $\\mathcal {R}(I^{\\beta }_\\theta ) \\cap \\mathcal {N}(P) = \\emptyset $ .", "For any $w \\in \\mathcal {R}(I^{\\beta }_\\theta ) \\cap \\mathcal {N}(P)$ , there exists $\\psi \\in H^{1-\\beta }_0(0,1)$ and $c, c^{\\prime } \\in \\mathbb {R}$ such that $w = I^{\\beta }_\\theta \\psi = c w_l^c + c^{\\prime } w_r^c.$ Then the same argument following (REF ) shows that $\\psi \\equiv 0$ and $c = c^{\\prime } = 0$ .", "That is, $w \\equiv 0$ .", "We thus prove the third equality in (REF ).", "We now prove the main result of this section.", "Let $0 < \\beta < 1/2$ and $0 \\le \\theta \\le 1$ .", "Let $w_l^c$ and $w_r^c$ be defined as in Theorem .", "Then for any $g \\in H^1(0,1)$ , the following integral equation $I^{\\beta }_\\theta \\phi + \\bigl (g(0) - (1-\\theta ){}_0^rI^{\\beta }_1\\phi \\big )w_l^c + \\big (g(1) - \\theta ({}_0^lI^{\\beta }_1 \\phi ) \\big )w_r^c = g$ has a unique solution $\\phi \\in H^{1-\\beta }_0(0,1)$ .", "Conversely, for any $\\phi \\in H^{1-\\beta }_0(0,1)$ $g := I^{\\beta }_\\theta \\phi - (1-\\theta )\\big ({}_0^rI^{\\beta }_1\\phi \\big ) w_l^c - \\theta \\big ({}_0^lI^{\\beta }_1 \\phi \\big ) w_r^c \\in H^1_0(0,1).$ Furthermore, if $g \\in H^1_0(0,1)$ , then there exist positive constants $C_3$ and $C_4$ such that $C_3 \\Vert \\phi \\Vert _{H^{1-\\beta }(0,1)} \\le \\Vert g\\Vert _{H^1(0,1)} \\le C_4 \\Vert \\phi \\Vert _{H^{1-\\beta }(0,1)}.$ For any given $g \\in H^1(0,1)$ , by (REF ) in Theorem , there exist unique $\\phi \\in H^{1-\\beta }_0(0,1)$ and constants $c$ and $c^{\\prime }$ such that $g = I^{\\beta }_\\theta \\phi + c w_l^c + c^{\\prime }w_r^c.$ We use the definition (REF ) of $I^{\\beta }_\\theta $ and note that ${}_0^lI^{\\beta }_0 \\phi = 0$ and ${}_1^rI^{\\beta }_1 \\phi = 0$ to obtain $ g(0) = (1-\\theta ) {}^r_0I^{\\beta }_1 \\phi + c, \\qquad g(1) = \\theta {}^l_0I^{\\beta }_1 \\phi + c^{\\prime }.", "$ We thus prove (REF ).", "By Theorem , $g$ defined in (REF ) is in $H^1(0,1)$ .", "It is clear that $g(0) = g(1) = 0$ .", "This concludes the proof of (REF ).", "To prove (REF ) we note that ${}_0^rI^{\\beta }_1 \\phi = - \\int ^1_0 D({}_s^rI^{\\beta }_1\\phi )ds, \\qquad {}_0^lI^{\\beta }_1 \\phi = \\int ^1_0 D({}_0^lI^{\\beta }_s\\phi )ds.$ Hence $\\theta \\bigl |{}_0^lI^{\\beta }_1 \\phi \\bigr | + (1-\\theta ) \\bigl |{}_0^r I^{\\beta }_1 \\phi \\bigr | \\le C | \\phi |_{J^{1-\\beta ,\\theta }(0,1)}.$ We combine this inequality with (REF ) and Corollary to arrive at $\\begin{aligned}\\Vert g\\Vert _{H^1(0,1)} &\\le C \\big (\\Vert D I^{\\beta }_\\theta \\phi \\Vert _{L^2(0,1)} + \\theta \\bigl |{}_0^l{D}^{1-\\beta }_1 \\phi \\bigr | + (1-\\theta ) \\bigl |{}_0^r D^{1-\\beta }_1 \\phi \\bigr | \\big )\\\\[0.025in]& \\le C|\\phi |_{J^{1-\\beta ,\\theta }(0,1)} \\le C \\Vert \\phi \\Vert _{H^{1-\\beta }(0,1)}.\\end{aligned}$ We thus prove the right inequality in (REF ).", "To prove the left inequality in (REF ), let $P$ be the projection operator from $H^1(0,1)$ onto $H^1_0(0,1)$ as defined in (REF ).", "Choosing $v = Pw$ in (REF ) yields $\\Vert D(Pw) \\Vert _{L^2(0,1)} \\le \\Vert Dw \\Vert _{L^2(0,1)}, \\quad \\forall w \\in H^1(0,1).$ Although $P$ is not one-to-one from $H^1(0,1)$ to $H^1_0(0,1)$ , we claim that $PI^{\\beta }_\\theta $ is a bounded linear bijection from $H^{1-\\beta }_0(0,1)$ onto $H^1_0(0,1)$ .", "As a matter of fact, by Theorems and and equation (REF ), we conclude that $PI^{\\beta }_\\theta $ is a bounded linear operator from $H^{1-\\beta }_0(0,1)$ onto $H^1_0(0,1)$ .", "It remains to prove that $PI^{\\beta }_\\theta $ is injective.", "Suppose not, then there exists a $0 \\ne \\phi \\in H^{1-\\beta }_0(0,1)$ such that $PI^{\\beta }_\\theta \\phi = 0$ .", "Thus, $0 = (D(PI^{\\beta }_\\theta \\phi ),Dv)_{L^2(0,1)} = (DI^{\\beta }_\\theta \\phi ,Dv)_{L^2(0,1)},\\qquad \\forall v \\in H^1_0(0,1).$ By Theorem , this problem has only the trivial solution $\\phi = 0$ .", "This concludes that $PI^{\\beta }_\\theta $ is injective and so is a bounded linear bijection from $H^{1-\\beta }_0(0,1)$ onto $H^1_0(0,1)$ .", "By Banach's bounded inverse theorem, $PI^{\\beta }_\\theta $ has a bounded inverse operator $(PI^{\\beta }_\\theta )^{-1}$ from $H^1_0(0,1)$ onto $H^{1-\\beta }_0(0,1)$ .", "Thus, for any $g \\in H^1_0(0,1)$ , there exists a unique $\\phi \\in H^{1-\\beta }_0(0,1)$ such that $PI^{\\beta }_\\theta \\phi = g$ and vice versa.", "Furthermore, $\\Vert \\phi \\Vert _{H^{1-\\beta }(0,1)} \\le C \\Vert g\\Vert _{H^1(0,1)}.$ We thus prove the left inequality in (REF ) and so the theorem." ], [ "A Petrov-Galerkin formulation for constant-coefficient problems", "Lemma shows that the Galerkin formulation (REF ) may lose its coercivity in the context of variable-coefficient FDEs.", "Numerical evidence also indicated the illposedness of the Galerkin formulation [40].", "We note that the governing equation (REF ) is obtained by incorporating the fractional Fick's law into a canonical conservation law.", "This motivates the following Petrov-Galerkin weak formulation for problem (REF ) with $0 < \\beta < 1/2$ : Given $f \\in H^{-1}(0,1)$ , seek $u \\in H^{1-\\beta }_0(0,1)$ such that $A(u,v) := \\bigl (KDI^{\\beta }_\\theta u,Dv \\big )_{L^2(0,1)} = \\langle f,v \\rangle , \\qquad \\forall v \\in H^1_0(0,1).$ In this section we study the wellposedness of the weak formulation (REF ) for problem (REF ) with $K \\equiv 1$ and the characterization of its solution.", "For problem (REF ) with $0 < \\beta < 1/2$ , $0 \\le \\theta \\le 1$ , and $K \\equiv 1$ , the bilinear form $A(\\cdot ,\\cdot )$ is bounded and weakly coercive on the space $H^{1-\\beta }_0(0,1) \\times H^1_0(0,1)$ $\\begin{array}{l}\\displaystyle \\inf _{w \\in H^{1-\\beta }_0(0,1)\\setminus \\lbrace 0\\rbrace } \\sup _{v \\in H^1_0(0,1)\\setminus \\lbrace 0\\rbrace }\\frac{A(w,v)}{\\Vert w\\Vert _{H^{1-\\beta }(0,1)}\\Vert v\\Vert _{H^1(0,1)}} \\ge \\kappa ,\\\\[0.2in]\\displaystyle \\sup _{w \\in H^{1-\\beta }_0(0,1)} A(w,v) > 0, \\qquad \\forall \\ v \\in H^1_0(0,1)\\setminus \\lbrace 0\\rbrace \\end{array}$ for a positive constant $\\kappa =\\kappa (\\beta ) >0$ .", "Hence, the Petrov-Galerkin formulation (REF ) has a unique solution $u \\in H^{1-\\beta }_0(0,1)$ for which $\\Vert u\\Vert _{H^{1-\\beta }(0,1)} \\le \\frac{1}{\\kappa }\\Vert f\\Vert _{H^{-1}(0,1)}.$ For each $w \\in H^{1-\\beta }_0(0,1)$ , by Theorem there exists a unique $v \\in H^1_0(0,1)$ such that $I^{\\beta }_\\theta w - (1-\\theta )({}_0^rI^{\\beta }_1 w)w_l^c - \\theta ({}_0^lI^{\\beta }_1w)w_r^c = v$ where $w_l^c$ and $w_r^c$ are defined in Theorem .", "We differentiate this equation and apply (REF ) to obtain $\\begin{aligned}A(w,v) &\\displaystyle := \\big (DI^{\\beta }_\\theta w,Dv \\big )_{L^2(0,1)} = (Dv,Dv)_{L^2(0,1)} =\\Vert Dv\\Vert ^2_{L^2(0,1)}\\\\& \\displaystyle \\ge \\min \\left\\lbrace {2},{2C_0} \\right\\rbrace ^2 \\Vert v\\Vert ^2_{H^1(0,1)} \\\\& \\displaystyle \\ge C_3 \\min \\left\\lbrace {2},{2C_0} \\right\\rbrace ^2 \\Vert v\\Vert _{H^1(0,1)}\\Vert w\\Vert _{H^{1-\\beta }(0,1)}.\\end{aligned}$ We thus prove the first inequality in (REF ) with $\\kappa = C_3 \\min \\lbrace 1/2,1/(2C_0) \\rbrace ^2$ .", "For each $ v \\in H^1_0(0,1) \\setminus \\lbrace 0\\rbrace $ , it follows from Theorem that there exists a unique $w \\in H^{1-\\beta }_0(0,1)$ such that (REF ) holds.", "Then (REF ) shows that the second inequality in (REF ) holds.", "We apply Babus̆ka-Lax-Milgram theorem [3], to finish the proof.", "The following corollary characterizes the weak solution of the Petrov-Galerkin formulation (REF ) in terms of the weak solution to a Galerkin formulation of a canonical second-order diffusion equation.", "Assume that the conditions of Theorem hold.", "Let $w_l^c$ and $w_r^c$ be defined as in Theorem .", "Then $u \\in H^{1-\\beta }_0(0,1)$ is the weak solution of the Petrov-Galerkin formulation (REF ) if and only if $w \\in H^1_0(0,1)$ defined by $I^{\\beta }_\\theta u - (1-\\theta )({}_0^rI^{\\beta }_1u)w_l^c - \\theta ({}_0^lI^{\\beta }_1u)w_r^c = w$ is the weak solution of the Galerkin formulation $(Dw,Dv)_{L^2(0,1)} = \\langle f,v \\rangle , \\quad \\forall v \\in H^1_0(0,1).$ By Theorem , $u \\in H^{1-\\beta }_0(0,1)$ if and only if there is a functions $w \\in H^1_0(0,1)$ such that (REF ) holds.", "The proof resides in the fact that for $K \\equiv 1$ , $A(u,v) = (Dw,Dv)_{L^2(0,1)}$ for any $v \\in H^1_0(0,1)$ .", "Remark 5.1 Despite that they are both formulated for a constant-coefficient analogue of problem (REF ), the Petrov-Galerkin formulation (REF ) is different from the Galerkin formulation (REF ) in that the latter is defined on the product space $H^{1-2}_0(0,1) \\times H^{1-\\frac{\\beta }{2}}_0(0,1)$ for any given $f \\in H^{-(1-\\frac{\\beta }{2})}(0,1)$ with $0 < \\beta < 1$ while the former is defined on a different product space $H^{1-\\beta }_0(0,1) \\times H^1_0(0,1)$ for a given $f \\in H^{-1}(0,1)$ with $0 < \\beta < 1/2$ .", "Hence, Theorem does not apply to the Petrov-Galerkin formulation (REF ), to which Theorem applies." ], [ "Wellposedness, regularity and characterization of weak solutions to variable-coefficient FDEs", "In this section we study the wellposedness of the Petrov-Galerkin formulation (REF ) and the characterization of the corresponding weak solutions to problem (REF ).", "We begin by letting $w_l$ , $w_r$ , and $w_f$ be the weak solutions to the following Galerkin formulation for second-order diffusion equations: $\\begin{aligned}& (K Dw_l,Dv)_{L^2(0,1)} = 0 \\ \\ \\forall \\ v \\in H^1_0(0,1), \\quad w_l(0)=1, ~w_l(1) = 0;\\\\[0.025in]& (K Dw_r,Dv)_{L^2(0,1)} = 0 \\ \\ \\forall \\ v \\in H^1_0(0,1), \\quad w_r(0)=0, ~w_r(1) = 1;\\\\[0.025in]& (KDw_f,Dv)_{L^2(0,1)} = \\langle f,v \\rangle , \\ \\ \\forall \\ v \\in H^1_0(0,1), \\quad w_f(0) = w_f(1) = 0\\end{aligned}$ for a given $f \\in H^{-1}(0,1)$ .", "It is well known that these problems have unique solutions [15].", "The solutions $w_l$ and $w_r$ can be solved in closed form as follows $w_l= \\left(\\int ^1_0\\frac{1}{K(s)}ds \\right)^{-1} \\int ^1_x\\frac{1}{K(s)}ds, \\qquad w_r= \\left(\\int ^1_0\\frac{1}{K(s)}ds \\right)^{-1}\\int ^x_0\\frac{1}{K(s)}ds.$ It is clear that $0 \\le w_l, \\ w_r \\le 1$ and $w_l+w_r \\equiv 1$ , and that $w_l$ and $w_r$ are variable extensions of $w_l^c$ and $w_r^c$ introduced in Theorem .", "Let $0 < \\beta < 1/2$ and $0 \\le \\theta \\le 1$ .", "Then $u$ is a weak solution to the Petrov-Galerkin formulation (REF ) if and only if $u$ satisfies the following integral equation $I^{\\beta }_\\theta u - (1-\\theta )\\big ({}_0^r I^{\\beta }_1u \\big )w_l - \\theta \\big ({}_0^lI^{\\beta }_1u \\big )w_r = w_f.$ Suppose that $u$ satisfies (REF ).", "Then we use equations (REF ) to deduce $\\begin{aligned}A(u,v) & = \\big (KD(I^{\\beta }_\\theta u),Dv \\big )_{L^2(0,1)} \\\\& = (KDw_f,Dv)_{L^2(0,1)} + (1-\\theta ) \\big ({}_0^r I^{\\beta }_1 u \\big ) (KDw_l,Dv)_{L^2(0,1)}\\\\[0.025in]& \\qquad + \\theta \\big ({}_0^l I^{\\beta }_1 u \\big ) (KDw_r,Dv)_{L^2(0,1)} \\\\[0.025in]& = (KDw_f,Dv)_{L^2(0,1)} = \\langle f,v\\rangle , \\quad \\forall v \\in H^1_0(0,1).\\end{aligned}$ Thus, $u$ is a weak solution to the Petrov-Galerkin formulation (REF ).", "Conversely, let $u$ be a weak solution to the Petrov-Galerkin formulation (REF ).", "We define $w := I^{\\beta }_\\theta u - (1-\\theta )({}_0^rI^{\\beta }_1u)w_l - \\theta ({}_0^lI^{\\beta }_1u)w_r.", "$ Theorem ensures that $w \\in H^1(0,1)$ .", "In addition, we have $w(0) = I^{\\beta }_\\theta u|_{x=0} - (1-\\theta )({}_0^r I^{\\beta }_1u) = (1-\\theta )({}_0^r I^{\\beta }_1u) - (1-\\theta )({}_0^r I^{\\beta }_1u) = 0.$ Similarly, we have $w(1) = 0$ .", "Thus, $w \\in H^1_0(0,1)$ .", "Furthermore, $\\begin{aligned}(KDw,Dv)_{L^2(0,1)} & = \\big (KD(I^{\\beta }_\\theta u-(1-\\theta )({}_0^rI^{\\beta }_1u)w_l - \\theta ({}_0^lI^{\\beta }_1u)w_r), Dv \\big )_{L^2(0,1)}\\\\& = \\big (KD(I^{\\beta }_\\theta u),Dv\\big )_{L^2(0,1)} = \\langle f,v \\rangle , \\quad \\forall \\ v \\in H^1_0(0,1).\\end{aligned}$ In other words, both $w$ and $w_f$ are the solution to the same Galerkin formulation in (REF ).", "By the uniqueness of the weak solution to the problem, we conclude that $w = w_f$ .", "Thus, (REF ) implies that (REF ) holds.", "We are now in a position to study the existence and uniqueness of the weak solution to Petrov-Galerkin formulation (REF ).", "We apply Theorem to $w_l - w_l^c, w_r - w_r^c \\in H^1_0(0,1)$ to conclude that there exist unique $u_l$ and $u_r$ in $H^{1-\\beta }_0(0,1)$ such that $\\left\\lbrace \\begin{aligned}& I^{\\beta }_\\theta u_l - (1-\\theta )\\big ({}_0^r I^{\\beta }_1 u_l \\big )w_l^c - \\theta \\big ({}_0^l I^{\\beta }_1 u_l \\big )w_r^c = w_l - w_l^c,\\\\& I^{\\beta }_\\theta u_r - (1-\\theta )\\big ({}_0^r I^{\\beta }_1 u_r \\big )w_l^c - \\theta \\big ({}_0^lI^{\\beta }_1u_r \\big )w_r^c = w_r - w_r^c.\\end{aligned}\\right.$ We note that $w_l^c$ and $w_r^c$ satisfy (REF ) to obtain $\\begin{aligned}& (D I^{\\beta }_\\theta u_l, Dv)_{L^2(0,1)} = (Dw_l,Dv)_{L^2(0,1)}, \\quad &\\forall v \\in H^1_0(0,1), \\\\& (D I^{\\beta }_\\theta u_r, Dv)_{L^2(0,1)} = (Dw_r,Dv)_{L^2(0,1)}, \\ \\ & \\forall v \\in H^1_0(0,1).\\end{aligned}$ Namely, $u_l, u_r \\in H^{1-\\beta }_0(0,1)$ are two particular solutions to problem (REF ) with $K \\equiv 1$ and $f = -D^2w_l$ and $-D^2w_r$ , respectively.", "The fact that $w_l + w_r \\equiv 1$ implies that $u_l + u_r$ satisfies the Galerkin formulation (REF ) with $K \\equiv 1$ and $f \\equiv 0$ .", "The existence and uniqueness of the weak solution to the Galerkin weak formulation (REF ) concludes that $u_l + u_r \\equiv 0$ .", "In particular, if $K$ is constant, then $w_l \\equiv w_l^c$ and $w_r \\equiv w_r^c$ .", "So the right-hand sides of the preceding equations vanish, which implies $u_l = u_r \\equiv 0$ .", "Introduce the following variable analogue to the space $\\mathcal {U}$ in (REF ) $\\begin{aligned}\\mathcal {V} := & \\bigl \\lbrace g \\in H^1_0(0,1): \\exists \\phi \\in H^{1-\\beta }_0(0,1), ~ I^{\\beta }_\\theta \\phi - (1-\\theta ) \\big ({}_0^r I^{\\beta }_1 \\phi \\big ) w_l - \\theta \\big ({}_0^l I^{\\beta }_1 \\phi \\big )w_r = g \\bigr \\rbrace .\\end{aligned}$ The space $\\mathcal {V} = H^1_0(0,1)$ , provided the following condition holds $1 + \\theta \\big ({}_0^l{I}^{\\beta }_1 u_l \\big ) - (1-\\theta ) \\big ({}_0^rI^{\\beta }_1u_l \\big ) \\ne 0, \\quad 0 < x < 1$ or equivalently $1 - \\big ({}_0^l{I}^{\\beta }_1 u_r \\big ) + (1-\\theta ) \\big ({}_0^rI^{\\beta }_1u_r \\big ) \\ne 0, \\quad 0 < x < 1.$ We conclude from equation (REF ) in Theorem that for each $g \\in H^1_0(0,1)$ , there exists a unique $\\phi \\in H^{1-\\beta }_0(0,1)$ such that $\\begin{aligned}I^{\\beta }_\\theta \\phi &= g + (1-\\theta ) \\big ({}_0^r I^{\\beta }_1 \\phi \\big ) w_l^c + \\theta \\big ({}_0^l I^{\\beta }_1 \\phi )w_r^c\\\\&= g + (1-\\theta ) c_l w_l + \\theta c_r w_r - (1-\\theta ) c_l (w_l-w_l^c) - \\theta c_r (w_r-w_r^c)\\\\&\\quad \\quad +(1-\\theta )({}_0^r I^{\\beta }_1 \\phi - c_l)w_l^c + \\theta ({}_0^l I^{\\beta }_1 \\phi - c_r)w_r^c,\\end{aligned}$ where $c_l$ and $c_r$ are constants to be determined.", "We then evaluate $I^{\\beta }_\\theta ((1-\\theta ) c_l u_l + \\theta c_r u_r)$ and replace the $I^{\\beta }_\\theta u_l$ and $I^{\\beta }_\\theta u_r$ in the expressions by the rest of the terms in the equations (REF ) to obtain $\\begin{array}{l}\\hspace{-3.61371pt} I^{\\beta }_\\theta ((1-\\theta ) c_l u_l + \\theta c_r u_r) \\\\~= (1-\\theta )c_l \\bigl [ (1-\\theta )({}_0^r I^{\\beta }_1 u_l)w_l^c + \\theta ({}_0^l I^{\\beta }_1 u_l)w_r^c + w_l - w_l^c \\bigr ]\\\\ [0.025in]\\quad + \\theta c_r \\bigl [ (1-\\theta )({}_0^r I^{\\beta }_1 u_r)w_l^c + \\theta ({}_0^lI^{\\beta }_1u_r)w_r^c + w_r - w_r^c \\bigr ]\\\\[0.025in]~= (1-\\theta ) c_l (w_l - w_l^c) + \\theta c_r (w_r - w_r^c) \\\\[0.025in]\\quad + (1-\\theta )\\big ({}_0^r I^{\\beta }_1 ( (1-\\theta ) c_l u_l + \\theta c_r u_r) \\big ) w_l^c + \\theta \\big ({}_0^l I^{\\beta }_1 ( (1-\\theta ) c_l u_l + \\theta c_r u_r) \\big ) w_r^c.\\end{array}$ We sum equations (REF ) and (REF ) and cancel the corresponding terms to get $\\begin{array}{l}\\hspace{-10.84006pt} I^{\\beta }_\\theta ( \\phi +(1-\\theta ) c_l u_l + \\theta c_r u_r ) \\\\[0.05in]= g + (1-\\theta ) c_l w_l + \\theta c_r w_r + (1-\\theta )\\big ({}_0^r I^{\\beta }_1 (\\phi + (1-\\theta ) c_l u_l + \\theta c_r u_r) - c_l \\big ) w_l^c \\\\[0.05in]~ + \\theta \\big ({}_0^l I^{\\beta }_1 ( \\phi + (1-\\theta ) c_l u_l + \\theta c_r u_r) - c_r \\big ) w_r^c.\\end{array}$ Comparing this equation with the definition of the space $\\mathcal {V}$ we conclude that to eliminate the last two terms on the right-hand side we shall choose $c_l$ and $c_r$ to satisfy the equations $\\left\\lbrace \\begin{aligned}& {}_0^r I^{\\beta }_1 (\\phi + (1-\\theta ) c_l u_l + \\theta c_r u_r) - c_l = 0,\\\\& {}_0^l I^{\\beta }_1 (\\phi + (1-\\theta ) c_l u_l + \\theta c_r u_r) - c_r = 0.\\end{aligned}\\right.", "$ Equivalently, $\\displaystyle \\left[ \\begin{array}{cc}1 - (1-\\theta ) ({}_0^r I^{\\beta }_1 u_l) & - \\theta ({}_0^r I^{\\beta }_1 u_r) \\\\[0.075in]-(1-\\theta ) ({}_0^l I^{\\beta }_1 u_l) & 1 - \\theta ({}_0^l I^{\\beta }_1 u_r)\\end{array} \\right]\\left[ \\begin{array}{l}c_l \\\\ [0.1in]c_r\\end{array} \\right]=\\left[ \\begin{array}{l}{}_0^r I^{\\beta }_1 \\phi \\\\ [0.075in]{}_0^l I^{\\beta }_1 \\phi \\end{array} \\right].", "$ This linear system has a unique solution for each given $\\phi \\in H^{1-\\beta }_0(0,1)$ if and only if its coefficient matrix is nonsingular, i.e., $\\big (1-(1-\\theta )({}_0^rI^{\\beta }_1u_l)\\big )\\big (1-\\theta ({}_0^l I^{\\beta }_1u_r)\\big )- (1-\\theta )\\theta ({}_0^r{I}^{\\beta }_1u_r)({}_0^lI^{\\beta }_1u_l) \\ne 0.", "$ We incorporate the condition $u_r = - u_l$ into this equation to arrive at (REF ).", "Under the condition (REF ), we let $w := \\phi + (1-\\theta ) c_l u_l + \\theta c_r u_r$ .", "Then equation (REF ) can be rewritten as $c_l = {}_0^r I^{\\beta }_1 w, \\qquad c_r = {}_0^lI^{\\beta }_1 w$ and (REF ) can be expressed in the form $I^{\\beta }_\\theta w = g + (1-\\theta ) ({}_0^r I^{\\beta }_1 w)w_l + \\theta ({}_0^lI^{\\beta }_1 w) w_r.$ In other words, for the given $g \\in H^1_0(0,1)$ we have found a function $w \\in H^{1-\\beta }_0(0,1)$ such that the preceding equation holds.", "Hence, $g \\in \\mathcal {V}$ .", "That is, $\\mathcal {V}=H^1_0(0,1)$ .", "Given $f \\in H^{-1}(0,1)$ , the Petrov-Galerkin formulation (REF ) has at least one weak solution, provided that the condition (REF ) holds.", "Furthermore, the solution is unique if and only if condition (REF ) holds.", "Assume that condition (REF ) holds.", "Let $w_f \\in H^1_0(0,1)$ be the weak solution to the third equation in (REF ).", "By Theorem , there exists at least one $u \\in H^{1-\\beta }_0(0,1)$ such that $I^{\\beta }_\\theta u - (1-\\theta ) ({}_0^r I^{\\beta }_1 u) w_l - \\theta ({}_0^l I^{\\beta }_1 u)w_r = w_f.$ Theorem concludes that $u$ is a weak solution to the Petrov-Galerkin formulation (REF ).", "The solution to the Petrov-Galerkin formulation (REF ) is unique if and only if the corresponding homogeneous formulation has only the trivial solution.", "By Theorem , this is equivalent to that the following integral equation $I^{\\beta }_\\theta u = (1-\\theta )({}_0^r I^{\\beta }_1 u)w_l + \\theta ({}_0^lI^{\\beta }_1u)w_r$ has only the trivial solution.", "We claim that any solution $u$ to the integral equation (REF ) can be expressed as a linear combination of $u_l$ and $u_r$ introduced in (REF ).", "In fact, (REF ) can be rewritten as $\\begin{aligned}I^{\\beta }_\\theta u & =(1-\\theta )({}_0^r I^{\\beta }_1 u)(w_l-w_l^c) + \\theta ({}_0^l I^{\\beta }_1 u)(w_r-w_r^c)\\\\[0.025in]& ~+(1-\\theta )({}_0^r I^{\\beta }_1 u)w_l^c + \\theta ( {}_0^lI^{\\beta }_1 u) w_r^c.\\end{aligned}$ We use (REF ) and (REF ) to find that $\\begin{aligned}& I^{\\beta }_\\theta (u-(1-\\theta )({}_0^r I^{\\beta }_1 u)u_l - \\theta ({}_0^lI^{\\beta }_1 u)u_r) \\\\& \\quad = I^{\\beta }_\\theta u-(1-\\theta )({}_0^r I^{\\beta }_1 u) I^{\\beta }_\\theta u_l - \\theta ({}_0^l I^{\\beta }_1 u) I^{\\beta }_\\theta u_r \\\\& \\quad = (1-\\theta )({}_0^r I^{\\beta }_1 u)(w_l-w_l^c) + \\theta ({}_0^l I^{\\beta }_1 u)(w_r-w_r^c)\\\\[0.025in]& \\qquad + (1-\\theta )({}_0^r I^{\\beta }_1 u)w_l^c + \\theta ( {}_0^lI^{\\beta }_1 u) w_r^c\\\\&\\qquad - (1-\\theta )({}_0^r I^{\\beta }_1 u) [(1-\\theta )({}_0^r I^{\\beta }_1 u_l)w_l^c + \\theta ({}_0^l I^{\\beta }_1 u_l)w_r^c + w_l - w_l^c ] \\\\&\\qquad - \\theta ({}_0^l I^{\\beta }_1 u) [(1-\\theta ) ({}_0^r I^{\\beta }_1 u_r)w_l^c + \\theta ({}_0^lI^{\\beta }_1u_r)w_r^c + w_r - w_r^c]\\\\&\\quad = (1-\\theta )({}_0^r I^{\\beta }_1 u)w_l^c + \\theta ( {}_0^lI^{\\beta }_1 u) w_r^c\\\\&\\qquad - (1-\\theta )({}_0^r I^{\\beta }_1 u) [(1-\\theta )({}_0^r I^{\\beta }_1 u_l)w_l^c + \\theta ({}_0^l I^{\\beta }_1 u_l)w_r^c ] \\\\&\\qquad - \\theta ({}_0^l I^{\\beta }_1 u) [(1-\\theta ) ({}_0^r I^{\\beta }_1 u_r)w_l^c + \\theta ({}_0^lI^{\\beta }_1u_r)w_r^c ]\\\\&\\quad = (1-\\theta ) \\big [\\ {}_0^r I^{\\beta }_1 \\big (u-(1-\\theta )({}_0^r I^{\\beta }_1 u)u_l - \\theta ({}_0^l I^{\\beta }_1 u)u_r \\big ) \\big ] w_l^c \\\\&\\qquad + \\theta \\big [{}_0^l I^{\\beta }_1 \\big ( u-(1-\\theta )({}_0^r I^{\\beta }_1 u)u_l - \\theta ({}_0^l I^{\\beta }_1 u)u_r \\big ) \\big ] w_r^c.\\end{aligned}$ In other words, $\\phi := u-(1-\\theta )({}_0^r I^{\\beta }_1 u)u_l - \\theta ({}_0^l I^{\\beta }_1 u)u_r$ satisfies the homogeneous integral equation $I^{\\beta }_\\theta \\phi - (1-\\theta )({}_0^rI^{\\beta }_1\\phi )w_l^c - \\theta ({}_0^lI^{\\beta }_1 \\phi )w_r^c = 0.$ By Theorem , i.e., (REF ) with $g=0$ , this equation has only the trivial solution $\\phi \\equiv 0$ .", "That is, $u = (1-\\theta ) ({}_0^r I^{\\beta }_1 u) u_l + \\theta ({}_0^l I^{\\beta }_1 u)u_r.$ Thus, we have proved the claim.", "We are now in a position to prove that (REF ) has only the trivial solution.", "Let $u$ be any linear combination of $u_l$ and $u_r$ .", "Recall that $u_l + u_r \\equiv 0$ , we have $u = c_l u_l$ with $c_l$ being an undetermined constant.", "We prove that $c_l$ must be zero under the condition (REF ).", "As a matter of fact, equations (REF ) and (REF ) respectively reduce to $I^{\\beta }_\\theta u = c_l [ (1-\\theta )({}_0^r I^{\\beta }_1 u_l)w_l^c + \\theta ({}_0^l I^{\\beta }_1 u_l)w_r^c + w_l - w_l^c]$ and $\\begin{aligned}I^{\\beta }_\\theta u & =c_l\\big [(1-\\theta )({}_0^r I^{\\beta }_1 u_l)(w_l-w_l^c)+ \\theta ({}_0^l I^{\\beta }_1 u_l)(w_r-w_r^c)\\\\[0.025in]& \\quad +(1-\\theta )({}_0^r I^{\\beta }_1 u_l)w_l^c + \\theta ( {}_0^lI^{\\beta }_1 u_l) w_r^c\\big ].\\end{aligned}$ Subtracting the first equation from the second yields $ c_l\\big [(1-\\theta )({}_0^r I^{\\beta }_1 u_l)(w_l-w_l^c)+ \\theta ({}_0^l I^{\\beta }_1 u_l)(w_r-w_r^c)\\big ] = c_l(w_l - w_l^c).$ We utilize the fact that $w_r+w_l=w^c_r+w^c_l=1$ to rewrite this equation as $\\bigl [1 + \\theta ({}_0^l I^{\\beta }_1 u_l) - (1-\\theta )({}_0^r I^{\\beta }_1 u_l) \\bigr ] (w_l-w_l^c)c_l= 0, \\qquad 0 \\le x \\le 1.$ This equation has only the trivial solution $(w_l-w_l^c)c_l = 0$ if and only if condition (REF ) holds.", "As we are considering the case of a variable diffusivity coefficient $K$ , $w_l \\lnot \\equiv w_l^c$ .", "Otherwise, it is clear from (REF ) and the fact that $w_l^c = x$ that $K$ would be a constant.", "Hence, $c_l = 0$ if and only if condition (REF ) holds.", "Consider problem (REF ) with $0 < \\beta < 1/2$ , $0 \\le \\theta \\le 1$ , and that $K \\in L^\\infty (0,1)$ .", "In addition, assume that condition (REF ) holds.", "Then the bilinear form $A(\\cdot ,\\cdot )$ in (REF ) is bounded and weakly coercive on the space $H^{1-\\beta }_0(0,1) \\times H^1_0(0,1)$ .", "That is, the inequalities in (REF ) hold.", "Then, for any $f \\in H^{-1}(0,1)$ , the Petrov-Galerkin formulation (REF ) has a unique solution $u \\in H^{1-\\beta }_0(0,1)$ such that the stability estimate (REF ) holds.", "It is clear that the bilinear form $A(\\cdot ,\\cdot )$ is bounded on the space $H^{1-\\beta }_0(0,1) \\times H^1_0(0,1)$ .", "To prove its weak coercivity, for any given $v \\in H^1_0(0,1)\\setminus \\lbrace 0\\rbrace $ , by Theorem there exists a unique $w \\in H^{1-\\beta }_0 (0,1)$ such that $v = I^{\\beta }_\\theta w - (1-\\theta )({}_0^rI^{\\beta }_1 w )w_l - \\theta ({}_0^lI^{\\beta }_1 w) w_r.$ A direct evaluation reveals $\\begin{aligned}A(w,v) &= (KD(I^{\\beta }_\\theta w),Dv)_{L^2(0,1)} \\\\& = (KDv,Dv)_{L^2(0,1)} + (1-\\theta )({}_0^rI^{\\beta }_1w ) (KDw_l,Dv)_{L^2(0,1)} \\\\& \\quad + \\theta ({}_0^lI^{\\beta }_1w ) (KDw_r,Dv)_{L^2(0,1)} \\\\& = (KDv,Dv)_{L^2(0,1)} \\ge K_m \\Vert Dv\\Vert _{L^2(0,1)}^2\\\\& \\ge \\kappa \\Vert v\\Vert _{H^1(0,1)} \\Vert w\\Vert _{H^{1-\\beta }_0(0,1)}\\end{aligned}$ similarly to the derivation of (REF ).", "Thus, the first estimate in (REF ) holds.", "To prove the second estimate in (REF ), let $P_K$ be the projection operator from $H^1(0,1)$ onto $H^1_0(0,1)$ defined as follows: for any $ w \\in H^1(0,1)$ , find $P_K w \\in H^1_0(0,1)$ such that $(KD(P_K w),Dv)_{L^2(0,1)} = (KDw,Dv)_{L^2(0,1)}, \\quad \\forall \\ v \\in H^1_0(0,1).$ Clearly we have $\\Vert D(P_K w) \\Vert _{L^2(0,1)} \\le \\sqrt{\\frac{K_M}{K_m}}\\Vert Dw\\Vert _{L^2(0,1)}, \\quad \\forall \\ w \\in H^1(0,1).$ Similarly to the proof in Theorem , we can prove that $P_K I^{\\beta }_\\theta $ is a bounded linear bijection from $H^{1-\\beta }_0(0,1)$ onto $H^1_0(0,1)$ and has a bounded inverse, provided that condition (REF ) holds.", "Thus, for any $v \\in H^1_0(0,1)$ , there exists a unique $w \\in H^{1-\\beta }_0(0,1)$ which satisfies (REF ) and $ \\Vert w \\Vert _{H^{1-\\beta }(0,1)} \\le C \\Vert v \\Vert _{H^1(0,1)}.$ We get from (REF ) that $A(w,v) = (KDv,Dv)_{L^2(0,1)} \\ge K_m \\Vert Dv\\Vert ^2_{L^2(0,1)} \\ge \\kappa \\Vert w \\Vert _{H^{1-\\beta }(0,1)} \\Vert v \\Vert _{H^1(0,1)}.$ We thus prove (REF ).", "The rest of the theorem is a direct application of Babus̆ka-Lax-Milgram theorem [3], .", "Assume that the conditions in Theorem hold.", "Furthermore, we assume that $K \\in C^{m+1}[0,1]$ and $f \\in H^m(0,1)$ with $m$ being a nonnegative integer, then the following regularity estimates hold for the weak solution $u \\in H^{1-\\beta }_0(0,1)$ to the Petrov-Galerkin formulation (REF ) $\\big \\Vert I^\\beta _\\theta u \\big \\Vert _{H^{m+2}(0,1)} \\le C \\big (\\Vert K\\Vert _{C^{m+1}[0,1]} + \\Vert f\\Vert _{H^m(0,1)} \\big ).$ In other words, the weighted high-order Riemann-Liouville derivatives $D^{m+2} I^\\beta _\\theta u$ exist and can be bounded by the high-order norms of the data of the variable-coefficient FDE (REF ).", "Let $u$ be the weak solution to the Petrov-Galerkin formulation (REF ).", "By Theorem , $u$ satisfies (REF ).", "Under the conditions of the theorem, it follows from the classical theory of second-order diffusion equations [15] that $w_f \\in H^{m+2}(0,1) \\cap H^1_0(0,1)$ satisfies $\\Vert w_f\\Vert _{H^{m+2}(0,1)} \\le C \\Vert f \\Vert _{H^m(0,1)}.$ A direct differentiation of (REF ) concludes that there is a positive constant $C$ such that $\\Vert w_l\\Vert _{H^{m+2}(0,1)} + \\Vert w_r\\Vert _{H^{m+2}(0,1)} \\le C \\Vert K\\Vert _{C^{m+1}[0,1]}.$ We differentiate (REF ) successively for $k$ times with $0 \\le k \\le m+2$ to obtain $\\begin{array}{rl}D^k I^{\\beta }_\\theta u &= (1-\\theta )({}_0^r I^{\\beta }_1u)D^k w_l+\\theta ({}_0^lI^{\\beta }_1u)D^k w_r + D^k w_f.\\end{array}$ We apply Young's inequality and Sobolev's embedding theorem [1], [14] and the estimate (REF ) to deduce that $\\bigl | {}_0^r I^{\\beta }_1u \\big | + \\big | {}_0^lI^{\\beta }_1u \\big | \\le C(\\beta ) \\Vert u\\Vert _{L^\\infty (0,1)} \\le C \\Vert u\\Vert _{H^{1-\\beta }(0,1)} \\le C\\Vert f\\Vert _{H^{-1}(0,1)}.$ We thus come up with the following bounds for $2 \\le k \\le m+2$ $\\begin{array}{l}\\Vert D^k I^{\\beta }_\\theta u\\Vert _{L^2(0,1)} \\\\[0.05in]\\qquad \\le C \\bigl ( (1-\\theta ) \\big |{}_0^r I^{\\beta }_1 u \\big | \\Vert D^k w_l\\Vert _{C^{k-1}[0,1]}+ \\theta \\big |{}_0^lI^{\\beta }_1u \\big | \\Vert w_r \\Vert _{C^{k-1}[0,1]} + \\Vert w_f\\Vert _{H^k(0,1)} \\big )\\\\[0.05in]\\qquad \\le C \\bigl (\\Vert K\\Vert _{C^{m-1}[0,1]} + \\Vert f\\Vert _{H^{k-2}(0,1)} \\big ).\\end{array}$ We thus finish the proof of the theorem." ], [ "Application of the theory", "In §6 we proved the existence, uniqueness, regularity and characterization of the weak solution to the homogeneous Dirichlet boundary-value problem of two-sided variable-coefficient conservative FDEs (REF ), the weak coercivity and stability of the corresponding Petrov-Galerkin formulation (REF ), under the condition that (REF ) holds.", "While (REF ) looks somewhat nonconventional, in this section we look at some important special cases of (REF ) for which (REF ) holds.", "The conclusions of Theorems – hold for the one-sided analogue of problem (REF ).", "We need only to prove that (REF ) holds.", "By symmetry, we need only consider the case of $\\theta =1$ when (REF ) reduces to $ 1+{}_0^lI^{\\beta }_1u_l \\ne 0, \\quad 0 < x < 1.", "$ On the other hand, the first equation in (REF ) reduces to $\\begin{aligned}{}_0^l I^{\\beta }_x u_l &= w_l - w_l^c +({}_0^lI^{\\beta }_1u_l)w_r^c = w_l - 1 + \\big (1 + {}_0^lI^{\\beta }_1u_l \\big )x \\\\&= - w_r + \\big (1+{}_0^lI^{\\beta }_1u_l \\big )x.\\end{aligned}$ We apply $D ~{}_0^lI^{1-\\beta }_x$ on both sides of the equation and use Lemmas and as well as (REF ) to obtain $\\begin{array}{rl}u_l & = - D ~{}_0^lI^{1-\\beta }_x w_r + \\bigl ( 1+({}_0^lI^{\\beta }_1u_l) \\bigr ) \\bigl (D ~{}_0^lI^{1-\\beta }_x x \\bigr ) \\\\[0.05in]& = - {}_0^lI^{1-\\beta }_x Dw_r + \\bigl (1+({}_0^lI^{\\beta }_1u_l) \\bigr ) \\bigl ({}_0^lI^{1-\\beta }_x 1 \\bigr )\\\\[0.05in]& \\displaystyle = -{\\Gamma (1-\\beta )} \\int _0^x \\left[ {(x-s)^\\beta }\\left(\\int _0^1 {K(\\theta )} d\\theta \\right)^{-1} {K(s)} \\right] ds \\\\[0.15in]& \\displaystyle \\qquad + \\bigl (1+({}_0^lI^{\\beta }_1u_l) \\bigr ) {\\Gamma (1-\\beta )} \\int _0^x {(x-s)^\\beta }ds.\\end{array}$ We let $s = xt$ to get $\\begin{aligned}\\displaystyle {\\Gamma (1-\\beta )} \\int _0^x {(x-s)^\\beta }ds &= {\\Gamma (1-\\beta )} \\int _0^1 {(1-t)^\\beta }dt \\\\& = {\\Gamma (1-\\beta )} B(1,1-\\beta ) = {\\Gamma (2-\\beta )}\\end{aligned}$ with $B(\\cdot ,\\cdot )$ being the Beta function.", "We note from (REF ) that $u_l \\in H^{1-\\beta }_0(0,1)$ and let $x=1$ in (REF ) and (REF ) to obtain $\\displaystyle 1+{}_0^lI^{\\beta }_1u_l = \\Gamma (1-\\beta ) \\int _0^x \\left[ {(x-s)^\\beta }\\left(\\int _0^1 {K(\\theta )} d\\theta \\right)^{-1}{K(s)} \\right] ds > 0.$ Thus, (REF ) holds for problem (REF ) with $\\theta = 1$ .", "We can similarly prove that (REF ) holds for problem (REF ) with $\\theta = 0$ .", "The following theorem shows that problem (REF ) is well posed if the diffusivity coefficient $K$ is a perturbation of a constant.", "If the diffusivity coefficient $K$ in (REF ) is a perturbation from a constant in the $L^2$ sense, i.e., there exists a constant $\\varepsilon _0 > 0$ for which $\\Big \\Vert \\Big (\\int ^1_0\\frac{1}{K(s)}ds \\Big )^{-1} \\frac{1}{K} -1 \\Big \\Vert _{L^2(0,1)} < \\varepsilon _0,$ then the conclusions of Theorems – hold.", "We notice that $Dw_r = \\big (\\int ^1_0\\frac{1}{K(s)}ds \\big )^{-1} \\frac{1}{K} $ satisfies that $\\int _0^1 Dw_r dx = 1$ .", "Hence, the minimizer $P^*_0$ of the approximation $\\displaystyle P_0^* = \\mathrm {argmin}_{P_0 \\in \\mathbb {R}} \\big \\Vert Dw_r - P_0 \\big \\Vert _{L^2(0,1)} $ must be given by its first Fourier coefficient or $L^2$ projection to the space of constant $P^*_0 = \\int _0^1 Dw_r 1 dx = 1$ .", "Furthermore, $(1, Dv)_{L^2(0,1)}=0$ for any $ v \\in H^1_0(0,1)$ .", "Hence, the second equation in (REF ) still holds when $Dw_r$ in its right-hand side is replaced by $Dw_r - 1$ for any constant $C$ .", "By the stability estimate (REF ) for Theorem there exists a constant $C_4 > 0$ such that $\\displaystyle \\big \\Vert u_r \\big \\Vert _{H^{1-\\beta }(0,1)} \\le C_4 \\Big \\Vert \\Big (\\int ^1_0\\frac{1}{K} dx \\Big )^{-1} \\frac{1}{K} -1 \\Big \\Vert _{L^2(0,1)}.", "$ Furthermore, we notice that $\\begin{aligned}&\\bigl | \\theta \\big ({}_0^l{I}^{\\beta }_1 u_l \\big ) - (1-\\theta ) \\big ({}_0^rI^{\\beta }_1u_l \\big ) \\big | \\\\&\\quad \\displaystyle = {\\Gamma (\\beta )} \\Bigl | \\theta \\int _0^1 \\frac{u_l(s)}{(1-s)^{1-\\beta }} ds - (1-\\theta ) \\int _0^1 \\frac{u_l(s)}{s^{1-\\beta }} ds \\Big | \\\\&\\quad \\displaystyle \\le {\\Gamma (\\beta +1)} \\Vert u_l \\Vert _{L^\\infty (0,1)} \\le C_5 \\Vert u_l \\Vert _{H^{1-\\beta }_0(0,1)}\\end{aligned}$ where we have used Sobolev's embedding theorem at the last step.", "Consequently, we obtain $\\begin{aligned}&1 + \\theta \\big ({}_0^l{I}^{\\beta }_1 u_l \\big ) - (1-\\theta ) \\big ({}_0^rI^{\\beta }_1u_l \\big ) \\\\& \\quad \\displaystyle \\ge 1 - \\big | \\theta \\big ({}_0^l{I}^{\\beta }_1 u_l \\big ) - (1-\\theta ) \\big ({}_0^rI^{\\beta }_1u_l \\big ) \\big | \\\\& \\quad \\displaystyle \\ge 1 - C_5 \\Vert u_l \\Vert _{H^{1-\\beta }_0(0,1)} = 1 - C_5 \\Vert u_r \\Vert _{H^{1-\\beta }_0(0,1)}\\\\& \\quad \\displaystyle \\ge 1 - C_4 C_5 \\Big \\Vert \\Big (\\int ^1_0\\frac{1}{K} dx \\Big )^{-1} \\frac{1}{K} -1 \\Big \\Vert _{L^2(0,1)}.\\end{aligned}$ We finish the proof of the theorem by selecting $\\varepsilon _0=1/(C_4C_5)$ ." ] ]
1606.04912
[ [ "On point interactions realised as Ter-Martirosyan-Skornyakov\n Hamiltonians" ], [ "Abstract For quantum systems of zero-range interaction we discuss the mathematical scheme within which modelling the two-body interaction by means of the physically relevant ultra-violet asymptotics known as the \"Ter-Martirosyan-Skornyakov condition\" gives rise to a self-adjoint realisation of the corresponding Hamiltonian.", "This is done within the self-adjoint extension scheme of Krein, Visik, and Birman.", "We show that the Ter-Martirosyan-Skornyakov asymptotics is a condition of self-adjointness only when is imposed in suitable functional spaces, and not just as a pointwise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature." ], [ "Introduction", "According to a nomenclature that has emerged in various physical and mathematical contexts, one refers to the so-called Ter-Martirosyan–Skornyakov (henceforth TMS) operators as a distinguished class of quantum Hamiltonians for systems of non-relativistic particles with two-body “zero-range” (or “contact”, or “point”) interaction.", "This terminology stems from early works in nuclear physics, where it was the nucleon-nucleon coupling to be initially modelled as a “contact” interaction.", "Nowadays the typical experimental realisation is that of ultra-cold atom systems where, by Feshbach resonance methods, the two-body scattering length is tuned to a magnitude that exceeds by many orders its nominal value, and the effective range of the interaction shrinks correspondingly to a very small scale, so that to an extremely good approximation the interaction can be considered to be of infinite scattering length and/or zero range.", "In Section we will provide a more diffuse context and references.", "Informally speaking, TMS Hamiltonians are qualified by the two characteristics of acting as the $N$ -body $d$ -dimensional free Hamiltonian on functions that are supported away from the “coincidence hyperplanes” $\\lbrace x_i=x_j\\rbrace $ , and of having a domain that consists of square-integrable functions $\\Psi (x_1,\\dots ,x_N)$ , possibly with fermionic or bosonic exchange symmetry, which satisfy specific asymptotics when $|x_i-x_j|\\rightarrow 0$ for some or for all particle couples $i,j$ .", "This models an interaction supported only on the hyperplanes $\\lbrace x_i=x_j\\rbrace $ .", "It is customary to refer to this ultra-violet asymptotics as the “TMS condition”.", "The explicit form for the TMS condition has various versions (see, e.g., (REF ), (REF ), (REF ), (REF ), (REF ), or (REF ) below), all essentially equivalent to each other.", "Noticeably, such asymptotics emerge from different contexts and languages: on the one side the physical heuristics for an “effective” low-energy two-body scattering due to an interaction of very short range, on the other side the mathematical theory of self-adjoint extensions of symmetric operators on Hilbert space.", "This is a fascinating history of reciprocal influence and mutual inspiration between “early days” nuclear physics, modern condensed matter physics, and mathematical operator theory and self-adjoint extension theory.", "Section below will partially survey it.", "TMS Hamiltonians represent the modern operator-theoretic approach to multi-particle quantum systems with two-body point interaction, and have an intimate connection to the alternative approach based on energy quadratic forms.", "They arise as natural effective models, based on stringent physical heuristics on the behaviour of the many-body wave-function when two particles come on top of each other.", "In many circumstances, however, which depend essentially on the mass of the particles and on possible additional symmetries of the system, a formal TMS Hamiltonian fails to be self-adjoint and each of its self-adjoint extensions accounts for a different behaviour of the system when three particles get closer and closer to the same point.", "The question is then to identify these extensions (if more than one) and to study their stability and spectral properties.", "This is even more so since quantum systems with zero-range interactions may exhibit two somewhat exotic phenomena, as compared with the case of ordinary finite range potentials: the so-called “Thomas effect”, namely the emergence of an infinite discrete sequence of bound states with negative energy diverging to $-\\infty $ and eigenfunction collapsing onto the barycentre, and the “Efimov effect”, that consists of an infinite sequence of bound states with negative energy arbitrarily close to zero and eigenfunctions extending on a larger and larger spatial scale, with a non-square-integrable limit.", "The TMS Hamiltonians account for such phenomena as well.", "In this work we put the emphasis on the mathematical scheme within which a two-body interaction modelled by means of a TMS condition (a physical requirement) does correspond to a self-adjoint Hamiltonian (a mathematical constraint that ensures the well-posedness and the correct interpretation of the quantum model).", "Here is how the general problem is posed, as we shall elaborate further in the historical review of Section and then in the concrete settings of Sections and 1.", "One starts (mostly in $d=3$ dimensions) with the operator $\\mathring{H}$ obtained by restricting the $N$ -body free Hamiltonian to the regular wave-functions that are supported away from the coincidence hyperplanes $\\Gamma _{ij}:=\\lbrace x_i=x_j\\rbrace $ .", "Additional partial or global symmetries for $\\mathring{H}$ are possible, such as rotational symmetries, or exchange bosonic or fermionic symmetries for some or all of the $N$ particles.", "$\\mathring{H}$ is clearly densely defined, symmetric, and positive, and any its self-adjoint extension is naturally interpreted as a model for an interaction supported at $\\Gamma _{ij}$ .", "2.", "The self-adjoint extensions of $\\mathring{H}$ are restrictions of $\\mathring{H}^*$ .", "Thus, one first characterises the domain and the action of $\\mathring{H}^*$ and then one selects a special class of restrictions of $\\mathring{H}^*$ , which are obtained by reducing the domain of $\\mathring{H}^*$ to only those functions $\\Psi $ satisfying, for some or all couples of variables $x_i,x_j$ , the condition that, if $y_{ij}:=x_i-x_j$ and $(x_1,x_2,\\dots ,x_n)\\mapsto (y_{ij},y_2,\\dots ,y_N)$ is a regular change of variables, then $\\!\\!\\int _{\\begin{array}{c}p_{ij}\\in \\mathbb {R}^d \\\\ |p_{ij}|\\leqslant R\\end{array}} \\;\\widehat{\\Psi }(p_{ij},p_2,\\dots ,p_N)\\,\\mathrm {d}p_{ij}\\;=\\;(R-\\frac{1}{a_{ij}})\\,\\xi _{ij}(p_2,\\dots ,p_N)+o(1)\\,,\\;\\; R\\rightarrow +\\infty \\,.$ In (REF ) $(p_{ij},p_2,\\dots ,p_N)$ are the conjugate Fourier variables to $(y_{ij},y_2,\\dots ,y_N)$ , the function $\\xi _{ij}$ depends on $\\Psi $ and on the considered couple $i,j$ , but not on $R$ , and the constant $a_{ij}\\in \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ is prescribed and is $R$ -independent too.", "Equation (REF ) above is one version of the so-called Ter-Martirosyan–Skornyakov condition.", "It is based on stringent physical heuristics (see the discussion in Section and (REF )-(REF ) below) that allow one to interpret it as an interaction in the $(i,j)$ -channel with zero range and $s$ -wave scattering length equal to $-a_{ij}^{-1}$ .", "3.", "The TMS condition selects a “physical” extension $H_\\alpha $ of $\\mathring{H}$ labelled by the conventional parametrisation $\\alpha \\equiv (-a_{ij}^{-1})_{ij}$ .", "The mathematical problem is then to recognise $H_\\alpha $ as a self-adjoint extension of $\\mathring{H}$ , or to identify and classify its self-adjoint extensions, and then to investigate its spectral and stability properties.", "The scheme above, which was made explicit for the first time by Minlos [31] in 1987 (based on an old seminal but very concise work of Minlos and Faddeev [38] in 1961), has been since then the object of numerous investigations that we will quote in Section The general question of the self-adjoint realisation of $H_\\alpha $ and of its stability and spectral properties is still open.", "An amount of partial information is available for special cases of $N$ -body systems.", "In this work, beyond placing the problem into a historical perspective (see Section ), we discuss how the TMS condition (REF ) can be proved to be a self-adjointness condition, based on the extension theory specifically tailored for semi-bounded symmetric operators, as developed by Kreĭn, Višik, and Birman.", "(We review the main results of this theory in Appendix and we refer to [28] for a comprehensive discussion.)", "The use of the Kreĭn-Višik-Birman theory is in fact at the basis of all recent studies on the self-adjoint realisation of the operator $H_\\alpha $ selected by the TMS condition (REF ): in this context, it allows one to reduce the problem of the self-adjointness of $H_\\alpha $ on the Hilbert space $L^2(\\mathbb {R}^{Nd})$ (with possible symmetries) to the problem of the self-adjointness of a suitable integral operator on the space of the functions $\\xi _{ij}$ 's appearing in the asymptotics (REF ).", "What we do here is on the first place to elaborate in full detail the precise application of the Kreĭn-Višik-Birman theory to the TMS condition, with a discussion and through intermediate results that to our knowledge are not present in the literature.", "Further, we put for the first time the emphasis on the crucial difference between the TMS condition as a point-wise identity, and the same condition interpreted as a suitable functional identity, as we now explain.", "In Section , which contains the first group of our results, we discuss the TMS condition for the simplest composite system possible, consisting of two particles with point interaction.", "This is a well-studied quantum system that is completely understood within the standard self-adjoint extension theory a la von Neumann.", "First we re-obtain the well-known Hamiltonian of the system solely by means of the Kreĭn-Višik-Birman theory, a procedure that we did not manage to find elsewhere in the literature.", "This yields an alternative equivalent characterisation of the whole class of self-adjoint extensions of the “away-from-hyperplanes” free Hamiltonian $\\mathring{H}$ that represented the starting point of the analysis – see step 1 of the general scheme above.", "We then show that imposing the TMS condition to functions in the domain of $\\mathring{H}^*$ reproduces, for all possible values of the scattering length $a$ , all the self-adjoint extensions of $\\mathring{H}$ .", "In particular this shows that the TMS condition is a self-adjointness condition.", "In Section , where the second group of our results is presented, we follow the same approach for a more complicated system consisting of three particles with point interaction – we develop our explicit discussion for the so-called “2+1”-fermionic system, two identical fermions coupled with a third particle of different nature.", "Exploiting again the general results of the Kreĭn-Višik-Birman theory, we find that the TMS condition is a self-adjointness condition, which selects a sub-class of extensions of $\\mathring{H}^*$ , only if it is given as a suitable functional identity.", "As a generic point-wise identity, instead, the TMS condition is in general not even a condition for a symmetric extension of $\\mathring{H}$ .", "Furthermore, we show that the issue of the self-adjoint realisation of the TMS condition does indeed boil down to the self-adjointness problem of a simpler integral operator (that acts on the so-called “space of charges”) as normally given for granted in the literature, but on different (more regular) functional spaces than those considered so far.", "This brings us to the natural follow-up of the present analysis, which was in fact our original motivation and that we intend now to develop in a future work.", "There are indeed not completely understood discrepancies in the literature between the ranges of the particle masses in which the TMS condition for three-body systems is shown to be a self-adjointness condition, or instead to give rise to a symmetric extension of $\\mathring{H}$ with its own family of self-adjoint extensions.", "Such discrepancies emerge between the operator-theoretic approach sketched above and an alternative approach through quadratic forms (i.e., the construction of a closed and semi-bounded quadratic form such that the function in the domain of self-adjoint operator that realises it display the TMS asymptotics (REF )).", "Re-visiting the operator-theoretic approach in view of our present findings is likely to account for an explanation.", "We briefly elaborate on this point in the final Section" ], [ "A retrospective on Ter-Martirosyan–Skornyakov Hamiltonians for point interactions", "In this Section we present the historical emergence of TMS operators in the physics and mathematics of point interactions.", "It therefore should not be regarded as a complete review on the history of point interactions!", "In the 1930's Quantum Mechanics began to be applied to the newly observed nuclear phenomena.", "At first, the decrease by a factor $10^{-5}$ from the atomic to the nuclear scale made it plausible to model the interaction among nucleons as a delta-like interaction.", "In 1932 Wigner [49] calculated that the nuclear forces interaction must be of very short range and very strong magnitude.", "This led three years later first Bethe and Peierls [6], [7] and then Thomas [47] to describe the neutron-proton scattering by means of the two-body Schrödinger equation in the approximation of a potential of very short range, an approach subsequently developed by Fermi [20] and Breit [10] with the introduction of the so-called “delta pseudo-potential”.", "Formulated in modern terms, the celebrated “Bethe-Peierls contact condition”, which is still today ubiquitous in many formal physical treatments, prescribes on the basis of physical heuristics that the wave-function $\\Psi (x_1,\\dots ,x_N)$ of $N$ three-dimensional particles subject to a two-body zero-range interaction of scattering length $a_{ij}$ among particles $i$ and $j$ behaves asymptotically as $\\Psi (x_1,\\dots ,x_N)\\;\\approx \\;\\Big (\\frac{1}{|x_i-x_j|}-\\frac{1}{a_{ij}}\\Big )\\qquad \\textrm { as }|x_i-x_j|\\rightarrow 0$ where the point-wise limit (REF ) is meant as $\\Psi (x_1,\\dots ,x_N)\\;=\\;\\Big (\\frac{1}{|x_i-x_j|}-\\frac{1}{a_{ij}}\\Big )\\,\\xi (Z)+o(1)$ for some function $\\xi $ of the variable $Z$ in the hyperplane $\\lbrace x_i=x_j\\rbrace $ .", "Clearly, what makes this approximation appealing, and computationally advantageous, is its dependence on few parameters only (the $a_{ij}$ 's), instead of the complete knowledge of the interaction.", "While Bethe and Peierls had studied the problem of two low-energy nucleons with contact interaction and obtained (REF ) for the two-body problem, Thomas had considered the three-body problem showing that as the range of the two-body forces tends to zero the ground state of the three-body system can approach $-\\infty $ , even when the ground state energies of all two-body subsystems remain constant.", "This effect, referred to since then as the “Thomas effect”, was the first evidence that the deceptively simple three-body problem with zero-range interaction has a much richer (and potentially much more complicated) phenomenology than the analogous two-body problem.", "The next extensive study of a system of three low-energy nucleons appeared some 20 years later, in 1955, due to Ter-Martirosyan and Skornyakov [44], two nuclear physicists who credited Landau for the ideas they exploited.", "They assumed that the Bethe-Peierls condition remains valid in each two-body channel and they used it as boundary condition for solving the eigenvalue problem for the three-body Schrödinger equation with formal delta-like two-body potentials.", "For brevity let us revisit their conclusion in the simplest case of spinless identical particles, for which they made use of standard centre-of-mass Jacobi coordinates $y_1=x_1-\\frac{1}{2}(x_2+x_3)$ , $y_{23}=x_2-x_3$ and introduced a formal delta-like potential $\\delta (y_1)\\delta (y_{23})$ for wave-functions $\\Psi (y_1,y_{23})$ .", "By suitably expressing $\\widehat{\\Psi }(p,q)$ in terms of an auxiliary function $\\widehat{\\xi }(p)$ (here $(p,q)$ and $(y_1,y_{23})$ are Fourier conjugate variables), they found that (in units $\\hbar =1$ and particle mass $=1$ ) $\\Psi $ is a bound state of energy $-E<0$ whenever $(\\widehat{\\xi },E)$ is a solution to the integral equation $\\alpha +2\\pi ^2\\sqrt{\\,\\frac{3}{4}p^2+E\\,}\\,\\widehat{\\xi }(p)+2\\int _{\\mathbb {R}^3}\\frac{\\widehat{\\xi }(q)}{p^2+q^2+p\\cdot q+E}\\,\\mathrm {d}q\\;=\\;0\\,,$ where $\\alpha :=-1/a$ and $a$ is the $s$ -wave scattering length in each two-body channel.", "Analogues to equation (REF ) were later found for other systems with point interactions, among which, to mention those that have received the largest attention, three-body systems with different symmetries (three distinguishable particles, two identical fermions plus a third particle of different nature, etc.", "), four-body systems with two distinct couples of identical fermions, and more generally $N+M$ systems with $N$ identical fermions of one type plus $M$ identical fermions of another type.", "Each equation of this class is today referred to as a “Ter-Martirosyan–Skornyakov equation”.", "The subject proved soon to be worth a deeper understanding, despite the effectiveness of the description.", "On the one hand the manipulations of Ter-Martirosyan and Skornyakov (as well as the previous ones by Bethe, Peierls, Thomas, Fermi, and Breit) were rather formal for mathematical standards and called for a more rigorous justification.", "On the other hand, an evidence of some sort of indeterminate physical description emerged when Danilov [14] in 1961 observed that the TMS equation (REF ) has a solution $\\widehat{\\xi }$ for arbitrary values of $E$ , with large momentum asymptotics $\\widehat{\\xi }(p)\\;=\\;\\frac{1}{p^2}\\,\\big (A_E\\sin (s_0 \\ln |p|)+B_E\\cos (s_0\\ln |p|)\\big )+o\\Big (\\frac{1}{p^{2}}\\Big )\\,,$ where $s_0>0$ is an explicit universal constant and $A_E,B_E>0$ are two further constants that depend on $E$ .", "Inspired by ideas (mainly of Gribov) by which some additional “experimental” parameter that cannot be computed using only two-body experimental data was needed for the full description of the three-body system, Danilov proposed an ad hoc removal of this non-physical continuum of eigenvalues on $(-\\infty ,0)$ by constraining the solutions to (REF ) to have the form (REF ) with $A_E\\;=\\;\\beta B_E$ for some additionally prescribed parameter $\\beta \\in \\mathbb {R}$ .", "$\\beta $ was in some vague sense given the meaning of a three-body parameter, as opposite to $\\alpha $ in (REF ) which is a parameter of the two-body problem for each couple of bosons.", "Under the restriction (REF ), equation (REF ) has a discrete and infinite set of solutions $(\\widehat{\\xi }_n,E_n)$ , with energies $-E_n\\rightarrow -\\infty $ as $n\\rightarrow +\\infty $ according to the asymptotics $-E_n\\;=-3 \\exp \\Big (\\frac{2\\pi n}{s_0}-\\frac{2}{s_0}\\arctan \\frac{1}{\\beta }\\Big )(1+o(1))\\,,$ a quantitative manifestation of the Thomas effect.", "In modern mathematical terms, the phenomenon noted by Danilov is understood as follows: the three-body point-interaction Hamiltonian implicitly identified by Ter-Martirosyan and Skornyakov by means of the condition (REF ) for its eigenstates at given two-body scattering length $-\\alpha ^{-1}$ , is not a self-adjoint operator and it admits a one-parameter family of self-adjoint extensions, labelled by $\\beta \\in \\mathbb {R}$ ; for each $\\beta $ , the corresponding self-adjoint Hamiltonian has a countable discrete spectrum accumulating exponentially to $-\\infty $ with corresponding eigenfunctions that collapse onto the barycentre; the union of the negative spectra of all such self-adjoint extensions is the whole negative real line.", "Motivated by the scheme of Ter-Martirosyan and Skornyakov for the three-body problem with point interaction and by Danilov's observation, Minlos and Faddeev [38], [39] in the same year 1961 provided essentially the whole explanation above, including the asymptotics (REF ) and (REF ), in the form of two beautiful short announcements, albeit with no proofs or further elaborations.", "Theirs can be considered as the beginning of the mathematics of quantum systems with zero-range interactions.", "This is even more so because for the first time the problem was placed within a general mathematical framework, the theory of self-adjoint extensions of semi-bounded symmetric operator, that Kreĭn, Višik, and Birman had developed between the mid 1940's and the mid 1950's (see Appendix ).", "A somewhat different approach characterised the start of the mathematical study of the two-body problem.", "In 1960-1961, a few months before the works of Minlos and Faddeev on the three-body problem, Berezin and Faddeev [5] published the first rigorous analysis of a three-dimensional model with two particles coupled by a delta-like interaction.", "The emphasis was put in realising the formal Hamiltonian $-\\Delta +\\delta (x)$ as a self-adjoint extension of the restriction $-\\Delta |_{C^\\infty _0(\\mathbb {R}^3\\setminus \\lbrace 0\\rbrace )}$ (in the relative variable $x=x_1-x_2$ between the two particles).", "Working in Fourier transform, they recognised that the latter operator has deficiency indices $(1,1)$ and they characterised the whole family $\\lbrace H_\\alpha \\,|\\,\\alpha \\in \\mathbb {R}\\rbrace $ of its self-adjoint extensions as the operators $\\widehat{(H_\\alpha \\psi )}(p)\\;=\\;p^2\\widehat{\\psi }(p)-\\lim _{R\\rightarrow \\infty }\\frac{1}{4\\pi R}\\int _{\\begin{array}{c}\\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}\\widehat{\\psi }(q)\\,\\mathrm {d}q$ defined on the domain of $L^2(\\mathbb {R}^3)$ -functions $\\psi $ such that, as $R\\rightarrow \\infty $ , $\\int _{\\begin{array}{c}\\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}\\widehat{\\psi }(q)\\,\\mathrm {d}q\\;=\\;c \\,(R+2\\pi ^2\\alpha )+o(1)\\quad \\textrm { and }\\quad \\int _{\\mathbb {R}^3}|H_\\alpha \\psi |^2\\mathrm {d}x<\\infty \\,.$ (For a more direct comparison – see (REF ) in the following – we have replaced here the parameter $\\alpha $ of the notation of [5] with $-(8\\pi ^3\\alpha )^{-1}$ .)", "As (REF )-(REF ) were only announced with no derivation, with a sole reference to the monograph [1] of Akhiezer and Glazman on linear operators in Hilbert space, we are to understand that Berezin and Faddeev came to their conclusion by methods of von Neumann's self-adjoint extension theory, as presented in [1], combined with explicit calculations in Fourier transform.", "This leaves the question open on why they did not approach the extension problem within the same language of Kreĭn, Višik, and Birman, as used by Minlos and Faddeev for the three-body case.", "In this language, as we work out in Section , (REF )-(REF ) would have emerged as a very clean application of the general theory and, most importantly, the asymptotics in (REF ) would have arisen with a natural and intimate connection with the TMS equation (REF ).", "Berezin and Faddeev rather focused on re-interpreting the action of the Hamiltonian $H_\\alpha $ as a renormalised rank-one perturbation of the free Laplacian, re-writing (REF ) in position coordinates as $H_\\alpha \\psi \\;=\\;-\\Delta \\psi -\\frac{1}{4\\pi \\alpha }\\,\\lim _{R\\rightarrow \\infty }\\frac{1}{\\,2\\pi ^2+R/\\alpha \\,}\\frac{\\sin R|x|}{|x|}\\int _{\\mathbb {R}^3}\\frac{\\sin R|y|}{|y|}\\psi (y)\\,\\mathrm {d}y\\,.$ We conjecture that they did not know the old work of Bethe and Peierls for two nucleons, or they did not consider it relevant in their context, for no word is spent in [5] to derive the singularity $\\psi (x)\\sim |x|^{-1}$ as $|x|\\rightarrow 0$ from their asymptotics (REF ).", "With the subsequent theoretical and experimental advances in nuclear physics – the initial playground for models of point interactions – it became clear that the assumption of zero range was only a crude simplification of no fundamental level.", "The lack of a physically stringent character for the idealisation of zero range in experimentally observed quantum-mechanical systems, and the somewhat obscure emergence of the unboundedness from below for the self-adjoint realisations of the three-body Hamiltonian, decreased the physical interest towards point interactions and left their rigorous study in a relatively marginal position, and the approach of Ter-Martirosyan and Skornyakov quiescent.", "Moreover, after Faddeev published in 1963 his fundamental work [19] on the three-body problem with regular two-body forces, the concern of the physicists switched over to the numerical solutions of the corresponding Faddeev equations.", "In the Russian physical literature, mainly under the input of Faddeev, methods and models of point interactions, albeit not fully rigorous, moved their applicability to atomic and molecular physics, a mainstream that ideally culminates with the late 1970's monograph of Demkov and Ostrovskii [17] on the “zero-range potentials” and their application to atomic physics.", "The use of formal delta-like potentials remained for some decades as a tool for a formal first-order perturbation theory; in addition, the Kreĭn-Višik-Birman self-adjoint extension theory lost ground to von Neumann's theory in the literature in English language on the mathematics for quantum mechanics – it rather evolved in more modern forms in application to boundary value problems for partial differential equations, mainly in the modern theory of boundary triplets.", "It is the merit of Albeverio, Gesztesy, and Høegh-Krohn, and their collaborators (among whom, Streit and Wu), in the end of the 1970's and throughout the 1980's, to have unified an amount of previous investigations by establishing a proper mathematical branch on rigorous models of point interactions, with a systematic study of two-body Hamiltonians and of one-body Hamiltonians with finite or infinitely many fixed centers of point interaction.", "We refer to the monograph [2] for a comprehensive overview on this production, and especially to the end-of-chapter notes in [2] for a detailed account of the previous contributions.", "The main tools in this new mainstream were: von Neumann's extension theory on the first place (hence with no reference any longer to the methods of Kreĭn-Višik-Birman), by which point interaction Hamiltonians were constructed as self-adjoint extensions of the restriction of the free Laplacian to functions that vanish in a neighbourhood of the point where the interaction is supported; resolvent identities (of Kreĭn and of Konno-Kuroda type, see [2]) by which these self-adjoint extensions were recognised to be finite-rank perturbations of the free Laplacian, in the resolvent sense, and were also re-obtained by resolvent limits of Schrödinger Hamiltonians with shrinking potentials; plus an amount of additional methods (Dirichlet quadratic forms, non-standard analysis methods, renormalisation methods) for specific problems.", "Let us emphasize, in particular, that the original heuristic arguments of Bethe and Peierls and their two-body contact condition find a rigorous ground based on the fact, which can be proved within von Neumann's extension theory (see, e.g., [2]), that any self-adjoint extension of $\\Delta |_{C^\\infty _0(\\mathbb {R}^3\\setminus \\lbrace 0\\rbrace )}$ on $L^2(\\mathbb {R}^3)$ has a domain whose elements behave as $\\psi (x)\\sim (|x|^{-1}+\\alpha )$ as $|x|\\rightarrow 0$ , as an $s$ -wave (hence a “low-energy”) boundary conditions, for some $\\alpha \\in (-\\infty ,+\\infty ]$ .", "As for the initial three-body problem with two-body point interaction, it finally re-gained centrality from the mathematical point of view (while physically a stringent experimental counterpart was still lacking) around the end of the 1980's and throughout the 1990's.", "This was first due to Minlos and his school [31], [40], [41], [26], [27], [32], [43] (among which Melnikov, Mogilner, and Shermatov), by means of the operator-theoretic approach used for three identical bosons by Minlos and Faddeev, and slightly later due to Dell'Antonio and his school [46], [15], [16] (among which Figari and Teta), with an approach based on quadratic forms, where the “physical” energy form is first regularised by means of an ultra-violet cut-off and a suitable renormalisation procedure, and then is shown to be realised by a self-adjoint Hamiltonian.", "An alternative direction was started further later by Pavlov and a school that included Kuperin, Makarov, Melezhik, Merkuriev, and Motovilov, [24], [25], by indroducing internal degrees of freedom, i.e., a spin-spin contact interaction, so as to realise semi-bounded below three-body Hamiltonians.", "After a further period of relative quiescence, the subject has been experiencing a new boost, due to the last decade's rapid progress in the manipulation techniques for ultra-cold atoms and, in particular, for tuning the effective $s$ -wave scattering length by means of a magnetically induced Feshbach resonance [42].", "This has made it possible, among others, to prepare and study ultra-cold gases in the so-called “unitary regime” [11], i.e., the case of negligible two-body interaction range and huge, virtually infinite, two-body scattering length (both lengths being compared to a standard reference length such as the Bohr radius).", "In such a regime, unitary gases show properties, including superfluidity, that have the remarkable feature of being universal in several respects [9], and are under active experimental and theoretical investigation.", "As we do not have space here for an outlook on such an active field, we refer to the overview given in the introductory sections of the works [30], [29] and to the references therein.", "Let us only underline that from the experimental point of view, zero-range interactions in ultra-cold atom physics are today far from being just an idealisation of real-world two-body potentials with small support and in many realisations the zero-range, delta-like character of the interaction turns out to be an extremely realistic and in fact an unavoidable description.", "In turn, all this has brought new impulse and motivations to the already developing mathematical research on the subject, with a series of fundamental contributions in the last few years [33], [34], [35], [21], [12], [37], [36], [13], many of which provide rigorous ground to experimental or numerical evidence on the physical side." ], [ "Two-body point interaction à la Ter-Martirosyan–Skornyakov", "The Hamiltonian of point interaction between two particles in three dimension is well known since the first rigorous attempt [5] by Berezin and Faddeev in 1961, which we have already mentioned in Section , and the seminal work [3] by Albeverio and Høegh-Krohn in 1981.", "In [2] one can find the complete discussion of the self-adjoint realisation of this operator, its explicit domain and action, its resolvent, its spectral properties, its approximation by short-range potentials, and its scattering theory.", "In the first part of this Section we shall re-obtain this Hamiltonian and its main properties within the self-adjoint extension scheme of Kreĭn-Višik-Birman, as opposite to von Neumann's scheme used in the above works.", "We follow this line both for general reference, because to our knowledge this approach has never been worked out in the literature, and above all because we need to establish the grounds for the second part of this Section, where we shall realise the point interaction à la Ter-Martirosyan–Skornyakov.", "For the tools from the Kreĭn-Višik-Birman theory we shall make use of, we refer to the Appendix and, more diffusely, to the work [28]." ], [ "Point interaction Hamiltonian through the Kreĭn-Višik-Birman theory", "The starting point is the operator $\\mathring{H}\\;=\\;-\\Delta \\,,\\qquad \\mathcal {D}(\\mathring{H})\\;=\\;H^2_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\,,$ which is clearly a densely defined, symmetric, closed, and positive operator on the Hilbert space $L^2(\\mathbb {R}^3)$ .", "The variable $x\\in \\mathbb {R}^3$ has the meaning of relative variable between the two particles: after removing the centre of mass of the two-body system, the only relevant problem is in the relative variable.", "$\\mathring{H}$ is the closure of the negative Laplacian restricted to the smooth functions compactly supported away from the origin, and $H^2_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\;=\\;\\overline{\\,C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\,}^{\\Vert \\,\\Vert _{H^2}}.$ The space above is clearly a closed subspace of $H^2(\\mathbb {R}^2)$ , and it is also proper, as shown in (REF ) and (REF ) below.", "The free Hamiltonian on $C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )$ is the natural starting point when one aims at constructing a singular interaction supported only at $x=0$ , and any self-adjoint extension of this operator has the natural interpretation of a “candidate” Hamiltonian for the point interaction.", "Throughout this discussion it will be convenient to work in Fourier transform.", "We therefore re-write (REF ) by means of the following simple Lemma: Lemma For the operator $\\mathring{H}$ defined in (REF ) one has (i) the domain and the action of $\\mathring{H}$ are given by $\\mathcal {D}(\\mathring{H})\\;=\\;\\Big \\lbrace f\\in H^2(\\mathbb {R}^3)\\,,\\int _{\\mathbb {R}^3}\\widehat{f}(p)\\,\\mathrm {d}p=0\\Big \\rbrace \\,,\\qquad \\widehat{(\\mathring{H}f)}(p)\\;=\\;p^2\\widehat{f}(p)\\,; $ (ii) the Friedrichs extension of $\\mathring{H}$ is given by $\\mathcal {D}(\\mathring{H}_F)\\;=H^2(\\mathbb {R}^3)\\;, \\qquad \\widehat{(\\mathring{H}_Ff)}(p)\\;=\\;p^2\\widehat{f}(p)\\,.$ By suitable approximation arguments (see Appendix ), we have $\\mathcal {D}(\\mathring{H})\\;\\equiv \\;{H^2_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )}\\;=\\;\\big \\lbrace f\\in H^2(\\mathbb {R}^3)\\,\\big |\\, f(0)=0\\big \\rbrace \\,.$ Moreover, since $(1+p^2)\\widehat{f}$ and $(1+p^2)^{-1}$ are in $L^2(\\mathbb {R}^3)$ , then $\\widehat{f}\\in L^1(\\mathbb {R}^3)$ .", "Hence $0\\;=\\;f(0)\\;=\\;\\int _{\\mathbb {R}^3}{\\widehat{f}(p) \\mathrm {d}p}$ which, together with $\\widehat{(-\\Delta f)}=p^2\\widehat{f}\\in H^2(\\mathbb {R}^3)$ , proves part (i).", "As for part (ii), we first observe that the form domain of $\\mathring{H}$ , which is the completion of $\\mathcal {D}(\\mathring{H})$ in the $H^1$ -norm, is precisely $\\mathcal {D}[\\mathring{H}]\\;=\\overline{\\,H^2_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\;}^{\\,\\Vert \\,\\Vert _{H^2}}=\\;\\;H^1_0({\\mathbb {R}}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\,.$ Since $H^2(\\mathbb {R}^3)\\;\\subset \\; H^1_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\;=\\;H^1(\\mathbb {R}^3)$ (see Appendix ) and $H^2(\\mathbb {R}^3)$ is the domain of a self-adjoint extension of $\\mathring{H}$ , namely the self-adjoint $-\\Delta $ on $\\mathbb {R}^3$ , we conclude that $-\\Delta $ must be the Friedrichs extension $\\mathring{H}_F$ of $\\mathring{H}$ , owing to the characterisation of $\\mathring{H}_F$ as the unique self-adjoint extension of $\\mathring{H}$ whose operator domain is contained in $\\mathcal {D}[\\mathring{H}]$ .", "As every semi-bounded and densely defined symmetric operator, $\\mathring{H}$ admits self-adjoint extensions.", "We are after the family of such extensions.", "The first step is to determine the adjoint of $\\mathring{H}$ .", "Proposition Let $\\lambda >0$ .", "(i) One has $\\begin{split}\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})\\;&=\\;\\Big \\lbrace u_\\xi \\in L^2(\\mathbb {R}^3)\\textrm { of the form }\\,\\widehat{u}_\\xi (p)=\\frac{\\xi }{p^2+\\lambda }\\,\\Big |\\,\\xi \\in \\mathbb {C}\\Big \\rbrace \\\\&=\\;\\mathrm {span}\\,\\big \\lbrace \\big (\\,p^2+\\lambda )^{-1}\\big ){\\textrm {\\LARGE \\check{\\,}\\normalsize }}\\, \\big \\rbrace \\end{split}$ (ii) The domain and the action of the adjoint of $\\mathring{H}$ are given by $\\mathcal {D}(\\mathring{H}^*)&\\!=\\!&\\left\\lbrace g\\in L^2(\\mathbb {R}^3)\\left|\\!\\begin{array}{c}\\widehat{g}(p)=\\displaystyle \\widehat{f}(p)+\\frac{\\eta }{(p^2+\\lambda )^2}+\\frac{\\xi }{p^2+\\lambda } \\\\f\\in \\mathcal {D}(\\mathring{H})\\,,\\quad \\eta ,\\xi \\in \\mathbb {C}\\end{array}\\!\\!\\!\\right.\\right\\rbrace \\\\(\\widehat{(\\mathring{H}^*+\\lambda ) g)}\\,(p)&\\!=\\!& (p^2+\\lambda ) \\,\\Big (\\widehat{f}(p)+\\frac{\\eta }{(p^2+\\lambda )^2}\\Big ) \\\\\\widehat{(\\mathring{H}^* g)}(p)&\\!=\\!& p^2 \\widehat{g}(p)-\\xi \\,.", "$ Remark The decomposition (REF ) of the generic element $g\\in \\mathcal {D}(\\mathring{H}^*)$ depends on the chosen $\\lambda $ , but of course $\\mathcal {D}(\\mathring{H}^*)$ does not, nor does the outcome of $\\mathring{H}^*$ applied to $g$ , as one sees from ().", "[Proof of Proposition REF ] In order to apply the general decomposition formulas of Lemma we need to deal with an operator with positive bottom.", "To this aim we introduce the auxiliary operator $\\mathring{H}_{\\lambda }\\;:=\\;\\mathring{H}+\\lambda \\mathbb {1}$ which is by construction densely defined, symmetric, and closed, and with bottom $m(S)=\\lambda $ .", "Clearly, $\\mathcal {D}(\\mathring{H}_{\\lambda })=\\mathcal {D}(\\mathring{H})$ and $\\mathring{H}_{\\lambda }^*=\\mathring{H}^*+\\lambda \\mathbb {1}$ .", "Since $\\ker (\\mathring{H}^*_{\\lambda })\\;=\\mathrm {ran}(\\mathring{H}_{\\lambda })^\\perp $ , then $u\\in \\ker (\\mathring{H}^*_{\\lambda })\\;$ if and only if $0\\;=\\;\\int _{\\mathbb {R}^3}{(\\mathring{H}_{\\lambda }f) \\,u\\, \\mathrm {d}x}\\;=\\;\\int _{\\mathbb {R}^3}{\\widehat{(\\mathring{H}_{\\lambda }f)}\\,\\widehat{u} \\,\\mathrm {d}p}=\\int _{\\mathbb {R}^3}{\\widehat{f}\\, (p^2+\\lambda )\\,\\widehat{u} \\,\\mathrm {d}p}\\qquad \\forall f\\in \\mathcal {D}(\\mathring{H})\\,,$ which by (REF ) and a standard localisation argument yields (REF ).", "Because of $(\\mathring{H}_\\lambda )_F=\\mathring{H}_F+\\lambda \\mathbb {1}$ and (REF ), we have that $\\widehat{(\\mathring{H}_\\lambda )_F^{-1}u}\\;=\\;(p^2+\\lambda )^{-1}\\widehat{u}\\,.$ This, together with the decomposition formula () discussed in Appendix (Lemma ), and the characterisation (REF ) of $\\ker \\mathring{H}_\\lambda ^*$ yield immediately (REF ).", "The decomposition () also implies that the action of $\\mathring{H}_\\lambda ^*$ on a generic element $f+(\\mathring{H}_\\lambda )_F^{-1}u_\\eta +u_\\xi \\in \\mathcal {D}(\\mathring{H}_\\lambda ^*)$ is the same as the action of $(\\mathring{H}_\\lambda )_F$ on the component $f+(\\mathring{H}_\\lambda )_F^{-1}u_\\eta \\in \\mathcal {D}((\\mathring{H}_\\lambda )_F)$ , while $\\mathring{H}_\\lambda ^* u_\\xi =0$ : this is precisely ().", "As for (), it follows directly from (REF ) and ().", "Remark The decomposition formula (), together with (REF ) above, gives $\\mathcal {D}((\\mathring{H}_\\lambda )_F)\\;=\\;\\left\\lbrace g\\in L^2(\\mathbb {R}^3)\\left|\\!\\begin{array}{c}\\widehat{g}(p)=\\displaystyle \\widehat{f}(p)+(p^2+\\lambda )^{-2}\\eta \\\\f\\in \\mathcal {D}(\\mathring{H})\\,,\\quad \\eta \\in \\mathbb {C}\\end{array}\\!\\!\\!\\right.\\right\\rbrace $ Therefore, expression (REF ) shows that a generic $g\\in \\mathcal {D}(\\mathring{H}^*)$ is in general less regular than $H^2(\\mathbb {R}^3)$ , for only the component $\\mathcal {F}^{-1}(\\widehat{f}+(p^2+\\lambda )^{-2}\\eta )=f+(\\mathring{H}_\\lambda )_F^{-1}u_\\eta $ is in $\\mathcal {D}((\\mathring{H}_\\lambda )_F)=\\mathcal {D}(\\mathring{H}_F)=H^2(\\mathbb {R}^3)$ , whereas the component $u_\\xi $ is not.", "Related to that, () shows that whereas $\\mathring{H}^*g\\in L^2(\\mathbb {R}^3)$ , none of the two distributions $-\\Delta g$ and $(2\\pi )^{3/2}\\xi \\delta (x)$ whose difference gives precisely $\\mathring{H}^*g$ is realised as a square-integrable function (thus, in the difference the two non-square-integrable singularities cancel out).", "With the above knowledge of $\\mathring{H}^*+\\lambda \\mathbb {1}$ and $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ the Kreĭn-Višik-Birman theory provides an explicit prescription to restrict $\\mathring{H}_\\lambda ^*$ so as to find the whole family of self-adjoint extensions of $\\mathring{H}_\\lambda $ , and hence of $\\mathring{H}$ .", "Theorem  (i) The self-adjoint extensions of the operator $\\mathring{H}$ on $L^2(\\mathbb {R}^3)$ constitute the one-parameter family $\\lbrace \\mathring{H}^{(\\tau )}\\,|\\,\\tau \\in \\mathbb {R}\\cup \\lbrace \\infty \\rbrace \\rbrace $ where $\\mathring{H}^{(\\infty )}$ is the Friedrichs extension $\\mathring{H}_F$ , that is, $\\mathcal {D}(\\mathring{H}^{(\\infty )})\\;=H^2(\\mathbb {R}^3)\\;, \\qquad \\widehat{(\\mathring{H}^{(\\infty )} f)}(p)\\;=\\;p^2\\widehat{f}(p)\\,,$ whereas, for $\\tau \\in \\mathbb {R}$ , $\\mathcal {D}(\\mathring{H}^{(\\tau )})\\!&=&\\!\\!\\left\\lbrace g\\in L^2(\\mathbb {R}^3)\\!\\left|\\!\\begin{array}{c}\\widehat{g}(p)=\\displaystyle \\widehat{f}(p)+\\frac{\\tau \\,\\xi }{(p^2+\\lambda )^2}+\\frac{\\xi }{p^2+\\lambda } \\\\\\xi \\in \\mathbb {C}\\,,\\quad f\\in \\mathcal {D}(\\mathring{H})\\end{array}\\!\\!\\!\\!\\right.\\right\\rbrace \\\\((\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1}) \\,g)^{\\widehat{\\;}}\\,(p)\\!\\!&=&\\!\\!", "(p^2+\\lambda ) \\,\\Big (\\widehat{f}(p)+\\tau \\,\\frac{\\xi }{(p^2+\\lambda )^2}\\Big ) \\\\\\widehat{(\\mathring{H}^{(\\tau )} g)}(p)\\!\\!&=&\\!\\!", "p^2 \\widehat{g}(p)-\\xi \\,, $ where $\\lambda >0$ is arbitrary.", "(ii) Each extension $\\mathring{H}^{(\\tau )}$ is semi-bounded below.", "In particular, for the bottom $m(\\mathring{H}^{(\\tau )})$ of $\\mathring{H}^{(\\tau )}$ one has $\\begin{split}m(\\mathring{H}^{(\\tau )})\\;\\geqslant \\;0\\quad &\\Leftrightarrow \\quad \\tau \\;\\geqslant \\; 0 \\\\m(\\mathring{H}^{(\\tau )})\\;> \\;0\\quad &\\Leftrightarrow \\quad \\tau \\;>\\; 0\\,.\\end{split}$ (iii) For each $\\tau \\in \\mathbb {R}$ the quadratic form of the extension $\\mathring{H}^{(\\tau )}$ is given by $\\mathcal {D}[\\mathring{H}^{(\\tau )}]\\!&=&\\!\\!", "H^1(\\mathbb {R}^3)\\dotplus \\ker (\\mathring{H}^*+\\lambda \\mathbb {1}) \\\\\\mathring{H}^{(\\tau )}[\\phi +u_\\xi ]\\!\\!&=&\\!\\!", "\\Vert \\nabla \\phi \\Vert _2^2-\\lambda \\Vert \\phi +u_\\xi \\Vert _2^2+\\lambda \\Vert \\phi \\Vert _2^2+\\tau \\frac{\\;\\pi ^2}{\\sqrt{\\lambda }}|\\xi |^2 $ for any $\\phi \\in H^1(\\mathbb {R}^3)$ and any $\\widehat{u}_\\xi (p)=(p^2+\\lambda )^{-1}\\xi $ , $\\xi \\in \\mathbb {C}$ , where $\\lambda >0$ is arbitrary.", "Fixed $\\lambda >0$ , by Theorem the self-adjoint extensions of $\\mathring{H}+\\lambda \\mathbb {1}$ are one-to-one with the self-adjoint operators on Hilbert subspaces of $\\ker (\\mathring{H}+\\lambda \\mathbb {1})^*$ , which is the one-dimensional space found in (REF ).", "The generic case is that of a self-adjoint operator acting on the whole $\\ker (\\mathring{H}+\\lambda \\mathbb {1})^*$ , that is, the map $T_\\tau : u_\\xi \\mapsto \\tau u_\\xi $ of multiplication by the scalar $\\tau \\in \\mathbb {R}$ .", "In this case the expression (REF ) for $\\mathcal {D}(\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1})$ ($=\\mathcal {D}(\\mathring{H}^{(\\tau )})$ ) reads, by means of (REF ) and (REF ), precisely as (REF ).", "Then () follows from (REF ) and the fact that $\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1}=(\\mathring{H}^*+\\lambda \\mathbb {1})\\!\\upharpoonright _{\\mathcal {D}(\\mathring{H}^{(\\tau )})}$ , and () is an immediate consequence of ().", "The case of the subspace $\\lbrace 0\\rbrace $ of $\\ker (\\mathring{H}+\\lambda \\mathbb {1})^*$ corresponds by Proposition to the Friedrichs extension $(\\mathring{H}+\\lambda \\mathbb {1})_F=\\mathring{H}_F+\\lambda \\mathbb {1}$ , where $\\mathring{H}_F$ has been determined in (REF ) of Lemma REF .", "Thus, re-writing the generic $g\\in \\mathcal {D}(\\mathring{H}^{(\\tau )})$ as $\\widehat{g}=\\widehat{f}+(p^2+\\lambda )^{-2}\\widetilde{\\xi }+\\tau ^{-1}(p^2+\\lambda )^{-1}\\widetilde{\\xi }$ for arbitrary $f\\in \\mathcal {D}(\\mathring{H})$ and $\\widetilde{\\xi }\\in \\mathbb {C}$ , and comparing it with (REF ), one recognises that $\\mathcal {D}(\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1})=\\mathcal {D}(\\mathring{H}_F+\\lambda \\mathbb {1})$ when $\\tau =\\infty $ .", "Therefore, $\\mathring{H}^{(\\infty )}=\\mathring{H}_F$ and (REF ) follows from (REF ).", "This concludes the proof of part (i).", "The semi-boundedness of each $\\mathring{H}^{(\\tau )}$ follows by Proposition , or also Proposition .", "Conditions (REF ) for $\\mathring{H}^{(\\tau )}$ follow by the general conditions (REF ), using the fact that $m(\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1})=m(\\mathring{H}^{(\\tau )})+\\lambda $ and that $\\lambda >0$ is arbitrary, thus also part (ii) is proved.", "Last, we observe that $\\mathcal {D}[\\mathring{H}^{(\\tau )}]=\\mathcal {D}[\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1}]$ and $\\mathring{H}^{(\\tau )}[g]=(\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1})[g]-\\lambda \\Vert g\\Vert _2^2$ , thus one deduces (REF )-() from (REF ) of Theorem applied to $\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1}$ : formula (REF ) is an immediate consequence of (REF ); concerning (), formula (REF ) prescribes the contributions $(\\mathring{H}_F+\\lambda \\mathbb {1})[\\phi +u_\\xi ]$ and $-\\lambda \\Vert \\phi +u_\\xi \\Vert _2^2$ to $\\mathring{H}^{(\\tau )}[\\phi +u_\\xi ]$ , which are the first three summands in the r.h.s.", "of (), plus the term $T_\\tau [u_\\xi ]\\;=\\;\\tau \\int _{\\mathbb {R}^3}\\Big |\\frac{\\xi }{p^2+\\lambda }\\Big |^2\\,\\mathrm {d}p\\;=\\;\\tau \\,\\frac{\\;\\pi ^2}{\\sqrt{\\lambda }}\\,|\\xi |^2$ which is the fourth summand.", "This completes the proof of part (iii).", "Corollary For each self-adjoint extension $\\mathring{H}^{(\\tau )}$ and for arbitrary $\\lambda >0$ one has $\\mathcal {D}(\\mathring{H}^{(\\tau )})\\!\\!\\!&=&\\!\\!\\!\\Big \\lbrace g=\\phi +\\phi (0)\\,\\frac{8\\pi \\sqrt{\\lambda }}{\\tau }\\,G_\\lambda \\,\\Big |\\,\\phi \\in H^2(\\mathbb {R}^3)\\,,\\;G_\\lambda (x)=\\frac{e^{-\\sqrt{\\lambda }|x|}}{4\\pi |x|}\\Big \\rbrace \\\\(\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1})\\,g\\!\\!\\!&=&\\!\\!\\!", "(-\\Delta +\\lambda )\\,\\phi $ and $\\mathcal {D}[\\mathring{H}^{(\\tau )}]\\!\\!\\!&=&\\!\\!\\!\\Big \\lbrace g=\\phi +\\eta \\,G_\\lambda \\,\\Big |\\,\\phi \\in H^1(\\mathbb {R}^3)\\,,\\;\\eta \\in \\mathbb {C}\\,,\\;G_\\lambda (x)=\\frac{e^{-\\sqrt{\\lambda }|x|}}{4\\pi |x|}\\Big \\rbrace \\\\\\mathring{H}^{(\\tau )}[\\phi +\\eta \\,G_\\lambda ]\\!\\!\\!&=&\\!\\!\\!", "-\\lambda \\Vert \\phi +\\eta \\,G_\\lambda \\Vert _2^2+\\Vert \\nabla \\phi \\Vert _2^2+\\lambda \\Vert \\phi \\Vert _2^2+\\frac{\\tau }{8\\pi \\sqrt{\\lambda }}\\,|\\eta |^2\\,.", "$ Since $\\Big (\\frac{1}{(p^2+\\lambda )^2}\\Big ){\\textrm {\\LARGE \\check{\\,}\\normalsize }}(x)\\;=\\;\\sqrt{\\frac{\\pi }{8\\lambda }}\\,e^{-\\sqrt{\\lambda }|x|}\\quad \\textrm { and }\\quad \\Big (\\frac{1}{p^2+\\lambda }\\Big ){\\textrm {\\LARGE \\check{\\,}\\normalsize }}(x)\\;=\\;(2\\pi )^{3/2}\\,\\frac{\\,e^{-\\sqrt{\\lambda }|x|}}{4\\pi |x|}\\,,$ the inverse Fourier transform of a generic $\\widehat{g}$ of the form (REF ) reads $g=\\phi +\\xi \\,(2\\pi )^{3/2}\\,G_\\lambda $ where $G_\\lambda (x)\\;:=\\;\\frac{e^{-\\sqrt{\\lambda }|x|}}{4\\pi |x|}\\,,\\quad \\widehat{\\phi }\\;:=\\;\\widehat{f}+\\frac{\\tau \\,\\xi }{(p^2+\\lambda )^2}\\,,\\quad \\textrm { i.e., }\\quad \\phi \\;=\\;f+\\tau \\,\\xi \\,\\sqrt{\\frac{\\pi }{8\\lambda }}\\,e^{-\\sqrt{\\lambda }|x|}\\,.$ From (REF ) one computes $\\phi (0)\\;=\\;\\tau \\,\\xi \\,\\sqrt{\\frac{\\pi }{8\\lambda }}$ and plugging this into (REF ) one obtains (REF ).", "() is the inverse Fourier transform of ().", "(REF ) follows by taking the inverse Fourier transform in (REF ), using the characterisation (REF ) for $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ and formulas (REF ) and (REF ).", "() is a straightforward re-writing of ().", "Remark In Corollary REF above we have re-derived, apart from an obvious re-definition of the extension parameter $\\tau $ , the well-known formulas for $\\mathring{H}^{(\\tau )}$ previously obtained in the literature by means of von Neumann's extension theory.", "Indeed, each $\\mathring{H}^{(\\tau )}$ is precisely the extension $H_\\alpha $ discussed in [2], where $\\alpha \\;=\\;\\frac{\\tau -2\\lambda }{\\,8\\pi \\sqrt{\\lambda }\\,}\\,;$ by means of (REF ), the expression (REF ) for $\\mathcal {D}(\\mathring{H}^{(\\tau )})$ takes the form of [2].", "In particular, the analysis of the extension parameter $\\alpha $ done in [2] shows that the two-body point interaction modelled by the Hamiltonian $H_\\alpha $ has $s$ -wave scattering length equal to $-(4\\pi \\alpha )^{-1}$ .", "Remark The function $\\phi $ in the decompositions (REF ) and (REF ) is customarily referred to as the “regular part” of the given $g$ of the operator domain or the form domain of $\\mathring{H}^{(\\tau )}$ , the difference $g-\\phi $ taking the name of the “singular part” of $g$ .", "Formulas (REF ) and (REF ) give, for a generic $g=\\phi +\\phi (0) (8\\pi \\sqrt{\\lambda })/\\tau \\,G_\\lambda \\in \\mathcal {D}(\\mathring{H}^{(\\tau )})$ , $\\langle \\, g,(\\mathring{H}^{(\\tau )}+\\lambda \\mathbb {1})\\,g\\rangle \\;=\\;\\langle \\, \\phi ,(-\\Delta +\\lambda \\mathbb {1})\\,\\phi \\rangle + \\frac{8\\pi \\sqrt{\\lambda }}{\\tau }\\,|\\phi (0)|^2$ which provides once more the interpretation of the point-like character of the interaction modelled by $\\mathring{H}^{(\\tau )}$ at $x=0$ .", "Remark Owing to the decomposition (REF ), a generic $g\\in \\mathcal {D}(\\mathring{H}^{(\\tau )})$ displays the characteristic asymptotics of TMS-type (see (REF ) above) when $x\\rightarrow 0$ .", "Indeed, using the continuity of $\\phi $ , one derives from (REF ) $g(x)\\;=\\;\\phi (0)\\,\\frac{2\\sqrt{\\lambda }}{\\tau }\\Big (\\frac{1}{|x|}+\\frac{\\tau }{2\\sqrt{\\lambda }}\\Big )+o(1)\\qquad \\textrm { as }x\\rightarrow 0\\,,$ and replacing $\\phi (0)$ with $\\xi $ according to (REF ) and $\\tau $ with $\\alpha $ given by (REF ) one obtains $g(x)\\;=\\;\\xi \\sqrt{\\frac{\\pi }{2}\\,}\\,\\Big (\\frac{1}{|x|}+4\\pi \\alpha +\\sqrt{\\lambda }\\Big )+o(1)\\qquad \\textrm { as }x\\rightarrow 0\\,.$ In the expression above $\\lambda >0$ is arbitrary (and at fixed $g$ the charge $\\xi $ is implicitly $\\lambda $ -dependent, see (REF )) and one can therefore read the asymptotics as $\\lambda \\rightarrow 0$ ; thus, in terms of the scattering length $a=-(4\\pi \\alpha )^{-1}$ of the interaction, one has $g(x)\\;=\\;\\xi \\sqrt{\\frac{\\pi }{2}\\,}\\Big (\\frac{1}{|x|}-\\frac{1}{a}\\Big )+o(1)\\qquad \\textrm { as }x\\rightarrow 0\\,,$ which has the form of the TMS asymptotics (REF ).", "Remark Another customary and equivalent expression for the action of $\\mathring{H}^{(\\tau )}$ in spatial coordinates is obtained by taking the inverse Fourier transform in () (while the inverse Fourier transform in () yielded ()).", "One finds $\\mathring{H}^{(\\tau )}g\\;=\\;-\\Delta g-(2\\pi )^{3/2}\\xi \\,\\delta \\,,\\qquad g\\in \\mathcal {D}(\\mathring{H}^{(\\tau )})\\,,$ where $\\delta $ is the Dirac distribution.", "The l.h.s.", "of (REF ) is an $L^2$ -function for each $g\\in \\mathcal {D}(\\mathring{H}^{(\\tau )})$ and the r.h.s.", "expresses this $L^2$ -function as the difference of two distributions.", "In general an element $g\\in \\mathcal {D}(\\mathring{H}^{(\\tau )})$ does not belong to $H^2(\\mathbb {R}^3)$ , in which case $-\\Delta g$ is only meant as a distributional derivative: this term has an $L^2$ -part plus a distributional (non-square-integrable) part $(2\\pi )^{3/2}\\xi \\,\\delta $ which is cancelled in the difference in the r.h.s.", "of (REF ).", "This is consistent with (REF ) above, where formally (as $x\\rightarrow 0$ ) one obtains a distributional contribution in $-\\Delta g$ given by $-\\Delta |x|^{-1}$ , which is precisely a $\\delta $ -distribution.", "The cancellation occurring in the r.h.s.", "of (REF ) can be regarded as the renormalisation of $-\\Delta g$ needed to give meaning to $\\mathring{H}^{(\\tau )}g$ when $g\\in \\mathcal {D}(\\mathring{H}^{(\\tau )})$ .", "In Remark REF below we shall complete this comment by showing that (REF ) expresses the very same normalisation (REF ) announced by Berezin and Faddeev." ], [ "Ter-Martirosyan–Skornyakov construction for the point interaction", "The functions in $\\mathcal {D}(\\mathring{H}^*)$ have the following asymptotic behaviour.", "Lemma Let $g$ be an arbitrary function in $\\mathcal {D}(\\mathring{H}^*)$ .", "For a fixed $\\lambda >0$ let $\\widehat{g}=\\widehat{f}+(p^2+\\lambda )^{-2}\\eta +(p^2+\\lambda )^{-1}\\xi $ be the decomposition of $g$ obtained in Proposition REF for some $f\\in \\mathcal {D}(\\mathring{H})$ and some $\\eta ,\\xi \\in \\mathbb {C}$ .", "Then $\\int _{\\begin{array}{c} \\\\ \\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}{\\:\\widehat{g}(p) \\,\\mathrm {d}p}\\;=\\;4\\pi \\xi R+\\Big (\\!-2\\pi ^2\\sqrt{\\lambda }\\,\\xi +\\frac{\\;\\pi ^2}{\\sqrt{\\lambda }}\\,\\eta \\Big )+o(1)\\qquad \\textrm {as}\\qquad R\\rightarrow +\\infty \\,.$ Owing to Lemma REF , equation (REF ), $\\int _{|p|<R}\\widehat{f}(p)\\,\\mathrm {d}p\\rightarrow 0$ as $R\\rightarrow \\infty $ , so this integral is a $o(1)$ contribution in (REF ).", "As for the two other summands in the decomposition of $\\widehat{g}$ , we have $\\begin{split}\\int _{\\begin{array}{c}\\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}{\\frac{\\eta }{(p^2+\\lambda )^2} \\,\\mathrm {d}p}&\\;=\\;4\\pi \\eta \\int _{0}^R{\\frac{\\rho ^2}{(\\rho ^2+\\lambda )^2} \\,\\mathrm {d}\\rho }\\;=\\;-\\frac{2\\pi \\eta R}{R^2+\\lambda }+\\frac{2\\pi \\eta \\,}{\\sqrt{\\lambda }}\\arctan \\Big (\\frac{R}{\\sqrt{\\lambda }}\\Big ) \\\\&\\;=\\;\\frac{\\;\\pi ^2}{\\sqrt{\\lambda }}\\,\\eta +o(1)\\qquad \\textrm {as}\\qquad R\\rightarrow +\\infty \\end{split}$ and $\\begin{split}\\int _{\\begin{array}{c}\\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}{\\frac{\\xi }{p^2+\\lambda } \\,\\mathrm {d}p}&\\;=\\;4\\pi \\xi \\int _{0}^R{\\frac{\\rho ^2}{\\rho ^2+\\lambda } \\,\\mathrm {d}\\rho }\\;=\\;4\\pi \\xi \\Big (R-\\sqrt{\\lambda }\\arctan \\Big (\\frac{R}{\\sqrt{\\lambda }}\\Big )\\Big ) \\\\&\\;=\\;4\\pi \\xi R-2\\pi ^2\\sqrt{\\lambda }\\,\\xi +o(1)\\qquad \\textrm {as}\\qquad R\\rightarrow +\\infty \\,,\\end{split}$ which complete the computation of the r.h.s.", "of (REF ).", "One defines the TMS extension $\\mathring{H}_{\\langle \\alpha \\rangle }$ of $\\mathring{H}$ , $\\alpha \\in \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ , to be the restriction of $\\mathring{H}^*$ to those $g$ 's of $\\mathcal {D}(\\mathring{H}^*)$ for which in the asymptotics (REF ) the coefficients of the $O(R)$ term and of the $O(1)$ term are proportional by a factor $\\alpha $ , more precisely $8\\pi ^3\\alpha \\,\\xi \\;=\\;-2\\pi ^2\\sqrt{\\lambda }\\,\\xi +\\frac{\\;\\pi ^2}{\\sqrt{\\lambda }}\\,\\eta \\,.$ Thus, $\\mathcal {D}(\\mathring{H}_{\\langle \\alpha \\rangle })$ consists of all $g$ 's of $\\mathcal {D}(\\mathring{H}^*)$ for which $\\int _{\\begin{array}{c}\\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}\\,{\\widehat{g}(p) \\,\\mathrm {d}p}\\;=\\;4\\pi \\xi \\,(R+2\\pi ^2\\alpha )+o(1) \\qquad \\textrm {as}\\qquad R\\rightarrow +\\infty \\,.$ Both (REF ) and (REF ) express a TMS condition; it is a constraint on the singular part of the $g$ 's of $\\mathcal {D}(\\mathring{H}^*)$ that restricts the choice of $\\xi ,\\eta \\in \\mathbb {C}$ to those such that $\\eta \\;=\\;2\\sqrt{\\lambda }\\,(4\\pi \\alpha +\\sqrt{\\lambda })\\,\\xi \\,.$ Condition (REF ) above is precisely of the form $\\eta =\\tau \\xi $ with $\\tau \\;=\\;2\\sqrt{\\lambda }\\,(4\\pi \\alpha +\\sqrt{\\lambda })\\,.$ Owing to (REF ) of Theorem REF , this implies at once that the operator $\\mathring{H}_{\\langle \\alpha \\rangle }$ selected by the TMS condition (REF )-(REF )-(REF ) above is precisely the self-adjoint extension $\\mathring{H}^{(\\tau )}$ of $\\mathring{H}$ qualified by the parameter $\\tau $ given by (REF ) in terms of $\\alpha $ .", "We also observe that (REF ) is the inverse of (REF ): therefore, by what recalled in Remark REF , the TMS extension $\\mathring{H}_{\\langle \\alpha \\rangle }$ models a two-body point interaction with $s$ -wave scattering length equal to $-(4\\pi \\alpha )^{-1}$ .", "Thus, imposing the TMS condition on $\\mathring{H}^*$ with parameter $\\alpha $ produces the self-adjoint extension of $\\mathring{H}$ that gives the point interaction with scattering length $-(4\\pi \\alpha )^{-1}$ ; the collection of all the TMS extensions obtained this way cover the whole family of self-adjoint extensions of $\\mathring{H}$ : all self-adjoint extensions of $\\mathring{H}$ are of TMS type.", "Remark With the analysis of this Subsection we can supplement Remark REF above with the following observation.", "First, one recognises that the condition (REF ) identified by Berezin and Faddeev for each self-adjoint extension of $\\mathring{H}$ is precisely of the TMS form (REF ).", "Furthermore, the asymptotics (REF ) shows that for each $g\\in \\mathcal {D}(\\mathring{H}_{\\langle \\alpha \\rangle })$ the corresponding charge $\\xi $ is given by $\\xi \\;=\\;\\lim _{R\\rightarrow +\\infty }\\frac{1}{4\\pi R}\\int _{\\begin{array}{c}\\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}\\,{\\widehat{g}(p) \\,\\mathrm {d}p}$ which plugged into () yields $\\widehat{(\\mathring{H}_{\\langle \\alpha \\rangle } g)}(p)\\;=\\;p^2 \\widehat{g}(p)-\\lim _{R\\rightarrow +\\infty }\\frac{1}{4\\pi R}\\int _{\\begin{array}{c}\\,p\\in \\mathbb {R}^3 \\\\ \\!", "|p|<R\\end{array}}\\,{\\widehat{g}(p) \\,\\mathrm {d}p}\\,.$ This is exactly the property (REF ) announced by Berezin and Faddeev, and it is the Fourier-transformed version of the identity (REF ) that expresses $\\mathring{H}_{\\langle \\alpha \\rangle } g$ by means of a suitable renormalisation of $-\\Delta g$ ." ], [ "TMS Hamiltonians for the three-body problem with point interaction", "As discussed in Section , the problem of a three-particle quantum system with two-body point interaction has been studied since long.", "However, one still has a relatively limited knowledge of the corresponding Hamiltonians, primarily their self-adjoint realisation (whereas the information on their stability and spectral properties is only partial).", "The novel difficulty, as opposed to the two-body case, is due to the fact that the Ter-Martirosyan–Skornyakov condition does not select in general a domain of self-adjointness, and one has to further study the self-adjoint extension of the resulting TMS operator.", "We restrict our discussion to the most studied case, that of two identical fermions in interaction with a third particle of different nature (the so called “2+1 fermionic system”)." ], [ "The “2+1 fermionic system”", "After removing the centre of mass, the free Hamiltonian of a three-dimensional system of two identical fermions of unit mass in relative positions $x_1,x_2$ with respect to a third particle of different species and with mass $m$ is the operator $-\\Delta _{x_1}-\\Delta _{x_2}-\\frac{2}{m+1}\\nabla _{x_1}\\cdot \\nabla _{x_2}$ acting on the Hilbert space $\\mathcal {H}\\;=\\;L^2_\\mathrm {f}(\\mathbb {R}^3\\times \\mathbb {R}^3,\\mathrm {d}x_1\\mathrm {d}x_2)\\,,$ the subscript `f' standing for the fermionic sector of the $L^2$ -space, i.e, the square-integrable functions that are anti-symmetric under exchange $x_1\\leftrightarrow x_2$ .", "Following the same path as in Subsection REF , one therefore starts with the operator $\\begin{split}\\mathring{H}\\;&:=\\;-\\Delta _{x_1}-\\Delta _{x_2}-\\frac{2}{m+1}\\nabla _{x_1}\\cdot \\nabla _{x_2} \\\\\\mathcal {D}(\\mathring{H})\\;&:=\\;H^2_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))\\cap \\mathcal {H}\\,,\\end{split}$ where $\\Gamma _j\\;:=\\;\\lbrace (x_1,x_2)\\in \\mathbb {R}^3\\times \\mathbb {R}^3\\,|\\,x_j=0\\rbrace \\,,\\qquad j=1,2,$ and $H^2_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))\\;=\\;\\overline{C^\\infty _0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))}^{\\Vert \\,\\Vert _{H^2}}\\,.$ $\\mathring{H}$ is a densely defined, closed, positive, and symmetric operator on $\\mathcal {H}$ .", "As such, $\\mathring{H}$ has equal deficiency indices, and in Proposition REF below we shall see that they are infinite.", "Any self-adjoint extension of $\\mathring{H}$ has a natural interpretation of Hamiltonian of point interaction between each fermion and the third particle.", "It is convenient first to characterise $\\mathcal {D}(\\mathring{H})$ in Fourier transform.", "Lemma  (i) The domain of $\\mathring{H}$ is given by $\\begin{split}\\mathcal {D}(\\mathring{H})\\;&=\\;\\left\\lbrace f\\in H^2_\\mathrm {f}(\\mathbb {R}^3\\times \\mathbb {R}^3)\\left|\\!\\begin{array}{c}\\iint \\widehat{f}(p_1,p_2)\\,\\widehat{\\eta }(p_1)\\,\\mathrm {d}p_1\\mathrm {d}p_2=0 \\\\\\forall \\eta \\in H^{-1/2}(\\mathbb {R}^3)\\end{array}\\!\\!\\right.\\right\\rbrace \\\\&=\\;\\big \\lbrace \\,f\\in H^2_\\mathrm {f}(\\mathbb {R}^3\\times \\mathbb {R}^3)\\,\\big |\\,\\langle \\widehat{f},h\\rangle =0\\;\\;\\forall h\\in \\mathcal {X}\\,\\rbrace \\,,\\end{split}$ where $\\mathcal {X}\\;:=\\;\\big \\lbrace \\,h\\,|\\,h(p_1,p_2)=\\widehat{\\eta }(p_1)-\\widehat{\\eta }(p_2)\\textrm { for some }\\eta \\in H^{-1/2}(\\mathbb {R}^3)\\,\\big \\rbrace $ and the duality product is meant between the spaces $L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,(1+p_1^2+p_2^2)^2\\mathrm {d}p_1\\mathrm {d}p_2)$ and $L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,(1+p_1^2+p_2^2)^{-2}\\mathrm {d}p_1\\mathrm {d}p_2)$ .", "The action of $\\mathring{H}$ is given by $\\widehat{(\\mathring{H}f)}(p_1,p_2)\\;=\\;(p_1^2+p_2^2+\\mu \\,p_1\\cdot p_2)\\,\\widehat{f}(p_1,p_2)\\,,$ where $\\mu \\;:=\\;\\frac{2}{m+1}\\,.$ In other words, $\\mathcal {D}(\\mathring{H})$ consists of the $H^2_\\mathrm {f}$ -functions $f$ such that $\\int _{\\mathbb {R}^3}\\widehat{f}(p_1,p_2)\\mathrm {d}p_j=0$ in $H^{1/2}(\\mathbb {R}^3)$ , where $j\\in \\lbrace 1,2\\rbrace $ , and $\\mathring{H}$ acts on such $f$ 's as the free two-body (negative) Laplacian.", "(ii) The Friedrichs extension of $\\mathring{H}$ is given by $\\mathcal {D}(\\mathring{H}_F)\\;=\\;H^2_\\mathrm {f}(\\mathbb {R}^3\\times \\mathbb {R}^3)\\,,\\qquad \\widehat{(\\mathring{H}_Ff)}(p_1,p_2)\\;=\\;(p_1^2+p_2^2+\\mu \\,p_1\\cdot p_2)\\,\\widehat{f}(p_1,p_2)\\,.$ Clearly, for $j\\in \\lbrace 1,2\\rbrace $ one has $\\Gamma _j=\\cap _{\\nu =1}^3\\Gamma _{j,\\nu }$ , where $x_j\\equiv (x_{j,1},x_{j,2},x_{j,3})\\in \\mathbb {R}^3$ and $\\Gamma _{j,\\nu }$ is the hyperplane $x_{j,\\nu }=0$ .", "For a generic $f\\in H^2(\\mathbb {R}^3\\times \\mathbb {R}^3)$ a standard trace theorem [45] asserts that $f|_{\\Gamma _{1,1}}\\in H^{2-\\frac{1}{2}}(\\mathbb {R}^2\\times \\mathbb {R}^3)$ , and if in addition $f\\in H^2_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!\\Gamma _{1,\\nu })$ , then $f|_{\\Gamma _{1,1}}=0$ .", "Thus, by repeated application of the trace theorem to a function $f\\in H^2_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!\\Gamma _{1})$ , one finds $f|_{\\Gamma _1}=0$ in $H^{2-\\frac{3}{2}}(\\mathbb {R}^3)$ .", "Summarising, $f\\in \\mathcal {D}(\\mathring{H})\\qquad \\Leftrightarrow \\qquad f|_{\\Gamma _j}\\in H^{\\frac{1}{2}}(\\mathbb {R}^3)\\quad \\textrm {and}\\quad f|_{\\Gamma _j}=0\\,,\\quad j\\in \\lbrace 1,2\\rbrace \\,.$ One therefore has that $f\\in \\mathcal {D}(\\mathring{H})$ is equivalent to $0\\;=\\;\\langle \\eta ,f|_{\\Gamma _2}\\rangle _{H^{-\\frac{1}{2}},H^{\\frac{1}{2}}}\\;=\\;\\int _{\\mathbb {R}^3}\\overline{\\widehat{\\eta }(p_1)}\\,(\\widehat{f|_{\\Gamma _2})}(p_1)\\,\\mathrm {d}p_1\\qquad \\forall \\eta \\in H^{-\\frac{1}{2}}(\\mathbb {R}^3)\\,.$ Let now $f\\in \\mathcal {D}(\\mathring{H})$ .", "The Fourier transforms that follow are all of $L^2$ -functions, therefore the corresponding integral expressions are meant as $L^2$ -norm limits.", "From $f|_{\\Gamma _2}(x_1)\\;=\\;f(x_1,0)\\;=\\;\\frac{1}{\\;(2\\pi )^{3/2}}\\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3}\\widehat{f}(p_1,p_2) \\,e^{\\mathrm {i}x_1 p_1} \\mathrm {d}p_1\\,\\mathrm {d}p_2$ for a.e.", "$x_1\\in \\mathbb {R}^3$ , and from the distributional identity $\\delta (k)=(2\\pi )^{-3}\\int _{\\mathbb {R}^3}\\mathrm {d}x\\,e^{\\mathrm {i}x k}$ , one finds $\\begin{split}\\widehat{(f|_{\\Gamma _2})}(p_1)\\;&=\\;\\frac{1}{\\;(2\\pi )^{3/2}}\\int _{\\mathbb {R}^3}f|_{\\Gamma _2}(x_1)\\,e^{-\\mathrm {i}p_1 x_1}\\,\\mathrm {d}x_1 \\\\&=\\;\\frac{1}{\\;(2\\pi )^3}\\int _{\\mathbb {R}^3}\\mathrm {d}x_1 \\,e^{\\mathrm {i}x_1 (q_1-p_1)}\\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3}\\widehat{f}(q_1,q_2) \\,\\mathrm {d}q_1\\,\\mathrm {d}q_2 \\;=\\;\\int _{\\mathbb {R}^3}\\widehat{f}(p_1,p_2)\\,\\mathrm {d}p_2\\,.\\end{split}$ From this and from (REF ) one deduces that the function $p_1\\mapsto \\int \\widehat{f}(p_1,p_2)\\,\\mathrm {d}p_2$ vanishes in $H^{1/2}(\\mathbb {R}^3)$ .", "Plugging the last identity into (REF ) yields $\\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3}\\widehat{f}(p_1,p_2)\\,\\widehat{\\eta }(p_1)\\,\\mathrm {d}p_1\\mathrm {d}p_2\\;=\\;0 \\qquad \\forall \\eta \\in H^{-1/2}(\\mathbb {R}^3)\\,,$ which proves the first line in (REF ).", "This result, together with the anti-symmetry of $f$ , implies that $\\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3}\\widehat{f}(p_1,p_2)\\,(\\widehat{\\eta }(p_1)-\\widehat{\\eta }(p_2))\\,\\mathrm {d}p_1\\mathrm {d}p_2\\;=\\;0 \\quad \\forall \\eta \\in H^{-1/2}(\\mathbb {R}^3)\\quad (f\\in \\mathcal {D}(\\mathring{H}))\\,.$ We now observe that $\\widehat{f}\\in L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,(1+p_1^2+p_2^2)^2\\mathrm {d}p_1\\mathrm {d}p_2)$ because $\\mathcal {D}(\\mathring{H})\\subset H^2(\\mathbb {R}^3\\times \\mathbb {R}^3)$ , and that the map $(p_1,p_2)\\mapsto \\widehat{\\eta }(p_1)-\\widehat{\\eta }(p_2)$ belongs to $L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,(1+p_1^2+p_2^2)^{-2}\\mathrm {d}p_1\\mathrm {d}p_2)$ , because $\\begin{split}\\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3}&\\frac{|\\widehat{\\eta }(p_1)-\\widehat{\\eta }(p_2)|^2}{(1+p_1^2+p_2^2)^2}\\,\\mathrm {d}p_1\\mathrm {d}p_2\\;\\leqslant \\;4\\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3}\\frac{|\\widehat{\\eta }(p_1)|^2}{(1+p_1^2+p_2^2)^2}\\,\\mathrm {d}p_1\\mathrm {d}p_2 \\\\&=4\\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3}\\frac{|\\widehat{\\eta }(p_1)|^2}{(p_1^2+1)^{1/2}}\\,\\frac{(p_1^2+1)^{1/2}}{(1+p_1^2+p_2^2)^2}\\,\\mathrm {d}p_1\\mathrm {d}p_2\\;\\leqslant \\;4\\pi ^2\\Vert \\eta \\Vert _{H^{-1/2}}^2\\;<\\;+\\infty \\,,\\end{split}$ where we used $\\int _{\\mathbb {R}^3}\\frac{\\mathrm {d}p_2}{(1+p_1^2+p_2^2)^2}\\;=\\;\\frac{\\pi ^2}{\\:(p_1^2+1)^{1/2}}\\,.$ Thus, one can regard (REF ) as the vanishing of a duality product between the spaces $L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,(1+p_1^2+p_2^2)^2\\mathrm {d}p_1\\mathrm {d}p_2)$ and $L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,(1+p_1^2+p_2^2)^{-2}\\mathrm {d}p_1\\mathrm {d}p_2)$ , and since $f\\in \\mathcal {D}(\\mathring{H})$ was arbitrary, one concludes the second line in (REF ).", "Equations (REF )-(REF ) are the Fourier-transformed version of (REF ), and this concludes the proof of part (i).", "Concerning part (ii), we first observe that the form domain of $\\mathring{H}$ , namely the completion of $H^2_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))\\cap \\mathcal {H}$ , is the space $D[\\mathring{H}]\\;=\\;H^1_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))\\cap \\mathcal {H}\\,.$ By a standard approximation with smooth and compactly supported functions (see Appendix ), $H^2_{\\mathrm {f}}(\\mathbb {R}^3\\times \\mathbb {R}^3)\\;\\subset \\; H^1_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))\\cap \\mathcal {H}\\;=\\;H^1_{\\mathrm {f}}(\\mathbb {R}^3\\times \\mathbb {R}^3)\\,,$ and $H^2_{\\mathrm {f}}(\\mathbb {R}^3\\times \\mathbb {R}^3)$ is the domain of the self-adjoint extension of $\\mathring{H}$ given by the free negative Laplacian on $\\mathcal {H}$ .", "Therefore, owing to the characterisation of the Friedrichs extension $\\mathring{H}_F$ as the unique self-adjoint extension of $\\mathring{H}$ whose operator domain is contained in $\\mathcal {D}[\\mathring{H}]$ , one deduces immediately (REF ).", "In the next Proposition we characterise the adjoint of $\\mathring{H}$ , which is the preliminary step in order to identify the self-adjoint extensions of $\\mathring{H}$ within the general scheme of the Kreĭn-Višik-Birman theory.", "Proposition Let $\\lambda >0$ .", "(i) One has $\\begin{split}\\!\\!\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})\\;&=\\;\\left\\lbrace u_\\xi \\in L^2_\\mathrm {f}(\\mathbb {R}^3\\!\\times \\!\\mathbb {R}^3)\\left|\\!\\begin{array}{l}\\widehat{u}_\\xi (p_1,p_2)=\\displaystyle \\frac{\\widehat{\\xi }(p_1)-\\widehat{\\xi }(p_2)}{p_1^2+p_2^2+\\mu \\,p_1\\cdot p_2+\\lambda } \\\\\\xi \\in H^{-1/2}(\\mathbb {R}^3)\\end{array}\\!\\!\\right.\\right\\rbrace \\end{split}$ (ii) There exist constants $c_1,c_2>0$ such that for a generic $u_\\xi \\in \\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ one has $c_1\\Vert \\xi \\Vert _{H^{-1/2}(\\mathbb {R}^3)}\\leqslant \\;\\Vert u_\\xi \\Vert _{\\mathcal {H}}\\;\\leqslant c_2\\Vert \\xi \\Vert _{H^{-1/2}(\\mathbb {R}^3)}\\,.$ (iii) The domain and the action of the adjoint of $\\mathring{H}$ are given by $\\mathcal {D}(\\mathring{H}^*)\\;=\\;\\left\\lbrace \\!\\!\\begin{array}{c}g\\in L^2_\\mathrm {f}(\\mathbb {R}^3\\!\\times \\!\\mathbb {R}^3)\\quad \\textrm {such that} \\\\\\widehat{g}(p_1,p_2)=\\displaystyle \\widehat{f}(p_1,p_2)+\\frac{\\widehat{u}_\\eta (p_1,p_2)}{p_1^2+p_2^2+\\mu \\,p_1\\cdot p_2+\\lambda }+\\widehat{u}_\\xi (p_1,p_2) \\\\\\textrm {for}\\quad f\\in \\mathcal {D}(\\mathring{H})\\,,\\quad \\eta ,\\xi \\in H^{-1/2}(\\mathbb {R}^3)\\end{array}\\!\\!\\right\\rbrace $ and $(\\widehat{(\\mathring{H}^*+\\lambda ) g)}\\,(p_1,p_2)\\!\\!\\!&=&\\!\\!\\!", "(p_1^2+p_2^2+\\mu \\,p_1\\cdot p_2+\\lambda ) \\,\\widehat{F}_\\lambda (p_1,p_2) \\\\\\widehat{(\\mathring{H}^* g)}(p_1,p_2)\\!\\!\\!&=&\\!\\!\\!", "(p_1^2+p_2^2+\\mu \\,p_1\\cdot p_2) \\widehat{g}(p_1,p_2)-(\\widehat{\\xi }(p_1)-\\widehat{\\xi }(p_2)), $ where $u_\\eta $ and $u_\\xi $ are defined as in (REF ) above, and $\\widehat{F}_\\lambda (p_1,p_2)\\;:=\\;\\widehat{f}(p_1,p_2)+\\frac{\\widehat{u}_\\eta (p_1,p_2)}{p_1^2+p_2^2+\\mu \\,p_1\\cdot p_2+\\lambda }\\,.$ In order to prove part (i) we use the fact that $u\\in \\ker (\\mathring{H}^*+\\lambda \\mathbb {1})=\\mathrm {ran}(\\mathring{H}+\\lambda \\mathbb {1})^{\\perp }$ if and only if for every $f\\in \\mathcal {D}(\\mathring{H}+\\lambda \\mathbb {1})$ $\\begin{split}0\\;&=\\;\\int _{\\mathbb {R}^3}{u(x_1,x_2)((\\mathring{H}+\\lambda \\mathbb {1})f)(x_1,x_2)\\,\\mathrm {d}x_1 \\mathrm {d}x_2} \\\\&=\\;\\int _{\\mathbb {R}^3}{\\widehat{u}(p_1,p_2)((\\mathring{H}+\\lambda \\mathbb {1})f)^{\\widehat{\\;}}(p_1,p_2)\\,\\mathrm {d}p_1 \\mathrm {d}p_2} \\\\&=\\; \\int _{\\mathbb {R}^3}{\\widehat{u}(p_1,p_2)(p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )\\widehat{f}(p_1,p_2)\\,\\mathrm {d}p_1 \\mathrm {d}p_2}\\,,\\end{split}$ where we applied (REF ) in the last step.", "Since $\\mu \\in (0,2)$ (owing to (REF ) with $m> 0$ ) and $\\lambda >0$ , then $(1+p_1^2+p_2^2)\\;\\sim \\;(p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )$ (in the sense that each quantity controls the other from above and from below), and hence the fact that $u\\in \\ker (\\mathring{H}^*+\\lambda \\mathbb {1})\\subset L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,\\mathrm {d}p_1\\mathrm {d}p_2)$ is equivalent to $\\big (\\,(p_1,p_2)\\mapsto (p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )\\,\\widehat{u}(p_1,p_2)\\,\\big )\\;\\in \\; L^2(\\mathbb {R}^3\\times \\mathbb {R}^3,(1+p_1^2+p_2^2)^{-2}\\mathrm {d}p_1\\mathrm {d}p_2)\\,.$ Therefore, the last identity above implies, owing to the second line of (REF ), that $(p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )\\,\\widehat{u}(p_1,p_2)\\;=\\;\\widehat{\\xi }(p_1)-\\widehat{\\xi }(p_2)$ for some $\\xi \\in H^{-1/2}(\\mathbb {R}^3)$ , and hence each $u\\in \\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ is of the form $u\\equiv u_\\xi $ given by (REF ).", "Part (ii) is taken directly from [13].", "Concerning part (iii), because of $(\\mathring{H}+\\lambda \\mathbb {1})_F=\\mathring{H}_F+\\lambda \\mathbb {1}$ and (REF ), we have that $((\\mathring{H}+\\lambda \\mathbb {1})_F^{-1}u)^{\\widehat{\\;}}\\;=\\;(p^2+\\lambda )^{-1}\\widehat{u}\\,.$ This, together with the decomposition formula () of Lemma and the characterisation (REF ) of $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ yield immediately (REF ).", "The decomposition () also implies that the action of $\\mathring{H}^*+\\lambda \\mathbb {1}$ on a generic element $f+(\\mathring{H}^*+\\lambda \\mathbb {1})_F^{-1}u_\\eta +u_\\xi \\in \\mathcal {D}(\\mathring{H}^*+\\lambda \\mathbb {1})$ is the same as the action of $\\mathring{H}_F+\\lambda \\mathbb {1}$ on the component $f+(\\mathring{H}^*+\\lambda \\mathbb {1})_F^{-1}u_\\eta \\in \\mathcal {D}(\\mathring{H}_F+\\lambda \\mathbb {1})$ , while $(\\mathring{H}^*+\\lambda \\mathbb {1}) u_\\xi =0$ : this is precisely (REF ).", "As for (), it is an immediate consequence of (REF ) and (REF )." ], [ "General scheme for self-adjoint realisations of the 2+1 fermionic model", "The space $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ determined in (REF ) is the Kreĭn space of the model (the “boundary value space”, in modern terminology).", "It is known by the Kreĭn-Višik-Birman theory that the self-adjoint extensions of $\\mathring{H}$ defined in (REF ) are parametrised by self-adjoint operators acting on Hilbert subspaces of $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ , according to the classification given by Theorem in Appendix .", "Unlike the two-body model discussed in the previous Section, where $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ was one-dimensional, in the three-body model this space is infinite-dimensional (compare (REF ) with (REF )), which makes the variety of the self-adjoint extensions of $\\mathring{H}$ much more complicated.", "In this respect, the extensions of Ter-Martirosyan–Skornyakov type form a proper sub-family (in the two-body case, all self-adjoint extensions were of TMS type).", "Eventually one introduces the TMS extensions of $\\mathring{H}$ , analogously to Section REF for the two-body model, as restrictions of $\\mathring{H}^*$ to domains characterised by special asymptotics of their wave-functions in the vicinity of the coincidence hyperplanes.", "Prior to that, in this Subsection we shall develop a general scheme for the identification of a generic self-adjoint extension of $\\mathring{H}$ , within which we will later select those of TMS type.", "To this aim, it is convenient first to define the expressions $\\widehat{(T_\\lambda \\,\\xi )}(p)\\;:=\\;2\\pi ^2\\sqrt{\\nu p^2+\\lambda }\\;\\widehat{\\xi }(p)+\\int _{\\mathbb {R}^3}\\frac{\\widehat{\\xi }(q)}{p^2+q^2+\\mu p\\cdot q+\\lambda }\\,\\mathrm {d}q$ and $\\widehat{(W_\\lambda \\,\\xi )}(p)\\;:=\\;\\frac{2\\pi ^2}{\\sqrt{\\nu p^2+\\lambda }\\,}\\,\\widehat{\\xi }(p)-2\\!\\int _{\\mathbb {R}^3}\\frac{\\widehat{\\xi }(q)}{(p^2+q^2+\\mu p\\cdot q+\\lambda )^2}\\,\\mathrm {d}q$ for fixed $\\lambda >0$ , where $\\mu $ is given by (REF ) and $\\nu \\;:=\\;1-\\frac{\\:\\mu ^2}{4}\\;=\\;\\frac{m(m+2)}{(m+1)^2}\\,.$ Since, for arbitrary $\\varepsilon >0$ and $\\xi \\in H^{-\\frac{1}{2}+\\varepsilon }(\\mathbb {R}^3)$ , $\\begin{split}\\Big |\\int _{\\mathbb {R}^3}\\frac{\\widehat{\\xi }(q)}{p^2+q^2+\\mu p\\cdot q+\\lambda }\\,\\mathrm {d}q\\Big |\\;\\leqslant \\;\\Vert \\xi \\Vert _{H^{-\\frac{1}{2}+\\varepsilon }}\\Big (\\int _{\\mathbb {R}^3}\\frac{(q^2+1)^{\\frac{1}{2}-\\varepsilon }}{(p^2+q^2+\\mu p\\cdot q+\\lambda )^2}\\,\\mathrm {d}q\\Big )^{1/2}\\;<\\;+\\infty \\end{split}$ (owing to a Schwartz inequality in the first step and (REF ) in the second one), we see that the integral in (REF ) is finite for any $\\xi \\in H^{-\\frac{1}{2}+\\varepsilon }(\\mathbb {R}^3)$ , $\\varepsilon >0$ , while in general it diverges when $\\xi \\in H^{-\\frac{1}{2}}(\\mathbb {R}^3)$ , as the example $\\widehat{\\xi }_0(q)\\;:=\\;\\mathbf {1}_{\\lbrace |q|\\geqslant 2\\rbrace }(|q|\\ln |q|)^{-1}$ shows.", "A similar argument shows that the integral in (REF ) is finite too at least for $\\xi \\in H^{-\\frac{1}{2}}(\\mathbb {R}^3)$ .", "Summarising, $(\\widehat{T_\\lambda \\,\\xi })(p)$ is well-defined point-wise for almost every $p\\in \\mathbb {R}^3$ for $\\xi \\in H^{-\\frac{1}{2}+\\varepsilon }(\\mathbb {R}^3)$ , $\\varepsilon >0$ , whereas $(\\widehat{W_\\lambda \\,\\xi })(p)$ is so (at least) for $\\xi \\in H^{-1/2}(\\mathbb {R}^3)$ .", "The relevance of functions of the form $\\widehat{T_\\lambda \\,\\xi }$ and $\\widehat{W_\\lambda \\,\\eta }$ is due to the fact that they arise in the asymptotic behaviour of the elements of $\\mathcal {D}(\\mathring{H}^*)$ .", "Lemma Let $g$ be an arbitrary function in $\\mathcal {D}(\\mathring{H}^*)$ .", "For a fixed $\\lambda >0$ consider the decomposition of $\\widehat{g}$ in terms of $\\widehat{f}$ , $\\widehat{u}_\\xi $ , $\\widehat{u}_\\eta $ given by (REF ).", "Then, as $R\\rightarrow +\\infty $ , $\\begin{split}\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\widehat{g}(p_1,p_2) \\,\\mathrm {d}p_2}\\;&=\\;4\\pi \\widehat{\\xi }(p_1) R+\\big (-\\widehat{(T_\\lambda \\,\\xi )}(p_1)+\\!\\!\\begin{array}{c}\\frac{1}{2}\\end{array}\\!\\!\\!\\widehat{(W_\\lambda \\,\\eta )}(p_1)\\big )+o(1)\\end{split}$ as a point-wise identity for almost every $p_1$ .", "Remark For a generic $g\\in \\mathcal {D}(\\mathring{H}^*)$ , and correspondingly for a generic charge $\\xi \\in H^{-\\frac{1}{2}}(\\mathbb {R}^3)$ , the quantity in the l.h.s.", "of (REF ) is infinite for every finite $R$ because, as remarked after the definitions (REF )-(REF ), the quantity $(\\widehat{T_\\lambda \\,\\xi })(p)$ is in general infinite when $\\xi \\in H^{-\\frac{1}{2}}(\\mathbb {R}^3)$ .", "Instead, when additionally $\\xi \\in H^{-\\frac{1}{2}+\\varepsilon }(\\mathbb {R}^3)$ , with $\\varepsilon >0$ , the r.h.s.", "of (REF ) is finite (for almost every $p_1\\in \\mathbb {R}^3$ ): this case corresponds to a dense set of $g$ 's in $\\mathcal {D}(\\mathring{H}^*)$ , and for such $g$ 's the quantity in the l.h.s.", "of (REF ) is finite for finite $R$ and only diverges, linearly in $R$ , as $R\\rightarrow +\\infty $ .", "[Proof of Lemma REF ] Splitting $\\widehat{g}$ according to (REF ) yields $\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\widehat{g} \\,\\mathrm {d}p_2}\\;=\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\widehat{f} \\,\\mathrm {d}p_2}+\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\widehat{u}_{\\xi } \\,\\mathrm {d}p_2}+\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\frac{\\widehat{u}_{\\eta }}{p_1^2+p_2^2+\\mu p_1\\cdot p_2 +\\lambda } \\,\\mathrm {d}p_2}\\,.$ The first summand in the r.h.s.", "of (REF ) is a $o(1)$ -contribution, owing to Lemma REF (i).", "For the second summand, which is re-written as $\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}&{\\:\\widehat{u}_{\\xi }(p_1,p_2) \\,\\mathrm {d}p_2}\\;=\\\\&=\\;\\widehat{\\xi }(p_1)\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}\\;\\frac{\\mathrm {d}p_2}{p_1^2+p_2^2+\\mu p_1\\cdot p_2 +\\lambda }-\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}\\,{\\frac{\\widehat{\\xi }(p_2)}{p_1^2+p_2^2+\\mu p_1 \\cdot p_2+\\lambda }\\,\\mathrm {d}p_2}\\,,$ one finds $\\begin{split}\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}&\\;\\frac{\\mathrm {d}p_2}{p_1^2+p_2^2+\\mu p_1\\cdot p_2 +\\lambda }\\;=\\;2\\pi \\int _0^Rs^2\\,\\mathrm {d}s\\int _{-1}^1\\frac{\\mathrm {d}y}{p_1^2+s^2+\\mu |p_1|sy+\\lambda } \\\\&=\\;\\frac{2\\pi }{\\mu |p_1|}\\int _0^Rs\\ln \\frac{s^2+p_1^2+\\mu |p_1|s+\\lambda }{s^2+p_1^2-\\mu |p_1|s+\\lambda }\\,\\mathrm {d}s \\\\&=\\;2\\pi R\\,\\Big (1+\\frac{R}{2\\mu |p_1|}\\ln \\frac{R^2+p_1^2+\\mu |p_1|R+\\lambda }{R^2+p_1^2-\\mu |p_1| R+\\lambda }\\Big ) \\\\&\\qquad \\quad +2\\pi \\sqrt{\\nu p_1^2+\\lambda }\\,\\Big (\\!\\arctan \\frac{\\mu |p_1|-2R}{2\\sqrt{\\nu p_1^2+\\lambda }}-\\arctan \\frac{\\mu |p_1|+2R}{2\\sqrt{\\nu p_1^2+\\lambda }}\\,\\Big ) \\\\&\\qquad \\quad +\\pi \\frac{(4\\nu -2)p_1^2+\\lambda }{4\\sqrt{\\nu p_1^2+\\lambda }}\\,\\ln \\frac{R^2+p_1^2+\\mu |p_1|R+\\lambda }{R^2+p_1^2-\\mu |p_1| R+\\lambda } \\\\&=\\;4\\pi R-2\\pi ^2\\sqrt{\\nu p_1^2+\\lambda }+o(1)\\,,\\end{split}$ whence $\\begin{split}\\int _{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\begin{array}{c}\\\\ \\\\ \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\widehat{u}_{\\xi }(p_1,p_2) \\,\\mathrm {d}p_2}\\;&=\\;4\\pi \\,\\widehat{\\xi }(p_1)R-2\\pi ^2\\widehat{\\xi }(p_1)\\sqrt{\\nu p_1^2+\\lambda }-\\!\\int _{\\mathbb {R}^3}{\\frac{\\widehat{\\xi }(p_2)}{p_1^2+p_2^2+\\mu p_1 \\cdot p_2+\\lambda }\\mathrm {d}p_2}+o(1)\\,.\\end{split}$ Analogously, for the third summand in the r.h.s.", "of (REF ) one has $\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}\\,&{\\:\\frac{\\widehat{u}_{\\eta }(p_1,p_2)}{p_1^2+p_2^2+\\mu p_1 \\cdot p_2+\\lambda } \\,\\mathrm {d}p_2}\\;=\\\\&=\\;\\widehat{\\eta }(p_1)\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}\\;\\frac{\\mathrm {d}p_2}{(p_1^2+p_2^2+\\mu p_1\\cdot p_2 +\\lambda )^2}-\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}\\,{\\frac{\\widehat{\\eta }(p_2)}{(p_1^2+p_2^2+\\mu p_1 \\cdot p_2+\\lambda )^2}\\,\\mathrm {d}p_2}$ and $\\begin{split}\\int _{\\mathbb {R}^3}&\\;\\frac{\\mathrm {d}p_2}{(p_1^2+p_2^2+\\mu p_1\\cdot p_2 +\\lambda )^2}\\;=\\;2\\pi \\int _0^{+\\infty }\\!\\!s^2\\,\\mathrm {d}s\\int _{-1}^1\\frac{\\mathrm {d}y}{(p_1^2+s^2+\\mu |p_1|sy+\\lambda )^2} \\\\&=\\;\\int _0^{+\\infty }\\!\\!\\frac{4\\pi s^2}{(p_1^2+s^2+\\lambda )^2-\\mu ^2p_1^2 s^2}\\,\\mathrm {d}s\\;=\\;\\frac{\\pi ^2}{\\sqrt{\\nu p_1^2+\\lambda }}\\,,\\end{split}$ whence $\\int _{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\begin{array}{c}\\\\ \\\\ \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}\\!", "{\\frac{\\widehat{u}_{\\eta }(p_1,p_2)}{p_1^2+p_2^2+\\mu p_1 \\cdot p_2+\\lambda } \\,\\mathrm {d}p_2}\\;=\\frac{\\pi ^2\\,\\widehat{\\eta }(p_1)}{\\sqrt{\\nu p_1^2+\\lambda }}-\\int _{\\mathbb {R}^3}\\!", "{\\frac{\\widehat{\\eta }(p_2)}{(p_1^2+p_2^2+\\mu p_1 \\cdot p_2+\\lambda )^2}\\,\\mathrm {d}p_2}+o(1)\\,.$ These findings, re-written with the definitions (REF ) and (REF ), show that the r.h.s.", "of (REF ) is precisely given by formula (REF ).", "Thus, functions in $\\mathcal {D}(\\mathring{H}^*)$ display completely analogous asymptotics to the two-body model.", "It is important to observe, however, that whereas (REF ) was an identity between scalars, here (REF ) is a point-wise almost everywhere identity between functions.", "This is a crucial difference to keep into account when one imposes the Ter-Martirosyan–Skornyakov condition in such asymptotics.", "To elaborate on this point further at a later stage, let us also interpret $\\xi \\mapsto T_\\lambda \\xi $ and $\\xi \\mapsto W_\\lambda \\xi $ as maps between suitable functional spaces.", "Proposition Let $\\lambda >0$ .", "(i) For each $s\\geqslant 1$ the expression (REF ) defines a densely defined and symmetric operator $T_\\lambda :\\mathcal {D}(T_\\lambda )\\subset L^2(\\mathbb {R}^3)\\rightarrow L^2(\\mathbb {R}^3)$ with domain $\\mathcal {D}(T_\\lambda ):=H^s(\\mathbb {R}^3)$ .", "Moreover, $T_\\lambda $ maps continuously $H^s(\\mathbb {R}^3)$ into $H^{s-1}(\\mathbb {R}^3)$ for each $s\\in (-\\frac{1}{2},\\frac{3}{2})$ .", "Instead, $T_\\lambda H^{3/2}(\\mathbb {R}^3)\\nsubseteq H^{1/2}(\\mathbb {R}^3)$ .", "(ii) The expression (REF ) defines a bounded, positive, and invertible linear operator $W_\\lambda :H^{-1/2}(\\mathbb {R}^3)\\rightarrow H^{1/2}(\\mathbb {R}^3)$ , and for generic $u_\\xi ,u_\\eta \\in \\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ one has $\\langle u_\\xi ,u_\\eta \\rangle _{\\mathcal {H}}\\;=\\;\\langle \\xi ,W_\\lambda \\eta \\rangle _{H^{-\\frac{1}{2}}(\\mathbb {R}^3),H^{\\frac{1}{2}}(\\mathbb {R}^3)}\\,.$ Remark The choice of $L^2(\\mathbb {R}^3)$ as the Hilbert space where to study $T_\\lambda $ is made here for consistency with the previous literature [40], [26], [27], [43], [33], [34], [35], [37], [36], but it has no fundamental reason.", "As we shall discuss in Subsection REF below, what is intrinsically fundamental for the self-adjoint extension theory of $\\mathring{H}$ is the operator $W_\\lambda ^{-1}T_\\lambda $ on the Hilbert space $H^{-1/2}(\\mathbb {R}^3)$ .", "[Proof of Proposition REF ] (i) We re-write (REF ) as $T_\\lambda =L_\\lambda +Q_\\lambda $ , where $\\widehat{(L_\\lambda \\xi })(p)\\;:=\\;2\\pi ^2\\sqrt{\\nu p^2+\\lambda }\\;\\widehat{\\xi }(p)\\,,\\qquad \\widehat{(Q_\\lambda \\xi )}(p)\\;:=\\;\\int _{\\mathbb {R}^3}\\frac{\\widehat{\\xi }(q)}{p^2+q^2+\\mu p\\cdot q+\\lambda }\\,\\mathrm {d}q\\,.$ The symmetry of $T_\\lambda $ on $L^2(\\mathbb {R}^3)$ is obvious, since $L_\\lambda $ is the multiplication by a real function and $Q_\\lambda $ is an integral operator with real and symmetric kernel, and so too is the fact that $\\mathcal {D}(T_\\lambda )$ is dense in $L^2(\\mathbb {R}^3)$ .", "It is also clear that $\\Vert L_\\lambda \\xi \\Vert _{H^{s-1}}\\sim \\Vert \\xi \\Vert _{H^s}$ , thus it only remains to prove that $\\Vert Q_\\lambda \\xi \\Vert _{H^{s-1}}\\lesssim \\Vert \\xi \\Vert _{H^s}$ , i.e., $\\Vert (1+p^2)^{\\frac{s-1}{2}}\\widehat{(Q_\\lambda \\xi )}\\Vert _2\\lesssim \\Vert (1+p^2)^{\\frac{s}{2}}\\widehat{\\xi }\\Vert _2$ .", "In turn, setting $h(p):=(1+p^2)^{\\frac{s}{2}}\\widehat{\\xi }(p)$ , the last inequality is equivalent to $\\Vert \\widetilde{Q}_\\lambda h\\Vert _2\\lesssim \\Vert h\\Vert _2$ , where $(\\widetilde{Q}_\\lambda h)(p)\\;:=\\;\\int _{\\mathbb {R}^3}K_\\lambda (p,q)\\,h(q)\\,\\mathrm {d}q\\,,\\qquad K_\\lambda (p,q)\\;:=\\;\\frac{\\;(1+p^2)^{\\frac{s-1}{2}}}{(p^2+q^2+\\mu p\\cdot q+\\lambda )(1+q^2)^{\\frac{s}{2}}}\\,.$ It is easily verified (using (REF ) to introduce the $(p^2+q^2+1)$ -factors) that for the positive function $f(p)\\;:=\\;(1+p^2)^{-\\frac{3}{4}}$ one has $\\begin{split}\\int _{\\mathbb {R}^3}K_\\lambda (p,q)f(p)\\,\\mathrm {d}p\\;&\\lesssim \\;\\frac{1}{(1+q^2)^{\\frac{s}{2}}}\\int _{\\mathbb {R}^3}\\frac{\\mathrm {d}p}{(p^2+q^2+1)(1+p^2)^{\\frac{5}{4}-\\frac{s}{2}}} \\\\&\\lesssim \\;\\frac{1}{(1+q^2)^{\\frac{s}{2}}}\\,\\frac{1}{(1+q^2)^{\\frac{3}{4}-\\frac{s}{2}}}\\;=\\;f(q)\\,,\\qquad {\\textstyle s\\in (-\\frac{1}{2},\\frac{3}{2})}\\,,\\end{split}\\qquad \\mathrm {(*)}$ and $\\begin{split}\\int _{\\mathbb {R}^3}K_\\lambda (p,q)f(q)\\,\\mathrm {d}q\\;&\\lesssim \\;(1+p^2)^{\\frac{s-1}{2}}\\int _{\\mathbb {R}^3}\\frac{\\mathrm {d}q}{(p^2+q^2+1)(1+q^2)^{\\frac{3}{4}+\\frac{s}{2}}} \\\\&\\lesssim \\;(1+p^2)^{\\frac{s-1}{2}}\\,\\frac{1}{(1+p^2)^{\\frac{1}{4}+\\frac{s}{2}}}\\;=\\;f(p)\\,,\\qquad \\qquad {\\textstyle s\\in (-\\frac{1}{2},\\frac{3}{2})}\\,.\\end{split}\\qquad \\mathrm {(**)}$ A standard Schur test based on (*) and (**) implies $\\Vert \\widetilde{Q}_\\lambda h\\Vert _2\\lesssim \\Vert h\\Vert _2$ and hence $\\Vert Q_\\lambda \\xi \\Vert _{H^{s-1}}\\lesssim \\Vert \\xi \\Vert _{H^s}$ for $s\\in (-\\frac{1}{2},\\frac{3}{2})$ and arbitrary $\\xi \\in H^{s}(\\mathbb {R}^3)$ .", "The function $\\widehat{\\xi }_0:=\\mathbf {1}_{\\lbrace |p|\\leqslant 1\\rbrace }$ is a counter-example showing that the same bound cannot hold for $s\\geqslant \\frac{3}{2}$ : indeed, clearly $\\xi _0\\in H^{s}(\\mathbb {R}^3)$ for arbitrary $s\\in \\mathbb {R}$ , but $\\widehat{(Q_\\lambda \\xi _0)}(p)\\;\\sim \\;\\int _{\\begin{array}{c} \\\\ \\,q\\in \\mathbb {R}^3 \\\\ \\!", "|q|\\leqslant 1\\end{array}}\\frac{\\mathrm {d}q}{p^2+q^2+1}\\;\\sim \\;\\frac{1}{\\;(1+p^2)}$ and hence $Q_\\lambda \\xi _0\\notin H^{1/2}(\\mathbb {R}^3)$ .", "(ii) For arbitrary $\\xi ,\\eta \\in H^{-1/2}(\\mathbb {R}^3)$ we compute $\\begin{split}\\langle u_{\\xi }, u_{\\eta }\\rangle _\\mathcal {H}\\;&=\\;\\int _{\\mathbb {R}^3\\times \\mathbb {R}^3}{\\frac{\\overline{\\,\\widehat{\\xi }(p_1)}-\\overline{\\,\\widehat{\\xi }(p_2)}}{p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda }\\:\\frac{\\widehat{\\eta }(p_1)-\\widehat{\\eta }(p_2)}{p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda }\\,\\mathrm {d}p_1\\mathrm {d}p_2} \\\\&=\\;2\\int _{\\mathbb {R}^3\\times \\mathbb {R}^3}{\\frac{\\overline{\\,\\widehat{\\xi }(p_1)}\\,\\widehat{\\eta }(p_1)}{(p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )^2}-\\frac{\\overline{\\,\\widehat{\\xi }(p_1)}\\,\\widehat{\\eta }(p_2)}{(p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )^2}\\,\\mathrm {d}p_1 \\mathrm {d}p_2} \\\\&=\\;2\\int _{\\mathbb {R}^3}{\\overline{\\,\\widehat{\\xi }(p_1)}\\,\\Big ( \\frac{\\pi ^2}{\\sqrt{\\nu p_1^2+\\lambda }}\\,\\widehat{\\eta }(p_1)-\\int _{\\mathbb {R}^3}{\\frac{\\widehat{\\eta }(p_2)}{(p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )^2}\\,\\mathrm {d}p_2} \\Big )\\mathrm {d}p_1} \\\\&=\\;\\int _{\\mathbb {R}^3}\\overline{\\widehat{\\xi }(p)}\\,\\widehat{(W_\\lambda \\xi )}(p)\\,\\mathrm {d}p\\,,\\end{split}$ where we used the symmetry under exchange $p_1\\leftrightarrow p_2$ in the second step, (REF ) in the third step, and (REF ) in the last step.", "Therefore, $\\begin{split}\\Vert W_\\lambda \\eta \\Vert _{H^{1/2}}\\;&=\\;\\sup _{\\Vert \\xi \\Vert _{H^{-1/2}=1}}\\Big |\\int _{\\mathbb {R}^3}\\overline{\\widehat{\\xi }(p)}\\,\\widehat{(W_\\lambda \\eta )}(p)\\,\\mathrm {d}p\\,\\Big |\\;=\\;\\sup _{\\Vert \\xi \\Vert _{H^{-1/2}=1}}\\big |\\langle u_\\xi ,u_\\eta \\rangle _{\\mathcal {H}}\\big | \\\\&\\leqslant \\;\\sup _{\\Vert \\xi \\Vert _{H^{-1/2}=1}}\\Vert u_\\xi \\Vert _{\\mathcal {H}}\\Vert u_\\eta \\Vert _{\\mathcal {H}}\\;\\leqslant \\;\\textrm {const}\\cdot \\Vert \\eta \\Vert _{H^{-1/2}}\\qquad \\forall \\eta \\in H^{-1/2}(\\mathbb {R}^3)\\,,\\end{split}$ where we used (REF ) in the last step, which shows that $W_\\lambda H^{-1/2}(\\mathbb {R}^3)\\subset H^{1/2}(\\mathbb {R}^3)$ , that the map $W_\\lambda :H^{-1/2}(\\mathbb {R}^3)\\rightarrow H^{1/2}(\\mathbb {R}^3)$ is bounded, and that (REF ) holds true.", "Owing to (REF ), one has $\\langle \\eta ,W_\\lambda \\eta \\rangle _{H^{-1/2},H^{1/2}}=\\Vert u_\\eta \\Vert _{\\mathcal {H}}^2\\geqslant 0$ , thus $W_\\lambda $ is positive.", "Furthermore, the following chain of implications holds: $W_\\lambda \\eta =0$ $\\Rightarrow $ $\\langle u_\\xi ,u_\\eta \\rangle _{\\mathcal {H}}=0$ $\\forall u_\\xi \\in \\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ $\\Rightarrow $ $u_\\eta =0$ $\\Rightarrow $ $\\eta =0$ , where we used (REF ) in the first implication and (REF ) in the last one; this proves that $W_\\lambda $ is injective and hence invertible on its range.", "For a generic $\\xi \\in H^{-1/2}(\\mathbb {R}^3)$ one has this chain of implications: $\\langle \\xi ,\\phi \\rangle _{H^{-1/2},H^{1/2}}=0$ $\\forall \\phi =W_\\lambda \\eta \\in \\mathrm {ran}\\,W_\\lambda $ $\\Rightarrow $ $\\langle u_\\xi ,u_\\eta \\rangle _{\\mathcal {H}}=0$ $\\forall u_\\eta \\in \\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ $\\Rightarrow $ $u_\\xi =0$ $\\Rightarrow $ $\\xi =0$ (again using (REF ) in the first implication and (REF ) in the last one), hence by duality $\\mathrm {ran}\\,W_\\lambda $ must be dense in $H^{1/2}(\\mathbb {R}^3)$ .", "Since $W_\\lambda $ is bounded, then $\\mathrm {ran}\\,W_\\lambda $ is also closed in $H^{1/2}(\\mathbb {R}^3)$ , thus $W_\\lambda $ is an invertible bijection $H^{-1/2}(\\mathbb {R}^3)\\rightarrow H^{1/2}(\\mathbb {R}^3)$ .", "As an immediate consequence of Proposition REF , $\\langle \\xi ,\\eta \\rangle _{W_\\lambda }\\;:=\\;\\langle \\xi ,W_\\lambda \\,\\eta \\rangle _{H^{-\\frac{1}{2}},H^{\\frac{1}{2}}}\\;=\\;\\langle u_\\xi ,u_\\eta \\rangle _{\\mathcal {H}}$ defines a scalar product in $H^{-\\frac{1}{2}}(\\mathbb {R}^3)$ .", "It is equivalent to the standard scalar product of $H^{-\\frac{1}{2}}(\\mathbb {R}^3)$ , as follows by combining (REF ) with (REF ).", "We shall denote by $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ the Hilbert space consisting of the $H^{-\\frac{1}{2}}(\\mathbb {R}^3)$ -functions and equipped with the scalar product $\\langle \\cdot ,\\cdot \\rangle _{W_\\lambda }$ .", "Then the map $\\begin{split}U_\\lambda \\,:\\,\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})\\;&\\;\\xrightarrow{}\\;H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)\\,,\\qquad u_\\xi \\longmapsto \\,\\xi \\end{split}$ is an isomorphism between Hilbert spaces, with $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ equipped with the standard scalar product inherited from $\\mathcal {H}$ .", "One can therefore equivalently parametrise the self-adjoint extensions of $\\mathring{H}$ in terms of self-adjoint operators acting on Hilbert subspaces of $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ or of $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ .", "The whole family of such extensions is given by the Kreĭn-Višik-Birman theory through the classification of Theorem ." ], [ "Ter-Martirosyan–Skornyakov Hamiltonians of $2+1$ point interaction", "The previous analysis brings us now to the class of operators on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ (or, also, operators on $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ ) which identify those self-adjoint extensions of $\\mathring{H}$ of Ter-Martirosyan–Skornyakov type.", "As it will emerge in the following, the crucial point is the possibility of reducing $T_\\lambda $ to a $L^2$ -closed invariant subspace with values into $H^{1/2}(\\mathbb {R}^3)$ and to define on the orthogonal complement another $H^{1/2}$ -valued symmetric operator $S_0$ .", "Given the resulting $\\mathcal {T}_\\lambda =S_0\\oplus T_\\lambda $ one has then to investigate the self-adjointness of $W_\\lambda ^{-1}\\mathcal {T}_\\lambda $ on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ .", "We observe that this is related to, but it is not the same question of the self-adjointness of $T_\\lambda $ on $L^2(\\mathbb {R}^3)$ .", "The study of the self-adjoint extensions of $T_\\lambda $ , as a densely defined and symmetric operator on $L^2(\\mathbb {R}^3)$ , has been carried on systematically in a series of works by Minlos and Shermatov [40], Melnikov and Minlos [26], [27], Shermatov [43], and Minlos [33], [34], [35], [37], [36].", "In the additional work [12] by one of us and co-workers, the Friedrichs extension of $T_\\lambda $ was studied (in those regime of masses $m$ in which $T_\\lambda $ itself is semi-bounded below).", "It is relevant to remark that in all those works $\\mathcal {D}(T_\\lambda )$ was fixed in spaces of various regularity, at least $H^1(\\mathbb {R}^3)$ .", "We recall from Proposition REF (i) that $T_\\lambda H^1(\\mathbb {R}^3)\\subset L^2(\\mathbb {R}^3)$ because both its multiplicative part $L_\\lambda $ and its integral part $Q_\\lambda $ map separately $H^1(\\mathbb {R}^3)$ into $L^2(\\mathbb {R}^3)$ : as discussed in [12], in the domain of a self-adjoint extension $\\widetilde{T}_\\lambda $ of $T_\\lambda $ there are elements $\\xi $ for which neither $L_\\lambda \\xi $ nor $Q_\\lambda \\xi $ is square-integrable, but their difference is, due to a cancellation of singularities in $L_\\lambda \\xi +Q_\\lambda \\xi $ .", "More precisely, $T_\\lambda $ commutes with the rotations in $\\mathbb {R}^3$ and, with respect to the canonical decomposition $L^2(\\mathbb {R}^3)\\;\\cong \\;\\bigoplus _{\\ell =0}^\\infty L^2(\\mathbb {R}^+,r^2\\,\\mathrm {d}r)\\otimes \\mathrm {span}\\lbrace Y_{\\ell ,-\\ell },\\dots ,Y_{\\ell ,\\ell }\\rbrace \\;\\equiv \\;\\bigoplus _{\\ell =0}^\\infty L^2_\\ell (\\mathbb {R}^3)$ (where the $Y_{\\ell ,m}$ 's are the spherical harmonics on $\\mathbb {S}^2$ ), $T_\\lambda $ leaves each $L^2_\\ell (\\mathbb {R}^3)$ invariant and is densely defined and symmetric on $L^2_\\ell (\\mathbb {R}^3)$ , thus $T_\\lambda \\;=\\;\\bigoplus _{\\ell =0}^\\infty \\,T_\\lambda ^{(\\ell )}\\qquad \\qquad T_\\lambda ^{(\\ell )}\\,\\textrm { symmetric on } L^2_\\ell (\\mathbb {R}^3)$ and $T_\\lambda ^{(\\ell )}\\;=\\;\\mathcal {T}_{\\lambda }^{(\\ell )}\\otimes \\mathbb {1}\\qquad \\textrm { on }\\qquad L^2_\\ell (\\mathbb {R}^3)\\;\\cong \\;L^2(\\mathbb {R}^+,r^2\\,\\mathrm {d}r)\\otimes \\mathrm {span}\\lbrace Y_{\\ell ,-\\ell },\\dots ,Y_{\\ell ,\\ell }\\rbrace \\,.$ Therefore, the study of the self-adjointness or of the self-adjoint extensions of $T_\\lambda $ boils down to the same study for each $T_\\lambda ^{(\\ell )}$ .", "It is today well-known from the works cited above that for even $\\ell $ 's $T_\\lambda ^{(\\ell )}$ is self-adjoint on $L^2_\\ell (\\mathbb {R}^3)$ , while for odd $\\ell $ 's there exist masses $m_1>m_3>m_5>\\cdots $ such that $T_\\lambda ^{(\\ell )}$ is self-adjoint on $L^2_\\ell (\\mathbb {R}^3)$ for $m>m_\\ell $ and it has instead a one-parameter family of self-adjoint extensions for $m\\in (0,m_\\ell ]$ .", "As seen in the proof of Proposition REF (i), $T_\\lambda $ fails to map $H^{3/2}(\\mathbb {R}^3)$ into $H^{1/2}(\\mathbb {R}^3)$ and the counter-example considered therein was a function in $L^2_{\\ell =0}(\\mathbb {R}^3)\\cap H^{3/2}(\\mathbb {R}^3)$ .", "In fact, that failure is exceptional and it does not occur for $H^{3/2}$ -functions with a sufficient amount of oscillations, as the following Proposition shows.", "Proposition For each $\\ell \\geqslant 1$ , and in terms of the notation of (REF )-(REF ), $\\Vert T_\\lambda \\xi \\Vert _{H^{1/2}}\\;\\lesssim \\Vert \\xi \\Vert _{H^{3/2}}\\qquad \\forall \\xi \\in H^{3/2}(\\mathbb {R}^3)\\cap L^2_\\ell (\\mathbb {R}^3)\\,,$ whence, in particular, $T_\\lambda ^{(\\ell )}\\big (H^{3/2}(\\mathbb {R}^3)\\cap L^2_\\ell (\\mathbb {R}^3)\\big )\\;\\subset \\; \\big (H^{1/2}(\\mathbb {R}^3)\\cap L^2_\\ell (\\mathbb {R}^3)\\big )\\,.$ As discussed already in the proof of Proposition REF (i), we only need to prove the statement for $Q_\\lambda $ , the integral part of $T_\\lambda $ – see (REF ).", "Analogously to (REF )-(REF ), $Q_\\lambda \\;=\\;\\bigoplus _{\\ell =0}^\\infty \\,Q_\\lambda ^{(\\ell )}\\,,\\qquad Q_\\lambda ^{(\\ell )}\\;=\\;\\mathcal {Q}_{\\lambda }^{(\\ell )}\\otimes \\mathbb {1}\\quad \\textrm { on }\\quad L^2_\\ell (\\mathbb {R}^3)\\,,$ where $\\mathcal {Q}_{\\lambda }^{(\\ell )}$ acts symmetrically on $L^2(\\mathbb {R}^+,r^2\\,\\mathrm {d}r)$ .", "It is standard to derive from (REF ) and (REF ) that $(\\mathcal {Q}_{\\lambda }^{(\\ell )}f)(r)\\;=\\;2\\pi \\int _{-1}^{+1}\\!\\mathrm {d}y P_\\ell (y)\\!\\int _0^{+\\infty }\\!\\!\\frac{f(r^{\\prime })}{r^2+r^{\\prime 2}+\\mu r r^{\\prime } y +\\lambda }\\,r^{\\prime 2}\\mathrm {d}r^{\\prime }\\,,$ where $P_\\ell (y)\\;=\\;\\frac{1}{2^\\ell \\ell !", "}\\,\\frac{\\mathrm {d}^\\ell }{\\mathrm {d}y^\\ell }(y^2-1)^\\ell $ is the $\\ell $ -th Legendre polynomial.", "Thus, proving (REF ) is equivalent to proving $\\Vert (1+r^2)^{\\frac{1}{4}}(\\mathcal {Q}_{\\lambda }^{(\\ell )}f)\\Vert _{ L^2(\\mathbb {R}^+,r^2\\,\\mathrm {d}r)}\\;\\lesssim \\;\\Vert (1+r^2)^{\\frac{3}{4}}f\\Vert _{ L^2(\\mathbb {R}^+,r^2\\,\\mathrm {d}r)}\\,,$ which is in turn equivalent to the boundedness in $L^2(\\mathbb {R}^+,\\mathrm {d}r)$ of the integral operator $h\\mapsto \\widetilde{\\mathcal {Q}}_{\\lambda }^{(\\ell )} h$ defined by $(\\widetilde{\\mathcal {Q}}_{\\lambda }^{(\\ell )}h)(r)\\;:=\\;\\int _{-1}^{+1}\\!\\mathrm {d}y P_\\ell (y)\\!\\int _0^{+\\infty }\\!\\!\\frac{rr^{\\prime }\\,(1+r^2)^{\\frac{1}{4}}\\,h(r^{\\prime })}{(r^2+r^{\\prime 2}+\\mu r r^{\\prime } y +\\lambda )(1+r^{\\prime 2})^{\\frac{3}{4}}}\\,\\mathrm {d}r^{\\prime }\\,.$ Using (REF ) and integrating by parts $\\ell \\geqslant 1$ times in $y$ yields $\\begin{split}(\\widetilde{\\mathcal {Q}}_{\\lambda }^{(\\ell )}h)(r)\\;=\\;\\frac{(-1)^\\ell }{2^\\ell \\ell !", "}\\int _0^{+\\infty }\\!\\!\\!\\mathrm {d}r^{\\prime }\\,\\frac{rr^{\\prime }\\,(1+r^2)^{\\frac{1}{4}}h(r^{\\prime })}{(1+r^{\\prime 2})^{\\frac{3}{4}}}\\int _{-1}^{+1}\\!\\mathrm {d}y\\,\\frac{(y^2-1)^\\ell (\\mu r r^{\\prime })^\\ell }{(r^2+r^{\\prime 2}+\\mu r r^{\\prime } y +\\lambda )^{\\ell +1}}\\,.\\end{split}$ Since $|y|\\leqslant 1$ , analogously to (REF ) $(r^2+r^{\\prime 2}+\\mu r r^{\\prime } y +\\lambda )\\;\\sim \\;(r_1^2+r_2^2+1)\\;\\geqslant 0\\,.$ Then $\\begin{split}|(\\widetilde{\\mathcal {Q}}_{\\lambda }^{(\\ell )}h)(r)|\\;&\\lesssim \\;\\int _0^{+\\infty }\\!\\!\\!\\mathrm {d}r^{\\prime }\\,\\frac{rr^{\\prime }\\,(1+r^2)^{\\frac{1}{4}}\\,|h(r^{\\prime })|}{(1+r^{\\prime 2})^{\\frac{3}{4}}}\\int _{-1}^{+1}\\!\\mathrm {d}y\\,\\frac{(\\mu r r^{\\prime })^\\ell }{(r^2+r^{\\prime 2}+\\mu r r^{\\prime } y +\\lambda )^{\\ell +1}} \\\\&=\\;\\int _0^{+\\infty }\\!\\!\\!\\mathrm {d}r^{\\prime }\\,\\frac{rr^{\\prime }\\,(1+r^2)^{\\frac{1}{4}}\\,|h(r^{\\prime })|}{\\ell \\,(1+r^{\\prime 2})^{\\frac{3}{4}}}\\,(\\mu r r^{\\prime })^{\\ell -1}\\,\\times \\\\&\\qquad \\qquad \\qquad \\times \\Big (\\frac{1}{(r^2+r^{\\prime 2}-\\mu r r^{\\prime } y +\\lambda )^{\\ell }}-\\frac{1}{(r^2+r^{\\prime 2}+\\mu r r^{\\prime } y +\\lambda )^{\\ell }}\\Big ) \\\\&\\lesssim \\;\\int _0^{+\\infty }\\!\\!\\!\\mathrm {d}r^{\\prime }\\,\\frac{rr^{\\prime }\\,(1+r^2)^{\\frac{1}{4}}\\,|h(r^{\\prime })|}{(1+r^{\\prime 2})^{\\frac{3}{4}}}\\,(\\mu r r^{\\prime })^{\\ell }\\,\\times \\\\&\\qquad \\qquad \\qquad \\times \\,\\frac{(r^2+r^{\\prime 2}+1)^{\\ell -1}}{(r^2+r^{\\prime 2}-\\mu r r^{\\prime } y +\\lambda )^{\\ell }(r^2+r^{\\prime 2}+\\mu r r^{\\prime } y +\\lambda )^{\\ell }} \\\\&\\lesssim \\;\\int _0^{+\\infty }\\!\\!\\!\\mathrm {d}r^{\\prime }\\,\\frac{(rr^{\\prime })^{\\ell +1}(1+r^2)^{\\frac{1}{4}}}{(1+r^{\\prime 2})^{\\frac{3}{4}}(r^2+r^{\\prime 2}+1)^{\\ell +1}}\\,|h(r^{\\prime })|\\;\\equiv \\;\\int _0^{+\\infty }\\!\\!\\mathcal {K}_\\lambda ^{(\\ell )}(r,r^{\\prime })\\,|h(r^{\\prime })|\\,\\mathrm {d}r^{\\prime }\\,,\\end{split}$ where we used (REF ) in the first and last step, and the formula $(a^\\ell -b^\\ell )=(a-b)\\sum _{j=0}^{n-1}a^{n-j-1}b^{\\,j}$ ($a,b\\geqslant 0$ ) in the third step.", "From $\\int _0^{+\\infty }\\!\\!\\frac{r^{\\ell +1}(1+r^2)^{\\frac{1}{4}}}{(r^2+r^{\\prime 2}+1)^{\\ell +1}}\\,\\mathrm {d}r\\;\\lesssim \\;(1+r^{\\prime 2})^{-\\frac{2\\ell -1}{4}}$ we deduce $\\sup _{r^{\\prime }>0}\\int _0^{+\\infty }\\!\\!\\mathcal {K}_\\lambda ^{(\\ell )}(r,r^{\\prime })\\,\\mathrm {d}r\\;=\\;\\sup _{r^{\\prime }>0}\\frac{r^{\\prime \\ell +1}}{(1+r^{\\prime 2})^{\\frac{3}{4}}}\\int _0^{+\\infty }\\!\\!\\frac{r^{\\ell +1}(1+r^2)^{\\frac{1}{4}}}{(r^2+r^{\\prime 2}+1)^{\\ell +1}}\\,\\mathrm {d}r\\;\\lesssim \\;1\\,,\\qquad \\mathrm {(*)}$ and from $\\int _0^{+\\infty }\\!\\!\\frac{r^{\\prime \\ell +1}}{(1+r^{\\prime 2})^{\\frac{3}{4}}(r^2+r^{\\prime 2}+1)^{\\ell +1}}\\,\\mathrm {d}r^{\\prime }\\;\\lesssim \\;(1+r^2)^{-\\frac{2\\ell +3}{4}}$ we deduce $\\begin{split}\\sup _{r>0}\\int _0^{+\\infty }\\!\\!&\\mathcal {K}_\\lambda ^{(\\ell )}(r,r^{\\prime })\\,\\mathrm {d}r^{\\prime }\\;= \\\\&=\\;\\sup _{r>0}\\:r^{\\ell +1}(1+r^2)^{\\frac{1}{4}}\\!\\int _0^{+\\infty }\\!\\!\\frac{r^{\\prime \\ell +1}}{(1+r^{\\prime 2})^{\\frac{3}{4}}(r^2+r^{\\prime 2}+1)^{\\ell +1}}\\,\\mathrm {d}r^{\\prime }\\;\\lesssim \\;1\\,.\\end{split}\\qquad \\mathrm {(**)}$ A standard Schur test based on (*) and (**) implies $\\Vert \\widetilde{\\mathcal {Q}}_{\\lambda }^{(\\ell )}h\\Vert _2\\lesssim \\Vert h\\Vert _2$ , thus concluding the proof.", "An immediate consequence of Propositions REF (i) and REF is the following.", "Corollary Let $T_\\lambda ^+$ be the operator acting as $T_\\lambda $ on the Hilbert space $L^2_{+}(\\mathbb {R}^3):=\\bigoplus _{\\ell =1}^\\infty L^2_{\\ell }(\\mathbb {R}^3)$ with domain $\\mathcal {D}(T_\\lambda ^+):=H^{3/2}(\\mathbb {R}^3)\\cap L^2_{+}(\\mathbb {R}^3)$ .", "Then $T_\\lambda ^+$ is densely defined and symmetric, and it maps continuously $H^{3/2}(\\mathbb {R}^3)\\cap L^2_{+}(\\mathbb {R}^3)$ (with the $H^{\\frac{3}{2}}$ -norm) into $H^{1/2}(\\mathbb {R}^3)\\cap L^2_{+}(\\mathbb {R}^3)$ (with the $H^{\\frac{1}{2}}$ -norm).", "In turn, the Corollary above, together with Proposition, show that although in general $T_\\lambda H^{3/2}(\\mathbb {R}^3)\\;\\;H^{1/2}(\\mathbb {R}^3)\\,,$ nevertheless the map $W_\\lambda ^{-1}T_\\lambda $ can be defined on parts of $H^{-1/2}(\\mathbb {R}^3)$ with values in itself.", "Elaborating further, we obtain the following result.", "Proposition The following be given: two constants $\\lambda >0$ and $\\alpha \\in \\mathbb {R}$ , the densely defined and symmetric operator $T_\\lambda ^+:=\\bigoplus _{\\ell =1}^\\infty \\,T_\\lambda ^{(\\ell )}$ on the Hilbert space $L^2_{+}(\\mathbb {R}^3):=\\bigoplus _{\\ell =1}^\\infty L^2_{\\ell }(\\mathbb {R}^3)$ which acts as $T_\\lambda $ with domain $\\mathcal {D}(T_\\lambda ^+):=H^{3/2}(\\mathbb {R}^3)\\cap L^2_{+}(\\mathbb {R}^3)$ , and a densely defined and symmetric operator $S_0$ on the Hilbert space $L^2_{\\ell =0}(\\mathbb {R}^3)$ with $\\mathrm {ran}S_0\\subset H^{1/2}(\\mathbb {R}^3)$ .", "With respect to the decomposition $L^2(\\mathbb {R}^3)\\cong L^2_{\\ell =0}(\\mathbb {R}^3)\\oplus L^2_+(\\mathbb {R}^3)$ , let $\\mathcal {T}_\\lambda \\;:=\\;S_0\\oplus T^+_\\lambda \\,.$ Then $\\mathcal {T}_\\lambda $ is a densely defined and symmetric operator on $L^2(\\mathbb {R}^3)$ and $\\mathcal {A}_{\\lambda ,\\alpha }\\;:=\\;2\\,W_\\lambda ^{-1}(\\mathcal {T}_\\lambda +\\alpha \\mathbb {1})\\,,\\qquad \\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha })\\;:=\\;\\mathcal {D}(\\mathcal {T}_\\lambda )$ is a densely defined and symmetric operator on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ .", "Moreover, if $\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha }$ is a self-adjoint extension of $\\mathcal {A}_{\\lambda ,\\alpha }$ on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ , then $A_{\\lambda ,\\alpha }\\;:=\\;U_\\lambda ^{-1}\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha } U_\\lambda $ (where $U_\\lambda $ is the isomorphism (REF )) is a self-adjoint operator on $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ .", "The statements for $\\mathcal {T}_\\lambda $ and $A_{\\lambda ,\\alpha }$ are obvious, and so too is the density of $\\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha })$ in $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ .", "The symmetry of $\\mathcal {A}_{\\lambda ,\\alpha }$ follows from the identity, valid for $\\eta ,\\xi \\in \\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha })=\\mathcal {D}(\\mathcal {T}_\\lambda )$ , $\\begin{split}\\!\\!\\begin{array}{c}\\frac{1}{2}\\end{array}\\!\\!\\!\\langle \\eta ,\\mathcal {A}_{\\lambda ,\\alpha }\\xi \\rangle _{W_\\lambda }\\;&=\\;\\langle \\eta ,W_\\lambda ^{-1}(\\mathcal {T}_\\lambda \\xi +\\alpha \\xi )\\rangle _{W_\\lambda } \\;=\\;\\langle \\eta ,(\\mathcal {T}_\\lambda \\xi +\\alpha \\xi )\\rangle _{L^2} \\\\&=\\;\\langle (\\mathcal {T}_\\lambda \\eta +\\alpha \\eta ),\\xi \\rangle _{L^2}\\;=\\;\\langle W_\\lambda W_\\lambda ^{-1}(\\mathcal {T}_\\lambda \\eta +\\alpha \\eta ),\\xi \\rangle _{H^{\\frac{1}{2}},H^{-\\frac{1}{2}}} \\\\&=\\;\\langle W_\\lambda ^{-1}(\\mathcal {T}_\\lambda \\eta +\\alpha \\eta ),W_\\lambda \\xi \\rangle _{H^{-\\frac{1}{2}},H^{\\frac{1}{2}}}\\;=\\;\\!\\!\\begin{array}{c}\\frac{1}{2}\\end{array}\\!\\!\\!\\langle \\mathcal {A}_{\\lambda ,\\alpha }\\eta ,\\xi \\rangle _{W_\\lambda }\\,,\\end{split}$ where we used the symmetry of $\\mathcal {T}_\\lambda $ in $L^2(\\mathbb {R}^3)$ , the fact that $\\alpha $ is real, and the properties of $W_\\lambda $ discussed in Proposition REF (ii).", "Remark If, in addition to the assumptions of Proposition REF above, one assumes also that $T_\\lambda ^+$ and $S_0$ are self-adjoint on their respective Hilbert spaces and hence $\\mathcal {T}_\\lambda $ is self-adjoint on $L^2(\\mathbb {R}^3)$ , then $\\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha }^*)\\cap L^2(\\mathbb {R}^3)\\;=\\;\\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha })\\qquad \\qquad (\\mathcal {T}_\\lambda =\\mathcal {T}_\\lambda ^*)\\,.$ To see this, we use the fact that for generic $\\chi \\in \\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha }^*)$ there exists $c_\\chi >0$ such that $\\big |\\langle \\chi ,\\mathcal {A}_{\\lambda ,\\alpha }\\xi \\rangle _{W_\\lambda }\\big |\\;\\leqslant \\;c_\\chi \\,\\Vert \\xi \\Vert _{W_\\lambda }\\qquad \\forall \\xi \\in \\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha })=\\mathcal {D}(\\mathcal {T}_\\lambda )\\,.\\qquad \\mathrm {(*)}$ Owing to (REF ), (REF ), and (REF ), $\\Vert \\xi \\Vert _{W_\\lambda }\\;=\\;\\Vert u_\\xi \\Vert _\\mathcal {H}\\;\\sim \\;\\Vert \\xi \\Vert _{H^{-1/2}}\\;\\leqslant \\;\\Vert \\xi \\Vert _{L^2}\\,,$ whereas, for $\\chi \\in \\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha }^*)\\cap L^2(\\mathbb {R}^3)$ , $\\langle \\chi ,\\mathcal {A}_{\\lambda ,\\alpha }\\xi \\rangle _{W_\\lambda }\\;=\\;2\\,\\langle \\chi ,(\\mathcal {T}_\\lambda \\xi +\\alpha \\xi )\\rangle _{L^2}\\,.$ Therefore, (*) reads $\\big |\\langle \\chi ,(\\mathcal {T}_\\lambda \\xi +\\alpha \\xi )\\rangle _{L^2}\\big |\\;\\lesssim \\;\\Vert \\xi \\Vert _{L^2}\\qquad \\forall \\xi \\in \\mathcal {D}(\\widetilde{\\mathcal {T}}_\\lambda )$ or also (using $|\\langle \\chi ,\\xi \\rangle _{L^2}|\\leqslant \\Vert \\chi \\Vert _{L^2}\\Vert \\xi \\Vert _{L^2}$ ) $\\big |\\langle \\chi ,\\mathcal {T}_\\lambda \\xi \\rangle _{L^2}\\big |\\;\\lesssim \\;\\Vert \\xi \\Vert _{L^2}\\qquad \\forall \\xi \\in \\mathcal {D}(\\widetilde{\\mathcal {T}}_\\lambda )\\,.$ Since $\\mathcal {T}_\\lambda $ is self-adjoint on $L^2(\\mathbb {R}^3)$ , the last bound implies $\\chi \\in \\mathcal {D}(\\mathcal {T}^*_\\lambda )=\\mathcal {D}(\\mathcal {T}_\\lambda )=\\mathcal {D}(\\mathcal {A}_{\\lambda ,\\alpha })$ , whence the conclusion.", "We observe, however, that this argument and the conclusion (REF ) are not enough to claim that the self-adjointness of $\\mathcal {T}_\\lambda $ implies the self-adjointness of $\\mathcal {A}_{\\lambda ,\\alpha }$ : the latter could still have a larger adjoint and admit self-adjoint extensions.", "We thus see that each self-adjoint extension $\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha }$ of $\\mathcal {A}_{\\lambda ,\\alpha }$ on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ identifies one self-adjoint extension of $\\mathring{H}$ by means of its unitarily equivalent version $A_{\\lambda ,\\alpha }=U_\\lambda ^{-1}\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha } U_\\lambda $ .", "This extension, call it $\\mathring{H}_{\\langle \\alpha \\rangle }$ , as prescribed by Theorem (namely the Kreĭn-Višik-Birman extension theory) is the restriction of $\\mathring{H}^*$ to the domain $\\mathcal {D}(\\mathring{H}_{\\langle \\alpha \\rangle })\\;:=\\;\\left\\lbrace g=f+(\\mathring{H}_F+\\lambda \\mathbb {1})^{-1}(A_{\\lambda ,\\alpha }u_\\xi )+u_\\xi \\left|\\!\\begin{array}{c}f\\in \\mathcal {D}(\\mathring{H}) \\\\u_\\xi \\in \\mathcal {D}(A_{\\lambda ,\\alpha })\\end{array}\\!\\!\\!\\right.\\right\\rbrace \\,.$ Indeed, in comparison with the general formula (REF ), $\\mathcal {D}(A_{\\lambda ,\\alpha })$ has a trivial orthogonal complement in $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ .", "For each $g\\in \\mathcal {D}(\\mathring{H}_{\\langle \\alpha \\rangle })$ one deduces from (REF ) that $((\\mathring{H}_{\\langle \\alpha \\rangle }+\\lambda \\mathbb {1})g)^{\\widehat{\\;}}(p_1,p_2)\\;=\\;(p_1^2+p_2^2+\\mu p_1\\cdot p_2+\\lambda )\\widehat{f}(p_1,p_2)+\\widehat{(A_{\\lambda ,\\alpha }u_\\xi )}(p_1,p_2)\\,.$ A comparison between (REF ) and (REF ) shows that $\\mathcal {D}(\\mathring{H}_{\\langle \\alpha \\rangle })$ is obtained as a restriction $\\mathcal {D}(\\mathring{H}^*)$ by imposing the condition $u_\\eta \\;=\\;A_{\\lambda ,\\alpha } u_\\xi $ as an identity in $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ which, by the unitary equivalence (REF ), is equivalent to $\\eta \\;=\\;\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha }\\,\\xi $ as an identity in $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ .", "The self-adjoint extension $\\mathring{H}_{\\langle \\alpha \\rangle }$ is given by the restriction of $\\mathring{H}^*$ to those elements of $\\mathcal {D}(\\mathring{H}^*)$ whose charges, in terms of the decomposition (REF ), instead of belonging generically to $H^{-1/2}(\\mathbb {R}^3)$ are such that $\\xi $ belongs to the domain of $\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha }$ and $\\eta $ is of the form $\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha }\\xi $ .", "We now see that $\\mathring{H}_{\\langle \\alpha \\rangle }$ is an operator of TMS type.", "Indeed, owing to the definition (REF ), condition (REF ) implies $\\alpha \\,\\xi \\;=\\;-\\mathcal {T}_\\lambda \\xi +\\!\\!\\!\\begin{array}{c}\\frac{1}{2}\\end{array}\\!\\!\\!W_\\lambda \\eta \\qquad \\qquad \\forall \\xi \\in \\mathcal {D}(\\mathcal {T}_\\lambda )$ as an identity in $L^2(\\mathbb {R}^3)$ .", "In turn, owing to the definition (REF ), (REF ) reads $\\alpha \\,\\xi \\;=\\;-T^+_\\lambda \\xi +\\!\\!\\!\\begin{array}{c}\\frac{1}{2}\\end{array}\\!\\!\\!W_\\lambda \\eta \\,,\\qquad & &\\xi \\in \\mathcal {D}(T^+_\\lambda )\\subset L^2_+(\\mathbb {R}^3) \\\\\\alpha \\,\\xi \\;=\\;-S_0\\xi +\\!\\!\\!\\begin{array}{c}\\frac{1}{2}\\end{array}\\!\\!\\!W_\\lambda \\eta \\,,\\qquad & &\\xi \\in \\mathcal {D}(S_0)\\subset L^2_{\\ell =0}(\\mathbb {R}^3)\\,.", "$ Plugging (REF )-() into (REF ) yields the following asymptotics for elements in $\\mathcal {D}(\\mathring{H}_{\\langle \\alpha \\rangle })$ as $R\\rightarrow \\infty $ : $\\begin{split}\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\widehat{g}(p_1,p_2) \\,\\mathrm {d}p_2}\\;&=\\;\\widehat{\\xi }(p_1)(4\\pi R+\\alpha )+o(1) \\qquad \\qquad (\\,\\xi \\in \\mathcal {D}(T^+_\\lambda )\\,)\\end{split}$ $\\begin{split}\\int _{\\begin{array}{c} \\\\ \\,p_2\\in \\mathbb {R}^3 \\\\ \\!", "|p_2|<R\\end{array}}{\\:\\widehat{g}(p_1,p_2) \\,\\mathrm {d}p_2}\\;&=\\;\\widehat{\\xi }(p_1)(4\\pi R+\\alpha )+(\\widehat{S_0\\xi })(p_1)-(\\widehat{T_\\lambda \\xi })(p_1)+o(1) \\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\!", "(\\,\\xi \\in \\mathcal {D}(S_0)\\,)\\,.\\end{split}$ The asymptotics (REF ) is completely analogous to (REF ) for the two-body system, except for the fact that it only holds for non-spherically symmetric charges.", "If the charge $\\xi $ is spherically symmetric, then (REF ) is modified with an additional $O(1)$ -term in $R$ as done in (REF ).", "In fact, both (REF ) and (REF ) express the same kind of ultra-violet asymptotics at the coincidence hyperplanes, for their formal inverse Fourier transformed version reads $g(x_1,x_2)\\;\\sim \\;\\xi (x_1)\\Big (\\frac{1}{|x_2|}+\\alpha \\Big )+\\chi (x_1)\\qquad \\textrm { as }x_2\\rightarrow 0$ where $\\chi $ is identically zero if the charge $\\xi $ associated to $g\\in \\mathcal {D}(\\mathring{H}_{\\langle \\alpha \\rangle })$ belongs to $H^{3/2}(\\mathbb {R}^3)\\cap L^2_{+}(\\mathbb {R}^3)$ , whereas $\\widehat{\\chi }=\\widehat{S_0\\xi }-\\widehat{T_\\lambda \\xi }$ if $\\xi \\in \\mathcal {D}(S_0)\\subset L^2_{\\ell =0}(\\mathbb {R}^3)$ .", "In either case the leading singularity in $g(x_1,x_2)$ as $x_2\\rightarrow 0$ is precisely of the form $|x_2|^{-1}$ .", "We summarise the above analysis by saying that the operator $\\mathring{H}_{\\langle \\alpha \\rangle }$ , defined by (REF )-(REF ) in terms of a self-adjoint extension $\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha }$ of $\\mathcal {A}_{\\lambda ,\\alpha }=2\\,W_\\lambda ^{-1}(\\mathcal {T}_\\lambda +\\alpha \\mathbb {1})$ on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ , where $\\mathcal {T}_\\lambda =S_0\\oplus T^+_\\lambda $ , $S_0$ is a densely defined and symmetric operator $S_0$ on $L^2_{\\ell =0}(\\mathbb {R}^3)$ with $\\mathrm {ran}S_0\\subset H^{1/2}(\\mathbb {R}^3)$ , and $T^+_\\lambda $ is the component of $T_\\lambda $ on $L^2_+(\\mathbb {R}^3)\\bigoplus _{\\ell =1}^\\infty L^2_{\\ell }(\\mathbb {R}^3)$ with domain $H^{3/2}(\\mathbb {R}^3)\\cap L^2_+(\\mathbb {R}^3)$ , is a self-adjoint extension of $\\mathring{H}$ because it satisfies the condition (REF ), which is a special case of the conditions of self-adjoint extension of the Kreĭn-Višik-Birman theory; and moreover $\\mathring{H}_{\\langle \\alpha \\rangle }$ is a Hamiltonian of Ter-Martirosyan–Skornyakov type because (REF ), through (REF )-(), implies the TMS asymptotics (REF )-(REF ).", "We observe, however, two fundamental differences with respect to the construction of the point interaction Hamiltonian for the two-body system.", "In the two-body case, the TMS condition (REF ), imposed in the asymptotics of the elements of $\\mathcal {D}(\\mathring{H}^*)$ , turns out to be a condition of self-adjoint extension.", "In the 2+1 fermionic system, instead, 1. the TMS condition (REF ) is found to hold only for a class of self-adjoint extensions of $\\mathring{H}$ , those identified a la Kreĭn-Višik-Birman by the functional constraint (REF ), 2. and if one imposes (REF ) as a point-wise identity valid for a generic class of $\\xi $ 's, one is not guaranteed yet to have identified a domain of self-adjointness for an extension of $\\mathring{H}$ , because in general such an Ansatz is not implemented by a self-adjoint operator on $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ .", "Remark A clarification in retrospective on the emergence and the meaning of the operators $S_0$ and $\\mathcal {T}_\\lambda $ introduced in Proposition REF above is surely beneficial at this point.", "The key issue that arises evidently in our discussion is that (REF ) defines a densely defined, symmetric operator $T_\\lambda $ on $L^2(\\mathbb {R}^3)$ which fails to map a linear space of certain spherically symmetric $H^{\\frac{3}{2}}$ -functions into $H^{\\frac{1}{2}}(\\mathbb {R}^3)$ , whereas it does map into the latter space all the $H^{\\frac{3}{2}}$ -functions that are $L^2$ -orthogonal to the spherically symmetric ones.", "As a consequence, unlike what is customarily given for granted at this point in the literature, there arises the issue to make the symmetric operator $W_\\lambda ^{-1}T_\\lambda $ densely defined on the space of charges $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ and to realise it self-adjointly.", "However, one would like to have a self-adjoint operator on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ precisely of the form $W_\\lambda ^{-1}T_\\lambda $ : indeed on the one hand by general facts (Krein-Vishik-Birman) a self-adjoint operator on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ identifies a self-adjoint Hamiltonian of point interaction for the three-body system, and on the other hand the structure $W_\\lambda ^{-1}T_\\lambda $ results in a Ter-Martirosyan–Skornyakov condition for the elements in the domain of such an Hamiltonian.", "The way to retrieve a TMS-like Hamiltonian was then to cure $W_\\lambda ^{-1}T_\\lambda $ on the sector of spherical symmetry, by replacing $T_\\lambda $ with a modified operator $\\mathcal {T}_\\lambda =S_0\\oplus T_\\lambda ^+$ , where $T_\\lambda ^+$ is the restriction of $T_\\lambda $ to the subspace $\\bigoplus _{\\ell =1}^\\infty L^2_{\\ell }(\\mathbb {R}^3)$ and $S_0$ is an arbitrary self-adjoint operator in $L^2_{\\ell =0}(\\mathbb {R}^3)$ with values in $H^{\\frac{1}{2}}(\\mathbb {R}^3)$ .", "The corresponding $W_\\lambda ^{-1}\\mathcal {T}_\\lambda $ is now symmetric and densely defined on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ and any its self-adjoint extension identifies a TMS Hamiltonian for the three-body systems.", "For such a Hamiltonian, the TMS condition (REF ) emerges only when the charge $\\xi $ has symmetry $\\ell \\geqslant 1$ ; on the sector $\\ell =0$ what holds instead is the weaker condition ().", "The latter still prescribes that the generic element $g(x_1,x_2)$ of the domain of the TMS Hamiltonian has a leading singularity $|x_j|^{-1}$ as $|x_j|\\rightarrow 0$ when the charge $\\xi $ associated to $g$ is spherically symmetric, however no $\\alpha $ -constraint is prescribed between singular and regular part of $g$ .", "This larger freedom in the unconstrained regular part reflects precisely the arbitrariness of $S_0$ ." ], [ "Applications and concluding remarks", "For the two-body system with point interaction, imposing the TMS asymptotics at scattering length $(-4\\pi \\alpha )^{-1}$ selects the whole one-parameter family of self-adjoint extensions of $\\mathring{H}$ , the formal free Hamiltonian defined away from the coincidence configurations.", "For larger systems, $\\mathring{H}$ has infinite deficiency indices and the TMS asymptotics emerge for a proper subclass of extensions of $\\mathring{H}$ , provided that the charges are taken in a domain of suitable regularity and symmetry.", "Indeed, except for the two-body case, the TMS condition expresses point-wise asymptotics, which per se is not enough to be a condition of self-adjointness: the latter has to be a suitable functional condition, such as (REF ) in the preceding discussion.", "Recognising a TMS condition as a self-adjointness condition, by means of the general classification of self-adjoint extensions given by the Kreĭn-Višik-Birman theory, is an idea that dates back to the original announcements [38], [39] by Minlos and Faddeev in 1961 and it has been exploited in a series of works by Minlos and collaborators [31], [40], [26], [27], [32], [43], [33], [34], [35], [37], [36].", "To our understanding, however, the issue of making the operator that in our notation reads $W_\\lambda ^{-1}T_\\lambda $ a well-defined map on the space of charges $H^{-1/2}(\\mathbb {R}^3)$ , more precisely the issue on whether $\\mathrm {ran}\\,T_\\lambda \\subset H^{1/2}(\\mathbb {R}^3)=\\mathrm {ran}W_\\lambda $ , was never addressed, nor was it noted that $T_\\lambda $ fails to map spherically symmetric functions of $H^{3/2}(\\mathbb {R}^3)$ into $H^{1/2}(\\mathbb {R}^3)$ (see Proposition REF (i) and its proof).", "In fact, in all recent works [33], [34], [35], [37], [36] the initial domain of $T_\\lambda $ is taken to be $H^1(\\mathbb {R}^3)$ and hence, owing to Proposition REF (i), in general $\\mathrm {ran}\\,T_\\lambda $ consists of $L^2$ -functions that cannot be pulled back to $H^{-\\frac{1}{2}}$ -functions by the inverse of $W_\\lambda $ .", "Also, the ubiquitous statement in the above-mentioned literature, according to which each self-adjoint realisation in $L^2(\\mathbb {R}^3)$ of the operator $T_\\lambda $ defined formally by (REF ) identifies (by general facts of the Kreĭn-Višik-Birman theory) a self-adjoint extension of $\\mathring{H}$ that displays the TMS asymptotics for the functions of its domain, need be made more precise in two crucial respects.", "First, one has to factor out the part of $T_\\lambda $ that acts on $L^2_{\\ell =0}(\\mathbb {R}^3)$ , the spherically symmetric functions, as we argued in Proposition REF and in the discussion that followed from it.", "Second, even when $T_\\lambda $ is self-adjoint on the subspace of higher momentum charges, the corresponding densely defined and symmetric operator $\\mathcal {A}_{\\lambda ,\\alpha }=2\\,W_\\lambda ^{-1}(\\mathcal {T}_\\lambda +\\alpha \\mathbb {1})$ on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ is not necessarily self-adjoint and may in turn admit a multiplicity of self-adjoint extensions, as we observed in Remark REF : only a self-adjoint extension $\\widetilde{\\mathcal {A}}_{\\lambda ,\\alpha }$ of $\\mathcal {A}_{\\lambda ,\\alpha }$ identifies, a la Kreĭn-Višik-Birman, the self-adjoint extension $\\mathring{H}_{\\langle \\alpha \\rangle }$ of $\\mathring{H}$ .", "It remains therefore unclear how to relate the range of masses $m$ (identified in [33], [34], [35], [37], [36]) in which $T_\\lambda $ , initially defined on $H^{1}(\\mathbb {R}^3)$ , is self-adjoint or has a family of self-adjoint extensions on $L^2(\\mathbb {R}^3)$ with the actual range of masses in which $\\mathring{H}$ admits one or more self-adjoint extensions displaying the TMS asymptotics.", "The fact that $\\mathcal {A}_{\\lambda ,\\alpha }=2\\,W_\\lambda ^{-1}(\\mathcal {T}_\\lambda +\\alpha \\mathbb {1})$ may be only symmetric on $H^{-1/2}_{W_\\lambda }(\\mathbb {R}^3)$ even when $\\mathcal {T}_\\lambda $ is self-adjoint on $L^2(\\mathbb {R}^3)$ should account for a larger range of masses in which $\\mathring{H}$ has a multiplicity of TMS-like self-adjoint extensions than the range in which $T_\\lambda $ has a multiplicity of self-adjoint extensions.", "It becomes of great interest now to re-read and understand, in terms of the general classification of self-adjoint extensions of $\\mathring{H}$ provided by the Kreĭn-Višik-Birman theory, those results that produced TMS self-adjoint extensions of $\\mathring{H}$ through an approach based on quadratic forms on Hilbert space (see [46], [15], [16], [21], [12] and above all the recent work [13]).", "Indeed, the quadratic form approach produces a single self-adjoint TMS Hamiltonian, or alternatively a family of self-adjoint TMS Hamiltonians, all extensions of $\\mathring{H}$ , in a regime of masses that differs from what is known from the operator-theoretic approach and that coincides instead to what is found in the physical literature – see [13], as well as the discussion around [13].", "Owing to the general picture of the self-adjoint extension theory, each such Hamiltonian must be selected by a condition on the charges realised by a self-adjoint map on $\\ker (\\mathring{H}^*+\\lambda \\mathbb {1})$ as in (REF ) or, more generally, in (REF ).", "One should identify such a map and to compare it to its analog in the operator-theoretic approach.", "Armed with the analysis and the discussion developed here, we plan to address these issues in a follow-up work." ], [ "Basics of the Kreĭn-Višik-Birman self-adjoint extension theory", "In this appendix we collect the main results of the Kreĭn-Višik-Birman theory of self-adjoint extensions of semi-bounded symmetric operators.", "This theory was developed by Kreĭn [23], Višik [48], and Birman [8]) between the mid 1940's and the mid 1950's.", "For the present formulation and the proof of all the statements that follow we refer to the comprehensive discussion [28], as well as to the expository works [22], [4].", "For any given symmetric operator $S$ with domain $\\mathcal {D}(S)$ , let $m(S)\\;:=\\;\\inf _{\\begin{array}{c}f\\in \\mathcal {D}(S) \\\\ f\\ne 0\\end{array}}\\frac{\\langle f,Sf\\rangle }{\\Vert f\\Vert ^2}\\,.$ be the “bottom” of $S$ , i.e., its greatest lower bound.", "Hereafter $S$ shall be semi-bounded below, meaning therefore $m(S)>-\\infty $ .", "It is not restrictive to assume henceforth $m(S)>0\\,,$ for in the general case one applies the discussion that follows to the strictly positive operator $S+\\lambda \\mathbb {1}$ , $\\lambda >-m(S)$ , and then re-express trivially the final results in terms of the original $S$ .", "When $S$ is densely defined, the choice $m(S)>0$ implies that the Friedrichs extension $S_F$ of $S$ is invertible with bounded inverse defined everywhere on $\\mathcal {H}$: this will allow $S_F^{-1}$ to enter directly the discussion.", "In the general case in which $S_F$ is not necessarily invertible, the role of $S_F^{-1}$ is naturally replaced by the inverse $\\widetilde{S}^{-1}$ of any a priori known self-adjoint extension $\\widetilde{S}$ of $S$ , which thus takes the role of given “datum” of the theory.", "Moreover, with the choice $m(S)>0$ , the level 0 becomes naturally the reference value with respect to which to express the other distinguished (canonically given) extension of $S$ , the Kreĭn-von Neumann extension $S_N$ .", "Lemma (Decomposition formulas) For a densely defined symmetric operator $S$ with positive bottom, one has $\\mathcal {D}(S^*) \\!\\!\\!& = &\\!\\!\\!", "\\mathcal {D}(S_F)\\dotplus \\ker S^* \\\\\\mathcal {D}(S^*) \\!\\!\\!& = &\\!\\!\\!", "\\mathcal {D}(\\overline{S})\\dotplus S_F^{-1} \\ker S^*\\dotplus \\ker S^* \\\\\\mathcal {D}(S_F) \\!\\!& = &\\!\\!\\!", "\\mathcal {D}(\\overline{S})\\dotplus S_F^{-1} \\ker S^*\\,.", "$ Theorem (Classification of self-adjoint extensions – operator version.)", "Let $S$ be a densely defined symmetric operator on a Hilbert space $\\mathcal {H}$ with positive bottom ($m(S)>0$ ).", "There is a one-to-one correspondence between the family of all self-adjoint extensions of $S$ on $\\mathcal {H}$ and the family of the self-adjoint operators on Hilbert subspaces of $\\ker S^*$ .", "If $T$ is any such operator, in the correspondence $T\\leftrightarrow S_T$ each self-adjoint extension $S_T$ of $S$ is given by $\\begin{split}S_T\\;&=\\;S^*\\upharpoonright \\mathcal {D}(S_T) \\\\\\mathcal {D}(S_T)\\;&=\\;\\left\\lbrace f+S_F^{-1}(Tv+w)+v\\left|\\!\\!\\begin{array}{c}f\\in \\mathcal {D}(\\overline{S})\\,,\\;v\\in \\mathcal {D}(T) \\\\w\\in \\ker S^*\\cap \\mathcal {D}(T)^\\perp \\end{array}\\!\\!\\right.\\right\\rbrace .\\end{split}$ Theorem(Characterisation of semi-bounded extensions.)", "Let $S$ be a densely defined symmetric operator on a Hilbert space $\\mathcal {H}$ with positive bottom.", "If, with respect to the notation of (REF ), $S_T$ is a self-adjoint extension of $S$ , and if $\\alpha <m(S)$ , then $\\begin{split}\\langle g,S_T g\\rangle \\;&\\geqslant \\;\\alpha \\,\\Vert g\\Vert ^2\\qquad \\forall g\\in \\mathcal {D}(S_T) \\\\& \\Updownarrow \\\\\\langle v,T v\\rangle \\;\\geqslant \\;\\alpha \\Vert v\\Vert ^2+\\:&\\alpha ^2\\langle v,(S_F-\\alpha \\mathbb {1})^{-1} v\\rangle \\qquad \\forall v\\in \\mathcal {D}(T)\\,.\\end{split}$ As an immediate consequence, $m(T)\\geqslant m(S_T)$ for any semi-bounded $S_T$ .", "In particular, positivity or strict positivity of the bottom of $S_T$ is equivalent to the same property for $T$ , that is, $\\begin{split}m(S_T)\\;\\geqslant \\;0\\quad &\\Leftrightarrow \\quad m(T)\\;\\geqslant \\; 0 \\\\m(S_T)\\;> \\;0\\quad &\\Leftrightarrow \\quad m(T)\\;>\\; 0\\,.\\end{split}$ Moreover, if $m(T)>-m(S)$ , then $m(T)\\;\\geqslant \\; m(S_T)\\;\\geqslant \\;\\frac{m(S) \\,m(T)}{m(S)+m(T)}\\,.$ Theorem(Characterisation of semi-bounded extensions – form version.)", "Let $S$ be a densely defined symmetric operator on a Hilbert space $\\mathcal {H}$ with positive bottom and, with respect to the notation of (REF ), let $S_T$ be a semi-bounded (not necessarily positive) self-adjoint extension of $S$ .", "Then $\\mathcal {D}[T]\\;=\\; \\mathcal {D}[S_T]\\,\\cap \\,\\ker S^*$ and $\\begin{split}\\mathcal {D}[S_T]\\;&=\\;\\mathcal {D}[S_F]\\,\\dotplus \\,\\mathcal {D}[T] \\\\S_T[f+v,f^{\\prime }+v^{\\prime }]\\;&=\\;S_F[f,f^{\\prime }]\\,+\\,T[v,v^{\\prime }] \\\\&\\forall f,f^{\\prime }\\in \\mathcal {D}[S_F],\\;\\forall v,v^{\\prime }\\in \\mathcal {D}[T]\\,.\\end{split}$ As a consequence, $S_{T_1}\\,\\geqslant \\,S_{T_2}\\qquad \\Leftrightarrow \\qquad T_1\\,\\geqslant \\,T_2$ and $T\\;\\geqslant \\;S_T\\,.$ Proposition(Parametrisation of $S_F$ and $S_N$ .)", "Let $S$ be a densely defined symmetric operator on a Hilbert space $\\mathcal {H}$ with positive bottom and let $S_T$ be a positive self-adjoint extension of $S$ , parametrised by $T$ according to Theorems and .", "(i) $S_T$ is the Friedrichs extension when $\\mathcal {D}[T]=\\lbrace 0\\rbrace $ (“ $T=\\infty $ ”).", "(ii) $S_T$ is the Kreĭn-von Neumann extension when $\\mathcal {D}(T)=\\mathcal {D}[T]=\\ker S^*$ and $Tu=0$ $\\forall u\\in \\ker S^*$ ( $T=\\mathbb {O}$ ).", "Proposition(Finite deficiency index.)", "If $S$ is a semi-bounded and densely defined symmetric operator on a Hilbert space $\\mathcal {H}$ with finite deficiency index, then (i) the semi-boundedness of $S_T$ is equivalent to the semi-boundedness of $T$ ; (ii) any self-adjoint extension of $S$ is bounded below.", "Proposition(“Finite-dimensional” extensions are always semi-bounded.)", "Given a semi-bounded and densely defined symmetric operator $S$ on a Hilbert space $\\mathcal {H}$ , whose bottom is positive, all the self-adjoint extensions of $S_T$ of $S$ for which the parameter $T$ , in the parametrisation (REF ) of Theorem , is a self-adjoint operator acting on a finite-dimensional subspace of $\\ker S^*$ are semi-bounded.", "For the occurrence of unbounded below self-adjoint extensions it is necessary that $\\dim \\overline{\\mathcal {D}(T)}=\\infty $ ." ], [ "Proof of the identity (", "We prove here that the $H^2$ -closure of $C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )$ , which is by definition the space $H^2_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )$ , coincides with the space $\\lbrace f\\in H^2(\\mathbb {R}^3)\\,|\\, f(0)=0\\rbrace $ , thus obtaining the identity (REF ).", "Clearly $\\overline{\\,C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\,}^{\\Vert \\,\\Vert _{H^2}}\\;\\subset \\;\\big \\lbrace f\\in H^2(\\mathbb {R}^3)\\,\\big |\\, f(0)=0\\big \\rbrace $ because the $H^2$ -convergence implies the point-wise convergence of continuous functions.", "Thus, given $f\\in H^2(\\mathbb {R}^3)$ with $f(0)=0$ , we only need to find for arbitrary $\\varepsilon >0$ a function $f_\\varepsilon \\in C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )$ such that $\\Vert f-f_\\varepsilon \\Vert _{H^2}\\;\\leqslant \\;\\varepsilon \\,.$ Given a cut-off function $\\chi \\in C^\\infty ([0,+\\infty ))$ such that $\\begin{split}\\chi (r)\\;=\\;0\\qquad & \\textrm { for }\\;r\\in [0,1] \\\\\\chi (r)\\;=\\;1\\qquad & \\textrm { for }\\;r\\in [2,+\\infty )\\,,\\end{split}$ set $\\phi _n(x)\\;:=\\;\\chi (n|x|)\\,,\\qquad \\quad n\\in \\mathbb {N}\\,,\\qquad x\\in \\mathbb {R}^3\\,.$ Then, for any $n\\in \\mathbb {N}$ , $\\phi _n\\in C^\\infty (\\mathbb {R}^3)$ and $\\begin{split}\\phi _n(x)\\;=\\;0\\qquad \\qquad & \\textrm { for }\\;|x|\\;\\leqslant \\;\\frac{1}{n} \\\\\\phi _n(x)\\;=\\;1\\qquad \\qquad & \\textrm { for }\\;|x|\\;\\geqslant \\;\\frac{2}{n} \\\\|\\phi _n(x)|\\;\\leqslant \\;c_\\chi \\qquad \\quad \\;\\; & \\\\|\\nabla \\phi _n(x)|\\;\\leqslant \\;n \\,c_\\chi \\qquad \\;\\;\\; & \\;\\;\\forall x\\in \\mathbb {R}^3 \\\\|\\Delta \\phi _n(x)|\\;\\leqslant \\;n^2 c_\\chi \\qquad \\;\\;&\\end{split}$ where $c_\\chi $ depends only on $\\Vert \\chi \\Vert _{\\sup }$ , $\\Vert \\chi ^{\\prime }\\Vert _{\\sup }$ , and $\\Vert \\chi ^{\\prime \\prime }\\Vert _{\\sup }$ .", "Correspondingly, each function $\\phi _nf$ belongs to $H^2(\\mathbb {R}^3)$ and vanishes when $|x|\\leqslant n^{-1}$ .", "Furthermore, we now show that $\\Vert \\phi _n\\,f-f\\Vert _{H^2}\\;\\xrightarrow{}\\;0\\,.$ Indeed, $\\Vert \\phi _n\\,f-f\\Vert _{L^2}\\rightarrow 0$ follows immediately by dominated convergence and (REF ), whereas $\\Vert \\Delta (\\phi _n\\,f)-\\Delta f\\Vert _{L^2}\\rightarrow 0$ is obtained with the following argument.", "First, one estimates $\\begin{split}\\Vert \\Delta (\\phi _n\\,f)-\\Delta f\\Vert _{L^2}\\;\\leqslant \\;\\Vert (\\phi _n-1)\\Delta f\\Vert _{L^2}+2\\Vert (\\nabla \\phi _n)\\cdot (\\nabla f)\\Vert _{L^2}+\\Vert f\\Delta \\phi _n\\Vert _{L^2}\\,.\\end{split}$ For the first summand in the r.h.s.", "of the inequality above one has $\\Vert (\\phi _n-1)\\Delta f\\Vert _2\\rightarrow 0$ as $n\\rightarrow \\infty $ again by dominated convergence and (REF ).", "For the second summand, $\\begin{split}\\Vert (\\nabla \\phi _n)\\cdot (\\nabla f)\\Vert ^2_{L^2}\\;&\\leqslant \\;n^2\\,c_\\chi ^2\\int _{|x|\\leqslant 2 n^{-1}}|\\nabla f(x)|^2\\,\\mathrm {d}x \\\\&\\leqslant \\;n^2\\,c_\\chi ^2\\,\\Big (\\frac{4\\pi \\,2^3}{3\\,n^3}\\Big )^{2/3}\\Vert \\nabla f\\Vert ^2_{L^6(B_{2/n})} \\\\&\\lesssim \\;\\Vert f\\Vert ^2_{H^2(B_{2/n})}\\;\\xrightarrow{}\\;0\\end{split}$ where we used (REF ) in the first step, a Hölder inequality in the second, the Sobolev embedding in the third, and dominated convergence in the last, $B_r$ denoting the closed ball of $\\mathbb {R}^3$ of radius $r$ centred at the origin.", "For the third summand, $\\begin{split}\\Vert f\\Delta \\phi _n\\Vert _{L^2(\\mathbb {R}^3)}^2\\;&\\leqslant \\;n^4\\,c_\\chi ^2\\int _{\\mathrm {supp}(\\Delta \\phi _n)}|f(x)|^2\\,\\mathrm {d}x \\\\&\\lesssim \\;n^3\\,\\Vert f\\Vert ^2_{H^2(B_{2/n})}\\,|B_{2/n}|\\\\&\\lesssim \\;\\Vert f\\Vert ^2_{H^2(B_{2/n})}\\;\\xrightarrow{}\\;0\\end{split}$ where in the first step we used (REF ), in the second we used the estimate $|f(x)|\\;=\\;|f(x)-f(0)|\\;\\leqslant \\;\\frac{C}{\\sqrt{n\\,}\\,}\\,\\Vert \\nabla f\\Vert _{L^6(B_{2/n})}\\;\\lesssim \\;\\frac{1}{\\sqrt{n\\,}\\,}\\Vert f\\Vert _{H^2(B_{2/n})}$ (where the constant $C$ does not depend on $n$ ) that follows from Morrey's inequality and the Sobolev embedding (see [18], and in the last step we used dominated convergence.", "Thus, (REF ) is proved.", "As a consequence of (REF ) above, for the arbitrary $\\varepsilon >0$ fixed at the beginning there is $N_\\varepsilon \\in \\mathbb {N}$ and $\\delta :=N_\\varepsilon ^{-1}$ such that $g_\\varepsilon :=\\phi _{N_\\varepsilon }f$ is a $H^2$ -function satisfying $g_\\varepsilon \\equiv 0\\;\\textrm { on }\\;B_\\delta \\qquad \\textrm {and}\\qquad \\Vert f-g_\\varepsilon \\Vert _{H^2}\\;\\leqslant \\;\\frac{\\varepsilon }{3}\\,.$ We consider now a standard mollification $j_n*g_\\varepsilon $ of $g_\\varepsilon $ for some $j\\in C^\\infty _0(\\mathbb {R}^3)$ with $\\int _{\\mathbb {R}^3} j\\,\\mathrm {d}x =1$ and $j_n(x):=n^{-3}j(nx)$ , $n\\in \\mathbb {N}$ , $x\\in \\mathbb {R}^3$ .", "Then $j_n*g_\\varepsilon \\in C^\\infty (\\mathbb {R}^3)\\cap H^2(\\mathbb {R}^3)$ and $j_n*g_\\varepsilon \\rightarrow g_\\varepsilon $ in $H^2$ as $n\\rightarrow \\infty $ .", "Therefore, there is $n_\\varepsilon \\in \\mathbb {N}$ large enough so that $h_\\varepsilon :=j_{n_\\varepsilon }*g_\\varepsilon $ satisfies $\\Vert g_\\varepsilon -h_\\varepsilon \\Vert _{H^2}\\;\\leqslant \\;\\frac{\\varepsilon }{3}$ and $\\mathrm {supp}(j_{n_\\varepsilon })\\subset B_{\\delta /2}$ (where $\\delta $ is the radius of the ball that contains the support of $g_\\varepsilon $ , see (REF ) above).", "As a consequence, $h_\\varepsilon $ vanishes in $B_{\\delta /2}$ : indeed, if $x\\in B_{\\delta /2}$ and $y\\in \\mathrm {supp}(j_{n_\\varepsilon })$ , that is, $|x|\\leqslant \\frac{\\delta }{2}$ and $y\\leqslant \\frac{\\delta }{2}$ , one has $|x-y|\\leqslant \\delta $ and hence $h_\\varepsilon (x)\\;=\\;(j_{n_\\varepsilon }*g_\\varepsilon )(x)\\;=\\;\\int _{\\mathrm {supp}(j_{n_\\varepsilon })}g_\\varepsilon (x-y)\\,j_{n_\\varepsilon }(y)\\,\\mathrm {d}y\\;=\\;0$ because $g_\\varepsilon $ vanishes in $B_\\delta $ .", "Summarising, we have found $h_\\varepsilon \\in C^\\infty (\\mathbb {R}^3)\\cap H^2(\\mathbb {R}^3)$ such that $h_\\varepsilon \\equiv 0\\;\\textrm { on }\\;B_{\\delta /2}\\qquad \\textrm {and}\\qquad \\Vert g_\\varepsilon -h_\\varepsilon \\Vert _{H^2}\\;\\leqslant \\;\\frac{\\varepsilon }{3}\\,.$ Last, we consider a cut-off function $\\zeta \\in C^\\infty _0([0,+\\infty ))$ such that $\\begin{split}\\zeta (r)\\;=\\;1\\qquad & \\textrm { for }\\;r\\in [0,1] \\\\\\zeta (r)\\;=\\;0\\qquad & \\textrm { for }\\;r\\in [2,+\\infty )\\,,\\end{split}$ and set $h_{\\varepsilon ,n}:=\\;\\zeta (n^{-1}x)h_\\varepsilon (x)\\qquad \\quad n\\in \\mathbb {N}\\,,\\qquad x\\in \\mathbb {R}^3\\,.$ Then, in complete analogy to the reasoning above, we see that for $n$ large enough each $h_{\\varepsilon ,n}$ belongs to $C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!B_{\\delta /2})\\subset C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )$ and $h_{\\varepsilon ,n}\\rightarrow h_\\varepsilon $ in $H^2$ as $n\\rightarrow \\infty $ .", "This implies the existence of $M_\\varepsilon \\in \\mathbb {N}$ such that the function $f_\\varepsilon :=h_{\\varepsilon ,M_\\varepsilon }$ satisfies $f_\\varepsilon \\;\\in \\;C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )\\qquad \\textrm {and}\\qquad \\Vert h_\\varepsilon -f_\\varepsilon \\Vert _{H^2}\\;\\leqslant \\;\\frac{\\varepsilon }{3}\\,.$ Using (REF ), (REF ), and (REF ) above in a triangular inequality we finally conclude $\\Vert f-f_\\varepsilon \\Vert _{H^2}\\;\\leqslant \\;\\Vert f-g_\\varepsilon \\Vert _{H^2}+\\Vert g_\\varepsilon -h_\\varepsilon \\Vert _{H^2}+\\Vert h_\\varepsilon -f_\\varepsilon \\Vert _{H^2}\\;\\leqslant \\;\\varepsilon $ for a function $f_\\varepsilon \\;\\in \\;C^\\infty _0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )$ , which completes the proof of (REF )." ], [ "Proof of the inclusions and the identities in (", "We prove here (REF ): the proof applies straightforwardly also to obtain (REF ).", "Let $f\\in H^2(\\mathbb {R}^3\\times \\mathbb {R}^3)$ .", "We want to show that $f$ belongs to $H_0^1((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))$ .", "Since $H_0^1((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))\\;=\\;\\overline{\\,C_0^\\infty ((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))\\,}^{\\,\\Vert \\;\\Vert _{H^1}}\\,,$ it is enough to find, for arbitrary $\\varepsilon >0$ , a function $f_\\varepsilon \\in C_0^\\infty ((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))$ such that $\\Vert f-f_\\varepsilon \\Vert _{H^1}\\;\\leqslant \\;\\varepsilon \\,.$ First, since $f\\in H^2(\\mathbb {R}^3\\times \\mathbb {R}^3)\\subset H^1(\\mathbb {R}^3\\times \\mathbb {R}^3)$ and $C_0^\\infty (\\mathbb {R}^3\\times \\mathbb {R}^3)$ is dense in $H^1(\\mathbb {R}^3\\times \\mathbb {R}^3)$ , there exists $g_\\varepsilon \\in C_0^\\infty (\\mathbb {R}^3\\times \\mathbb {R}^3)$ such that $\\Vert f-g_\\varepsilon \\Vert _{H^1}\\;\\leqslant \\;\\frac{\\varepsilon }{2}\\,.$ Given a cut-off function $\\chi \\in C^\\infty ([0,+\\infty ))$ such that $\\begin{split}\\chi (r)\\;=\\;0\\qquad & \\textrm { for }\\;r\\in [0,1] \\\\\\chi (r)\\;=\\;1\\qquad & \\textrm { for }\\;r\\in [2,+\\infty )\\,,\\end{split}$ set $\\phi _n(x,y)\\;:=\\;\\chi (n|x|)\\chi (n|y|)\\,,\\qquad \\quad n\\in \\mathbb {N}\\,,\\qquad (x,y)\\in \\mathbb {R}^3\\times \\mathbb {R}^3\\,.$ Then, for any $n\\in \\mathbb {N}$ , $\\phi _n\\in C^\\infty (\\mathbb {R}^3\\times \\mathbb {R}^3)$ and $\\begin{split}\\phi _n(x,y)\\;=\\;0\\qquad \\qquad & \\textrm { for }\\;|x|\\;\\leqslant \\;\\frac{1}{n}\\textrm { or }|y|\\;\\leqslant \\;\\frac{1}{n} \\\\\\phi _n(x,y)\\;=\\;1\\qquad \\qquad & \\textrm { for }\\;|x|\\;\\geqslant \\;\\frac{2}{n}\\textrm { and }|y|\\;\\geqslant \\;\\frac{2}{n} \\\\|\\phi _n(x,y)|\\;\\leqslant \\;c_\\chi \\qquad \\quad \\;\\; & \\;\\;\\forall (x,y)\\in \\mathbb {R}^3\\times \\mathbb {R}^3 \\\\|\\nabla \\phi _n(x,y)|\\;\\leqslant \\;n\\,c_\\chi \\qquad \\;\\;\\: & \\;\\;\\forall (x,y)\\in \\mathbb {R}^3\\times \\mathbb {R}^3 \\,,\\end{split}$ where here and henceforth $\\nabla $ denotes the 6-dimensional gradient and $c_\\chi $ depends only on $\\Vert \\chi \\Vert _{\\sup }$ and $\\Vert \\chi ^{\\prime }\\Vert _{\\sup }$ .", "Correspondingly, each function $g_{\\varepsilon ,n}\\;:=\\;\\phi _n \\,g_\\varepsilon $ belongs to $C^\\infty _0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))$ and we now show that $\\Vert g_{\\varepsilon ,n}-g_\\varepsilon \\Vert _{H^1}\\;\\xrightarrow{}\\;0\\,.$ Indeed, $\\Vert g_{\\varepsilon ,n}-g_\\varepsilon \\Vert _{L^2}\\rightarrow 0$ follows immediately by dominated convergence and (REF ), whereas $\\Vert \\nabla g_{\\varepsilon ,n}-\\nabla g_\\varepsilon \\Vert _{L^2}\\rightarrow 0$ follows from the vanishing of both summands in the r.h.s.", "of the inequality $\\Vert \\nabla g_{\\varepsilon ,n}-\\nabla g_\\varepsilon \\Vert _{L^2}\\;\\leqslant \\;\\Vert \\phi _n\\nabla g_{\\varepsilon }-\\nabla g_\\varepsilon \\Vert _{L^2}+\\Vert g_{\\varepsilon }\\nabla \\phi _n\\Vert _{L^2}\\,.$ Explicitly, $\\Vert \\phi _n\\nabla g_{\\varepsilon }-\\nabla g_\\varepsilon \\Vert _{L^2}\\rightarrow 0$ by dominated convergence, owing to (REF ), whereas $\\begin{split}\\Vert g_{\\varepsilon }\\nabla \\phi _n\\Vert _{L^2(\\mathbb {R}^3)}^2\\;&\\leqslant \\;n^2\\,c_\\chi ^2\\iint _{\\mathrm {supp}(\\nabla \\phi _n)\\cap \\mathrm {supp}(g_\\varepsilon )} |g_\\varepsilon |^2\\,\\mathrm {d}x\\,\\mathrm {d}y \\\\&\\leqslant \\;n^2\\,c_\\chi ^2\\Big (\\frac{\\,C_{g_\\varepsilon }}{n^3}\\Big )^{\\!\\frac{2}{3}}\\Vert g_\\varepsilon \\Vert ^2_{L^6(B_{\\varepsilon ,n})} \\\\&\\lesssim \\;\\Vert g_\\varepsilon \\Vert ^2_{H^2(B_{\\varepsilon ,n})}\\;\\xrightarrow{}\\;0\\,,\\end{split}$ where we used (REF ) in the first step, a Hölder inequality and the estimate $\\begin{split}B_{\\varepsilon ,n}\\;:=&\\;\\;\\mathrm {supp}(\\nabla \\phi _n)\\cap \\mathrm {supp}(g_\\varepsilon ) \\\\|B_{\\varepsilon ,n}|\\;\\leqslant &\\;\\;\\big |\\big (\\lbrace |x|\\leqslant 2/n\\rbrace \\cup \\lbrace |y|\\leqslant 2/n\\rbrace \\big )\\cap \\mathrm {supp}(g_\\varepsilon )\\big |\\;\\leqslant C_{g_\\varepsilon } n^{-3}\\end{split}$ in the second step, where $C_{g_\\varepsilon }$ depends only on the radius of $\\mathrm {supp}(g_\\varepsilon )$ , the continuous embedding $H^2(B_{\\varepsilon ,n})\\subset L^6(B_{\\varepsilon ,n})$ in the third step, and dominated convergence in the last step.", "As a consequence of (REF ) above, for the arbitrary $\\varepsilon >0$ fixed at the beginning there is $N_\\varepsilon \\in \\mathbb {N}$ such that $f_\\varepsilon :=g_{\\varepsilon ,N_{\\varepsilon }}\\in C^\\infty _0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))$ satisfies $\\Vert g_\\varepsilon -f_\\varepsilon \\Vert _{H^1}\\;\\leqslant \\;\\frac{\\varepsilon }{2}\\,.$ A triangular inequality based on (REF ) and (REF ) then yields (REF ), thus completing the proof of (REF ).", "Clearly, (REF ) also shows that to any function in $C^\\infty _0(\\mathbb {R}^3\\times \\mathbb {R}^3)$ there is a function in $C^\\infty _0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))$ arbitrarily close in the $H^1$ -norm, which implies that the spaces $H^1(\\mathbb {R}^3\\times \\mathbb {R}^3)$ and $H^1_0((\\mathbb {R}^3\\times \\mathbb {R}^3)\\!\\setminus \\!", "(\\Gamma _1\\cup \\Gamma _2))$ coincide.", "This yields the identity in (REF ).", "The arguments above apply virtually unchanged both for the inclusion $H^2(\\mathbb {R}^3)\\subset H^1_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )$ and for the identity $H^1_0(\\mathbb {R}^3\\!\\setminus \\!\\lbrace 0\\rbrace )=H^1(\\mathbb {R}^3)$ , with the obvious removal of one variable, thus proving (REF )." ], [ "Acknowledgements", "We warmly thank G. Dell'Antonio and L. Guerini for enlightening discussions on the subject." ] ]
1606.05222
[ [ "Spherical Accretion of Matter by Charged Black Holes on f(T) Gravity" ], [ "Abstract We studied the spherical accretion of matter by charged black holes on $f(T)$ Gravity.", "Considering the accretion model of a isentropic perfect fluid we obtain the general form of the Hamiltonian and the dynamic system for the fluid.", "We have analysed the movements of an isothermal fluid model with $p=\\omega e$ and where $p$ is the pressure and $e$ the total energy density.", "The analysis of the cases shows the possibility of spherical accretion of fluid by black holes, revealing new phenomena as cyclical movement inside the event horizon." ], [ "Introduction", "Gravitation and Cosmology modern are based on General Relativity (GR), which in turn is based on the riemannian Differential Geometry.", "The structure of the so-called space-time, a four-dimensional differentiable manifold, is completely characterized by a metric, because the connection, Riemann and Ricci tensors and scalar curvature depend only of the metric components, its inverse and their derivatives [1].", "It is a fact that this structure can only describe the current evolution of our universe, being consistent with the observational data, introducing a fluid model where the pressure must necessarily be negative today, model known by $\\Lambda $ CDM [2].", "An alternative description for the accelerated evolution of our universe is one in which modifies the Einstein equation, for a generalized such that at some threshold parameters theory falls in GR.", "The simplest way to do this is to directly modify the Einstein-Hilbert action to another which has the limit of GR, such as of the $f(R)$ Gravity [3], which can be made the limit for small curvatures and get $f(R)\\sim a_0+a_1R+O(R^2)$ , for example.", "It is common generalize GR by modifying the Einstein-Hilbert action, using a function of a scalar theory, as the theories $f(R,\\mathcal {T})$ Gravity [4], $f(G)$ [5] and $f(R,G)$ [6] Gravities, for example, where $\\mathcal {T}$ is the trace of the energy-momentum tensor and $G$ the Gauss-Bonnet term.", "An alternative description, but equivalent, of gravity can be done now using the torsion of space-time, rather than curvature.", "Considering now the identically zero curvature, we can assign the gravitational interaction effects to torsion.", "The theory that represents such an alternative is commonly called Teleparallel Theory (TT) [7], where are now tetrads that make the role of dynamic fields, which completely determining the geometric objects such as connection, torsion, contorsion and Torsion scalar.", "As this theory is equivalent to GR, the equations of motion can only describe the accelerated evolution of the current phase of our universe through again the introduction of dark energy.", "Then we can think analogously to generalize this theory to one that contains terms of order higher torsion scalar, as $f(T)$ Gravity [8].", "We also have the same possibilities for generalization for $f(T,\\mathcal {T})$ [9] and $f(T,T_G)$ [10] Gravities, or more generally as in [11].", "In this paper, we will restrict ourselves to $f(T)$ Gravity.", "In the case of the study of Cosmology by $f(T)$ Gravity, we have the most varied results.", "The study by local phenomena such as black holes solutions, we still have much work ahead of us.", "The first hole black solution theory is charged solutions on $3D$ [13] and $D$ dimensions [12].", "Later appears more charged solutions in $4D$ , re-obtaining thus solutions of Reisnner-Nordstrom, Reisnner-Nordstrom-AdS (dS), Schwarzshild-AdS (dS) e Schwarzshild [14].", "Another paper discusses the formulation of the Kerr solution for $f(T)$ Gravity [15].", "The solutions of charged black holes, resulting in non-linear electrodynamics are obtained in [16], which we will study here in this work.", "We also have regular black holes solutions in [17].", "Even with few solutions found in this theory, we have very few studies of local phenomena such as bending of light [18] and solar tests [19].", "The study of accretion of matter by a black hole was first performed by Bondi [20], but in a newtonian form.", "Generalization to account for relativistic effects was formulated by Michel on 1971 [21], so is called accretion type-Michel for this approach.", "In this paper we will use the accretion type-Michel.", "The description of spherical accretion made by Michel is the phenomenon critically, having a certain radius value where the system becomes critical.", "Considering effects of curvature, we have several results of spherical accretion by black holes [22], and some are for modified gravity [23].", "The only pioneer work in considering the zero curvature and the torsion effects on $f(T)$ Gravity was recently published [24].", "We will follow the same methodology of this work but generalizing the accretion analysis for a general spherically symmetric metric.", "So our motivation is to treat the spherical accretion of matter by charged black hole of the $f(T)$ theory.", "In Section we discussed the spherical accretion of matter by a black hole, giving the expression of the Hamiltonian of the fluid and dynamic system.", "In Section we restrict our analysis to the model of isothermal isentropic fluids, particularizing for ultra-stiff cases, ultra-relativistic, radiation and sub-relativistic fluids.", "In Section we present our final considerations." ], [ "Spherical Accretion", "The spherical accretion study of matter by black holes is usually based on the movement of fluid in the neighboring region to the event horizon.", "This can be addressed as follows.", "Let's take two equations that completely characterizes the accretion (ejection) spherical of a perfect isotropic fluid; one is the equation of conservation of source material and the other is the equation that describes the conservation of energy.", "Let's start by the equation similar to that of continuity in fluid mechanics, which is described by [25] $\\nabla _{\\mu }J^{\\mu }\\equiv 0\\,,$ where $J^{\\mu }=nu^{\\mu }$ , $n=\\rho $ is the number of the baryonic density and $u^{\\mu }=dx^{\\mu }/d\\tau $ is the four-speed fluid.", "Analogously to the theorem of Noether of fluid mechanics, Bernoulli's theorem in hydrodynamics relativistic have to spherical symmetry, a certain amount that is conserved in the co-moving frame to fluid flow.", "This law is represented by the equation [25] $u^{\\nu }\\nabla _{\\nu }\\left[hu_{\\mu }\\zeta ^{\\mu }\\right]\\equiv 0\\,,$ where $h$ is the enthalpy of the fluid and $\\zeta ^{\\mu }$ a Killing vector of the temporal symmetry of space-time.", "To simplify our study and make the equations having analytical solutions, we will restrict ourselves to the case where the space-time has spherical symmetry, which implies in the following general form of the line element $dS^2=A(r)dt^2-B(r)dr^2-C(r)\\left(d\\theta ^2+\\sin ^2\\theta d\\phi ^2\\right)\\,.$ For the line element (REF ) we have the following metrics determinant $g=-ABC^2$ .", "The fluid movement follows the same symmetries of space-time, ie spherical symmetry.", "So we can then take a fluid that moves in temporal and radial directions, and its four-speed given by $[u^{\\mu }]=[u^0,u,0,0]$ .", "Thus, because of the spherical symmetry, the continuity equation (REF ), integrated, gives us $C\\sqrt{AB}nu=c_1\\,.$ This is one of the equations principles that govern the movement of the fluid.", "We can then normalize the four-speed for $g_{\\mu \\nu }u^{\\mu }u^{\\nu }=1$ , which gives us the following identities $u^{0}=\\sqrt{A^{-1}\\left(1+Bu^2\\right)}\\,,\\,u_{0}=g_{00}u^{0}=\\sqrt{A\\left(1+Bu^2\\right)}\\,.$ Because of spherical symmetry, it is common we take the following vector of Killing $[\\zeta ^{\\mu }]=[1,0,0,0]$ to integrate (REF ), resulting in $h\\sqrt{A\\left(1+Bu^2\\right)}=c_2\\,.$ We may relate the three-velocity of the fluid with the radial and temporal components of the four-velocity through the line element to the equatorial plane $\\theta =\\pi /2$ $dS^2(\\theta =\\pi /2)=\\left(\\sqrt{A}dt\\right)^2-\\left(\\sqrt{B}dr\\right)^2\\,.$ Now we define the three-velocity of the fluid by $v=\\frac{\\sqrt{B}dr}{\\sqrt{A}dt}\\,,$ that replacing $u=dr/d\\tau $ and $u^{0}=dt/d\\tau $ , we have $v^2=\\frac{B}{A}\\left(\\frac{u}{u^{0}}\\right)^2\\,.$ Using (REF ) and isolating $u^2$ , we have $u^2=\\frac{v^2}{B(1-v^2)}\\,,\\,u_{0}^2=\\frac{A}{1-v^2}\\,.$ Now through (REF ) we can rewrite our equation (REF ) as $\\frac{A\\left(Cnv\\right)^2}{1-v^2}=c_1^2\\,.$ Bernoulli's theorem of relativistic hydrodynamics says that every symmetry of space-time we have a conserved quantity associated with this symmetry.", "For the temporal translational symmetry, the conserved quantity is the square of the integration constant $c_2$ in (REF ), and should be proportionate to the Hamiltonian of the fluid.", "using again (REF ), we define the Hamiltonian fluid as $\\mathcal {H}(r,v)=\\frac{h^2(r,v)A(r)}{1-v^2}\\,.$ The intensive and extensive thermodynamic quantities of the fluid are related by equations $dp=n\\left(dh-Tds\\right)\\,,\\,de=hdn+nTds\\,,$ where $T$ and $s$ are specific entropy and the temperature of the fluid, and $e$ is the total energy density of the fluid.", "In general, the study of fluid motion becomes very complicated, and it is due to this reason that we must assume two simplifications for perfect fluid our case.", "The first is considered that a fluid does not heat exchange with the outside, then the fluid is classified as adiabatic fluid, and $u^{\\mu }\\nabla _{\\mu }s\\equiv 0$ .", "The second is when the fluid keeping its entropy constant during movement, then the fluid thus characterized by isentropic where $\\nabla _{\\mu }s\\equiv 0$ and therefore $s=s_0\\in \\Re ^{+}$ .", "Considering that our case study is a fluid adiabatic and isentropic, this implies $ds\\equiv 0$ and then $dp=ndh\\,,\\,de=hdn\\,.$ We can divide $dp$ by $de$ to define the speed of sound as $a^2=\\frac{dp}{de}=\\frac{d\\ln h}{d\\ln n}\\,.$ Now we will establish a dynamic system through Hamiltonian (REF ) $\\frac{dr}{dt}=\\frac{\\partial \\mathcal {H}}{\\partial v}\\,,\\,\\frac{dv}{dt}=-\\frac{\\partial \\mathcal {H}}{\\partial r}$ Now do the same steps contained in [24], remembering that our metric is generalized to (REF ), what gives us the following dynamic system $&&\\dot{r}=\\frac{2Ah^2}{v(1-v^2)^2}(v^2-a^2)\\,,\\\\&&\\dot{v}=-\\frac{h^2}{1-v^2}\\left[\\frac{dA}{dr}-2a^2A\\frac{d\\ln (\\sqrt{A}C)}{dr}\\right]\\,.$ We have here some possibilities of critical points for this dynamic system, but we will focus only on single physical possibility, where $v_c=a\\,,\\,A^{\\prime }=2a^2A(\\ln \\sqrt{A}C)^{\\prime }\\,,$ which defines a critical point for the system (REF )-().", "We will now specify a famous model for the perfect fluid for the next section." ], [ "Isothermal Fluids", "An important model that is well in agreement with the reality is when the fluid keeps a constant temperature in the thermodynamic process and thus the speed of sound is a constant, $a^2=\\omega =dp/de$ .", "This is another very important simplification to analytically solve the equations of motion of the fluid.", "Thus, integrating, we have what the pressure is proportional to total energy density $p=\\omega e\\,.$ We define the enthalpy of the fluid by [25], [24] $h=\\frac{e+p}{n}\\,,$ Here we can find an ordinary differential equation for the total density of the fluid, for this, we take (REF ), (REF ) and (REF ), we integrate with respect to $n$ , which provides us $e(n)=\\frac{e_c}{n_c^{\\omega +1}}n^{1+\\omega }\\,.$ The enthalpy (REF ) is then given by $h=\\frac{(1+\\omega )e_c}{n_c^{\\omega +1}}n^{\\omega }\\,.$ We want to work with the thermodynamic variables $(r,v)$ , then we must put the other thermodynamic quantities on the basis of these two variables.", "We then replace the dependence on $n$ for $n(r,v)$ , for this, we will use (REF ) $n=\\frac{c_1}{vC}\\sqrt{\\frac{1-v^2}{A}}\\,.$ We now take (REF ) and (REF ), considering (REF ), to rewrite as $\\mathcal {H}=\\mathcal {H}_0\\frac{1}{\\left(vC\\right)^{2\\omega }}\\left[\\frac{A}{1-v^2}\\right]^{1-\\omega }\\,,\\,\\mathcal {H}_0=\\left[\\frac{(1+\\omega )e_cc_1^{\\omega }}{n_c^{1+\\omega }}\\right]^2$ We can then define $\\mathcal {H}$ by a transformation in the time coordinate (Killing vector of this symmetrical), or $t\\rightarrow \\mathcal {H}_0t$ , resulting in $\\mathcal {H}\\rightarrow \\mathcal {H}/\\mathcal {H}_0$ , then we redefine the Hamiltonian for $\\mathcal {H}[r,v]=\\frac{1}{\\left[vC(r)\\right]^{2\\omega }}\\left[\\frac{A(r)}{1-v^2}\\right]^{1-\\omega }\\,.$ In the next subsection we will specify the cases between the relationship of pressure and total energy density of the fluid." ], [ "Ultra-stiff fluid", "The model in which $\\omega =1$ is called Ultra-stiff fluid.", "Thus, the Hamiltonian (REF ), for this case becomes $\\mathcal {H}=\\frac{1}{r^4v^2}\\,,\\,v_c=\\sqrt{\\omega }$ The behaviour of the dynamic system (REF )-() is the same for each specified solution.", "We will take here the following solutions of charged black hole for $f(T)$ Gravity [16] $&&dS_1^2=\\left(1-\\frac{2r_0}{r^2}\\right)dt^2-\\left(1-\\frac{2r_0}{r^2}\\right)^{-2}dr^2-r^2\\left(d\\theta ^2+\\sin ^2\\theta d\\phi ^2\\right)\\,,\\,^{(1)}F^{10}=-\\frac{4r_0^2}{qr^3}\\sqrt{1-\\frac{2r_0}{r^2}}\\,,\\\\&&dS_2^2=\\frac{1}{r^2}\\sqrt{r^4-4e^{r_0}}dt^2-\\frac{r^8}{(r^4-4e^{r_0})^2}dr^2-r^2\\left(d\\theta ^2+\\sin ^2\\theta d\\phi ^2\\right)\\,,\\,^{(2)}F^{10}=-\\frac{16e^{2r_0}}{qr^8}\\left(r^4-4e^{r_0}\\right)^{3/4}\\,,\\\\&&dS_3^2=\\frac{1}{r^2}\\@root 3 \\of {r^6-2r_0}dt^2-\\frac{r^{12}}{(r^6-2r_0)^2}dr^2-r^2\\left(d\\theta ^2+\\sin ^2\\theta d\\phi ^2\\right)\\,,\\,^{(3)}F^{10}=-\\frac{4r_0^2}{qr^{12}}\\left(r^6-2r_0\\right)^{5/6}\\,,\\\\&&dS_4^2=\\left(e^{r_0/2}r-1\\right)^{-2}\\left(dt^2-dr^2\\right)-r^2\\left(d\\theta ^2+\\sin ^2\\theta d\\phi ^2\\right)\\,,\\\\&&^{(4)}F^{10}=-\\frac{r(e^{r_0/2}-2)^2}{q(e^{r_0/2}r-2)^3}\\left(2-4e^{r_0/2}r+e^{r_0}r^2\\right)\\left(e^{r_0/2}r-1\\right)^2e^{r_0/2}\\,,$ where $^{(i)}F^{10}$ with $i=1,...,4$ are the Maxwell tensor components.", "The event horizons are determined by $B^{-1}(r)=0$ , that through (REF )-() are given by $r_{h(1)}=\\sqrt{2r_0}\\,,\\,r_{h(2)}=\\sqrt{2}e^{r_0/4}\\,,\\,r_{h(3)}=\\left(2r_0\\right)^{1/6}\\,,\\,r_{h(4)}=e^{-r_0/2}\\,.$ These are four solutions of charged black holes arising from $f(T)$ theory, with non-linear electrodynamics source obtained in [16].", "In the case of ultra-stiff fluid all solutions behave likewise, this results already indicated in [24].", "So we take specific values of energy to the Hamiltonian $\\mathcal {H}$ and make the diagram in the phase space of the dynamical system.", "The figure REF shows the phase space of the solution (REF ) for the values $\\mathcal {H}=\\lbrace \\mathcal {H}_c-10^{-1},\\mathcal {H}_c,\\mathcal {H}_c+3\\times 10^{-1}\\rbrace $ , with $\\mathcal {H}_c=0.25$ .", "The values of the critical radius and critical velocity are obtained by the equation (REF ), resulting in $r_c=\\sqrt{2r_0}=r_H$ e $v_c=1$ , with $r_0=1$ .", "We note here that the dynamic system (REF )-() does not have really a critical point in the horizon, it is clear from the figure REF , where we have no intersection between the curves.", "The figure REF represent then the phase space of the dynamic system of the first solution, showing the fluid flow near the black hole event horizon.", "We observed from the figure that the fluid motion begins as purely subsonic accretion ($v>-v_c\\equiv -1$ ), where in red curves represent the energy $\\mathcal {H}_c-10^{-1}$ , the blue curves the energy $\\mathcal {H}_c$ , and the green curves the energy $\\mathcal {H}_c+3\\times 10^{-1}$ .", "This accretion indicates that how much further away from the horizon is the fluid, more slower the radial velocity of accretion.", "We see that the top of the diagram represents the ejection movement of the fluid.", "We also note that the closer the horizon is the fluid, faster it is ejects.", "Figure: Representation of the phase space of the solution () for ω=1\\omega =1.The behavior is exactly the same for all solutions reviewed here.", "It is easy to see that the Hamiltonian (REF ) admits global solution to the radial velocity $v$ , with value $v_{\\infty }=0$ ($r\\rightarrow \\infty $ )." ], [ "Ultra-relativistic fluid", "The ultra-relativistic fluid can be classified by $p=(1/2)e$ , then $\\omega =1/2$ .", "The Hamiltonian (REF ), for the solutions (REF )-() is given by $&&\\mathcal {H}_1(r,v)=\\frac{1}{r^2v}\\sqrt{\\frac{1}{(1-v^2)}\\left(1-\\frac{2r_0}{r^2}\\right)}\\,,\\,\\mathcal {H}_2(r,v)=\\frac{1}{r^2v}\\sqrt{\\frac{\\sqrt{r^4-4e^{r_0}}}{(1-v^2)r^2}}\\,,\\\\&&\\mathcal {H}_3(r,v)=\\frac{1}{r^2v}\\sqrt{\\frac{(r^6-2r_0)^{1/3}}{(1-v^2)r^2}}\\,,\\,\\mathcal {H}_4(r,v)=\\frac{1}{r^2v}\\sqrt{\\frac{1}{(1-v^2)(e^{r_0/2}r-1)^2}}\\,.$ The critical speed for the four solutions is always given by $v_c=\\sqrt{\\omega }=0.707107$ .", "The horizons are given by (REF ).", "The critical radius is determined by the equation (REF ), which for the solutions (REF )-() we have $r_{c(1)}=\\sqrt{3r_0}\\,,\\,r_{c(2)}=\\left(6e^{r_0}\\right)^{1/4}\\,,\\,r_{c(3)}=\\left(3r_0\\right)^{1/6}\\,,\\,r_{c(4)}=\\frac{2}{3}e^{-r_0/2}\\,.$ Critical values for the Hamiltonian are obtained for $\\mathcal {H}(r_{c(i)},v_c)=\\mathcal {H}_{c(i)}$ in (REF ) and (), with $i=1,...,4$ .", "We represent the contour, or phase space, to the solutions taking fixed the Hamiltonian.", "For the first and fourth solutions we take $\\mathcal {H}^2_{1}(r,v)=\\lbrace \\mathcal {H}_{c(1)}^2-10^{-2},\\mathcal {H}_{c(1)}^2,\\mathcal {H}_{c(1)}^2+3\\times 10^{-2}\\rbrace $ and $\\mathcal {H}^2_{4}(r,v)=\\lbrace \\mathcal {H}_{c(4)}^2-1000,\\mathcal {H}_{c(4)}^2,\\mathcal {H}_{c(4)}^2+1200\\rbrace $ .", "Already the contours for the solution () are obtained for $\\mathcal {H}^4_{2}(r,v)=\\lbrace \\mathcal {H}_{c(2)}^4-10^{-3},\\mathcal {H}_{c(2)}^4,\\mathcal {H}_{c(2)}^4+3\\times 10^{-3}\\rbrace $ , and for solution () by the expression $\\mathcal {H}^6_{3}(r,v)=\\lbrace \\mathcal {H}_{c(3)}^6-10^{-1},\\mathcal {H}_{c(3)}^6,\\mathcal {H}_{c(3)}^6+2\\rbrace $ .", "We represent the phase space of the solutions in figure REF .", "In the diagram at the top of the left side is the phase space of the first solution (REF ), we have the following types of movement of the fluid: a) red curves.", "Those who are before the critical radius, $r_{c(1)}=1.73205$ , it begins with a supersonic ejection movement ($1>v>v_c$ ), going to subsonic ejection ($v_c>v>0$ ), coming close to the horizon with zero speed, then passing to the movement of a subsonic accretion ($0<v<-v_c$ ), ending in a supersonic accretion ($-v_c>v>-1$ ).", "The curve in the lower part of the diagram begins as an supersonic accretion, passing the accretion for subsonic, ending with zero velocity away from the event horizon.", "The curve at the top of the diagram begins far from the horizon like a subsonic ejection, reaching supersonic ejection away from the horizon to near the speed of light.", "b) the blue curves.", "We have three possibilities here.", "The first is the curve of the region smaller than the critical radius.", "This movement begins as a supersonic ejection, it passes through a critical point of bifurcation unstable, becoming subsonic ejection, which reaches a zero speed on the horizon, becoming subsonic accretion, passing again to an unstable bifurcation, finishing as accretion supersonic.", "The second is that the bottom of the diagram, which begins as a supersonic accretion, through a bifurcation ending as a subsonic accretion.", "The third is the curve of the upper diagram, which begins as a subsonic ejection, through a bifurcation and ending as a supersonic ejection.", "c) the green curves.", "We have three types of movements here.", "At the bottom we have a purely super sonic accretion movement.", "At the top we have a purely supersonic ejection movement.", "Far from the horizon, near zero speed, fluid movement begins as subsonic ejection, increasing the speed to maximum, then decreasing the radial velocity to zero by passing the motion to subsonic accretion, and the farther away the horizon speed tends to zero.", "Figure: Representation of the phase space for solutions () (top left), () (top right), () (lower left) and () (bottom right).", "The sound speed value is given by a 2 =ω=1/2a^2=\\omega =1/2.The movement of fluid to the solutions () and () is identical to the first solution.", "The only differences are the values of the horizon and the critical radius, which are $\\lbrace r_{c(2)}=2.00961,r_{h(2)}=1.81589\\rbrace $ and $\\lbrace r_{c(3)}=1.20094,r_{h(3)}=1.12246\\rbrace $ .", "The movement of the fluid for solution () is quite different from other solutions.", "The phase space for this solution is shown in figure REF at the bottom right.", "The critical radius and the horizon are located in $\\lbrace r_{c(4)}=0.404354,r_{h(4)}=0.606531\\rbrace $ .", "We note that the value of the horizon this time is greater than the critical radius, making it clear in the diagram.", "We can see that we have the following movements for the fluid: a) the red curves, the fluid begins with supersonic accretion, passing for a subsonic accretion.", "Then, the top of the diagram begins as subsonic ejection, and ending supersonic ejection.", "b) the blue curves, whose movement is identical to the red curves.", "c) the green curves.", "The curves in the region $r>r_h$ have the same movement of fluid to the curves in red.", "But the fluid exhibits a cyclic motion within the horizon, where there is a stable critical point.", "This movement can not be observed by an observer outside the event horizon.", "This is the first time it appears such a phenomenon." ], [ "Radiation", "An important model for a fluid is when its total energy density is equivalent to one third pressure, then classify as radiation, and $\\omega =1/3$ .", "The Hamiltonians, arising from (REF ), for this case are $&&\\mathcal {H}_1(r,v)=\\left[\\frac{(r^2-2r_0)}{r^4v(1-v^2)}\\right]^{2/3}\\,,\\,\\mathcal {H}_2(r,v)=\\left[\\frac{\\sqrt{r^4-4e^{r_0}}}{r^4v(1-v^2)}\\right]^{2/3}\\,,\\\\&&\\mathcal {H}_3(r,v)=\\left[\\frac{(r^6-2r_0)^{1/3}}{r^4v(1-v^2)}\\right]^{2/3}\\,,\\,\\mathcal {H}_4(r,v)=\\left[\\left(e^{r_0/2}r-1\\right)^2(1-v^2)r^2v\\right]^{-2/3}\\,.$ The horizons are given by (REF ).", "The critical rays are determined by (REF ) $r_{c(1)}=2\\sqrt{r_0}\\,,\\,r_{c(2)}=\\left(8e^{r_0}\\right)^{1/4}\\,,\\,r_{c(3)}\\left(2\\sqrt{r_0}\\right)^{1/3}=\\,,\\,r_{c(4)}=e^{-r_0/2}/2\\,.$ The critical speed of sound is given by $v_c=\\sqrt{\\omega }=0.57735$ .", "The movements of the fluid for the solutions () and () are similar to the previous section, obtained the contour of the equations $\\mathcal {H}_2^3(r,v)=\\left\\lbrace \\mathcal {H}_{c(2)}^3-10^{-2},\\mathcal {H}_{c(2)}^3,\\mathcal {H}_{c(2)}^3+3\\times 10^{-2}\\right\\rbrace $ and $\\mathcal {H}_4^3(r,v)=\\left\\lbrace \\mathcal {H}_{c(4)}^3-10^{4},\\mathcal {H}_{c(4)}^3,\\mathcal {H}_{c(4)}^3+11\\times 10^{3}\\right\\rbrace $ , with $\\mathcal {H}_{c(2)}^3=0.155199,\\mathcal {H}_{c(4)}^3=12768.3$ .", "The figure REF shows the phase space of these solutions on the right side in the top and lower.", "As the movements are similar to the previous section, we will not comment on them here.", "The main difference is the critical radius for each movement.", "Figure: Representation of the phase space for solutions () (top left), () (top right), () (lower left) and () (bottom right).", "The sound speed value is given by a 2 =ω=1/3a^2=\\omega =1/3.In the case of the left side of the phase diagram at the top of the figure REF , concerning the equation $\\mathcal {H}_1^3(r,v)=\\left\\lbrace \\mathcal {H}_{c(1)}^3-10^{-1},\\mathcal {H}_{c(1)}^3,\\mathcal {H}_{c(1)}^3+3\\times 10^{-1}\\right\\rbrace $ where $\\mathcal {H}_{c(1)}^3=0.105469$ , the movements are divided as follows: a) curves on red.", "The movement begins as supersonic ejection, passing to subsonic ejection, reducing the ejection speed to zero, then going to subsonic accretion, and ending as supersonic accretion.", "Interestingly, there is a movement inside the event horizon inversely to the outside, starting as supersonic accretion and ending as ejection (inside) supersonic.", "b) curves in blue, we have the beginning as supersonic accretion, through an unstable critical point, becoming accretion subsonic.", "At the top is similar.", "The movement begins as subsonic ejection, goes through an unstable critical point, ends as ejection supersonic.", "A novelty is the movement that appears inside the event horizon.", "This movement started as supersonic accretion, it passes subsonic accretion, speed reduces to zero, becoming subsonic ejection, and ending supersonic ejection.", "c) curves on green.", "There is an accretion of movement and other purely supersonic ejection.", "Two other movements, an accretion and other ejection purely subsonic.", "Similarly to other curves, there is a movement within the horizon that begins as supersonic accretion and ends as supersonic ejection.", "In the case of movements to the equation $\\mathcal {H}_3^9(r,v)=\\left\\lbrace \\mathcal {H}_{c(3)}^9-1,\\mathcal {H}_{c(3)}^9,\\mathcal {H}_{c(3)}^9+2\\right\\rbrace $ , where $\\mathcal {H}_{c(3)}^9=4.80542$ , we have a lot of similarity with the previous case, in which the main difference is the existence of motion where the radial coordinate is greater than the critical radius, for curves in red, as shown in figure REF ." ], [ "Sub-relativistic fluid", "Now finally, we discuss the case where the pressure is equal to $p=\\rho /4$ , then $a^2=\\omega =1/4$ .", "This value of the speed of sound characterizes fluid as sub-relativistic.", "In this case, the Hamiltonian for the solutions (REF )-() are given by $&&\\mathcal {H}_1(r,v)=\\frac{1}{r^2\\sqrt{v}}\\left[\\frac{(r^2-2r_0)}{r^2(1-v^2)}\\right]^{3/4}\\,,\\,\\mathcal {H}_2(r,v)=\\frac{1}{r^2\\sqrt{v}}\\left[\\frac{\\sqrt{r^4-4e^{r_0}}}{r^2v(1-v^2)}\\right]^{3/4}\\,,\\\\&&\\mathcal {H}_3(r,v)=\\frac{1}{r^2\\sqrt{v}}\\left[\\frac{(r^6-2r_0)^{1/3}}{r^2v(1-v^2)}\\right]^{3/4}\\,,\\,\\mathcal {H}_4(r,v)=\\frac{1}{r^2\\sqrt{v}}\\left[\\left(e^{r_0/2}r-1\\right)^2(1-v^2)\\right]^{-3/4}\\,.$ The horizons are given by (REF ).", "The critics rays are determined by (REF ) $r_{c(1)}=\\sqrt{5r_0}\\,,\\,r_{c(2)}=\\left(10e^{r_0}\\right)^{1/4}\\,,\\,r_{c(3)}=\\left(5\\sqrt{r_0}\\right)^{1/6}\\,,\\,r_{c(4)}=\\frac{2}{5}e^{-r_0/2}\\,.$ The critical speed of sound is given by $v_c=\\sqrt{\\omega }=0.5$ .", "The figure REF shows the phase space of Hamiltonians (REF ) and ().", "The contours were obtained for the equations $\\mathcal {H}_1^4(r,v)=\\left\\lbrace \\mathcal {H}_{c(1)}^4-10^{-2},\\mathcal {H}_{c(1)}^4,\\mathcal {H}_{c(1)}^4+3\\times 10^{-2}\\right\\rbrace $ where $\\mathcal {H}_{c(1)}^4=0.08192$ ; $\\mathcal {H}_2^8(r,v)=\\left\\lbrace \\mathcal {H}_{c(2)}^8-10^{-3},\\mathcal {H}_{c(2)}^8,\\mathcal {H}_{c(2)}^8+3\\times 10^{-2}\\right\\rbrace $ where $\\mathcal {H}_{c(2)}^8=0.0262795$ ; $\\mathcal {H}_3^4(r,v)=\\left\\lbrace \\mathcal {H}_{c(3)}^4-0.8,\\mathcal {H}_{c(3)}^4,\\mathcal {H}_{c(3)}^4+2\\right\\rbrace $ where $\\mathcal {H}_{c(3)}^4=1.94557$ ; and $\\mathcal {H}_4^4(r,v)=\\left\\lbrace \\mathcal {H}_{c(4)}^4-5\\times 10^{4},\\mathcal {H}_{c(4)}^4,\\mathcal {H}_{c(4)}^4+51\\times 10^{3}\\right\\rbrace $ where $\\mathcal {H}_{c(4)}^4=58656.7$ .", "We can clearly see the figure REF , that the movements of the fluid are similar to those described in subsection REF , so we will not specify them here.", "Figure: Representation of the phase space for solutions () (top left), () (top right), () (lower left) and () (bottom right).", "The sound speed value is given by a 2 =ω=1/4a^2=\\omega =1/4.The spherical accretion of matter by charged black holes on $ f (T) $ Gravity is studied and the conclusion is that it is possible, resulting in several movements for the fluid.", "Considering a perfect isentropic and isothermal fluid, we analyzed the cases of ultra-stiff, ultra-relativistic, radiation and sub-relativistic fluids.", "We consider the contour for each specific value of the Hamiltonian as a constant movement, and represent the phase space for the four analyzed solutions.", "The conclusion follows that it is perfectly possible accretion and ejection of matter spherical type-fluid by the black holes studied.", "The movement of the fluid more deserves emphasis here is the cyclic appearing within the horizon for a given energy of the fluid.", "Our perspective is that these solutions are stable for both a thermodynamic system (thermodynamic stability) and for small perturbations in the geometry (geometric stability).", "This should be the next test for these solutions in future work.", "Acknowledgement: MER thanks UFPA, Edital 04/2014 PROPESP, and CNPq, Edital MCTI/CNPQ/Universal 14/2014, for partial financial support." ] ]
1606.04918
[ [ "Towards Geometric D6-Brane Model Building on non-Factorisable Toroidal\n $\\mathbb{Z}_4$-Orbifolds" ], [ "Abstract We present a geometric approach to D-brane model building on the non-factorisable torus backgrounds of $T^6/\\mathbb{Z}_4$, which are $A_3 \\times A_3$ and $A_3 \\times A_1 \\times B_2$.", "Based on the counting of `short' supersymmetric three-cycles per complex structure {\\it vev}, the number of physically inequivalent lattice orientations with respect to the anti-holomorphic involution ${\\cal R}$ of the Type IIA/$\\Omega\\cal{R}$ orientifold can be reduced to three for the $A_3 \\times A_3$ lattice and four for the $A_3 \\times A_1 \\times B_2$ lattice.", "While four independent three-cycles on $A_3 \\times A_3$ cannot accommodate phenomenologically interesting global models with a chiral spectrum, the eight-dimensional space of three-cycles on $A_3 \\times A_1 \\times B_2$ is rich enough to provide for particle physics models, with several globally consistent two- and four-generation Pati-Salam models presented here.", "We further show that for fractional {\\it sLag} three-cycles, the compact geometry can be rewritten in a $(T^2)^3$ factorised form, paving the way for a generalisation of known CFT methods to determine the vector-like spectrum and to derive the low-energy effective action for open string states." ], [ "Introduction", "Ever since the finding in 1985 that string theory constitutes a framework for unifying quantum field theory and gravity, which - when compactified on a Calabi-Yau-threefold or some singular limit thereof such as a toroidal orbifold - leads to ${\\cal N}=1$ supersymmetry in four dimensions [1], the search for vacuum configurations with not only the Standard Model spectrum but also its interactions has been intensively pursued.", "Starting from sporadic models of the $E_8 \\times E_8$ heterotic string such as [2], [3], by implementing systematic computer searches, large classes of vacua with particle physics spectra on $T^6/\\mathbb {Z}_N$ and $T^6/\\mathbb {Z}_N \\times \\mathbb {Z}_M$ orbifolds [4], [5], [6] (see also [7] for the heterotic $SO(32)$ string theory) and Calabi-Yau manifolds [8], [9], [10], [11], [12] could be constructed.", "With the identification of D-branes as dynamical objects in Type I and II string theory [13] in 1995, model building also opened up in these theories, which are conjectured to be related by S-duality [14], [15] and M-/F-theory duality [16], [17] to the vacua of the heterotic string.", "The virtue of D-brane model building lies in the fact that physical quantities, like e.g.", "the number of particle generations, are expressed in terms of topology and geometry of the compact space, which is particularly intuitive for the case of D6-branes on three-cycles in Type IIA orientifold compactifications, see e.g.", "[18] for a broad recent overview.", "This intuition, however, comes at the cost of the relatively little explored symplectic structure of generic Calabi-Yau threefolds.", "Since supersymmetric - or mathematically expressed special Lagrangian (sLag) - three-cycles constitute a largely unexplored area (except for [19], [20] and [21]), intersecting D6-brane models in Type IIA orientifolds have focussed on tori and toroidal orbifolds, and more specifically on backgrounds which are factorised into two-tori, $T^6=(T^2)^3$ , see e.g.", "[22], [23], [24], and $(T^2)^3/\\Gamma $ with Abelian point groups $\\Gamma = \\mathbb {Z}_N$ or $\\mathbb {Z}_N \\times \\mathbb {Z}_M$ .", "In the latter case, models with all three-cycles inherited from the underlying torus have been constructed for $\\Gamma =\\mathbb {Z}_2 \\times \\mathbb {Z}_2$  [25], [26], [27], [28], [29], [30] and $\\mathbb {Z}_2 \\times \\mathbb {Z}_4$  [31], [32], [33] without discrete torsion, while fractional three-cycles consisting of components inherited from the torus plus exceptional divisors at orbifold singularities have been employed for $\\Gamma =\\mathbb {Z}_4$  [34], $\\mathbb {Z}_6$  [35], [33], [36], [37], $\\mathbb {Z}_6^{\\prime }$  [38], [39], [40], [41], [42], [37] and $\\mathbb {Z}_{12-II}$  [43] with a single $\\mathbb {Z}_2$ sector and $\\Gamma =\\mathbb {Z}_2 \\times \\mathbb {Z}_2$  [44], [45], $\\mathbb {Z}_2 \\times \\mathbb {Z}_6^{\\prime }$  [45], [46], [47] and $\\mathbb {Z}_2 \\times \\mathbb {Z}_6$  [45], [48], [49], [50] with discrete torsion with two $\\mathbb {Z}_2$ subgroups.", "All of the above mentioned types of string vacua contain a plethora of scalar fields with flat directions.", "While the above models with a $\\mathbb {Z}_2 \\times \\mathbb {Z}_2$ subsymmetry allow for rigid fractional three-cycles which, in the open string spectrum, provide gauge groups without brane recombination/splitting moduli in the adjoint representation, it has recently been noticed that (most) twisted complex structure moduli associated to deformations of singularities are in fact stabilised by the existence of D-branes with $U(1)$ symmetries [51], [52], [50], [53].", "To further stabilise the dilaton and untwisted complex structure moduli, one usually argues that closed string background NS-NS fluxes (see [54], [55], [56] for reviews) provide a non-trivial scalar potential, see also [57], [58] for attempts to incorporate NS-NS fluxes on the factorisable $T^6/\\mathbb {Z}_4$ orbifold and [59] for the factorisable $T^6/\\mathbb {Z}_6^{\\prime }$ orbifold.", "However, incorporating a non-trivial NS-NS flux $H_3$ will in general violate the factorisation into two-tori and instead lead to so-called non-factorisable torus backgrounds [60], [61], [62].", "Orbifolds of non-factorisable tori have, to our best knowledge, scarcely been considered in the literature.", "Within Type IIA orientifolds, one of the first studies of non-factorisable $\\mathbb {Z}_N$ orbifolds can be found in [63], where special configurations of D6-branes on top of the O6-planes lead to a local cancellation of the RR tadpoles within the compact space.", "In [64] a similar analysis for $\\mathbb {Z}_N\\times \\mathbb {Z}_M$ orbifolds was performed.", "In [65], [66] orientifolds of $T^6/(\\mathbb {Z}_2\\times \\mathbb {Z}_2)$ with (non-)factorisable lattices were considered, including D6-branes which are not parallel to the O6-planes, and in [43], three-cycles on the $D_4 \\times A_2$ and $D_4 \\times A_1 \\times A_1$ lattices with $\\mathbb {Z}_{12-I}$ and $\\mathbb {Z}_{12-II}$ orbifold symmetry, respectively, were studied.", "Finally, in [67] the Yukawa couplings for a torus generated by a $D_6$ lattice were computed.", "Here, we will for the first time perform a thorough study of all possible sLag three-cycles on the two different non-factorisable lattice backgrounds $A_3 \\times A_3$ and $A_3 \\times A_1 \\times B_2$ of $T^6/\\mathbb {Z}_4$ , for which we briefly provided some preliminary results in [68].", "Besides from being able to classify sLag cycles on toroidal orbifolds, these geometrically simple backgrounds are equipped with the non-negligible power of allowing for an explicit string quantisation and thus Conformal Field Theory (CFT) techniques, which do not only reproduce the RR tadpole cancellation conditions and chiral spectrum, but are indispensable for distinguishing gauge group enhancements $U(N) \\hookrightarrow SO(2N)$ or $USp(2N)$ for D-branes wrapped on orientifold invariant three-cycles and for deriving the low-energy effective action (for particle physics models based on powerful RCFT techniques see e.g.", "also [69], [70], [71], [72]).", "So far, CFT results within the geometrically intuitive approach to Type II string model building with D-branes have been obtained only for gauge couplings at one-loop and $n$ -point couplings at tree-level using bulk cycles on the factorisable six-torus $(T^2)^3$  [73], [74], [75], [76], [77], [78].", "For fractional cycles on $(T^2)^3/\\mathbb {Z}_N$ and $\\mathbb {Z}_N \\times \\mathbb {Z}_M$ orbifolds [79], [37], [80], [81], the one-loop corrections to gauge couplings and the Kähler potential at leading order could be derived, while the one-loop corrections to the open string Kähler potential are only known for bulk cycles on such orbifolds [82], see also [45], [46], [83], [48] for the distinction of $U(N) \\hookrightarrow SO(2N)$ versus $USp(2N)$ gauge group enhancement using one-loop gauge threshold computations.", "The aim of the present work is not only to generalise the geometric methods of deriving the chiral spectrum from topological intersection numbers, but also to initiate the generalisation of CFT techniques to so-called non-factorisable lattice backgrounds.", "The outline of this article is as follows: in section  we first study the geometry of $T^6/\\mathbb {Z}_4$ on non-factorisable tori, then implement anti-holomorphic involutions on the lattices to study orientifolds of Type IIA string theory in section , after which we proceed to discuss first hints on physical equivalences among different choices, and finally we study supersymmetric D6-branes wrapped on (fractional) three-cycles.", "In section , we argue that any supersymmetric fractional three-cycle on $T^6/\\mathbb {Z}_4$ can be written in a factorised form, which paves the way for implementing CFT methods in order to distinguish gauge group enhancements $U(N) \\hookrightarrow SO(2N)$ versus $USp(2N)$ and to derive the vector-like matter spectrum.", "We then proceed to provide some explicit Pati-Salam models with two and four particle generations in section .", "Section  contains our conclusions and outlook, and appendix  contains some further explicit examples of globally consistent D6-brane configurations with chiral matter on the $A_3 \\times A_1 \\times B_2$ lattice." ], [ "Non-factorisable $T^6/\\mathbb {Z}_4$ orbifold geometries", "In this section, we discuss the three-cycle geometry on the two non-factorisable background lattices, $A_3\\times A_3$ and $A_3\\times A_1\\times B_2$ , of $T^6/\\mathbb {Z}_4$ .", "The $\\mathbb {Z}_4$ -action is usually encoded in the shift vector $\\vec{\\zeta }=\\frac{1}{4}(1,-2,1)$ if the six-torus is parameterized by three complex coordinates, and more generally the $\\mathbb {Z}_4$ -action is generated by the Coxeter element $Q$ which acts on the root lattice of the corresponding orbifold spanned by the simple roots $\\lbrace e_i\\rbrace _{i=1,\\dots ,6}$ .", "Furthermore, we denote the six toroidal one-cycles along the directions $\\lbrace e_i\\rbrace $ by $\\pi _i$ and toroidal two- and three-cycles by $\\pi _{ij}:=\\pi _i\\wedge \\pi _j$ and $\\pi _{ijk}:=\\pi _i\\wedge \\pi _j\\wedge \\pi _k$ , respectively.", "The Hodge numbers of all three possible lattice backgrounds - one factorisable and two non-factorisable ones - are summarized in table REF (cf.", "e.g.", "[84]).", "Table: Summary of the Hodge numbers per untwisted and twisted sector of the T 6 /ℤ 4 T^6/\\mathbb {Z}_4 orbifolds on factorisable and non-factorisable tori.We are in particular interested in characterization of the three-homology $H_3(T^6/\\mathbb {Z}_4,\\,\\mathbb {Z})$ of each background lattice.", "This homology class contains in general the $\\mathbb {Z}_4$ -invariant bulk $\\pi ^\\text{bulk}$ and exceptional three-cycles $\\pi ^\\text{exc}$ as well as fractional linear combinations thereof, the so-called fractional three-cycles $\\pi ^\\text{frac}$ .", "The bulk three-cycles are inherited from the underlying torus and can be computed by taking the $\\mathbb {Z}_4$ -orbits thereof: $\\pi ^\\text{bulk}:=\\sum _{i=0}^{i=3}Q^i\\pi ^\\text{torus}\\,.$ The exceptional three-cycles arise for the $T^6/\\mathbb {Z}_4$ orbifolds only in the $\\mathbb {Z}_2$ -twisted sector.", "They stem from the resolution of the $\\mathbb {Z}_2$ -invariant two-tori and can be written as a product of an exceptional two-cycle $\\mathbf {e}_{\\alpha \\beta }$ (with $\\alpha \\beta $ labelling the location of the cycle) and a $\\mathbb {Z}_2$ -invariant toroidal one-cycle (plus some $\\mathbb {Z}_4$ -image).", "Finally, fractional three-cycles are either one-half of a bulk cycle or linear combinations of one-half of some bulk and exceptional three-cycles with the combinatorics depending on the corresponding singularities traversed by the bulk cycles as well as sign factors associated to the choice of some discrete Wilson line, or geometrically speaking the orientation the corresponding exceptional three-cycle is wrapped.", "Based on the discussion in this section, we will proceed to discuss (supersymmetric) Type IIA orientifolds with O6-planes and D6-branes and the associated anti-holomorphic involution on the background geometry in section ." ], [ "$B_2 \\times (A_1)^2 \\times B_2$", "Before investigating the three-cycle geometry of the non-factorisable lattices, we briefly review three-cycles on the factorisable background on the group lattice $B_2 \\times (A_1)^2 \\times B_2$ as first discussed in [34], see e.g.", "also the appendix of [45] for the Hodge numbers per twist sector displayed in table REF .", "Although the $B_2$ -torus is a square-torus, we take the positive simple roots of the $B_2$ -Lie algebra as basis of the torus lattice (see figure REF ).", "The $\\mathbb {Z}_4$ -action is then generated by the Coxeter element $Q$ : $Q:=\\begin{pmatrix}1&-1\\\\2&-1\\end{pmatrix}\\oplus \\begin{pmatrix}-1&0\\\\0&-1\\end{pmatrix}\\oplus \\begin{pmatrix}1&-1\\\\2&-1\\end{pmatrix}\\,.$ acting on three two-tori $T^2_{(i)}$ .", "A basis of bulk three-cycles on the factorisable lattice is given by $\\begin{aligned}\\gamma _1 := \\sum _{k=0}^3 Q^k (\\pi _{235}+\\pi _{236})= 2 \\, (\\pi _{136} +\\pi _{235}+2\\pi _{236}),\\quad \\gamma _2 := -\\sum _{k=0}^3 Q^k \\pi _{236} = 2 \\, (\\pi _{135}+\\pi _{136}+ \\pi _{235}),\\\\\\bar{\\gamma }_1 := \\sum _{k=0}^3 Q^k (\\pi _{245}+\\pi _{246})= 2 \\, (\\pi _{146} +\\pi _{245}+2\\pi _{246}),\\quad \\bar{\\gamma }_2 := -\\sum _{k=0}^3 Q^k \\pi _{246} =2 \\, (\\pi _{145}+\\pi _{146}+ \\pi _{245}).\\end{aligned}$ An arbitrary bulk three-cycle can be represented by the (pairwise co-prime) toroidal wrapping numbers $(n^i,m^i)_{i=1,2,3}$ , $\\begin{aligned}\\pi ^{\\text{bulk}} =& \\sum _{k=0}^3 Q^k \\left(\\bigwedge _{i=1}^3 (n^i \\pi _{2i-1} + m^i \\pi _{2i} )\\right)\\\\=& A n^2 \\, \\gamma _1 + A m^2 \\, \\bar{\\gamma }_1 + B n^2 \\, \\gamma _2 + B m^2 \\, \\bar{\\gamma }_2,\\end{aligned}$ where on the second line the $\\mathbb {Z}_4$ invariant bulk wrapping numbers with $A := n^1m^3+m^1n^3-2n^1n^3,\\qquad B := n^1 m^3 + m^1n^3-m^1m^3,$ have been used.", "The basic non-vanishing bulk intersection numbers are computed from $\\pi _a^{\\text{bulk}} \\circ \\pi _b^{\\text{bulk}} \\equiv \\frac{1}{4} \\left(\\sum _{i=0}^3 Q^i \\pi ^{\\text{torus}}_a \\right) \\circ \\left(\\sum _{i=0}^3 Q^i \\pi ^{\\text{torus}}_b \\right)= \\pi ^{\\text{torus}}_a \\circ \\left(\\sum _{i=0}^3 Q^i \\pi ^{\\text{torus}}_b \\right)$ and read $\\gamma _i \\circ \\bar{\\gamma }_j = 2 \\, \\delta _{ij}.$ The bulk three-cycles thus do not form an unimodular basis.", "In addition to the bulk three-cycles, there exist twelve exceptional cycles with basis $\\varepsilon _i = (e_{\\alpha \\beta } - e_{Q(\\alpha )Q(\\beta )}) \\wedge \\pi _3,\\qquad \\bar{\\varepsilon }_i = (e_{\\alpha \\beta } - e_{Q(\\alpha )Q(\\beta )}) \\wedge \\pi _4$ (where $\\alpha $ and $\\beta $ denote the $\\mathbb {Z}_2$ -invariant points on the four-torus $T^2_{(1)}\\times T^2_{(3)}$ and $Q(\\alpha )$ and $Q(\\beta )$ their $\\mathbb {Z}_4$ images), and with non-vanishing intersection numbers $\\varepsilon _i \\circ \\bar{\\varepsilon }_j = - 2 \\, \\delta _{ij}.$ Fractional three-cycles of the form $\\pi ^{\\text{frac}} = \\frac{1}{2} \\left(\\pi ^{\\text{bulk}} +(-1)^{\\tau ^{\\mathbb {Z}_2}} \\sum _{\\text{fixed set of } i} [\\pm (n^2 \\varepsilon _i + m^2 \\bar{\\varepsilon }_i)]\\right),$ with the $\\mathbb {Z}_2$ eigenvalue $(-1)^{\\tau ^{\\mathbb {Z}_2}}$ parametrised by $\\tau ^{\\mathbb {Z}_2} \\in \\lbrace 0,1\\rbrace $ and the sum over four $i$ such that the product of the signs $\\pm $ gives +1, then generate the unimodular sixteen dimensional basis of three-cycles." ], [ "$A_3 \\times A_3$", "We start the discussion of non-factorisable $\\mathbb {Z}_4$ -orbifolds with the lattice of the type $A_3\\times A_3$ (see figure REF ).", "Figure: T 6 /ℤ 4 T^6/\\mathbb {Z}_4-orbifold on the A 3 ×A 3 A_3\\times A_3-lattice and its ℤ 2 \\mathbb {Z}_2-fixed lines (red in red).The $\\mathbb {Z}_4$ group acts by the Coxeter element $Q$ on the vectors $\\lbrace e_i\\rbrace _{i=1,\\dots ,6}$ , which span the six-torus: $ \\begin{array}{cccc}&Qe_1=e_2\\,,&Qe_2=e_3\\,,&Qe_3=-e_1-e_2-e_3\\,,\\\\&Qe_4=e_5\\,,&Qe_5=e_6\\,,&Qe_6=-e_4-e_5-e_6\\,,\\end{array}$ which can be written in the matrix form $ Q:=\\begin{pmatrix}0&0&-1&0&0&0\\\\1&0&-1&0&0&0\\\\0&1&-1&0&0&0\\\\0&0&0&0&0&-1\\\\0&0&0&1&0&-1\\\\0&0&0&0&1&-1\\end{pmatrix}\\,.$ This action forms a discrete subgroup of $SU(3)$ and thus preserves $\\mathcal {N}=2$ supersymmetry in four dimensions when compactifying Type II string theory.", "The Hodge numbers of this orbifold are (see e.g.", "[84]) $h_{21}=h^\\text{untw}_{21}+h_{21}^{\\mathbb {Z}_2}=1+0$ and $h_{11}=h^\\text{untw}_{11}+h_{11}^{\\mathbb {Z}_4}+h_{11}^{\\mathbb {Z}_2}=5+16+4$ as displayed in the middle row in table REF .", "Due to $h_{21}^{\\mathbb {Z}_2}=0$ , the three-homology of this orbifold contains only (fractions of) bulk three-cycles and is four-dimensional.", "For the $Q$ -action to be an isometry of the lattice, one has to require the invariance of the scalar product $Q^tgQ=g$ [85], which restricts the shape of the metric of the six-torus to $g=\\left(\\begin{array}{cccccc}R_1^2 & a R_1^2 & -(2 a+1) R_1^2 & e R_1 R_2 & s R_1 R_2 & c R_1R_2 \\\\a R_1^2 & R_1^2 & a R_1^2 & d R_1 R_2 & e R_1 R_2 & s R_1 R_2 \\\\-(2 a+1) R_1^2 & a R_1^2 & R_1^2 & c R_1 R_2 & d R_1 R_2 & e R_1 R_2 \\\\e R_1 R_2 & d R_1 R_2 & c R_1 R_2 & R_2^2 & b R_2^2 & -(2 b+1) R_2^2 \\\\s R_1 R_2 & e R_1 R_2 & d R_1 R_2 & b R_2^2 & R_2^2 & b R_2^2 \\\\c R_1 R_2 & s R_1 R_2 & e R_1 R_2 & -(2 b+1) R_2^2 & b R_2^2 &R_2^2 \\\\\\end{array}\\right)$ with $s:=-(c+d+e)$ .", "The moduli $R_1$ and $R_2$ describe the radii of $A_3\\times A_3$ whereas $a$ $(b)$ specifies the cosine of the angle between the vectors $e_1$ and $e_2$ ($e_4$ and $e_5$ ) and $c,d$ and $e$ specify the cosines of the angles between $e_4$ and the vectors $e_3$ , $e_2$ and $e_1$ , respectively.", "To describe the three-cycles on the six-torus $T^6$ we use the usual notation $\\pi _{ikl}=\\pi _i\\wedge \\pi _k\\wedge \\pi _l$ .", "Due to the action of $Q$ , it suffices to consider the three-cycles which wrap a two-cycle on one $A_3$ -torus and a one-cycle on the other one.", "Such cycles can a priori be described by twelve wrapping numbers $(m^i,n^i,p^i,q^i,r^i,s^i)^{i=1,2}$ : $ \\pi ^\\text{torus}:=(m^1\\pi _1+n^1\\pi _2+p^1\\pi _3)\\wedge (m^2\\pi _1+n^2\\pi _2+p^2\\pi _3+q^2\\pi _4+r^2\\pi _5+s^2\\pi _6)\\wedge (q^1\\pi _4+r^1\\pi _5+s^1\\pi _6)\\,.$ By taking orbits of the $Q$ -action, the basis of $\\mathbb {Z}_4$ -invariant three-cycles is given by $\\begin{array}{cll}\\gamma _1&:=&\\sum _{i=0}^3Q^i\\pi _{124}=-\\pi _{125}-\\pi _{126}-\\pi _{134}-\\pi _{135}+\\pi _{235}+\\pi _{236}\\,,\\\\\\gamma _2&:=&\\sum _{i=0}^3Q^i\\pi _{125}=\\pi _{124}+\\pi _{125}-\\pi _{135}-\\pi _{136}-\\pi _{234}-\\pi _{235}\\,,\\\\\\bar{\\gamma }_1&:=&\\sum _{i=0}^3Q^i\\pi _{145}=-\\pi _{146}-\\pi _{245}-\\pi _{246}+\\pi _{256}-\\pi _{345}+\\pi _{356}\\,,\\\\\\bar{\\gamma }_2&:=&\\sum _{i=0}^3Q^i\\pi _{245}=\\pi _{145}-\\pi _{156}+\\pi _{245}-\\pi _{246}-\\pi _{256}-\\pi _{346}\\,.\\end{array}$ Using the ansatz $(\\ref {2.toroidal_3-cycle})$ for a toroidal three-cycle, we can compute the corresponding bulk cycle with the orbifold map $\\sum _{i=0}^3Q^i$ and decompose it in the basis $(\\ref {2.bulk_basis1})$ : $\\begin{split}\\pi ^\\text{bulk}&=\\left(A_1(q^1-s^1)-A_2(q^1+r^1-s^1)+A_3r^1\\right)\\gamma _1\\\\&+\\left(A_1r^1+A_2(q^1-r^1-s^1)-A_3(q^1-s^1)\\right)\\gamma _2\\\\&+\\left(B_1(m^1-p^1)-B_2(m^1+n^1-p^1)+B_3n^1\\right)\\bar{\\gamma }_1\\\\&+\\left(B_1n^1+B_2(m^1-n^1-p^1)-B_3(m^1-p^1)\\right)\\bar{\\gamma }_2\\end{split}$ with $\\begin{array}{ccc}A_1:=m^1n^2-n^1m^2\\,,\\quad &A_2:=m^1p^2-p^1m^2\\,,\\quad &A_3:=n^1p^2-p^1n^2\\,,\\\\B_1:=q^2r^1-r^2q^1\\,,\\quad &B_2:=q^2s^1-s^2q^1\\,,\\quad &B_3:=r^2s^1-s^2r^1\\,.\\end{array}$ It is easy to verify that the non-vanishing intersection numbers between the basis elements (REF ) are given by $\\gamma _i\\circ \\bar{\\gamma }_j=-\\delta _{ij},$ where we used analogously to the factorised lattice background in section REF that we can define the intersection number between two toroidal three-cycles $\\pi ^{\\text{torus}}_a$ and $\\pi ^{\\text{torus}}_b$ on $T^6$ as $\\pi ^{\\text{torus}}_a\\circ \\pi ^{\\text{torus}}_b=\\pi ^{\\text{torus}}_a \\wedge \\pi ^{\\text{torus}}_b/\\text{Vol}(T^6)$ to arrive at the intersection number between two $\\mathbb {Z}_4$ -invariant bulk three-cycles $\\pi _a\\circ \\pi _b=\\frac{1}{4}\\bigl (\\sum _{i=0}^{3}Q^i\\pi ^{\\text{torus}}_a\\bigr )\\circ \\bigl (\\sum _{i=0}^{3}Q^i\\pi ^{\\text{torus}}_b\\bigr ).$ Therefore, $\\lbrace \\gamma _1,\\,\\gamma _2,\\,\\bar{\\gamma }_1,\\,\\bar{\\gamma }_2\\rbrace $ already builds the unimodular basis.", "Despite this and the fact that this orbifold does not have any exceptional three-cycles, in order to compare with the other non-factorisable lattice in section REF below, we can consider the special class of three-cycles, for which the toroidal building blocks are $\\mathbb {Z}_2$ -invariant, i.e.", "fractional three-cyclesThis construction will be used in the next sections.", "We will see that if both $A_3$ -tori are orthogonal to each other, any fractional three-cycle is Lagrangian.", "Note also that for $T^6/(\\mathbb {Z}_2 \\times \\mathbb {Z}_2)$ without discrete torsion, the unimodular basis is constructed analogously using $\\mathbb {Z}_2 \\times \\mathbb {Z}_2$ -invariant three-cycles $\\pi ^{\\text{torus}}=\\frac{1}{4}\\pi ^{\\text{bulk}}$ , cf.", "e.g. [86].", "However, since $Q$ as defined in (REF ) permutes toroidal one-cycles on $T^6/\\mathbb {Z}_4$ non-trivially, here we have to restrict to a special subclass of all a priori allowed bulk three-cycles.", "$\\pi ^\\text{frac}=\\frac{1}{2}\\pi ^\\text{bulk}.$ Using the $Q$ -transformation of the wrapping numbers $\\left(\\begin{array}{ccc}m^i\\,,&n^i\\,,&p^i\\\\q^i\\,,&r^i\\,,&s^i\\end{array}\\right)^{i=1,2}\\quad \\stackrel{Q}{\\longrightarrow } \\quad \\left(\\begin{array}{ccc}-p^i\\,,&m^i-p^i\\,,&n^i-p^i\\\\-s^i\\,,&q^i-s^i\\,,&r^i-s^i\\end{array}\\right)^{i=1,2},$ we obtain the following condition.", "Any toroidal three-cycle $\\pi ^\\text{torus}$ of the form $(\\ref {2.toroidal_3-cycle})$ is $\\mathbb {Z}_2$ -invariant if and only if $\\begin{array}{ccc}r^1(A_1+A_3)=0\\,,\\quad &n^1(B_1+B_3)=0\\,,\\\\(s^1-r^1)A_3=q^1A_1\\,,\\quad &(p^1-n^1)B_3=m^1B_1\\,,\\\\(A_1+A_3)(s^1-q^1)=0\\,,\\quad &(B_1+B_3)(m^1-p^1)=0\\,,\\\\(A_1+A_3)s^1=(q^1-r^1+s^1)A_2\\,,\\quad &(B_1+B_3)p^1=(m^1-n^1+p^1)B_2.\\end{array}$ Among this special class of three-cycles from $\\mathbb {Z}_2$ -invariant toroidal cycles, there are also those that satisfy $Q\\pi ^\\text{torus}=-\\pi ^\\text{torus}$ .", "They do not contribute to the bulk cycles, or in other words are trivial in the $H_3(T^6/\\mathbb {Z}_4,\\mathbb {Z})$ homology.", "Taking into account this fact, we can reduce the conditions (REF ) to $\\begin{array}{cccc}A_1+A_3=0\\,,\\quad &B_1+B_3=0\\,,&\\\\(q^1-r^1+s^1)A_i=0\\,,\\quad &(m^1-n^1+p^1)B_i=0& \\text{for all }i\\,.\\end{array}$ Note that due to the above $\\mathbb {Z}_2$ -invariance constraints on the wrapping numbers, the basis of these fractional three-cycles coincides with the basis $\\lbrace \\gamma _1,\\,\\gamma _2,\\,\\bar{\\gamma }_1,\\,\\bar{\\gamma }_2\\rbrace $ .", "Let us for example consider the toroidal three-cycle $\\pi =(\\pi _1+\\pi _2)\\wedge (\\pi _1+\\pi _3)\\wedge (\\pi _4-\\pi _6)$ .", "Obviously, it is $\\mathbb {Z}_2$ -invariant and gives rise to the fractional three-cycle $\\pi ^\\text{frac}=\\pi +Q\\pi =-2\\gamma _1$ on $T^6/\\mathbb {Z}_4$ .", "Although the original cycle has coprime wrapping numbers and no other toroidal three-cycle $\\pi ^\\prime $ exists such that $\\pi =2\\pi ^\\prime $ , the corresponding bulk cycle $-2\\gamma _1$ is non-coprime.", "This will play a role for computing the gauge group in the section REF ." ], [ "$A_3 \\times A_1\\times B_2$", "Now we consider three-cycles on $T^6/\\mathbb {Z}_4$ with the lattice of type $A_3\\times A_1\\times B_2$ .", "The $\\mathbb {Z}_4$ -action is generated by the Coxeter element which acts on the root lattice spanned by the simple roots $\\lbrace e_i\\rbrace _{i=1,\\dots ,6}$ in the following way $ \\begin{array}{llll}&Qe_1=e_2\\,,&Qe_2=e_3\\,,&Qe_3=-e_1-e_2-e_3\\,,\\\\&Qe_4=-e_4\\,,&Qe_5=e_5+2e_6\\,,&Qe_6=-e_5-e_6\\,,\\end{array}$ which can be cast in the matrix form $Q:=\\left(\\begin{array}{cccccc}0 & 0 & -1 & 0 & 0 & 0 \\\\1 & 0 & -1 & 0 & 0 & 0 \\\\0 & 1 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & -1 \\\\0 & 0 & 0 & 0 & 2 & -1 \\\\\\end{array}\\right)\\,.$ Again, this action corresponds to a discrete subgroup of $SU(3)$ , and thus preserves $\\mathcal {N}=2$ supersymmetry in four dimensions when considering Type II string theory compactifications.", "The Hodge numbers of this orbifold are [84] $h_{21}=h^\\text{untw}_{21}+h_{21}^{\\mathbb {Z}_2}=1+2$ and $h_{11}=h^\\text{untw}_{11}+h_{11}^{\\mathbb {Z}_4}+h_{11}^{\\mathbb {Z}_2}=5+16+6$ (see table REF ).", "Thus, we expect four bulk and four exceptional three-cycles on this orbifold.", "From solving the equation $Q^tgQ=g$ we obtain $g:=e_i\\cdot e_j=\\begin{pmatrix}R_3^2&aR_3^2&-(1+2a)R_3^2&dR_3R_1&bR_3R_2&cR_3R_2\\\\aR_3^2&R_3^2&aR_3^2&-dR_3R_1&-(b+2c)R_3R_2&(b+c)R_3R_2\\\\-(1+2a)R_3^2&aR_3^2&R_3^2&dR_3R_1&-bR_3R_2&-cR_3R_2\\\\dR_3R_1&-dR_3R_1&dR_3R_1&R_1^2&0&0\\\\bR_3R_2&-(b+2c)R_3R_2&-bR_3R_2&0&2R_2^2&-R_2^2\\\\cR_3R_2&(b+c)R_3R_2&-cR_3R_2&0&-R_2^2&R_2^2\\end{pmatrix}.$ The real positive moduli $R_3$ , $R_1$ and $R_2$ describe the radii of $A_3\\times A_1 \\times B_2$ , respectively, and $a$ , $b$ , $c$ and $d$ specify the cosines of angles between the vectors of the lattice.", "More precisely, $a$ is the cosine of the angle between the vectors $e_1$ and $e_2$ , $d$ the cosine of the angle between $e_1$ and $e_4$ , $b$ ($c$ ) is the cosine of the angle between $e_1$ and $e_5$ ($e_6$ ).", "Bulk three-cycles We make the ansatz that any toroidal three-cycle is factorisable in the sense that it can be characterised a priori by ten wrapping numbers $(m^1,n^1,p^1,q^1)\\times (m^2,n^2,p^2,q^2)\\times (m^3,n^3)$ and written as $ \\pi ^{\\text{torus}}:=\\bigwedge _{i=1}^2(m^i\\pi _1+n^i\\pi _2+p^i\\pi _3+q^i\\pi _4) \\wedge (m^3\\pi _5+n^3\\pi _6)\\,.$ The last doublet $(m^3,n^3)$ gives us the one-cycle on the $B_2$ -torus.", "The two quadruplets $(m^i,n^i,p^i,q^i)$ parametrise the two-cycle on $A_3\\times A_1$ .", "It is easy to see that the representation of three-cycles by this ansatz is not unique, i.e.the same three-cycle can be described by different wrapping numbers, e.g.", "when permuting the indices $i=1$ and 2.", "By taking orbits of the $Q$ -action, we can define a basis of the $\\mathbb {Z}_4$ -invariant bulk three-cycles $ \\begin{split}\\gamma _1&:=-\\sum _{i=0}^3 Q^i\\pi _{136}=2(\\pi _{125}+\\pi _{126}-\\pi _{136}-\\pi _{235}-\\pi _{236})\\,,\\\\\\gamma _2&:=-\\sum _{i=0}^3 Q^i\\pi _{125}=2(\\pi _{126}+\\pi _{135}+\\pi _{136}-\\pi _{236})\\,,\\\\\\bar{\\gamma }_1&:=\\sum _{i=0}^3 Q^i\\pi _{146}=\\pi _{145}+2\\pi _{146}+2\\pi _{245}+2\\pi _{246}+\\pi _{345}\\,,\\\\\\bar{\\gamma }_2&:=\\sum _{i=0}^3 Q^i\\pi _{246}=-\\pi _{145}+2\\pi _{246}+\\pi _{345}+2\\pi _{346}\\,.\\end{split}$ Note that here the linear combinations $\\frac{1}{2}(\\gamma _1\\pm \\gamma _2)$ are also bulk cycles.", "The decomposition in the basis $\\lbrace \\gamma _{1,2},\\,\\bar{\\gamma }_{1,2}\\rbrace $ of any bulk three-cycle inherited from a toroidal one of the type $(\\ref {2.torus_cycle})$ is given by $ \\begin{split}\\pi ^{\\text{bulk}}=& P\\gamma _1+Q\\gamma _2+\\bar{P}\\bar{\\gamma }_1 +\\bar{Q}\\bar{\\gamma }_2\\\\=&[(A_2-A_3)m^3+\\frac{1}{2}(A_1-2A_2+A_3)n^3]\\gamma _1+[(A_2-A_1)m^3+\\frac{1}{2}(A_1-A_3)n^3]\\gamma _2+\\\\&[(-B_1+B_2+B_3)m^3+(B_1-B_3)n^3]\\bar{\\gamma }_1+[(-B_1-B_2+B_3)m^3+B_2n^3]\\bar{\\gamma }_2\\, ,\\end{split}$ with $\\begin{split}A_1&:=m^1n^2-n^1m^2,\\quad B_1:=m^1q^2-q^1m^2,\\\\A_2&:=m^1p^2-p^1m^2,\\quad \\: B_2:=n^1q^2-q^1n^2,\\\\A_3&:=n^1p^2-p^1n^2,\\quad \\;\\;\\: B_3:=p^1q^2-q^1p^2\\,.\\end{split}$ Using the formula for bulk intersection numbers (REF ), we obtain for the bulk basis of $(\\ref {2.bulk_basis})$ $ \\gamma _i\\circ \\bar{\\gamma }_j=-2\\delta _{ij},\\quad \\gamma _i\\circ \\gamma _j=\\bar{\\gamma }_i\\circ \\bar{\\gamma }_j=0\\,,$ which shows that the integral basis of bulk three-cycles is not unimodular.", "Exceptional three-cycles Besides the four bulk three-cycles $\\gamma _i,\\bar{\\gamma }_i$ (with $i=1,2$ ), there are also four exceptional three-cycles appearing in the $\\mathbb {Z}_2$ twisted sector of the orbifold.", "It is easy to see that $Q^2$ acts trivially on the sub-manifold $(\\pi _1+\\pi _3)\\wedge \\pi _4$ .", "One can show that there are eight such $Q^2$ -invariant sub-manifolds, which are indicated in red in figure $\\ref {fig:Fig2}$ .", "We numerate them by ${i \\bar{j}}$ where the first index denotes the $\\mathbb {Z}_2$ -invariant two-tori (1,2) on the $A_3\\times A_1$ -torus and the second one the $\\mathbb {Z}_2$ -invariant points ($\\color {red}\\bar{1},\\,\\bar{2},\\,\\bar{3},\\,\\bar{4}$ ) on the $B_2$ -torus.", "These $Q^2$ -invariant sub-manifolds can be arranged in six congruence classes (under the $Q$ -action): $\\lbrace 1\\bar{1}\\rbrace ,\\,\\lbrace 1\\bar{2}\\rbrace ,\\,\\lbrace 1\\bar{3},\\,1\\bar{4}\\rbrace ,\\,\\lbrace 2\\bar{1}\\rbrace ,\\,\\lbrace 2\\bar{2}\\rbrace ,\\,\\lbrace 2\\bar{3},\\,2\\bar{4}\\rbrace .", "$ Figure: T 6 /ℤ 4 T^6/\\mathbb {Z}_4-orbifold on the A 3 ×A 1 ×B 2 A_3\\times A_1\\times B_2-lattice and its ℤ 2 \\mathbb {Z}_2-fixed lines (redin red).The resolution of these six $\\mathbb {Z}_2$ -singular sub-manifolds gives rise to six four-dimensional sub-manifolds with the topology $S^2\\times T^2$ in accordance with $h_{22}^{\\mathbb {Z}_2}\\stackrel{\\text{Poincaré}}{=}h_{11}^{\\mathbb {Z}_2}=6$ in table REF , where the exceptional two-cycle $\\mathbf {e}_{i\\bar{j}}$ describes the $S^2$ -part, and the two one-cycles $\\pi _1+\\pi _3$ and $\\pi _4$ span a two-torus.", "The index of $\\mathbf {e}_{i\\bar{j}}$ is inherited from the numeration of the $\\mathbb {Z}_2$ -invariant two-tori.", "Finally by splitting $T^2$ into one-cycles $\\pi _1+\\pi _3$ and $\\pi _4$ , we construct $Q$ -invariant exceptional three-cycles.", "Due to the anti-symmetric action of $Q$ on the one-cycles, only the exceptional two-cycles $\\mathbf {e}_{1\\bar{3}},\\, \\mathbf {e}_{1\\bar{4}},\\,\\mathbf {e}_{2\\bar{3}},\\,\\mathbf {e}_{2\\bar{4}}$ provide non-trivial results in the construction.", "Thus, the exceptional three-cycles are: $ \\begin{split}\\gamma _3&:=(\\mathbf {e}_{1\\bar{3}}-\\mathbf {e}_{1\\bar{4}})\\wedge (\\pi _1+\\pi _3)\\,,\\qquad \\bar{\\gamma }_3:=(\\mathbf {e}_{1\\bar{3}}-\\mathbf {e}_{1\\bar{4}})\\wedge \\pi _4\\,,\\\\\\gamma _4&:=(\\mathbf {e}_{2\\bar{3}}-\\mathbf {e}_{2\\bar{4}})\\wedge (\\pi _1+\\pi _3)\\,,\\qquad \\bar{\\gamma }_4:=(\\mathbf {e}_{2\\bar{3}}-\\mathbf {e}_{2\\bar{4}})\\wedge \\pi _4,\\end{split}$ with the intersection numbers $ \\gamma _i\\circ \\bar{\\gamma }_j=2\\delta _{ij}\\,,\\quad \\gamma _i\\circ \\gamma _j=\\bar{\\gamma }_i\\circ \\bar{\\gamma }_j=0\\qquad i=3,4\\,.$ Since the intersection form of the $\\gamma _i$ 's, and $\\bar{\\gamma }_i$ 's $(i=1,2,3,4)$ is not unimodular, these three-cycles do not form the minimal integral basis.", "We thus have to consider fractional three-cycles, which can consist of half a bulk cycle and simultaneously of half an exceptional cycle.", "Fractional three-cycles and their integral basis In order to write down an integral basis for the three-cycles such that the intersection form is unimodular, we start by specifying the construction of fractional cycles, which are $\\mathbb {Z}_2$ -invariant, analogously to the factorisable case reviewed in section REF .", "The fractional cycles can wrap half a bulk cycle and half an exceptional one $\\pi ^{\\text{frac}}=\\frac{1}{2}\\pi ^{\\text{bulk}}+\\frac{1}{2}\\pi ^{\\text{exc}}\\,.$ Motivated by the factorisable orbifold, we apply the well-known techniques of construction of the fractional cycles to our case.", "The non-factorisable structure of the lattice gives rise to some differences and therefore some modification of these techniques is needed.", "In the case of the factorisable torus $T^2_1\\times T^2_2\\times T^2_3$ , every $\\mathbb {Z}_2$ -invariant three-cycle passes through two $\\mathbb {Z}_2$ -invariant points per $T^2$ (here $T^2_1 \\times T^2_3$ ) and contains a $\\mathbb {Z}_2$ -invariant one-cycle on the remaining two-torus (here $T^2_2$ ).", "In the $A_3\\times A_3$ -case we saw that this does not hold true and we had to generalise this condition to $Q^2\\pi ^\\text{torus}=\\pi ^\\text{torus}$ .", "The same happens in the present case of the $A_3 \\times A_1 \\times B_2$ -lattice.", "Let us for example consider the toroidal cycle $(1,0,0,0)\\times (0,0,0,1)\\times (1,0)\\equiv \\pi _{145}$ through the origin.", "Despite the fact that this three-cycle contains the $\\mathbb {Z}_2$ -invariant one-cycle $\\pi _4$ and passes through a $\\mathbb {Z}_2$ -invariant point, $\\pi _{145}$ is not $\\mathbb {Z}_2$ -invariant and therefore cannot be fractional.", "Indeed, one can check that $Q^2\\pi _{145}=-\\pi _{345}$ .", "To impose the additional constraint on the numbers $(m^i,n^i,p^i,q^i)_{i=1,2}\\times (m^3,n^3)$ ensuring the $Q^2$ -invariance of the toroidal three-cycle we use the $Q$ -action on the wrapping numbers $\\begin{pmatrix}m^i,&n^i,&p^i,&q^i\\\\&m^3,&n^3&\\end{pmatrix}\\;\\;\\stackrel{Q}{\\longrightarrow }\\;\\;\\begin{pmatrix}-p^i,&m^i-p^i,&n^i-p^i,&-q^i\\\\&m^3-n^3,&2m^3-n^3&\\end{pmatrix}.$ For any toroidal three-cycle $\\pi ^{\\text{torus}}$ of the form $(\\ref {2.torus_cycle})$ , it can be shown that the following statements are equivalent: $\\pi ^{\\text{torus}} \\text{ is $\\mathbb {Z}_2$-invariant}\\Longleftrightarrow \\;Q^2\\pi ^{\\text{torus}}=\\pi ^{\\text{torus}}\\;\\Longleftrightarrow \\; A_1+A_3=0\\;\\;\\text{and }\\;B_1-B_2+B_3=0.$ Only such $\\mathbb {Z}_2$ -invariant toroidal three-cycles can be used for the construction of the fractional cycles.", "From the factorisable case we know that the toroidal three-cycles giving rise to the fractional cycles have to contain $\\mathbb {Z}_2$ -invariant one-cycles.", "Indeed, it can be verified that any cycle which satisfies the conditions $(\\ref {2.fractional_condition})$ can be written as $(m^1,n^1,p^1,q^1)\\times (m^2,n^2,p^2,q^2)\\times (m^3,n^3)=(m^1,n^1,p^1,0)\\times (\\tilde{p}^2,0,\\tilde{p}^2,\\tilde{q}^2)\\times (m^3,n^3)+\\text{R},$ where $\\tilde{p}^2:=p^2-\\frac{n^2}{n^1}p^1$ and $\\tilde{q}^2:=q^2-\\frac{n^2}{n^1}q^1$ for $n^1\\ne 0$ .$n^1=0$ gives rise to a similar result.", "The remaining term $\\text{R}:=(0,0,0,q^1)\\times (\\tilde{p}^2,0,\\tilde{p}^2,\\tilde{q}^2)\\times (m^3,n^3)$ does not contribute to the bulk three-cycle and can be neglected.", "Therefore, any $\\mathbb {Z}_2$ -invariant three-cycle contains a linear combination of the $\\mathbb {Z}_2$ -invariant one-cycles $\\pi _1+\\pi _3$ and $\\pi _4$ .", "The next step to construct the fractional cycles is to determine the exceptional part.", "In the factorisable case we identify which $\\mathbb {Z}_2$ -invariant points the toroidal three-cycle passes through.", "On the $A_3\\times A_1\\times B_2$ -orbifold also the $\\mathbb {Z}_2$ -invariant lines are involved and so we have to calculate their intersection with the toroidal three-cycle.", "Another important difference is that for the factorisable lattice $(T^2)^3$ , the co-prime condition on the wrapping numbers $(n^i,m^i)$ per $T^2_i$ is necessary and sufficient for the fractional cycle to have a co-prime decomposition in the unimodular basis.", "On the contrary, in the non-factorisable case the sufficiency condition gets lost and some further restrictions on the wrapping numbers $(m^i,n^i,p^i,q^i)$ are needed.", "It turns out that it is convenient to neglect these constraints at first and better to select the co-prime cycles after they are expressed in the unimodular basis.", "Thus, in this case, for the computation of the exceptional part we have to know not just which $\\mathbb {Z}_2$ -invariant lines the toroidal three-cycle intersects but also how many times.", "That requires some combinatorics.", "Using the property (REF ) for the $\\mathbb {Z}_2$ -invariant three-cycles, we have only to compute how many times the cycle with the wrapping numbers $(m^1,n^1,p^1,0)$ intersects the one-cycle $\\pi _1+\\pi _3$ .", "We find that the number $N$ of intersection points is $N=\\text{g.c.d.", "}(m^1-p^1,n^1)\\,.$ Furthermore, we make the following considerations and set several notations: The three-cycle can pass through both fixed lines 1 and 2 on the $A_3$ -torus if and only if $\\frac{m^1-p^1}{N}$ and $\\frac{n^1}{N}$ are odd.", "We introduce three new parameters counting the different fixed lines traversed by the three-cycle $ \\tau :={\\left\\lbrace \\begin{array}{ll}\\displaystyle 2 &\\text{if } \\frac{m^1-p^1}{N} \\text{ and } \\frac{n^1}{N} \\text{ are odd },\\\\1& \\text{ otherwise }\\end{array}\\right.", "}$ and $\\tau ^1,\\tau ^2\\in \\lbrace 0,1\\rbrace \\quad \\text{with }\\;\\tau ^1+\\tau ^2=\\tau .$ On the $B_2$ -torus, a one-cycle with even wrapping number $n^3$ passes through the fixed points $\\bar{1}$ and $\\bar{2}$ (both $\\mathbb {Z}_4$ -invariant), or $\\bar{3}$ and $\\bar{4}$ (both $\\mathbb {Z}_2$ -invariant).", "In this case, the fractional three-cycle contains either none or two exceptional three-cycles.", "If $n^3$ is odd, the one-cycle on $B_2$ intersects one $\\mathbb {Z}_4$ - and one $\\mathbb {Z}_2$ -invariant fixed point, which in any case gives rise to one corresponding exceptional cycle.", "Thus, we define the parameter $\\sigma $ , which counts the number of exceptional three-cycles contributing to a fractional cycle for given bulk part, by $\\sigma :={\\left\\lbrace \\begin{array}{ll}\\displaystyle 0,2&\\quad \\text{if }n^3\\;\\;\\text{is even,}\\\\1&\\quad \\text{if }n^3\\;\\;\\text{is odd}.\\end{array}\\right.", "}$ Furthermore, we introduce two parameters $s^1,s^2=\\pm 1$ which describe the winding directions of the exceptional cycles.", "Now we are able to write down the final form of the fractional three-cycle which is stemmed from the $\\mathbb {Z}_2$ -invariant toroidal three-cycle with the wrapping numbers $(m^1,n^1,p^1,0)\\times (\\tilde{p}^2,0,\\tilde{p}^2,\\tilde{q}^2)\\times (m^3,n^3)$ .", "It has the following form: $\\pi ^{\\text{frac}}=\\frac{1}{2}\\pi ^{\\text{bulk}}+N\\frac{\\tilde{p}^2\\sigma }{2}(s^1\\tau ^1\\gamma _3+s^2\\tau ^2\\gamma _4)+N\\frac{\\tilde{q}^2\\sigma }{2}(s^1\\tau ^1\\bar{\\gamma }_3+s^2\\tau ^2\\bar{\\gamma }_4).$ It is easy to check that the coefficients of $\\pi ^{\\text{bulk}}$ , stemming from the toroidal cycle with the wrapping numbers $(m^1,n^1,p^1,0)\\times (\\tilde{p}^2,0,\\tilde{p}^2,\\tilde{q}^2)\\times (m^3,n^3)$ , have $N$ as defined in (REF ) as a common divisor too.", "Naively one can expect the restriction to $N=1$ for the co-prime condition.", "However, this is not necessarily true.", "Let us for example consider the toroidal cycle with wrapping numbers $(1,0,-1,0)\\times (0,0,0,1)\\times (0,1)$ passing through the fixed point 1.", "It gives rise to the fractional cycle $\\frac{1}{2}(2\\bar{\\gamma }_1\\pm 2\\bar{\\gamma }_3)$ .", "Although $N=2$ this cycle is co-prime because the cycle $\\frac{1}{2}(\\bar{\\gamma }_1\\pm \\bar{\\gamma }_3)$ does not exist due to the condition $(\\ref {2.fractional_condition})$ .", "As a consequence of the considerations above we can summarise: Any fractional cycle containing only the one-cycle factor $\\pi _1+\\pi _3$ ($\\tilde{q}^2=0$ ) within the exceptional part can be expressed as a sum over the unbarred cycles $\\gamma _i$ , $ \\pi :=v^1\\gamma _1+v^2\\gamma _2+v^3\\gamma _3+v^4\\gamma _4,$ with an even number of the coefficients $v^i$ with half-integer values.", "Any fractional cycle containing only the one-cycle factor $\\pi _4$ ($\\tilde{p}^2=0$ ) within the exceptional part can be expanded as a sum over the barred cycles $\\bar{\\gamma }_i$ , $ \\bar{\\pi }:=\\bar{v}^1\\bar{\\gamma }_1+\\bar{v}^2\\bar{\\gamma }_2-\\bar{v}^3\\bar{\\gamma }_3-\\bar{v}^4\\bar{\\gamma }_4,$ where $\\bar{v}^i$ are either all half-integer or all integer.", "In order to determine an integral symplectic basis for the three-cycles, we search for pairs of cycles $\\pi $ and $\\bar{\\pi }$ such that $\\pi \\circ \\bar{\\pi }=-2$ , with the coefficients $v^i$ and $\\bar{v}^i$ satisfyingThe signum function is given by sgn$(x):={\\left\\lbrace \\begin{array}{ll}\\displaystyle -1&\\text{if }x<0\\,,\\\\0&\\text{if }x=0\\,,\\\\1&\\text{if }x>0\\,.\\end{array}\\right.", "}$ sgn$v^i=\\,$ sgn$\\bar{v}^i$ for all $i$ .", "All possible cycles can be combined in three sets: $(v^1,v^2,v^3,v^4)=(\\underline{\\pm \\frac{1}{2},\\pm \\frac{1}{2},0,0})$ and $(\\bar{v}^1,\\bar{v}^2,\\bar{v}^3,\\bar{v}^4)=(\\underline{\\pm 1,\\pm 1,0,0})$ where the underlying denotes all permutations of entries.", "$(v^1,v^2,v^3,v^4)=(\\underline{\\pm 1,0,0,0})$ for the (un-)barred cycles.", "$(v^1,v^2,v^3,v^4)=(\\pm \\frac{1}{2},\\pm \\frac{1}{2},\\pm \\frac{1}{2},\\pm \\frac{1}{2})$ for the (un-)barred cycles.", "Altogether there are 48 unbarred and 48 barred cycles.", "It is not difficult to specify a basis of these cycles.", "For the unbarred part we obtain $ \\begin{split}\\alpha _1:=&(0,\\frac{1}{2},-\\frac{1}{2},0)\\,,\\\\\\alpha _2:=&(0,0,\\frac{1}{2},-\\frac{1}{2})\\,,\\\\\\alpha _3:=&(0,0,0,1)\\,,\\\\\\alpha _4:=&(\\frac{1}{2},-\\frac{1}{2},-\\frac{1}{2},-\\frac{1}{2}),\\end{split}$ and for the barred one, $ \\begin{split}\\bar{\\alpha }_1:=&(0,1,1,0)\\,,\\\\\\bar{\\alpha }_2:=&(0,0,-1,1)\\,,\\\\\\bar{\\alpha }_3:=&(0,0,0,-1)\\,,\\\\\\bar{\\alpha }_4:=&(\\frac{1}{2},-\\frac{1}{2},\\frac{1}{2},\\frac{1}{2})\\,.\\end{split}$ Therefore, the fractional cycles form the $F_4\\oplus F_4$ -lattice, and the intersection matrix takes the form $ \\alpha _i\\circ \\bar{\\alpha }_j=\\begin{pmatrix}-2&1&0&0\\\\1&-2&2&0\\\\0&1&-2&1\\\\0&0&1&-2\\end{pmatrix}\\,.$ This is the Cartan matrix for $F_4$ , and it is unimodular.", "Indeed, it is easy to verify that any bulk, exceptional or fractional cycle can be expanded in the basis $(\\ref {2.basis_unbarred_cycles})$ and $(\\ref {2.basis_barred_cycles})$ with integer coefficients.", "In other words, since the determinant of the intersection matrix (REF ) is 1, the $\\alpha _i$ 's and $\\bar{\\alpha }_i$ 's form an integral basis of the homology lattice $H_3(M,\\mathbb {Z})$ ." ], [ "Intersecting brane worlds", "The aim of this article is not only the study of non-factorisable $\\mathbb {Z}_4$ -orbifolds but also model building with $\\mathbb {Z}_4$ -orientifolds of Type IIA superstring theory.", "We are interested in global supersymmetric models with a semi-realistic chiral spectrum, in particular Pati-Salam-models as D6-brane realisations with only three visible stacks, which have on other lattices been the most simple kind of global GUT model to be found, see e.g.", "[34], [31], [87], [88], [89], [40], [46], [48].", "The introduction of an anti-holomorphic involution $\\mathcal {R}$ on the $\\mathbb {Z}_4$ -orbifold gives rise to orientifold six-planes (O6-planes), which wrap the fixed loci of $\\mathcal {R}Q^k$ which together form some element of $H_3(T^6/\\mathbb {Z}_4,\\mathbb {Z})$ .", "In the following, we denote this homological cycle by $\\pi _\\text{O6}$ .", "The ($\\mathbb {Z}_4$ -orbits of) O6-planes have negative RR charge, which has to be canceled by introducing stacks of $N_a$ space-time filling D6-branes which wrap a three-cycle $\\pi _a$ on the orbifold.", "The RR tadpole cancellation condition is given by [86] $\\sum _aN_a(\\pi _a+\\pi _a^\\prime )-4\\pi _\\text{O6}=0\\,,$ where $\\pi _a^\\prime $ is the $\\mathcal {R}$ -image of the three-cycle $\\pi _a$ with in general $\\pi _a^{\\prime } \\ne \\pi _a$ .", "The resulting gauge group is then generically $\\prod _aU(N_a)$ .", "The case $\\pi _a^\\prime =\\pi _a$ gives rise to the rank-preserving symmetry enhancement: $U(N_a)\\,\\hookrightarrow \\,USp(2N_a)$ or $SO(2N_a)$ .", "We call a model with the RR tadpole condition (REF ) implemented global, otherwise local.", "The chiral massless spectrum can be computed from topological intersection numbers [86].", "For the gauge group $\\prod _aU(N_a)$ it is given in the table REF .", "Table: Chiral spectrum for intersecting D6-branes with gauge group ∏ a U(N a )\\prod _a U(N_a).Furthermore, the ($\\mathbb {Z}_4$ -orbits of the) O6-planes preserve $\\mathcal {N}=1$ .", "For semi-realistic models to be supersymmetric, we have to require that all D6-branes preserve the same supersymmetry, i.e., that they are wrapped on special Lagrangian three-cycles with the same calibration as the O6-planes.", "These additional constraints on the wrapped three-cycles will be considered in more details in the section REF ." ], [ "Anti-holomorphic involutions $\\mathcal {R}$ of {{formula:fbd0e4e3-4187-4f21-8668-5282ae4c519c}}", "Before we start with the construction of the ($\\mathbb {Z}_4$ -orbit of) O6-plane(s), we have to calculate how the anti-holomorphic involution acts on the real lattice in each case.", "It is known [90], [91] that there exist a set of complex coordinates $\\lbrace z^i\\rbrace _{i=1,2,3}$ on which the twist $Q$ $(\\ref {2.coxeter_2})$ acts diagonally $ Q^tz^i=e^{2\\pi i\\zeta _i}z^i\\;$ with the eigenvalues $(i,-1,i)$ for the shift vector $\\vec{\\zeta }=\\frac{1}{4}(1,-2,1)$ .", "In these coordinates, the anti-holomorphic involution (including $\\mathbb {Z}_4$ -twists) is simply given by complex conjugation $ \\mathcal {R}(Q^t)^n:\\;\\;z^i\\rightarrow e^{i\\theta _{n_i}}\\bar{z}^i$ for some real parameter $\\theta _{n_i}$ .", "We will now discuss the a priori different possible choices of $(\\theta _{n_1},\\theta _{n_2},\\theta _{n_3})$ for each $\\mathbb {Z}_4$ -invariant background lattice and argue that some of the different choices lead to physically equivalent vacua, at least based on the allowed ranks of gauge groups in the RR tadpole cancellation conditions as well as on the counting of supersymmetric bulk cycles per given length." ], [ "$B_2 \\times (A_1)^2 \\times B_2$", "In the factorisable case, there exist a priori two choices of orientifold axes per two-torus [92], [93], [94] denoted by A (reflection along the short one-cycle, here $\\pi _{2/6}$ or some $\\mathbb {Z}_4$ -images thereof, cf.", "figure REF ) and B (reflection along the long one-cycle, here $\\pi _{1/5}$ ) for the $B_2$ -tori and two lattice orientations a (rectangular lattice) and b (tilted lattice) for the $(A_1)^2$ torus.", "Due to the permutation symmetry $T^2_1 \\leftrightarrow T^2_3$ , here the combinatorics provides at most six inequivalent choices of phases (REF ), denoted by AaA, AaB, BaB and AbA, AbB, BbB, for which the O6-planes are displayed in table REF .", "Table: Fixed planes for the different lattice orientations of B 2 ×(A 1 ) 2 ×B 2 B_2 \\times (A_1)^2 \\times B_2, weighted with the number N O6 =2(1-b)N_{O6}=2(1-b) of parallel O6-planes along T 2 2 T^2_{2} withb=0,1 2b =0, \\frac{1}{2} for a and b, respectively.However, the massless closed string spectrum - encoded in the orientifold-even and -odd Hodge numbers $(h_{11}^+,h_{11}^-,h_{21})$ counting vectors, Kähler and complex structure moduli as derived for generic Calabi-Yau backgrounds in [95] -, which was derived using CFT techniques in [34], [45] suggests that there are pairwise relations ${\\bf AaA} \\leftrightarrow {\\bf BaB}$ and ${\\bf AbA} \\leftrightarrow {\\bf BbB}$ , reducing the number of physically inequivalent backgrounds to four.", "This assumption is supported by the analogy to the relations among different lattice backgrounds derived for $\\mathbb {Z}_6^{(\\prime )}$ and $\\mathbb {Z}_2 \\times \\mathbb {Z}_6^{(\\prime )}$ in [42], [46], [48].", "Since D6-branes on the factorisable $T^6/(\\mathbb {Z}_4 \\times \\Omega \\mathcal {R})$ orientifold have been considered at length in [34] with the result that at most two generations of chiral particles can be engineered by two supersymmetric intersecting D6-brane orbits, we will from now on concentrate on backgrounds with non-factorisable tori, where we will determine all a priori different choices of lattice orientations and then search for physical equivalences." ], [ "$A_3 \\times A_3$", "The eigenvectors of $Q^t$ with $Q$ defined in (REF ) give rise to the complex coordinates [84] $\\begin{split}z^1:=&\\frac{1}{\\sqrt{2}}\\left(x^1+ix^2-x^3\\right)\\,,\\\\z^2:=&\\frac{1}{\\sqrt{8u_2}}\\left((x^1-x^2+x^3+\\mathcal {U}(x^4-x^5+x^6)\\right)\\,,\\\\z^3:=&\\frac{1}{\\sqrt{2}}\\left(x^4+ix^5-x^6\\right) \\, ,\\end{split}$ where the complex structure is $\\mathcal {U}:=u_1+iu_2=-\\frac{R_2}{2aR_1}\\left(c+e+i\\sqrt{-(c+e)^2+4ab}\\right)$ .", "With the transformation from the complex coordinates to the real ones, we are able to write down the action of $\\mathcal {R}$ on the real lattice.", "There are a priori four possible choices of angles in the anti-holomorphic involution: $\\vec{\\theta }=(0,0,0)\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_1e_1=e_1\\,, &\\mathcal {R}_1e_2=-e_1-e_2-e_3\\,,\\\\&\\mathcal {R}_1e_3=e_3\\,,&\\mathcal {R}_1e_4=u_1(e_1+e_3)-e_6\\,,\\\\&\\mathcal {R}_1e_5=-u_1(e_1+e_3)-e_5\\,,&\\mathcal {R}_1e_6=u_1(e_1+e_3)-e_4\\,,\\end{array}\\right\\rbrace \\quad \\textbf {AAA}$ $\\vec{\\theta }=(\\frac{\\pi }{2},0,0)\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_2e_1=e_1+e_2+e_3\\,,&\\mathcal {R}_2e_2=-e_3\\,,\\\\&\\mathcal {R}_2e_3=-e_2\\,,&\\mathcal {R}_1e_4=u_1(e_1+e_3)-e_6\\,,\\\\&\\mathcal {R}_1e_5=-u_1(e_1+e_3)-e_5\\,,&\\mathcal {R}_1e_6=u_1(e_1+e_3)-e_4\\,,\\end{array}\\right\\rbrace \\quad \\textbf {BAA}$ $\\vec{\\theta }=(0,0,\\frac{\\pi }{2})\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_3e_1=e_1\\,,&\\mathcal {R}_3e_2=-e_1-e_2-e_3\\,,\\\\&\\mathcal {R}_3e_3=e_3\\,,&\\mathcal {R}_3e_4=u_1(e_1+e_3)+e_5\\,,\\\\&\\mathcal {R}_3e_5=u_1(e_1+e_3)+e_4\\,,&\\mathcal {R}_3e_6=u_1(e_1+e_3)-e_4-e_5-e_6\\,,\\end{array}\\right\\rbrace \\quad \\textbf {AAB}$ $\\vec{\\theta }=(\\frac{\\pi }{2},0,\\frac{\\pi }{2})\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_4e_1=e_1+e_2+e_3\\,,&\\mathcal {R}_4e_2=-e_3\\,,\\\\&\\mathcal {R}_4e_3=-e_2\\,,&\\mathcal {R}_4e_4=u_1(e_1+e_3)+e_5\\,,\\\\&\\mathcal {R}_4e_5=u_1(e_1+e_3)+e_4\\,,&\\mathcal {R}_4e_6=u_1(e_1+e_3)-e_4-e_5-e_6.\\end{array}\\right\\rbrace \\quad \\textbf {BAB}$ Here we took the notation for the lattices from [63], which we will now explain.", "The complex coordinates (REF ) parametrise the three-planesThese three-planes are closely related to the “factorisation” of the non-factorisable torus we will discuss in the next section.", "where $Q$ acts as a rotation.", "A means that, in their choice of basis, the orientifold plane lies along the horizontal axis in the corresponding plane, while B corresponds to the angle of the orientifold plane with respect to the horizontal axis being $\\pi /4$ .", "Note also that in contrast to the $B_2\\times (A_1)^2\\times B_2$ -orbifold, the BAA- and AAB-lattices are really different geometrically and cannot be related by exchanging $T^3_1 \\leftrightarrow T^3_2$ .", "This will be further specified in the sections REF and .", "Since only a crystallographic action on the lattice is allowed $(\\mathcal {R}_ir=r+\\Lambda $ for arbitrary lattice vectors $r\\in \\Lambda )$ , the real part of the complex structure can take only the value $u_1=0$ ($\\Rightarrow \\;c+e=0$ in (REF ) and only A as middle entry of the lattice orientation).", "The orientifold projections act then on the homology classes of three-cycles (REF ) in the following way: $ \\mathcal {R}_1:\\qquad \\gamma _1\\leftrightarrow -\\gamma _2\\,,\\qquad \\bar{\\gamma }_1\\leftrightarrow \\bar{\\gamma }_2\\,.$ $ \\mathcal {R}_{2/3}:\\qquad {\\left\\lbrace \\begin{array}{ll}\\gamma _1\\rightarrow \\gamma _1\\,,\\quad &\\quad \\bar{\\gamma }_1\\rightarrow -\\bar{\\gamma }_1\\,,\\\\\\gamma _2\\rightarrow -\\gamma _2\\,,\\quad &\\quad \\bar{\\gamma }_2\\rightarrow \\bar{\\gamma }_2\\,.\\end{array}\\right.", "}$ $ \\mathcal {R}_4:\\qquad \\gamma _1\\leftrightarrow \\gamma _2\\,,\\quad \\quad \\bar{\\gamma }_1\\leftrightarrow -\\bar{\\gamma }_2\\,.$ Furthermore, we calculate the fixed point set for the orientifold involutions $ \\lbrace \\mathcal {R}_i\\,,\\, \\mathcal {R}_iQ\\,,\\,\\mathcal {R}_iQ^2\\,,$ $\\mathcal {R}_iQ^3\\,\\rbrace $ .", "The results are listed in tables REF -REF for the respective $\\mathcal {R}_i$ .", "Table: Toroidal cycles wrapped by the O6-planes on the AAA lattice orientation of A 3 ×A 3 A_3 \\times A_3.The fixed point set under $\\mathcal {R}_iQ^n$ is computed in the following way, e.g.for $\\mathcal {R}_1$ : First, we calculate the three eigenvectors of $\\mathcal {R}_1$ corresponding to the eigenvalue 1, in this case $e_1\\,,\\,e_3$ and $e_4-e_6$ .", "The to these vectors corresponding one-cycles $\\pi _i$ span a three-cycle $\\pi _{134}-\\pi _{136}$ .", "The next step is to determine how many three-cycles of the same homology class we have which are point-wise $\\mathcal {R}_1$ -invariant.", "In the case at hand, we obtain two such submanifolds, which go through the $\\mathcal {R}_1$ fixed points $(0,0,0,0,\\frac{m}{2},0)$ with $m\\in \\lbrace 0,1\\rbrace $ .", "The projection $\\mathcal {R}_1Q$ leads to similar results.", "The fixed point sets of $\\mathcal {R}Q^2$ and $\\mathcal {R}Q^3$ can be calculated by acting with $Q$ on the $\\mathcal {R}$ - and $\\mathcal {R}Q$ -fixed point sets, respectively.", "After all fixed point sets have been computed, from purely geometric considerations there still remains an ambiguity in both the global sign of all sets and the relative sign between the $\\mathcal {R}$ - and $\\mathcal {R}Q$ -sets which needs to be fixed, i.e., the three-cycle $\\pm (\\text{Fix}(\\mathcal {R}+\\mathcal {R}Q^2)\\pm \\text{Fix}(\\mathcal {R}Q+\\mathcal {R}Q^3))$ is invariant for any choice of the signs.", "Each of the two sets of fixed points corresponds to a different bulk cycle.", "Asking both of them to have the same calibration will fix the relative sign between them.", "The global sign is fixed once we choose one of the two possible calibration conditions (either $\\int (e^{i\\varphi }\\Omega _3) >0$ or $\\int (e^{i\\varphi +i\\pi }\\Omega _3) >0$ ); alternatively, choosing the global sign will fix the calibration condition that has to be used.", "This calibration will be explained in more detail in section REF .", "Table: Toroidal cycles wrapped by the O6-planes for the BAA lattice orientation of A 3 ×A 3 A_3 \\times A_3.Table: Toroidal cycles wrapped by the O6-planes for the AAB lattice orientation of A 3 ×A 3 A_3 \\times A_3.Table: Toroidal cycles wrapped by the O6-planes for the BAB lattice orientation of A 3 ×A 3 A_3 \\times A_3.Adding all contributions, we can express the corresponding O6-planes as elements of $H_3(T^6/\\mathbb {Z}_{4},\\mathbb {Z})$ : $\\pi _{\\text{O}6_1}&:=2(\\gamma _2-\\gamma _1)-2(\\bar{\\gamma }_1+\\bar{\\gamma }_2)\\,,\\\\\\pi _{\\text{O}6_2}&:=-4\\gamma _1-4\\bar{\\gamma }_2\\,,\\\\\\pi _{\\text{O}6_3}&:=-2\\gamma _1-2\\bar{\\gamma }_2\\,,\\\\\\pi _{\\text{O}6_4}&:=2(\\gamma _1+\\gamma _2)+2(\\bar{\\gamma }_2-\\bar{\\gamma }_1)\\,.$ Obviously, the O6-planes (REF ) are invariant under the corresponding orientifold projections given in equations (REF ), (REF ) and (REF )." ], [ "$A_3 \\times A_1\\times B_2$", "We can approach the $A_3\\times A_1\\times B_2$ -case in an analogous way.", "Due to (REF ), the complex coordinates are: $ \\begin{split}z^1&=\\frac{1}{\\sqrt{2}}(x^1+ix^2-x^3)\\,,\\\\z^2&=\\frac{1}{2\\sqrt{2\\text{Im}(\\mathcal {U})}}(x^1-x^2+x^3+2\\mathcal {U}x^4)\\,,\\\\z^3&=x^5-\\frac{x^6}{2}+i\\frac{x^6}{2} \\, ,\\end{split}$ with the complex structure $ \\mathcal {U}:=u_1+iu_2:=-\\frac{R_1}{2aR_3}(d+i\\sqrt{-a-d^2})\\,.$ For this non-factorisable lattice, there are four possible orientations: $\\vec{\\theta }=(0,0,0)\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_1e_1=e_1\\,, &\\mathcal {R}_1e_2=-e_1-e_2-e_3\\,,\\\\&\\mathcal {R}_1e_3=e_3\\,,&\\mathcal {R}_1e_4=2u_1(e_1+e_3)-e_4\\,,\\\\&\\mathcal {R}_1e_5=e_5\\,,&\\mathcal {R}_1e_6=-e_5-e_6\\,,\\end{array}\\right\\rbrace \\quad \\textbf {AAB}$ $\\vec{\\theta }=(0,0,-\\frac{\\pi }{2})\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_2e_1=e_1\\,,&\\mathcal {R}_2e_2=-e_1-e_2-e_3\\,,\\\\&\\mathcal {R}_2e_3=e_3\\,,&\\mathcal {R}_2e_4=2u_1(e_1+e_3)-e_4\\,,\\\\&\\mathcal {R}_2e_5=-e_5-2e_6\\,,&\\mathcal {R}_2e_6=e_6\\,,\\end{array}\\right\\rbrace \\quad \\textbf {AAA}$ $\\vec{\\theta }=(-\\frac{\\pi }{2},0,-\\frac{\\pi }{2})\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_3e_1=-e_2\\,,&\\mathcal {R}_3e_2=-e_1\\,,\\\\&\\mathcal {R}_3e_3=e_1+e_2+e_3\\,,&\\mathcal {R}_3e_4=2u_1(e_1+e_3)-e_4\\,,\\\\&\\mathcal {R}_3e_5=-e_5-2e_6\\,,&\\mathcal {R}_3e_6=e_6\\,,\\end{array}\\right\\rbrace \\quad \\textbf {ABA}$ and $\\vec{\\theta }=(-\\frac{\\pi }{2},0,0)\\Rightarrow \\left\\lbrace \\begin{array}{cll}&\\mathcal {R}_4e_1=-e_2\\,,&\\mathcal {R}_4e_2=-e_1\\,,\\\\&\\mathcal {R}_4e_3=e_1+e_2+e_3\\,,&\\mathcal {R}_4e_4=2u_1(e_1+e_3)-e_4\\,,\\\\&\\mathcal {R}_4e_5=e_5\\,,&\\mathcal {R}_4e_6=-e_5-e_6.\\end{array}\\right\\rbrace \\quad \\textbf {ABB}$ Here we also took again the notation for the lattices from [63].", "Since only the crystallographic action on the lattice is allowed $(\\mathcal {R}_ir=r+\\Lambda $ for arbitrary lattice vectors $r\\in \\Lambda )$ , the real part of the complex structure can take only two values $u_1=0,\\frac{1}{2}$ .", "We adapt the above lattice notation for different values of $u_1$ in the following way: We write a small a and b as a subscript after the first A for $u_1=0$ and $u_1=\\frac{1}{2}$ , respectively.", "The b-type lattice is to our best knowledge investigated here for the first time, while the D6-brane configurations, which cancel the bulk RR tadpoles locally on top of the O6-planes, in  [63] correspond to the a-type choice $u_1=0$ .", "The orientifold action on the homological three-cycles is given by $ \\mathcal {R}_1:\\qquad {\\left\\lbrace \\begin{array}{ll}\\gamma _1\\rightarrow \\gamma _2\\,,\\qquad \\bar{\\gamma }_1\\rightarrow -\\bar{\\gamma }_2-2u_1\\gamma _2\\,,\\\\\\gamma _2\\rightarrow \\gamma _1\\,,\\qquad \\bar{\\gamma }_2\\rightarrow -\\bar{\\gamma }_1-2u_1\\gamma _1\\,,\\\\\\gamma _3\\rightarrow \\gamma _3\\,,\\qquad \\bar{\\gamma }_3\\rightarrow -\\bar{\\gamma }_3+2u_1\\gamma _3\\,,\\\\\\gamma _4\\rightarrow \\gamma _4\\,,\\qquad \\bar{\\gamma }_4\\rightarrow -\\bar{\\gamma }_4+2u_1\\gamma _4\\,.\\end{array}\\right.", "}$ $ \\mathcal {R}_2:\\qquad {\\left\\lbrace \\begin{array}{ll}\\gamma _1\\rightarrow \\gamma _1\\,,\\quad &\\quad \\bar{\\gamma }_1\\rightarrow -\\bar{\\gamma }_1-2u_1\\gamma _1\\,,\\\\\\gamma _2\\rightarrow -\\gamma _2\\,,\\quad &\\quad \\bar{\\gamma }_2\\rightarrow \\bar{\\gamma }_2+2u_1\\gamma _2\\,,\\\\\\gamma _3\\rightarrow -\\gamma _3\\,,\\quad &\\quad \\bar{\\gamma }_3\\rightarrow \\bar{\\gamma }_3-2u_1\\gamma _3\\,,\\\\\\gamma _4\\rightarrow -\\gamma _4\\,,\\quad &\\quad \\bar{\\gamma }_4\\rightarrow \\bar{\\gamma }_4-2u_1\\gamma _4\\,.\\end{array}\\right.", "}$ $ \\mathcal {R}_3:\\qquad {\\left\\lbrace \\begin{array}{ll}\\gamma _1\\rightarrow -\\gamma _2\\,,\\qquad \\bar{\\gamma }_1\\rightarrow \\bar{\\gamma }_2+2u_1\\gamma _2\\,,\\\\\\gamma _2\\rightarrow -\\gamma _1\\,,\\qquad \\bar{\\gamma }_2\\rightarrow \\bar{\\gamma }_1+2u_1\\gamma _1\\,,\\\\\\gamma _3\\rightarrow -\\gamma _3\\,,\\qquad \\bar{\\gamma }_3\\rightarrow \\bar{\\gamma }_3-2u_1\\gamma _3\\,,\\\\\\gamma _4\\rightarrow -\\gamma _4\\,,\\qquad \\bar{\\gamma }_4\\rightarrow \\bar{\\gamma }_4-2u_1\\gamma _4\\,.\\end{array}\\right.", "}$ $ \\mathcal {R}_4:\\qquad {\\left\\lbrace \\begin{array}{ll}\\gamma _1\\rightarrow \\gamma _1\\,,\\quad &\\quad \\bar{\\gamma }_1\\rightarrow -\\bar{\\gamma }_1-2u_1\\gamma _1\\,,\\\\\\gamma _2\\rightarrow -\\gamma _2\\,,\\quad &\\quad \\bar{\\gamma }_2\\rightarrow \\bar{\\gamma }_2+2u_1\\gamma _2\\,,\\\\\\gamma _3\\rightarrow \\gamma _3\\,,\\quad &\\quad \\bar{\\gamma }_3\\rightarrow -\\bar{\\gamma }_3+2u_1\\gamma _3\\,,\\\\\\gamma _4\\rightarrow \\gamma _4\\,,\\quad &\\quad \\bar{\\gamma }_4\\rightarrow -\\bar{\\gamma }_4+2u_1\\gamma _4\\,.\\end{array}\\right.", "}$ Furthermore, we can calculate the fixed point sets for the orientifold projections $\\lbrace \\mathcal {R}_i\\,,\\; \\mathcal {R}_iQ\\,,$ $\\mathcal {R}_iQ^2\\,,\\;\\mathcal {R}_iQ^3\\,\\rbrace $ .", "The results are listed in tables REF -REF for the respective $\\mathcal {R}_i$ .", "In figures REF -REF we illustrate the O6-planes for the lattices A$_\\textbf {a}$AB and A$_\\textbf {a}$BB.We will see in the next section that the other a-type lattices are related to these.", "Table: Toroidal cycles wrapped by the O6-planes for the AAB orientation of A 3 ×A 1 ×B 2 A_3 \\times A_1 \\times B_2.Table: Toroidal cycles wrapped by the O6-planes for the AAA orientation of A 3 ×A 1 ×B 2 A_3 \\times A_1 \\times B_2.Table: Toroidal cycles wrapped by the O6-planes for the ABA orientation of A 3 ×A 1 ×B 2 A_3 \\times A_1 \\times B_2.Table: Toroidal cycles wrapped by the O6-planes for the ABB orientation of A 3 ×A 1 ×B 2 A_3 \\times A_1 \\times B_2.Figure: ℛ 1 \\mathcal {R}_1- and ℛ 1 Q 2 \\mathcal {R}_1Q^2-contributions to the O6-planes of the A 3 ×A 1 ×B 2 A_3\\times A_1\\times B_2-orientifold with A 𝐚 {}_{\\bf a}AB-lattice (u 1 =0u_1=0).Figure: ℛ 1 Q\\mathcal {R}_1Q- and ℛ 1 Q 3 \\mathcal {R}_1Q^3-contributions to the O6-plane of the A 3 ×A 1 ×B 2 A_3\\times A_1\\times B_2-orientifold with A 𝐚 {}_{\\bf a}AB-lattice (u 1 =0u_1=0).Figure: ℛ 4 \\mathcal {R}_4-contribution to the O6-plane of the A 3 ×A 1 ×B 2 A_3\\times A_1\\times B_2-orientifold with A 𝐚 {}_{\\bf a}BB-lattice (u 1 =0u_1=0).", "The second fixed point set is located at π 4 =1 2\\pi _4=\\frac{1}{2} and denoted by the orange point in the picture.Figure: ℛ 4 Q 2 \\mathcal {R}_4Q^2-contributions to the O6-plane of the A 3 ×A 1 ×B 2 A_3\\times A_1\\times B_2-orientifold with A 𝐚 {}_{\\bf a}BB-lattice (u 1 =0u_1=0).", "The second fixed point set is located at π 4 =1 2\\pi _4=\\frac{1}{2} and denoted by the green point in the picture.Figure: ℛ 4 Q\\mathcal {R}_4Q- and ℛ 4 Q 3 \\mathcal {R}_4Q^3-contributions to the O6-plane of the A 3 ×A 1 ×B 2 A_3\\times A_1\\times B_2-orientifold with A 𝐚 {}_{\\bf a}BB-lattice (u 1 =0u_1=0).Adding up all contributions we obtain the corresponding O6-planes: $\\pi _{\\text{O}6_1}&:=(1-2u_1)\\gamma _1+(1+2u_1)\\gamma _2-2(\\bar{\\gamma }_1-\\bar{\\gamma }_2)\\,,\\\\\\pi _{\\text{O}6_2}&:=-2\\gamma _1-2u_1\\gamma _2-2\\bar{\\gamma }_2\\,,\\\\\\pi _{\\text{O}6_3}&:=2(\\gamma _2-\\gamma _1)-4u_1\\gamma _2-2(\\bar{\\gamma }_1+\\bar{\\gamma }_2)\\,,\\\\\\pi _{\\text{O}6_4}&:=(2-2u_1)\\gamma _1+4u_1\\gamma _2+4\\bar{\\gamma }_2\\,.$ It is easy to check that the resulting O6-planes are invariant under corresponding orientifold projections defined in equations (REF ) to  (REF )." ], [ "Supersymmetric cycles", "From the phenomenological point of view (like chirality and stability at low energies) $\\mathcal {N}=1$ supersymmetric models are particularly interesting.", "This requires the D6-branes to preserve some supersymmetry, which leads to additional geometrical conditions on the allowed three-cycles; namely, the three-cycles have to be special Lagrangian.", "A three-cycle $\\pi $ is called special Lagrangian (sLag) if it satisfies $J\\mid _{\\pi }=0\\,,\\\\\\text{Im}(e^{i\\varphi }\\Omega _3)\\mid _{\\pi }=0\\,,\\\\\\text{Re}(e^{i\\varphi }\\Omega _3)\\mid _{\\pi }>0\\,,$ where $\\varphi $ is an arbitrary constant phase, and $J$ and $\\Omega _3$ are the covariantly constant Kähler two-form and holomorphic three-form, respectively, that always exist on a Calabi-Yau threefold (and, in particular, on the $T^6/\\mathbb {Z}_4$ orbifold we consider here).", "They can be defined locally by $ \\Omega _3=dz^1\\wedge dz^2\\wedge dz^3,\\quad J=i\\sum _{i,\\bar{j}}g_{i\\bar{j}}dz^i\\wedge d\\bar{z}^{\\bar{j}}.$ If two Lagrangian three-cycles are calibrated by different values of the phase $\\varphi $ , the corresponding D6-branes preserve different supersymmetries.", "Since the three-cycle wrapped by the O6-plane is automatically sLag, the phase $\\varphi $ is fixed to $\\varphi =\\varphi _\\text{O6}$ , and we need to search for the Lagrangian three-cycles which are calibrated by $\\text{Re}(e^{i\\varphi _\\text{O6}}\\Omega _3)$ .", "In terms of the $\\vec{\\theta }$ , the angle $\\varphi _\\text{O6}$ is given by $\\varphi _\\text{O6}=-\\frac{1}{2}\\sum _i\\theta _i\\ (+\\pi ).$ The possible $+\\pi $ in the previous equation arises from the freedom we have to choose either $\\int (e^{i\\varphi }\\Omega _3) >0$ or $\\int (e^{i\\varphi +i\\pi }\\Omega _3) >0$ as the calibration condition, or, alternatively, the global sign of the three-cycle wrapped by the O6-plane.", "$B_2 \\times (A_1)^2 \\times B_2$ Due to the factorisable structure of the $B_2 \\times (A_1)^2 \\times B_2$ -lattice, any three-cycle with the wrapping numbers $(n^i,m^i)_{i=1,2,3}$ is automatically Lagrangian.", "The condition that a D6-brane wrapping such a three-cycle preserves the same supersymmetry as the O6-plane (i.e.", "is sLag with identical calibration) is $\\sum _i\\phi _i=0\\ \\text{mod }2\\pi $ where $\\phi _i$ is an angle between the three-cycle and the O6-plane on the two-torus $T^2_{(i)}$ .", "Fractional three-cycles with the same calibration arise when (fractions of) exceptional three-cycles are added through which a given bulk cycle passes with the one restriction on the relative prefactor discussed below equation (REF ); for more details see [34].", "$A_3\\times A_3$ With respect to the complex coordinates (REF ), the $Q$ -invariant metric $g$ of equation (REF ) is given by $g_{i\\bar{j}}=\\left(\\begin{array}{ccc}2R_1^2(1+a)&0&R_1R_2(e-c-i(c+2d+e))\\\\0&-8u_2aR_1^2&0\\\\R_1R_2(e-c+i(c+2d+e))&0&2R_2^2(1+b)\\\\\\end{array}\\right) \\, .$ The Kähler-form $J$ on the $A_3\\times A_3$ -lattice background is given by $J:=2R_1^2(1+a)\\omega _1-2u_2aR_1^2\\omega _2+2R_2^2(1+b)\\omega _3+R_1R_2\\left((c+2d+e)\\omega _4+(c-e)\\omega _5\\right)$ with real $\\mathbb {Z}_4$ -invariant two-forms $\\omega _i$ : $\\begin{split}\\omega _1&:=dx^{12}+dx^{23}\\,,\\\\\\omega _2&:=dx^{14}-dx^{15}+dx^{16}-dx^{24}+dx^{25}-dx^{26}+dx^{34}-dx^{35}+dx^{36}\\,,\\\\\\omega _3&:=dx^{45}+dx^{56}\\,,\\\\\\omega _4&:=dx^{14}-dx^{16}+dx^{25}-dx^{34}+dx^{36}\\,,\\\\\\omega _5&:=-dx^{15}+dx^{24}-dx^{26}+dx^{35}\\,,\\end{split}$ where $dx^{ij} := dx^i\\wedge dx^j$ .", "The Kähler two-form is negative under the orientifold projection, i.e.", "$\\mathcal {R}J=-J$ .", "This means that the $\\mathcal {R}$ -even part of $J$ has to vanish, which fixes part of the moduli, in addition to the fixing $u_1=0$ (i.e.", "$c+e=0$ ) mentioned in the previous subsection.", "The additional moduli fixing for the different orientifold projections is given in table REF .", "Table: Additional moduli fixing and calibration arising from the condition ℛJ=-J\\mathcal {R}J=-J for the different choices of A 3 ×A 3 A_3 \\times A_3 lattice orientations.While, in the factorisable case, i.e.", "$T^6=(T^2)^3$ , all factorised three-cycles are Lagrangian, this does not hold true anymore on the orbifolds with some non-factorisable lattice.", "However, if we fix the moduli $c=d=e=0$ , i.e.", "both $A_3$ -tori are orthogonal to each other, and use the relation $dx^i(\\pi _j)=\\delta ^i_j$ , we can verify that any toroidal $\\mathbb {Z}_2$ -invariant three-cycle (REF ) is automatically Lagrangian.", "The next step is to specify the sLag condition on our orientifolds $T^6/(\\mathbb {Z}_4\\times \\Omega \\mathcal {R}_i)$ in order to find the supersymmetric three-cycles.", "The holomorphic three-form $\\Omega _3=\\text{Re}\\Omega _3+i\\text{Im}\\Omega _3$ has the decomposition $ \\begin{split}\\text{Re}(\\Omega _3)&=\\frac{1}{8\\sqrt{u_2}}\\bigl (\\rho _1-u_2\\rho _4\\bigr )\\,,\\\\\\text{Im}(\\Omega _3)&=\\frac{1}{8\\sqrt{u_2}}\\bigl (\\rho _2+u_2\\rho _3\\bigr )\\,,\\end{split}$ with real $\\mathbb {Z}_4$ -invariant three-forms $ \\begin{split}\\rho _1&:=-dx^{124}+dx^{125}+dx^{126}+2dx^{134}-2dx^{136}-dx^{234}-dx^{235}+dx^{236}\\,,\\\\\\rho _2&:=-dx^{124}-dx^{125}+dx^{126}+2dx^{135}+dx^{234}-dx^{235}-dx^{236}\\,,\\\\\\rho _3&:=dx^{145}-2dx^{146}+dx^{156}-dx^{245}+dx^{256}-dx^{345}+2dx^{346}-dx^{356}\\,,\\\\\\rho _4&:=dx^{145}-dx^{156}+dx^{245}-2dx^{246}+dx^{256}-dx^{345}+dx^{356}\\,.\\end{split}$ Because this case is not so interesting from the phenomenological point of view, we do not go into detail here and only give the supersymmetry parameters $\\varphi _{\\text{O6}_i}$ in table REF for each involution.", "$A_3\\times A_1\\times B_2$ With respect to the complex coordinates $(\\ref {A3A1B2complex_coord})$ , the $A_3\\times A_1\\times B_2$ -orbifold has the hermitian metric $g_{i\\bar{j}}=\\left(\\begin{array}{ccc}2R_3^2(1+a)&0&\\frac{R_2R_3}{\\sqrt{2}}(b+i(b+2c))\\\\0&-8u_2aR_3^2&0\\\\\\frac{R_2R_3}{\\sqrt{2}}(b-i(b+2c))&0&2R_2^2\\\\\\end{array}\\right)\\,,$ which gives rise to the Kähler-form $ J=2R_3^2(1+a)\\omega _1+2R_2^2\\omega _2-4R_3^2au_2\\omega _3-R_3R_2(b+2c)\\omega _4-R_3R_2b\\omega _5$ with the following $\\mathbb {Z}_4$ -invariant two-forms: $ \\begin{split}\\omega _1&:=dx^{12}+dx^{23}\\,,\\\\\\omega _2&:=dx^{56}\\,,\\\\\\omega _3&:=dx^{14}-dx^{24}+dx^{34}\\,,\\\\\\omega _4&:=2dx^{15}-dx^{16}+dx^{26}-2dx^{35}+dx^{36}\\,,\\\\\\omega _5&:=-dx^{16}+2dx^{25}-dx^{26}+dx^{36}\\,.\\end{split}$ The anti-symmetry of the orientifold projection on $J$ leads here again to the fixing of some moduli, as shown in the table REF .", "Table: Moduli fixing due to ℛJ=-J\\mathcal {R}J=-J and calibration for different choices of the anti-holomorphic involution on the A 3 ×A 1 ×B 2 A_3 \\times A_1 \\times B_2 lattice.As in the $A_3\\times A_3$ -case, we verify that any toroidal $\\mathbb {Z}_2$ -invariant three-cycle (REF ) is automatically Lagrangian for the choice of fixing the moduli to $b=c=0$ .", "Using the complex coordinates $(\\ref {A3A1B2complex_coord})$ , we can write $\\Omega _3$ in real coordinates $ \\begin{split}\\text{Re}(\\Omega _3)&=\\frac{1}{4\\sqrt{u_2}}\\bigl (\\rho _1+u_1\\rho _3-u_2\\rho _4\\bigr )\\,,\\\\\\text{Im}(\\Omega _3)&=\\frac{1}{4\\sqrt{u_2}}\\bigl (\\rho _2+u_2\\rho _3+u_1\\rho _4\\bigr )\\,,\\end{split}$ where the $\\mathbb {Z}_4$ -invariant three-forms are: $ \\begin{split}\\rho _1&:=-dx^{125}+dx^{126}-dx^{136}+2dx^{135}-dx^{235}\\,,\\\\\\rho _2&:=-dx^{125}+dx^{136}+dx^{235}-dx^{236}\\,,\\\\\\rho _3&:=2dx^{145}-dx^{146}-dx^{246}-2dx^{345}+dx^{346}\\,,\\\\\\rho _4&:=dx^{146}+2dx^{245}-dx^{246}-dx^{346}\\,.\\end{split}$ The next step is to find the sLags on the orientifolds $T^6/(\\mathbb {Z}_4\\times \\Omega \\mathcal {R}_i)$ with $A_3 \\times A_1 \\times B_2$ lattice.", "The sLag condition for any three-cycle (REF ) inherited from the torus can be expressed as follows, $&\\text{Im}(\\Omega _3)\\mid _{\\pi ^{\\text{torus}}}=\\frac{1}{4\\sqrt{u_2}} ([Q-P] +u_1 [\\bar{P}-\\bar{Q}] -u_2 [\\bar{P}+\\bar{Q}] )\\, ,\\\\ &\\text{Re}(\\Omega _3)\\mid _{\\pi ^{\\text{torus}}}=\\frac{1}{4\\sqrt{u_2}} ([P+Q] - u_1 [\\bar{P}+\\bar{Q}] - u_2 [\\bar{P}-\\bar{Q}] )\\, ,\\ $ where $P,\\,Q,\\,\\bar{P},\\,\\bar{Q}$ are the bulk wrapping numbers defined in (REF ).", "Using this decomposition of a (fraction of a) bulk three-cycle into real and imaginary part in dependence of the complex structure $u$ and the bulk wrapping numbers for the corresponding O6-planes we can calculate the supersymmetry parameters $\\varphi _{\\text{O}6_i}$ per lattice orientation displayed in table REF ." ], [ "Pairwise relations between choices of orientifold axes", "We know that in the factorisable case a priori six choices of the anti-holomorphic involution are possible, but that there are pairwise relations between them so that only four are physically inequivalent.", "This means that different but equivalent orientifold projections give rise to the same global semi-realistic particle models as discussed in section REF .", "The same observation can be made in the non-factorisable cases.", "In order to find these relations between the lattices, we compute all supersymmetric three-cycles which do not overshoot the bulk RR tadpole cancellation condition in (REF ).", "Together with the supersymmetry conditions (REF ) this gives rise to the restriction on the toroidal and corresponding bulk wrapping numbers, e.g.", "for the A$_\\textbf {a}$AB-lattice of the $A_3\\times A_1\\times B_2$ lattice, they are bounded by $\\begin{array}{cl}A_1n^3-A_2n^3+2A_2m^3&\\leqslant 8\\,,\\\\2B_3n^3-2B_2m^3&\\leqslant 16\\,,\\end{array}\\quad \\Longleftrightarrow \\quad \\begin{array}{rl}P+Q&\\leqslant 8\\,,\\\\\\bar{Q}-\\bar{P}&\\leqslant 16.\\end{array}$ The restrictions on the other lattice orientations take a similar form.", "The required O6-plane bulk wrapping numbers entering (REF ) are given in (REF ) for the $A_3\\times A_3$ lattice, and in (REF ) for the $A_3\\times A_1\\times B_2$ lattice.", "$A_3\\times A_3$ For this orbifold we found a priori four possible involutions.", "However, due to the relation $\\mathcal {R}_4Q^3=-\\mathcal {R}_1$ between the involutions $\\mathcal {R}_1$ and $\\mathcal {R}_4$ the AAA- and BAB-lattices give physically identical models on the $A_3\\times A_3$ -orientifold.", "But the relation between the corresponding complex structure values cannot be verified more precisely at this point.", "There are two possibilities $\\text{or}\\;\\begin{array}{cc}&\\text{\\textbf {AAA} dual to \\textbf {BAB} and }\\; u_2=u_2^\\prime \\,, \\\\&\\text{\\textbf {AAA} dual to \\textbf {BAB} and }\\;u_2=\\frac{1}{u_2^\\prime }\\,.\\end{array}$ Moreover, the orientifold projections $\\mathcal {R}_2$ and $\\mathcal {R}_3$ - corresponding to the BAA and AAB lattice orientation, respectively, according to table REF - act on the three-cycles in the same way, and therefore the supersymmetric three-cycles on the corresponding orientifolds are the same.", "At first sight one might be tempted to identify these orientifolds, but the distinction of the length of the O6-planes, $\\pi _{\\text{O6}_2}=2\\pi _{\\text{O6}_3}$ , gives in principle rise to different allowed ranks and lengths of bulk cycles in the RR tadpole cancellation conditions and consequently to more possible models for the BAA-orientifold.", "In conclusion, by investigating the structure of bulk three-cycles and their RR tadpole cancellation conditions, we arrive at three physically inequivalent $A_3 \\times A_3$ -lattice orientations AAA, BAA and AAB.", "$A_3\\times A_1\\times B_2$ In this case there are a priori eight possible lattice orientations.", "The number of the corresponding fractional cycles not overshooting the bulk RR tadpole cancellation conditions and the number of possible complex structure values $u_2$ are presented in the table REF .", "Table: The number of supersymmetric fractional cycles bounded by the bulk RR tadpole cancellation condition and the number of possible complex structure values u 2 u_2 for different choices of orientifold axes.Furthermore, we can easily verify the following pairwise relations between the different lattice orientations: $\\begin{split}\\text{\\textbf {A}$_\\textbf {a}$\\textbf {AA} dual to \\textbf {A}$_\\textbf {a}$\\textbf {AB} and }\\; u_2=\\frac{1}{2u_2^\\prime }\\,, \\\\\\text{\\textbf {A}$_\\textbf {a}$\\textbf {BA} dual to \\textbf {A}$_\\textbf {a}$\\textbf {BB} and }\\; u_2=\\frac{1}{2u_2^\\prime }\\,, \\\\\\text{\\textbf {A}$_\\textbf {b}$\\textbf {AA} dual to \\textbf {A}$_\\textbf {b}$\\textbf {BB} and }\\; u_2=\\frac{1}{4u_2^\\prime }\\,, \\\\\\text{\\textbf {A}$_\\textbf {b}$\\textbf {BA} dual to \\textbf {A}$_\\textbf {b}$\\textbf {AB} and }\\; u_2=\\frac{1}{4u_2^\\prime }\\,.\\end{split}$ Altogether, we have thus four physically inequivalent lattices (two with $u_1=0$ and two with $u_1=\\frac{1}{2}$ ) and can restrict our further considerations to the lattice orientations AAB and ABB." ], [ "Cross-check: D6-branes on top of O6-planes", "Non-factorisable $T^6/\\mathbb {Z}_4$ orbifolds have been briefly studied in the past.", "For instance, in [63] only D6-branes on top of the orientifold planes were considered, and only for a particular choice of the moduli in (REF ) and (REF ), i.e., the radii of all tori being equal and all the tori being orthogonal to each other.", "Our goal in this section is twofold.", "At first, we will reproduce and extend the CFT methods and results of [63] to arbitrary values of the moduli, in particular providing the Kaluza-Klein and winding modes for generic D6-brane configuration.", "Secondly, we will compare this result with the geometric method described earlier on, i.e.", "by requiring that the sLag three-cycles wrapped by generic D6-branes satisfy the RR tapdole cancellation condition (REF ) - in the case at hand with only two stacks of D6-branes.", "Let us start by reviewing the method used in [63].", "We denote a basis of the torus lattice by $\\lbrace e_i\\rbrace $ .", "Let $\\lbrace e_i^*\\rbrace $ be a basis of the dual lattice such that $e_i\\cdot e_j^*=\\delta _{ij}$ .", "If the lattice vectors $e_i$ form the $Q$ -invariant metric $g_{ij}$ , the dual vectors $\\lbrace e_i^*\\rbrace $ form the metric $g^*_{ij}=e_i^*\\cdot e_j^*=g_{ij}^{-1}$ .", "Note that the dual vectors transform under $Q^t$ and $\\mathcal {R}_{m\\;(m=1,2,3,4)}^t$ .", "In general, insertions of $\\Omega \\mathcal {R}_m Q^{2k}$ and $\\Omega \\mathcal {R}_m Q^{2k+1}$ in the Klein bottle trace (and strings starting on the $6_{2k}$ and $6_{2k+1}$ branes in the annulus and Möbius strip) give different lattice contributions, so we need to compute both cases separately.", "In the rest of this section we will only consider the first case, but the second one is obtained analogously, by replacing $\\mathcal {R}_m$ by $\\mathcal {R}_mQ$ throughout the elaboration.", "Let $\\textbf {v}_i$ ($i=1,2,3$ ) be the lattice vectors that span the (fraction of the) bulk three-cycle wrapped by the O6-plane.", "They satisfy $\\mathcal {R}_m\\textbf {v}_i=\\textbf {v}_i$ .", "The winding modes are described by vectors $\\textbf {w}_i$ , $i=1,2,3$ , satisfying $\\mathcal {R}_m\\textbf {w}_i=-\\textbf {w}_i$ .", "The momentum modes $\\textbf {p}_i$ appearing in the Klein bottle amplitude in sectors where there are fixed tori correspond to vectors in the dual lattice invariant under $\\mathcal {R}_m$ .", "Using all these vectors we define the matrices $(M_{KB})_{ij}&:=&\\textbf {p}_i\\cdot \\textbf {p}_j,\\\\(M_{A})_{ij}=(M_{MS})_{ij}&:=& \\textbf {v}_i\\cdot \\textbf {v}_j,\\\\(W_{KB})_{ij}=(W_{MS})_{ij}&:=& \\textbf {w}_i\\cdot \\textbf {w}_j.$ The lattice mode contributions (in the corresponding sector) for the Klein bottle, annulus, and Möbius strip are $KB&=&\\frac{4^n}{(\\det M_{KB}\\det W_{KB})^{1/2}},\\\\A&=&\\frac{\\det M_A}{(\\det g)^{1/2}},\\\\MS&=&\\frac{4^n(\\det M_{MS})^{1/2}}{(\\det W_{MS})^{1/2}},$ where $n=\\dim (M)=\\dim (W)$ , ($n=3$ in the untwisted sector, which is the one we are interested in).", "The tadpole cancellation condition is then given by $KB+\\frac{M^2\\cdot A}{16}-\\frac{M\\cdot MS}{16}=0$ where $M$ is the number of identical branes (in the $\\mathcal {R}_mQ$ case we will denote it by $N$ ).", "$A_3\\times A_3$ Let us start with the $A_3\\times A_3$ lattice.", "For concreteness, we will focus on the orientation AAB.", "We will show that the number of supersymmetric D6-branes needed to cancel the (bulk) RR tadpole depends on the angle-moduli between the two $A_3$ -tori.", "Recall that for the orientifold AAB we have the involution $\\mathcal {R}_3$ (REF ), which fixes the moduli to $c+e=0$ (as consequence of $u_1=0$ ) and $e+d=0$ (see table REF ).", "Using these constraints and the $Q$ -invariant metric $g$ , we obtain the following momentum modes p$_i$ : $\\textbf {p}_1&=&2e_1^*-e_2^*,\\\\\\textbf {p}_2&=&e_1^*-e_3^*,\\\\\\textbf {p}_3&=&e_4^*+e_5^*-e_6^*.$ The winding modes are: $\\textbf {w}_1&=&e_1+2e_2+e_3,\\\\\\textbf {w}_2&=&e_4-e_5,\\\\\\textbf {w}_3&=&e_4+e_6.$ The $\\textbf {v}_i$ vectors spanning the O6-planes are: $\\textbf {v}_1&=&e_1,\\\\\\textbf {v}_2&=&e_3,\\\\\\textbf {v}_3&=&e_4+e_5.$ The determinants of the corresponding matrices $M_{KB}$ , $M_{A}$ and $W_{KB}$ are $\\begin{split}\\det M_{KB}&=-\\frac{2}{a (1+a+b+ab-2e^2) R_1^4R_2^2}\\,,\\\\\\det M_{A}&=-8 a (1+a+b+ab-2e^2) R_1^4R_2^2\\,,\\\\\\det W_{KB}&=-32b(1+a+b+ab-2e^2) R_1^2R_2^4\\,.\\end{split}$ They give rise to the lattice mode contributions to the untwisted sector $KB=8\\sqrt{\\frac{a}{b}}\\frac{R_1}{R_2},\\quad A=2\\sqrt{\\frac{a}{b}}\\frac{R_1}{R_2},\\quad MS=32\\sqrt{\\frac{a}{b}}\\frac{R_1}{R_2}.$ The untwisted RR tadpole cancellation condition is $KB+\\frac{M^2\\cdot A}{16}-\\frac{M\\cdot MS}{16}=0\\Rightarrow \\left(M-8\\right)^2=0.$ The $\\mathcal {R}_3Q$ -case leads to the analogous result $(N-8)^2=0$ .", "Due to the action of the $\\mathbb {Z}_2$ symmetry on the corresponding Chan-Paton factors, the gauge group will have rank $\\frac{M}{2} \\times \\frac{N}{2}$ , and the gauge group for generic values of moduli is $U(4)\\times U(4)$ (with a possible gauge symmetry enhancement $U(4) \\hookrightarrow USp/SO(8)$This is a shorthand notation for either $USp(8)$ or $SO(8)$ ., for special choices of geometric moduli - in particular for the choice $c=d=e=0$ of the two $A_3$ lattices orthogonal to each other in (REF ) -, where determining the appropriate type of symmetry enhancement requires the development of CFT techniques at one-loop so far only available for $(T^2)^3$ factorisable backgrounds, see e.g.", "[73], [96], [79], [37], [80]).", "The complementary purely geometric considerations are as follows: The cycle wrapped by the O6-plane is $-2\\gamma _1-2\\bar{\\gamma }_2$ .", "For $e=0$ , its contribution to the tadpole can be cancelled by a stack of $N_1=4$ branes wrapping the cycle $-\\gamma _1$ and a second stack of $N_2=4$ branes wrapping $-\\bar{\\gamma }_2$ , giving rise to a gauge group $USp/SO(8)\\times USp/SO(8)$ in agreement with the CFT result.", "The results for the remaining orientifold projections are summarised in the table REF .", "Table: IIA Orientifolds of ℤ 4 \\mathbb {Z}_4-orbifolds with the lattice of the type A 3 ×A 3 A_3\\times A_3 with D-branes on O-plane.", "For generic values of the metric moduli, the gauge group is U(M 2)×U(N 2)U(\\frac{M}{2}) \\times U(\\frac{N}{2}).$A_3\\times A_1 \\times B_2$ Let us now consider the $A_3\\times A_1 \\times B_2$ lattice.", "For simplicity we will present the analysis of the case where the $B_2$ -torus is orthogonal to $A_3\\times A_1$ .", "This means that the moduli $b,c$ vanish.For an arbitrary choice of these moduli (up to the moduli fixing in table REF ), we obtain the same results.", "For the explicit computations, we will focus on the $\\mathcal {R}_1$ involution (REF ).", "A$_\\textbf {a}$AB The condition $u_1=0$ restricts the moduli $d$ and $a$ .", "From the definition of the complex structure modulus (REF ) we obtain that $d$ is fixed to 0 and $a$ has to be negative.", "Since the dual basis vectors $e_i^*$ transform under the transposed orientifold action $\\mathcal {R}_1^t$ , the momentum modes $\\textbf {p}_i$ are the eigenvectors of $\\mathcal {R}_1^t$ to the eigenvalue $+1$ : $\\begin{split}\\textbf {p}_1&=e_1^*-e_3^*,\\\\\\textbf {p}_2&=2e_1^*-e_2^*+e_3^*,\\\\\\textbf {p}_3&=e_5^*+e_6^*.\\end{split}$ The winding modes are $\\begin{split}\\textbf {w}_1&=e_1+2e_2+e_3,\\\\\\textbf {w}_2&=e_4,\\\\\\textbf {w}_3&=e_5-e_6,\\end{split}$ and the $\\textbf {v}_i$ vectors describing the bulk part of a fractional cycle parallel to some O6-plane are $\\begin{split}\\textbf {v}_1&=e_1,\\\\\\textbf {v}_2&=e_3,\\\\\\textbf {v}_3&=e_5+e_6.\\end{split}$ The determinants of the different matrices are $\\det M_{KB}:=-\\frac{2}{a (1 + a) R_3^4R_2^2}\\,,\\quad \\det M_{A}:=-8 a (1 + a) R_3^4R_2^2\\,,\\quad \\det W_{KB}:=8 (1 + a) R_3^2R_2^2R_1^2\\,.$ The lattice mode contributions to the untwisted sector are $KB=16\\sqrt{-a}\\frac{R_3}{R_1},\\quad A=4\\sqrt{-a}\\frac{R_3}{R_1},\\quad MS=64\\sqrt{-a}\\frac{R_3}{R_1}.$ and the corresponding untwisted RR tadpole cancellation condition reads $KB+\\frac{M^2\\cdot A}{16}-\\frac{M\\cdot MS}{16}=0\\Rightarrow (M-8)^2=0.$ Thus, we get $M=8$ which agrees with [63].", "In a similar way one shows that the tadpole condition from the $\\mathcal {R}_1Q$ -part gives rise to $N=16$ .", "The resulting gauge group is thus $U(4) \\times U(8)$ or some rank preserving gauge enhancement to $SO(2M)$ or $USp(2M)$ (for one or both gauge factors).", "The CFT calculation is complemented by purely geometric considerations as follows: The cycle wrapped by the O6-planes is $\\gamma _1+\\gamma _2-2(\\bar{\\gamma }_1-\\bar{\\gamma }_2)$ .", "Its contribution to the tadpole can be cancelled by a stack of $N_1=4$ branes wrapping the fractional cycle $\\frac{1}{2}\\gamma _1+\\frac{1}{2}\\gamma _2$ and a stack of $N_2=8$ branes wrapping $-\\frac{1}{2}\\bar{\\gamma }_1+\\frac{1}{2}\\bar{\\gamma }_2\\pm \\frac{1}{2}\\bar{\\gamma }_3\\pm \\frac{1}{2}\\bar{\\gamma }_4$ (any choice of sign for the exceptional part is allowed).", "Since the first stack is invariant under the orientifold action, the actual gauge group is $SO/USp(8)\\times U(8)$ .", "Our result - which agrees for both (CFT and cycle homology) methods - differs slightly from [63] in the fact that the first gauge factor experiences a gauge group enhancement, which they do not mention.", "A$_\\textbf {b}$AB In this case, from the condition $u_1=\\frac{1}{2}$ we obtain $\\begin{split}d=-\\frac{aR_3}{R_1}\\,,\\quad a(1+a\\frac{R_3^2}{R_1^2})<0\\,.\\end{split}$ The momentum modes $\\textbf {p}_i$ are $\\begin{split}\\textbf {p}_1&=e_1^*-e_3^*,\\\\\\textbf {p}_2&=2e_1^*-e_2^*+e_4^*,\\\\\\textbf {p}_3&=e_6^*.\\end{split}$ The winding modes $\\textbf {w}_i$ and lattice vectors $\\textbf {v}_i$ are $\\begin{split}\\textbf {w}_1&=e_1+2e_2+e_3,\\\\\\textbf {w}_2&=e_2+e_4,\\\\\\textbf {w}_3&=e_5,\\end{split}$ $\\begin{split}\\textbf {v}_1&=e_1,\\\\\\textbf {v}_2&=e_3,\\\\\\textbf {v}_3&=e_5+2e_6.\\end{split}$ The determinants of the different matrices are thus given by $\\det M_{KB}=\\frac{-2}{(a+a^2)R_2^2R_3^4}\\,,\\quad \\det M_{A}=-8(1+a)aR_2^2R_3^4\\,,\\quad \\det W_{KB}=8(1+a)(R_1^2+aR_3^2)R_2^2R_3^2\\,,$ and the resulting untwisted RR tadpole cancellation condition reads $KB+\\frac{M^2\\cdot A}{16}-\\frac{M\\cdot MS}{16}=0\\Rightarrow (M-8)^2=0,$ i.e.", "we obtain $M=8$ .", "In a similar manner, the $\\mathcal {R}_1Q$ -insertion gives rise to $N=8$ .", "Therefore, the gauge group is $U(4)\\times U(4)$ or some rank preserving gauge group enhancement thereof.", "The complementary considerations in terms of cycle homologies are as follows: The cycle wrapped by the O6-planes is $2\\gamma _2-2(\\bar{\\gamma }_1-\\bar{\\gamma }_2)$ .", "Its contribution to the tadpole can be cancelled, for instance, by a stack of $N_1=4$ branes wrapping the fractional cycle $\\frac{1}{2}\\gamma _1+\\frac{1}{2}\\gamma _2$ and a stack of $N_2=4$ branes wrapping $-\\frac{1}{2}\\gamma _1+\\frac{1}{2}\\gamma _2-\\bar{\\gamma }_1+\\bar{\\gamma }_2\\pm (\\frac{1}{2}\\gamma _3-\\bar{\\gamma }_3)\\pm (\\frac{1}{2}\\gamma _4-\\bar{\\gamma }_4)$ (any choice of sign for the exceptional part is allowed).", "Naively, this gives rise to the gauge group $U(4)\\times U(4)$ , but the first stack maps to itself under the orientifold action, which enhances the symmetry to $USp/SO(8)\\times U(4)$ .", "The results for the remaining orientifold projections can be derived analogously and are summarised in the table REF .", "Table: IIA Orientifolds of ℤ 4 \\mathbb {Z}_4-orbifolds with the lattice of the type A 3 ×A 1 ×B 2 A_3\\times A_1\\times B_2 with D-branes on top of the O-planes canceling the RR tadpoles.The column $u_1=0$ agrees mostly with the result [63], while the results for $u_1=\\frac{1}{2}$ are presented here for the first time.", "We can also see the relation between the lattices (REF ) if we interchange $M$ and $N$ ." ], [ "Factorisation of non-factorisable orbifolds", "In this chapter we will show that both non-factorisable orbifolds can be written in a factorised form.", "This identification between the lattices was already detected in [97] by consideration of factorisable orbifolds.", "Here we want to explain which conclusions this identification means from the point of view of non-factorisable orbifolds.", "We make the restriction that in the first non-factorisable orbifold background, both $A_3$ lattices are orthogonal to each other, and in the second non-factorisable orbifold background, the $A_3\\times A_1$ lattice is orthogonal to the $B_2$ lattice.", "This fixes the moduli to $c=d=e=0$ and $b=c=0$ , respectively.", "On both orbifolds we can find a real basis such that the non-factorisable structure of the lattice decomposes into three two-tori, but an additional $\\mathbb {Z}_2$ shift symmetry appears.", "Any three-cycle, written now as a product of three one-cycles on each two-torus, is automatically $\\mathbb {Z}_2$ -invariant.", "Moreover, only $\\mathbb {Z}_2$ -invariant three-cycles on the non-factorisable lattice can be expressed as three-cycles with respect to the new basis.", "Therefore, the number of wrapping numbers is reduced from twelve ($A_3\\times A_3$ ) or ten ($A_3\\times A_1\\times B_2$ ) to six in agreement with the naive expectation from the known factorisable orbifold backgrounds $(T^2)^3/\\mathbb {Z}_N$ or $(T^2)^3/(\\mathbb {Z}_N \\times \\mathbb {Z}_M)$ .Note, however, that due to the additional symmetry some of the new wrapping numbers can now also be half-integer.", "Because any three-cycle, written in the new basis, is $\\mathbb {Z}_2$ -invariant, it follows that it is automatically Lagrangian.", "Furthermore, it can be verified that the sLag condition can be expressed in the same form as for the usual factorisable orbifolds: the sum of the angles between the one-cycles wrapped by supersymmetric D6-branes on each two-torus and the ${\\cal R}_i$ -invariant O6-plane has to vanish (mod $2\\pi $ ).", "$A_3\\times A_3$ In this case, we introduce new coordinates along the directions $\\begin{split}v_1&:=\\pi _1+\\pi _2\\,,\\quad v_2:=\\pi _2+\\pi _3\\,,\\quad v_3:=\\pi _1+\\pi _3\\,,\\\\v_4&:=\\pi _4+\\pi _6\\,,\\quad v_5:=\\pi _4+\\pi _5\\,,\\quad v_6:=\\pi _5+\\pi _6\\,.\\end{split}$ With respect to this basis, the metric $g$ and the Coxeter element $Q$ take a factorised form, $g=\\text{diag}\\,\\left(2(1+a)R_1^2,\\,2(1+a)R_1^2,\\,-4aR_1^2,-4bR_2^2,\\,2(1+b)R_2^2,\\,2(1+b)R_2^2\\right)\\,,\\\\Q=\\text{diag}\\left(\\begin{array}{cc}0 & -1 \\\\1 & 0 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}-1 & 0 \\\\0 & -1 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}0 & -1 \\\\1 & 0 \\\\\\end{array}\\right)\\,,$ and also the orientifold projections $\\mathcal {R}_i$ become factorised, e.g.", ": $\\mathcal {R}_4=\\text{diag}\\left(\\begin{array}{cc}1 & 0 \\\\0 & -1 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}1 & 0 \\\\0 & -1 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}1 & 0 \\\\0 & -1 \\\\\\end{array}\\right)\\,.$ But as already mentioned, the basis change gives rise to additional symmetries, which identify points of the factorised torus by the following shifts $\\begin{split}&p \\simeq p+\\frac{v_1+v_2+v_3}{2}\\,,\\\\&p \\simeq p+\\frac{v_4+v_5+v_6}{2} \\, ,\\end{split}$ for any point $p$ on the torus, as depicted in figure REF .", "Figure: Shift identifications of points on the factorised form of the A 3 ×A 3 A_3\\times A_3-lattice by v 1 +v 2 +v 3 2\\frac{v_1+v_2+v_3}{2} (red) and v 4 +v 5 +v 6 2\\frac{v_4+v_5+v_6}{2} (green).In the new coordinates, only six wrapping numbers occur, $\\pi ^{\\text{torus}}:=(\\tilde{n}^1v_1+\\tilde{m}^1v_2)\\wedge (\\tilde{n}^2v_3+\\tilde{m}^2v_4)\\wedge (\\tilde{n}^3\\pi _5+\\tilde{m}^3\\pi _6)\\,,$ in terms of which the bulk wrapping numbers defined in (REF ) and (REF ) can be rewritten as follows: $\\begin{array}{ll}A_1q^1=-A_3q^1=-(\\tilde{n}^1+\\tilde{m}^1)\\tilde{n}^2\\tilde{n}^3\\,,&B_1m^1=-B_3m^1=\\tilde{n}^1\\tilde{m}^2(\\tilde{n}^3+\\tilde{m}^3)\\,,\\\\A_1r^1=-A_3r^1=-(\\tilde{n}^1+\\tilde{m}^1)\\tilde{n}^2(\\tilde{n}^3+\\tilde{m}^3)\\,,&B_1n^1=-B_3n^1=(\\tilde{n}^1+\\tilde{m}^1)\\tilde{m}^2(\\tilde{n}^3+\\tilde{m}^3)\\,,\\\\A_1s^1=-A_3s^1=-(\\tilde{n}^1+\\tilde{m}^1)\\tilde{n}^2\\tilde{m}^3\\,,&B_1p^1=-B_3p^1=\\tilde{m}^1\\tilde{m}^2(\\tilde{n}^3+\\tilde{m}^3)\\,,\\\\A_2q^1=(\\tilde{n}^1-\\tilde{m}^1)\\tilde{n}^2\\tilde{n}^3\\,,&B_2m^1=\\tilde{n}^1\\tilde{m}^2(-\\tilde{n}^3+\\tilde{m}^3)\\,,\\\\A_2r^1=(\\tilde{n}^1-\\tilde{m}^1)\\tilde{n}^2(\\tilde{n}^3+\\tilde{m}^3)\\,,&B_2n^1=(\\tilde{n}^1+\\tilde{m}^1)\\tilde{m}^2(-\\tilde{n}^3+\\tilde{m}^3)\\,,\\\\A_2s^1=(\\tilde{n}^1-\\tilde{m}^1)\\tilde{n}^2\\tilde{m}^3\\,,&B_2p^1=\\tilde{m}^1\\tilde{m}^2(-\\tilde{n}^3+\\tilde{m}^3)\\,.\\end{array}$ Note that due to the shift symmetry (REF ), the wrapping numbers $\\tilde{n}^2$ and $\\tilde{m}^2$ can also have half-integer values.", "On the factorised lattice, the geometric difference between the length of the O6-planes on the AAB- and BAA-orientations becomes clear: for the AAB-lattice, the ${\\cal R}$ -invariant O6-plane is placed along the axes $v_1-v_2$ and $v_3$ on $T^2_{(1)} \\times T^2_{(2)}$ and therefore it passes through the points which are identified by the shift symmetry (REF ), but this is no longer the case for the BAA-lattice.", "$A_3\\times A_1\\times B_2$ Here it suffices to introduce new directions only on $A_3\\times A_1$ : $\\begin{split}v_1&:=\\pi _1+\\pi _2\\,,\\quad v_3:=\\pi _1+\\pi _3\\,,\\\\v_2&:=\\pi _2+\\pi _3\\,,\\quad v_4:=\\pi _4\\,.\\end{split}$ This basis factorises the torus, such that the metric becomes $g=\\text{diag}\\left(\\begin{array}{cc}2(1+a)R_3^2& 0 \\\\0 & 2(1+a)R_3^2 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}-4aR_3^2 & -4au_1R_3^2 \\\\-4au_1R_3^2 & R_1^2 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}2R_2^2 & -R_2^2 \\\\-R_2^2 & R_2^2 \\\\\\end{array}\\right)\\,.$ For the Coxeter element $Q$ and the orientifold projections $\\mathcal {R}_i$ we obtain $Q=\\text{diag}\\left(\\begin{array}{cc}0 & -1 \\\\1 & 0 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}-1 & 0 \\\\0 & -1 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}1 & -1 \\\\2 & -1 \\\\\\end{array}\\right)\\,,\\\\\\mathcal {R}_1=\\text{diag}\\left(\\begin{array}{cc}0 & -1 \\\\-1 & 0 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}1 & 2u_1 \\\\0 & -1 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}1 & -1 \\\\0 & -1 \\\\\\end{array}\\right)\\,,\\\\\\mathcal {R}_4=\\text{diag}\\left(\\begin{array}{cc}-1 & 0 \\\\0 & 1 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}1 & 2u_1 \\\\0 & -1 \\\\\\end{array}\\right)\\oplus \\left(\\begin{array}{cc}1 & -1 \\\\0 & -1 \\\\\\end{array}\\right)\\,.$ Also in this case the basis change gives rise to an additional shift symmetry displayed in figure REF , $p\\simeq p+\\frac{v_1+v_2+v_3}{2}\\quad \\text{for any point $p$ on the torus}.$ Instead of ten wrapping numbers which we need to describe a fractional three-cycle in non-factorisable coordinates, in the $v$ -basis the usual six wrapping numbers are sufficient: $\\pi ^{\\text{torus}}:=(\\tilde{n}^1v_1+\\tilde{m}^1v_2)\\wedge (\\tilde{n}^2v_3+\\tilde{m}^2v_4)\\wedge (m^3\\pi _5+n^3\\pi _6)\\,.$ The relation between the non-factorised toroidal and bulk wrapping numbers in (REF ) and (REF ) and the new factorised ones is given by $\\begin{array}{ll}A_1=-A_3=-(\\tilde{n}^1+\\tilde{m}^1)\\tilde{n}^2\\,,&A_2=(\\tilde{n}^1-\\tilde{m}^1)\\tilde{n}^2\\,,\\\\B_1=\\tilde{n}^1\\tilde{m}^2\\,,&B_3=\\tilde{m}^1\\tilde{m}^2 = B_2 - B_1\\,.\\end{array}$ Notice that here, due to the shift symmetry (REF ), the wrapping number $\\tilde{n}^2$ can be half-integer if $\\tilde{m}^1+\\tilde{n}^1$ is even.", "Figure: Shift symmetry of the factorised form of the A 3 ×A 1 A_3\\times A_1-lattice along v 1 +v 2 +v 3 2\\frac{v_1+v_2+v_3}{2} (in red)." ], [ "Concrete Pati-Salam Models on $A_3 \\times A_1 \\times B_2$", "In this chapter we present the construction of some local and global semi-realistic supersymmetric Pati-Salam (PS) models using intersecting D6-branes.Notice that for generic configurations of gauge groups, global models do not only have to satisfy the RR tadpole cancellation conditions, but also the K-theory constraints, which are usually derived by scanning through all possible probe D-branes supporting $USp(2)$ gauge factors [98].", "For PS models with all gauge groups (including `hidden' ones) of even rank, the K-theory constraints are, however, trivially fulfilled.", "In section REF we have seen that there exist duality relations between the different orientifold projections ${\\cal R}_i$ ($i=1,2,3,4$ ) and that only four lattice orientations are independent.", "For the construction of particle spectra, we choose the AAB- and ABB-types lattices.", "In addition to the RR tadpole cancellation and sLag conditions, we require that the $U(4)_a$ stack is free from (anti-)symmetric representations.The systematic computer search on the $A_3 \\times A_1 \\times B_2$ lattices actually showed that all two- and four-generation models of PS-type satisfy this condition without a priori imposing it.", "An intensive computer search showed that only global supersymmetric PS-models with two or four generations are possible.", "The supersymmetry conditions give rise to restrictions on the allowed values of the imaginary part $u_2$ of the complex structure modulus $\\mathcal {U}$ , and only a small number of values can be used for the construction of global PS-models without overshooting the bulk RR tadpole cancellation condition.", "The allowed values of $u_2$ are illustrated in the table REF .", "For some branes, some bulk wrapping numbers turn out to be zero, and thus the supersymmetry conditions do not fix the complex structure.", "If PS-models can be constructed just with such branes, we will list the arbitrariness in $\\text{Im}(\\mathcal {U})$ as $\\infty $ in table REF .", "Table: The possible values of Im(𝒰)\\text{Im}(\\mathcal {U}) for global supersymmetric Pati-Salam models, with ∞\\infty denoting that it can be choosen arbitrarily.Two generations We start the search for global PS-models with the case of two particle generations and note some general properties of the models obtained.", "All models with $U(4)\\times U(2)\\times U(2)$ -gauge group in the visible sector contain chiral particles which transform in the (anti-)symmetric representation of the $U(2)$ on the $b$ and/or $c$ -stack.", "The spectrum of the models with gauge symmetry enhancement $U(1) \\hookrightarrow USp/SO(2)$ on both the $b$ - and $c$ -stack, on the other hand, lacks the bifundamental representations in the $bc$ -sector, i.e., there is no Higgs field in the chiral spectrum of such models.", "In table REF , we provide an explicit example of a PS-model with $U(4)\\times U(2)\\times USp/SO(2)$ gauge group in the visible sector.", "Its chiral spectrum is given in the table REF .", "In appendix , we provide an explicit example for each other type of visible gauge group, i.e.", "$U(4)\\times U(2)\\times U(2)$ and $U(4)\\times USp/SO(2)\\times USp/SO(2)$ .", "Table: D6-branes for a two generation supersymmetric PS-model with A 𝐚 _\\textbf {a}BB-lattice and u 2 =1/2u_2=1/2.Table: Chiral spectrum for the two generation U(4)×U(2)×USp/SO(2)×USp/SO(2)×U(2)U(4)\\times U(2)\\times USp/SO(2)\\times USp/SO(2)\\times U(2)-PS model with D6-brane configuration given in table .Four generations In a similar way, we can realise global supersymmetric PS-models with four generations with different gauge groups (without/with some enhancement of the `left' and `right' stack) in the visible sector.", "In general, only the $U(4)\\times USp/SO(2)\\times USp/SO(2)$ -models do not contain chiral matter in the (anti-)symmetric representation on the $b$ - and $c$ - stacks.", "But these models also do not contain chiral states in the bifundamental representation of the $bc$ -sector, which could serve as a (chiral) Higgs multiplet.", "In appendix , we collect several examples of global D6-brane configurations of such types together with their chiral spectrum.", "Three generations As mentioned above, there are no global supersymmetric Pati-Salam models with three generations.", "But it is possible to construct models where the bulk part of the tadpole vanishes and only the exceptional part remains.", "Such local models are only realisable on the ABB-orientifolds.", "More precisely, for the A$_\\textbf {a}$BB-lattice only $u_2=3\\,,\\,\\frac{1}{3}$ provides such models, and the gauge group is enhanced on the $b$ - and $c$ -stacks.", "For the A$_\\textbf {b}$BB-lattice, it is possible to construct local models with $u_2=\\frac{3}{2}$ with symmetry enhancement on the $b$ - and/or $c$ -stack.", "A general property of all such models is the appearance of chiral particles which transform in the (anti-)symmetric representation of $U(4)_a$ ." ], [ "Discussion and Conclusions", "In this article, we explored the full three-cycle geometry of the non-factorisable $T^6/\\mathbb {Z}_4$ orbifold on the two possible root lattices $A_3 \\times A_3$ and $A_3 \\times A_1 \\times B_2$ and compared it with the factorisable $B_2 \\times (A_1)^2 \\times B_2$ case.", "We found that, under the anti-holomorphic involution ${\\cal R}$ of Type IIA/$\\Omega \\mathcal {R}$ orientifolds, there exist a priori four different lattice orientations for $A_3\\times A_3$ , and eight for $A_3\\times A_1\\times B_2$ .", "However, a closer look at the number of supersymmetric (i.e.", "sLag) fractional three-cycles bounded in their length by the RR tadpole cancellation conditions and by the allowed values for the complex structure moduli, which encode the relative angles between the different root lattices as well as between the generators within each $A_3$ lattice, reveals - in analogy to the factorisable cases [42], [46], [48] - the existence of several dualities, which lead to identical physics for different choices of lattice orientations under ${\\cal R}$ .", "More precisely, on $A_3\\times A_3$ we found a duality relation between the AAA and BAB lattices, leaving three independent choices.", "For the $A_3\\times A_1\\times B_2$ lattice, there are four pairwise duality relations (A$_\\textbf {a}$ AA and A$_\\textbf {a}$ AB, A$_\\textbf {a}$ BA and A$_\\textbf {a}$ BB, A$_\\textbf {b}$ AA and A$_\\textbf {b}$ BB, A$_\\textbf {b}$ BA and A$_\\textbf {b}$ AB), leaving only four independent choices.", "After computing the sLag three-cycles, in order to ascertain our results, we cross-checked agreement of the RR tadpole cancellation conditions among our new purely geometric derivation as well as via the `old' CFT method, in particular for the special D6-brane configurations on top of the O6-planes in [63], compared to which we generalised the CFT results to arbitrary values of the angles inside the $A_3$ lattices and the angle between the $A_3$ and the $A_1$ lattices.", "For the $A_3\\times A_3$ lattices and the a-lattices of $A_3\\times A_1\\times B_2$ , our results derived in a twofold, mutually agreeing way (mostly) coincide with the results listed in [63], while those corresponding to the ${\\bf b}$ -type lattices of $A_3 \\times A_1 \\times B_2$ are newly found here.", "With the full list of allowed sLag three-cycles at hand, we proceeded to search for local and global semi-realistic Pati-Salam models on both types of lattices.", "The $A_3\\times A_3$ lattices happened to be very restricted - e.g.", "by the small number of available three-cycles - and therefore unsuitable for model building.", "But the $A_3\\times A_1\\times B_2$ lattice with different orientations provided a very rich class of backgrounds with ample potential for model building.", "Although the search for global Pati-Salam models - to which we restricted ourselves here since the K-theory constraints are trivially fulfilled in that case - with three generations did not bear any fruit, many models with two and four generations were found.", "From a qualitative point of view, our results are in agreement with [99] - which studied these non-factorisable orbifolds in the context of the heterotic string - in the sense that the $A_3\\times A_1\\times B_2$ lattice is the most promising non-factorisable one for model building.", "Our first model searches typically lead to globally consistent models, where one or more stacks of D-branes wrap three-cycles invariant under the anti-holomorphic involution ${\\cal R}$ .", "While it is well known that in such cases an enhancement of the gauge group $U(N) \\hookrightarrow USp(2N)$ or $SO(2N)$ occurs, the correct distinction can - with the generally available model building techniques to date - only be done by computing open string CFT amplitudes such as the gauge thresholds for factorisable backgrounds in [45], [46], [83], [48] and reading off the correct sign for the orientifold projection from the Möbius strip contribution to either the one-loop beta function coefficient or to the RR tadpoles.", "Our finding in section , that fractional three-cycles can be rewritten in a factorised form, suggests that the CFT methods can be straightforwardly generalised to the non-factorisable backgrounds of $T^6/\\mathbb {Z}_4$ discussed here from a purely geometric viewpoint.", "By fully developing the corresponding CFT, not only the compete chiral plus vector-like matter spectrum can be determined, but also the low-energy effective action can (in principle) be derived.", "Identifying all probe D-branes supporting $USp(2)$ gauge groups by means of CFT techniques is furthermore necessary to determine the K-theory constraints for all future Standard Model or GUT model building.", "Last but not least, it will be interesting to extend the methods for studying non-factorisable orbifolds to other point groups $\\mathbb {Z}_{N \\ne 4}$ , and to incorporate closed string fluxes and study if particle physics models with (nearly) complete stabilisation of the closed string moduli are within reach.", "Acknowledgements: This work is partially supported by the Cluster of Excellence `Precision Physics, Fundamental Interactions and Structure of Matter' (PRISMA) DGF no.", "EXC 1098, the DFG research grant HO 4166/2-2, and the DFG Research Training Group `Symmetry Breaking in Fundamental Interactions' GRK 1581." ], [ "More Pati-Salam models", "In this appendix, we provide additional explicit examples of global Pati-Salam models with two and four generations on the $A_3 \\times A_1 \\times B_2$ lattice." ], [ "2 generations", "In the main text, we presented a global model where one of the left- or right-symmetric groups of the Pati-Salam gauge group is provided by an enhanced $SO(2)_{L/R}$ or $USp(2)_{L/R}$ symmetry.", "We also found global models where none or both left- and/or right-symmetric groups are replaced by an enhanced gauge symmetry; we will show an example of each of type in the following.", "Tables REF and REF show the D6-branes and chiral spectrum, respectively, of a global Pati-Salam model with visible gauge group $U(4)\\times U(2)\\times U(2)$ .", "Table: D6-branes for a global two generation PS-model on the A 𝐚 _\\textbf {a}BB-lattice with u 2 =1u_2=1.Table: Chiral spectrum of the two generation U(4)×U(2)×U(2)×U(2)×USp/SO(4)U(4)\\times U(2)\\times U(2)\\times U(2)\\times USp/SO(4)-PS model with D-brane configuration displayed in table .Tables REF and REF show the D6-branes and chiral spectrum, respectively, of a global Pati-Salam model with visible gauge group $U(4)\\times USp/SO(2)\\times USp/SO(2)$ .", "Table: D6-branes for a global two generation PS-model on the A 𝐚 _\\textbf {a}BB-lattice with non-fixed u 2 u_2.Table: Chiral spectrum of the two generation U(4)×USp/SO(2)×USp/SO(2)×USp/SO(2)×U(4)U(4)\\times USp/SO(2)\\times USp/SO(2)\\times USp/SO(2)\\times U(4)-PS model with D6-brane configuration displayed in table ." ], [ "4 generations", "As in the previous case, we also found four-generation global Pati-Salam models where none, one or both left- and/or right-symmetric groups are replaced by an enhanced symmetry $SO(2)_{L/R}$ or $Usp(2)_{L/R}$ .", "Here we will show a concrete example for each of these three cases.", "Tables REF and REF show the D6-brane configuration and chiral spectrum, respectively, of a Pati-Salam model with visible sector $U(4)\\times U(2)\\times U(2)$ .", "Table: D6-branes for a global four generation PS-model on the A 𝐚 _\\textbf {a}BB-lattice with u 2 =1u_2=1.Table: Chiral spectrum of the four generation U(4)×U(2)×U(2)×USp/SO(4)×U(2)U(4)\\times U(2)\\times U(2)\\times USp/SO(4)\\times U(2)-PS model with D6-brane configuration displayed in table '.Tables REF and REF show the D6-brane configuration and chiral spectrum, respectively, of a Pati-Salam model with visible sector $U(4)\\times U(2)\\times USp/SO(2)$ .", "Table: D6-branes for a global four generation PS-model on the A 𝐚 _\\textbf {a}BB-lattice with u 2 =1u_2=1.Table: Chiral spectrum of the four generation U(4)×U(2)×USp/SO(2)×U(3)×USp/SO(2)U(4)\\times U(2)\\times USp/SO(2)\\times U(3)\\times USp/SO(2)-PS model with D6-brane configuration given in table .Tables REF and REF show the D6-brane configuration and chiral spectrum, respectively, of a Pati-Salam model with visible sector $U(4)\\times USp/SO(2)\\times USp/SO(2)$ .", "Table: D6-branes for a global four generation PS-model one the A 𝐚 _\\textbf {a}AB-lattice with non-fixed u 2 u_2.Table: Chiral spectrum of the four generation U(4)×USp/SO(2)×USp/SO(2)×USp/SO(4)U(4)\\times USp/SO(2)\\times USp/SO(2)\\times USp/SO(4)-PS model with D6-brane configuration displayed in table .tocsectionReferences" ] ]
1606.04926
[ [ "Suppression of lattice thermal conductivity by mass-conserving cation\n mutation in multi-component semiconductors" ], [ "Abstract In semiconductors almost all heat is conducted by phonons (lattice vibrations), which is limited by their quasi-particle lifetimes.", "Phonon-phonon interactions represent scattering mechanisms that produce thermal resistance.", "In thermoelectric materials, this resistance due to anharmonicity should be maximised for optimal performance.", "We use a first-principles lattice-dynamics approach to explore the changes in lattice dynamics across an isostructural series where the average atomic mass is conserved: ZnS to CuGaS$_2$ to Cu$_2$ZnGeS$_4$.", "Our results demonstrate an enhancement of phonon interactions in the multernary materials, and confirm that lattice thermal conductivity can be controlled independently of the average mass and local coordination environments." ], [ "Suppression of lattice thermal conductivity by mass-conserving cation mutation in multi-component semiconductors Taizo Shibuya Department of Mechanical Engineering, Keio University, Yokohama 223-8522, Japan Jonathan M. Skelton Adam J. Jackson Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom Kenji Yasuoka Department of Mechanical Engineering, Keio University, Yokohama 223-8522, Japan Atsushi Togo Elements Strategy Initiative for Structural Materials, Kyoto University, Kyoto 606-8501, Japan Isao Tanaka Elements Strategy Initiative for Structural Materials, Kyoto University, Kyoto 606-8501, Japan Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan Aron Walsh Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom Global E$^3$ Institute and Department of Materials Science and Engineering, Yonsei University, Seoul 120-749, Korea [][email protected] In semiconductors almost all heat is conducted by phonons (lattice vibrations), which is limited by their quasi-particle lifetimes.", "Phonon-phonon interactions represent scattering mechanisms that produce thermal resistance.", "In thermoelectric materials, this resistance due to anharmonicity should be maximised for optimal performance.", "We use a first-principles lattice-dynamics approach to explore the changes in lattice dynamics across an isostructural series where the average atomic mass is conserved: ZnS to CuGaS2 to Cu2ZnGeS4.", "Our results demonstrate an enhancement of phonon interactions in the multernary materials, and confirm that lattice thermal conductivity can be controlled independently of the average mass and local coordination environments.", "Direct heat to electricity conversion in a thermoelectric device represents an important process for energy generation and efficiency.", "[1] The physical principles and applications of thermoelectric power conversion are well established; however, current commercial devices are based on heavy metal tellurides, which have issues associated with cost, toxicity and element availability.", "A new generation of materials are required to support growth of this technology, ideally with performance superior to existing compounds.", "[2] There has already been decades of research focused on discovering materials with enhanced performance.", "A set of standard guidelines have emerged in the field and are widely adopted.", "These include: high atomic weight (which led to PbTe and Bi2Te3);[3] loosely bound “rattling\" atoms (e.g.", "clathrates);[4]; low dimensionality (e.g.", "nanorods and quantum dots);[5] and phonon localization (e.g.", "superlattices and alloys).", "[6] Since hundreds of materials have been systematically screened, tried and tested, new ideas are required to move this field forward.", "[7], [8], [9] A critical factor underpinning thermoelectric performance is thermal conductivity, which represents a thermal loss mechanism that should be minimised for high-performance.", "All crystals have phonons (extended lattice vibrations), but the thermal conductivity is linked to their lifetime, which can vary by several orders of magnitude between materials.", "[10] Both electrons and phonons can play a role in thermal conductivity, but for dielectrics and intrinsic semiconductors the latter are usually dominant.", "An ideal thermoelectric material will offer “phonon-glass electron-crystal\" behaviour:[11] a crystalline system where phonons have short lifetimes (low thermal conductivity), while electron carriers have long lifetimes (high electrical conductivity).", "In this study we attempt to answer a simple question: can the lattice thermal conductivity of tetrahedral semiconductors be tuned without changing the local structure or average mass of the compound?", "The topic was inspired by the early work of Pamplin[12] and Goodman[13] on cation cross-substitution to produce multi-component semiconductors, which was later applied to screening materials for photovoltaics,[14] topological conductivity[15] and spintronics[16].", "We consider the mutation from ZnS $\\rightarrow $ CuGaS2 $\\rightarrow $ Cu2ZnGeS4 where the average atomic mass and crystal structures are conserved.", "A systematic analysis, from direct computation of the phonon energies and lifetimes, demonstrates a large decrease in phonon lifetimes on transition from the binary to ternary compounds, but no further enhancement is found in the quaternary system.", "Analysis of the results suggests avenues for lowering the thermal conductivity of semiconducting crystals.", "Lattice thermal conductivity from first-principles.", "Within the harmonic approximation, phonons propagate indefinitely and the lattice thermal conductivity is formally infinite.", "Even in the absence of structural defects and chemical impurities, the interaction between phonon modes leads to scattering processes, of which anharmonic three-phonon interactions are considered to be dominant.", "As stated by Ziman in his seminal monograph: “Knowledge of the magnitude of the anharmonic terms which generate the [phonon-phonon scattering] processes is scanty, and can only be obtained by roundabout arguments from other general macroscopic phenomena\" [17].", "Fifty years later, the study of many-phonon processes remains a daunting task.", "By solving the phonon Boltzmann equation within the relaxation-time approximation (RTA), the lattice thermal conductivity tensor ($\\kappa $ ) can be expressed succinctly as a sum over phonons of band index $\\lambda $ and wavevector $q$ : $\\kappa = \\sum _{q \\lambda } C_{V, q \\lambda } \\nu _{q \\lambda } \\otimes \\nu _{q \\lambda } \\tau _{q \\lambda }$ where $C_V$ is the isochoric modal heat capacity, $\\nu $ is the group velocity, and $\\nu _{q \\lambda } \\tau _{q \\lambda } = \\Lambda _{q \\lambda }$ , the phonon mean free path.", "The single-mode relaxation time $\\tau $ can be deduced from the imaginary part of the phonon self-energy, computed within many-body perturbation theory.", "At high temperatures, $C_V$ approaches a constant, so the two critical parameters for lattice thermal conductivity are $\\nu $ and $\\Lambda $ .", "The velocity $\\nu $ is determined simply by the phonon dispersion with respect to reciprocal-space wavevector ($\\frac{\\partial \\omega }{\\partial q}$ ).", "$\\Lambda $ depends on $\\nu $ , but also on the relaxation time, which is closely related to the phonon-phonon interaction strength and the distribution of frequencies in the phonon density of states, the latter determining the number of possible energy-conserving scattering events.", "It has only recently become possible to directly compute many-phonon interactions from first-principles.", "[18], [19], [20] Even though a robust infrastructure now exists, there is a large computational cost to performing the simulations for complex crystals.", "By calculating the three-phonon interaction strength ($\\phi _{\\lambda \\lambda ^{\\prime }\\lambda ^{\\prime \\prime }}$ ), $\\tau $ can be computed up to second-order within many-body perturbation theory.", "We employ the Phonopy and Phono3py packages[20], [21] using VASP[22] as the force calculator and the PBEsol[23] exchange-correlation treatment within Kohn-Sham density functional theory (DFT).", "PBEsol was chosen as it has been found to be an excellent compromise between accuracy and computational cost for lattice dynamics.", "[24] For each material we consider the harmonic phonon dispersion, thermal expansion within the quasi-harmonic approximation (QHA), and finally lattice thermal conductivity within the RTA.", "The technical set-up and procedure is provided in the computational details section.", "It should be noted that an alternative approach to capture anharmonic interactions is through molecular dynamics simulations where, for example, the Green-Kubo method can be used to probe phonon lifetimes and thermal conductivity.", "[25] Figure: Phonon dispersion of ZnS, CuGaS2 and Cu2ZnGeS4 ascalculated within the harmonic approximation (PBEsol/DFT) at the equilibrium (athermal)lattice constants.", "LO-TO splitting of the Γ\\Gamma point modes is included based on the calculateddielectric tensors.For ZnS, the dispersion determined from neutron scattering measurements is overlaid.Illustrations of the crystal structures show the polyhedra coloured according to the cation sublattice.Zincblende ZnS.", "ZnS can crystallise in two polymorphs, but here we consider only zincblende (sphalerite), the most common face-centered-cubic form with space group type $F\\bar{4}3m$ ($T_d$ symmetry).", "The primitive cell contains two lattice sites, and hence there are 6 phonon modes.", "At the zone-centre (q=0), the $T_2$ (IR and Raman active) optic branch is split into lower-energy transverse (TO) and higher-energy longitudinal (LO) modes.", "In the first reported Raman measurement of ZnS crystals, the room temperature TO and LO modes were measured at 276 and 351 cm$^{-1}$ , respectively.", "[27] The calculated values from the harmonic phonon dispersion (Figure REF ) are 275 and 335 cm$^{-1}$ , in good agreement.", "In addition, the phonon dispersion across the Brillouin zone agrees very well with neutron scattering measurements.", "[26] The thermal expansion, calculated within the QHA, is plotted in Figure REF .", "The known negative thermal expansion at low temperatures is reproduced and the room temperature value of 2 $\\times 10^{-6}$ K is in good agreement with both experiment and reported lattice-dynamics calculations.", "[28] Figure: (a) Temperature dependence of volumetric thermal expansion of ZnS, CuGaS2 and Cu2ZnGeS4 ascalculated within the quasi-harmonic approximation (PBEsol/DFT).Note that similar negative thermal expansion at low temperatures is found in each material.", "(b) Comparison between measurements (Browder and Smith) and simulation for ZnS.From binary to ternary: CuGaS2.", "The crystal structure of CuGaS2 is a simple $1a\\times 1a\\times 2a$ supercell expansion of sphalerite with the Cu and Ga atoms aligned along the (201) planes.", "The chalcopyrite mineral structure with space group type $I\\bar{4}2d$ ($D_{2d}$ symmetry) has eight atoms in the primitive cell and 24 phonon modes.", "Even at the Brillouin zone centre, the phonon structure is complicated with $A_1 + 2A_2 + 3B_1 + 4B_2 +7E$ vibrational branches, of which only the $A_2$ modes are IR and Raman inactive.", "The harmonic phonon dispersion (Figure REF ) is in good agreement with previous calculations[31], [32].", "The measured upper optic branch runs from 261 to 387 cm$^{-1}$ (collated in [31]), which matches well with our calculated values of 255 to 377 cm$^{-1}$ .", "The thermal expansion behaviour is found to be similar to ZnS, including the low-temperature lattice contraction.", "The linear expansion coefficient $\\alpha $ = 25.4$\\times 10 ^{-6}$ K$^{-1}$ at 300 K is in agreement with the experimental measurements of 23 – 27$\\times 10 ^{-6}$ K$^{-1}$[33], [34].", "From ternary to quaternary: Cu2ZnGeS4.", "The kesterite crystal structure adopted by Cu2ZnGeS4 is closely related to chalcopyrite, with the addition of alternating Cu-Zn and Cu-Ge (001) planes.", "The space group type is $I\\bar{4}$ ($S_4$ symmetry) and there are again eight atoms in the primitive, giving rise to 24 phonon modes.", "The zone-centre irreducible representations are $3A + 7B + 7E$ .", "All modes Raman active, while only the B and E modes are IR active.", "The phonon dispersion is shown in Figure REF , and the optic branches run from 78 to 373 cm$^{-1}$ .", "The measured Raman modes run from 72 to 409 cm$^{-1}$ , which were also found to be in good agreement with prior DFT calculations using the PBE functional.", "[35] Figure: Density functions, DD, showing the contribution of modes with different frequencies to the accumulated thermal conductivity in ZnS, CuGaS2 and Cu2ZnGeS4 (a-c).These are compared to similar functions computed from the trace of the group-velocity tensor products (d-f), the phonon lifetimes (g-i), and the averaged phonon-phonon interaction strengths (j-l).On each plot, the functions are overlaid on the background phonon density of states (DoS) curves.", "A large value of in a frequency band with a small DoS implies that those modes have a high value of the accumulated quantity (e.g.", "higher group velocity, longer lifetime).Anharmonicity and thermal conductivity.", "To assess the modal contributions to the thermal transport, the room-temperature (300 K) modal thermal-conductivity tensors $\\kappa _{\\lambda }$ were integrated over the phonon Brillouin zone using the linear tetrahedron method.", "The frequency derivatives of the resulting accumulation functions were then computed, yielding a function analogous to a density of states (DoS) that quantifies the contribution of modes in different frequency bands to the overall transport.", "A similar procedure was followed for integrating the tensor products $\\nu _{q \\lambda } \\otimes \\nu _{q \\lambda }$ , the phonon lifetimes $\\tau _{\\lambda }$ , and the averaged phonon-phonon interaction strengths $P_{qj,\\lambda }$ (see Ref.", "[20] for further details).", "With reference to the phonon DoS, these functions allow the modal contributions to the thermal conductivities to be interpreted in terms of mode lifetimes and group velocities (c.f.", "Eq.", "REF ).", "The four sets of calculated functions are overlaid on the phonon DoS curves in Figure REF .", "In all three materials, the majority of the heat transport is through modes up to $\\sim $ 3 THz, which is due to a combination of long lifetimes and high group velocities.", "All three compounds also show a smaller secondary contribution to the thermal conductivity from modes between 3–4 THz with a comparable group velocity, but relatively shorter lifetimes.", "We note that the modal heat capacities are saturated at 300 K, and the contribution of this term to $\\kappa $ essentially mirrors the phonon DoS.", "The data in Figure REF shows that the reduction in thermal conductivity on going from ZnS to CGS and CZGS is attributable both to a reduction in the mode lifetimes, and a reduction in the group velocity, the latter of which can be attributed to the weaker bonding.", "For the most part, long-lived modes are associated with regions of the DoS in which the interaction strength is small.", "It is worth noting that the interaction strength is considerably larger in ZnS than in CGS and CZGS, despite its longer phonon lifetimes and high thermal conductivity.", "This can be explained in terms of the conservation of energy: the more complex structures of CGS and CZGS lead to a broadening of the phonon DoS, particularly the high-frequency optic branches, which in turn provides more energy-conserving scattering channels that outweigh the weaker phonon-phonon interactions.", "Figure REF shows the calculated thermal-conductivity curves for the three materials.", "The measured thermal conductivities for ZnS range from 360 Wm$^{-1}$ K$^{-1}$ at 30 K to 27 Wm$^{-1}$ K$^{-1}$ at 300 K,[36] the latter of which is within a factor of two of our calculated value.", "For CuGaS2 and Cu2ZnGeS4, the isotropic average of the thermal conductivity $\\kappa = \\frac{1}{3}(\\kappa _{xx} + \\kappa _{yy} + \\kappa _{zz})$ is presented in Figure REF .", "The anisotropy in the thermal conductivity of these tetragonal crystals is small, but non-negligible.", "A higher conductivity is found in the ab plane ($\\kappa _{xx}^{CGS}$ = $\\kappa _{xx}^{CZGS}$ = 10.2 Wm$^{-1}$ K$^{-1}$ at T = 300 K) than along the c axis ($\\kappa _{zz}^{CGS}$ = 8.8 Wm$^{-1}$ K$^{-1}$ ; $\\kappa _{xx}^{CZGS}$ = 8.0 Wm$^{-1}$ K$^{-1}$ at T = 300 K).", "Although average mass is conserved for the three compounds considered here, the mass variance caused by the 2Zn $\\rightarrow $ Cu + Ga mutation may result in additional scattering analogous to the anharmonic scattering found in disordered alloys.", "We have tested this hypothesis by modifying ZnS with heavy and light isotopes in the same arrangement as the ternary and quaternary structures (labelled “mv\" in Figure REF ).", "The 300 K thermal conductivity is reduced by 18 % for the natural isotope variation, 38 % for CuGaS2-type variation, and 46 % for Cu2ZnGeS4-type variation.", "This mass variance contribution is significant, but a much weaker effect than the changes in the force constants caused by the chemical substitutions.", "In the actual calculations of CuGaS2 and Cu2ZnGeS4, the thermal conductivity is reduced by 78 % in comparison to the value for ZnS at 300 K. As noted above, acoustic phonon modes are responsible for conducting most of the heat, owing to their large group velocity and longer lifetime.", "While optical phonons (less disperse bands with lower $\\nu $ ) do not directly contribute to thermal transport, they are important indirectly in determining mode lifetimes by their involvement in scattering processes.", "A clear change between the phonon DoS on transition from the binary to multernary materials is that the optic branch of the DoS is widened.", "These modes mostly involve motion of the anion sub-lattice — animations of the modes are provided as supplementary information — and the distribution of environments found when multiple cations are introduced causes a spread in frequency.", "The result is a higher probability of three-phonon interactions, limiting the lifetimes, and this, together with a reduction in the group velocity, serves to reduce the thermal conductivity.", "Figure: (a) Calculated lattice thermal conductivity of ZnS, CuGaS2 (CGS) and Cu2ZnGeS4 (CZGS) using harmonic force constants calculated within PBEsol/DFT.The isotropic average is shown for CGS and CZGS.For ZnS, the effects of natural isotope variation (mv-Nat) and artificial isotope variation mirroring the ternary (mv-CGS) and quaternary (mv-CZGS) systems are also shown.", "(b) Comparison of the measured and calculated thermal conductivity of ZnS.In the 1970 overview by Spitzer[10] on the thermal conductivity of semiconductors, an important observation was made: “It is found that lattice thermal conductivity may be correlated rather reliably with crystal structure.", "In general, increasing coordination of the ions is associated with decreasing thermal conductivity\".", "We have further shown that even for a fixed coordination number, the composition can have a large impact.", "Now that the underlying contributions to thermal transport can be separated and quantified, deeper insights can be obtained into the structure-property relationships that can be used towards the rational engineering of thermal transport.", "Computational details: For the density functional theory calculations, a kinetic energy cut-off of 450 eV for the plane wave-basis set was combined with a reciprocal-space sampling equivalent to $8 \\times 8 \\times 8$ $k$ -points for the zincblende primitive cell (i.e.", "$4 \\times 4 \\times 4$ for CuGaS2 and Cu2ZnGeS4, respectively).", "Projector augmented-wave (PAW) pseudopotentials[37] were employed.", "A tolerance of 10$^{-8}$ eV was applied during the electronic minimization, and geometry optimisations were carried out to a force threshold of 10$^{-1}$ eV/Å.", "The precision of the charge-density grids was automatically chosen to avoid aliasing errors, and an additional support grid with $8 \\times $ the number of points was used to evaluate the forces during the single-point force evaluations for the lattice-dynamics calculations.", "The PAW projection was applied in reciprocal space (LREAL = .FALSE.", "in VASP).", "For the lattice-dynamics calculations, second- and third-order force constants were computed in $2 \\times 2 \\times 2$ expansions of the primitive cells.", "Additional calculations were carried out on cells with expansions and contractions of $\\pm 3 \\%$ about the the athermal equilibrium volume, in steps of 1 %.", "The third-order calculations were performed using the 300 K volumes predicted from the quasi-harmonic approximation.", "Phonon DoS curves and thermodynamic partition functions were computed using $\\Gamma $ -centred $q$ -point grid with $48 \\times 48 \\times 48$ subdivisions to integrate the phonon Brillouin zones.", "The phonon lifetimes for ZnS and CuGaS2/Cu2ZnGeS4 were calculated using $\\Gamma $ -centred $q$ -point grids with $48 \\times 48 \\times 48$ and $16 \\times 16 \\times 16$ subdivisions, respectively, and the tetrahedron method was used for interpolation.", "Data Access Statement The crystal structures and phonon data reported in this work are available in an online repository at https://github.com/WMD-group/Phonons, which can be processed using the Phonopy and Phono3py packages available from http://atztogo.github.io/phonopy and http://atztogo.github.io/phono3py.", "The animations were made using ascii-phonons, available from https://github.com/ajjackson/ascii-phonons.", "The calculations in this work used the facilities of the Supercomputer Center at thInse titute for Solid State Physics, University of Tokyo.", "Some calculations were also performed on the UK Archer HPC facility, accessed through membership of the UK HPC Materials Chemistry Consortium, which is funded by EPSRC Grant No.", "EP/L000202.", "We also made use of the Balena HPC facility at the University of Bath, which is maintained by Bath University Computing Services.", "J.M.S.", "is funded by an EPSRC Programme Grant (grant no.", "EP/K004956/1).", "A.W.", "acknowledges support from the Royal Society and the ERC (grant no.", "277757)." ] ]
1606.04914
[ [ "The winter day as a constraint for human activity in Western Europe" ], [ "Abstract Time use surveys in Denmark, Spain, France, Ireland, Italy and United Kingdom are analyzed to provide start, noon and end times for the main activities of a society: labor (the focus of this preprint), sleeping and eating.", "Also, the location at home is analyzed.", "Local times are converted into mean solar times and compared to latitude.", "Observed trends allow to unveil the winter day as a restriction for the human activity.", "Alternatively, apparently large time differences set forth by clocks, becomes smaller when observed as a time distance to winter sunrise or sunset." ], [ "Introduction", "Time use surveys characterize the behavior of societies by computing the amount of time individuals spent in labor, household activities, leisure activities and sleeping on an average day.", "They also provide valuable information on how these activities are shared by gender or age.", "Most of the information disseminated by Time use surveys concerns intervals of time expressed as an average number of hours per day.", "Yet this work focuses on the times at which main activities (eating, sleeping and working) occur.", "Time use surveys also locate where an activity is being held from which it is possible to track the binary condition of being at home or out of home.", "This work is a follow up of a previous preprint[1] focused on Time Use Surveys in Italy[2], Spain[3] and United Kingdom[4].", "Here surveys from Denmark[5], France[6] and Ireland[7] will also be analyzed.", "The methodology (see Section ) slightly differs from the previous work which helps improving the analysis of the afternoon.", "As in the previous work the microdata of the surveys here analyzed were retrieved freely from the Internet (Spain and Denmark) or were obtained after contacting with the institution.", "That was the single criterion to pick a survey.", "Irish National Time Use Survey slices the day in ninety six slots of fifteen minutes in duration.", "Every other survey slices the day in one hundred and forty four slots of ten minutes in duration.", "The respondent is compelled to fill the relevant information in every spot.", "That is: which activity he was doing, where, and with whom, among others.", "The questionnaires also collect a myriad of information as for instance week day and month of the diary, gender of the respondent, incomes or number of individuals in the household.", "The burden of this paper is linking the social activity to the solar activity in Central and Western Europe.", "For so doing we will compare solar times at which activities occur rather than local times which are computed from the surveys.", "It will be shown that the winter day, which yields the latest sunrise and the earliest sunset of the year, is a powerful restriction for the social activities and provide an explanation for the way in which they occur all the year round.", "Notice that winter day length depends solely on the latitude.", "Only respondents who reported labor activity at some point of the day in a week day (Monday to Friday) will be analyzed.", "Daily reports will be grouped along the First Level of the Nomenclature of Territorial Units for Statistics (NUTS-1) which is described elsewhere.", "[8] When diaries are geolocated to the NUTS-2 level, data will be weight-averaged by population to get NUTS-1 level statistics.", "Finally, the weight-averaged value of NUTS-1 level statistics will be the country level mean result.", "This work analyzes daily activity plot which, essentially, computes the shares of people doing an activity as a function of time and during an average week day.", "This work will not thoroughly describe labor activities and their differences from region to region or from country to country, yet a brief description of the labor activity is worth to quote.", "Labor activity soars after the minimum level in the morning.", "Then, somewhere close to noon the activity attains a maximum and starts decreasing, that is lunch time.", "Then labor activity is resumed, and gets a relative maximum in the afternoon.", "Then it starts decreasing to the ground level.", "Notice that the area enclosed by the labor activity plot has units of time and indeed is the mean daily amount of time spent at work by an employed person.", "Every labor activity plot displays this simple behavior with forenoon and afternoon peaks, sharp up-rise in the morning and a slower decay in the afternoon.", "Not surprisingly, magnitudes like the strength of the peaks or their relative position regionally changes.", "Labor activity plot will then be characterized by three times: start time, noon time and end time.", "Noon time is easily computed as the time when half the total area has been consumed.", "The inset of Fig.", "REF display the position of these times for the labor activity displayed.", "Start and end times essentially requires the definition of a threshold level above which one may reasonable say that the region is “working” and below which one may reasonable say that the region is not “working”.", "Notice that this approach slightly differs from statistically analyzing the distribution of times when individuals get to work or leave working.", "[1] A very simple choice for a threshold is the 50% level, but this choice jeopardizes the analysis in some regions (notably in Spain and Southern Italy) because labor activity is not resumed back to the 50% in the afternoon.", "Also, it is never the case that 100% of the employed persons are on duty.", "Labor activity only reaches the 90% level in some regions of Italy.", "In contrast some regions only reaches the 70% level.", "A naive election for the threshold is then half the value of strongest peak.", "Which is to say the 50% of the peak.", "Notice that this threshold, though variable, is consistently set for every region.", "Also notice that this threshold is always weaker than the choice of the 50% level, thus leading to earlier start times and later end times.", "The threshold allows us to introduce the labor start time —computed as the time when labor activity overshoots the threshold in the morning— and the labor end —computed as the time when labor activity undershoots the threshold in the afternoon.", "Likewise, activity plots can be retrieved for the location at-home (see Appendix ), the eating activity (see Appendix ) and the sleeping activity (see Appendix ).", "For the sleeping and at-home analysis the threshold level is effectively set to the 50% since peaks gets close to the 100% level: almost every employee is awaken or out of home at some specific time.", "On the other hand, the eating activity is different since it does not smoothly change on time.", "Instead, it occurs in bursts: in the morning —breakfast—, noon —lunch— and afternoon —dinner.", "Eating times will be ascribed to the time position of the peak for every burst." ], [ "Comparing times", "The burden of this work is to provide a comprehensive comparison of the times when the main activities occur.", "Results obtained from Time Use Survey only display local times.", "This is a very awful quantity to compare because it lacks a common time basis.", "To be more specific: they depend on the local time zone.", "Reducing data to a common time basis (say, for instance, UTC) will only display Earth's rotation with eastern regions displaying earlier times than western regions.", "The only comprehensive way of comparing social time over a wide geographical areas is through the local mean solar time which accounts for the local time zone $\\Delta _i$ and Earth's rotation.", "Local mean solar time $t^{\\star }_i$ given by: $t^{\\star }_i=t_i - \\Delta _i + \\lambda _i \\omega ^{-1}$ where $t_i$ is any time defined in Sec.", "REF expressed in local time, $\\Delta _i$ is the local time zone expressed as an offset to UTC time, $\\lambda _i$ is the mean longitude of the region, expressed as an offset to the UTC prime meridian (Greenwich meridian) and $\\omega $ is Earth's angular rotation speed (one revolution per day, conveniently expressed as ${15}{}$ in this context).", "The quantity $\\tau _i=\\Delta _i-\\lambda _i\\omega ^{-1},$ is the time offset between local civil time and local solar time.", "It generally ranges in $\\pm {30}{}$ .", "This range is exceeded when local time zone is not the geographical time zone, as in the case of Spain and France since end of World War Two.", "Mean solar local times $t^{\\star }_i$ can be directly linked to solar activity.", "The most simple case is solar noon which, by definition, occur at $\\tau ={12}{}$ everywhere but it is also the case for properties like sunrises and sunsets." ], [ "Geolocation", "As it was mentioned earlier on Sec.", "statistic data from Time Use Survey will be grouped along NUTS-1 regions.", "In Eq.", "(REF ) the longitude of the region is needed.", "In the forthcoming discussion its latitude will also be employed.", "This geographical data will be retrieved from database after weight-averaging geographical coordinates of cities whose population is large than 1000 people and exists on the database.", "This will suffice for the purpose of this paper.", "Fig.", "REF shows the mean values of the geographical data obtained from the database and the weighted-average values (open symbols) corresponding to the country level longitude and latitude.", "Notice that for computing this country level mean values ES7 is not considered in Spain, and overseas departments are not considered in France.", "Figure: A map of Central and Western Europe.", "Shaded NUTS-1 regions are analyzed in this work.", "Darker regions (Ireland, United Kingdom and Canary Islands (ES7)) set clocks to Western European Time zone (WET=UTC+00), lighter regions set clocks to Central European Time zone (CET=UTC+01).The top panel of Fig.", "REF shows the latitude in the vertical axis versus start, noon and end labor local times for the set of regions analyzed in this work.", "Figure: Top figure: labor start, noon and end local times plotted versus latitude.", "Morning times looks like similar across surveys with 08am as a popular choice in Europe.", "Labor end times shows a trend with later end times as latitude decreases.", "The inset shows an labor activity plot and the definition of the labor start, noon, and end times.", "Labor middle times (not shown in the figure) are mostly independent of latitude.", "Bottom figure: the same as before but with the xx-axis displaying solar times instead of local times.", "Solar activity and borders can be added to the plot, including winter sunrise and sunset lines.", "Labor times correlated with the winter day.In the morning most of data are grouped around 08:00am local time.", "In the afternoon labor end times display a downward trend with end times increasing with decreasing latitude.", "At noon data points seem rather scattered.", "Similar results can be observed for the eating activity and the at-home location.", "As shown in the appendixes.", "In the search of an explanation one may think that social activity arises from a fundamental rest state at midnight when people mostly sleptAlbeit for a non-zero background level corresponding to a “social watch”.", "Hence, wake-up time and, more generally, the start times arise from a common initial condition.", "Hereafter, cultural and legal factors may influence the behavior of the activities and would trigger differences at the end of the day.", "Even if that were the case, two fundamental questions would arise.", "The first one is why is eight o'clock the preferred value of the labor start time.", "The second one is why labor end times follow this specific trend characterized by a unique slope.", "A better insight to the problem and an effective answer to these questions is obtained when the local mean solar time (see Eq.", "(REF )) is plotted instead of the local time as observed in the bottom panel of Fig.", "REF .", "The reader will observe in this plot geopolitical borders, which are plotted at the country-level mean value.After setting $t_k$ and since $\\Delta _k$ is also set, horizontal axis display longitude while vertical axis display latitude; hence borders can be displayed for this specific $t_k$ .", "Upon this choice of $x$ -axis, solar properties can be plotted as well.", "Fig.", "REF (bottom) shows lines of constant solar elevation angles $z$ including the null condition $z={0}{}$ which approximately sets the sunrise and sunset lines.", "[9] The reader should notice that the winter day is an extremal condition for the solar activity since the sun is apparently traveling around the tropic which is opposite to the observer, yielding to the latest sunrise, the earliest sunset and, also, the smallest solar elevation angle at any time of the day.", "As a result in the trapeze bounded by parallels (top and bottom) and the sunrise and sunset lines (sides) the solar elevation angle is always positive irrespective of the calendar day.", "Say, there is always daylight in this area and human activity can be developed at ease.", "It should be stressed however that data points do not display winter properties of the distribution of working times but yearly averaged properties.", "This analysis provides a simple explanation for the main properties of the labor activity.", "Labor start time —year round— is chosen close to the winter sunrise line.", "That way, employed persons never get to work too dark at dawn.", "Labor end times —year round— is mostly coincidental to the winter sunset line.", "Differences may well appear here because of lack of common legal regulations or differences in the shares of economic sectors through survey.", "Nonetheless, evidence suggests that the winter sunset is also an attractor for the end of the labor activity.", "That way, as the year progresses, employed persons will mostly enjoy later sunsets in their leisure time." ], [ "Sunrise, sunset, time zone", "In understanding this analysis first notice that, in the morning, labor start solar times have a trend with latitude: the more latitude, the later the labor start solar time.", "In the afternoon the trend is just the opposite.", "The significant point here is the fact that yearly averaged labor times do not follow yearly averaged solar activity, which is independent of latitude.For instance yearly averaged solar sunrise and sunset times are 06am and 06pm, irrespective of the latitude —if daylight saving time is not considered.", "Thus yearly averaged sunrise and sunset local times only depends on longitude for a given time zone.", "Modern societies do not filter out the solar seasonal cycle, they have memory.", "The trends are easily explained by the solar activity at the winter day which sets the latest sunrise of the year, the earliest sunset of the year and the shortest daylight length.", "In fact, Fig.", "REF (top) panel just shows up an interesting natural phenomenon: the way in which the terminator lineThe line separating day from night.", "sweeps Europe in winter.", "This can be easily understood by watching a animation of the winter day in Europehttp://bit.ly/1U8enMa where it can be seen how winter sunrise line sweeps Europe from Southeast to Northwest (the earliest sunrise coming at Southeastern Italy, the latest sunrise at the Scottish Highlands) and it is mostly coincidental over Northwestern Italy, Germany, the French hexagon, Denmark, Netherlands, Belgium and the Iberian Peninsula.Winter sunrise in Berlin and Barcelona are coincidental.", "Also in Madrid and Cologne, in Paris and Bilbao, or in Lisbon and Amsterdam.", "After that, sunrise line sweeps the British islands.", "In the afternoon, the terminator line sweeps Europe from Northeast to Southwest; the sun leaves Europe through the Southwestern tip of the Iberian Peninsula).", "The preceding paragraph is the description of a natural phenomenon which is immutable.", "Contrastingly, societies must decide which time zone will clocks display and this decision changes winter sunrise and sunset local times, and then alter Fig.", "REF (top).", "Fig.", "REF shows in the horizontal axis the time offset $\\tau _i$ (see Eq.", "(REF )) —the time distance from local solar noon to local civil noon— thus takes into account the time zone of regions.", "Hence the map shown in Fig.", "REF is slightly different from a standard map, since $\\Delta _i$ does not get the same value everywhere.", "Notice England comes over East Germany, Scotland mostly over Denmark and Ireland is over Netherlands.", "Figure: A collapsed version of Fig.", "where time zone is taken into account and the horizontal axis display time offset instead of longitude.", "The main consequence of that is the shifting of WET regions (Portugal, United Kingdom, Ireland and Canary Islands relative to CET regions.", "Larger open symbols display the weight-averaged latitude and offset for a country.", "Vertical axis displays time distance from local civil noon to local solar noon.", "Slanted grid display winter sunrise axis and winter sunset axis.", "The terminator at winter sunrise or sunset is perpendicular to its respective axes.Horizontal axis essentially display noon properties, the slanted grid shown on the plot display the solstitial sunrises and sunsets, which are opposite (winter sunrise matches to summer sunset).Notice we are defining sunrises and sunsets by $z={0}{}$ , which is a simplification, and we are not taking into account daylight saving time.", "Fig.", "REF then shows the unique consequences of any time zone choice: it sets noon and it sets the extremal sunrises and sunsets.", "Fig.", "REF shows that with the present choice of time zones in Europe, winter sunrise progresses in Western Europe from 08am to 09am (local time), thus making eight o'clock (local time) a popular choice for getting to work.", "On the contrary, winter sunset lacks of a common local time in Western Europe.", "The terminator sweeps Europe from Northeast to Southwest and darkness steadily covers Denmark and United Kingdom first, then Ireland, then France, Italy and, finally, Spain.", "Fig.", "REF mimics the top panel of Fig.", "REF but shows local times for winter sunrise, noon, and winter sunset instead.", "The correspondence between these two figures are evident.", "Figure: Winter sunrise, noon, and winter sunset local times plotted along with latitude as vertical coordinate.", "The similarities with Fig.", "(top) are evident.The interesting point here is that had France and Spain turned back to Western Europe Time zone at the end of World War Two, they would be placed over Italy, Austria and the Czech Republic in Fig.", "REF .", "Hence, the latest sunrise time would happen one hour earlier and seven o'clock may have been a popular choice for the labor start time in Spain and, probably, in France.", "That is the choice in the United States of America —which share latitude with the Iberian PeninsulaIn Fig.", "REF the East Coast of the United States would lie east of Italy, and winter sunrise occur at seven o'clock local time.", "The reader should be aware of the trick: 07WET is exactly the same as 08CET.", "The bottom line here is that by setting clocks to CET zone, France and Spain have prompted a common winter sunrise in Western Europe, Germany and the British Islands.", "Ultimately that choice leads to common labor start times (as observed in Fig.", "REF (top)).", "As a consequence they have fuelled different winter sunset and labor end times (see also Fig.", "REF (top) and Fig.", "REF ).As a example, winter sunset is almost coincidental in Paris and London.", "Yet it is 03:45pm local time in London, and 04:45pm local time in Paris.", "However these differences, though noticeable, are unimportant.", "On the contrary they are convenient as apparently, they only highlights the way human activity fits to solar activity (see Fig.", "REF (bottom)).", "To put it shortly they tell us about a natural phenomenon.", "Finally it should be stressed that only at temperate latitudes can society fit the human activity to the solar activity this way.", "To be specific, only if the winter day length is close to the average daily labor time —which roughly speaking ranges in seven to eight hours per day and employed person—could society fit that way.", "If day length is too short —at high enough latitude— employed people have the necessity of starting and/or ending their duties before sunrise or after sunset.", "On the contrary, if day length is too long —at low enough latitude— there is no need to fit labor start and end times to winter sunrises or sunsets." ], [ "Noon, lunch break and day length", "The preceding discussion focused on labor start and end times and the way they fit to the winter sunrise/sunset lines.", "Noon also deserves an explanation.", "The reader should notice that labor activity fits to winter sunrise in the morning and to winter sunset in the afternoon.", "And they have opposite trends.", "There must be a cross-over somewhere in the day: it is at noon.", "Labor noon time —the time when half the total labor activity of a region has been already consumed— correlates well with the winter sunrise, see Fig.", "REF (bottom).", "Labor noon time comes anywhere from four to five hours after winter sunrise.", "This is likely telling us that morning labor is relatively similar across societies.Albeit some minor differences which are evident in Fig.", "REF , for instance Italy is delayed with respect to France in the morning, but advanced at noon.", "By comparison afternoon labor must be different.", "This comes out by simply considering that at labor noon times follow the sunrise trend —the higher the latitude, the later the time— while labor end times follow the opposite trend.", "The only way this is achieved by slowing down labor activity at low latitude regions.", "One way of so doing is having longer lunch break the longer your winter day length is.", "The rationale here is that winter day is long enough so as to exceed the average time spent at work.", "Hence longer intraday breaks are possible without resuming labor after any sunset of the year.", "Another way of so doing is just by burning work in the morning, so that employed persons need not to come back to work in the afternoon.", "This scenario is only possible at low latitude since as latitude decreases, winter day length increases and the time distance from sunrise to noon increases as well.", "Longer mornings can accommodate larger amounts of work.", "In fact, data in Fig.", "REF and Fig.", "REF shows that only in Italy and Spain (the southern most countries of the analysis) does the labor noon time go well before lunch time.", "Contrastingly in Denmark, Ireland and United Kingdom labor noon time and lunch time differ in few minutes, while in France lunch time comes well before labor noon time." ], [ "Hours ticked by", "Clocks and longitude are linked to each other since the latter can only be measured with the help of the former.", "Clocks were then linked to noon in modern times.", "Noon, very conveniently, has a regular period of twenty four hours year round.", "But nobody is perfect.", "As we set our clocks to noon —the most regular phenomenon— we lost track of sunrises and sunsets —the most sensible phenomena— which primarily set human activity.", "Hence, the vast myriad of labor end times observed in the top panel of Fig.", "REF (top) can easily hide the natural phenomenon observed in the bottom panel.", "This is also speaking about a fundamental issue when dealing with clocks.", "Although meridians are very convenient, it should not be forgotten that points along a meridian are different to each other in a fundamental time property: winter day length.", "As a running example we take the cities of London and Castellon (in Eastern Spain), which share the prime meridian.", "Yet, noon at Castellon comes at least 4h45m after sunrise; while at London noon can come as low as 3h45m after sunrise, a phenomenon impossible to be observed in the Spanish city.", "That makes a difference when social time schedules are build up.", "Castellon and London do share daily noon; but they do not share winter sunrise and hence, they will not share labor start time.", "On the contrary, points sharing a latitude are all identical in some way.", "For sure they differ in the longitude.", "But longitude is just a matter of time: sooner or later whatever happened somewhere, will happen somewhere else along a parallel.", "For instance, whatever happened in Naples, will happen one hour later in Castellon.", "Physicists, geographers, astronomers are fond of meridians, transits and noons which remain highly unnoticeable for the average person.", "People are quite more sensible to sunrise and sunset." ], [ "Conclusion", "Winter day provides an explanation for the way some European societies develop their daily activities.", "The rationale is if you survive to the worst case scenario of the year —the shortest day of the year— you will manage to survive to any other day.", "That would be true at temperate latitudes where winter day-length is similar to the daily average working time.", "In the morning, differences are rare because societies come from a common fundamental rest state, and the only activities an employed person does are related to working.", "In the afternoon, employed persons develop activities non related to work.", "The winter sunset apparently triggers a myriad of social processes which include the end of labor —somehow near to the winter sunset—, coming back home —which happens some two to three hours after winter sunset— or having dinner —three to four hours after winter sunset—.", "Since distance to winter sunrise or winter sunset can not be easily measured by clocks, this facts mostly remain unnoticed.", "People usually do not care why they get to work at eight o'clock in the morning, or why they have dinner at eight o'clock in the morning.", "From a scientific point of view the point here to emphasize is that times as provided by clock —say, local times $t_i$ — must be compared to each other once time zone $\\Delta _i$ , longitude $\\lambda _i$ and latitude $\\phi _i$ are taken into account.", "These parameters enter —together with Earth's angular speed of rotation $\\omega $ and obliquity $\\varepsilon $ — in the equation that provides time distance to winter sunrise (positive branch) and winter sunset (negative branch) $\\delta _i$ : $\\delta _i=t_i-\\Delta _i-{12}{}+\\frac{1}{\\omega }\\left(\\lambda \\pm \\cos ^{-1}(\\tan (|\\phi _k|)\\tan (|\\varepsilon |))\\right)$" ], [ "Acknowledgments", "The author acknowledges the institutions of Denmark, Spain, France, Ireland, Italy and United Kingdom that provided the surveys and, thanks them for releasing the microdata to the scientific community.", "The author also acknowledges to Politikon (http://www.politikon.es/) —a think tank based in Spain— for hosting most of the results and discussions related to this work.", "The author also wants to thank to his editor, Octavio Medina.", "The author thanks Dr. José Fernández-Albertos at Centro de Ciencias Humanas y Sociales from the Spanish Centro Superior de Investigaciones Científicas for some fruitful discussions through Twitter social network." ], [ "At home", "Time Use Surveys not only track daily activities but also track locations.", "As simple analysis of the location where activities occur allows us to track whether the employee is at home or not and an “activity” which computes the shares of employed persons at home.", "From this statistics it is possible to compute times ascribed to leaving home or coming home.", "It should be mentioned that leaving home in the morning is mostly related to going to work as soon as possible, while coming back home in the afternoon is not necessarily.", "People often develop leisure activities —or more generally non-working activities— only in the afternoon.", "Fig.", "REF shows the local (top) and solar (bottom) times corresponding to this “activity”.", "They show similar trends to those observed in Fig.", "REF .", "Figure: Top, local times for leaving home, coming home and the noon condition.", "As previously observed morning time are rather similar in Europe while the afternoon times are variable.", "Bottom the same plot but with the solar time in the xx-axis.", "Data are comprehensively described with the coming home condition located from two to three hours after winter sunset.As observed in the bottom panel, employed persons are back at home some two-three hours after winter sunset." ], [ "Eating times", "Fig.", "REF shows the eating local (top) and solar (bottom) times.", "Local times display the usual trend observed in Fig.", "REF and explained in Sec. .", "Figure: Top, local times for the breakfast, lunch and dinner determined by the positions of the bursts observed in the inset.", "As in Fig.", "breakfast occurs in a narrow range of times, while lunch and dinner times progressively get wider ranges.", "Bottom, solar times for breakfast, lunch and dinner and their relation to the winter day.", "Summer sunset line is delayed by one hour due to daylight saving time.Solar times comprehensively describe data.", "Lunch time occur in Europe three hours before winter sunset, except in France, and dinner time mostly in the range three to four hours after winter sunset.", "An interesting question to answer is what does lunch mean?", "Two answers seem possible: (1) eating at noon, (2) eating well before (winter) sunset.", "A possible explanation for the French anomaly deals with the time zone change enforced in 1940 amidst World War Two.", "The point here is that lunch times could have not changed since 1940 and thus they are located off the main trend.", "On the contrary, in Spain —which carries the same history of time zone changes— lunch times likely got delayed after clocks were set one hour ahead in 1940, thus offsetting the time zone change.", "This idea can be tested by removing the time zone change in France and Spain (say, making their time zone equal to Western European Time $\\Delta _k=0$ when computing $t^{\\star }_k$ ).", "In so doing French and Spanish data will move one hour rightward.", "French data would then lie on the main trend thus suggesting that their lunch times are those prior to World War Two.", "Contrastingly, Spanish data would lie way off the main trend and anomalously close to the winter sunset.", "Bottom panel of Fig.", "REF also display the summer sunset —shifted by one hour due to daylight saving time— which is almost coincidental with the dinner time at Southeastern Italy and Spain.", "Hence with little effort, dinner times in Spain and Italy can be delayed so that dinner may become “eating at night”.", "Of course it is not possible to achieve this task in France, United Kingdom, Ireland, Denmark or Northwestern Spain and Italy where having dinner before sunset is usual." ], [ "Sleep times", "Times related to the sleeping activity can be observed in Fig.", "REF where the top panel displays local times and the bottom panel displays solar times.", "Figure: Top, local times for the sleeping activity (bedtimes, noon and wake-up time) and latitude.", "Bottom, the corresponding solar times.", "Sleep times do not follow the trends observed elsewhere in this work.", "They play the role of a crossover from the positively slanted morning times to the negatively slanted afternoon times.The interesting point here is that sleep times can hardly be related to winter sunrise or winter sunset, albeit the trivial idea that wake-up time are close to winter sunrise.", "The issue here is that trends must be reversed again.", "So bedtimes play the same role of crossing-over as lunch-break was playing at noon.", "For instance northern countries exhibit earlier dinner solar times as compared to southern countries (see Fig.", "REF (bottom)) but their bed times are not that early.", "Nonetheless sleeping times are similar from survey to survey and range from 7h25m per day and employed person in Denmark to 7h45m per day and employed person in United Kingdom." ] ]
1606.04903
[ [ "Legendre-type relations for generalized complete elliptic integrals" ], [ "Abstract Legendre's relation for the complete elliptic integrals of the first and second kinds is generalized.", "The proof depends on an application of the generalized trigonometric functions and is alternative to the proof for Elliott's identity." ], [ "Introduction", "Let $k \\in [0,1)$ .", "The complete elliptic integrals of the first kind $K(k)=\\int _0^1 \\frac{dt}{\\sqrt{(1-t^2)(1-k^2t^2)}}$ and of the second kind $E(k)=\\int _0^1 \\sqrt{\\frac{1-k^2t^2}{1-t^2}}\\,dt$ play important roles in classical analysis.", "In this paper, we consider generalizations of $K(k)$ and $E(k)$ as $K_{p,q,r}(k):=\\int _0^1 \\frac{dt}{(1-t^q)^{1/p} (1-k^qt^q)^{1/r}}$ and $E_{p,q,r}(k):=\\int _0^1 \\frac{(1-k^qt^q)^{1/r^*}}{(1-t^q)^{1/p}}\\,dt,$ where $p \\in \\mathbb {P}^*:=(-\\infty ,0) \\cup (1,\\infty ],\\ q,\\ r \\in (1,\\infty )$ and $1/s+1/s^*=1$ .", "For $p=\\infty $ we regard $K_{p,q,r}$ and $E_{p,q,r}$ as $K_{\\infty ,q,r}(k):=\\int _0^1\\frac{dt}{(1-k^qt^q)^{1/r}},\\quad E_{\\infty ,q,r}(k):=\\int _0^1(1-k^qt^q)^{1/r^*}\\,dt.$ Under the convention that $1/\\infty =0$ and $1/0=\\infty $ , we should note that $s \\in \\mathbb {P}^*$ if and only if $s^* \\in (0,\\infty )$ , particularly, $\\infty ^*=1$ .", "In case $p=q=r=2$ , $K_{p,q,r}(k)$ and $E_{p,q,r}(k)$ are reduced to the classical $K(k)$ and $E(k)$ , respectively.", "There is a lot of literature about the generalized complete elliptic integrals.", "$K_{p,q,p}$ is introduced in [11] with a generalization of the Jacobian elliptic function with a period of $4K_{p,q,p}$ to study a bifurcation problem of a bistable reaction-diffusion equation involving $p$ -Laplacian.", "Relationship between $K_{p,q,p}$ and $E_{p,q,p^*}$ has been observed in [3], [15].", "Regarding $K_{p,q,p^*}$ , another generalization of Jacobian elliptic function with a period of $K_{p,q,p^*}$ is given and the basis properties for the family of these functions are shown in [12].", "Moreover, $K_{p,q,p^*}$ is also applied to a problem on Bhatia-Li's mean and a curious relation between $K_{p,q,p^*}$ and $E_{p,q,p^*}$ is given in [9].", "It is well known that $K(k)$ and $E(k)$ satisfy the famous Legendre's relation (see, for example, [2], [4], [6]): $E(k)K(k^{\\prime })+K(k)E(k^{\\prime })-K(k)K(k^{\\prime })=\\frac{\\pi }{2},$ where $k^{\\prime }=\\sqrt{1-k^2}$ .", "Our purpose in the present paper is to generalize Legendre's relation (REF ) to the generalized complete elliptic integrals above.", "To state the results, we will give some notations.", "For $p \\in \\mathbb {P}^*$ and $q \\in (1,\\infty )$ , let $\\pi _{p,q}:=2\\int _0^1 \\frac{dt}{(1-t^q)^{1/p}}=\\frac{2}{q}B\\left(\\frac{1}{q},\\frac{1}{p^*}\\right),$ where $B$ denotes the beta function.", "In particular, $\\pi _{\\infty ,q}=2$ for any $q \\in (1,\\infty )$ .", "We write $K_{p,q}:=K_{p,q,q^*},\\ E_{p,q}:=E_{p,q,q^*}$ for $p \\in \\mathbb {P}^*$ and $q \\in (1,\\infty )$ ; $K_p:=K_{p,p,p^*},\\ E_p:=E_{p,p,p^*},\\ \\pi _p:=\\pi _{p,p}$ for $p \\in (1,\\infty )$ .", "Theorem 1.1 Let $p \\in \\mathbb {P}^*,\\ q,\\ r \\in (1,\\infty )$ and $k \\in (0,1)$ .", "Then $E_{p,q,r^*}(k)K_{p,r,q^*}(k^{\\prime })+K_{p,q,r^*}(k)E_{p,r,q^*}(k^{\\prime })\\\\-K_{p,q,r^*}(k)K_{p,r,q^*}(k^{\\prime })=\\frac{\\pi _{p,q}\\pi _{s,r}}{4},$ where $k^{\\prime }:=(1-k^q)^{1/r}$ and $1/s=1/p-1/q$ .", "Corollary 1.2 (Case $q=r$ ) Let $p \\in \\mathbb {P}^*,\\ q \\in (1,\\infty )$ and $k \\in (0,1)$ .", "Then $E_{p,q}(k)K_{p,q}(k^{\\prime })+K_{p,q}(k)E_{p,q}(k^{\\prime })\\\\-K_{p,q}(k)K_{p,q}(k^{\\prime })=\\frac{\\pi _{p,q}\\pi _{s,q}}{4},$ where $k^{\\prime }:=(1-k^q)^{1/q}$ and $1/s=1/p-1/q$ .", "Corollary 1.3 ([13], Case $p=q=r$ ) Let $p \\in (1,\\infty )$ and $k \\in (0,1)$ .", "Then $E_{p}(k)K_{p}(k^{\\prime })+K_{p}(k)E_{p}(k^{\\prime })\\\\-K_{p}(k)K_{p}(k^{\\prime })=\\frac{\\pi _{p}}{2},$ where $k^{\\prime }:=(1-k^p)^{1/p}$ .", "Remark 1.4 Using (REF ), the author establishes computation formulas of $\\pi _p$ for $p=3$ in [13]; for $p=4$ in [14].", "In fact, (REF ) is equivalent to Elliott's identity (REF ) below.", "The advantage of our result lies in the facts that it is understandable without acknowledge of hypergeometric functions and that its proof gives an alternative proof for Elliott's identity with straightforward calculations." ], [ "Proof of Theorem ", "The following property immediately follows from the definitions of $K_{p,q,r}$ and $E_{p,q,r}$ .", "Proposition 2.1 Let $p \\in \\mathbb {P}^*,\\ q,\\ r \\in (1,\\infty )$ .", "Then, $K_{p,q,r}(k)$ is increasing on $[0,1)$ and $K_{p,q,r}(0)&=\\frac{\\pi _{p,q}}{2},\\\\\\lim _{k \\rightarrow 1-0}K_{p,q,r}(k)&={\\left\\lbrace \\begin{array}{ll}\\infty & \\mbox{if}\\ 1/p+1/r \\ge 1,\\\\\\pi _{u,q}/2\\ (1/u=1/p+1/r) & \\mbox{if}\\ 1/p+1/r<1;\\end{array}\\right.", "}$ and $E_{p,q,r}(k)$ is decreasing on $[0,1]$ and $E_{p,q,r}(0)=\\frac{\\pi _{p,q}}{2},\\quad E_{p,q,r}(1)=\\frac{\\pi _{v,q}}{2}\\ (1/v=1/p-1/r^*).$ For $p \\in \\mathbb {P}^*$ and $q \\in (1,\\infty )$ , the generalized trigonometric function $\\sin _{p,q}{x}$ is the inverse function of $\\sin _{p,q}^{-1}{x}:={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\int _0^x \\frac{dt}{(1-t^q)^{1/p}} & \\mbox{if}\\ p \\ne \\infty ,\\\\x & \\mbox{if}\\ p=\\infty .\\end{array}\\right.", "}$ Clearly, $\\sin _{p,q}{x}$ is increasing function from $[0,\\pi _{p,q}/2]$ onto $[0,1]$ .", "For $p=q=2$ , $\\sin _{p,q}{\\theta }$ and $\\pi _{p,q}=2\\sin _{p,q}^{-1}{1}$ are identical to the classical $\\sin {\\theta }$ and $\\pi $ , respectively.", "Moreover, $\\sin _{p,q}{\\theta }$ and $\\pi _{p,q}$ play important roles to express the solutions $(\\lambda ,u)$ of inhomogeneous eigenvalue problem of $p$ -Laplacian $-(|u^{\\prime }|^{p-2}u^{\\prime })^{\\prime }=\\lambda |u|^{q-2}u,\\ p,\\ q \\in (1,\\infty )$ , with a boundary condition (see [5], [10], [11] and the references given there).", "For $p \\ne \\infty $ and $x \\in (0,\\pi _{p,q}/2)$ , we also define $\\cos _{p,q}{x}:=(\\sin _{p,q}{x})^{\\prime }$ .", "It is easy to check that for $x \\in (0,\\pi _{p,q}/2)$ , $\\cos _{p,q}^p{x}+\\sin _{p,q}^q{x}=1,\\quad (\\cos _{p,q}{x})^{\\prime }=-\\frac{q}{p}\\sin _{p,q}^{q-1}{x}\\cos _{p,q}^{2-p}{x}.$ Now, we apply the generalized trigonometric function to the generalized complete elliptic integrals.", "For $p \\in \\mathbb {P}^*$ and $q,\\ r \\in (1,\\infty )$ , using $\\sin _{p,q}{\\theta }$ and $\\pi _{p,q}$ , we can express $K_{p,q,r}(k)$ and $E_{p,q,r}(k)$ as follows.", "$K_{p,q,r}(k)&=\\int _0^{\\pi _{p,q}/2}\\frac{d\\theta }{(1-k^q\\sin _{p,q}^{q}{\\theta })^{1/r}},\\\\E_{p,q,r}(k)&=\\int _0^{\\pi _{p,q}/2}(1-k^q\\sin _{p,q}^{q}{\\theta })^{1/r^*}\\,d\\theta .$ Then, we see that the functions $K_{p,q,r}(k)$ and $E_{p,q,r}(k)$ satisfy a system of linear differential equations.", "Proposition 2.2 Let $p \\in \\mathbb {P}^*,\\ q,\\ r \\in (1,\\infty )$ .", "Then, $\\frac{dE_{p,q,r}}{dk}&=\\frac{q(E_{p,q,r}-K_{p,q,r})}{r^*k},\\\\\\frac{dK_{p,q,r}}{dk}&=\\dfrac{aE_{p,q,r}-(a-k^q)K_{p,q,r}}{k(1-k^q)},$ where $a:=1+q/r^*-q/p$ .", "We consider the case $p \\ne \\infty $ .", "Differentiating $E_{p,q,r}(k)$ we have $\\frac{dE_{p,q,r}}{dk}&=\\frac{q}{r^*}\\int _0^{\\pi _{p,q}/2}\\dfrac{-k^{q-1}\\sin _{p,q}^q{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}}\\,d\\theta =\\frac{q}{r^*k} (E_{p,q,r}-K_{p,q,r}).$ Next, for $K_{p,q,r}(k)$ $\\frac{dK_{p,q,r}}{dk}=\\frac{q}{r}\\int _0^{\\pi _{p,q}/2}\\dfrac{k^{q-1}\\sin _{p,q}^q{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1+1/r}}\\,d\\theta .$ Here we see that $& \\frac{d}{d\\theta }\\left(\\frac{-\\cos _{p,q}^{p/r}{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}}\\right)=\\frac{q(1-k^q)\\sin _{p,q}^{q-1}{\\theta }\\cos _{p,q}^{1-p/r^*}{\\theta }}{r(1-k^q\\sin _{p,q}^q{\\theta })^{1+1/r}},\\\\& \\lim _{\\theta \\rightarrow \\pi _{p,q}/2} \\cos _{p,q}^{p-1}{\\theta }=\\lim _{\\theta \\rightarrow \\pi _{p,q}/2} (1-\\sin _{p,q}^q{\\theta })^{1/p^*}=0;$ so that we use integration by parts as $\\frac{dK_{p,q,r}}{dk}&=\\frac{k^{q-1}}{1-k^q}\\int _0^{\\pi _{p,q}/2}\\frac{d}{d\\theta }\\left(\\frac{-\\cos _{p,q}^{p/r}{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}}\\right)\\sin _{p,q}{\\theta }\\cos _{p,q}^{p/r^*-1}{\\theta }\\,d\\theta \\\\&=\\frac{k^{q-1}}{1-k^q} \\left[\\frac{-\\sin _{p,q}{\\theta }\\cos _{p,q}^{p-1}{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}}\\right]_0^{\\pi _{p,q}/2}\\\\& \\qquad +\\frac{k^{q-1}}{1-k^q} \\int _0^{\\pi _{p,q}/2}\\frac{\\cos _{p,q}^{p/r}{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}}\\left(\\cos _{p,q}^{p/r^*}{\\theta }-\\frac{(q/r^*-q/p)\\sin _{p,q}^q{\\theta }}{\\cos _{p,q}^{p/r}{\\theta }}\\right)\\,d\\theta \\\\&=\\frac{k^{q-1}}{1-k^q} \\int _0^{\\pi _{p,q}/2}\\frac{\\cos _{p,q}^p{\\theta }-(q/r^*-q/p)\\sin _{p,q}^q{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}}\\,d\\theta \\\\&=\\frac{k^{q-1}}{1-k^q} \\int _0^{\\pi _{p,q}/2}\\frac{(1+q/r^*-q/p)(1-k^q\\sin _{p,q}^q{\\theta })-(1+q/r^*-q/p-k^q)}{k^q(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}}\\,d\\theta \\\\&=\\frac{(1+q/r^*-q/p)E_{p,q,r}-(1+q/r^*-q/p-k^q)K_{p,q,r}}{k(1-k^q)}.$ The case $p=\\infty $ is proved similarly.", "Indeed, $\\frac{dE_{\\infty ,q,r}}{dk}&=\\frac{q}{r^*}\\int _0^1\\dfrac{-k^{q-1}\\theta ^q}{(1-k^q\\theta ^q)^{1/r}}\\,d\\theta =\\frac{q}{r^*k} (E_{\\infty ,q,r}-K_{\\infty ,q,r})$ and $\\frac{dK_{\\infty ,q,r}}{dk}&=\\frac{q}{r}\\int _0^1\\dfrac{k^{q-1}\\theta ^q}{(1-k^q\\theta ^q)^{1+1/r}}\\,d\\theta \\\\&=\\frac{k^{q-1}}{1-k^q}\\int _0^1\\frac{d}{d\\theta }\\left(-\\left(\\frac{1-\\theta ^q}{1-k^q\\theta ^q}\\right)^{1/r}\\right)\\theta (1-\\theta ^q)^{1/r^*}\\,d\\theta \\\\&=\\frac{k^{q-1}}{1-k^q} \\left[\\frac{-\\theta (1-\\theta ^q)}{(1-k^q\\theta ^q)^{1/r}}\\right]_0^1\\\\& \\qquad +\\frac{k^{q-1}}{1-k^q} \\int _0^1\\left(\\frac{1-\\theta ^q}{1-k^q\\theta ^q}\\right)^{1/r}\\left((1-\\theta ^q)^{1/r^*}-\\frac{(q/r)\\theta ^q}{(1-\\theta ^q)^{1/r}}\\right)\\,d\\theta \\\\&=\\frac{k^{q-1}}{1-k^q} \\int _0^1\\frac{1-\\theta ^q-(q/r)\\theta ^q}{(1-k^q\\theta ^q)^{1/r}}\\,d\\theta \\\\&=\\frac{k^{q-1}}{1-k^q} \\int _0^1\\frac{(1+q/r^*)(1-k^q\\theta ^q)-(1+q/r^*-k^q)}{k^q(1-k^q\\theta ^q)^{1/r}}\\,d\\theta \\\\&=\\frac{(1+q/r^*)E_{\\infty ,q,r}-(1+q/r^*-k^q)K_{\\infty ,q,r}}{k(1-k^q)}.$ This completes the proof.", "Proposition REF now yields Theorem REF .", "[Proof of Theorem REF ] Let $k^{\\prime }:=(1-k^q)^{1/r},\\ E^{\\prime }_{p,r,q^*}(k):=E_{p,r,q^*}(k^{\\prime })$ and $K^{\\prime }_{p,r,q^*}(k):=K_{p,r,q^*}(k^{\\prime })$ .", "As $dk^{\\prime }/dk=-(q/r)k^{q-1}/(k^{\\prime })^{r-1}$ , Proposition REF gives $\\frac{dE_{p,q,r^*}}{dk}&=\\frac{q(E_{p,q,r^*}-K_{p,q,r^*})}{rk},\\\\\\frac{dK_{p,q,r^*}}{dk}&=\\dfrac{aE_{p,q,r^*}-(a-k^q)K_{p,q,r^*}}{k(k^{\\prime })^r},\\\\\\frac{dE^{\\prime }_{p,r,q^*}}{dk}&=\\frac{k^{q-1}(-E^{\\prime }_{p,r,q^*}+K^{\\prime }_{p,r,q^*})}{(k^{\\prime })^r},\\\\\\frac{dK^{\\prime }_{p,r,q^*}}{dk}&=\\frac{q(-bE^{\\prime }_{p,r,q^*}+(b-(k^{\\prime })^r)K^{\\prime }_{p,r,q^*})}{rk(k^{\\prime })^r},$ where $a:=1+q/r-q/p$ and $b:=1+r/q-r/p$ .", "We denote the left-hand side of (REF ) by $L(k)$ .", "A direct computation shows that $\\frac{d}{dk} & L(k)\\\\& =\\frac{q(E_{p,q,r^*}-K_{p,q,r^*})}{rk} \\cdot K^{\\prime }_{p,r,q^*}+E_{p,q,r^*} \\cdot \\frac{q(-bE^{\\prime }_{p,r,q^*}+(b-(k^{\\prime })^r)K^{\\prime }_{p,r,q^*})}{rk(k^{\\prime })^r}\\\\& \\quad +\\dfrac{aE_{p,q,r^*}-(a-k^q)K_{p,q,r^*}}{k(k^{\\prime })^r} \\cdot E^{\\prime }_{p,r,q^*}+K_{p,q,r^*} \\cdot \\frac{k^{q-1}(-E^{\\prime }_{p,r,q^*}+K^{\\prime }_{p,r,q^*})}{(k^{\\prime })^r} \\\\& \\qquad -\\dfrac{aE_{p,q,r^*}-(a-k^q)K_{p,q,r^*}}{k(k^{\\prime })^r} \\cdot K^{\\prime }_{p,r,q^*}-K_{p,q,r^*} \\cdot \\frac{q(-bE^{\\prime }_{p,r,q^*}+(b-(k^{\\prime })^r)K^{\\prime }_{p,r,q^*})}{rk(k^{\\prime })^r}\\\\& =\\left(\\frac{q}{rk}+\\frac{q(b-(k^{\\prime })^r)}{rk(k^{\\prime })^r}-\\frac{a}{k(k^{\\prime })^r}\\right)E_{p,q,r^*}K^{\\prime }_{p,r,q^*}\\\\& \\quad +\\left(-\\frac{q}{rk}+\\frac{k^{q-1}}{(k^{\\prime })^r}+\\frac{a-k^q}{k(k^{\\prime })^r}-\\frac{q(b-(k^{\\prime })^r)}{rk(k^{\\prime })^r}\\right)K_{p,q,r^*}K^{\\prime }_{p,r,q^*}\\\\& \\qquad +\\left(-\\frac{qb}{rk(k^{\\prime })^r}+\\frac{a}{k(k^{\\prime })^r}\\right)E_{p,q,r^*}E^{\\prime }_{p,r,q^*}\\\\& \\qquad \\quad +\\left(-\\frac{a-k^q}{k(k^{\\prime })^r}-\\frac{k^{q-1}}{(k^{\\prime })^r}+\\frac{qb}{rk(k^{\\prime })^r}\\right)K_{p,q,r^*}E^{\\prime }_{p,r,q^*}\\\\& =\\frac{qb-ra}{rk(k^{\\prime })^r}(E_{p,q,r^*}K^{\\prime }_{p,r,q^*}-K_{p,q,r^*}K^{\\prime }_{p,r,q^*}-E_{p,q,r^*}E^{\\prime }_{p,r,q^*}+K_{p,q,r^*}E^{\\prime }_{p,r,q^*}).$ Since $qb-ra=0$ , we see that $dL/dk=0$ .", "Thus $L(k)$ is a constant $C$ .", "We will evaluate $C$ as follows.", "Since $|(K_{p,q,r^*} & -E_{p,q,r^*})K^{\\prime }_{p,r,q^*}| \\\\&=\\int _0^{\\pi _{p,q}/2}\\left(\\frac{1}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r^*}}-(1-k^q\\sin _{p,q}^q{\\theta })^{1/r}\\right)\\,d\\theta \\\\& \\qquad \\times \\int _0^{\\pi _{p,r}/2}\\frac{d\\theta }{(1-(k^{\\prime })^r\\sin _{p,r}^r{\\theta })^{1/q^*}}\\\\&=\\int _0^{\\pi _{p,q}/2}\\frac{k^q\\sin _{p,q}^q{\\theta }}{(1-k^q\\sin _{p,q}^q{\\theta })^{1/r^*}}\\,d\\theta \\cdot \\int _0^{\\pi _{p,r}/2}\\frac{d\\theta }{(\\cos _{p,r}^p{\\theta }+k^q\\sin _{p,r}^r{\\theta })^{1/q^*}}\\\\& \\le k^q K_{p,q,r^*}(k) \\cdot \\frac{1}{k^{q-1}}\\frac{\\pi _{p,r}}{2}\\\\& =\\frac{\\pi _{p,r}}{2}kK_{p,q,r^*}(k),$ we obtain $\\lim _{k \\rightarrow +0}(K_{p,q,r^*}-E_{p,q,r^*})K^{\\prime }_{p,r,q^*}=0$ .", "Therefore, from Proposition REF $C=\\lim _{k \\rightarrow +0}K_{p,q,r^*}E^{\\prime }_{p,r,q^*}=K_{p,q,r^*}(0)E_{p,r,q^*}(1)=\\frac{\\pi _{p,q}\\pi _{s,r}}{4},$ where $1/s=1/p-1/q$ .", "Thus, we conclude the assertion.", "Finally, we will give a remark for Theorem REF .", "From the series expansion and the termwise integration, it is possible to express the generalized complete elliptic integrals by Gaussian hypergeometric functions $K_{p,q,r}(k)&=\\frac{\\pi _{p,q}}{2}F\\left(\\frac{1}{q},\\frac{1}{r};\\frac{1}{p^*}+\\frac{1}{q};k^q\\right),\\\\E_{p,q,r}(k)&=\\frac{\\pi _{p,q}}{2}F\\left(\\frac{1}{q},-\\frac{1}{r^*};\\frac{1}{p^*}+\\frac{1}{q};k^q\\right).$ By these expressions and letting $1/p=1/2-b,\\ 1/q=1/2+a,\\ 1/r=1/2-c$ and $k^q=x$ in (REF ), we obtain Elliott's identity (see Elliott [7]; see also [1], [2] and [8]): $F\\left({{1/2+a,-1/2-c}\\atop {a+b+1}} ;x\\right)F\\left({{1/2-a,1/2+c}\\atop {b+c+1}} ;1-x\\right)\\\\+F\\left({{1/2+a,1/2-c}\\atop {a+b+1}};x\\right)F\\left({{-1/2-a,1/2+c}\\atop {b+c+1}};1-x\\right)\\\\-F\\left({{1/2+a,1/2-c}\\atop {a+b+1}};x\\right)F\\left({{1/2-a,1/2+c}\\atop {b+c+1}};1-x\\right)\\\\=\\frac{\\Gamma (a+b+1)\\Gamma (b+c+1)}{\\Gamma (a+b+c+3/2)\\Gamma (b+1/2)}$ for $|a|,\\,|c|<1/2$ and $b \\in (-1/2,\\infty )$ , where $\\Gamma $ denotes the gamma function.", "Also, letting $1/p=2-c-a$ and $1/q=1-a$ in (REF ) of Corollary REF , we have the identity of [1] for $a \\in (0,1)$ and $c \\in (1-a,\\infty )$ .", "A series of Vuorinen's works on Elliott's identity with his coauthors starting from [1] deals with the concavity/convexity properties of certain related functions to the left-hand side of (REF )." ] ]
1606.05115
[ [ "Accumulate then Forward: An Opportunistic Relaying Protocol for\n Wireless-Powered Cooperative Communications" ], [ "Abstract This paper investigates a wireless-powered cooperative communication network consisting of a source, a destination and a multi-antenna decode-and-forward relay.", "We consider the relay as a wireless-powered node that has no external power supply; but it is equipped with an energy harvesting (EH) unit and a rechargeable battery such that it can harvest and accumulate energy from radio-frequency signals broadcast by the source.", "By fully incorporating the EH feature of the relay, we develop an opportunistic relaying protocol, termed accumulate-then-forward (ATF), for the considered WPCCN.", "We then adopt the discrete Markov chain to model the dynamic charging and discharging behaviors of the relay battery.", "Based on this, we derive a closed-form expression for the exact outage probability of the proposed ATF protocol.", "Numerical results show that the ATF scheme can outperform the direct transmission one, especially when the amount of energy consumed by relay for information forwarding is optimized." ], [ "Introduction", "Radio-frequency (RF) energy harvesting technique has recently been regarded as a new viable solution to extend the lifetime of energy-constrained wireless networks [1].", "This technique has opened up new opportunities for cooperative communications as it enables a new cooperative manner for wireless devices (see [2] and references therein).", "In particular, the relay in conventional cooperative networks, can now harvest energy from the information source and then use the harvested energy to assist the source's transmission.", "In this sense, the relay is more willing to cooperate since it does not need to consume its own energy.", "In this paper, we refer to a cooperative communication network with wireless powered relay as a wireless-powered cooperative communication network (WPCCN).", "In fact, the design and analysis of WPCCNs have become a hot research topic in wireless communication area very recently (see, e.g., [3], [4], [5], [6], [7], [8]).", "It is worth pointing out that the design of wireless-powered networks is inherently different from that of conventional wireless networks powered by solar/wind energy harvesting (e.g., [9]).", "Specifically, in wireless-powered networks, the amount of energy harvested by wireless-powered nodes highly depends on the network operation modes, while these two events are independent in conventional energy harvesting networks.", "In practice, due to the propagation loss of RF signals, the amount of energy harvested by wireless-poweredThroughout this paper, we use the terms “wireless-powered\" and “energy harvesting\" interchangeably.", "nodes during one transmission block are normally very limited.", "Thus, it is necessary for these nodes to equip with energy storage (e.g., a rechargeable battery) such that they can accumulate enough amount of harvested energy before performing one round of information transmission.", "However, to the best knowledge of the authors, only a few papers in open literature incorporated the energy accumulation process in the design/analysis of WPCCNs.", "In [3], Krikidis et al.", "studied a classical three-node relay network with an energy harvesting relay, where the relay was assumed to have a discrete and finite-capacity rechargeable battery and the energy accumulation process at the relay was characterized by a finite-state Markov chain.", "Recently, [5], [6], [7] extended [3] to a more general scenario with multiple energy harvesting relays, in which several relay selection schemes were proposed and analyzed.", "Besides, a continuous battery model was adopted in [8] to analyze the throughput performance of a three-node relay network with energy accumulation.", "However, an infinite capacity of the relay battery was assumed in [8], which makes the resulting analysis somewhat ideal.", "A common assumption in [3], [5], [6], [7], [8] is that no direct link exists between source and destination.", "However, the direct link of a WPCCN actually plays a crucial role in practice.", "This is because the amount of energy harvested by the wireless-powered relay is generally limited, which means the signal-to-noise ratio (SNR) of the source-destination link may be comparable or even larger than that of the relay-destination link.", "In other word, relying solely on an energy harvesting relay to accomplish the information delivery from source to destination may lead to poor system performance.", "Thus, the direct link should be incorporated in the designs of WPCCNs.", "When the direct link is available, a natural question that arises is “how often should the source cooperate with the energy harvesting relay?\".", "This is actually a non-trivial problem.", "Specifically, the relay can only accumulate little amount of energy for information forwarding when the cooperation is too intensive, which may lead to even worse performance than the direct transmission (i.e., no cooperation) scheme.", "On the other side, few cooperation will yield the under-utilization of the relay and then make the cooperation gain insignificant.", "Motivated by this open problem, in this paper we focus on the design of a three-node WPCCN consisting of one source, one energy harvesting relay implementing decode-and-forward protocol and one destination, where the direct link between source and destination exists.", "We consider that the relay is equipped with a rechargeable battery and multiple antennas, which includes single-antenna relay scenarios studied in [3], [5], [6], [8] as special cases.", "Towards the appropriate usage of the energy harvesting relay, we develop an opportunistic relaying protocol, termed accumulate-then-forward (ATF), for the considered WPCCN, where the relay cooperates with the source in an opportunistic manner.", "Particularly, the cooperation between source and relay is activated only when the accumulated energy at relay exceeds a predefined energy threshold and the decoding of source's information at relay is successful.", "Otherwise, the source has to transmit information to destination by itself, while the relay accumulates the energy harvested from source's signals.", "By modeling dynamic charging/discharging behaviors of relay battery as a finite-state Markov chain, we then analyze the outage probability of the proposed ATF protocol over mixed Rician-Rayleigh fading channels.", "Specifically, in contrast to the Rayleigh fading model used in [3], [5], [6], [8], we adopt Rician fading to characterize the channel fading between source and relay.", "This is more practical by considering that the line-of-sight (LoS) path is very likely to exist between source and relay as the current wireless energy harvesting techniques can only support a relatively short distance.", "Numerical results are finally performed to verify the theoretical analysis and illustrate the effects of several system parameters on the network outage probability." ], [ "System Model and Protocol Design", "We consider a WPCCN consisting of one single-antenna source $S$ , one decode-and-forward (DF) relay $R$ equipped with $N$ antennas, and one single-antenna destination $D$ .", "We assume that $S$ and $D$ have embedded power supplies, while $R$ is a wireless-powered device.", "But, $R$ is equipped with an wireless energy harvesting unit and a rechargeable battery such that it can accumulate the energy harvested from RF signals broadcast by $S$ .", "Furthermore, the relay is equipped with separate energy and information receivers [10].", "As such, it can flexibly switch its received signal to one of these two receivers to realize energy harvesting or information decoding.", "Let $h_{SD}$ denote the complex channel coefficient of $S$ -$D$ link.", "Also let $\\mathbf {h}_{SR}$ and $\\mathbf {h}_{RD}$ denote the $N\\times 1$ channel vectors of $S$ -$R$ and $R$ -$D$ link, respectively.", "As the up-to-date wireless energy harvesting techniques could only be operated within a relatively short distance, the line-of-sight (LoS) path is very likely to exist between $S$ and $R$ .", "Motivated by this fact, we consider an asymmetric scenario for the fading distributions of $S$ -$R$ link and $R$ -$D$ link.", "In particular, the elements of $\\mathbf {h}_{SR}$ are subject to independent and identically distributed (i.i.d.)", "Rician fading, while the elements of $\\mathbf {h}_{RD}$ are subject to i.i.d.", "Rayleigh fading.", "Besides, the channel coefficient of direct link $h_{SD}$ follows Rayleigh distribution.", "Furthermore, all channels between $S$ , $R$ and $D$ are assumed to experience slow, independent and frequency flat fading such that the channel gains remain unchanged within each transmission block but change independently from one block to the other.", "We now propose an accumulate-then-forward (ATF) protocol for WPCCN.", "In the proposed ATF protocol, the energy harvesting relay accumulates the energy harvested from signals broadcast by $S$ and assists its information transmission in an opportunistic manner.", "Specifically, the relay opportunistically switches between the energy harvesting mode and the information cooperation mode based on both whether its residual energy in battery exceeds a predefined energy threshold and whether its decoding of source's information is correct.", "Here, we use $T$ to denote the duration of each transmission block, which is further divided into two time slots with equal length $T/2$ when $R$ attempts to decode source's information (i.e., the residual energy at $R$ exceeds the predefined energy threshold).", "Mode I: This mode corresponds to the case when the current residual energy in relay battery, denoted by $E_R$ , is less than the energy threshold $E_T$ .", "In this situation, $R$ chooses to perform energy harvesting operation to further accumulate its energy and the source has to transmit its information to $D$ by itself during the whole transmission block.", "Mode II: In this mode, the current residual energy $E_R$ at $R$ is not less than $E_T$ , but the decoding of the source's information at $R$ during the first time slot is unsuccessful.", "Thus, $R$ still cannot cooperate with $S$ in this mode.", "During the second time slot, $S$ is motivated to re-transmit its information to enhance the received SNR at $D$ , while $R$ can harvest energy from this signal to charge its battery.", "Mode III: Here, $R$ has accumulated enough energy (i.e., $E_R \\ge E_T$ ) and its information decoding of the first hop is also correct.", "As such, $R$ will work in information cooperation mode by helping forward the source's information to $D$ during the second time slot, while $S$ can keep in silence during this period.", "In the following, we express the harvested energy at $R$ and the received SNR at $D$ of the proposed ATF for three possible modes mathematically.", "Without lose of generality, we consider a normalized transmission block (i.e., $T=1$ ) hereafter.", "Moreover, we use $P_S$ to denote the transmit power of $S$ and $H_{xy}=\\Vert \\mathbf {h}_{xy}\\Vert ^2$ to denote the channel power gain, where $x,y\\in \\left\\lbrace S,R,D\\right\\rbrace $ and $\\Vert \\mathbf {x}\\Vert $ denotes the Euclidean norm of a vector $\\mathbf {x}$ ." ], [ "Mode I", "In this mode, $R$ harvests energy during the whole transmission block.", "Thus, the amount of harvested energy at $R$ can be expressed as $E_{H}^{\\rm {I}} = \\eta P_S H_{SR},$ where $\\eta \\in \\left(0,1\\right]$ is the energy conversion efficiency.", "In (REF ), we ignore the amount of energy harvested from the noise as it is normally below the sensitivity of energy harvesting circuit.", "Let $x_S$ denote the transmitted signal by $S$ with unit energy.", "In Mode I, the received signal at $D$ comes only from the direct link.", "Thus, the received SNR at $D$ in Mode I is given by $\\gamma _{D}^{\\rm {I}} = \\gamma _{SD}= P_S H_{SD}/N_0,$ where $N_0$ is the power of the additive Gaussian white noise (AWGN)." ], [ "Mode II", "Since $R$ has stored sufficient energy in Mode II, during the first time slot, it will try to perform information decoding based on the signal received from $S$ .", "We assume that the maximum ratio combining (MRC) technique is adopted at $R$ to maximize the received SNR.", "In this case, the received SNR at the $R$ is given by $\\gamma _{SR} = P_S H_{SR}/N_0.$ Recall that in this mode the source's information is not decoded correctly at $R$ .", "In the second time slot, $S$ has to re-transmit its information to $D$ as $R$ cannot help, while $R$ can harvest energy from this re-transmitted signal.", "The harvested energy during the second time slot is given by $E_{H}^{\\rm {II}} = \\eta P_S H_{SR}/2.$ On the other hand, $D$ receives two copies of the same information from $S$ .", "With the MRC technique, the received SNR at $D$ in Mode II can be expressed as $\\gamma _{D}^{\\rm {II}} = \\gamma _{SD} + \\gamma _{SD} = 2 P_S H_{SD}/N_0.$" ], [ "Mode III", "In this mode, $R$ utilizes the received signal from $S$ to decode the information in the first hop and the decoding is correct.", "Thus, $R$ can cooperate with $S$ by forwarding the source's information to $D$ .", "Here, we consider that $R$ will spend $E_T$ amount of energy to perform information forwarding and the fixed transmit power is assumed to be $P_R = 2 E_T$ .", "Moreover, for simplicity, it employs transmit antenna selection scheme [11].", "That is, the antenna with maximum channel power gain is selected to forward information.", "We define $h_{RD}^* = \\max \\lbrace \\mathbf {h}_{RD}\\rbrace $ .", "In this case, the received SNR at $D$ during the second time slot is given by $\\gamma _{RD} = P_R H_{RD}/N_0,$ where $H_{RD}=|h_{RD}^*|^2$ is the channel power gain of $R$ -$D$ link.", "Note that the received SNR at $D$ during the first time slot is same as (2).", "Using the MRC technique, the resulting SNR at $D$ in Mode III can be characterized as $\\gamma _{D}^{\\rm {III}} = \\gamma _{SD} + \\gamma _{RD} = \\left(P_S H_{SD} + P_R H_{RD}\\right)/{N_0}.$ In this section, we analyze the outage probability of the proposed ATF protocol over mixed Rician-Rayleigh fading channels.", "To this end, we first model the dynamic behaviors of the relay battery by a finite-state Markov chain (MC) [3]." ], [ "Markov Chain Description", "We assume that $R$ is equipped by a $L$ discrete-level battery with a capacity $C$ .", "The $i$ th energy level is defined as $\\varepsilon _i = i C / L$ , $i\\in \\lbrace 0,1,2,\\ldots ,L \\rbrace $ .", "We define the state $S_i$ as the state of relay's residual battery being $\\varepsilon _i$ .", "$P_{i,j}$ is defined as the state transition probability from $S_i$ to $S_j$ .", "Let $\\lambda [m]\\in \\lbrace \\lambda _{{\\rm {I}}},\\lambda _{{\\rm {II}}},\\lambda _{{\\rm {III}}}\\rbrace $ denote the system operation mode in the $m$ -th transmission block, where $\\lambda _X$ , $X\\in {\\left({\\rm {I}},{\\rm {II}},{\\rm {III}}\\right)}$ , represents the event that the $X$ -th mode is operated.", "Considering the discrete battery model adopted in this paper, the discretized amount of energy $\\varepsilon _H$ harvested by $R$ should be re-calculated as $\\small {\\varepsilon _H^X \\triangleq \\varepsilon _i,~{\\rm {where}}~ i = \\arg \\max \\nolimits _{j\\in \\lbrace 0,1,\\ldots ,L\\rbrace }\\left\\lbrace \\varepsilon _j:\\varepsilon _j<E_H^X\\right\\rbrace ,}$ where $X\\in \\lbrace {\\rm {I}},{\\rm {II}}\\rbrace $ .", "Similarly, the actual amount of energy consumed by $R$ for information forwarding should be defined by $\\varepsilon _T \\triangleq \\varepsilon _i,~{\\rm {where}}~ i = \\arg \\min \\nolimits _{j\\in \\lbrace 1,\\ldots ,L\\rbrace }\\left\\lbrace \\varepsilon _j:\\varepsilon _j\\ge E_T\\right\\rbrace .$ In this paper, we assume that $R$ can decode the information correctly if its received SNR exceeds a predetermined threshold.", "Let $\\mathbb {R}$ denote the transmit rate of $S$ .", "The SNR threshold of $S$ -$R$ link in Mode II or III can then be defined as $\\gamma _0 = 2^{2\\mathbb {R}}-1$ .", "We now can describe the three possible operations of the proposed ATF protocol during the $m$ -th transmission block mathematically as follows $\\lambda [m] ={\\left\\lbrace \\begin{array}{ll}\\lambda _{\\rm {I}}, &\\mbox{if ~$\\varepsilon _T>\\varepsilon [m]$},\\\\\\lambda _{\\rm {II}}, &\\mbox{if ~$\\varepsilon _T\\le \\varepsilon [m]~\\&~\\gamma _{SR}<\\gamma _0$},\\\\\\lambda _{\\rm {III}}, &\\mbox{if ~$\\varepsilon _T\\le \\varepsilon [m]~\\&~\\gamma _{SR}\\ge \\gamma _0$},\\end{array}\\right.", "}$ where $\\varepsilon [m]$ denotes the relay's residual energy at the beginning of the $m$ -th transmission block.", "Moreover, the residual energy at the beginning of the $(m+1)$ th transmission block can thus be expressed as $\\varepsilon [m+1] ={\\left\\lbrace \\begin{array}{ll}\\min \\lbrace \\varepsilon [m] + \\varepsilon _H^{\\rm {I}}, C \\rbrace , &\\mbox{if $\\lambda [m] = \\lambda _{\\rm {I}}$}\\\\\\min \\lbrace \\varepsilon [m] + \\varepsilon _H^{\\rm {II}}, C \\rbrace , &\\mbox{if $\\lambda [m] = \\lambda _{\\rm {II}}$}\\\\\\varepsilon [m] - \\varepsilon _T, &\\mbox{if $\\lambda [m] = \\lambda _{\\rm {III}}$}\\\\\\end{array}\\right.", "}.$ Based on the above mathematical description, we now derive the state transition probabilities of the formulated MC for the relay's battery.", "Inspired by [3], the state transition of the MC can be generally split into the following eight cases.", "We consider the MC starts with the state $S_0$ , i.e., the battery of $R$ is empty.", "It is obvious that Mode I will be activated in this case.", "Furthermore, the amount of harvested energy during the current block should be discretized to zero, which indicates that the condition $E_H^{\\rm {I}} < \\varepsilon _1=C/L$ holds.", "The transition probability of this case is characterized as $\\small {P_{0,0} = \\Pr \\left\\lbrace E_{H}^{\\rm {I}} < \\frac{C}{L} \\right\\rbrace =F_{H_{SR}}\\left(\\frac{C}{\\eta P_S L}\\right),}$ where $F_{H_{SR}}(\\cdot )$ denotes the cumulative distribution function (CDF) of $H_{SR}$ .", "According to [12], we can write the CDF of $H_{SR}$ as $F_{H_{SR}}\\left(x\\right) = 1-Q_N\\left(\\sqrt{2NK},\\sqrt{\\frac{2(K+1)}{\\Omega _{SR}}x}\\right)$ , where $Q_N\\left(\\cdot ,\\cdot \\right)$ is the generalized ($N$ th-order) Marcum $Q$ -function [13], $K$ is the Rician $K$ -factor defined as the ratio of the powers of the LoS component to the scattered components and $\\Omega _{SR} = \\mathbb {E}\\lbrace \\left|h_{SR,i}\\right|^2\\rbrace ,~\\forall i\\in \\left\\lbrace 1,\\ldots ,N\\right\\rbrace $ , with $\\mathbb {E}\\lbrace \\cdot \\rbrace $ denoting the statistical expectation and $h_{SR,i}$ denoting the $i$ -th element of ${\\bf h}_{SR}$ ." ], [ "The empty battery is partially charged ($S_0$ to {{formula:c0f63a55-81e5-4f09-be0f-efe8febf3eea}} with {{formula:29e28bcf-5799-4c43-9b3f-dfdd47f37a7b}} )", "Mode I is activated to charge the battery.", "We can also deduce that the effective amount of harvested energy should be expressed as $\\varepsilon _H^{\\rm {I}} = i C/L$ , which means $E_H^{\\rm {I}}$ falls between the battery levels $i$ and $i+1$ .", "Thus, the transition probability is $\\small {\\begin{split}P_{0,i} &= \\Pr \\left\\lbrace \\frac{i C}{L} \\le E_{H}^{\\rm {I}} < \\frac{\\left(i+1\\right)C}{L}\\right\\rbrace \\\\&= F_{H_{SR}}\\left(\\frac{\\left(i+1\\right)C}{\\eta P_S L}\\right) - F_{H_{SR}}\\left(\\frac{i C}{\\eta P_S L}\\right).\\end{split}}$" ], [ "The empty battery is fully charged ($S_0$ to {{formula:e69c0450-d608-4c08-a475-42aed4f3bfad}} )", "Similar to the previous two cases, the transition probability can be calculated as $\\small {P_{0,L} = \\Pr \\left\\lbrace E_{H}^{\\rm {I}} \\ge C \\right\\rbrace = 1 - F_{H_{SR}}\\left(\\frac{C}{\\eta P_S}\\right).", "}$" ], [ "The non-empty and non-full battery remains unchanged ($S_i$ to {{formula:eff16853-f020-42d6-95ac-928b22ac2f23}} with {{formula:5305a1a1-f2a1-406a-a185-e86f2ed91ec5}} )", "The battery stays at the same level, which indicates that $R$ either operates in Mode I or Mode II with zero effective harvested energy (i.e.", "$E_{H}^{\\rm {I}}$ or $E_{H}^{\\rm {II}}$ is discretized to zero).", "The transition probability of this case is characterized as $\\small {\\begin{split}&P_{i,i} = \\Pr \\left\\lbrace \\left[ \\left(E_T > \\frac{i C}{L}\\right) \\cap \\left( E_{H}^{\\rm {I}} < \\frac{C}{L}\\right) \\right]\\right.\\\\&~~~~~\\left.\\cup \\left[\\left(E_T \\le \\frac{i C}{L}\\right) \\cap \\left(\\gamma _{SR} < \\gamma _0\\right) \\cap \\left( E_{H}^{\\rm {II}}< \\frac{C}{L}\\right)\\right] \\right\\rbrace \\\\&={\\left\\lbrace \\begin{array}{ll}F_{H_{SR}}\\left(\\frac{C}{\\eta P_S L}\\right),~~ \\mbox{if ~$E_T > \\frac{i C}{L}$};\\\\F_{H_{SR}}\\left(\\frac{\\gamma _0 N_0}{P_S}\\right) ,~~\\mbox{if ~$E_T \\le \\frac{i C}{L}~\\&~\\gamma _0 N_0 < \\frac{2 C}{\\eta L}$}; \\\\F_{H_{SR}}\\left(\\frac{2 C}{\\eta P_S L}\\right), ~~\\mbox{if ~$E_T \\le \\frac{i C}{L}~\\&~\\gamma _0 N_0 \\ge \\frac{2 C}{\\eta L}$} .", "\\\\\\end{array}\\right.", "}\\end{split}}$" ], [ "The non-empty battery is partially charged ($S_i$ to {{formula:8516504f-2861-409e-ae4b-04c8e8962f50}} with {{formula:5ca42648-f028-4a7e-bd88-e5c11d152af1}} )", "Similar as the previous case, the battery is partially charged from level $i$ to $j$ (i.e., $\\varepsilon _H^X = \\left(j-i\\right)C/L$ ).", "Thus, the transition probability can be derived as (REF ) shown on the top of next page.", "Figure: NO_CAPTION" ], [ "The non-empty and non-full battery is fully charged ($S_i$ to {{formula:1473845a-d140-4647-ba88-d9f7aa8699a2}} with {{formula:88908499-1f5c-43ce-8721-654e0459e359}} )", "In this case, the effective harvested energy $\\varepsilon _H^X$ , either from Mode I or Mode II, is greater than the residual space of the battery.", "The transition probability is thus given by $\\small {\\begin{split}&P_{i,L} = \\Pr \\left\\lbrace \\left[ \\left(E_T > \\frac{i C}{L}\\right)\\cap \\left( E_{H}^{\\rm {I}} \\ge \\frac{\\left(L-i\\right)C}{L}\\right) \\right]\\right.\\\\&\\left.\\cup \\left[\\left(E_T \\le \\frac{i C}{L}\\right) \\cap \\left(\\gamma _{SR} < \\gamma _0\\right) \\cap \\left( E_{H}^{\\rm {II}}\\ge \\frac{\\left(L-i\\right)C}{L}\\right)\\right] \\right\\rbrace \\\\&={\\left\\lbrace \\begin{array}{ll}1-F_{H_{SR}}\\left(\\frac{\\left(L-i\\right)C}{\\eta P_S L}\\right),~\\mbox{if $E_T > \\frac{i C}{L}$;} \\\\0,~~~~~~~~~~~~~~~~~~~~~~~~\\mbox{if $E_T \\le \\frac{i C}{L}~\\&~\\gamma _0 < \\frac{2\\left(L-i\\right) C}{\\eta N_0 L}$;}~~~~~~~~~~~~~~~~~~~ \\\\F_{H_{SR}}\\left(\\frac{\\gamma _0 N_0}{P_S}\\right)-F_{H_{SR}}\\left(\\frac{\\left(L-i\\right)C}{\\eta P_S L}\\right),\\\\~~~~~~~~~~~~~~~~~~~~~~~~~~~\\mbox{if $E_T \\le \\frac{i C}{L}~\\&~\\gamma _0 \\ge \\frac{2\\left(L-i\\right) C}{\\eta N_0 L}$.}\\\\\\end{array}\\right.", "}\\end{split}}$" ], [ "The battery remains full ($S_L$ to {{formula:0f58ed64-9bbd-47f4-9f68-16b3525e1064}} )", "In this case, the battery of $R$ certainly has enough energy to support information forwarding in the second hop.", "Thus, only Mode II can be performed so that the battery level is not reduced.", "Since the battery is full at the beginning of the transition, $\\varepsilon _H^{\\rm {II}}$ can be any value.", "The transition probability can be evaluated as $\\small {\\begin{split}P_{L,L} = \\Pr \\left\\lbrace \\gamma _{SR} < \\gamma _0 \\right\\rbrace = F_{H_{SR}}\\left(\\frac{\\gamma _0 N_0}{P_S}\\right)\\end{split}}.$" ], [ "The non-empty battery discharged ($S_j$ to {{formula:f4eeac08-7927-4429-adc3-ca69e0ffaa0b}} with {{formula:350f8ede-5e76-4b71-8029-1fb80b4a210c}} )", "According to the principle of the proposed ATF scheme described in Sec.", ", the battery level is decreased only when Mode III is operated.", "The transition probability can thus be evaluated as $\\small {\\begin{split}P_{j,i} =& \\Pr \\left\\lbrace \\left(\\gamma _{SR} > \\gamma _0\\right) \\cap \\left(E_T = \\frac{\\left(j-i\\right)C}{L}\\right)\\right\\rbrace \\\\=&{\\left\\lbrace \\begin{array}{ll}1-F_{H_{SR}}\\left(\\frac{\\gamma _0 N_0}{P_S}\\right),&\\mbox{if ~$E_T = \\frac{\\left(j-i\\right)C}{L}$;} \\\\0,&\\mbox{if ~$E_T \\ne \\frac{\\left(j-i\\right)C}{L}$.}", "\\\\\\end{array}\\right.", "}\\end{split}}$ We are now ready to derive the steady state distribution of the relay battery.", "Let $\\mathbf {M} = \\left[P_{i,j}\\right]_{(L+1)\\times (L+1)}$ denote the state transition matrix of the formulated MC.", "It is easy to verify that $\\mathbf {M}$ is irreducible and row stochastic.", "Thus, there should exists a unique solution $\\pi $ that satisfies the following equation [14] $\\mathbf {\\pi } = \\left(\\pi _0,\\pi _1,\\ldots ,\\pi _L\\right)^T=\\mathbf {M}^T \\pi .$ This $\\pi $ is actually the discrete distribution of the relay residual energy and can be calculated as $\\pi = \\left(\\mathbf {M}^T - \\mathbf {I} + \\mathbf {B}\\right)^{-1}\\mathbf {b},$ where $\\mathbf {M}^T$ denotes the transpose matrix of $\\mathbf {M}$ , $\\mathbf {I}$ is the identity matrix, $B_{i,j} = 1, \\forall {i,j}$ , and $\\mathbf {b} = \\left(1,1,\\ldots ,1\\right)^T$[14]." ], [ "Outage Probability", "Based on the steady state of the relay battery derived in the previous subsection, we now analyze the outage probability of the proposed ATF scheme.", "Let $\\Phi _X$ , $X\\in \\left({\\rm {I}}, {\\rm {II}}, {\\rm {III}}\\right)$ denote the outage event of Mode I, II, and III, respectively.", "According to the full probability theory, we can express the outage probability of the considered WPCCN as $\\small {\\begin{split}P_{\\rm {out}}&=\\left(1-P_E\\right) \\Pr \\left\\lbrace \\Phi _{\\rm {I}}\\right\\rbrace + P_E \\Pr \\left\\lbrace \\gamma _{SR} < \\gamma _0 \\right\\rbrace \\Pr \\left\\lbrace \\Phi _{\\rm {II}}\\right\\rbrace \\\\&~~~+ P_E \\Pr \\left\\lbrace \\gamma _{SR}\\ge \\gamma _0\\right\\rbrace \\Pr \\left\\lbrace \\Phi _{\\rm {III}}\\right\\rbrace ,\\end{split}}$ where $P_E$ denotes the probability that the residual energy at $R$ is no less than the energy threshold $E_T$ , which can be expressed as $P_E = \\sum \\nolimits _{i=k}^{L} \\pi _i,~{\\rm {s.t.", "}}~k = \\arg \\min \\nolimits _{k\\in 1,\\ldots ,L}\\left\\lbrace \\varepsilon _k \\ge E_T\\right\\rbrace .$ In Mode I, $S$ sends the information to $D$ during the whole block without the help of $R$ .", "Thus, we have $\\Pr \\left\\lbrace \\Phi _{\\rm {I}}\\right\\rbrace = \\Pr \\left\\lbrace \\gamma _{D}^{\\rm {I}} < \\gamma _1\\right\\rbrace = F_{H_{SD}}\\left(\\frac{\\gamma _1 N_0}{P_S}\\right),$ where $\\gamma _1 = 2^\\mathbb {R}-1$ is the outage threshold without cooperation and $F_{H_{SD}}(\\cdot )$ is the CDF of $H_{SD}$ .", "Since the $S$ -$D$ link suffers from Rayleigh fading, we have $F_{H_{SD}}\\left(y\\right) = 1- \\exp \\left({-\\frac{y}{\\Omega _{SD}}}\\right)$ , where $\\Omega _{SD} = \\mathbb {E}\\lbrace \\left|h_{SD}\\right|^2\\rbrace $ is the mean of $H_{SD}$[15].", "For Mode II, the outage probability can be characterized as $\\Pr \\left\\lbrace \\Phi _{\\rm {II}}\\right\\rbrace =\\Pr \\left\\lbrace \\gamma _{D}^{\\rm {II}}<\\gamma _0\\right\\rbrace = F_{H_{SD}}\\left(\\frac{\\gamma _0 N_0}{2 P_S}\\right).$ Similarly, for Mode III, we can evaluate the outage probability as follows $\\small {\\Pr \\left\\lbrace \\Phi _{\\rm {III}}\\right\\rbrace = \\Pr \\left\\lbrace \\gamma _{D}^{\\rm {III}} < \\gamma _0 \\right\\rbrace = \\Pr \\left\\lbrace \\gamma _{RD} + \\gamma _{SD} < \\gamma _0 \\right\\rbrace .", "}$ With the aid of [11], we can express the term $\\Pr \\left\\lbrace \\gamma _{RD} + \\gamma _{SD} < \\gamma _0 \\right\\rbrace $ in closed-form as (REF ) on top of next page, in which $\\bar{\\gamma }_{SD}=P_S \\Omega _{SD}/N_0$ and $\\bar{\\gamma }_{RD}=2E_T \\Omega _{RD}/N_0$ and $\\Omega _{RD} = \\mathbb {E}\\lbrace \\left|h_{RD,i}\\right|^2\\rbrace ,~\\forall i\\in \\left\\lbrace 1,\\ldots ,N\\right\\rbrace $ with $h_{RD,i}$ denoting the $i$ -th element of ${\\bf h}_{RD}$ .", "By substituting (REF ), (REF ), (REF ) and (REF ) into (REF ), we now have obtained a closed-form expression for the outage probability of the proposed ATF protocol.", "Figure: NO_CAPTION" ], [ "Numerical Results", "In this section, we provide some simulation results to verify the above theoretical analysis and illustrate the impacts of several parameters on system performance.", "We adopt the channel model $\\Omega _{ij}=\\left(1+d_{ij}^{\\alpha }\\right)^{-1}$ to capture the path-loss effect, where $d_{ij}$ denotes the distance between nodes $i$ and $j$ , $\\alpha \\in \\left[2,5\\right]$ is the path-loss exponent.", "In the following simulations, we set $d_{SD}=50$ m, $d_{SR}=5$ m, $d_{RD}=45$ m, the path-loss factor for all paths $\\alpha =3$ , the Rician-factor $K=10$ , the noise power $N_0=-60$ dBm, the energy conversion efficiency $\\eta =0.5$ , and the transmission rate of the system $\\mathbb {R}=1$ .", "Figure: Outage probability of the proposed ATF scheme versus the source transmit power for different antenna numbers and battery levels, where ℝ=1\\mathbb {R}=1, C=5×10 -3 C=5\\times 10^{-3}, E T =1×10 -4 E_T=1\\times 10^{-4}, N=[2,4,6]N=[2, 4, 6].Figure: Outage probability of the proposed ATF scheme versus the consumed energy for information forwarding at the relay with different transmit power at the source, where ℝ=1\\mathbb {R}=1, C=5×10 -3 C=5\\times 10^{-3}, P S =[20,30]P_S = [20,30]dBm, N=[2,4,6]N=[2,4,6]We first compare the analytical outage probability of the considered system with its associated Monte Carlo simulation, which corresponds to the practical case that the charging of the relay batteries is continuous (i.e., $\\ L \\rightarrow \\infty $ ).", "To this end, in Fig.", "REF , we plots the outage probability curves of the proposed ATF scheme versus the source transmit power for different antenna numbers and battery levels.", "It can be seen from this figure that the derived analytical expressions of outage probability approach its corresponding Monte Carlo simulation results as the discrete battery level $L$ increases.", "Particularly, when $L = 100$ , the analytical expression coincides well with its corresponding simulation, which verifies the effectiveness of the adopted MC model and the correctness of our theoretical analysis in Sec.", ".", "We can also observe from Fig.", "REF that when the source transmit power $P_S$ is small, the outage probability of the proposed ATF scheme is similar to that of the direct transmission without cooperation.", "This is because the relay cannot accumulate sufficient energy to forward information and $R$ keeps operating in Mode I.", "However, as $P_S$ increases, the proposed ATF gradually outperforms the direct transmission scheme as the relay can accumulate enough energy such that it can assist the source's transmission opportunistically.", "Furthermore, the more the antennas of the relay, the larger the performance gap introduced by larger battery levels.", "The outage probability of the ATF scheme versus the transmit energy at the relay (i.e, $E_T$ ) is drawn for different source transmit powers and antenna numbers in Fig.", "REF .", "This figure is a stair-stepping plot due to the adopted discrete battery model.", "It can be observed from the figure that there exists an optimal value of $E_T$ , which minimizes the system outage probability.", "Moreover, when the source transmit power increases from 20dBm to 30dBm, the optimal transmit energy of the relay slightly shifts to the right as the relay can harvest more energy.", "Furthermore, for a fixed source transmit power, the relay with more antennas requires a smaller optimal transmit energy.", "Figure: Outage probability of the proposed ATF scheme with optimal transmit energy at the relay, where ℝ=1\\mathbb {R}=1, C=5×10 -3 C=5\\times 10^{-3}, N=[2,4,6]N=[2, 4, 6].Fig.", "REF performs the outage performance comparison between the proposed ATF protocol with optimal transmit energy at the relay and the direct transmission scheme without cooperation.", "Note that the optimal transmit energy at the relay for a certain setup could be readily obtained via an exhaustive search of all discrete energy levels.", "We can see from this figure that with the optimal $E_T$ , the proposed ATF protocol is significantly superior to the direct transmission scheme, especially when the source transmit power is high enough.", "In addition, this performance gain can be further enlarged by increasing the number of antennas equipped at the relay.", "This is because equipping more antennas at relay can not only effectively increase the amount of harvested energy in the first hop but also efficiently improve the received SNR at the destination in the second hop." ], [ "Conclusion", "In this paper we developed an accumulate-then-forward (ATF) protocol for cooperative communications via a multi-antenna energy harvesting relay.", "By modeling the charging/discharging behaviors of the relay battery as a finite-state Markov chain, we derived a closed-form expression for the exact outage probability of the considered network over mixed Rician-Rayleigh fading channels.", "Numerical results showed that the system outage probability decreases with the increase of source transmit power and number of antennas at relay.", "Furthermore, the proposed ATF protocol can outperform the direct transmission scheme, especially when the relay consumes the optimal amount of energy for information forwarding." ] ]
1606.05057
[ [ "New variable stars discovered in the fields of three Galactic open\n clusters using the VVV Survey" ], [ "Abstract This project is a massive near-infrared (NIR) search for variable stars in highly reddened and obscured open cluster (OC) fields projected on regions of the Galactic bulge and disk.", "The search is performed using photometric NIR data in the $J$-, $H$- and $K_s$- bands obtained from the Vista Variables in the V\\'ia L\\'actea (VVV) Survey.", "We performed in each cluster field a variability search using Stetson's variability statistics to select the variable candidates.", "Later, those candidates were subjected to a frequency analysis using the Generalized Lomb-Scargle and the Phase Dispersion Minimization algorithms.", "The number of independent observations range between 63 and 73.", "The newly discovered variables in this study, 157 in total in three different known OCs, are classified based on their light curve shapes, periods, amplitudes and their location in the corresponding color-magnitude $(J-K_s,K_s)$ and color-color $(H-K_s,J-H)$ diagrams.", "We found 5 possible Cepheid stars which, based on the period-luminosity relation, are very likely type II Cepheids located behind the bulge.", "Among the newly discovered variables, there are eclipsing binaries, $\\delta$ Scuti, as well as background RR Lyrae stars.", "Using the new version of the Wilson & Devinney code as well as the \"Physics Of Eclipsing Binaries\" (PHOEBE) code, we analyzed some of the best eclipsing binaries we discovered.", "Our results show that these studied systems turn out to be ranging from detached to double-contact binaries, with low eccentricities and high inclinations of approximately $80^{\\circ}$.", "Their surface temperatures range between $3500$K and $8000$K." ], [ "Introduction", "Star clusters are important building blocks of galaxies so knowledge of their individual and statistical properties is of great astrophysical importance.", "These populations, composed of stars sharing the same age and initial chemical composition, have allowed us to test theories of stellar evolution and have also helped to reveal the structure of their host galaxies.", "In particular, Galactic open clusters (OCs) have long been considered excellent targets not only to probe the Galactic disk population [19], [5] but also to trace its chemical evolution [10].", "Estimates indicate that the Milky Way currently hosts a total of about 2.5x10$^4$ or more OCs [30].", "However, in the catalogue by [26], only 2808 Galactic OCs have reasonable estimates of basic cluster parameters such as distance and age, as well as an estimation of the interstellar extinction.", "This number clearly represents a lower limit to the possible amount of clusters belonging to the Milky Way, if we take into account the recently found clusters and cluster candidates [6], [17], [7], [8], [32], [31], [11], [4] and the “still unseen” OCs, which are deeply embedded in obscured regions or are just too faint to be detected.", "Distances, masses, and ages for OCs are generally determined from color-magnitude diagrams (CMDs).", "However, observations of OCs projected on the Galactic bulge region are strongly affected by the effects of both interstellar reddening and high field stellar density [40], [2].", "In these objects, the cluster main sequences appear not to be clearly visible, which reduces the accuracy of the relevant physical parameters derived from their CMDs.", "In these cases, it is very helpful to identify cluster member variable stars since these stars can provide a more precise measurement of the clusters' parameters (particularly their distances).", "The current project, based on near-infrared (NIR) photometric data, is a search for stellar variability in the fields of Galactic OCs which lie in the highly reddened and obscured regions of the Galactic bulge and disk.", "As part of a massive search for variable stars in OCs, we characterize for the first time new variable stars detected in three Galactic OC regions located toward the inner Galaxy.", "For this purpose, we make use of the VISTA Variables in the Vía Láctea (VVV) Survey, which is an ESO Public NIR time-domain survey of the inner Milky Way [29].", "VVV aims to map the Galactic bulge as well as an adjacent section of the mid-plane, covering stellar populations all the way to the Galactic center, including regions of intense star formation.", "This survey has been performing time-domain observations in the $K_s$ -band for over 5 years [34] and provides an atlas of 562 square degrees of the sky in 5 wavebands ($ZYJHK_s$ ), encompassing about a billion objects.", "The VVV Survey is discovering hundreds of new clusters, many of them being very distant and deeply embedded objects toward the inner Galaxy [7], [8], [11], [32], [31].", "The relevant physical parameters (reddenings, distances, masses, luminosities, sizes and metallicities) for these new clusters are still poorly known or unknown.", "Our project aims at focusing on all OCs in the VVV survey area.", "In this first approach, we analyze a few known OCs with previously identified variables.", "We selected, on the one hand, two moderately young and extended OCs with a relatively high amount of catalogued variables in their fields [47] and a considerable amount of new possible variable member candidates.", "On the other hand, we also selected an intermediate-age more compact OC with only a few known variable stars and some newly discovered variable candidates detected in our study.", "Because the clusters are projected onto high density stellar regions of the Galactic bulge, we are at present mainly focused on searching probable NIR counterparts of the already catalogued variable stars as well as searching new variables only discovered using the VVV data.", "The subsequent goals will be related to the clusters' analysis, i.e., to comparing the advantages and disadvantages of analyzing extended and compact OCs and selecting close and faint OCs to warrant their non saturation and usefulness for a future analysis of their proper motions.", "In addition, with the help of our new data, we aim at improving some of these clusters' parameters that appeared to be uncertain.", "In this work we present the first results obtained for three OCs projected on the inner parts of the Galactic bulge, namely Antalova 1, ASCC 90 and ESO 393-15.", "We describe them in the next section together with the data collected from the VVV Survey.", "Section 3 details the variable stars found in the fields of these three OCs and their classification based on the obtained light curves.", "Data analysis and discussion as well as some future work is described in Section 4." ], [ "Data acquisition and selected targets", "The observations were made as part of the VVV Survey.", "The VIRCAM camera on the 4.1 m VISTA telescope is an array of 16 NIR detectors which produce a combined image of 11.6'$\\times $ 11.6' with a pixel size of 0.34” [13].", "The photometry and data reduction have been described in detail elsewhere [34], [14], [1].", "We briefly mention here that the individual VVV images were reduced, astrometrized and stacked by the Cambridge Astronomy Survey Unit (CASU) using the VISTA Data Flow System (VDFS) pipeline [18], [23], [22], and the photometry has been calibrated onto the VISTA filter system.", "The aperture photometry has been made by CASU on the individual processed images, and generated light curves were then analyzed for variability [14], [1].", "PSF photometry for each OC in the different available images was extracted and later cross-matched to create the CMDs.", "A brief description of the selected targets as well as a summary of the previous results for the fields under investigation is given below." ], [ "Antalova 1", "Antalova 1 (IAU designation C1725-315) is catalogued as a moderately young and metal-poor OC located in Scorpius at $\\alpha _{2000}$ = 17h 28m 57s, $\\delta _{2000}$ = -31$^{\\circ }$ 34' 48”; l = 355.86$^{\\circ }$ , b = +1.64$^{\\circ }$ (K13).", "It is projected on the inner bulge in the area named b345 in [34].", "Antalova 1 has been classified as IV2pn in the Trumpler system [3], i.e., as an OC with the fourth highest concentration degree, a medium range of brightness of its stars and a scanty population.", "[25] published a catalogue of astrophysical data for 520 Galactic OCs - among them Antalova 1 - which could be identified in their All-Sky Compiled Catalogue of 2.5 million stars (ASCC-2.5).", "By applying homogeneous procedures and algorithms, K05 determined angular sizes and fundamental astrophysical parameters for their cluster sample.", "For Antalova 1, they estimated an angular radius of 35', and obtained the following results: E(B-V) = 0.25, d = 850 pc and 316 Myr.", "It should be noted, however, that owing to the relatively bright limiting magnitude (V $\\sim $ 12.5) of the ASCC-2.5, K05's sample does not include faint and generally remote or highly obscured OCs.", "More recently, using a combination of uniform kinematic and NIR photometric data gathered in the all-sky catalogue PPMXL [33] and the 2MASS catalogue [35], K13 reported exact positions, apparent radii, proper motions, reddenings, distances and ages for more than 2800 mostly confirmed OCs.", "For Antalova 1, they provided the parameters listed in Table REF .", "[12] identified Antalova 1 in the Radial Velocity Experiment [36] and determined its mean cluster metallicity as [M/H] = -0.66 $\\pm $ 0.19 (Table REF ) from a cleaned working sample.", "A total of 43 variable stars have been catalogued by [47] in the cluster field.", "Figure: Color- magnitude and color-color diagrams of Antalova 1.", "Grey dots indicate all stars within the clusters' radii, while black dots represent stars that remain after the decontamination procedure.", "In the bottom diagrams, we subtracted also the clear bulge main-sequence background and fitted with a red line the isochrone corresponding to the parameters estimated by K13.", "The corresponding solar metallicity isochrone is represented by the purple line.", "Dark grey points represent the bulge red giant branch.", "We also superimposed on the diagrams the variable stars found in the cluster field.", "Different symbols and colors represent different types of variables." ], [ "ASCC 90", "ASCC 90, also known as KPR 90 [24], has so far received little attention.", "This cluster is also situated in Scorpius at equatorial coordinates $\\alpha _{2000}$ = 17h 39m 07s, $\\delta _{2000}$ = -34$^{\\circ }$ 48' 54” and Galactic coordinates l = 354.29$^{\\circ }$ and b = -1.91$^{\\circ }$ .", "It is projected into the region named b301 in [34].", "K05 obtained for ASCC 90 the following results: $E$ ($B-V$ ) = 0.30, d = 500 pc and age = 646 Myr, which are nearly identical to those recently provided by K13.", "Z12 reported a total of 62 variable stars in the field of ASCC 90.", "No metallicity has been determined for this cluster up to now.", "Figure: Same as Figure but for ASCC 90." ], [ "ESO 393-15", "ESO 393-15 (C1740-342) is an intermediate-age (1.4 Gyr) OC located in a rich stellar field in Scorpius at $\\alpha _{2000}$ = 17h 43m 35s, $\\delta _{2000}$ = -34$^{\\circ }$ 13' 38\" and l = 355.26$^{\\circ }$ and b = -2.40$^{\\circ }$ .", "This is a [39] class III3 OC [3], i.e.", "a moderately populated cluster with no noticeable concentration and a medium range in the star brightness.", "ESO 393-15 lies in the area b302 of [34].", "According to K13, this cluster is a small sized object with an angular diameter of 5.4', located at 2471 pc from the Sun and affected by $E$ ($B-V$ ) = 1.67.", "Z12 reported five variable stars in the cluster's field.", "No metallicity estimate has so far been obtained for this cluster.", "Figure: Same as Figure but for ESO 393-15.Table REF lists the coordinates of the clusters and their locations in the VVV area, while Table REF summarizes the fundamental parameters of the selected targets reported by K13, including their apparent radii, distances, reddenings, ages, proper motions, radial velocities and metallicities, when available.", "Table: Open cluster coordinatesTable: Open cluster known parameters taken from .The $(J-K_s, K_s)$ CMDs as well as $(H-K_s, J-H)$ color-color diagrams were built for each OC.", "As can be seen in Figures REF - REF , the clusters are projected onto high stellar density regions where the Galactic disk and bulge populations greatly contaminate the clusters' CMDs (grey dots).", "A decontamination procedure was required in order to detect and separate clusters from their background stars.", "With this in mind, we applied a statistical method consisting in defining a ring region surrounding each cluster located 5' away from its field and considered it as a comparison field and counted the stars lying within different intervals of magnitude-color $[\\delta K_s, \\delta (J-K_s)]$ in the CMD of each selected region.", "Finally, the number of stars counted for each interval $[\\delta K_s, \\delta (J-K_s)]$ in the CMD of the comparison fields was subtracted from the number of stars of the corresponding cluster regions.", "This procedure, applied successfully to other more compact clusters, failed to yield the expected results.", "It could be argued that at least in two of the three studied objects, the failure of the cleaning method may be due to the fact that these two OCs are not concentrated and are also very extended, so the separation between cluster and field stars is more confusing.", "In Figures REF - REF we marked with black dots the stars that remain after the decontamination procedure.", "As can be seen in the upper-right panel, a estrange X-shape appears as resembling two different stellar populations.", "We believe that the bulge main sequence seen in the color-magnitude diagrams (star concentration at $K_s >$ 16 approximately) is the responsable for such feature.", "By subtracting this stellar population, the color-magnitude and color-color diagrams of the two lower panels in Figues REF - REF are obtained.", "Dark grey points most probably represent the red giant branch of the background bulge older stellar population.", "Isochrone fittings from [9] were superimposed on the CMDs (red lines) based on cluster ages, distances, reddenings, and metallicities (when published) from K13.", "We compare with the reddening maps of [20] and [21] obtained with the VVV data, but since those maps were calculated for the bulge distances, the values represent an upper limit for the possible clusters reddenings.", "For each cluster field, we extracted and analyzed VVV data for objects that best matched the positions of previously reported variable stars (Z12), and also performed a blind variability search.", "Objects with putative light variations were selected using Stetson's variability statistics [38].", "The preselected candidates ($\\sim $ 10% of the objects) were then subjected to a frequency analysis.", "Signal detection was performed using the Generalized Lomb-Scargle [46] and the Phase Dispersion Minimization [37] algorithms.", "Phase-folded light curves with the resulting preliminary periods were then visually inspected to select the best solution and to reject spurious signals resulting from various systematics, e.g., rotating diffraction spikes of nearby saturated stars.", "We refined the periods and optimized the light-curve fit by an iterative procedure including the steps of outlier rejection, aperture optimization, determination of the optimal order of the best-fitting Fourier sum, and refining the period by a non-linear least squares method.", "The mean apparent $K_s$ magnitudes of the stars and the total amplitudes of the light curves were computed from the final Fourier solutions.", "At the end of the process, we identified a total of 157 variable stars with the number of independent observations ranging between 63 and 73.", "Figure: Distribution of the new variables in the period-amplitude plane.", "Different symbols and colors represent the various types of variables.The classification of periodic variable stars in this study was based on the periods and amplitudes, on visual appearance of the light curves and on the objects' color indices.", "Classifying NIR light curves is not a straightforward procedure because $K_s$ -band light-curves of pulsating variable stars have typically smaller amplitudes and are much more featureless than in the optical region, making it difficult to distinguish different types of objects on the basis of NIR data alone.", "The catalogue of Z12 reports variable stars found in the fields of the three selected clusters.", "However, by cross-matching their coordinates with the VVV source catalog, we have not found any variable star with NIR counterpart in any of the three fields.", "We attribute that to the fact that most of the catalogued variables are bright stars, so they are saturated on the VVV $K_s$ -band.", "Other variable stars might also have amplitudes too low to be detected in our procedure as possible variables.", "In the field of Antalova 1, we detected 58 new variable stars from which 7 are newly possible RR Lyrae type stars, 3 were classified as possible Cepheids and 18 as probable eclipsing binary (EB) systems.", "The remaining ones are reliably classified as undefined types of variable stars (denoted as MISC or marked with asteriks) or stars whose periods could not be well determined.", "Their coordinates, periods, amplitudes, $<K_s>$ weighted mean magnitudes from the Fourier fit, $J-K_s$ and $H-K_s$ colors computed from the first $K_s$ -band epoch, and classifications are given in Table .", "In the field of ASCC 90, we have detected 81 new variable stars.", "Out of these, 10 are found to be possible RR Lyrae, 2 suspected Cepheids and 1 possible $\\delta $ Scuti, while 43 are found to have similar characteristics to EB systems.", "Their basic properties are detailed in Table .", "In the field of ESO 393-15, we identified 18 new variable stars from which 2 are classified as RR Lyrae stars, 1 as possible $\\delta $ Scuti star, 1 as ellipsoidal and 8 as binary systems.", "We show in Table the corresponding parameters and classifications for all the variable stars found in this field.", "Figure REF shows a schematic view in the period-amplitude plane (Bailey diagram) of the whole sample of newly identified variables in the CMDs of the three studied OCs.", "Symbols are the same as in Fig.", "REF .", "Figures REF -REF show the light curves obtained for all the well classified variables, i.e., variables with Alias periods are not included.", "Figure: Light curves of the variable stars found in the field of Antalova 1Figure: Light curves of the variable stars found in the field of ASCC 90Figure: Continued.Figure: Light curves of the variable stars found in the field of ESO 393-15lcccccccccc List of variable stars found in Antalova 1 ID $\\alpha _{2000}$ $\\delta _{2000}$ Dist $^{a}$ Period Amp $^{b}$ $<K_s>$ $J-K_s$ $H-K_s$ Class Comments [hms] [dms] [arcmin] [days] [mag] [mag] [mag] [mag] (other types/periods) Continued.", "ID $\\alpha _{2000}$ $\\delta _{2000}$ Dist $^{a}$ Period Amp $^{b}$ $<K_s>$ $J-K_s$ $H-K_s$ Class Comments [hms] [dms] [arcmin] [days] [mag] [mag] [mag] [mag] (other types/periods) V1 17:28:26.70 -31:29:39.0 5.17 0.60804 0.530 14.24 0.82 0.24 MISC .", ".", ".", "V2 17:28:43.23 -31:30:50.9 10.97 6.82885 0.077 13.43 1.49 0.44 Bin Ellipsoidal V3 17:28:46.35 -31:25:00.9 9.79 0.33013 0.602 13.88 0.96 0.26 Bin .", ".", ".", "V4 17:28:47.57 -31:28:55.0 5.89 0.63813 0.187 14.58 1.22 0.38 RR* Bin V5 17:28:50.93 -31:28:55.2 5.88 1.48478 0.118 14.28 1.07 0.33 Bin RR; P=0.74298 d V6 17:28:51.09 -31:28:37.2 6.18 0.48991 0.305 13.92 0.90 0.32 RR .", ".", ".", "V7 17:28:53.84 -31:33:35.8 1.20 5.40317 0.148 14.18 0.81 0.27 MISC .", ".", ".", "V8 17:28:53.87 -31:31:45.9 3.04 5.44742 0.084 13.52 1.08 0.26 MISC Spheroidal* V9 17:28:54.68 -31:23:55.8 10.87 21.73366 0.120 14.06 1.48 0.46 MISC .", ".", ".", "V10 17:28:58.37 -31:22:58.6 11.82 0.53609 0.352 14.89 0.91 0.23 RR Bin V11 17:29:00.89 -31:21:39.0 13.15 7.77096 0.091 13.77 1.60 0.48 Cep* MISC V12 17:29:01.74 -31:26:39.5 8.14 0.24126 0.313 15.21 0.85 0.57 Bin .", ".", ".", "V13 17:29:06.70 -31:31:22.3 3.43 0.22554 0.302 15.39 0.93 0.36 Bin .", ".", ".", "V14 17:29:11.64 -31:33:26.8 1.38 0.41652 0.164 14.04 0.75 0.27 MISC .", ".", ".", "V15 17:29:12.51 -31:23:24.4 11.40 0.16319 0.100 13.21 0.67 0.25 Bin .", ".", ".", "V16 17:29:13.03 -31:30:21.8 4.44 23.20854 0.091 13.73 1.60 0.50 Bin .", ".", ".", "V17 17:29:14.26 -31:27:32.2 7.27 1.57594 0.189 14.37 0.98 0.36 Bin .", ".", ".", "V18 17:29:14.47 -31:35:31.0 0.77 0.48816 0.252 14.47 1.00 0.35 RR .", ".", ".", "V19 17:29:16.09 -31:27:11.2 7.62 38.92414 0.081 12.39 1.58 0.51 Bin .", ".", ".", "V20 17:29:16.24 -31:17:19.1 17.48 0.21118 0.282 14.35 0.74 0.27 RR .", ".", ".", "V21 17:29:16.38 -31:26:55.8 7.88 6.58358 0.102 12.99 1.19 0.37 Cep* .", ".", ".", "V22 17:29:16.87 -31:16:55.7 17.87 0.52663 0.252 14.65 0.92 0.28 MISC .", ".", ".", "V23 17:29:17.45 -31:32:09.3 2.67 13.66544 0.080 12.85 1.40 0.44 MISC .", ".", ".", "V24 17:29:18.23 -31:25:38.5 9.17 2.07537 0.415 14.33 1.17 0.38 Bin .", ".", ".", "V25 17:29:18.29 -31:35:42.0 0.97 0.60952 0.204 RR 14.41 0.97 0.29 Bin V26 17:29:18.75 -31:21:26.7 13.36 0.17437 0.354 13.87 0.69 0.19 MISC Bin; P=0.34874 V27 17:29:20.22 -31:35:36.5 0.90 1.06129 0.603 14.96 1.07 0.33 Bin P=0.53066 V28 17:29:21.23 -31:21:43.8 13.08 0.18109 0.353 13.36 0.59 0.16 MISC Bin; P=0.36216 V29 17:29:22.00 -31:31:07.1 3.71 0.55032 0.398 14.97 1.27 0.38 MISC .", ".", ".", "V30 17:29:22.17 -31:35:01.1 0.47 5.70666 0.113 13.52 1.32 0.38 Bin .", ".", ".", "V31 17:29:24.16 -31:28:27.7 6.35 0.53211 0.317 14.59 0.95 0.31 RR/Bin .", ".", ".", "V32 17:29:26.25 -31:34:25.8 0.61 0.83882 0.079 14.17 1.41 0.42 MISC .", ".", ".", "V33 17:29:29.30 -31:37:16.6 2.53 0.48178 0.244 14.12 1.08 0.34 MISC .", ".", ".", "V34 17:29:29.90 -31:30:52.7 3.96 0.28913 0.222 15.31 0.79 0.27 Bin MISC V35 17:29:31.07 -31:25:01.1 9.80 0.51328 0.480 13.82 0.80 0.26 Bin .", ".", ".", "V36 17:29:31.57 -31:36:04.7 1.40 27.07240 0.090 14.21 1.40 0.40 Bin MISC V37 17:29:34.09 -31:33:05.4 1.82 2.40947 0.400 14.95 1.26 0.35 Bin .", ".", ".", "V38 17:29:35.26 -31:34:58.1 0.66 9.85270 0.119 13.11 1.87 0.61 Cep* .", ".", ".", "V39 17:28:29.67 -31:29:05.4 5.73 .", ".", ".", "0.236 13.93 1.86 0.61 MISC .", ".", ".", "V40 17:28:32.97 -31:30:24.2 4.41 .", ".", ".", "0.305 14.43 1.80 0.63 MISC .", ".", ".", "V41 17:28:53.86 -31:28:31.8 6.27 .", ".", ".", "0.227 14.90 1.40 0.43 MISC .", ".", ".", "V42 17:28:58.92 -31:21:40.4 13.13 .", ".", ".", "0.162 12.74 0.69 0.34 MISC .", ".", ".", "V43 17:29:01.23 -31:34:01.9 0.77 .", ".", ".", "0.306 14.19 1.23 0.44 MISC .", ".", ".", "V44 17:29:02.51 -31:33:52.3 0.93 .", ".", ".", "0.298 12.99 1.61 0.57 MISC .", ".", ".", "V45 17:29:03.44 -31:20:31.7 14.27 .", ".", ".", "0.110 12.99 1.46 0.50 MISC .", ".", ".", "V46 17:29:03.67 -31:33:37.5 1.18 .", ".", ".", "0.387 14.72 1.25 0.50 MISC .", ".", ".", "V47 17:29:05.37 -31:30:22.0 4.44 .", ".", ".", "0.303 14.34 1.73 0.59 MISC .", ".", ".", "V48 17:29:06.48 -31:31:57.1 2.85 .", ".", ".", "0.278 14.28 1.08 0.31 MISC .", ".", ".", "V49 17:29:09.50 -31:28:01.6 6.78 .", ".", ".", "0.346 14.67 1.43 0.59 MISC .", ".", ".", "V50 17:29:11.58 -31:34:18.9 0.54 .", ".", ".", "0.561 13.77 2.06 0.71 MISC .", ".", ".", "V51 17:29:13.84 -31:21:55.6 12.88 .", ".", ".", "0.257 13.76 1.36 0.39 MISC .", ".", ".", "V52 17:29:2.84 -31:34:28.6 0.57 .", ".", ".", "0.132 14.44 1.27 0.45 MISC .", ".", ".", "V53 17:29:25.01 -31:31:54.8 2.92 .", ".", ".", "0.230 14.45 1.29 0.41 MISC .", ".", ".", "V54 17:29:25.11 -31:34:52.0 0.47 .", ".", ".", "0.443 15.01 1.65 .", ".", ".", "MISC .", ".", ".", "V55 17:29:29.68 -31:21:45.2 13.06 .", ".", ".", "0.205 14.48 .", ".", ".", ".", ".", ".", "MISC .", ".", ".", "V56 17:29:31.32 -31:22:33.1 12.26 .", ".", ".", "0.219 13.95 0.85 0.41 MISC .", ".", ".", "V57 17:29:34.02 -31:38:55.2 4.17 .", ".", ".", "0.130 14.14 1.08 0.36 MISC .", ".", ".", "V58 17:29:35.60 -31:31:19.2 3.54 .", ".", ".", "0.430 14.24 0.96 0.27 MISC .", ".", ".", "$^{a}$ Distance to the cluster center.", "$^{b} K_s$ total amplitude.", "lcccccccccc List of variable stars found in ASCC 90 ID $\\alpha _{2000}$ $\\delta _{2000}$ Dist $^{a}$ Period Amp $^{b}$ <$K_s$ > $J-K_s$ $H-K_s$ Class Comments [hms] [dms] [arcmin] [days] [mag] [mag] [mag] [mag] (other types/periods) Continued.", "ID $\\alpha _{2000}$ $\\delta _{2000}$ Dist $^{a}$ Period Amp $^{b}$ <$K_s$ > $J-K_s$ $H-K_s$ Class Comments [hms] [dms] [arcmin] [days] [mag] [mag] [mag] [mag] (other types/periods) V1 17:37:56.45 -34:49:51.1 17.71 0.17500 0.273 15.02 0.75 0.17 Bin $\\delta $ Sct V2 17:37:57.42 -34:50:46.3 17.55 0.60235 0.244 12.21 0.59 0.18 RR/Bin .", ".", ".", "V3 17:38:00.24 -34:43:10.0 17.69 0.41059 0.185 13.65 1.02 0.28 RR/Bin .", ".", ".", "V4 17:38:04.41 -34:49:15.8 15.70 69.23252 0.136 13.91 1.39 0.43 MISC .", ".", ".", "V5 17:38:06.23 -34:44:11.2 15.96 0.13602 0.071 14.64 0.81 0.22 MISC .", ".", ".", "V6 17:38:09.52 -34:41:55.3 16.02 0.13791 0.303 15.45 1.17 0.28 MISC .", ".", ".", "V7 17:38:12.52 -34:50:02.1 13.72 0.31308 0.260 15.00 0.69 0.21 MISC Bin; P=0.62616 V8 17:38:16.80 -34:41:22.3 14.68 0.16621 0.199 15.05 1.40 0.40 MISC .", ".", ".", "V9 17:38:16.81 -34:40:58.5 7.07 0.14247 0.055 13.75 1.65 0.48 MISC .", ".", ".", "V10 17:38:23.72 -34:48:19.2 10.89 0.47946 0.445 15.43 0.72 0.21 MISC Bin; P=0.95893 V11 17:38:27.14 -34:35:38.8 16.62 7.06969 0.088 13.68 2.00 0.61 Cep* .", ".", ".", "V12 17:38:28.74 -34:40:22.0 12.86 5.26004 0.083 14.10 1.35 0.36 MISC .", ".", ".", "V13 17:38:30.78 -34:44:38.8 10.05 0.44373 0.376 14.90 1.00 0.24 MISC .", ".", ".", "V14 17:38:32.66 -34:35:32.7 15.91 9.52668 0.048 12.45 1.78 0.54 MISC Bin; P=19.04586 V15 17:38:34.82 -34:33:26.9 17.44 0.56796 0.567 14.63 1.02 0.32 Bin .", ".", ".", "V16 17:38:35.83 -34:38:45.9 12.81 0.14247 0.148 14.15 1.04 0.32 RR $\\delta $ Sct* V17 17:38:41.52 -34:45:55.7 7.07 4.73548 0.150 14.88 1.35 0.36 Bin .", ".", ".", "V18 17:38:41.67 -34:33:26.0 16.73 0.31084 0.608 15.72 1.20 0.32 Bin .", ".", ".", "V19 17:38:42.87 -34:49:42.6 16.73 0.14247 0.290 14.52 1.14 0.33 MISC .", ".", ".", "V20 17:38:43.33 -34:36:14.5 14.00 0.19380 0.080 13.96 1.03 0.29 MISC Bin; P=0.38759 V21 17:38:43.55 -34:46:41.7 6.31 0.17915 0.295 15.36 0.85 0.22 MISC .", ".", ".", "V22 17:38:46.21 -34:32:11.7 17.51 33.44160 0.110 14.37 1.39 0.36 MISC .", ".", ".", "V23 17:38:49.38 -34:44:55.2 5.97 0.56232 0.302 14.88 1.13 0.36 RR* .", ".", ".", "V24 17:38:49.64 -34:44:42.2 6.07 0.33683 0.109 12.45 1.46 0.42 Bin P=0.16841 V25 17:38:51.49 -34:43:57.4 6.31 0.75759 0.648 14.87 0.30 1.08 Bin .", ".", ".", "V26 17:38:52.06 -34:46:41.9 11.29 0.37722 0.220 14.00 0.95 0.50 Bin P=0.18861 V27 17:38:52.21 -34:34:44.1 14.65 0.55657 0.340 14.73 0.95 0.28 RR* Bin V28 17:38:54.31 -34:49:44.8 3.33 0.40672 0.390 15.85 0.68 0.17 Bin .", ".", ".", "V29 17:38:57.99 -34:49:25.8 2.36 0.14247 0.049 13.46 1.34 0.36 Bin* .", ".", ".", "V30 17:38:59.59 -34:34:20.5 14.68 0.16144 0.348 15.27 .", ".", ".", "0.26 Bin .", ".", ".", "V31 17:39:01.16 -34:33:12.2 15.77 0.23383 0.322 15.26 0.78 0.22 RR* .", ".", ".", "V32 17:39:01.84 -34:49:41.6 15.68 0.23723 0.474 15.73 0.84 0.24 MISC .", ".", ".", "V33 17:39:03.03 -34:34:15.3 14.68 0.52639 0.093 13.14 0.81 0.24 Bin .", ".", ".", "V34 17:39:06.21 -34:41:41.7 7.21 25.96282 0.118 13.95 1.61 0.47 MISC .", ".", ".", "V35 17:39:10.17 -34:31:43.1 17.20 7.36927 0.092 13.67 1.38 0.40 Bin .", ".", ".", "V36 17:39:12.84 -34:44:37.6 4.50 8.91203 0.096 14.86 1.50 0.41 Bin .", ".", ".", "V37 17:39:13.47 -34:40:25.7 8.61 0.19258 0.430 13.13 0.89 0.26 Bin .", ".", ".", "V38 17:39:14.85 -34:41:43.9 7.42 1.19758 0.272 12.19 0.33 0.03 Bin P=0.59881 V39 17:39:15.71 -34:31:38.1 17.40 0.12466 0.553 13.62 0.91 0.27 Bin* .", ".", ".", "V40 17:39:21.22 -34:40:02.3 9.53 1.11931 0.086 14.10 1.23 0.39 Bin MISC; P=0.55964 V41 17:39:21.62 -34:44:08.5 5.97 0.84096 0.189 13.63 0.70 0.24 Bin .", ".", ".", "V42 17:39:22.93 -34:43:02.0 7.06 0.20557 0.172 14.95 0.92 0.25 RR* .", ".", ".", "V43 17:39:24.41 -34:36:38.5 12.99 0.41902 0.270 14.87 1.02 0.31 Bin .", ".", ".", "V44 17:39:29.49 -34:47:04.7 5.86 0.59093 0.192 14.62 0.93 0.29 RR Bin V45 17:39:31.18 -34:40:08.5 10.61 0.45602 0.362 14.88 1.04 0.32 RR* .", ".", ".", "V46 17:39:31.58 -34:43:44.3 7.99 0.82798 0.227 13.03 0.59 0.13 Bin P=0.41489 d V47 17:39:33.43 -34:48:00.6 6.62 0.46938 0.334 15.08 0.97 0.34 RR .", ".", ".", "V48 17:39:34.52 -34:37:47.0 13.05 0.28486 0.235 14.66 1.64 0.54 Bin .", ".", ".", "V49 17:39:39.15 -34:39:26.3 12.38 31.21711 0.098 14.09 1.50 0.42 MISC .", ".", ".", "V50 17:39:39.48 -34:35:06.6 15.98 0.21359 0.301 14.97 0.86 0.27 Bin .", ".", ".", "V51 17:39:43.53 -34:43:29.1 10.57 0.54353 0.551 15.16 0.83 0.26 Bin RR V52 17:39:47.16 -34:40:15.5 13.21 11.67808 0.144 12.98 1.37 0.37 MISC Bin; P=23.34019 V53 17:39:47.51 -34:43:31.5 11.42 0.21146 0.350 16.07 1.24 0.36 Bin .", ".", ".", "V54 17:39:47.64 -34:48:30.7 10.12 0.66579 0.233 15.34 0.81 0.24 Bin .", ".", ".", "V55 17:39:48.19 -34:46:30.2 10.53 0.14445 0.627 14.77 0.86 0.22 MISC Bin; P=0.28891 V56 17:39:48.47 -34:38:39.6 14.53 0.18105 0.233 14.52 14.52 0.34 MISC Bin; P=0.36210 V57 17:39:48.60 -34:39:40.6 13.86 0.14247 0.288 15.20 .", ".", ".", ".", ".", ".", "MISC .", ".", ".", "V58 17:39:49.25 -34:38:31.3 14.77 6.62427 0.116 13.97 1.63 0.47 Cep* .", ".", ".", "V59 17:39:51.82 -34:49:31.2 11.17 0.21663 0.390 14.38 0.80 0.24 MISC .", ".", ".", "V60 17:39:57.00 -34:45:38.3 8.61 5.32970 0.282 13.94 1.27 0.36 Bin .", ".", ".", "V61 17:39:57.11 -34:44:32.7 13.21 30.33833 0.058 13.24 3.29 2.50 MISC .", ".", ".", "V62 17:39:58.25 -34:38:45.5 16.30 0.18358 0.217 14.61 1.10 0.37 MISC .", ".", ".", "V63 17:39:59.22 -34:47:07.6 13.22 0.61524 0.163 14.83 0.97 0.28 Bin .", ".", ".", "V64 17:40:01.28 -34:40:45.6 15.78 0.31585 0.271 14.78 0.82 0.25 Bin .", ".", ".", "V65 17:40:01.57 -34:36:49.8 18.18 6.85206 0.114 13.62 1.32 0.40 Bin .", ".", ".", "V66 17:40:04.50 -34:45:33.9 14.70 76.75488 0.084 14.02 1.26 0.35 Bin .", ".", ".", "V67 17:40:04.78 -34:39:55.4 16.96 0.44270 0.161 13.96 0.76 0.25 Bin P=0.22135 V68 17:40:04.89 -34:45:57.9 14.72 0.18533 0.078 13.13 0.91 0.25 Bin .", ".", ".", "V69 17:40:05.48 -34:45:31.4 14.96 0.27276 0.377 14.51 0.61 0.16 MISC .", ".", ".", "V70 17:40:07.25 -34:41:46.2 16.62 0.37462 0.106 13.63 0.72 0.24 Bin .", ".", ".", "V71 17:40:07.59 -34:40:28.1 17.21 0.15041 0.433 15.38 0.93 0.31 $\\delta $ Sct .", ".", ".", "V72 17:40:09.37 -34:43:47.9 16.36 2.91559 0.109 13.30 1.14 0.33 MISC Bin; P=5.83089 V73 17:40:13.58 -34:43:44.7 17.38 5.83096 0.183 14.84 1.44 0.46 Bin P=22.45679 V74 17:38:22.24 -34:21:45.7 11.01 .", ".", ".", "0.170 14.62 1.11 0.28 MISC .", ".", ".", "V75 17:38:22.95 -34:46:38.9 11.29 .", ".", ".", "0.066 13.51 1.19 0.31 MISC .", ".", ".", "V76 17:38:45.39 -34:27:47.4 18.22 .", ".", ".", "0.594 15.47 1.96 0.58 MISC .", ".", ".", "V77 17:38:55.04 -34:30:07.7 14.04 .", ".", ".", "0.315 15.01 1.39 0.36 MISC .", ".", ".", "V78 17:38:57.45 -34:29:18.0 14.72 .", ".", ".", "0.238 15.16 .", ".", ".", "0.56 MISC .", ".", ".", "V79 17:39:22.98 -34:29:16.7 16.41 .", ".", ".", "0.250 14.19 1.29 .", ".", ".", "MISC .", ".", ".", "V80 17:39:26.48 -34:40:26.5 7.84 .", ".", ".", "0.464 15.03 1.58 0.41 MISC .", ".", ".", "V81 17:39:28.21 -34 43:05.3 14.71 .", ".", ".", "0.080 13.52 1.33 0.40 MISC .", ".", ".", "$^{a}$ Distance to the cluster center.", "$^{b} K_s$ total amplitude.", "lcccccccccc List of variable stars found in ESO 393-15 ID $\\alpha _{2000}$ $\\delta _{2000}$ Dist $^{a}$ Period Amp $^{b}$ <$K_s$ > $J-K_s$ $H-K_s$ Class Comments [hms] [dms] [arcmin] [days] [mag] [mag] [mag] [mag] (other types/periods) V1 17:43:14.58 -34:13:19.2 5.19 0.58904 0.228 14.47 0.79 0.26 RR* .", ".", ".", "V2 17:43:19.36 -34:10:48.6 4.88 0.18203 0.221 14.89 0.70 0.20 Bin .", ".", ".", "V3 17:43:21.49 -34:14:23.1 3.53 0.16302 0.395 15.13 0.82 0.24 $\\delta $ Sct .", ".", ".", "V4 17:43:22.09 -34:12:10.9 3.61 9.26665 0.108 13.67 0.95 0.26 Bin* .", ".", ".", "V5 17:43:22.45 -34:11:42.9 3.74 12.98331 0.075 13.63 1.20 0.34 MISC .", ".", ".", "V6 17:43:24.76 -34:13:08.6 2.68 3.47711 0.058 14.64 1.08 0.29 MISC .", ".", ".", "V7 17:43:25.80 -34:16:42.2 3.88 0.61680 0.199 14.55 0.77 0.24 Bin/RR .", ".", ".", "V8 17:43:25.81 -34:13:31.4 2.38 0.39748 0.267 14.97 0.73 0.22 Bin .", ".", ".", "V9 17:43:31.30 -34:10:07.5 3.65 35.12639 0.072 14.50 1.19 0.30 Ellipsoidal* .", ".", ".", "V10 17:43:33.39 -34:13:43.8 0.49 0.52252 0.333 12.91 0.60 0.22 MISC Bin; P=1.04504 V11 17:43:36.01 -34:15:55.0 2.29 0.16241 0.466 15.73 1.66 0.31 Bin .", ".", ".", "V12 17:43:36.62 -34:14:35.0 1.01 0.13454 0.367 14.95 0.89 0.25 RR Bin; P=0.26908 V13 17:43:36.74 -34:17:06.4 3.49 9.95217 0.133 13.52 1.08 0.31 Bin P=4.97596 V14 17:43:39.01 -34:12:08.9 1.75 14.83956 0.084 13.65 0.99 0.31 MISC .", ".", ".", "V15 17:43:40.77 -34:16:20.5 3.03 0.59805 0.254 15.24 0.68 0.19 Bin .", ".", ".", "V16 17:43:44.46 -34:11:08.7 3.38 19.94113 0.103 14.60 1.07 0.32 MISC .", ".", ".", "V17 17:43:29.27 -34:15:50.0 2.67 .", ".", ".", "0.261 13.89 1.19 0.39 MISC .", ".", ".", "V18 17:43:33.04 -34:09:41.2 3.99 .", ".", ".", "0.227 14.59 1.22 0.49 MISC .", ".", ".", "$^{a}$ Distance to the cluster center.", "$^{b} K_s$ total amplitude." ], [ "Discussion", "Antalova 1 and ASCC 90 are moderately young OCs, while ESO 393-15 is an intermediate-age OC (Table REF ).", "Therefore, it is quite clear that the RR Lyrae stars found in their respective fields must be background stars projected in the direction of the clusters.", "However, since our project aims at searching variable stars as a whole and in the Galactic OC fields in particular, the classification of such new variables as RR Lyrae stars enlarges not only the sample of RR Lyrae variables currently known but also the background stellar properties to be studied from them." ], [ "Cepheids analysis", "We have detected a total of 5 Cepheid candidates in the fields of Antalova 1 and ASCC 90.", "In order to determine their probability of being cluster members, we calculated their distances and interstellar extinctions by using the period-luminosity (PL) relations in the $H$ and $K_s$ passbands.", "To determine the type of each Cepheid, we used the PL relations for classical and type II Cepheids.", "Assuming at first they are classical Cepheids, we used the following PL relations of [15] adapted to the VVV Survey passbands: $M_H = -3.228\\,[\\pm 0.06] \\times (\\log P - 1) - 5.617\\,[\\pm 0.048],$ $M_{K_s} = -3.269\\,[\\pm 0.05] \\times (\\log P - 1) - 5.663\\,[\\pm 0.048],$ where $M_H$ and $M_{K_s}$ are the absolute magnitudes in the $H$ and $K_s$ bands, respectively.", "The color excesses are calculated from: $E(H-K_s) = < H-K_s> - (M_H-M_{K_s}),$ where $<H-K_s>$ is the mean $(H-K_s)$ color index as defined in [15].", "The total absorption in the $Ks$ -band and the individual distances R for both Cepheid types were derived from the following relations given by [15]: $A(K_s) = 1.634 \\times E(H-K_s),$ $\\log R = 1 + 0.2 \\times (<K_s> - A(K_s)-M_{K_s}),$ where the mean $<K_s>$ magnitudes of the stars were computed from the optimized Fourier fits of the light curves.", "Secondly, assuming the stars are type II Cepheids, the following PL relations from [28] were employed: $M_H = -2.340\\,[\\pm 0.05] \\times (\\log P - 1.2) + 14.760\\,[\\pm 0.017],$ $M_{K_s} = -2.410\\,[\\pm 0.05] \\times (\\log P - 1.2) + 14.617\\,[\\pm 0.015].$ The results obtained after an analysis of the possible types of Cepheids are shown in Table REF .", "Since types I and II Cepheids have similar periods and NIR light curves, it is difficult to distinguish their right types in a first approach.", "Taking into account the $K_s$ reddening map of the VVV bulge obtained by [20] and [21], a detailed analysis was recently carried out by [16] examining the coherence in the results obtained for the extinction and distances.", "We analyzed our results in a similar way and conclude that the possible Cepheids are most likely to be type II from the background fields.", "These are old population stars lying behind the Galactic bulge.", "Table: Observed and derived parameters for the 5 Cepheid candidates of the two possible types" ], [ "Eclipsing binaries analysis", "From the whole sample of binary system candidates, we selected the most probable ones and determined the physical parameters of the eclipsing components.", "We first used the Wilson & Devinney (WD) code [45], which operates in two steps while fitting the light curves, i.e., the LC (a subjective iteration) and the DC (an objective iteration) processes.", "The LC procedure is based on parameters previously determined from theory or observation [42], [41], [43], [44].", "On the other hand, the DC is the differential calculus which aims at better determining the geometrical and physical parameters of the systems by reducing possible associated errors.", "Light curves were first analyzed by the LC procedure using known parameters, such as the period.", "The output file was then used as input for the DC procedure.", "The effectiveness of the WD code is not particularly favorable since it is time consuming so, as we intend to study many objects, we searched for another method.", "Currently, there are several graphical user interface programs that can be used for a scientific study of these objects.", "Among them, we tested the PHOEBE (PHysics Of Eclipsing BinariEs) program.", "This is a tool for modeling eclipsing binary stars based on photometric and spectroscopic (radial velocity) data, which is also based on the WD code.", "PHOEBE can determine the parameters associated with the physical and geometrical conditions of the system, like in the WD code.It can also deal with the parameters of the binary components to which we have access once the values associated with the obtained light-curves were determined.", "We used both WD and PHOEBE codes for our analysis and probed different classification modes.", "Taking into account the shape of the light curves, we conclude that the systems involved are detached, semi-detached or close contact binary systems.", "The best fittings of the corresponding light curves are shown in Figures REF - REF .", "Table REF shows the resulting parameters that could be obtained from the current analysis.", "$M_2/M_1$ represents the mass ratio between the components.", "We would like to point out that, in general terms, the studied systems exhibit orbits that are almost circular, their eccentricities being about $5\\times 10^{-4}$ .", "The stars involved seem to have solar or higher than solar surface temperatures.", "Table: Model and fixed parameters for the selected eclipsing binary stars.Figure: Light curves of the best binaries found in the field of Antalova 1.", "Solid lines represent the best fittings of the observed light curves for the eclipsing binary stars.", "Diamonds stand for the observed <Ks><Ks> magnitudes taken from the VVV Survey.Figure: Light curves of the best binaries found in the field of ASCC 90.", "Lines and symbols represent the same as in Figure .Figure: Light curves of the best binaries found in the field of ESO 393-15.", "Lines and symbols represent the same as in Figure .Since the cross-match is purely positional, most of the newly discovered variable stars are not cluster members but part of the bulge population projected onto the clusters' fields, such as RR Lyrae stars and the Cepheids found.", "According to the CMDs, we can obtain possible candidates for the clusters' members.", "In ASCC 90 we found 43 possible EBs.", "However, only 8 of these binaries might be cluster members.", "In Antalova 1, from 18 new found EBs, only 6 candidates could be cluster members and in ESO 393-15, from 18 BEs found in its region, only 3 of them might be cluster member candidates.", "The rest of the EBs found in ESO 393-15 are probably background objects because they appear to have much redder colors than the corresponding cluster isochrone.", "Besides the detected variability and deep photometry, the good spatial resolution of the VVV Survey also allows for proper motion studies to be carried out in the future when a longer baseline becomes available.", "A further spectroscopic study is planned to obtain radial velocities which will help us to confirm or deny the physical association of the variables found to the corresponding clusters.", "The search in both cases, extended and compact OCs, present benefits as well as disadvantages.", "While the probabilities of finding variable stars in extended clusters are higher, their decontamination processes are more complicated, and the probabilities of containing field stars existing in the cluster regions are also higher.", "Our project intends to find the possible variables belonging to the host clusters, as well as to complete the search in the NIR bands and characterize the field variable stars.", "In our view, determining cluster membership is a difficult issue, contamination by background variable stars is a major problem and we believe that most of the variables found in the fields of the OCs studied here appear not to be associated with such OCs.", "A definite way to secure membership would be the use of kinematic data.", "Unfortunately, the radial velocities of these three clusters are largely unknown and quite telescope time consuming to obtain.", "However, their proper motions may be measured instead using the same VVV data.", "The accuracy of the proper motions measured by VVV astrometry has been estimated to be $\\sim $ 2 mas/yr [27].", "Such precision is not enough to secure cluster membership.", "We need to increase the time baseline for some more years in order to obtain good and reliable (at least statistically) proper motions for the OCs of our study." ], [ "Conclusions", "We have presented a new search for variable stars in the fields of three studied OCs using the NIR database of the VVV Survey.", "We have defined the procedures for variable stars search in VVV OCs.", "In this final section, a brief summary of our findings is presented.", "We found 5 new Cepheids in the cluster fields, none of which appear to be cluster members.", "A large number of background RR Lyrae stars in the fields of the three studied clusters have also been found.", "We identified a large number of EBs in the cluster fields.", "A total of 17, 42, and 8 EBs were found in the fields of Antalova 1, ASCC 90 and ESO 393-15, respectively.", "Based on NIR CMDs, we recognized only 8, 6, and 6 of such EBs as probable members of Antalova 1, ASCC 90 and ESO 393-15, respectively.", "We obtained fundamental parameters for some selected EBs.", "They were classified as detached, semi-detached and/or double-contact binaries, with low eccentricities and high inclinations.", "Future spectroscopic follow-up of some of these binary systems may help to confirm cluster membership and also to obtain heliocentric distances to the clusters.", "Our experience shows that for clusters projected onto highly reddened and obscured regions of the bulge, it is very hard to decontaminate the CMDs.", "Another major problem is to discriminate cluster members from field stars.", "Lastly, it has not been possible to determine the variable type of some objects.", "That is mainly due to their lack of periodicity and to our inability to phase them using the current data.", "Therefore, they remain unclassified." ], [ "Acknowledgements", "We gratefully acknowledge the use of data from the ESO Public Survey program 179.B-2002 taken with the VISTA 4.1 m telescope and data products from the Cambridge Astronomical Survey Unit.", "Support for TP, DM, ID and JAG is provided by the Ministry of Economy, Development, and Tourism's Millennium Science Initiative through grant IC120009, awarded to the Millennium Institute of Astrophysics, MAS.", "TP, JJC and LVG acknowledge financial support from the Argentinian institutions FONCYT, CONICET and SECYT (Universidad Nacional de Córdoba).", "DM is also supported by the Center for Astrophysics and Associated Technologies PFB-06, and Fondecyt Project No.", "1130196.", "JAG acknowledges support from the FIC-R Fund, allocated to the project 30321072, by CONICYT's PCI program through grant DPI20140066 and from Fondecyt Iniciación 11150916.", "This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France; also the SAO/NASA Astrophysics data (ADS)." ] ]
1606.05028
[ [ "The multiplex dependency structure of financial markets" ], [ "Abstract We propose here a multiplex network approach to investigate simultaneously different types of dependency in complex data sets.", "In particular, we consider multiplex networks made of four layers corresponding respectively to linear, non-linear, tail, and partial correlations among a set of financial time series.", "We construct the sparse graph on each layer using a standard network filtering procedure, and we then analyse the structural properties of the obtained multiplex networks.", "The study of the time evolution of the multiplex constructed from financial data uncovers important changes in intrinsically multiplex properties of the network, and such changes are associated with periods of financial stress.", "We observe that some features are unique to the multiplex structure and would not be visible otherwise by the separate analysis of the single-layer networks corresponding to each dependency measure." ], [ "Introduction", "In the last decade network theory has been extensively applied to the analysis of financial markets.", "Financial markets and complex systems in general are comprised of many interacting elements, and understanding their dependency structure and its evolution with time is essential to capture the collective behaviour of these systems, to identify the emergence of critical states, and to mitigate systemic risk arising from the simultaneous movement of several factors.", "Network filtering is a powerful instrument to associate a sparse network to a high-dimensional dependency measure and the analysis of the structure of such a network can uncover important insights on the collective properties of the underlying system.", "Following the line first traced by the preliminary work of Mantegna [1], a set of time series associated with financial asset values is mapped into a sparse complex network whose nodes are the assets and whose weighted links represent the dependencies between the corresponding time series.", "Filtering correlation matrices has been proven to be very useful for the study and characterization of the underlying interdependency structure of complex datasets [1], [2], [3], [4], [5].", "Indeed, sparsity allows to filter out noise, and sparse networks can then be analyzed by using standard tools and indicators proposed in complex networks theory to investigate the multivariate properties of the dataset [6], [7].", "Further, the filtered network can be used as a sparse inference structure to construct meaningful and computationally efficient predictive models [7], [8].", "Complex systems are often characterized by non-linear forms of dependency between the variables, which are hard to capture with a single measure and are hard to map into a single filtered network.", "A multiplex network approach, which considers the multi-layer structure of a system in a consistent way, is thus a natural and powerful way to take into account simultaneously several distinct kinds of dependency.", "Dependencies among financial time series can be described by means of different measures, each one having its own advantages and drawbacks, and this has lead to the study of different type of networks, namely correlation networks, causality networks, etc.", "The most common approach uses Pearson correlation coefficient to define the weight of a link, because this is a quantity that can be easily and quickly computed.", "However, the Pearson coefficient measures the linear correlation between two time series [9], and this is quite a severe limitation, since nonlinearity has been shown to be an important feature of financial markets [10].", "Other measures can provide equally informative pictures on assets relationships.", "For instance, the Kendall correlation coefficient takes into account monotonic non-linearity [11] [12], while others measures, such as the tail dependence, quantify dependence in extreme events.", "It is therefore important to describe quantitatively how these alternative descriptions relate but also differ from the Pearson correlation coefficient, and also to monitor how these differences change in time, if at all.", "In this work we exploit the power of a multiplex approach to analyse simultaneously different kinds of dependencies among financial time series.", "The theory of multiplex network is a recently introduced framework that allows to describe real-world complex systems consisting of units connected by relationships of different kinds as networks with many layers, and where the links at each layer represent a different type of interaction between the same set of nodes [13], [14].", "A multiplex network approach, combined with network filtering, is the ideal framework to investigate the interplay between linear, non-linear and tail dependencies, as it is specifically designed to take into account the peculiarity of the patterns of connections at each of the layers, but also to describe the intricate relations between the different layers [15].", "The idea of analyzing multiple layers of interaction was introduced initially in the context of social networks, within the theory of frame analysis [16].", "The importance of considering multiple types of human interactions has been more recently demonstrated in different social networks, from terrorist organizations [14] to online communities; in all these cases, multilayer analyses unveil a rich topological structure [17], outperforming single-layer analyses in terms of network modeling and prediction as well [18][19].", "In particular multilayer community detection in social networks has been shown to be more effective than single-layer approaches [20]; similar results have been reported for community detection on the World Wide Web [21][22] and citation networks [23].", "For instance, in the context of electrical power grids, multilayer analyses have provided important insight into the role of synchronisation in triggering cascading failures [24][25].", "Similarly, the analyses on transport networks have highlighted the importance of a multilayer approach to optimise the system against nodes failures, such as flights cancellation [26].", "In the context of economic networks, multiplex analyses have been applied to study the World Trade Web [27].", "Moreover, they have been extensively used in the context of systemic risk, where graphs are used to model interbank and credit networks [28][29].", "Here, we extend the multiplex approach to financial market time series, with the purpose of analysing the role of different measures of dependences namely the Pearson, Kendall, Tail and Partial correlation.", "In particular we consider the so-called Planar Maximally Filtered Graph (PMFG) [2] [3] [4] [7] as filtering procedure to each of the four layers.", "For each of the four unfiltered dependence matrices, the PMFG filtering starts from the fully connected graph and uses a greedy procedure to obtain a planar graph that connects all the nodes and has the largest sum of weights [3] [4].", "The PMFG is able to retain a higher number of links, and therefore a larger amount of information, than the Minimum Spanning Tree (MST) and can be seen as a generalization of the latter which is always contained as a proper sub-graph [2].", "The topological structures of MST and PMFG have been shown to provide meaningful economic and financial information [30], [31], [32], [33], [34] that can be exploited for risk monitoring [35], [36], [37] and asset allocation [38], [39].", "The advantage of adopting a filtering procedure is not only in the reduction of noise and dimensionality but more importantly in the possibility to generate of sparse networks, as sparsity is a requirement for most of the multiplex network measures that will be used in this paper [14].", "It is worth mentioning that the filtering of the partial correlation layer requires an adaptation of the PMFG algorithm to deal with asymmetric relations.", "We have followed the approach suggested in [40], that rules out double links between nodes.", "The obtained planar graph corresponding to partial correlations has been then converted into an undirected graph, and included in the multiplex.", "We have constructed a time-varying multiplex network with $M=4$ layers and a varying number of nodes.", "Nodes represent stocks, selected from a data set of $N_{tot}=1004$ US stocks which have appeared at least once in S&P500 in the period between 03/01/1993 and 26/02/2015.", "The period under study has been divided into 200 rolling time windows, each of $\\theta = 1000$ trading days.", "The network at time $T=1,2,\\ldots , 200$ can be described by the adjacency matrix $a^{\\alpha }_{ij}(T)$ , with $i,j=1,\\ldots ,N(T)$ and $\\alpha =1,2,3,4$ .", "The network at time window $T$ has $N(T) < N_{tot}$ nodes, representing those stocks which were continuously traded in time window $T$ .", "The links at each of the four layers are constructed by means of the PMFG procedure from Pearson, Kendall, Tail and Partial dependencies (see Materials and Methods for details).", "Figure: The multiplex nature of dependenceamong financial assets.", "The plots report the networkanalysis of a multiplex whose four layers are Planar MaximallyFiltered Graphs (PMFGs) obtained from four classical dependencemeasures, namely Pearson, Kendall, Tail and Partial correlation,computed on rolling time windows of 23 trading days between 1993and 2015.", "Each of the four layers provides different information onthe dependency structure of a market.", "Although market events andtrends have a somehow similar effect on the average dependence〈w ij 〉\\langle w_{ij} \\rangle between nodes at the different layers(panel (a)), each layer has a distinct local structure.", "This ismade evident by the plots of the average edge overlap 〈O〉\\langle O\\rangle (panel (b)) and of the fraction U [α] U^{[\\alpha ]} of edgesunique to each layer, an edge exists on average on less than twolayers, and up to 70%70\\% of the edges of a layer are not present onany other layer.", "Moreover, the same node can have different degreesacross the four layers, as indicated by the relatively low valuesof the pairwise inter-layer degree correlation coefficientρ [α,β] \\rho ^{[\\alpha ,\\beta ]} reported in panel (d) for two pairs oflayers over the whole observation interval.Fig.", "REF (a) shows how the average link weight of each of the four dependency networks changes over time.", "These results indicate an overall increase of the typical weights in the examined period 1993-2015 and reveals a strongly correlated behaviour of the four curves (with linear correlation coefficients between the curves range in $[0.91,0.99]$ ).", "This strong correlation in the temporal patterns of the four measures of dependence may lead to the wrong conclusion that the four networks carry very similar information about the structure of financial systems.", "Conversely, we shall see that even basic multiplex measures suggest otherwise.", "In Fig.", "REF (b) we report the average edge overlap $\\langle O\\rangle $ , that is the average number of layers of the financial multiplex network where a generic pair of nodes $(i,j)$ is connected by an edge (see Materials and Methods for details).", "Since our multiplex network consists of four layer, $\\langle O \\rangle $ takes values in $[1,4]$ , and in particular we have $\\langle O \\rangle =1$ when each edge is present only in one layer, while $\\langle O\\rangle =4$ when the four networks are identical.", "The relatively low values of $\\langle O \\rangle $ observed in this case reveal the complementary role played by the different dependency indicators.", "It is interesting to note that the edge overlap $\\langle O \\rangle $ displays a quite dynamic pattern, and its variations seem to be related to the main financial crises highlighted by the vertical lines in Fig.", "REF (b).", "The first event that triggers a sensible decrease in the average edge overlap is the Russian crisis in 1998, which corresponds to the overall global minimum of $\\langle O \\rangle $ in the considered interval.", "Then, $\\langle O \\rangle $ starts increasing towards the end of year 2000 and reaches its global maximum at the beginning of 2002, just before the market downturn of the same year.", "We observe a marked decrease in 2005, in correspondence with the second phase of the housing bubble, which culminates in the dip associated to the credit crunch at the end of 2007.", "A second, even steeper drop occurs during the Lehman Brothers default of 2008.", "After that, the signal appears more stable and weakly increasing, especially towards the end of 2014.", "Since each edge is present, on average, in less than two layers, each of the four layers effectively provides a partial perspective on the dependency structure of the market.", "This fact is made more evident by the results reported in Fig.", "REF (c), where we show, for each layer $\\alpha =1,\\ldots , 4$ , the fraction of edges $U^{[\\alpha ]}$ that exist exclusively in that layer (see Materials and Methods for details).", "We notice that, at any point in time, from $30\\%$ to $70\\%$ of the edges of each of the four layers are unique to that layer, meaning that a large fraction of the dependence relations captured by a given measure are not captured by the other measures.", "Another remarkable finding is that also the relative importance of a stock in the network, measured for instance by its centrality in terms of degree [41], [39], varies a lot across layers.", "This is confirmed by the degree correlation coefficient $\\rho ^{[\\alpha ,\\beta ]}$ for pairs of layers $\\alpha $ and $\\beta $ .", "In general, high values of $\\rho ^{[\\alpha , \\beta ]}$ signal the presence of strong correlations between the degree of the same node in the two layers (see Materials and Methods for details).", "Fig.", "REF (d) shows $\\rho ^{[\\alpha , \\beta ]}$ as a function of time for two pairs of dependence measures, namely Pearson–Kendall and Kendall–Tail.", "Notice that the degrees of the layers corresponding to Pearson and Kendall exhibit a relatively large correlation, which remains quite stable over the whole time interval.", "Conversely, the degrees of nodes in the Kendall and Tail layers are on average less correlated, and the corresponding values of $\\rho ^{[\\alpha ,\\beta ]}$ exhibit larger fluctuations.", "For example, in the tenth time window we find that General Electric stock (GE US) is a hub in Kendall layer with 71 connections, but it has only 16 connections in the Tail layer: therefore the relevance of this stock in the dependence structure depends sensitively on the layer.", "The presence of temporal fluctuations in $\\langle O \\rangle $ , in particular the fact that $\\langle O \\rangle $ reaches lower values during financial crises, together with the unique patterns of links at each layer, testified by high values of $U^{[\\alpha ]}$ and by relatively weak inter-layer degree-degree correlations for some pairs of layers, confirm that an analysis of relations among stocks simply based on one dependence measure can neglet relevant information which can however be captured by other measures.", "As we will show below, a multiplex network approach which takes into account at the same time all the four dependence measures, but without aggregating them into a single-layer network, is able to provide a richer description of financial markets.", "Figure: Multidegrees reveal the different role of industrial sectors during crises.The plots of the average multidegree of the nodes of the same industrial sector restricted to edges existing exclusively on the (a) Pearson,(b) Kendall, (c) Tail, and (d) Partial layers, clearly show thatsome dependence measures can reveal structures which are unnoticedby other measures.", "In particular, the plot of the averagemultidegree associated to edges existing on at least one layer amongKendall, Tail and Partial, but not on Pearson (panel (e)), reveals that Pearson correlation does not capture many important features such as the prominent role of Basic Materials, Financial, Consumer Goods and Industrials during crises and the increasing importance of Technology and Consumer Services after the 2007-2008 crisis.As a first example of useful quantities that can be investigated in a multiplex network, we have computed the so-called multidegree $k_i^{\\vec{m}}$ for each node $i$ in the network, corresponding to different multiedges (see Materials and Methods) [42].", "In particular, we have normalised the multidegree of node $i$ dividing it by the corresponding node overlapping degree $o_i$ , so that the resulting $k_i^{\\vec{m}}/o_i$ is the fraction of multiplex edges of node $i$ that exist only on a given subset of layers.", "In Fig.", "REF we report the average normalised multidegree of each of the 10 industry sectors listed in the ICB classification [43].", "We focus on the edges existing exclusively in one of the four layers and on the combination of multi-edges associated to edges existing in either of the Kendall, Tail, or Partial layer, but not in the Pearson layer.", "As shown in Fig.", "REF , the multidegree exhibits strong variations in time and high heterogeneity across different industries.", "Industries such as Oil $\\&$ Gas, Utilities, and Basic Materials, show low values of normalised multidegree in all the four panels (a)-(d).", "Conversely, the edges of nodes corresponding to Industrials, Finance, Technology, Telecommunications, and Consumer Services tend to concentrate in one or in a small subset of layers only.", "For instance, we observe a relatively high concentration of edges at the Kendall layer for nodes corresponding to Finance, Industrials and Consumer Goods stocks in the period preceding the Dot-com bubble and the 2002 downturn, a feature visible in the Pearson layer in Fig.", "REF (a).", "Analogously, we notice a sudden increase of edges unique to the Tail layer for nodes in Consumer Goods, Consumer Services and Health Care after the 2007-2008 crisis.", "The presence of large heterogeneity and temporal variations in the relative role of different industrial sectors confirms the importance of using a multiplex network approach to analyse dependence among assets.", "From this perspective it is particularly interesting to discuss the plot of multidegree restricted to edges that are present on either Kendall, Partial or Tail layer, but are not present in the Pearson layer as reported in Fig.", "REF (e).", "Despite the Pearson correlation coefficient is the most used measure to study dependencies, the plot reveals that until 2002 an analysis of the financial network based exclusively on Pearson correlations would have missed from 40$\\%$ up to 60$\\%$ of the edges of assets in sectors such as Basic Materials, Financial, Consumer Goods and Industrials.", "The study of evolution with time in Fig.", "REF (e) reveals that the relative role of such industrial sectors in Kendall, Tail and Partial layers becomes relatively less important between the two crises in 2002 and in 2007, but then such sectors become central again during the 2007-2008 crisis and beyond.", "This prominent role is quite revealing but it would not had been evident from the analysis of the Pearson layer alone.", "Let us also note that, the period following the 2007-2008 crisis is also characterised by a sensible and unprecedented increase of the normalised multidegree on Kendall, Partial and Tail layers of stocks belonging to Technology and Telecommunications sectors, whose importance in the market dependence structure has been therefore somehow underestimated over the last ten years by the studies based exclusively on Pearson correlation.", "To better quantify the relative importance of specific nodes and groups of nodes we computed the overlapping degree and partecipation coefficient, respectively measuring the total number of edges of a node and how such edges are distributed across the layers (see Materials and Methods for details).", "We started by computing the average degree $k_I^{[\\alpha ]}$ at layer $\\alpha $ of nodes belonging to each ICB industry sector $I$ , defined as $k_I^{[\\alpha ]}=\\frac{1}{N_I}\\sum _{i\\in I} k_i^{[\\alpha ]}\\delta (c_i, I)$ , where by $c_i$ we denote the industry of node $i$ and $N_I$ is the number of nodes belonging to industry sector $I$ .", "Figs.", "REF a)-d) show $k_I^{[\\alpha ]}$ as a function of time for each of the four layers.", "Figure: Average node degree as aproxy of the importance of an industry.", "The plots of averagedegree of the nodes belonging to the different industrial sectorsrestricted to the (a) Pearson, (b) Kendall, (c) Tail, and (d)Partial layers, and of the average overlapping degree reported inpanel (e) confirm the relative importance of Financials.", "However,the average participation coefficient (panel (f)) suggests that thedependence structure of some sectors such as Basic Materials,Industrials, and Oil & Gas, has become more heterogeneous, i.e.,focused only on a subset of the four layers, after the 2007-2008crisis.Notice that nodes in the Financial sector exhibit a quite high average degree, no matter the dependence measure used, with a noticeable peak before the Dot-com bubble in 2002.", "After that, the average degree of Financials has dropped sensibly, with the exception of the 2007-2008 crisis.", "Apart from the existence of similarities in the overall trend of Financials across the four layers, the analysis of the average degree suggests again the presence of high heterogeneity across sectors and over time.", "In the Pearson layer, Basic Materials is the second most central industry throughout most of the observation interval, whereas Industrials and Oil $\\&$ Gas acquired more connections in the period following the 2007-2008 crisis.", "The degree in the Kendall layer is distributed more homogeneously among the sectors than in the Pearson layer.", "Interestingly, the plot of degree on the Tail layer looks similar to that of the Pearson layer.", "Finally, in the Partial layer we observe the highest level of concentration of links in Finance (consistently to what found in [40]) and, after the 2007-08 crisis, in Basic Materials.", "We have also calculated for each industry $I$ the average overlapping degree $o_I \\equiv \\langle o_i \\rangle _{i \\in I}$ , where $o_i$ is the overlapping degree of node $i$ , which quantifies the overall importance of each industrial sector in the multiplex dependence network.", "The average overlapping degree of each industry is shown as a function of time in Fig.", "REF (e).", "As we can see, $o_I$ is able to highlight the prominent role played in the multiplex network by Financials, Basic Materials, Oil & Gas, and Industrials sectors, revealing also the presence of four different phases between 1997 and 2015.", "The first phase, during which Financials is the only prominent industry, covers the period between 1997 and 2000.", "The second phase, between 2000 and the 2007-08 crisis, is characterised by the emergence of Basic Materials as the second most central sector.", "In the third phase, between 2009 and 2014, Financials looses its importance in favour of Industrials, Oil & Gas and Basic Materials (that becomes the most central one).", "Finally, in 2014 a new equilibrium starts to emerge, with Financials and Industrials gaining again a central role in the system.", "The participation coefficient complements the information provided by the overlapping degree, quantifying how the edges of a node are distributed over the layers of the multiplex.", "In particular, the participation coefficient of node $i$ is equal to 0 if $i$ has edges in only one of the layers, while it is maximum and equal to 1 when the edges of node $i$ are equally distributed across the layers (see Materials and Methods for details).", "In Fig.", "REF (f) we report, as a function of time, the average participation coefficient $P_I$ for each ICB industry $I$ .", "Interestingly, the plot reveals that the increase of the overlapping degrees of Financials, Basic Materials, Industrials, and Oil & Gas sectors shown in Fig.", "REF (e) are normally accompanied by a substantial decrease of the corresponding participation coefficients.", "This indicates that those sectors accumulated degree on just one or two layers, confirming what we found in multidegree analysis.", "A somehow more detailed analysis of the temporal evolution of participation coefficient for each sector is reported in Appendix.", "By using filtered networks from different correlation measures we have demonstrated that a multiplex network approach can reveal features that would have otherwise been invisible to the analysis of each dependency measure in isolation.", "Although the layers produced respectively from Pearson, Kendall, Tail and Partial correlations show a certain overall similarity, they exhibit distinct features that are associated with market changes.", "For instance, we observed that average edge overlap between the first three layers, drops significantly during periods of market stress revealing that non-linear effects are more relevant during crisis periods.", "The analysis of the average multidegree associated to edges not present on the Pearson layer, but existing on at least one of the three remaining layers, indicates that Pearson correlations alone can miss to detect some important features.", "We observed that the relative importance of non-linearity and tails on market dependence structure, as measured by mean edge overlap between the last three layers, has dropped significantly in the first half of 2000s and then risen steeply between 2005 and the 2007-08 crisis.", "Overall, financial crises trigger remarkable drops in the edge overlap, widening therefore the differences among the measures of dependence just when evaluation of risk becomes of the highest importance.", "Different industry sectors exhibit different structural overlaps.", "For instance, Financials, Industrials and Consumer Goods show an increasing number of connections only on Kendall layer in the late 90s/early 2000, at the edge of the Dot-Com bubble.", "After the 2007-08 crisis these industries tend to have many edges on the Kendall, Tail and Partial which are not present on the Pearson layer.", "This observation questions whether we can rely on the Pearson estimator alone, when analysing correlations between stocks.", "A study of the overlapping degree and of the participation coefficient shows that asset centrality, an important feature for portfolio optimization [39], [45], changes considerably across layers with largest desynchronized changes occurring during periods of market distress.", "Overall our analysis indicates that different dependency measures provide complementary informations about the structure and evolution of markets, and that a multiplex network approach can reveal useful to capture systemic properties that would otherwise go unnoticed.", "The original dataset consists of the daily prices of $N_{tot}=1004$ US stocks traded in the period between 03/01/1993 and 26/02/2015.", "Each stock in the dataset has been included in S&P500 at least once in the period considered.", "Hence the stocks considered provide a representative picture of the US stock market over an extended time window of 22 years, and cover all the 10 industries listed in the Industry Classification Benchmark (ICB) [43] (Fig.REF ).", "It is important to notice that most of the stocks in this set are not traded over the entire period.", "This is a major difference with respect to the majority of the works on dynamic correlation-based networks, in which only stocks continuously traded over the period under study are considered, leading to a significant “survival bias”.", "For each asset $i$ we have calculated the series of log-returns, defined as $r_i(t) = log(P_i(t)) - log(P_i(t-1))$ , where $P_i(t)$ is stock price at day $t$ .", "The construction of the time-varying multiplex networks is based on log-returns and has been performed in moving time windows of $\\theta = 1000$ trading days (about 4 years), with a shift of $dT=23$ trading days (about one month), adding up to 200 different multiplex networks, one for each time window.", "For each time window $T$ , four different $N(T) \\times N(T)$ dependence matrices have been computed, respectively based on the four different estimators illustrated below.", "Since the number of active stocks changes with time, dependence matrices at different times can have different number of stocks $N(T)$ , as shown in Fig.REF .", "In the figure is also shown the ICB industry composition of our dataset in each time window, confirming that we have a representative sample of all market throughout the period.", "We have verified that the results we are discussing in the following are robust against change of $\\theta $ and $dT$ .", "Figure: Number of stocks in each ICBindustry in time.", "Number of stocks that are continuouslytraded in each time window together with their partition in termsof ICB industries.We have considered four different measures of dependence between two time series $r_i(u)$ and $r_j(u)$ , $i, j =1,2,\\ldots ,N$ , $u=1,2,\\ldots , \\theta $ , indicated in the following respectively as Pearson, Kendall, Tail and Partial.", "– Pearson dependence –   It is a measure of linear dependence between two time series, and is based on the evaluation of the Pearson correlation coefficient [46].", "We have used the exponentially smoothed version of this estimator [47], in order to mitigate excessive sensitiveness to outliers in remote observations: $\\rho ^w_{ij} = \\frac{\\sum _{u=1}^{\\theta } w_u(r_i(u) - \\bar{r_i}^w)(r_j(u)-\\bar{r_j}^w)}{\\sqrt{\\sum _{u=1}^{\\theta } w_u(r_i(u)-\\bar{r_i}^w)^2}\\sqrt{\\sum _{u=1}^{\\theta } w_u(r_j(u)-\\bar{r_j}^w)^2}} ~~,$ with $w_u = w_0 \\exp \\left(\\frac{u-\\theta }{T^*} \\right) ~~,$ where $T^*$ is the weight characteristic time ($T^*>0$ ) that controls the rate at which past observations lose importance in the correlation, and $w_0$ is a constant connected to the normalisation constraint $\\sum _{u=1}^{\\theta } w_u = 1$ .", "We have chosen $T^* = \\theta / 3$ according to previously established criteria [47].", "– Kendall dependence –    It is a measure of dependence that takes into account the nonlinearity of a time series.", "It is based on the evaluation of the so-called Kendall's $\\tau $ rank correlation coefficient, starting from the quantities $d_k(u,v) \\equiv sgn(r_k(u) - r_k(v))$ .", "The estimator counts the number of concordant pairs, i.e.", "pairs of observations such that $d_i(u,v)$ and $d_j(u,v)$ have equal signs, minus the number of discordant pairs [11].", "As for the case of the Pearson dependence, we have used the exponentially smoothed version of the estimator [47]: $\\tau ^w_{ij} = \\sum ^{\\theta }_{u=1} \\sum ^{\\theta }_{v=u+1} w_{u,v} ~d_i(u,v) d_j (u,v) ~~,$ with $w_{u,v} = w_0 \\exp \\left(\\frac{u-\\theta }{T^*} \\right) \\exp \\left(\\frac{v-\\theta }{T^*} \\right) ~~,$ where $T^*$ is again the weight characteristic time.", "– Tail dependence –    It is a non-parametric estimator of tail copula that provides a measure of dependence focused on extreme events.", "It is based on the evaluation of the following estimator [48]: $C_{ij}(p_1,p_2) = \\frac{\\sum _{u=1}^{\\theta } \\mathbb {1}_{ \\lbrace F^i(r_i(u))< p_1 \\wedge F^j(r_j(u))< p_2 \\rbrace }}{\\sum _{u=1}^{\\theta } \\mathbb {1}_{ \\lbrace F^i(r_i(u))< p_1 \\vee F^j(r_j(u))< p_2 \\rbrace }}$ where $F^i$ and $F^j$ are the empirical cumulative probabilities of returns $r_i(u)$ and $r_j(u)$ respectively, $p_1$ and $p_2$ are two parameters representing the percentiles above which an observation is considered (lower) tail.", "We focus on lower tails since we are interested in risk management applications, where the attention is on losses.", "It can be shown that this is a consistent estimator of tail copula [48].", "In this work we have chosen $p_1=p_2=0.1$ (i.e.", "we consider tail every observation below the 10th percentile), as a trade-off between the need of statistic and the interest in extreme events.", "– Partial dependence–   It is a measure of dependence that quantifies to what extent each asset affects other assets correlation.", "The Partial correlation $\\rho _{ik|j}$ , or correlation influence, between assets $i$ and $k$ based on $j$ , is the Pearson correlation between the residuals of $r_i(u)$ and $r_k(u)$ obtained after regression against $r_j(u)$ [49].", "It can be written in terms of a Pearson correlation coefficient as follows [40]: $\\rho _{ik|j} = \\frac{\\rho _{ik} - \\rho _{ij} \\rho _{kj}}{\\sqrt{[1-\\rho _{ij}^2][1-\\rho _{kj}^2]}}$ This measure represents the amount of correlation between $i$ and $k$ that is left once the influence of $j$ is subtracted.", "Following [40], we define the correlation influence of $j$ on the pair $i,k$ as: $d(i,k|j)= \\rho _{ik} - \\rho _{ik|j}$ $d(i,k|j)$ is large when a significant fraction of correlation between $i$ and $k$ is due to the influence of $j$ .", "Finally, in order to translate this into a measure between $i$ and $j$ , the so-called Partial dependence, we average it over the index $k$ : $d(i|j) = \\langle d(i,k|j) \\rangle _{k \\ne i,j}$ $d(i|j)$ is the measure of influence of $j$ on $i$ based on Partial correlation.", "It is worth noting that, unlike the other measures of dependence, $d(i|j)$ provides a directed relation between assets (as in general $d(i|j) \\ne d(j|i)$ ).", "In the rest of the paper we refer to this indicator as “Partial dependence”, even though strictly speaking we are analysing the Correlation influence based on Partial correlation.", "For each of the 200 time windows we have then constructed a multiplex network with $M=4$ layers obtained respectively by means of the four dependence indicators.", "In order to reduce the noise and the redundance contained in each dependence matrix we have applied the Planar Maximally Filtered Graph [2] [3] [4] [7].", "It is worth mentioning that the filtering of the correlation influence layer requires an adaptation of the PMFG algorithm to deal with asymmetric relations.", "We have followed the approach suggested in [40] that rules out double links between nodes.", "The obtained planar graphs have been then converted into an undirected graphs and included in the multiplex.", "Let us consider a weighted multiplex network $\\mathcal {M}$ on $N$ nodes, defined by the $M$ -dimensional array of weighted adjacency matrices $\\mathcal {W} = \\lbrace W^{[1]}, W^{[2]}, \\ldots , W^{[M]}\\rbrace $ , where $W^{[\\alpha ]}=\\lbrace w_{ij}^{[\\alpha ]}\\rbrace $ are the matrices of weights that determine the topology of the $\\alpha $ -th layer though the PMFG filtering.", "Here the weight $w_{ij}^{[\\alpha ]}$ represents the strength of the correlation between node $i$ and node $j$ on layer $\\alpha $ , where the different layers are obtained through different correlation measures.", "In the following we will indicate by $W^{[\\alpha ]}$ the weighted adjacency matrix of the PMFG associated to layer $\\alpha $ , and by $A^{[\\alpha ]}$ the corresponding unweighted adjacency matrix, where $a_{ij}^{[\\alpha ]} = 1$ if and only if $w_{ij}^{[\\alpha ]} \\ne 0$ .", "We denote by $K^{[\\alpha ]}=\\frac{1}{2}\\sum _{ij}a_{ij}^{[\\alpha ]}$ the number of edges on layer $\\alpha $ , and by $K = \\frac{1}{2}\\sum _{i,j} \\left[1 -\\prod _{\\alpha }(1-a_{ij}^{[\\alpha ]})\\right]$ the number of pairs of nodes which are connected by at least one edge on at least one of the $M$ layers.", "Notice that since the network at each layer is a PMFG, then we have $K^{[\\alpha ]} = 3(N-2)\\: \\forall \\alpha $ by construction.", "We consider some basic quantities commonly used to characterise multiplex networks [42], [14].", "The first one is the mean edge overlap, defined as the average number of layers on which an edge between two randomly chosen nodes $i$ and $j$ exists: $\\langle O \\rangle = \\frac{1}{2K}\\sum _{i,j} \\sum _{\\alpha }a_{ij}^{[\\alpha ]}.$ Notice that $\\langle O \\rangle =1$ only when all the $M$ layers are identical, i.e.", "$A^{[\\alpha ]}\\equiv A^{[\\beta ]}\\: \\forall \\alpha ,\\beta =1,\\ldots ,M$ , while $\\langle O \\rangle =0$ if no edge is present in more than one layer, so that the average edge overlap is in fact a measure of how much similar is the structure of the layers of a multiplex network.", "A somehow dual quantity is the fraction of edges of layer $\\alpha $ which do not exist on any other layer: $U^{[\\alpha ]} = \\frac{1}{2K^{[\\alpha ]}}\\sum _{i,j}a_{ij}^{[\\alpha ]}\\prod _{\\beta \\ne \\alpha } \\left(1-a_{ij}^{[\\beta ]}\\right)$ which quantifies how peculiar is the structure of a given layer $\\alpha $ , since $U^{[\\alpha ]}$ is close to zero only when almost all the edges of layer $\\alpha $ are also present on at least one of the other $M-1$ layers.", "More accurate information about the contribution of each node to a layer (or to a group of layers) can be obtained by the so-called multidegree of a node $i$ .", "Let us consider the vector $\\vec{m} =(m_1,m_2,...,m_M)$ , with $M$ equal to the number of layers, where each $m_{\\alpha }$ can take only two values $\\lbrace 1,0\\rbrace $ .", "We say that a pair of nodes $i,j$ has a multilink $\\vec{m}$ if they are connected only on those layers $\\alpha $ for which $m_{\\alpha }=1$ in $\\vec{m}$  [42].", "The information on the $M$ adjacency matrices $a_{ij}^{\\alpha }$ ($\\alpha =1,..,M$ ) can then be aggregated in the multiadjacency matrix $A_{ij}^{\\vec{m}}$ , where $A_{ij}^{\\vec{m}}=1$ if and only if the pair $i,j$ is connected by a multilink $\\vec{m}$ .", "Formally [42], [13]: $A_{ij}^{\\vec{m}} \\equiv \\prod _{\\alpha =1}^M [a_{ij}^{\\alpha }m_{\\alpha }+(1-a_{ij}^{\\alpha })(1-m_{\\alpha })].$ From the multiadjacency matrix we can define the multidegree $\\vec{m}$ of a node $i$ , as the number of multilinks $\\vec{m}$ connecting $i$ : $k_{i}^{\\vec{m}} = \\sum _{j} A_{ij}^{\\vec{m}}.$ This measure allows us to calculate e.g.", "how many edges node $i$ has on layer 1 only ($k_{i}^{\\vec{m}}$ choosing $m_1=1$ , $m_{\\alpha }=0 ~~\\forall \\alpha \\ne 1$ ), integrating the global information provided by $U^{[\\alpha ]}$ .", "The most basic measure to quantify the importance of single nodes on each layer is by means of the node degree $k_i^{[\\alpha ]} =\\sum _{j}a_{ij}^{[\\alpha ]}$ .", "However, since the same node $i$ is normally present at all layers, we can introduce two quantities to characterise the role of node $i$ in the multiplex[14], namely the overlapping degree $o_i = \\sum _{\\alpha }k_i^{[\\alpha ]}$ and the multiplex participation coefficient: $P_i = \\frac{M}{M-1}\\left[1 -\\sum _{\\alpha }\\left(\\frac{k_i^{[\\alpha ]}}{o_i}\\right)\\right].$ The overlapping degree is just the total number of edges incident on node $i$ at any layer, so that node are classified as hubs if they habve a relatively large value of $o_i$ .", "The multiplex participation coefficient quantifies the dispersion of the edges incident on node $i$ across the layers.", "In fact, $P_i=0$ if the edges of $i$ are concentrated on exactly one of the $M$ layers (in this case $i$ is a focused node), while $P_i=1$ if the edges of $i$ are uniformly distributed across the $M$ layers, i.e.", "when $k_i^{[\\alpha ]} = \\frac{o_i}{M}\\: \\forall \\alpha $ (in which case $i$ is a truly multiplex node).", "The scatter plot of $o_i$ and $P_i$ is called multiplex cartography and has been used as a synthetic graphical representation of the overall heterogeneity of node roles observed in a multiplex.", "In a multiplex network is important also to look at the presence and sign of inter-layer degree correlations.", "This can be done by computing the inter-layer degree correlation coefficient [15]: $\\rho ^{[\\alpha , \\beta ]} = \\frac{\\sum _{i}\\left( R^{[\\alpha ]}_i -\\overline{ R^{[\\alpha ]} } \\right) \\left( R^{[\\beta ]}_i -\\overline{ R^{[\\beta ]} } \\right)}{\\sqrt{\\sum _i\\left(R^{[\\alpha ]}_i - \\overline{ R^{[\\alpha ]} } \\right)^2\\sum _j\\left( R^{[\\beta ]}_j - \\overline{ R^{[\\beta ]} } \\right)^2}}$ where $R_i^{[\\alpha ]}$ is the rank of node $i$ according to its degree on layer $\\alpha $ and $\\overline{R^{[\\alpha ]}}$ is the average rank by degree on layer $\\alpha $ .", "In general $\\rho ^{[\\alpha ,\\beta ]}$ takes values in $[-1,1]$ , where values close to $+1$ (respectively, $-1$ ) indicate the of strong positive (resp.", "negative) correlations, while $\\rho ^{[\\alpha ,\\beta ]}\\simeq 0$ if the degrees at the two layers are uncorrelated.", "In Fig.", "REF we plot the time evolution of the average participation coefficient $P_I$ (x-axis) of stocks in the industrial sector $I$ against the average overlapping degree $o_I$ (y-axis).", "Each circle corresponds to one of the 200 time windows, while the size and colour of each circles represent different time windows.", "Each panel corresponds to one industrial sector $I$ .", "The diagrams reveal that in the last 20 years the role of different sectors has changed radically, and in different directions.", "For instance, stocks in the Financials sector evolved from a relatively large overlapping degree and a small participation coefficient in the late 1990s, to a smaller number of edges, distributed more homogeneously across the layers, towards the end of the observation period.", "Conversely, Industrials stocks have acquired degree on some of the layers, resulting in a considerable decrease of participation coefficient.", "This is another indication of the importance of monitoring all the layers together, as an increase in the structural role of an industry (as measured by the overlapping degree) is typically due to only a subset of layers (as indicated by the corresponding decrease of partecipation coefficient).", "Figure: Industries evolution in theoverlapping degree/partecipation coefficient plane.", "Fixed anindustry II, we have plotted for each time window a circle whoseyy coordinate is the average overlapping degree o I o_I and whose xxcoordinate is the average partecipation coefficient P I P_I.", "Pointsat different times are characterized with different sizes (small tolarge) and colours (legend on the right).", "In a) - l) we show theresults respectively for Basic Materials, Consumer Goods, ConsumerServices, Financials, Health Care, Industrials, Oil & Gas,Technology, Telecommunications and Utilities.The authors wish to thank Alessandro Fiasconaro for useful discussions at the beginning of this project.", "V.L.", "acknowledges support from the EPSRC project GALE, EP/K020633/1.", "The authors wish to thank Bloomberg for providing the data.", "TDM wishes to thank the COST Action TD1210 for partially supporting this work.", "N.M. and V. N. contributed equally to this work.", "All the authors devised the study, performed the experiments and simulations, analysed the results, wrote the paper, and approved the final draft.", "Competing financial interests.", "The authors declare no competing financial interest." ] ]
1606.04872
[ [ "Reproducing Kernel Hilbert Space vs. Frame Estimates" ], [ "Abstract We consider conditions on a given system $\\mathcal{F}$ of vectors in Hilbert space $\\mathcal{H}$, forming a frame, which turn $\\mathcal{H}$ into a reproducing kernel Hilbert space.", "It is assumed that the vectors in $\\mathcal{F}$ are functions on some set $\\Omega$.", "We then identify conditions on these functions which automatically give $\\mathcal{H}$ the structure of a reproducing kernel Hilbert space of functions on $\\Omega$.", "We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism.", "Applications are given to Hilbert spaces associated to families of Gaussian processes." ], [ "Introduction", "A reproducing kernel Hilbert space (RKHS) is a Hilbert space $\\mathcal {H}$ of functions on a set, say $\\Omega $ , with the property that $f(t)$ is continuous in $f$ with respect to the norm in $\\mathcal {H}$ .", "There is then an associated kernel.", "It is called reproducing because it reproduces the function values for $f$ in $\\mathcal {H}$ .", "Reproducing kernels and their RKHSs arise as inverses of elliptic PDOs, as covariance kernels of stochastic processes, in the study of integral equations, in statistical learning theory, empirical risk minimization, as potential kernels, and as kernels reproducing classes of analytic functions, and in the study of fractals, to mention only some of the current applications.", "They were first introduced in the beginning of the 20ties century by Stanisław Zaremba and James Mercer, Gábor Szegö, Stefan Bergman, and Salomon Bochner.", "The subject was given a global and systematic presentation by Nachman Aronszajn in the early 1950s.", "The literature is by now vast, and we refer to the following items from the literature, and the papers cited there [4], [1], [16], [12], [15], [8].", "Our aim in the present paper is to point out an intriguing use of reproducing kernels in the study of frames in Hilbert space." ], [ "An Explicit Isomorphism", "Let $\\mathcal {H}$ be a separable Hilbert space, and let $\\lbrace \\varphi _{n}\\rbrace _{n \\in \\mathbb {N}}$ be a system of vectors in $\\mathcal {H}$ .", "Then we shall study relations of $\\mathcal {H}$ as a reproducing kernel Hilbert space (RKHS) subject to properties imposed on the system $\\lbrace \\varphi _{n}\\rbrace _{n \\in \\mathbb {N}}$ .", "A RKHS is a Hilbert space $\\mathcal {H}$ of functions on some set $\\Omega $ such that for all $t \\in \\Omega $ , there is a (unique) $K_{t} \\in \\mathcal {H}$ with $f(t)=\\langle K_{t}, f \\rangle _{\\mathcal {H}}$ , for all $t \\in \\Omega $ , for all $f \\in \\mathcal {H}$ .", "In the theorem below we study what systems of functions $\\varphi _{n} \\in \\mathcal {H} \\cap \\lbrace \\text{functions on some set } \\Omega \\rbrace $ yield RKHSs; i.e., if $\\lbrace \\varphi _{n}\\rbrace _{n \\in \\mathbb {N}}$ satisfies (REF ), what additional conditions are required to guarantee that $\\mathcal {H}$ is a RKHS?", "Given $\\lbrace \\varphi _{n}\\rbrace _{n \\in \\mathbb {N}} \\subset \\mathcal {H}$ , we shall introduce the Gramian $G=(\\langle \\varphi _{i}, \\varphi _{j}\\rangle _{\\mathcal {H}})$ considered as an $\\infty \\times \\infty -$ matrix.", "Under mild restrictions on $\\lbrace \\varphi _{n}\\rbrace _{n \\in \\mathbb {N}}$ , it turns out that $G$ defines an unbounded (generally) selfadjoint linear operator $l^{2} \\overset{G}{\\rightarrow } l^{2}$ $(G(c_{j}))_{k}=\\sum _{j \\in \\mathbb {N}} \\langle \\varphi _{k}, \\varphi _{j}\\rangle _{\\mathcal {H}}c_{j}.$ Let $\\mathcal {F}$ denote finitely supported sequence with (REF ) defined on all finitely supported sequence $(c_{j})$ $\\mathcal {F}$ in $l^{2}$ , i.e., $(c_{j}) \\in \\mathcal {F}$ if and only if there exists $n \\in \\mathbb {Z}_{+}$ such that $c_{j}=0$ , for all $j \\ge n$ ; but note that $n$ depends the sequences.", "Denoting $\\delta _{j}$ the canonical basis in $l^{2}$ , $\\delta _{j}(j)=\\delta _{i,j}$ , note $\\mathcal {F}=span\\lbrace \\delta _{j}|j \\in \\mathbb {N}\\rbrace $ .", "Further, note that the RHS in (REF ) is well defined when $\\sum _{j}\\left|\\langle \\varphi _{k}, \\varphi _{j} \\rangle _{\\mathcal {H}}\\right|^{2}<\\infty $ , for all $k \\in \\mathbb {N}$ .", "Suppose $\\mathcal {H}$ , $\\lbrace \\varphi _{n}\\rbrace $ are given.", "Assume that each $\\varphi _{n}$ is a function on $\\Omega $ where $\\Omega $ is a given set $\\lbrace \\varphi _{n}\\rbrace $ is a frame in $\\mathcal {H}$ , see (REF ) and (REF ), and that $\\lbrace \\varphi _{n}(t)\\rbrace \\in l^{2}$ , for all $t \\in \\Omega $ then $\\mathcal {H}$ is a reproducing kernel Hilbert space (RKHS) with kernel $K^{G}(s,t)=\\langle l(s), G^{-1}l(t) \\rangle _{2}=l(s)^{*}G^{-1}l(t),$ where $l(t)=\\lbrace \\varphi _{n}(t)\\rbrace \\in l^{2}$ , and where $G$ is the Gramian of $G=(\\langle \\varphi _{n}, \\varphi _{m} \\rangle _{\\mathcal {H}})$ .", "Moreover, $G$ defines selfadjoint operator in $l^{2}$ with dense domain, and we get an isometric isomorphism $\\mathcal {H}_{RK} \\overset{T_{G}}{\\rightarrow } \\mathcal {H}$ $T_{G}(\\sum _{n}c_{n}\\varphi _{n})=T^{*}c$ where $T$ is the frame operator.", "Overview: Since $\\lbrace \\varphi _{n}\\rbrace \\subset \\mathcal {H}$ is a frame, the Gramian $G_{mn}:=\\langle \\varphi _{m}, \\varphi _{n}\\rangle _{\\mathcal {H}}$ , is an $\\infty \\times \\infty $ matrix defining a bounded operator $l^{2} \\rightarrow l^{2}$ , invertible with $(G^{-1})_{mn}$ such that $\\sum _{k=1}^{\\infty }(G^{-1})_{mk}\\langle \\varphi _{k}, \\varphi _{n} \\rangle _{\\mathcal {H}}=\\delta _{m, n}$ and the reproducing kernel of $\\mathcal {H}$ is $\\sum _{m}\\sum _{n}\\overline{\\varphi _{m}(s)}G_{mn}^{-1}\\varphi _{n}(t)=\\langle l(s), G^{-1}l(t) \\rangle _{2}$ (details) By (REF ) and Lemma , the frame operators $T$ and $T^{*}$ are as follows: Given $\\mathcal {H}$ , $\\lbrace \\varphi _{n}\\rbrace $ , set ${\\left\\lbrace \\begin{array}{ll}T:\\mathcal {H} \\rightarrow l^{2} \\\\T^{*}:l^{2} \\rightarrow \\mathcal {H}\\end{array}\\right.", "}$ to be the two linear operators $Tf=(\\langle \\varphi _{n}, f\\rangle _{\\mathcal {H}})$ and adjoint $T^{*}$ as follows: $T^{*}c=\\sum _{n}c_{n}\\varphi _{n}.$ We have $\\langle Tf, c \\rangle _{l^{2}}=\\langle f, T^{*}c \\rangle _{\\mathcal {H}}, \\quad \\text{and} \\quad T^{*}Tf=\\sum \\langle \\varphi _{n}, f \\rangle \\varphi _{n}, \\quad \\forall f \\in \\mathcal {H}, \\quad \\forall c \\in l^{2},$ $(TT^{*}c)_{n}=(Gc)_{n}=\\sum _{m}G_{nm}c_{m}, \\quad \\forall c \\in l^{2}.$ Do the real case first, then it is easy to extend to complex valued functions.", "Note that $TT^{*}$ is an operator in $l^{2}$ , i.e., $l^{2} \\overset{TT^{*}}{\\rightarrow } l^{2}.$ It has a matrix-representation as follows $(TT^{*})_{i,j}=\\langle \\delta _{i}, TT^{*}\\delta _{j}\\rangle _{l^{2}}$ We have $(TT^{*})_{i,j}=G_{i,j}=\\langle \\varphi _{i}, \\varphi _{j}\\rangle _{\\mathcal {H}},\\quad \\forall (i,j)\\in \\mathbb {N} \\times \\mathbb {N}.$ By (REF ), we have $(TT^{*})_{i,j}&=\\langle \\delta _{i}, TT^{*}\\delta _{j} \\rangle _{l^{2}} \\\\&=\\langle T^{*}\\delta _{i}, T^{*}\\delta _{j} \\rangle _{\\mathcal {H}} \\\\&=\\langle \\varphi _{i}, \\varphi _{j}\\rangle _{\\mathcal {H}} = G_{i,j}$ which is the desired conclusion (REF ).", "Both $T^{*}T$ and $TT^{*}$ are self-adjoint: If $B_{i}$ , $i=1,2$ are the constants from the frame estimates, then: $B_{1}\\Vert c\\Vert _{2}^{2} \\le \\Vert T^{*}c\\Vert _{\\mathcal {H}}^{2} \\le B_{2}\\Vert c\\Vert _{2}^{2} \\quad \\forall c \\in l^{2}, \\quad \\text{and}$ $B_{1}\\Vert f\\Vert _{\\mathcal {H}}^{2} \\le \\Vert Tf\\Vert _{l^{2}}^{2} \\le B_{2}\\Vert f\\Vert _{\\mathcal {H}}^{2} \\quad \\forall f \\in \\mathcal {H};$ equivalently $B_{1}\\Vert f\\Vert _{\\mathcal {H}}^{2} \\le \\sum _{n}| \\langle \\varphi _{n}, f \\rangle _{\\mathcal {H}}|^{2}\\le B_{2}\\Vert f\\Vert _{\\mathcal {H}}^{2}.$ Set $K(s,t)=\\sum _{n=1}^{\\infty }\\varphi _{n}(s)^{*}\\varphi _{n}(t)=l(s)^{*}l(t)=\\langle l(s), l(t) \\rangle _{2}$ We have $B_{1}I_{l^{2}} \\le TT^{*} \\le B_{2}I_{l^{2}}, \\quad \\text{and}$ $B_{1}I_{\\mathcal {H}} \\le T^{*}T \\le B_{2}I_{\\mathcal {H}}.$ If $B_{1}=B_{2}=1$ , then we say that $\\lbrace \\varphi _{n}\\rbrace _{n\\in \\mathbb {N}}$ is a Parseval frame.", "For the theory of frames and some of their applications, see e.g., [7], [6], [5] and the papers cited there.", "By the polar-decomposition theorems, see e.g., [11] we conclude that there is a unitary isomorphism $u: \\mathcal {H} \\rightarrow l^{2}$ such that $T=u(T^{*}T)^{1/2}=(TT^{*})^{1/2}u$ ; and so in particular, the two s.a. operators $T^{*}T$ and $TT^{*}$ are unitarily equivalent.", "$l(t)=(\\varphi _{n}(t))\\in l^{2}.$ Therefore $(T^{*}T)^{-1/2}$ is well defined $\\mathcal {H} \\rightarrow \\mathcal {H}$ .", "Now (REF ) holds if and only if $f=\\sum \\langle (T^{*}T)^{-1/2}\\varphi , f \\rangle (T^{*}T)^{-1/2}\\varphi _{n}$ or equivalently: $f=\\sum \\langle \\psi _{n}, f \\rangle _{\\mathcal {H}}\\psi _{n},$ where $\\psi _{n}:=(T^{*}T)^{-1/2}\\varphi _{n}.$ Here we used that $T^{*}T$ is a selfadjoint operator in $\\mathcal {H}$ , and it has a positive spectral lower bound; where $\\lbrace \\varphi _{j}\\rbrace _{j \\in \\mathbb {N}}$ is assumed to be a frame.", "There is an operator $L:\\mathcal {H} \\rightarrow \\mathcal {H}$ (the Lax-Milgram operator) such that $\\sum _{n=1}^{\\infty }\\langle f, \\varphi _{n} \\rangle _{\\mathcal {H}}\\langle \\varphi _{n} Lg \\rangle _{\\mathcal {H}}=\\langle f,g \\rangle _{\\mathcal {H}}$ holds for all $f \\in \\mathcal {H}$ .", "We shall apply the Lax-Milgram lemma [11], p. 57 to the sesquilinear form $\\mathcal {B}(f,g)=\\sum _{n=1}^{\\infty }\\langle f, \\varphi _{n} \\rangle _{\\mathcal {H}}\\langle \\varphi _{n}, g \\rangle _{\\mathcal {H}}, \\quad \\forall f,g \\mathcal {H}.$ Since $\\lbrace \\varphi _{n}\\rbrace _{n=1}^{\\infty }$ is given to be a frame in $\\mathcal {H}$ , then our frame-bounds $B_{1}>0$ and $B_{2}<\\infty $ such that (REF ) holds.", "Introducing $\\mathcal {B}$ from (REF ) this into $B_{1}\\Vert f\\Vert _{\\mathcal {H}}^{2} \\le \\mathcal {B}(f,f) \\le B_{2}\\Vert f\\Vert _{\\mathcal {H}}^{2},\\quad \\forall f \\mathcal {H}.$ The existence of the operator $L$ as stated in (REF ) now follows from the Lax-Milgram lemma.", "Let $\\mathcal {H}$ , $\\lbrace \\varphi _{n}\\rbrace $ , $T$ , $T^{*}$ be as in Lemma ; and let $L$ be the Lax-Milgram operator; then $L=(T^{*}T)^{-1}$ .", "The kernel $K^{G}(\\cdot , \\cdot )$ on $\\Omega \\times \\Omega $ from (REF ) is well-defined and positive definite.", "We must show that all the finite double summations $\\sum _{i}\\sum _{j}\\overline{c_{i}}c_{j}K^{G}(t_{i},t_{j})$ are $\\ge 0$ , whenever $(c_{i})$ is a finite system of coefficients, and $(t_{i})$ is a finite sample of points in $\\Omega $ .", "Now fix $(c_{i})$ and $(t_{i})$ as specified, and, for $n \\in \\mathbb {N}$ , set $F_{n}:=\\sum _{i}c_{i}\\varphi _{n}(t_{i});$ then we have the following: $\\sum _{i}\\sum _{j}\\overline{c_{i}}c_{j}K^{G}(t_{i},t_{j})&= \\sum _{i}\\sum _{j}\\overline{c_{i}}c_{j}\\langle l(t_{i}), G^{-1}l(t_{j})\\rangle _{l^{2}} \\\\&= \\sum _{i}\\sum _{j}\\overline{c_{i}}c_{j}\\sum _{m}\\sum _{n}\\overline{\\varphi _{m}(t_{i})}G^{-1}_{m,n}\\varphi _{n}(t_{j}) \\\\&=\\sum _{m}\\sum _{n}\\overline{F_{m}}G^{-1}_{m,n}F_{n} \\ge 0.$ We have the following: $\\psi _{n}(t)=(G^{-1/2}\\varphi )_{n}(t)=\\sum _{m=1}^{\\infty }(G_{nm}^{-1/2}\\varphi _{m})(t)=G^{-1/2}l(t)_{n},$ and these functions are in the RKHS of the kernel $K^{G}$ from (REF ).", "Begin with (the frame identity): $(T^{*}T)\\varphi _{n}\\underset{by (\\ref {eq:2.2})}{=}\\sum _{m=1}^{\\infty } \\langle \\varphi _{m}, \\varphi _{n} \\rangle \\varphi _{m}=(Gl)_{n},\\quad \\forall n \\in \\mathbb {Z}_{+}, \\text{ where } l=\\begin{bmatrix}\\varphi _{1} \\\\\\varphi _{2} \\\\\\vdots \\end{bmatrix}$ if and only if $(T^{*}T)l(t)=G(l(t)).$ Now approximate $\\sqrt{x}$ with polynomials (Weierstrass), and we get $(T^{*}T)^{-1/2}l(t)=G^{-1/2}l(t).$ Recall, $\\psi _{n}=(T^{*}T)^{-1/2}\\varphi _{n}$ .", "$\\psi _{n}(t)=(G^{-1/2}l(t))_{n}$ .", "Now rewrite (REF ) as $f(t)=\\sum _{n=1}^{\\infty }\\langle \\psi _{n},f \\rangle _{\\mathcal {H}}\\psi _{n}(t)=\\sum _{n=1}^{\\infty }\\langle G^{-1/2}\\varphi _{n}, f\\rangle _{\\mathcal {H}}(G^{-1/2}\\varphi )_{n}(t)=\\langle K_{t}^{G},f\\rangle $ where $K_{t}^{G}&=\\sum _{n=1}^{\\infty }G^{-1/2}\\varphi _{n}(\\cdot )(G^{-1/2}\\varphi _{n})(t) \\\\&\\underset{by (\\ref {eq:2.12})}{=} K^{G}(s,t)=\\sum _{n=1}^{\\infty }(G^{-1/2}\\varphi _{n})(s)(G^{-1/2}\\varphi _{n})(t) \\\\&=\\langle G^{-1/2}l(s), G^{-1/2}l(t)\\rangle _{2} \\\\&=\\langle l(s), G^{-1}l(t)\\rangle _{2}$ For the complex case, the result still holds, mutatis mutandis; one only needs to add the complex conjugations.", "Note that (REF ) is the reproducing property.", "The function $(\\psi _{n}(t))$ from (REF ) in Lemma satisfy $\\sum _{n\\in \\mathbb {N}}\\overline{\\psi _{n}(s)}\\psi _{n}(t)=K^{G}(s,t), \\quad \\forall (s,t) \\in \\Omega \\times \\Omega .$ $LHS_{(\\ref {eq:2.12.2})}&=\\langle G^{-1/2}l(s), G^{-1/2}l(t)\\rangle _{2} \\\\&= \\langle l(s), (G^{-1/2})^{2}l(t)\\rangle _{2} \\\\&=\\langle l(s), G^{-1}(l(t)),\\rangle _{2} \\\\&= K^{G}(s,t).$ The following isometric property holds: $\\left\\Vert \\sum _{n=1}c_{n}\\varphi _{n}(\\cdot ) \\right\\Vert _{\\mathcal {H}}^{2}&= \\sum \\sum c_{m}c_{n}\\langle \\varphi _{n}, \\varphi _{m} \\rangle _{\\mathcal {H}} \\\\&= c^{T}Gc=\\langle c,Gc \\rangle _{2} \\\\&=\\langle c, TT^{*}c\\rangle _{2}=\\left\\Vert T^{*}c\\right\\Vert _{\\mathcal {H}}^{2},\\quad T^{*}c \\in \\mathcal {H}, \\quad c \\in l^{2}$ where $T$ and $T^{*}$ are the frame operators $\\mathcal {H} \\overset{T}{\\underset{T^{*}}{\\longleftrightarrow }}l^{2}$ , i.e., $Tf=(\\langle \\varphi _{n}, f\\rangle _{\\mathcal {H}})_{n}\\in l^{2}$ The Lax operator $L$ satisfies $Lf:=\\sum _{n}({T^{*}}^{-1}f)_{n}\\varphi _{n}(\\cdot )$ , for all $f \\in \\mathcal {H}$ and it is isometric $\\mathcal {H} \\rightarrow \\mathcal {H}$ .", "In the theorem, we assume that the given Hilbert space $\\mathcal {H}$ has a frame $\\lbrace \\varphi _{n}\\rbrace \\subset \\mathcal {H}$ consisting of functions on a set $\\Omega $ .", "So this entails a lower, and an upper frame bound, i.e., $0<B_{1}\\le B_{2}<\\infty $ .", "The following example shows that the conclusion in the theorem is false if there is not a positive lower frame-bound.", "Set $\\mathcal {H}=L^{2}(0,1)$ , $\\Omega =(0,1)$ the open unit-inbound, and $\\varphi _{n}(t)=t^{n}$ , $n \\in \\lbrace 0\\rbrace \\cup \\mathbb {N}=\\mathbb {N}_{0}$ .", "In this case, the Gramian $G_{nm}=\\int _{0}^{1}x^{n+m}dx=\\frac{1}{n+m+1}$ is the $\\infty \\times \\infty $ Hilbert matrix, see ([13], [10], [14]).", "In this case it is known that there is an upper frame bound $B_{2}=\\pi $ , i.e., $\\sum _{n=0}^{\\infty }\\left|\\int _{0}^{1}f(x)x^{n}dx\\right|^{2} \\le \\pi \\int _{0}^{1}|f(x)|^{2}dx;$ in fact, for the operator-norm, we have $\\Vert G\\Vert _{l^{2}\\rightarrow l^{2}}=\\pi ;$ but there is not a lower frame bound.", "Moreover, $G$ define a selfadjoint operator in $l^{2}(\\mathbb {N}_{0})$ with spectrum $[0, \\pi ]=$ the closed interval.", "This implies that there cannot be a positive lower frame-bound.", "Moreover, it is immediate by inspection that $\\mathcal {H}=L^{2}(0,1)$ is not a RKHS." ], [ "Frames and Gaussian Processes", "In [2] and [3], it was shown that for every positive Borel measure $\\sigma $ on $\\mathbb {R}$ such that $\\int _{\\mathbb {R}}\\frac{d\\sigma (u)}{1+u^{2}}< \\infty ,$ there is a unique (up to measure isomorphism) Gaussian proess $X$ as follows: $X=X_{\\varphi }$ is indexed by the Schwartz-space $\\mathcal {S}=\\mathcal {S}(\\mathbb {R})$ of function $\\varphi $ on $\\mathbb {R}$ , i.e., $\\varphi \\in \\mathcal {S} \\iff $ $\\varphi \\in \\mathbb {C}^{\\infty }$ , and for all $N, M \\in \\mathbb {N}$ we have $\\underset{m \\le M}{max} \\underset{x \\in \\mathbb {R}}{sup}\\left|x^{N}\\left(\\frac{\\partial }{\\partial x}\\right)^{m}\\varphi (x)\\right|< \\infty $ with $\\mathbb {E}(X_{\\varphi })=0$ , and $\\mathbb {E}(X_{\\varphi }^{2})=\\int _{\\mathbb {R}}|\\hat{\\varphi }|^{2}d\\sigma $ for all $\\varphi \\in \\mathcal {S}$ .", "Let $\\Omega :=\\mathcal {S}^{\\prime }=$ the dual $=$ the Schwartz space of all tempered distribution, then $X_{\\varphi }$ is defined on $\\mathcal {S}^{\\prime }$ , by $X_{\\varphi }(w)=w(\\varphi ), \\quad \\varphi \\in \\mathcal {S}, \\quad w \\in \\mathcal {S}^{\\prime }.$ It is real valued Gaussian random variable.", "[2], [3] there is a unique measure $\\mathbb {P}=\\mathbb {P}_{\\sigma }$ on $\\mathcal {S}^{\\prime }$ such that $X_{\\varphi }$ is Gaussian for all $\\varphi \\in \\mathcal {S}$ , $\\mathbb {E}(X_{\\varphi })=0, \\quad \\text{and}$ $\\mathbb {E}_{\\sigma }(e^{iX_{\\varphi }})=\\int _{\\mathcal {S}^{\\prime }}e^{iX_{\\varphi }}d\\mathbb {P}_{\\sigma }=e^{-\\frac{1}{2}\\Vert \\widehat{\\varphi }\\Vert _{\\sigma }^{2}}=e^{-\\frac{1}{2}\\int _{\\mathbb {R}}|\\widehat{\\varphi }(u)|^{2}d\\sigma (u)}$ where $\\widehat{\\varphi }=$ the Fourier transform, $\\widehat{\\varphi }(u)=\\int _{\\mathbb {R}}e^{ixu}\\varphi (x)dx.$ Let $\\lbrace f_{n}\\rbrace _{n\\in \\mathbb {N}}$ be a real-valued frame in $L^{2}(\\mathbb {R}, \\sigma )$ $(=\\lbrace f \\text{ on } \\mathbb {R}, \\text{ such that } \\Vert f\\Vert _{\\sigma }^{2}:=\\int _{\\mathbb {R}}|f(u)|^{2}d\\sigma (u)<\\infty \\rbrace )$ with frame bounds $a$ , $b$ such that $0<a\\le b<\\infty $ , so $a\\Vert f\\Vert _{\\sigma }^{2} \\le \\sum _{n \\in \\mathbb {N}}\\left|\\int _{\\mathbb {R}}f(u)f_{n}(u)d\\sigma (u)\\right|^{2} \\le b\\Vert f\\Vert _{\\sigma }^{2}.$ Let $\\lbrace B_{n}\\rbrace _{n \\in \\mathbb {N}}$ be a system of i.i.d.", "(independent identically distributed) $N(0,1)$ Gaussian random variables on $(\\Omega , \\mathcal {F}, \\mathbb {P}_{\\sigma })$ , and set $Y_{\\varphi }(\\cdot )=\\sum _{n\\in \\mathbb {N}}\\langle f_{n}, \\widehat{\\varphi }\\rangle _{L^{2}(\\sigma )}B_{n}(\\cdot ), \\quad \\text{(Karhunen-Loève);}$ then $a\\text{ }\\mathbb {E}_{\\sigma }\\left(|X_{\\varphi }|^{2} \\right) \\le \\mathbb {E}_{\\sigma }\\left(|Y_{\\varphi }|^{2} \\right) \\le b\\text{ }\\mathbb {E}_{\\sigma }\\left(|X_{\\varphi }|^{2} \\right).$ Using the i.i.d.", "$N(0,1)$ property of $\\lbrace B_{n}\\rbrace _{n\\in \\mathbb {N}}$ , we get $\\mathbb {E}_{\\sigma }\\left(|Y_{\\varphi }|^{2} \\right)=\\sum _{n\\in \\mathbb {N}}|\\langle f_{n}, \\widehat{\\varphi } \\rangle _{L^{2}(\\sigma )}|^{2}.$ The desired conclusion (REF ) now follows from (REF ) combined with $\\mathbb {E}_{\\sigma }\\left(|X_{\\varphi }|^{2} \\right)=\\Vert \\widehat{\\varphi }\\Vert _{L^{2}(\\sigma )}^{2}, \\quad \\text{see \\cite {AlJo12},\\cite {AlJoLe11}}$ while (REF ) is immediate from (REF ).", "The property for $\\lbrace Y_{\\varphi }\\rbrace _{\\varphi \\in \\mathcal {S}}$ in (REF ) agrees with $\\lbrace X_{\\varphi }\\rbrace _{\\varphi \\in \\mathcal {S}}$ if and only if $\\lbrace f_{n}\\rbrace _{n \\in \\mathbb {N}}$ is a Parseval frame in $L^{2}(\\sigma )$ .", "This follows from the Karhunen-Loève theorem; see [9], [2], [3]." ] ]
1606.04868
[ [ "Cosmic signatures in earth's seismic tremor?" ], [ "Abstract Even in absence of earthquakes, each site on earth experiences continuous elastic vibrations which are mostly traced to the non-linear interactions of sea waves.", "However, the fine structure of the spectrum at mHz frequencies shows hundreds of highly significant narrow bandwidth peaks, with a persistence and a coincidence with solar acoustic eigenmodes which are incompatible with any geophysical origin.", "The feasibility of a common cosmic origin is evaluated through an estimate of the gravitational wave cross-section of the earth, combined with its elastic response and with the stochastic amplification produced by the interference of the cosmic signal with tremor of oceanic origin.", "The measured spectral peaks appear compatible with a gravitational monochromatic illumination at strains $h \\gtrsim 10^{-20} $.", "We analize in detail the band around 2.614 mHz, where the binary white dwarf J0651+2844 - which is the second strongest known gravitational stellar source - is expected to emit.", "Compatible spectral tremor peaks are found for both the earth and the sun, but their amplitude is 3 o.m.", "larger than independent estimates, so that a gravitational source attribution would call for a variety of unknown non-luminous sources with definite mass-distance ratios." ], [ "Cosmic signatures in earth's seismic tremor? ", "Francesco Mulargia Dipartimento di Fisica e Astronomia, Università di Bologna, viale Berti Pichat 8 40127 Bologna, Italy ([email protected]) Even in absence of earthquakes, each site on earth experiences continuous elastic vibrations which are mostly traced to the non-linear interactions of sea waves.", "However, the fine structure of the spectrum at mHz frequencies shows hundreds of highly significant narrow bandwidth peaks, with a persistence and a coincidence with solar acoustic eigenmodes which are incompatible with any geophysical origin.", "The feasibility of a common cosmic origin is evaluated through an estimate of the gravitational wave cross-section of the earth, combined with its elastic response and with the stochastic amplification produced by the interference of the cosmic signal with tremor of oceanic origin.", "The measured spectral peaks appear compatible with a gravitational monochromatic illumination at strains $h \\gtrsim 10^{-20} $ .", "We analize in detail the band around 2.614 mHz, where the binary white dwarf J0651+2844 - which is the second strongest known gravitational stellar source - is expected to emit.", "Compatible spectral tremor peaks are found for both the earth and the sun, but their amplitude is 3 o.m.", "larger than independent estimates, so that a gravitational source attribution would call for a variety of unknown non-luminous sources with definite mass-distance ratios.", "Keywords: gravitational wave detectors and experiments; noise; seismicity; surface waves and free oscillations; 04.80.Nn, 05.40.Ca, 91.30.Dk, 91.30.Fn Seismic tremor, i.e., the background motion of the earth in absence of earthquakes, is an elastic \"noise\" wavefield present always and everywhere, extending over more than seven frequency decades from $ < 10^{-6}$ Hz to the acoustic band.", "It consists of the stochastic interference of the elastic waves produced by a variety of sources, both natural and anthropic.", "Its amplitude varies largely according to site, time of the day and season, but its power spectral pattern remains unchanged, with a strong monotonical decrease with frequency (in displacement), interrupted by a \"hump\" around $\\ 200$ mHz and a lesser one around 60 mHz (Fig.", "REF ).", "Seismic tremor is traditionally attributed to oceanic waves, and in particular to two nonlinear interactions – the shallow sea wave-sloping bottom interaction and the deep ocean wave-wave interaction – which, by displacement continuity, excite seismic waves in the solid ocean bottom[14, 12].", "Numerical models applied to a realistic global bathymetry confirm that these two mechanisms provide an adequate approximation to the general spectral pattern[24].", "However, the recordings of seismographic and gravimetric stations reveal a fine structure with many narrow highly significant spectral peaks, particularly in the range 1-10 mHz and with large annual cycles in amplitude[18].", "These go under the name of earth's hum, and are commonly identified with the elastic response - i.e., the earth's normal modes - to the continuous broadband stochastic excitation from infragravity sea waves[24].", "Such an explanation leads to several inconsistencies: 1) most spectral peaks occur at frequencies within 2% - but not in exact coincidence - with low order earth spherical and toroidal eigenmodes[23]; 2) some peaks are extraneous to earth elastic eigenmodes, but are in surprising coincidence (beyond chance at $6 \\sigma $ ) with solar acoustic eigenmodes[23]; 3) atmospheric turbulence is not a tenable alternative mechanism, since its persistence is much too short[24]; 4) an instrumental artefact origin, produced by magnetic spurious sensitivity of the seismometers, is implausible[23]; 5) several peaks show diurnal and semidiurnal cycles[9], inconsistent with any geophysical origin[24]; 6) last, but by no means least, multitaper high resolution spectral analysis reveals that most peaks have a very narrow relative bandwidth $R_B \\simeq \\Delta \\omega / \\omega $ , standing for quality factors $Q \\gg 300$ [23]; this is incompatible with a broadband noise response origin, since it would primarily excite (cf.", "e.g., [16]) the spheroidal and toroidal modes of the Earth with their related $Q \\lesssim 300 $ [8]; the same bandwidth incompatibility a fortiori applies to the broadband noise excitation of atmospheric and oceanic eigenmodes, for which $Q \\sim 1$ [5].", "The above inconsistencies lead to attribute the narrow spectral tremor peaks to their source, which must be monochromatic and with a high $Q$ factor.", "Since no credible geophysical candidate exists, a source external to the earth must be hypothesized, and one proposition along this line was advanced, identifying them with the seismo-magnetic excitation of the earth by the solar acoustic eigenmodes[23].", "While the still unspecified acoustic-magneto-elastic coupling at its basis is worth further investigation, we explore here an alternative non-terrestrial origin, i.e., that the narrow tremor peaks are produced on both the earth and the sun by the elastic excitation from cosmic gravitational wave monochromatic sources.", "Note how under this perspective the gravitational signature is not a signal above noise, as it is commonly assumed[1], but rather a constituent of seismic noise itself (cf.", "the Feynman diagrams in Fig.REF ).", "Choosing a rectilinear coordinate system, as justified by the relative smallness of the earth's mass, and assuming local elastic isotropy, gravitational waves (from now on GW) induce in solid bodies a displacement $\\eta $ [7, 17].", "$\\frac{\\partial }{\\partial t} \\big ( \\rho \\frac{\\partial \\eta _i }{\\partial t} \\big ) = \\frac{\\partial }{\\partial x_j} \\big [\\lambda \\delta _{ij} \\epsilon _{kk} + 2 \\frac{\\partial \\mu }{\\partial x_j} (\\epsilon _{ij} + \\frac{1}{2} h_{ij})+ 2 \\mu \\frac{\\partial \\epsilon _{ij}}{\\partial x_j} \\big ]$ where $\\rho $ is the density and $\\lambda $ , $\\mu $ the Lamé constants.", "According to equation (REF ), GW produce elastic strains only at rigidity discontinuities, where ${\\partial \\mu } /{\\partial x} \\ne 0$ .", "The largest elastic body on earth is the earth itself.", "Since rigidity is tied to shear wave velocity $v_s$ and density $\\rho $ as $\\mu = \\rho v^2_s$ , there exist two major rigidity discontinuities in the earth, located respectively at the surface and at the core–mantle boundary, where there is a transition from the solid silicate mantle to the liquid iron outer core.", "In terms of rigidity, the largest of these discontinuities is the one at depth, where $\\mu $ drops from $2.9 \\times 10^{11}$ N m$^{-2}$ to 0, while at the surface it goes from 0 to $2.7 \\times 10^{10}$ N m$^{-2}$ [8].", "However, since the free surface effect provides a factor of $\\sim 2$ amplitude increase (cf.", "e.g., [5]), and since we just aim at order of magnitude estimates, it will suffice to simply account for the total area.", "Let us consider a monochromatic GW cosmic source emitting at frequency $f$ .", "The GW induce gravitational elastic waves (from now on GEW) with displacement $u_j^{GEW}$ dependent on the incidence angle $\\theta ^{GW}$ to the normal of the surface of the $\\mu $ discontinuity $u_x & = & \\Psi \\sin \\theta ^{GW} \\cos \\theta ^{GW} \\nonumber \\\\u_y & = & \\Psi \\sin \\theta ^{GW} \\\\u_z & = & \\Psi \\sin ^2 \\theta ^{GW} \\nonumber $ where $\\Psi $ is the gravitational power emitted by the specific cosmic source.", "The interaction of GW with the inner and outer rigidity discontinuities of the earth generates $P$ and $S$ waves, the wave-guide interference of which induces, in turn, also Rayleigh and Love waves at the surface and Stoneley waves at the mantle-core interface.", "The major difference between the GEW peaks and those excited by noise lies in bandwidth, since $Q \\rightarrow \\infty $ for an ideal monochromatic GEW source, while $Q \\sim 300$ for broadband seismic noise exciting the earth normal modes[8, 5].", "In fact, such a difference is experimentally apparent between the measured narrow hum peaks - which occur at frequencies not coincident but close to earth normal modes - and the latter (see Fig.REF ).", "Since hum peaks occur within a few percent of the frequency of either a spherical or toroidal earth mode (see Fig.REF and cf.", "[23]) of the earth, which behaves at this frequency as a damped harmonic oscillator with $Q = Q_d \\sim 300$ , this implies a response amplification by a factor $q \\sim 100$ .", "Under a single pulse, this would result in a compatible decay, accompanied by a progressive spectral broadening and decay by energy transfer to the nearby bands.", "However, under a persistent monochromatic input, equilibrium will be reached after $\\sim Q_d$ cycles, and thereafter no further decay will occur, resulting in a narrow bandwidth peak ruled by the sampling frequency, the record length and the spectral technique adopted (see Fig.REF -top).", "Combining the above response argument with the stationary state of a quadrupole GW source, the gravitational cross section $\\Xi $ at the frequency $f$ of a GW detector with mass $M$ and geometric area $A$ can be written as (cf.", "[25, 7, 6] $\\Xi = \\frac{4 \\pi ^2}{15} \\big ( \\frac{r_g f}{c} \\big ) q A$ where $r_g$ is the Schwarzschild radius, $ r_g ={2 GM}/ {c^2} $ , $G$ is the gravitational constant and $q$ is the elastic response at the frequency $f$ .", "Given the radii at core–mantle boundary and at the surface, $A \\simeq 6.6 \\times 10^{14}$ m$^2$ , while the mass of the detector is that of the mantle + crust, i.e.", "$ 5 \\times 10^{24}$ kg, the Schwarzschild radius is $ r_g \\sim 7 \\times 10^{-3}$ m. Hence, considering that $ Q_d\\approx 300$ [8, 5], i.e., $ q \\sim 100$ , the gravitational cross section has a quite large extension linearly dependent on frequency: for example, at $f=1$ mHz, $\\Xi \\simeq 4 \\times 10^3 $ m$^2$ , i.e., the size of football pitch, while at $f=10$ mHz it compares with Place de la Concorde in Paris.", "Following the above discussion on response, let us take the earth as a set of weakly damped harmonic oscillators, which reach thermodynamic equilibrium when the rate of energy dissipation is balanced by the absorption of incoming energy.", "This occurs (cf.", "Fig.REF -top) in a time $ \\tau = Q_d/(2 \\pi f)$ , in which approximately 40% of the energy is dissipated, while the remaining 60% is converted into mechanical energy $E$ .", "At equilibrium, equipartition assigns a Boltzmann energy $kt$ to each degree of freedom and the latter consists of equal quantities of potential and kinetic energy, while linearity warrants that the excited earth is equivalent to a single oscillator with mass $M$ , amplitude $x$ and energy $E = \\pi ^2 f^2 M x^2$ .", "After \"loading up\" for the time $\\tau $ , the system will (cf.", "[6, 20]) absorb GW power $\\Psi $ and re-emit GEW of amplitude $x$ as $\\Psi ~ \\Xi ~ \\tau = \\pi ^2 f^2 M x^2$ where $x$ accounts for the excitation signal $u$ and any amplification this is subject to.", "Another amplification beyond the earth's elastic response is likely to occur.", "The oceanic wave-wave and wave bottom interactions at the basis of tremor (Fig.", "REF ) are highly nonlinear and act upon an excitable system dominated by noise[22].", "A particular type of amplification is known to operate in such systems[13]: stochastic resonance.", "This can be summarized as the statistical facilitation of the transition to a higher energy state by the addition of random noise[11].", "Stochastic resonance occurs when the escape time from a potential well - the Kramers escape time - matches on average the periodicity of a weak forcing, which acts then as a trigger for the transition to another nearby potential minimum, resulting in an amplification of the forcing itself.", "Formally amenable to a Fokker-Planck equation, stochastic resonance is observed in a variety of systems, ranging from mammalian brain neural excitation, bistable ring lasers, semiconductor devices, chemical reactions, mechanoreceptor cells, etc., and can easily lead to $\\gtrsim 30 $ dB amplifications[11].", "An example of this effect, relative to a Duffing nonlinear oscillator, is shown in Fig.", "(REF )-bottom, illustrating how a small monochromatic excitation can be strongly amplified by adding substantial amounts of \"tuned\" noise.", "In the interaction of oceanic waves, stochastic resonance is likely to operate promoting the ensemble excitation of the system into a local higher energy state - i.e., a seiche - with transitions stochastically sympathetic with the forcing signal.", "This occurs at different geographic scales, and it is observed in harbours[19] as well as in oceans[2].", "Consistently with the typical values observed for this effect[19], we assume that a further seasonally variable amplification $K$ , up to $ \\sim 10^2$ , is provided by the interaction with oceanic noise, which acts as a modulator.", "Summing up, the measured tremor monochromatic displacement $x$ at the frequency $f_0$ is the original GEW signal $u$ amplified as $x = u q K$ so that, from equation (REF ), the gravitational power flux taking into account the earth's cross section results $\\Psi = \\frac{15 c^3}{ G} \\frac{(fx)^2}{ A q^2}= \\frac{15 c^3}{ G} \\frac{(f u K)^2}{ A}$ Let us now estimate the GEW by evaluating the GW power flux required to generate a tremor peak like the ones observed, which in the 1-10 mHz band have power spectral amplitudes from $P(x) \\sim 10^{-11} $ m$^2$ /Hz to $P(x)\\simeq 10^{-14} $ m$^2$ /Hz (see Fig.REF ).", "The measured displacement amplitudes at $f_0$ can then be written as (cf.", "[17]) $x = \\sqrt{2 P(x) S_{BW}}$ where $S_{BW} = f_u - f_l$ is the spectral bandwidth, and $f_u$ , $f_l$ respectively the upper and lower corner frequencies around $f_0$ .", "Now, $S_{BW}$ and is primarily a function of the source, i.e., of its monochromatic character and persistence, but also of the sampling rate and of the techniques used in the spectral analysis.", "Since after adequate corrections most cosmic monochromatic sources are stable in excess of $\\sim 10^{-8}$ s, GPS accuracy together with record length and the specific spectral technique adopted become the main limiting factors for spectral bandwidth[17].", "Hundreds of tremor narrow spectral peaks, many of which in correspondence with solar acoustic modes, are found at a statistical confidence level $> 99$ % in the frequency band below 10 mHz [18, 23].Each of these peaks, identified in both gravimetric and seismic recordings through high resolution multitaper spectral spectral techniques, can be hypothesized as originated by GEW.", "As a specific example, let us consider in detail the frequency band around 2.614 mHz, which is interesting because it coincides with the supposed GW emission of the binary white dwarf J0651+2844, presently the second strongest supposed stellar GW emitter.", "A narrow spectral peak is apparent at this frequency in the waveforms of the Black Forest Observatory, Schiltach, Germany (BFO) seismic station in the January-July 2004 period (see Fig.4 of [23]).", "Chosing at random among seismic stations distant from the ocean and known for low environmental noise, we consider 51 days of continuous seismic recordings at Obninsk (OBN), Russia, and Tamanrasset (TAM), Algeria, starting January 1 2013 at 00.00.00.", "A winter period is chosen to have the largest amplitude of the narrow spectral peaks for stations located in the northern hemisphere[18].", "Only the vertical component is considered in order to avoid the complication of the response to the earth gravity field induced by tilt on the horizontal components[21].", "Tremor spectral peaks compatible with such a GW illumination are apparent in the recordings (Figs.", "REF and REF ), both in correspondence of the central peak at $f=2.614$ mHz, and of the related Zeeman splitting[7] into the four singlets $ f \\pm F , f \\pm 2 F$ , produced by the perturbation of the incoming GW signal by earth daily rotation at the frequency $F=0.01157$ mHz.", "Note that such a splitting is routinely observed right after large earthquakes, which make the whole earth \"ring\"[5].", "The spectral pattern shows an obvious difference between the narrow peaks and the much broader earth elastic eigenmodes (Fig.", "REF ).", "While at mHz frequencies the GPS timing accuracy ($\\sim 100$ ns) could provide $S_{BW} \\ll 10^{-7}$ Hz, Fig.", "(REF ) suggests a spectral bandwidth $\\sim 10^{-6}$ Hz, most likely due to the comparatively short record length analized.", "Note that extending record length could tighten bandwidth, but would also increase the peak height, since the signal amplitude has obviously to remain the same.", "Entering the latter value with the measured spectral amplitude $P(x) \\sim 8 \\times 10^{-11} $ m$^2$ /Hz in equation (REF ) yields $x \\sim 10^{-8}$ m. The 2.614 mHz peak occurs at a frequency within 2% of the $_0S_{17}$ (at 2.5687 mHz) and $_0S_{18}$ (at 2.6747 mHz) earth spheroidal modes - and of the solar $P_{0,18}$ pressure mode (at 2.629 mHz)[3] - as well as within a few percent of the earth spherical modes $_0S_{17}$ , $_0S_{18}$ , $_3S_{6}$ , $_7S_{2}$ , $_5S_{5}$ (see Fig.REF ), providing a response amplification of the incoming GEW by a factor $q \\sim 100$ .", "Since we consider a winter period and stations in the northern hemisphere, a factor of $ K\\sim 100 $ stochastic amplification is assumed, resulting from eq.", "(REF ) in a GEW amplitude $u \\sim 10^{-12}$ m. Hence, considering an average seismic velocity of $\\sim 10$ km/sec[8], a gravitational strain $h \\sim 10^{-19}$ is obtained over one cycle.", "This is $\\sim 3$ o.m.", "above the gravitational emission estimated for J0651[4], suggesting that an attribution of the 2.614 mHz tremor peak to this $GW$ source seems presently unrealistic.", "Therefore, considering that this source should be among the strongest GW known emitters, the hypothesis of a GEW origin for the narrow tremor spectral peaks would generally call for non-luminous monochromatic gravitational sources.", "From the above arguments and Kepler's 3rd law we may define the set of binary systems of total mass $M_b$ illuminating the earth with a monochromatic GW power flux $\\Psi $ at the frequency $f$ from a distance $d$ as $\\frac{({f M_b})^{10/3}}{d^2} \\simeq \\Psi \\frac{c^5}{(2 \\pi G )^{7/3}}$ which provides the total mass - distance source ratios compatible with a measured power flux at a given frequency.", "For example, a tremor peak at 10 mHz, where the measured power density at $S_{BW} \\sim 10^{-6} $ Hz is $P(x)\\simeq 10^{-14} $ m$^2$ /Hz, stands for a GW strain $h\\sim 10^{-20}$ and, according to equation (REF ), for a GW flux $\\Psi \\sim 10^{-5}$ Joule/m$^2$ s. According to equation REF , this is compatible, for example, with a binary system of rotating black holes of total mass $\\sim 8$ solar masses, located at $\\sim 1000$ parsec from the earth.", "Such an object would have a $\\sim 10^5$ km orbital radius and output more than $10^{33}$ J/m$^2$ s in GW, hardly giving any other sign of its existence.", "It would also be compatible with a system 1 o.m.", "larger in terms of mass at a $\\sim 1$ o.m.", "larger distance, i.e., at the border of our galaxy.", "A gravitational attribution to the tremor peaks would first of all account for the coincidence with solar peaks, which would have the same origin, since the estimated gravitational cross section of the sun[15] essentially coincides with the present estimates for the earth.", "It would also be consistent with a cyclic annual dependence, since the stochastic amplification by oceanic tremor has a strong seasonality.", "Finally, for non-polar GW sources it would account for a diurnal and semidiurnal amplitude variation.", "In fact, considering tremor recordings at $N\\ge 5$ points - and provided that site-related effects of stochastic amplification can be sorted - the five-fold degeneracy of the gravitational tremor (together with its two polarizations) allows in principle to determine the amplitude and the celestial position of each source[10].", "The seismic tremor waveforms are web available thanks to the IRIS - Incorporated Research Institutions for Seismology.", "The author is indebted to the colleagues of the Dipartimento di Fisica e Astronomia dell'Universitá di Bologna and of the Istituto Nazionale di Astrofisica di Bologna, and in particular to Sasha Kamenshchick, Luca Ciotti, Roberto Casadio, Michele Cicoli, Andrea Comastri and Alberto Sesana for stimulating discussions.", "This work was performed with a RFO contribution of the Università di Bologna, Italy.", "[1] B. 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Lopes and J.", "Silk, Helioseismology and asteroseismology: looking for gravitational waves in acoustic oscillations, Astrophys.", "J., 794 (2014) 32-39.", "[16] F. Mulargia and S. Castellaro, Passive imaging in nondiffuse acoustic wavefields, Phys.", "Rev.", "Lett., 100 (2008) 218501.", "[17] F. Mulargia, and A. Kamenshchik, Global seismic network as a GW Antenna, Phys.", "Lett.", "A, 380 (2016) 1503-1507.", "[18] K. Nawa, N. Suda, Y. Fukao, T. Sato, Y. Aoyama, and K. Shibuya, 1998.Incessant excitation of the Earth’s free oscillations, Earth Plan.", "Space, 50 (1998) 3-18.", "[19] M. Okihiro, R. T. Guza, and R. J. Seymour, Excitation of Seiche Observed in a Small Harbor, J. Geophys.", "Res., 98 (1993) 18,201-18,211.", "[20] K. Riles, Gravitational Waves: Sources, Detectors and Searches, arXiv:1209.0667 [21] P.W.", "Rodgers, The response of the horizontal pendulum seismometer to Rayleigh and Love waves, tilt, and free oscillations of the earth, Bull.", "Seismol.", "Soc.", "Am., 58 (1968) 1384-1406.", "[22] L. W. Schwartz, and J. D. Fenton, Strongly nonlinear waves, Ann.", "Rev.", "Fluid Mech., 14 (1982) 39-60.", "[23] D. J. Thomson and F. L. Vernon, III, Unexpected, high-Q, low-frequency peaks in seismic spectra, Geophys.", "J.", "Int., 202 (2015) 1690-1710.", "[24] S.C. Webb, The Earth's hum:the excitation of Earth normal modes by ocean waves, Geophys.", "J.", "Int., 174 (2008) 542-566.", "[25] J. Weber, Phys.", "Rev.", "Lett., 117 (1960) 306-313." ] ]
1606.04863
[ [ "Production of the $X_b$ in $\\Upsilon(5S, 6S)\\to \\gamma X_b$ radiative\n decays" ], [ "Abstract In this work, we investigate the production of $X_b$ in the process $\\Upsilon(5S,6S)\\to \\gamma X_b$, where $X_b$ is assumed to be the counterpart of $X(3872)$ in the bottomonium sector as a $B {\\bar B}^*$ molecular state.", "We use the effective Lagrangian based on the heavy quark symmetry to explore the rescattering mechanism and calculate their production ratios.", "Our results have shown that the production ratios for the $\\Upsilon(5S,6S) \\to \\gamma X_b$ are orders of $10^{-5}$ with reasonable cutoff parameter range $\\alpha \\simeq 2\\sim 3$.", "The sizeable production ratios may be accessible at the future experiments like forthcoming BelleII, which will provide important clues to the inner structures of the exotic state $X_b$." ], [ "Introduction", "In the past decades, many so called XYZ have been observed by the Belle, BaBar, CDF, D0, CMS, LHCb, and BESIII collaborations [1].", "Some of them cannot fit into the conventional heavy quarkonium in the quark model [2], [3], [4], [5].", "Up to now, many studies on the production and decay of these XYZ states have been carried out in order to understand its nature (for a recent review, see Refs.", "[6], [7], [8]).", "In 2003, the Belle Collaboration discovered an exotic candidate $X(3872)$ in the process $B^+\\rightarrow K^++ J/\\psi \\pi ^+\\pi ^-$  [9] which was subsequently confirmed by the BaBar Collaboration [10] in the same channel.", "It was also discovered in proton-proton/antiproton collisions at the Tevatron [11], [12] and LHC [13], [14].", "The $X(3872)$ is a particularly intriguing state because on the one hand its total width $\\Gamma <1.2$ MeV [1] is tiny compared to typical hadronic widths; on the other hand the closeness of its mass to the $D^0\\overline{D}^{*0}$ threshold ($M_{X(3872)}-M_{D^0}-M_{D^{*0} }=(-0.12\\pm 0.24)$  MeV) and its prominent decays to $D^0\\overline{D}^{*0}$  [1] suggest that it may be an meson-meson molecular state [15], [16].", "Many theoretical works have been carried out in order to understand the nature of $X(3872)$ since the first observation of $X(3872)$ .", "It is also natural to look for the counterpart with $J^{PC}=1^{++}$ (denoted as $X_b$ hereafter) in the bottom sector.", "These two states are related by heavy quark symmetry which should have some universal properties.", "The search for $X_b$ may provide us important information on the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule configuration.", "Since the mass of $X_b$ may be very heavy and its $J^{PC}$ is $1^{++}$ , it is less likely for a direct discovery at the current electron-positron collision facilities, though the Super KEKB may provide an opportunity in $\\Upsilon (5S,6S)$ radiative decays [17].", "In Ref.", "[18], a search for $X_b$ in the $\\omega \\Upsilon (1S)$ final states has been presented and no significant signal is observed for such a state.", "The production of $X_b$ at the LHC and the Tevatron [19], [20] and other exotic states at hadron colliders [21], [23], [24], [22], [25], [26] have been extensively investigated.", "In the bottomonium system, the isospin is almost perfectly conserved, which may explain the escape of $X_b$ in the recent CMS search [27].", "As a result, the radiative decays and isospin conserving decays will be of high priority in searching for $X_b$  [28], [29], [30].", "In Ref.", "[28], we have studied the radiative decays of $X_b \\rightarrow \\gamma \\Upsilon (nS)$ ($n=1, 2, 3$ ), with $X_b$ being a candidate for the $B{\\bar{B}}^*$ molecular state, and found that the partial widths into $\\gamma X_b$ are about 1 keV.", "In Ref.", "[29], we studied the rescattering mechanism of the isospin conserving decays $X_b\\rightarrow \\Upsilon (1S)\\omega $ , and our results show that the partial width for the $X_b\\rightarrow \\Upsilon (1S)\\omega $ is about tens of keVs.", "In this work, we will further investigate the $X_b$ production in $\\Upsilon (5S,6S) \\rightarrow \\gamma X_b$ with $X_b$ being a $B{\\bar{B}}^*$ molecule candidate.", "To investigate this process, we calculate the intermediate meson loop (IML) contributions.", "As well know, IML transitions have been one of the important nonperturbative transition mechanisms been noticed for a long time [31], [32], [33].", "Recently, this mechanism has been used to study the production and decays of ordinary and exotic states [51], [53], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [52], [54], [55], [56], [57], [58], [59], [60] and B decays [61], [62], [63], [66], [68], [67], [65], [64], and a global agreement with experimental data were obtained.", "Thus this approach may be suitable for the process $\\Upsilon (5S,6S) \\rightarrow \\gamma X_b$ .", "The paper is organized as follows.", "In Sec.", ", the effective Lagrangians for our calculation.", "Then in Sec.", ", we present our numerical results.", "Finally we give the summary in Sec.", "." ], [ "Effective Lagrangians", "Based on the heavy quark symmetry, we can write out the relevant effective Lagrangian for the $\\Upsilon (5S)$  [68], [69] $\\mathcal {L}_{\\Upsilon (5S) B^{(*)} B^{(*)}} &=&ig_{\\Upsilon BB} \\Upsilon _{\\mu } (\\partial ^\\mu B \\bar{B}- B\\partial ^\\mu \\bar{B})-g_{\\Upsilon B^* B} \\varepsilon _{\\mu \\nu \\alpha \\beta }\\partial ^{\\mu } \\Upsilon ^{\\nu } (\\partial ^{\\alpha } B^{*\\beta } \\bar{B}+ B \\partial ^{\\alpha }\\bar{B}^{*\\beta })\\nonumber \\\\&&-ig_{\\Upsilon B^* B^*} \\big \\lbrace \\Upsilon ^\\mu (\\partial _{\\mu } B^{* \\nu } \\bar{B}^*_{\\nu }-B^{* \\nu } \\partial _{\\mu }\\bar{B}^*_{\\nu })+ (\\partial _{\\mu } \\Upsilon _{\\nu } B^{* \\nu } -\\Upsilon _{\\nu }\\partial _{\\mu } B^{* \\nu }) \\bar{B}^{* \\mu } \\nonumber \\\\&& +B^{* \\mu }(\\Upsilon ^\\nu \\partial _{\\mu } \\bar{B}^*_{\\nu } -\\partial _{\\mu } \\Upsilon ^\\nu \\bar{B}^*_{\\nu })\\big \\rbrace , $ where ${{B}^{(*)}}=\\left(B^{(*)+},B^{(*)0}\\right)$ and ${\\bar{B}^{(*)T}}=\\left(B^{(*)-},\\bar{B}^{(*)0}\\right)$ correspond to the bottom meson isodoublets.", "$\\epsilon _{\\mu \\nu \\alpha \\beta }$ is the anti-symmetric Levi-Civita tensor and $\\epsilon _{0123}= +1$ .", "Since $\\Upsilon (5S)$ is above the threshold of $B^{(*)} {\\bar{B}}^{(*)}$ , the coupling constants between $\\Upsilon (5S)$ and $B^{(*)} {\\bar{B}}^{(*)}$ can be determined via experimental data for $\\Upsilon (5S) \\rightarrow B^{(*)} {\\bar{B}}^{(*)}$  [1].", "The experimental branching ratios and the corresponding coupling constants are listed in Table REF .", "Since there is no experimental information on $\\Upsilon (6S) \\rightarrow B^{(*)} {\\bar{B}}^{(*)}$  [1], we choose the coupling constants between $\\Upsilon (6S)$ and $B^{(*)} {\\bar{B}}^{(*)}$ the same values as that of $\\Upsilon (5S)$ .", "Table: The coupling constants of Υ(5S)\\Upsilon (5S) interacting with B (*) B ¯ (*) B^{(*)}{\\bar{B}}^{(*)}.", "Here, we list the corresponding branching ratios of Υ(5S)→B (*) B ¯ (*) \\Upsilon (5S)\\rightarrow B^{(*)}{\\bar{B}}^{(*)}.In order to calculate the process depicted in Fig.", "REF , we also need the photonic coupling to the bottomed mesons.", "The magnetic coupling of the photon to heavy bottom meson is described by the Lagrangian [72], [73] ${\\cal L}_\\gamma = \\frac{e\\beta Q_{ab}}{2} F^{\\mu \\nu } {\\rm Tr}[H_b^\\dagger \\sigma _{\\mu \\nu } H_a ] + \\frac{e Q^\\prime }{2m_{Q}} F^{\\mu \\nu } {\\rm Tr}[H_a^\\dagger H_a \\sigma _{\\mu \\nu }], $ with $H&=&\\left( \\frac{1+ \\unknown.", "{/}{v} }{2} \\right)[\\mathcal {B}^{*\\mu }\\gamma _\\mu -\\mathcal {B}\\gamma _5],$ where $\\beta $ is an unknown constant, $Q= {\\rm diag}\\lbrace 2/3, -1/3, -1/3\\rbrace $ is the light quark charge matrix, and $Q^\\prime $ is the heavy quark electric charge (in units of $e$ ).", "$\\beta \\simeq 3.0$ GeV$^{-1}$ is determined in the nonrelativistic constituent quark model and has been adopted in the study of radiative $D^*$ decays [73].", "In the $b$ and $c$ systems, the $\\beta $ value is the same due to heavy quark symmetry [73].", "In Eq.", "(REF ), the first term is the magnetic moment coupling of the light quarks, while the second one is the magnetic moment coupling of the heavy quark and hence is suppressed by $1/m_Q$ .", "At last, assuming that $X_b$ is an $S$ -wave molecule with $J^{PC}=1^{++}$ given by the superposition of $B^0 {\\bar{B}}^{*0}+c.c$ and $B^- {\\bar{B}}^{*+}+c.c$ hadronic configurations as $|X_b\\rangle = \\frac{1}{2} [ (|B^0{\\bar{B}}^{*0}\\rangle - |B^{*0} {\\bar{B}}^0\\rangle ) + (| B^+ B^{*-}\\rangle - | B^- B^{*+}\\rangle ) ].$ As a result, we can parameterize the coupling of $X_b$ to the bottomed mesons in terms of the following Lagrangian ${\\cal L} = \\frac{1}{2} X_{b\\mu }^{\\dagger } [x_1(B^{*0\\mu } {\\bar{B}}^0 - B^{0} {\\bar{B}}^{*0\\mu })+x_2(B^{*+\\mu } B^- - B^+ B^{*-\\mu })] + h.c.,$ where $x_i$ denotes the coupling constant.", "Since the $X_b$ is slightly below the $S$ -wave $B{\\bar{B}}^*$ threshold, the effective coupling of this state is related to the probability of finding the $B{\\bar{B}}^*$ component in the physical wave function of the bound states and the binding energy, $\\epsilon _{X_b}=m_B+m_{B^*}-m_{X_b}$  [70], [71], [36] $x_i^2 \\equiv 16\\pi (m_B+ m_{B^*})^2 c_i^2 \\sqrt{\\frac{2\\epsilon _{X_b}}{\\mu }} ,$ where $c_i=1/{\\sqrt{2}}$ , $\\mu =m_B m_{B^*}/(m_B+m_{B^*})$ is the reduced mass.", "Here, we should also notice that the coupling constant $x_i$ in Eq.", "(REF ) is based on the assumption that $X_b$ is a shallow bound state where the potential binding the mesons is short-ranged.", "Based on the relevant Lagrangians given above, the decay amplitudes in Fig.", "REF can be generally expressed as follows, $M_{fi}=\\int \\frac{d^4 q_2}{(2\\pi )^4} \\sum _{B^* \\ \\mbox{pol.", "}}\\frac{T_1T_2T_3}{D_1 D_2 D_3}{\\cal F}(m_2,q_2^2)$ where $T_i$ and $D_i = q_i^2-m_i^2 \\ (i=1,2,3)$ are the vertex functions and the denominators of the intermediate meson propagators.", "For example, in Fig.", "REF (a), $T_i \\ (i=1,2,3)$ are the vertex functions for the initial $\\Upsilon (5S)$ , final $X_b$ and photon, respectively.", "$D_i \\ (i=1,2,3)$ are the denominators for the intermediate $B^+$ , $B^{-}$ and $B^{*+}$ propagators, respectively.", "Since the intermediate exchanged bottom mesons in the triangle diagram Fig.", "REF are off-shell, in order to compensate this off-shell effects arising from the intermediate exchanged particle and also the non-local effects of the vertex functions [74], [75], [76], we adopt the following form factors, ${\\cal F}(m_{2}, q_2^2) \\equiv \\left(\\frac{\\Lambda ^2-m_{2}^2}{\\Lambda ^2-q_2^2}\\right)^n,$ where $n=1,2$ corresponds monopole and dipole form factor, respectively.", "$\\Lambda \\equiv m_2+\\alpha \\Lambda _{\\rm QCD}$ and the QCD energy scale $\\Lambda _{\\rm QCD} = 220$ MeV.", "This form factor is supposed and many phenomenological studies have suggested $\\alpha \\simeq 2\\sim 3$ .", "These two form factors can help us explore the dependence of our results on the form factor.", "The explicit expression of transition amplitudes can be found in Appendix (A.2) in Ref.", "[77], where radiative decays of charmonium are studied extensively based on effective Lagrangian approach." ], [ "Numerical Results", "Before proceeding the numerical results, we first briefly review the predictions on mass of $X_b$ .", "The existence of the $X_b$ is predicted in both the tetraquark model [78] and those involving a molecular interpretation [79], [81], [80].", "In Ref.", "[78], the mass of the lowest-lying $1^{++}$ $\\bar{b} \\bar{q} bq$ tetraquark is predicated to be 10504 MeV , while the mass of the $B\\bar{B}^*$ molecular state is predicated to be a few tens of MeV higher [79], [81], [80].", "For example, in Ref.", "[79], the mass was predicted to be 10562 MeV, which corresponds to a binding energy to be 42 MeV, while the mass was predicted to be $(10580^{+9}_{-8})$  MeV, which corresponds to a binding energy $(24^{+8}_{-9})$ MeV in Ref. [81].", "As can be seen from the theoretical predictions, it might be a good approximation and might be applicable if the binding energy is less than 50 MeV.", "In order to cover the range the previous molecular and tetraquark predictions on Ref.", "[78], [79], [81], [80], we present our results up to a binding energy of 100 MeV, and we will choose several illustrative values: $\\epsilon _{X_b} = (5,10,25,50,100)$ MeV.", "In Table REF , we list the predicted branching ratios by choosing the monopole and dipole form factors and three values for the cutoff parameter in the form factor.", "As a comparison, we also list the predicted branching ratios in NREFT approach.", "From this table, we can see that the branching ratios for $\\Upsilon (5S) \\rightarrow \\gamma X_b$ are orders of $10^{-5}$ .", "The results are not sensitive to both the form factors and the cutoff parameter we choose.", "In Fig.", "REF (a), we plot the the branching ratios for $\\Upsilon (5S) \\rightarrow \\gamma X_b$ in terms of the binding energy $\\epsilon _{X_b}$ with the monopole form factors $\\alpha =2.0$ (solid line), $2.5$ (dashed line), and $3.0$ (dotted line), respectively.", "The coupling constant of $X_b$ in Eq.", "(REF ) and the threshold effects can simultaneously influence the binding energy dependence of the branching ratios.", "With the increasing of the binding energy $\\epsilon _{X_b}$ , the coupling strength of $X_b$ increases, and the threshold effects decrease.", "Both the coupling strength of $X_b$ and the threshold effects vary quickly in the small $\\epsilon _{X_b}$ region and slowly in the large $\\epsilon _{X_b}$ region.", "As a result, the behavior of the branching ratios is relatively sensitive at small $\\epsilon _{X_b}$ , while it becomes smooth at large $\\epsilon _{X_b}$ .", "Results with the dipole form factors $\\alpha =2.0$ , $2.5$ , and $3.0$ are shown in Fig.", "REF (b) as solid, dash, and dotted curves, respectively.", "The behavior is similar to that of Fig.", "REF (a).", "We also predict the branching ratios of $\\Upsilon (6S) \\rightarrow \\gamma X_b$ and present the relevant numerical results in Table REF and Fig.", "REF with the monopole and dipole form factors.", "At the same cutoff parameter $\\alpha $ , the predicted rates for $\\Upsilon (6S)\\rightarrow \\gamma X_b$ are a factor of 2-3 smaller than the corresponding rates for $\\Upsilon (5S) \\rightarrow \\gamma X_b$ .", "It indicates that the intermediate $B$ -meson loop contribution to the process $\\Upsilon (6S)\\rightarrow \\gamma X_b$ is smaller than that to $\\Upsilon (5S) \\rightarrow \\gamma X_b$ .", "This is understandable since the mass of $\\Upsilon (6S)$ is more far away from the thresholds of $B^{(*)}B^{(*)}$ than the $\\Upsilon (5S)$ .", "But their branching ratios are also about orders of $10^{-5}$ with a reasonable cutoff parameter $\\alpha =2\\sim 3$ .", "In Ref.", "[51], authors introduced a nonrelativistic effective field theory method to study the meson loop effects of $\\psi ^\\prime \\rightarrow J/\\psi \\pi ^0$ .", "Meanwhile they proposed a power counting scheme to estimate the contribution of the loop effects, which is used to judge the impact of the coupled-channel effects.", "For the diagrams in Fig.", "REF , the vertex involving the initial bottomonium is in $P$ -wave.", "The momentum in this vertex is contracted with the final photon momentum $q$ , and thus should be counted as $q$ .", "The decay amplitude scales as follows, $\\frac{v^5}{(v^2)^3} q^2 \\sim \\frac{q^2}{v},$ where $v$ is understood as the average velocity of the intermediate bottomed mesons.", "As a cross-check, we also present the branching ratios of the decays in the framework of NREFT.", "The relevant transition amplitudes are similar to that given in Ref.", "[36] with only different masses and coupling constants.", "The obtained numerical results for $\\Upsilon (5S) \\rightarrow \\gamma X_b$ and $\\Upsilon (6S) \\rightarrow \\gamma X_b$ in terms of the binding energy are listed in the last column of Table REF and REF , respectively.", "As shown in Table REF , except for the largest binding energy $\\epsilon _{X_b}=100$ MeV, the NREFT predictions of $\\Upsilon (5S) \\rightarrow \\gamma X_b$ are about 1 order of magnitude smaller than the ELA results at the commonly accepted range.", "For $\\Upsilon (6S) \\rightarrow \\gamma X_b$ shown in Table REF , the NREFT predictions are several times smaller than the ELA results in small binding energy range, while the predictions of these two methods are comparable at large binding energy.", "These difference may give some sense of the theoretical uncertainties for the predicted rates and indicates the viability of our model to some extent.", "Here we should notice, for the isoscalar $X_b$ , the pion exchanges might be nonperturbative and produce sizeable effects [81], [82], [83].", "In Ref.", "[81], their calculations show that the relative errors of $C_{0X}$ are about 20% for the $X_b$ .", "Even if we take into account this effect, the estimated order of the magnitude for the branching ratio $\\Upsilon (5S,6S) \\rightarrow \\gamma X_b$ may also be sizeable, which may be measured in the forthcoming BelleII experiments." ], [ "Summary", "In this work, we have investigated the production of $X_b$ in the radiative decays of $\\Upsilon (5S,6S)$ .", "Based on the $B {\\bar{B}}^*$ molecular state picture, we considered its production through the mechanism with intermediate bottom meson loops.", "Our results have shown that the production ratios for the $\\Upsilon (5S,6S) \\rightarrow \\gamma X_b$ are about orders of $10^{-5}$ with a commonly accepted cutoff range $\\alpha =2\\sim 3$ .", "As a cross-check, we also calculated the branching ratios of the decays in the framework of NREFT.", "Except for the large binding energy, the NREFT predictions of $\\Upsilon (5S) \\rightarrow \\gamma X_b$ are about 1 orders of magnitude smaller than the ELA results.", "The NREFT predictions of $\\Upsilon (6S) \\rightarrow \\gamma X_b$ are several times smaller than the ELA results in small binding energy range, while the predictions of these two methods are comparable at large binding energy.", "In Ref.", "[28], [29], we have studied the radiative decays and the hidden bottomonium decays of $X_b$ .", "If we consider that the branching ratios of the isospin conserving process $X_b \\rightarrow \\omega \\Upsilon (1S)$ are relatively large, a search for $\\Upsilon (5S)\\rightarrow \\gamma X_b\\rightarrow \\gamma \\omega \\Upsilon (1S)$ may be possible for the updated BelleII experiments.", "These studies may help us investigate the $X_b$ deeply.", "The experimental observation of $X_b$ will provide us with further insight into the spectroscopy of exotic states and is helpful to probe the structure of the states connected by the heavy quark symmetry." ], [ "Acknowledgements", "This work is supported in part by the National Natural Science Foundation of China (Grant Nos.", "11275113, 11575100, 11505104) and the Natural Science Foundation of Shandong Province (Grant No.", "ZR2015JL001)." ] ]
1606.05118
[ [ "Automatic quality evaluation and (semi-) automatic improvement of OCR\n models for historical printings" ], [ "Abstract Good OCR results for historical printings rely on the availability of recognition models trained on diplomatic transcriptions as ground truth, which is both a scarce resource and time-consuming to generate.", "Instead of having to train a separate model for each historical typeface, we propose a strategy to start from models trained on a combined set of available transcriptions in a variety of fonts.", "These \\emph{mixed models} result in character accuracy rates over 90\\% on a test set of printings from the same period of time, but without any representation in the training data, demonstrating the possibility to overcome the typography barrier by generalizing from a few typefaces to a larger set of (similar) fonts in use over a period of time.", "The output of these mixed models is then used as a baseline to be further improved by both fully automatic methods and semi-automatic methods involving a minimal amount of manual transcriptions.", "In order to evaluate the recognition quality of each model in a series of models generated during the training process in the absence of any ground truth, we introduce two readily observable quantities that correlate well with true accuracy.", "These quantities are \\emph{mean character confidence C} (as given by the OCR engine OCRopus) and \\emph{mean token lexicality L} (a distance measure of OCR tokens from modern wordforms taking historical spelling patterns into account, which can be calculated for any OCR engine).", "Whereas the fully automatic method is able to improve upon the result of a mixed model by only 1-2 percentage points, already 100-200 hand-corrected lines lead to much better OCR results with character error rates of only a few percent.", "This procedure minimizes the amount of ground truth production and does not depend on the previous construction of a specific typographic model." ], [ "=1 [pages=1-last]2016-10-20-histocr.pdf" ] ]
1606.05157
[ [ "Edge-fault-tolerant edge-bipancyclicity of balanced hypercubes" ], [ "Abstract The balanced hypercube, $BH_n$, is a variant of hypercube $Q_n$.", "R.X.", "Hao et al.", "$(2014)$ \\cite{R.X.Hao} showed that there exists a fault-free Hamiltonian path between any two adjacent vertices in $BH_n$ with $(2n-2)$ faulty edges.", "D.Q.", "Cheng et al.", "$(2015)$ \\cite{Dongqincheng2} proved that $BH_n$ is $6$-edge-bipancyclic after $(2n-3)$ faulty edges occur for all $n\\ge2$.", "In this paper, we improve these two results by demonstrating that $BH_n$ is $6$-edge-bipancyclic even when there exist $(2n-2)$ faulty edges for all $n\\ge2$.", "Our result is optimal with respect to the maximum number of tolerated edge faults." ], [ "Introduction", "In the field of parallel and distributed systems, interconnection networks are an important research area.", "Typically, the topology of a network can be represented as a graph in which the vertices represent processors and the edges represent communication links.", "The hypercube network has been proved to be one of the most popular interconnection networks as it possesses many excellent properties such as a recursive structure, regularity, and symmetry.", "It is well known that no network typically meets all the aspects of a given set of requirements.", "Thus, a number of hypercube variants have been proposed, such as folded hypercubes [6], crossed cubes [5], Möbius cubes [4], twisted cubes [8], and shuffle cubes [10] and so on(see [14]).", "The balanced hypercube, proposed by Huang and Wu [9], is also a hypercube variant.", "Similar to hypercubes, balanced hypercubes are bipartite graphs [9] that are vertex-transitive [13] and edge-transitive [19].", "Balanced hypercubes are superior to hypercubes in that they have a smaller diameter as compared to hypercubes.", "Studies on balanced hypercubes can be found in [2], [3], [7], [9], [13], [11], [12], [15], [16], [17], [18], [19].", "For graph definitions and notations, we follow [1].", "A graph $G$ consists of a vertex set $V(G)$ and an edge set $E(G)$ , where an edge is an unordered pair of distinct vertices of $G$ .", "A graph $G$ is called bipartite if its vertex set can be partitioned into two parts $V_1, V_2$ such that every edge has one endpoint in $V_1$ and one in $V_2$ .", "A vertex $v$ is a neighbor of $u$ if $(u, v)$ is an edge of $G$ , and $N_{G}(u)$ denotes the set of all the neighbors of $u$ in $G$ .", "A path $P$ of length $\\ell $ from $x$ to $y$ , denoted by $\\ell $ -path $P$ , is a finite sequence of distinct vertices $\\langle v_0, v_1, \\cdots , v_{\\ell }\\rangle $ such that $x=v_0, y=v_{\\ell }$ , and $(v_i, v_{i+1})\\in E$ for $0\\le i\\le \\ell -1$ .", "We also denote the path $P$ as $\\langle v_0, v_1, \\cdots , v_i, Q, v_j, v_{j+1}, \\cdots , v_{\\ell }\\rangle $ , where $Q$ is the path $\\langle v_i, v_{i+1}, \\cdots , v_j\\rangle $ .", "A cycle $C$ of length $\\ell +1$ is a closed path$\\langle v_0, v_1, \\cdots , v_{\\ell }, v_0\\rangle $ , denoted by $(\\ell +1)$ -cycle $C$ .", "In an interconnection network, the problem of simulating one network by another is modeled as a graph embedding problem.", "In all embedding problems, the cycle embedding problem is one of the most common problem; it refers to finding a cycle of a given length in a graph.", "A graph $G$ of order $|V(G)|$ is $m$ -pancyclic, if it contains every $\\ell $ -cycle for $m\\le \\ell \\le |V(G)|$ .", "A bipartite graph $G$ is $m$ -bipancyclic, if it contains every even $\\ell $ -cycle for $m\\le \\ell \\le |V(G)|$ .", "A graph $G$ is pancyclic (resp.", "bipancyclic) if it is $g$ -pancyclic ($g$ -bipancyclic), where $g=g(G)$ is the girth of $G$ .", "A graph $G$ is vertex-pancyclic (resp.", "edge-pancylic) if every vertex (resp.", "edge) lies on various $\\ell $ -cycles for all $g\\le \\ell \\le V(G)$ .", "A path is called a Hamiltonian path if it contains all the vertices of $G$ .", "A graph $G$ is said to be Hamiltonian connected if there exists a Hamiltonian path between any two vertices of $G$ .", "A bipartite graph is Hamiltonian laceable if there is a Hamiltonian path between any two vertices in different bipartite sets.", "A bipartite graph $G$ is $k$ -fault-tolerant hamiltonian laceable (resp.", "bipancyclic, vertex-bipancyclic, and edge-bipancyclic ) if $G-F$ remains Hamiltonian laceable (resp.", "bipancyclic, vertex-bipancyclic, and edge-bipancyclic ) for $F\\subseteq V(G)\\cup E(G)$ , $|F|\\le k$ .", "A bipartite graph $G$ is $k$ -edge-fault-tolerant Hamiltonian laceable (resp.", "bipancyclic, vertex-bipancyclic, and edge-bipancyclic ) if $G-F$ remains Hamiltonian $lacelabe$ (resp.", "bipancyclic, vertex-bipancyclic, and edge-bipancyclic ) for $F\\subseteq E(G), |F|\\le k$ .", "The balanced hypercube, $BH_n$ , has been studied by many researchers.", "Xu et al.", "[15] proved that $BH_n$ is edge-bipancyclic and Hamiltonian laceable.", "Yang [16] proved that $BH_n$ is bipanconnected.", "Yang [17] also demonstrated that the super connectivity of $BH_n$ is $(4n-4)$ and the super edge-connectivity of $BH_n$ is $(4n-2)$ for $n\\ge 2$ .", "Lü et al.", "[12] proved that $BH_n$ is hyper-Hamiltonian laceable.", "Cheng et al.", "[2] proved that $BH_n$ is $(n-1)$ -vertex-fault-tolerant edge-bipancyclic.", "Hao et al.", "[7] showed that there exists a fault-free Hamiltonian path between any two adjacent vertices in $BH_n$ with $(2n-2)$ faulty edges.", "Zhou et al.", "[18] proved that $BH_n$ is $(2n-2)$ -edge-fault-tolerant Hamiltonian laceable.", "Cheng et al.", "[3] proved that $BH_n$ is $(2n-3)$ edge-fault-tolerant 6-edge-bipancyclic for all $n\\ge 2$ .", "In this paper, we improve the results of Hao et al.", "[7] and Cheng et al.", "[3] by demonstrating that $BH_n$ is $(2n-2)$ edge-fault-tolerant 6-edge-bipancyclic for all $n\\ge 2$ .", "Our result is optimal with respect to the maximum number of tolerated edge faults.", "The rest of this paper is organized as follows.", "In Section 2, we introduce two equivalent definitions of balanced hypercubes and discuss some of their properties.", "In Section 3, we investigate edge-bipancyclic of $BH_n$ with faulty edges.", "Finally, we conclude this paper in Section 4." ], [ "Balanced hypercubes", "Wu and Huang [9] presented two equivalent definitions of $BH_n$ as follows: Definition 2.1 An $n$ -dimensional balanced hypercube $BH_n$ has $2^{2n}$ vertices, each labeled by an $n$ -bit string $(a_0, a_1, \\cdots , a_{n-1})$ , where $a_i\\in \\lbrace 0, 1, 2, 3\\rbrace $ for all $0\\le i\\le n-1$ .", "A arbitrary vertex $(a_0, a_1, \\cdots , a_{i-1}, a_{i}$ , $a_{i+1}, \\cdots , a_{n-1})$ is adjacent to the following $2n$ vertices: $\\begin{array}{l}(1)~~ ((a_0\\pm 1)\\mod {\\ }4, a_1, \\cdots , a_{i-1}, a_{i}, a_{i+1}, \\cdots , a_{n-1})~where~1\\le i\\le n-1, \\vspace{2.125pt} \\\\(2)~~ ((a_0\\pm 1)\\mod {\\ }4, a_1, \\cdots , a_{i-1}, (a_{i}+(-1)^{a_0})\\mod {\\ }4, a_{i+1}, \\cdots , a_{n-1})~where~1\\le i\\le n-1.\\end{array}$ In $BH_n$ , the first coordinate $a_0$ of vertex $(a_0, a_1, \\cdots , a_{n-1})$ is called the inner index, and the second coordinate $a_i(1\\le i\\le n-1)$ is called the $i$ -dimension index.", "From the definition, we have that $N_{BH_n}((a_0, a_1, \\cdots , a_{n-1}))=N_{BH_n}((a_0+2, a_1, \\cdots , a_{n-1}))$ .", "Figure REF shows two balanced hypercubes of dimensional one and two.", "Figure: Illustration of BH 1 BH_1 and BH 2 BH_2Briefly, we assume that `$+, -$ ' for the coordinate of a vertex is an operation with mod 4 in the remainder of the paper.", "Let $X_{j, i}=\\lbrace (a_0, a_1, \\cdots , a_{j-1}, a_{j}, a_{j+1}, \\cdots , a_{n-1})\\mid a_k\\in \\lbrace 0, 1, 2, 3\\rbrace , 0\\le k\\le n-1, a_j=i\\rbrace $ for $1\\le j\\le n-1$ and $ i\\in \\lbrace 0, 1, 2, 3\\rbrace $ and let $BH^{j, i}_{n-1}=BH_n[X_{j, i}]$ .", "Then, $BH_n$ can be divided into four copies: $BH^{j, 0}_{n-1}, BH^{j, 1}_{n-1}, BH^{j, 2}_{n-1}, BH^{j, 3}_{n-1}$ where $BH^{j, i}_{n-1}\\cong BH_{n-1}$ for $i=0, 1, 2, 3$ [2].", "We use $BH^i_{n-1}$ to denote $BH^{n-1, i}_{n-1}$ for $i=0,1, 2, 3$ .", "Definition 2.2 The balanced hypercube $BH_n$ can be constructed recursively as follows: $BH_1$ is a 4-cycle with vertex-set $\\lbrace 0, 1, 2, 3\\rbrace $ .", "$BH_{n}$ is a construct from four copies of $BH_{n-1}: BH^0_{n-1}, BH^1_{n-1}, BH^2_{n-1}, BH^3_{n-1}$ .", "Each vertex $(a_0, a_1, \\cdots $ , $ a_{n-2}, i)$ has two extra adjacent vertices: $\\begin{array}{l}~(1)~ In~ BH^{i+1}_{n-1}: (a_0 \\pm 1, a_1, \\cdots , a_{n-2}, i+1)~if~a_0~is~ even\\vspace{4.25pt}.", "\\\\~(2)~ In~ BH^{i-1}_{n-1}: (a_0 \\pm 1, a_1, \\cdots , a_{n-2}, i-1)~if~a_0~is~ odd.\\end{array}$ Since $BH_n$ is a bipartite graph, then $V(BH_n)$ can be divided into two disjoint parts.", "Obviously, the vertex-set $V_1=\\lbrace a=(a_0, a_1, \\cdots , a_{n-1})\\mid a\\in V(BH_n)$ and $ a_0$ is odd$\\rbrace $ and $V_2=\\lbrace a=(a_0, a_1, \\cdots , a_{n-1})\\mid a\\in V(BH_n)$ and $ a_0$ is even$\\rbrace $ form the desired partition.", "We use black nodes to denote the vertices in $V_1$ and white nodes to denote the vertices in $V_2$ .", "Let $(u, v)$ be an edge of $BH_n$ , if $u$ and $v$ differ only with regard to the inner index, then $(u, v)$ is said to be a 0-dimensional edge.", "If $u$ and $v$ differ not only in terms of the inner index but also with regard to the $i$ -dimension index, then $(u, v)$ is called the $i$ -dimensional edge.", "We use $\\partial D_d(0\\le d\\le n-1)$ to denote the set of all $d$ -dimensional edges.", "There are some known properties about $BH_n$ .", "Lemma 2.3 ([13], [19]) The balanced hypercube $BH_n$ is vertex-transitive and edge-transitive.", "Lemma 2.4 ([18]) The balanced hypercube $BH_n$ is $(2n-2)$ -edge-fault-tolerant Hamiltonian laceable for $n\\ge 2$ .", "Lemma 2.5 ([15]) The balanced hypercube $BH_n$ is edge-bipancyclic for $n\\ge 2$ .", "Lemma 2.6 ([2]) Let $e=(x, y)$ be an arbitrary edge in $BH^{j, 0}_{n-1}$ .", "Then, there exist two internal vertex-disjoint paths $\\langle x, x_1, y_1, x_2, y_2, x_3, y_3, y\\rangle $ and $\\langle x, x^{\\prime }_1, y^{\\prime }_1, x^{\\prime }_2, y^{\\prime }_2, x^{\\prime }_3, y^{\\prime }_3, y\\rangle $ in $BH_n$ such that $(x_i, y_i), (x^{\\prime }_i, y^{\\prime }_i)$$\\in E(BH^{j, i}_{n-1})$ where $1\\le j\\le n-1$ and $i=1, 2, 3$ .", "Lemma 2.7 ([11]) Let $n\\ge 2$ be an integer.", "Then, $BH_n-\\partial D_0$ has four components, and each component is isomorphic to $BH_{n-1}$ .", "Remark.", "The above Lemma shows that one can divide $BH_n$ into four $BH_n$ s by deleting $\\partial D_d$ for any $d\\in \\lbrace 0, 1, \\cdots , n-1\\rbrace $ .", "The four components of $BH_n$ through the deletion of $\\partial D_j$ are $BH^{j, 0}_{n-1}$ , $BH^{j, 1}_{n-1}$ , $BH^{j, 2}_{n-1}$ , and $BH^{j, 3}_{n-1}$ for $1\\le j\\le n-1$ .", "For convenience, we use $BH^{0, 0}_{n-1}, BH^{0, 1}_{n-1}, BH^{0, 2}_{n-1}, and BH^{0, 3}_{n-1}$ to denote the components of $BH_n-\\partial D_0$ throughout this paper.", "A graph $G$ is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for an arbitrary vertex $v$ in $V_{i}$ where $i\\in \\lbrace 0, 1\\rbrace $ , there exists a Hamiltonian path in $G-v$ joining any two different vertices in $V_{1-i}$ .", "Lü et al.", "obtained the following result.", "Lemma 2.8 ([12]) The balanced hypercube $BH_n$ is hyper-Hamiltonian laceable for $n\\ge 1$ .", "In the following, we discuss some properties that are used in the proof of our main results.", "Lemma 2.9 For an arbitrary vertex $u$ in $BH^{j, i}_{n-1}$ where $0\\le j\\le n-1$ , $0\\le i\\le 3$ .", "Suppose that $F\\subseteq E(BH_n), |F|\\le 2n-2$ and $|F\\cap BH^{j, i}_{n-1}|\\le 2n-3$ .", "Then, there exists a 2-path $\\langle u, v, w\\rangle \\subseteq BH_n\\setminus F$ where $u, v\\in BH^{j, i}_{n-1}, w\\in BH_n\\setminus BH^{j,i}_{n-1}$ .", "Proof: Without loss of generality, we can assume that $u=(0, 0, \\cdots , 0)\\in BH^{0}_{n-1}$ .", "Note that $N_{BH^{0}_{n-1}}(u)=2n-2$ and $u$ is a white vertex, there exist $2(2n-2)$ different edges from $N_{BH^{0}_{n-1}}(u)$ to $BH^{3}_{n-1}$ .", "Suppose that $|F\\cap BH^{0}_{n-1}|=k, |F\\cap (BH_n\\setminus BH^{0}_{n-1})|=t$ .", "We have                                                              $\\left\\lbrace \\begin{aligned}&k+t\\le 2n-2; \\\\& k\\le 2n-3.\\end{aligned}\\right.$ Hence, there exists at least one 2-path $\\langle u, v, w\\rangle \\subseteq BH_n\\setminus F$ where $u, v\\in BH^{0}_{n-1}, w\\in BH^{3}_{n-1}$ owing to $2((2n-2)-k)-t\\ge 2(2n-2)-(k+t)-k\\ge 2n-2-k\\ge 1$ .", "See figure REF for illustration.", "Figure: The fault-free path 〈u,v,w〉\\langle u, v, w\\rangle of Lemma Lemma 2.10 Suppose that $e=(u, v)$ is an edge between $BH^{j, i}_{n-1}$ and $BH^{j, i+1}_{n-1}$ where $0\\le j\\le n-1, 0\\le i\\le 3$ for $n\\ge 2$ .", "Then, there exists a cycle $C$ of length 8 in $BH_n\\setminus F$ where $F\\subseteq E(BH_n)$ , $|F|\\le 2n-2$ and $|F\\cap \\partial D_{j}|\\ge 1$ such that $|E(C)\\cap BH^{j, i}_{n-1}|=1$ .", "Proof: By Lemma REF , $BH_n$ is edge-transitive, Without loss of generality, let $j=n-1$ and $u=(0, 0, \\cdots , 0), v=(1, 0, \\cdots , 0, 1)$ .", "There exist $4(n-1)$ edge disjoint paths of length 5 from $N_{BH^{0}_{n-1}}(u)$ to $N_{BH^{1}_{n-1}}(v)$ such that each path has an edge in $BH^2_{n-1}$ and $BH^{3}_{n-1}$ .", "We list them as follows (see figure REF ): $\\begin{array}{l}P_{0, 1}=\\langle (1, 0, \\cdots , 0), (2, 0, \\cdots , 0, 3), (3, 0, \\cdots , 0, 3), (0, 0, \\cdots , 0, 2), (1, 0, \\cdots , 2), (2, 0, \\cdots , 1)\\rangle ; \\\\P_{0, 2}= \\langle (1, 0, \\cdots , 0), (0, 0, \\cdots , 0, 3), (1, 0, \\cdots , 0, 3), (2, 0, \\cdots , 0, 2), (3, 0, \\cdots , 2), (0, 0, \\cdots , 1)\\rangle ;\\\\P_{0, 3}= \\langle (3, 0, \\cdots , 0), (2, 0, \\cdots , 0, 3), (1, 0, \\cdots , 0, 3), (0, 0, \\cdots , 0, 2), (3, 0, \\cdots , 2), (2, 0, \\cdots , 1)\\rangle ;\\\\P_{0, 4}=\\langle (3, 0, \\cdots , 0), (0, 0, \\cdots , 0, 3), (3, 0, \\cdots , 0, 3), (2, 0, \\cdots , 0, 2), (1, 0, \\cdots , 2), (0, 0, \\cdots , 1)\\rangle ;\\\\P_{k, 1}=\\langle (1, \\overbrace{0, \\cdots , 0}^{k-1}, 1, \\overbrace{0, \\cdots , 0, 0}^{n-k-1}), (2, 0, \\cdots , 0, 1, 0, \\cdots , 0, 3), (3, 0, \\cdots , 0, 2, 0, \\cdots , 0, 3), \\vspace{4.25pt}\\\\~~~~~~~~~(0, 0, \\cdots , 0, 2, 0, \\cdots , 0, 2), (1, 0, \\cdots , 0, 3, 0, \\cdots , 0, 2), (2, 0, \\cdots , 0, 3, 0, \\cdots , 0, 1) \\rangle ; \\\\P_{k, 2}= \\langle (1, \\overbrace{0, \\cdots , 0}^{k-1}, 1, \\overbrace{0, \\cdots , 0, 0}^{n-k-1}), (0, 0, \\cdots , 0, 1, 0, \\cdots , 0, 3), (1, 0, \\cdots , 0, 2, 0, \\cdots , 0, 3), \\vspace{4.25pt}\\\\~~~~~~~~~(2, 0, \\cdots , 0, 2, 0, \\cdots , 0, 2), (3, 0, \\cdots , 0, 3, 0, \\cdots , 0, 2), (0, 0, \\cdots , 0, 3, 0, \\cdots , 0, 1)\\rangle ;\\end{array}$ $\\begin{array}{l}P_{k, 3}= \\langle (3, \\overbrace{0, \\cdots , 0}^{k-1}, 1, \\overbrace{0, \\cdots , 0, 0}^{n-k-1}), (2, 0, \\cdots , 0, 1, 0, \\cdots , 0, 3), (1, 0, \\cdots , 0, 2, 0, \\cdots , 0, 3), \\vspace{4.25pt}\\\\~~~~~~~~~(0, 0, \\cdots , 0, 2, 0, \\cdots , 0, 2), (3, 0, \\cdots , 0, 3, 0, \\cdots , 0, 2), (2, 0, \\cdots , 0, 3, 0, \\cdots , 0, 1)\\rangle ;\\\\P_{k, 4}= \\langle (3, \\overbrace{0, \\cdots , 0}^{k-1}, 1, \\overbrace{0, \\cdots , 0, 0}^{n-k-1}), (0, 0, \\cdots , 0, 1, 0, \\cdots , 0, 3), (3, 0, \\cdots , 0, 2, 0, \\cdots , 0, 3), \\vspace{4.25pt}\\\\~~~~~~~~~(2, 0, \\cdots , 0, 2, 0, \\cdots , 0, 2), (1, 0, \\cdots , 0, 3, 0, \\cdots , 0, 2), (0, 0, \\cdots , 0, 3, 0, \\cdots , 0, 1)\\rangle \\end{array}$        where $1\\le k\\le n-1.$ Figure: The 5-paths in Lemma Suppose that $|F\\cap (BH^{n}-\\partial D_{n-1})|=k$ , $|F\\cap \\partial D_{n-1}|=t$ , then $k+t\\le 2n-2$ and $t\\ge 1$ .", "Hence, there exists at least one desired 8-cycle owing to $2((2n-2)-k)-t\\ge 2(2n-2)-(k+t)-k\\ge 2n-2-k\\ge 1$ .", "$\\Box $" ], [ " Edge-bipancyclicity of $BH_n$ under edge faults", "In this section, we consider the edge-bipancyclicity of $BH_n$ for at most $(2n-2)$ faulty edges.", "Let $e=(x, y)$ be an edge between $BH^0_{n-1}$ and $BH^1_{n-1}$ and suppose that $x^{\\prime }, y^{\\prime }\\in BH_n$ such that $N_{BH_n}(x)=N_{BH_n}(x^{\\prime })$ and $N_{BH_n}(y)=N_{BH_n}(y^{\\prime })$ .", "Let $F=\\lbrace (x, y^{\\prime }), (x^{\\prime }, y)\\rbrace $ .", "From the concluding remarks of [3], we have that there does not exists a cycle of length 4 in $BH_n\\setminus F$ that contains $e$ .", "Thus, in the following, we prove that $BH_n$ is $(2n-2)$ edge-fault-tolerant 6-bipancyclic.", "Lemma 3.1 The balanced hypercube $BH_2$ is 2-edge-fault-tolerant 6-bipancyclic.", "Proof: The proof is rather long, and we therefore provide it in Appendix A. Theorem 3.2 The balanced hypercube $BH_n$ is $(2n-2)$ -edge-fault-tolerant edge 6-bipancyclic for $n\\ge 2$ .", "Proof: We prove this theorem by induction on $n$ .", "By Lemma REF , the theorem holds for $n=2$ .", "Assume that it is true for $2\\le k< n$ .", "Let $F$ be any subset of $E(BH_n)$ with $|F|\\le 2n-2$ and $F_i=\\partial D_i\\cap F$ for $0\\le i\\le n-1$ .", "We get $|F|=\\sum _{i=0}^{n-1}|F_i|$ .", "Accordingly, without loss of generality, we can assume that $|F_{n-1}|\\ge |F_{n-2}|\\ge \\cdots \\ge |F_0|$ .", "Let $F^i=F\\cap E(BH^{i}_{n-1})$ for $0\\le i\\le 3$ .", "We obtain $F=F^0\\cup F^1\\cup F^2\\cup F^3\\cup F_{n-1}$ and $|F^0\\cup F^1\\cup F^2\\cup F^3|\\le 2n-4$ .", "Let $e$ be any edge in $BH_n\\setminus F$ and $\\ell $ be any even integer with $6\\le \\ell \\le 2^{2n}$ .", "We need to construct an $\\ell $ -cycle in $BH_n\\setminus F$ containing $e$ .", "Case 1: $e=(u, v)\\notin \\partial D_{n-1}$ .", "Without loss of generality, we can assume that $e\\in BH^0_{n-1}$ .", "Subcase 1.1: $6\\le \\ell \\le 2^{2n-2}$ .", "Since $|F^0|\\le |F^0\\cup F^1\\cup F^2\\cup F^3|\\le 2n-4$ , by induction hypothesis, it holds .", "Subcase 1.2: $2^{2n-2}+2\\le \\ell \\le 2^{2n-1}+6$ .", "By induction hypothesis, there exists a fault-free Hamiltonian cycle $C$ in $BH^0_{n-1}$ containing $e$ , say $\\langle c^1, c^2, \\cdots $ , $c^{2^{2n-2}}, c^1\\rangle $ where $c^1=u, c^{2^{2n-2}}=v$ .", "We can observe that $C\\setminus \\lbrace e\\rbrace $ is a $(2^{2n-2}-1)$ -path.", "Then, $M=\\lbrace (c^1, c^{2^{2n-3}+6}), \\cdots $ , $(c^i, c^{2^{2n-3}+i+5}), \\cdots $ , $(c^{2^{2n-3}-5}$ , $c^{2^{2n-2}})\\rbrace $ is a set with $2^{(2n-3)}-5$ pairs of distinct vertices of $BH^0_{n-1}$ such that $d_{C\\setminus \\lbrace e\\rbrace }(c^i, c^{2^{2n-3}+i+5})=2^{2n-3}+5$ for all $1\\le i\\le 2^{2n-3}-5$ .", "Thus, $c^i$ and $c^{2^{2n-3}+i+5}$ are in different partite sets.", "There exists at least one pair $(c^t, c^{2^{2n-3}+t+5})$ in $M$ such that $\\begin{array}{l}|F\\cap \\lbrace e_1, e_2\\mid e_1, e_2 {\\rm ~are~ two~} (n-1) {\\rm \\text{-}dimensional ~edges ~incident ~with~} c^t\\rbrace |\\le 1~{\\rm and}\\vspace{4.25pt} \\\\|F\\cap \\lbrace e_3, e_4\\mid e_3, e_4 {\\rm ~are~ two~} (n-1) {\\rm \\text{-}dimensional ~edges ~incident ~with~} c^{2^{2n-3}+t+5}\\rbrace |\\le 1\\vspace{4.25pt}\\end{array}$ owing to $2\\cdot (2^{2n-3}-5)>2n-2$ for all $n\\ge 3$ .", "Without loss of generality, let $c^t$ be a white vertex and $c^{2^{2n-3}+t+5}$ be a black vertex.", "Then, there exist two fault-free $(n-1)$ -dimensional edges $(c^t, v^1), (c^{2^{2n-3}+t+5}, u^3)$ where $v^1\\in BH^1_{n-1}$ and $u^3\\in BH^3_{n-1}$ .", "Let $P_0=\\langle c^{2^{2n-3}+t+5}, c^{2^{2n-3}+t+6}, \\cdots $ , $c^{2^{2n-2}-1}, v, u, c^2, \\cdots , c^t\\rangle $ .", "Thus, $P_0$ is a $(2^{2n-3}-5)$ -path that contains $(u, v)$ .", "By Lemma REF , there exists a fault-free 2-path $\\langle u^3, v^3, u^2\\rangle $ and a fault-free 2-path $\\langle v^1, u^1, v^2\\rangle $ where $u^i, v^i\\in BH^{i}_{n-1}$ for $1\\le i\\le 3$ .", "Since $|F^2|\\le 2n-4$ , by induction hypothesis, there exists a Hamiltonian cycle $C_2$ in $BH^2_{n-1}\\setminus F$ .", "Thus, there exist two fault-free path $P^{\\prime }_2, P^{\\prime \\prime }_2$ in $BH^2_{n-1}$ joining $u^2$ and $v^2$ with length $|V(P^{\\prime }_2)|$ and $2^{2n-2}-|(V(P^{\\prime }_2))|$ , respectively, where $1\\le |V(P^{\\prime }_2)|\\le 2^{2n-3}-1$ .", "Subcase 1.2.1: $|V(P^{\\prime }_2)|=2^{2n-3}-1$ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ , where $\\ell _i$ satisfies one of the following conditions for $i=0, 1, 2, 3$ .", "$\\begin{array}{lllll}\\ell _0=2^{2n-3}-5,& \\ell _1=1,& \\ell _2=2^{2n-3}+1, &\\ell _3=1~&\\rm {or} \\\\\\ell _0=2^{2n-3}-5, &5\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=2^{2n-3}-1, & \\ell _3=1~&\\rm {or} \\\\\\ell _0=2^{2n-3}-5,& 5\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=2^{2n-3}-1, &5\\le \\ell _3\\le 2^{2n-2}-1.\\end{array}$ Since $|F^i|\\le 2n-4$ for $i=1, 3$ , by the induction hypothesis, there exists an $(\\ell _i+1)$ -cycle $C_i$ in $BH^i_{n-1}\\setminus F$ containing $(u^i, v^i)$ if $5\\le \\ell _i\\le 2^{2n-2}-1$ for $i=1, 3$ .", "Let $\\begin{array}{lll}P_0=\\langle c^{2^{2n-3}+t+5}, c^{2^{2n-3}+t+6}, \\cdots , c^{2^{2n-2}-1}, v, u, c^2, \\cdots , c^t\\rangle {\\rm ~ with~ length}~ \\ell _0 ,\\vspace{4.25pt} \\\\P_1=\\left\\lbrace \\begin{aligned}& (v^1, u^1)&\\rm {if} &~~\\ell _1=1,\\\\& C_1-(v^1, u^1)&\\rm {if} &~~5\\le \\ell _1\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_2=\\left\\lbrace \\begin{aligned}& P^{\\prime }_2& \\rm {if}&~~\\ell _2=2^{2n-3}-1,\\\\& P^{\\prime \\prime }_2&\\rm {if}&~~\\ell _2=2^{2n-3}+1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_3=\\left\\lbrace \\begin{aligned}& (v^3, u^3)&\\rm {if}&~~\\ell _3=1,\\\\& C_3-(v^3, u^3)&\\rm {if}&~~5\\le \\ell _3\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt}\\end{array}$ Then, $C=\\langle c^t, v^1, P_1, u^1, v^2, P_2, u^2, v^3, P_3, u^3, c^{2^{2n-3}+t+5}, P_0, c^t\\rangle $ ( see figure REF ) is the desired cycle.", "Figure: Illustration for the cycle CC of subcase 1.2.1 and subcase 1.2.2 in theorem .Subcase 1.2.2: $1\\le |V(P^{\\prime }_2)|\\le 2^{2n-3}-3$ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ , where $\\ell _i$ satisfies one of the following conditions for $i=0, 1, 2, 3$ .", "$\\begin{array}{lllll}\\ell _0=2^{2n-3}-5, &5\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=|V(P^{\\prime }_2)|, &\\ell _3=1~&\\rm {or} \\\\\\ell _0=2^{2n-3}-5,& 5\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=|V(P^{\\prime }_2)|, &5\\le \\ell _3\\le 2^{2n-2}-1.\\vspace{4.25pt}\\end{array}$ Since $|F^i|\\le 2n-4$ for $i=1, 3$ , by the induction hypothesis, there exists an $(\\ell _i+1)$ -cycle $C_i$ in $BH^i_{n-1}\\setminus F$ containing $(u^i, v^i)$ if $5\\le \\ell _i\\le 2^{2n-2}-1$ for $i=1, 3$ .", "Let $\\begin{array}{lll}P_0=\\langle c^{2^{2n-3}+t+5}, c^{2^{2n-3}+t+6}, \\cdots , c^{2^{2n-2}-1}, v, u, c^2, \\cdots , c^t\\rangle {~\\rm ~with~length~} \\ell _0,\\vspace{4.25pt} \\\\P_1=\\left\\lbrace \\begin{aligned}& (v^1, u^1)&\\rm {if} &~~\\ell _1=1,\\\\& C_1-(v^1, u^1)&\\rm {if} &~~5\\le \\ell _1\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_2=P^{\\prime }_2, \\vspace{4.25pt}\\\\P_3=\\left\\lbrace \\begin{aligned}& (v^3, u^3)&\\rm {if}&~~\\ell _3=1,\\\\& C_3-(v^3, u^3)&\\rm {if}&~~5\\le \\ell _3\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt}\\end{array}$ Then, $C=\\langle c^t, v^1, P_1, u^1, v^2, P_2, u^2, v^3, P_3, u^3, c^{2^{2n-3}+t+5}, P_0, c^t\\rangle $ ( see figure REF ) is the desired cycle.", "Subcase 1.3: $2^{2n-1}+8\\le \\ell \\le 2^{2n}$ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ , where $\\ell _i$ satisfies one of the following conditions for $i=0, 1, 2, 3$ .", "$\\begin{array}{lllll}\\ell _0=2^{2n-2}-1, & 5\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=2^{2n-2}-1, &\\ell _3=1~&\\rm {or} \\\\\\ell _0=2^{2n-2}-1,& 5\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=2^{2n-2}-1, &5\\le \\ell _3\\le 2^{2n-2}-1.\\vspace{4.25pt}\\end{array}$ By the induction hypothesis, there exists a fault-free Hamiltonian cycle $C_0$ in $BH^0_{n-1}$ containing $e$ , say $\\langle c^1, c^2, \\cdots , c^{2^{2n-2}}, c^1\\rangle $ with $c^1=u, c^{2^{2n-2}}=v$ .", "Let $M=\\lbrace (c^1, c^2), \\cdots , (c^{2i-1}, c^{2i}), \\cdots $ , $(c^{2^{2n-2}-1}, c^{2^{2n-2}}) \\rbrace $ , then $M$ is a set with $2^{2n-3}$ mutually disjoint edges.", "There exists an edge $(c^{2t-1}, c^{2t})$ in $M$ such that $ \\begin{array}{l}|F\\cap \\lbrace e_1, e_2\\mid e_1, e_2 {\\rm ~are~ two~} (n-1) {\\rm \\text{-}dimensional ~edges ~incident ~with~} c^{2t-1}\\rbrace |\\le 1 {\\rm ~and} \\vspace{4.25pt} \\\\|F\\cap \\lbrace e_3, e_4\\mid e_3, e_4 {\\rm ~are~ two~} (n-1) {\\rm \\text{-}dimensional ~edges ~incident ~with~} c^{2t}\\rbrace |\\le 1\\vspace{4.25pt}\\end{array}$ since $2\\cdot (2^{2n-3})>2n-2$ for all $n\\ge 3$ .", "Let $(c^{2t-1}, v^1), (c^{2t}, u^3)$ be two fault-free $(n-1)$ -dimensional edges where $v^1\\in BH^1_{n-1}$ , $u^3\\in BH^3_{n-1}$ .", "By Lemma REF , there exists a fault-free 2-path $\\langle u^3, v^3, u^2\\rangle $ and a fault-free 2-path $\\langle v^1, u^1, v^2\\rangle $ where $v^i, u^i\\in BH^i_{n-1}$ for $i=1, 2, 3$ .", "By Lemma REF , there exists a Hamiltonian path $P_2$ in $BH^2_{n-1}\\setminus F$ joining $v^2$ to $u^2$ .", "Note that $|F^i|\\le 2n-4$ , by the induction hypothesis, there exists an $(\\ell _i+1)$ -cycle $C_i$ in $BH^i_{n-1}\\setminus F$ containing $(u^i, v^i)$ where $5\\le \\ell _i\\le 2^{2n-2}-1$ for $i=1, 3$ .", "Let $\\begin{array}{lll}P_0=C_0-(c^{2t-1}, c^{2t}),\\vspace{4.25pt} \\\\P_1=\\left\\lbrace \\begin{aligned}& (v^1, u^1)&\\rm {if} &~~\\ell _1=1,\\\\& C_1-(v^1, u^1)&\\rm {if} &~~5\\le \\ell _1\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_2 {\\rm ~be~ the~ Hamiltonian~ path~ of~} BH^2_{n-1} {\\rm ~joining~} v^2 {\\rm ~to~} u^2, \\vspace{4.25pt}\\\\P_3=\\left\\lbrace \\begin{aligned}& (v^3, u^3)&\\rm {if}&~~\\ell _3=1,\\\\& C_3-(v^3, u^3)&\\rm {if}&~~5\\le \\ell _3\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt}\\end{array}$ Then, $C=\\langle c^{2t-1}, v^1, P_1, u^1, v^2, P_2, u^2, v^3, P_3, u^3, c^{2t}, P_0, c^{2t-1}\\rangle $ (see figure REF ) is the desired cycle.", "Figure: Illustration for the cycle CC of subcase 1.3 in theorem .Case 2: $e=(u, v)\\in \\partial D_{n-1}$ .", "Subcase 2.1: $|F_{n-1}|\\le 2n-3$ .", "We divide $BH_n$ into four parts, $BH^{n-2, 0}_{n-1}, BH^{n-2, 1}_{n-1}, BH^{n-2, 2}_{n-1}, and BH^{n-2, 3}_{n-1}$ .", "If $|F|\\le 2n-3$ , then, $|F\\cap (\\cup _{i=0}^{3}BH^{n-2, i}_{n-1})|\\le 2n-3$ .", "If $|F|=2n-2$ , note that $|F_{n-1}|\\ge |F_{n-2}|\\ge \\cdots \\ge |F_0|$ , $|F_{n-1}|\\le 2n-3$ , we have $|F_{n-2}|= 1$ .", "As a result, $|F\\cap (\\cup _{i=0}^{3}BH^{n-2, i}_{n-1})|\\le 2n-3$ .", "Subcase 2.1.1: $|F\\cap BH^{n-2, i}_{n-1}|\\le 2n-4$ for all $i=0, 1, 2, 3$ .", "By a similar discussion as case 1, we obtain the result.", "Subcase 2.1.2: There exists an $i\\in \\lbrace 0, 1, 2, 3\\rbrace $ such that $|F\\cap BH^{n-2, i}_{n-1}|=2n-3$ .", "Without loss of generality, we can assume that $|F\\cap BH^{n-2, 0}_{n-1}|=2n-3$ .", "Thus, $|F\\cap (BH_n\\setminus BH^{n-2, 0}_{n-1})|\\le 1$ and $|F\\cap BH^{n-2, i}_{n-1}|=0$ for $i=1, 2, 3$ .", "Subcase 2.1.2.1: $e\\in BH^{n-2, 0}_{n-1}$ .", "Subcase 2.1.2.1.1: $\\ell =6$ .", "Note that $e$ is a fault-free edge and there are $(4n-6)$ different 2-paths in $BH^{n-2, 0}_{n-1}$ containing $e$ .", "Since $4n-6-(2n-3)=2n-3\\ge 1$ , there exists at least one fault-free 2-path in $BH^{n-2, 0}_{n-1}$ containing $e$ , say $\\langle u, v, w\\rangle $ .", "Without loss of generality, let $v$ be a black vertex and $u, w$ be two white vertices.", "Notice that $|F_{n-2}|\\le 1$ , we obtain that there exists two fault-free $(n-2)$ -dimensional edges $(u, u^1), (w, w^1)$ where $u^1, w^1\\in BH^{n-2, 1}_{n-1}$ .", "It is easy to verify that $d(u^1, w^1)=2$ .", "Suppose that $v^1$ is the vertex that is adjacent to both $w^1$ and $u^1$ .", "Since $|F\\cap BH^{n-2, 1}_{n-1}|=0$ , then $C=\\langle u, v, w, w^1, v^1, u^1, u\\rangle $ (see figure REF ) is the desired cycle.", "Figure: Illustration for the cycle CC of subcase 2.1.2.1.1 in theorem Subcase 2.1.2.1.2: $\\ell =8$ .", "By Lemma REF , there are two 8-cycles $C_1, C_2$ in $BH_n$ containing $e$ such that $E(C_1)\\cap E(C_2)=e$ and $|E(C_i)\\cap BH^{n-2, j}_{n-1}|=1$ for $i=1, 2, j=0, 1, 2, 3$ .", "Note that $|F\\cap (BH_n\\setminus BH^{n-2, 0}_{n-1})|\\le 1$ .", "There exists at least one fault-free 8-cycle that contains $e$ , say $\\langle u, v^1, u^1, v^2, u^2, v^3, u^3, v, u\\rangle $ where $u^i, v^i\\in BH^{n-2, i}_{n-1}$ for $1\\le i\\le 3$ .", "Subcase 2.1.2.1.3: $10\\le \\ell \\le 3\\cdot 2^{2n-2}+2$ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ , where $\\ell _i$ satisfies one of the following conditions for $i=0, 1, 2, 3$ .", "$\\begin{array}{llll}\\ell _0=1, 3\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=1, &\\ell _3=1~&\\rm {or} \\\\\\ell _0=1, 3\\le \\ell _1\\le 2^{2n-2}-1, &3\\le \\ell _2\\le 2^{2n-2}-1, & \\ell _3=1 &\\rm {or}\\\\\\ell _0=1, 3\\le \\ell _1\\le 2^{2n-2}-1, &3\\le \\ell _2\\le 2^{2n-2}-1, &3\\le \\ell _3\\le 2^{2n-2}-1.\\vspace{4.25pt}\\end{array}$ Let $\\langle u, v^1, u^1, v^2, u^2, v^3, u^3, v, u\\rangle $ be a fault-free 8-cycle where $u^i, v^i\\in BH^{n-2, i}_{n-1}$ for $1\\le i\\le 3$ .", "Since $|F\\cap BH^{n-2, i}_{n-1}|=0$ for $i=1, 2, 3$ .", "By Lemma REF , there exists an $(\\ell _i+1)$ -cycle $C_i$ in $BH^{n-2, i}_{n-1}$ containing $(u^i, v^i)$ where $3\\le \\ell _i\\le 2^{2n-2}-1$ for $1\\le i\\le 3$ .", "Let $\\begin{array}{lll}P_1=\\left\\lbrace \\begin{aligned}& (v^1, u^1)&\\rm {if} &~~\\ell _1=1,\\\\& C_1-(v^1, u^1)&\\rm {if} &~~3\\le \\ell _1\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_2=\\left\\lbrace \\begin{aligned}& (v^2, u^2)&\\rm {if} &~~\\ell _2=1,\\\\& C_2-(v^2, u^2)&\\rm {if} &~~3\\le \\ell _2\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_3=\\left\\lbrace \\begin{aligned}& (v^3, u^3)&\\rm {if}&~~\\ell _3=1,\\\\& C_3-(v^3, u^3)&\\rm {if}&~~3\\le \\ell _3\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt}\\end{array}$ Then, $C=\\langle u, v^1, P_1, u^1, v^2, P_2, u^2, v^3, P_3, u^3, v, u\\rangle $ (see figure REF ) forms the desired cycle.", "Figure: Illustration for the cycle CC of subcase 2.1.2.1.3 in theorem Subcase 2.1.2.1.4: $3\\cdot 2^{2n-2}+4\\le \\ell \\le 2^{2n} $ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ , where $\\ell _0= 2^{2n-2}-1, 3\\le \\ell _i\\le 2^{2n-2}-1$ for $i=1, 2, 3$ .", "Let $\\bar{e}=(u^0, v^0)$ be any faulty edge in $BH^{n-2, 0}_{n-1}$ .", "By the induction hypothesis, there exists a Hamiltonian cycle $C_0$ in $BH^{n-2, 0}_{n-1}-F+\\lbrace \\bar{e}\\rbrace $ containing $e$ .", "Obviously, $|F\\cap E(C_0)|\\le 1$ .", "If $|F\\cap E(C_0)|= 1$ , then $\\bar{e}\\in E(C_0)$ , we can assume that $(a^0, b^0)=\\bar{e}$ .", "If $|F\\cap E(C_0)|= 0$ , let $(a^0, b^0)$ be any edge in $E(C_0)\\setminus \\lbrace e\\rbrace $ .", "Note that $|F\\cap (BH_n\\setminus BH^{n-2, 0}_{n-1})|\\le 1$ , by Lemma REF , there exists a fault-free 8-cycle $\\langle a^0, b^1, a^1, b^2, a^2, b^3, a^3, b^0, a^0\\rangle $ in $BH_n$ where $a^i, b^i\\in BH^{n-2, i}_{n-1}$ for $i=0, 1, 2, 3$ .", "Note that $|F\\cap BH^{n-2, i}_{n-1}|=0$ for $1\\le i\\le 3$ , by Lemma REF , there exists an $(\\ell _i+1)$ -cycle $C_i$ in $BH^{n-2, i}_{n-1}$ containing $(a^i, b^i)$ where $3\\le \\ell _i\\le 2^{2n-2}-1$ for $1\\le i\\le 3$ .", "Let $P_i=C_i-(b^i, a^i)$ for $i=0, 1, 2, 3$ , then $C=\\langle a^0, b^1, P_1, a^1, b^2, P_2, a^2, b^3, P_3, a^3, b^0, P_0, a^0\\rangle $ (see figure REF ) forms the desired cycle.", "Figure: Illustration for the cycle CC of subcase 2.1.2.1.4 in theorem Subcase 2.1.2.2: $e\\in BH^{n-2, i}_{n-1}$ where $i=1, 2, 3$ .", "Without loss of generality, we assume that $e\\in BH^{n-2, 1}_{n-1}$ .", "Subcase 2.1.2.2.1: $6\\le \\ell \\le 2^{2n-2}$ .", "Since $|F\\cap BH^{n-2, 1}_{n-1}|=0$ , by the induction hypothesis, it holds.", "Subcase 2.1.2.2.2: $2^{2n-2}+2\\le \\ell \\le 2^{2n-1}-2$ .", "We can represent $\\ell =\\ell _1+\\ell _2+2$ , where $2\\le \\ell _1\\le 2^{2n-2}-2, \\ell _2=2^{2n-2}-2$ .", "Since $|F\\cap BH^{n-2, 1}_{n-1}|=0$ , by Lemma REF , there exists a Hamiltonian cycle $C_1$ in $BH^{n-2, 1}_{n-1}$ containing $e$ , say $\\langle c^0, c^1, \\cdots , c^{2^{2n-2}-1}, c^0\\rangle $ , where $c^0=u, c^1=v$ .", "Let $\\ell _1$ be an even integer.", "Then, $c^{\\ell _1}$ is a white vertex and $\\langle u, c^1, c^2, \\cdots , c^{\\ell _1}\\rangle $ is an $\\ell _1$ -path in $BH^{n-2, 1}_{n-1}$ containing $e$ where $2\\le \\ell _1\\le 2^{2n-2}-2$ .", "Notice that $|F_{n-2}|\\le 1$ .", "We can assume that $(c^{\\ell _1}, u^2), (u, v^2)$ are two fault-free $(n-2)$ -dimensional edges where $u^2, v^2\\in BH^{n-2, 2}_{n-1}$ since every vertex has two extra neighbors.", "By Lemma REF , there exists a $(2^{2n-2}-2)$ -path in $BH^{n-2, 2}_{n-1}$ joining $u^2$ to $v^2$ .", "Let $\\begin{array}{l}P_1=\\langle u, v, c^2, c^3, \\cdots , c^{\\ell _1}\\rangle , \\\\P_2~{\\rm be~} {\\rm the~ path~of~length~}2^{2n-2}-2~ {\\rm in}~ BH^{n-2, 2}_{n-1}~{\\rm joining~}u^2~{\\rm and~} v^2.\\vspace{4.25pt}\\end{array}$ Then, the cycle $C=\\langle u, P_1, c^{\\ell _1}, u^2, P_2, v^2, u\\rangle $ (see figure REF )forms the desired cycles.", "Figure: Illustration for the cycle CC of subcase 2.1.2.2.2 in theorem Subcase 2.1.2.2.3: $2^{2n-1}\\le \\ell \\le 2^{2n-1}+8$ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ , where $\\ell _0=1, \\ell _1=5, \\ell _2=2^{2n-2}-1, 2^{2n-2}-9\\le \\ell _3\\le 2^{2n-2}-1$ .", "Let $\\langle u, v, w^1, x^1, y^1, z^1\\rangle $ be a fault-free 5-path of $BH^{n-2, 1}_{n-1}$ and $(z^1, u^0), (u, v^2)$ be two fault-free $(n-2)$ -dimensional edges where $u^0\\in BH^{n-2, 0}_{n-1}, v^2\\in BH^{n-2, 2}_{n-1}$ .", "By Lemma REF , there exists a 2-path $\\langle u^0, v^0, u^3\\rangle $ and a 2-path $\\langle u^3, v^3, u^2\\rangle $ where $u^i, v^i\\in BH^{n-2, i}_{n-1}$ .", "By Lemma REF , there exists a $(\\ell _3+1)$ -cycle of $BH^{n-2, 3}_{n-1}$ containing $(u^3, v^3)$ where $2^{2n-2}-5\\le \\ell _3\\le 2^{2n-2}-1$ .", "By Lemma REF , there exists a Hamiltonian path $P_2$ in $BH^{n-2, 2}_{n-1}$ joining $u^2$ and $v^2$ .", "Let $P_1= \\langle u, v, w^1, x^1, y^1, z^1\\rangle , P_3=C_3-(u_3, v_3)$ .", "Then, $C=\\langle u, P_1, z^1, u^0, v^0, u^3, P_3, v^3, u^2, P_2, v^2, u\\rangle $ (see figure REF ) is the desired cycle.", "Figure: Illustration for the cycle CC of subcase 2.1.2.2.3 in theorem Subcase 2.1.2.2.4: $2^{2n-1}+10\\le \\ell \\le 2^{2n}$ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ where $5\\le \\ell _0\\le 2^{2n-2}-1, \\ell _1= 2^{2n-2}-1, \\ell _2=2^{2n-2}-1, 3\\le \\ell _3\\le 2^{2n-2}-1.\\vspace{4.25pt}$ Let $(a^0, b^0)$ be a faulty edge in $BH^{n-2, 0}_{n-1}$ , where $a^0$ is a white vertex.", "Assume that $(a^0, b^1), (b^0, a^3)$ , $(a^3, b^3)$ , and $(b^3, a^2)$ are fault-free edges where $a^i, b^i\\in BH^{n-2, i}_{n-1}$ for $i=0, 1, 2, 3$ .", "Note that $|F\\cap BH^{n-2, 1}_{n-1}|=0$ , by Lemma REF , there exits a Hamiltonian cycle $C_1$ in $BH^{n-2, 1}_{n-1}$ containing $e$ .", "Suppose that $N_{C_1}(b^1)=\\lbrace a^1, c^1\\rbrace $ .", "Thus, $(b^1, a^1)\\ne e$ or $(b^1, c^1)\\ne e$ .", "Without loss of generality, assume that $(b^1, a^1)\\ne e$ .", "Note that $|N_{BH^{n-2, 2}_{n-1}}(a^1)|=2$ and $|F_{n-2}|\\le 1$ .", "Suppose that $(a^1, b^2)$ is a fault-free edge where $b^2\\in BH^{n-2, 2}_{n-1}$ .", "By Lemma REF , there exists a fault-free Hamiltonian path $P_2$ in $BH^{n-2, 2}_{n-1}$ joining $a^2$ and $b^2$ .", "By the induction hypothesis, there exists an $(\\ell _0+1)$ -cycle $C_0$ in $BH^{n-2, 0}_{n-1}-F+{(a^0, b^0)}$ containing $(a^0, b^0)$ where $5\\le \\ell _0\\le 2^{2n-2}-1$ .", "By Lemma REF , there exists an $(\\ell _3+1)$ -cycle $C_3$ in $BH^{n-2, 3}_{n-1}$ containing $a^3, b^3$ where $3\\le \\ell _3\\le 2^{2n-2}-1$ .", "Let $\\begin{array}{lll}P_0= C_0-(a^0, b^0), \\\\P_1=C_1-(b_1, a_1), \\\\P_2 ~{\\rm be~ the~ Hamiltonian~path~joining~}a^2~{\\rm and~}b^2,~\\\\P_3= C_3-(a^3, b^3).\\vspace{4.25pt}\\end{array}$ Then, $C=\\langle a^0, P_0, b^0, a^3, P_3, b^3, a^2, P_2, b^2, a^1, P_1, b^1, a^0\\rangle $ (see figure REF ) is the desired cycle.", "Figure: Illustration for the cycle CC of subcase 2.1.2.2.4 in theorem Subcase 2.2: $|F_{n-1}|= 2n-2$ .", "Subcase 2.2.1: $\\ell =6$ .", "Without loss of generality, we can assume that $e\\in BH^{n-2, 0}_{n-1}$ .", "If $|F\\cap BH^{n-2, 0}_{n-1}|\\le 2n-4$ , by the induction hypothesis, there exists a 6-cycle in $BH^{n-2, 0}_{n-1}$ .", "Thus, we assume that $|F\\cap BH^{n-2, 0}_{n-1}|= 2n-3$ or $2n-2$ .", "Note that $e$ is a fault-free edge and there are $(4n-6)$ different 2-paths in $BH^{n-2, 0}_{n-1}$ containing $e$ .", "Since $4n-6-(2n-2)=2n-4\\ge 1$ , there exists at least one fault-free 2-path in $BH^{n-2, 0}_{n-1}$ containing $e$ , say $\\langle u, v, w\\rangle $ .", "Without loss of generality, let $v$ be a black vertex and $u, w$ be two white vertices.", "Notice that $|F_{n-2}|=0$ , we can assume that $(u, u^1), (w, w^1)$ are two fault-free $(n-2)$ -dimensional edges where $u^1, w^1\\in BH^{n-2, 1}_{n-1}$ .", "It is easy to check that $d(u^1, w^1)=2$ .", "Suppose that $v^1$ is the vertex that is adjacent to both $w^1$ and $u^1$ .", "Let $\\bar{v}^1$ be the vertex such that $v^1$ and $\\bar{v}^1$ differ in only the inner index.", "Then, $(w^1, \\bar{v}^1), (u^1, \\bar{v}^1)\\in E(BH_n)$ .", "Since $|F\\cap BH^{n-2, 1}_{n-1}|\\le 1$ , then $\\langle u, v, w, w^1, v^1, u^1, u\\rangle $ or $\\langle u, v, w, w^1, \\bar{v}^1, u^1, u\\rangle $ is the desired cycle.", "Subcase 2.2.2: $\\ell =8$ .", "By Lemma REF , there exists a fault-free 8-cycle $\\langle u=u^0, v=v^1, u^1, v^2, u^2, v^3, u^3, v^0, u\\rangle $ .", "Subcase 2.2.3: $10\\le \\ell \\le 2^{2n}$ .", "We can represent $\\ell =\\ell _0+\\ell _1+\\ell _2+\\ell _3+4$ , where $\\ell _i$ satisfies one of the following conditions for $i=0, 1, 2, 3$ .", "$\\begin{array}{lllll}3\\le \\ell _0\\le 2^{2n-2}-1, & \\ell _1=1, &\\ell _2=1, &\\ell _3=1~&{\\rm or}\\\\3\\le \\ell _0\\le 2^{2n-2}-1, &3\\le \\ell _1\\le 2^{2n-2}-1, &\\ell _2=1, &\\ell _3=1~&{\\rm or}\\\\3\\le \\ell _0\\le 2^{2n-2}-1, &3\\le \\ell _1\\le 2^{2n-2}-1, &3\\le \\ell _2\\le 2^{2n-2}-1, &\\ell _3=1~&{\\rm or}\\\\3\\le \\ell _0\\le 2^{2n-2}-1, &3\\le \\ell _1\\le 2^{2n-2}-1, &3\\le \\ell _2\\le 2^{2n-2}-1, &3\\le \\ell _3\\le 2^{2n-2}-1.\\vspace{4.25pt}\\end{array}$ Note that $F_{n-1}=2n-2$ , we have $F\\cap BH^{i}_{n-1}=0$ for all $i=0, 1, 2, 3$ .", "By Lemma REF , there exists an $(\\ell _i+1)$ -cycle $C_i$ in $BH^i_{n-1}$ containing $(u^i, v^i)$ where $3\\le \\ell _i\\le 2^{2n-2}-1$ for $i=0, 1, 2, 3$ .", "Let $\\begin{array}{lll}P_0=\\left\\lbrace \\begin{aligned}& (v^0, u^0)&\\rm {if} &~~\\ell _0=1,\\\\& C_0-(v^0, u^0)&\\rm {if} &~~3\\le \\ell _0\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_1=\\left\\lbrace \\begin{aligned}& (v^1, u^1)&\\rm {if} &~~\\ell _1=1,\\\\& C_1-(v^1, u^1)&\\rm {if} &~~3\\le \\ell _1\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_2=\\left\\lbrace \\begin{aligned}& (v^2, u^2)&\\rm {if} &~~\\ell _2=1,\\\\& C_2-(v^2, u^2)&\\rm {if} &~~3\\le \\ell _2\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt} \\\\P_3=\\left\\lbrace \\begin{aligned}& (v^3, u^3)&\\rm {if}&~~\\ell _3=1,\\\\& C_3-(v^3, u^3)&\\rm {if}&~~3\\le \\ell _3\\le 2^{2n-2}-1,\\end{aligned}\\right.\\vspace{4.25pt}\\end{array}$ Then, $C=\\langle v^0, P_0, u^0, v^1, P_1, u^1, v^2, P_2, u^2, v^3, P_3, u^3, v^0\\rangle ~$ (see figure REF ) forms the desired cycle.", "$\\Box $ Figure: Illustration for the cycle CC of subcase 2.2.3 in theorem" ], [ "Appendix A. Proof of Lemma ", "Lemma REF The balanced hypercube $BH_2$ is 2-edge-fault-tolerant 6-bipancyclic.", "By Lemma REF , for an arbitrary fault-free edge $(u, v)$ , there exists a fault-free Hamiltonian path $P$ that joins $u$ and $v$ , then $\\langle u, P, v, u\\rangle $ is the fault-free 16-cycle.", "Hence, we only need to construct a fault-free $\\ell $ -cycle in $BH_2$ containing $(u, v)$ where $6\\le \\ell \\le 14$ .", "Suppose that $|F|=2$ , without loss of generality, we can assume that $|F\\cap \\partial D_1|\\ge |F\\cap \\partial D_0|$ .", "Case 1: $|F\\cap \\partial D_1|=2, |F\\cap \\partial D_0|=0$ .", "Subcase 1.1: $e=(u, v)\\in \\partial D_0$ .", "Without loss of generality, we can assume that $e=(u, v)\\in BH^{0}_1$ .", "Suppose that $u=(a_0, 0)$ is a white vertex, $v=(b_0, 0)$ is a black vertex.", "Subcase 1.1.1: $6\\le \\ell \\le 14$ .", "There are three $\\ell $ -cycles $C_1, C_2, C_3$ in $BH_n$ containing $e$ where $6\\le \\ell \\le 14$ , such that $(C_i\\cap \\partial D_1)\\cap (C_j\\cap \\partial D_1)=\\emptyset $ for all $1\\le i\\ne j\\le 3$ .", "We list them as follows: Three 6-cycles: $~~\\left\\lbrace \\begin{aligned}&\\langle (a_0, 0), (b_0, 0), (a_0, 3), (b_0+2, 3), (a_0+2, 3), (b_0+2, 0), (a_0, 0)\\rangle ; \\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 3), (b_0, 3), (a_0, 3), (b_0+2, 0), (a_0, 0)\\rangle ;\\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 0), (b_0, 1), (a_0, 1), (b_0+2, 1), (a_0, 0)\\rangle .\\end{aligned}\\right.$ Three 8-cycles: $~~\\left\\lbrace \\begin{aligned}&\\langle (a_0, 0), (b_0, 0), (a_0, 3), (b_0, 3), (a_0, 2), (b_0+2, 3), (a_0+2, 3), (b_0+2, 0), (a_0, 0)\\rangle ; \\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 3), (b_0, 3), (a_0+2, 2), (b_0+2, 3), (a_0, 3), (b_0+2, 0), (a_0, 0)\\rangle ;\\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 0), (b_0, 1), (a_0+2, 1), (b_0, 2), (a_0, 1), (b_0+2, 1), (a_0, 0)\\rangle \\vspace{4.25pt}.\\end{aligned}\\right.$ Three 10-cycles: $~~\\left\\lbrace \\begin{aligned}&\\langle (a_0, 0), (b_0, 0), (a_0, 3), (b_0, 3), (a_0, 2), (b_0, 2), (a_0+2, 2), (b_0+2, 3), (a_0+2, 3), (b_0+2, 0), (a_0, 0)\\rangle ; \\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 3), (b_0, 3), (a_0+2, 2), (b_0, 2), (a_0, 2), (b_0+2, 3),(a_0, 3), (b_0+2, 0), (a_0, 0)\\rangle ;\\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 0), (b_0, 1), (a_0+2, 1), (b_0+2, 2), (a_0+2, 2), (b_0, 2), (a_0, 1), (b_0+2, 1), (a_0, 0)\\rangle \\vspace{4.25pt}.\\end{aligned}\\right.$ Three 12-cycles: $~~\\left\\lbrace \\begin{aligned}&\\langle (a_0, 0), (b_0, 0), (a_0, 3), (b_0, 3), (a_0+2, 3), (b_0+2, 3), (a_0+2, 2), (b_0, 2), (a_0, 2), (b_0+2, 2), (a_0, 1),\\\\ & (b_0, 1), (a_0, 0)\\rangle ; \\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 3), (b_0+2, 3), (a_0, 3), (b_0, 3), (a_0, 2), (b_0+2, 2), (a_0+2, 2), (b_0, 2) , (a_0+2, 1),\\\\ & (b_0+2, 1), (a_0, 0)\\rangle ;\\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 0), (b_0, 1), (a_0+2, 1), (b_0+2, 2), (a_0+2, 2), (b_0, 2), (a_0, 2), (b_0+2, 3), (a_0, 3), \\\\ & (b_0+2, 0), (a_0, 0)\\rangle \\vspace{4.25pt}.\\end{aligned}\\right.$ Three 14-cycles: $~~\\left\\lbrace \\begin{aligned}&\\langle (a_0, 0), (b_0, 0), (a_0, 3), (b_0, 3), (a_0+2, 3), (b_0+2, 3), (a_0+2, 2), (b_0, 2), (a_0, 2), (b_0+2, 2), (a_0+2, 1),\\\\ & (b_0, 1), (a_0, 1), (b_0+2, 1), (a_0, 0)\\rangle ; \\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 3), (b_0+2, 3), (a_0, 3), (b_0, 3), (a_0+2, 2), (b_0+2, 2), (a_0, 2), (b_0, 2), (a_0+2, 1), \\\\ & (b_0+2, 1), (a_0, 1), (b_0, 1), (a_0, 0)\\rangle ;\\\\& \\langle (a_0, 0), (b_0, 0), (a_0+2, 0), (b_0, 1), (a_0, 1), (b_0+2, 2), (a_0+2, 2), (b_0, 2), (a_0, 2), (b_0+2, 3), (a_0, 3),\\\\ & (b_0, 3), (a_0+2, 3), (b_0+2, 0), (a_0, 0)\\rangle \\vspace{4.25pt}.\\end{aligned}\\right.$ Notice that $|F\\cap \\partial D_1|=2, |F\\cap \\partial D_0|=0$ , there exists at least one fault-free $\\ell $ -cycle in $BH_2$ containing $e$ where $6\\le \\ell \\le 14$ .", "Subcase 1.2: $e=(u, v)\\in \\partial D_1$ .", "Without loss of generality, we can assume that $e=(u, v)=(u^0, v^1)$ is an edge between $BH^0_1$ and $BH^1_1$ where $u^0=(a_0, 0), v^1=(b_0, 1)$ .", "Subcase 1.2.1: $\\ell =6, 8$ .", "There exist three $\\ell $ -cycles $C_1, C_2, C_3$ in $BH_2$ containing $e$ where $\\ell =6$ or 8, such that $(C_i\\cap \\partial D_1)\\cap (C_j\\cap \\partial D_1)=\\lbrace e\\rbrace $ for $1\\le i\\ne j\\le 3$ .", "We list them as follows: Three 6-cycles: $~~~~~~\\left\\lbrace \\begin{aligned}&\\langle (a_0, 0), (b_0, 1), (a_0, 1), (b_0+2, 1), (a_0+2, 0), (b_0, 0), (a_0, 0)\\rangle ; \\\\& \\langle (a_0, 0), (b_0, 1), (a_0, 1), (b_0+2, 2), (a_0+2, 1), (b_0+2, 1), (a_0, 0)\\rangle ;\\\\& \\langle (a_0, 0), (b_0, 1), (a_0+2, 0), (b_0+2, 0), (a_0, 3), (b_0, 0), (a_0, 0)\\rangle .\\end{aligned}\\right.$ Three 8-cycles: $~~~~~~\\left\\lbrace \\begin{aligned}&\\langle (a_0, 0), (b_0, 1), (a_0, 1), (b_0, 2), (a_0, 2), (b_0, 3), (a_0, 3), (b_0, 0), (a_0, 0)\\rangle ; \\\\& \\langle (a_0, 0), (b_0, 1), (a_0+2, 1), (b_0+2, 2), (a_0+2, 2), (b_0+2, 3), (a_0+2, 3), (b_0+2, 0), (a_0, 0)\\rangle ;\\\\& \\langle (a_0, 0), (b_0, 1), (a_0, 1), (b_0+2, 2), (a_0, 2), (b_0+2, 3), (a_0, 3), (b_0+2, 0), (a_0, 0)\\rangle .\\end{aligned}\\right.$ Since $|F\\cap \\partial D_1|=2, |F\\cap \\partial D_0|=0$ , and $e$ is a fault-free edge, then there exists at least one fault-free 6-cycle and one fault-free 8-cycle in $BH_2$ containing $e$ .", "Subcase 1.2.2: $10\\le \\ell \\le 14$ .", "By the proof of subcase 1.2.1, there exists a fault-free 8-cycle $C$ that contains $e$ such that $|C\\cap BH^{i}_1|=1$ for $0\\le i\\le 3$ , say $\\langle u^0, v^0, u^3, v^3, u^2, v^2, u^1, v^1, u^0\\rangle $ where $u^i, v^i\\in BH^{i}_{n-1}$ for $i=0, 1, 2, 3$ .", "Since $|F\\cap \\partial D_0|=0$ .", "It is easy to check that there exists an $\\ell _i$ -path $P_i$ in $BH^i_1$ joining $u^i$ to $v^i$ where $\\ell _i=1$ or 3 for $i=1, 2, 3$ .", "Then, the cycle $\\langle u^0, v^0, u^3, P_3, v^3, u^2, P_2, v^2, u^1, P_1, v^1, u^0\\rangle $ with length $\\ell =5+\\ell _1+\\ell _2+\\ell _3$ forms the desired cycle.", "Case 2: $|F\\cap \\partial D_1|=1, |F\\cap \\partial D_0|=1$ .", "Subcase 2.1: $e=(u, v)\\in \\partial D_1$ .", "Without loss of generality, we can assume that $e=(u, v)=(u^0, v^1)$ is an edge between $BH^0_1$ and $BH^1_1$ where $u^0=(a_0, 0)$ , $v=(b_0, 1)$ .", "Subcase 2.1.1: $\\ell =6$ .", "If $((a_0+2, 0), (b_0+2, 1))$ is a fault-free edge.", "Let $\\begin{array}{l}C_1=\\langle u^0, v^1, (a_0, 1), (b_0+2, 1), (a_0+2, 0), (b_0, 0), u^0\\rangle ;\\vspace{4.25pt} \\\\C_2=\\langle u^0, v^1, (a_0+2, 1), (b_0+2, 1), (a_0+2, 0), (b_0+2, 0), u^0\\rangle .\\end{array}$ Then, $C_1, C_2$ are two cycles in $BH_2$ containing $e$ and $C_1\\cap C_2=\\lbrace e, ((a_0+2, 0), (b_0+2, 1))\\rbrace $ is the fault-free edge set.", "Thus, $C^1$ or $C^2$ is a fault-free 6-cycle.", "If $((a_0+2, 0), (b_0+2, 1))$ is a faulty edge.", "Then, $(u^0, (b_0+2, 1))$ is a fault-free edge.", "Let $\\begin{array}{l}C_3=\\langle u^0, v^1, (a_0, 1), (b_0, 2), (a_0+2, 1), (b_0+2, 1), u^0\\rangle ;\\vspace{4.25pt} \\\\C_4=\\langle u^0, v^1, (a_0+2, 1), (b_0+2, 2), (a_0, 1), (b_0+2, 1), u^0\\rangle .\\end{array}$ Then, $C_3, C_4$ are two cycles in $BH_2$ containing $e$ and $C_3\\cap C_4=\\lbrace e, (u^0, (b_0+2, 1))\\rbrace $ is the fault-free edge set.", "Thus, $C^3$ or $C^4$ is a fault-free 6-cycle.", "Subcase 2.1.2: $\\ell =8$ .", "By Lemma REF , it holds.", "Subcase 2.1.3: $10\\le \\ell \\le 14$ .", "By the proof of subcase 2.1.2, there exists a fault-free 8-cycle $C$ that contains $e$ such that $|C\\cap BH^{i}_1|=1$ for $0\\le i\\le 3$ , say $\\langle u^0, v^0, u^3, v^3, u^2, v^2, u^1, v^1, u^0\\rangle $ where $u^i, v^i\\in BH^{i}_{n-1}$ for $i=0, 1, 2, 3$ .", "Note that $|F\\cap \\partial D_0|=1$ .", "Without loss of generality, let $|F\\cap BH^0_{1}|=1$ .", "It is easy to check that there exists an $\\ell _i$ -path in $BH^i_1$ joining $u^i$ to $v^i$ where $\\ell _i=1$ or 3 for $i=1, 2, 3$ .", "Then, $\\langle u, v^0, u^3, P_3, v^3, u^2, P_2, v^2, u^1, P_1, v, u\\rangle $ with length $\\ell =5+\\ell _1+\\ell _2+\\ell _3$ forms the desired cycle.", "Subcase 2.2: $e=(u, v)\\in \\partial D_0$ .", "We divide $BH_2$ into four $BH_1$ s, denoted by $\\overline{BH^0_1}, \\overline{BH^1_1}, \\overline{BH^2_1}, and \\overline{BH^3_1}$ , by deleting all 1-dimensional edges.", "Then, $e$ is an edge between $\\overline{BH^i_1}$ and $\\overline{BH^{i+1}_1}$ for $0\\le i\\le 3$ .", "By a similar discussion for subcase 2.1, we obtain the result.", "$\\Box $" ], [ "Conclusion", "In this paper, we consider the edge-bipancyclicity of $BH_n$ for at most $(2n-2)$ faulty edges and prove that each fault-free edge lies on a fault-free cycle of any even length from 6 to $2^{2n}$ .", "Our result improves the results of Hao et al.", "[7] and Cheng et al.", "[3] and it is optimal with respect to the maximum number of tolerated edge faults.", "In addition, it is of interest to consider the problem of fault-tolerant embedding cycles with each vertex incident to at least two non-faulty edges." ], [ "Acknowledgement", "The authors would like to express their gratitude to the anonymous referees for their kind suggestions and useful comments on the original manuscript, which resulted in this final version.", "This research is supported by the National Natural Science Foundation of China (11571044, 61373021, 11461004), the Fundamental Research Funds for the Central University of China ." ] ]
1606.05152
[ [ "On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck\n system with cosmological scalar field" ], [ "Abstract The Einstein-Vlasov-Fokker-Planck system describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity.", "We study the Cauchy problem for spatially homogeneous and isotropic solutions and prove the existence of both global solutions and solutions that blow-up in finite time depending on the size of certain functions of the initial data.", "We also derive information on the large-time behavior of global solutions and toward the singularity for solutions which blow-up in fine time.", "Our results entail the existence of a phase of decelerated expansion followed by a phase of accelerated expansion, in accordance with the physical expectations in cosmology." ], [ "Introduction", "The purpose of this paper is to study spatially homogeneous and isotropic solutions of the Einstein-Vlasov-Fokker-Planck system.", "The model describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity.", "It is assumed that diffusion takes place in a cosmological scalar field, which can be identified with the dark energy source responsible for the phase of accelerated expansion of the Universe.", "When relativistic effects are neglected the motion of self-gravitating kinetic particles undergoing diffusion is described by the frictionless Vlasov-Poisson-Fokker-Planck system in the gravitational case, which is given by $&\\partial _tf+p\\cdot \\nabla _xf-\\nabla U\\cdot \\nabla _pf=\\sigma \\Delta _pf\\\\&\\Delta U=4\\pi \\rho ,\\quad \\rho (t,x)=\\int _{\\mathbb {R}^3}f(t,x,p)\\,dp.$ Here $f(t,x,p)$ is the phase-space density of a system of unit mass particles, $U(t,x)$ is the Newtonian gravitational potential generated by the particle system, and $\\sigma >0$ is the diffusion constant; the remaining physical constants have been set to one.", "The mathematical properties of the system () have been extensively studied in the literature.", "In particular, it is known that the Vlasov-Poisson-Fokker-Planck system admits a global unique solution for given initial data [4], [11], whose asymptotic behavior for large times has been studied in [6].", "The proof of both these results makes use of the explicit form of the fundamental solution to the linear Fokker-Planck equation.", "Similar results have also been obtained for the related Vlasov-Maxwell-Fokker-Planck system [12], [13].", "A relativistic generalization of the Vlasov-Poisson-Fokker-Planck system in the gravitational case has been introduced recently in [5] and the purpose of this paper is to initiate its mathematical study.", "When dealing with the relativistic model one is faced with many new difficulties, including the hyperbolic and nonlinear character of the Einstein field equations (compared to the linearity of the Poisson equation in ()), the non-uniform ellipticity of the diffusion operator, the time-dependence of the diffusion matrix, and the absence of an explicit formula for the fundamental solution to the linear relativistic Fokker-Planck equation.", "The further notorious complexity of the Einstein equations suggests that one begins by studying solutions with symmetries.", "In this paper we consider solutions which are spatially homogeneous and isotropic.", "Under these symmetry assumptions the Einstein equations reduce to a system of nonlinear ordinary differential equations.", "In the absence of diffusion our model reduces to the Einstein-Vlasov system with cosmological constant.", "The regularity and asymptotic behavior of spatially homogeneous solutions to the latter system have been studied in [8].", "The generalization of these results to spatially inhomogeneous solutions (with surface symmetry) is given in [3].", "When the cosmological constant is set to zero one obtains the Einstein-Vlasov system.", "Some results available for this system are summarized in [1].", "It has been shown recently in [7], [9] that the Cauchy problem for the Einstein-Vlasov system is globally well-posed for small data with no symmetry restrictions.", "In [2] we have studied a similar problem when the Einstein equation is replaced by a nonlinear wave equation for a scalar field.", "It turns out that for spatially homogeneous and isotropic solutions, the Fokker-Planck equation considered in [2] is the same as the one derived in the present paper and thus the analysis of the matter equation does not pose any new difficulty.", "Contrastingly, the Einstein equations behave quite differently than the field equation considered in [2] The remainder of the paper is organized as follows.", "In the next section we derive the Einstein-Vlasov-Fokker-Planck system for spatially homogeneous and isotropic solutions.", "In Section  we study the Cauchy problem and prove that, depending on the size of certain functions of the the initial data, regular solutions either exist globally or blow-up in finite time.", "In Section  we derive information on the asymptotic behavior as $t\\rightarrow \\infty $ of global solutions and toward the singularity for solutions which blow-up in finite time.", "We also show that for an open set of initial data which describe an initially decelerating expanding Universe, global solutions will eventually give rise to a phase of accelerated expansion, in agreement with the expectations in Cosmology [14]." ], [ "Derivation of the model", "We begin with a short description of the general relativistic kinetic theory of diffusion, see [5] for more details.", "In the following, Greek indices run from 0 to 3, while Latin indices run from 1 to 3.", "Moreover all indices are raised and lowered with the matrix $g_{\\mu \\nu }$ and its inverse $g^{\\mu \\nu }:=(g^{-1})_{\\mu \\nu }$ , and the Einstein summation rule is applied (e.g., $g^{\\mu \\nu }g_{\\nu \\alpha }=\\delta ^\\mu _{\\ \\alpha }$ ), unless otherwise stated.", "Let $(M,g)$ be a spacetime.", "Let $x^\\mu $ denote local coordinates on $M$ , with $t=x^0$ being timelike, and let $(x^\\mu ,p^\\nu )$ be the induced canonical local coordinates on the tangent bundle of the spacetime.", "The mass-shell for particles with unit mass is the 7-dimensional submanifold of the tangent bundle obtained by imposing $g_{\\mu \\nu }p^\\mu p^\\nu =-1$ and $p^0>0$ , so that $(p^0,p^1,p^2,p^3)$ can be interpreted as the components of the four-momentum of a unit mass particle.", "The mass-shell condition is used to express $p^0$ in terms of the spatial components of the four-momentum, namely $p^0=-g_{00}^{-1}\\big [g_{0j}p^j+\\sqrt{(g_{0j}p^j)^2-g_{00}(1+g_{ij}p^ip^j)}\\big ].$ It follows that $(x^\\mu ,p^i)$ define a local system of coordinates on the mass-shell.", "The particle distribution function $f$ is defined on the mass-shell and is therefore a function of the coordinates $(t,x^i,p^j)$ .", "The Fokker-Planck equation for $f$ is the PDE $L f:=p^0\\partial _t f+p^i\\partial _{x^i}f-\\Gamma ^i_{\\ \\mu \\nu }p^\\mu p^\\nu \\partial _{p^i}f=\\sigma \\mathcal {D}_p f,$ where $L$ is the Liouville (or Vlasov) operator, $\\Gamma ^{\\alpha }_{\\ \\mu \\nu }$ denote the Christoffel symbols of the metric $g$ , $\\sigma $ is the (positive) diffusion constant, and $\\mathcal {D}_p$ is the diffusion operator, which is the Laplace-Beltrami operator associated to the Riemannian metric $h=h_{ij}dp^i dp^j,\\quad h_{ij}=g_{ij}+g_{00}\\frac{p_ip_j}{(p_0)^2}-\\frac{p_i}{p_0}g_{0j}-\\frac{p_j}{p_0}g_{0i}.$ We have $\\mathcal {D}_pf=\\frac{1}{\\sqrt{\\det h}}\\partial _{p^i}\\left(\\sqrt{\\det h}\\,(h^{-1})^{ij}\\partial _{p^j}f\\right),$ where $(h^{-1})^{ij}$ is the inverse matrix of $h_{ij}$ (i.e., $(h^{-1})^{ij}h_{jk}=\\delta ^i_{\\ k}$ ).", "From the particle distribution $f$ we can construct the particle current density and the energy-momentum tensor by $T^{\\mu \\nu }=\\sqrt{-\\det g}\\int f\\, \\frac{p^\\mu p^\\nu }{-p_0}\\,dp,\\quad J^{\\mu }=\\sqrt{-\\det g}\\int f\\, \\frac{p^\\mu }{-p_0}\\,dp,$ where $dp=dp^1\\wedge dp^2\\wedge dp^3$ and the integration is over the fibers of the mass-shell.", "It can be shown that the tensors $T^{\\mu \\nu }$ and $J^\\mu $ verify $\\nabla _\\mu T^{\\mu \\nu }=3\\sigma J^\\nu ,\\quad \\nabla _\\mu J^\\mu =0.$ To close the system we require the metric $g$ to solve the Einstein equations with cosmological scalar field $\\phi $ , which, in units $8\\pi G=c=1$ , is given by $R_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }R+\\phi g_{\\mu \\nu }=T_{\\mu \\nu }.$ By (REF ) and the Bianchi identity $\\nabla ^\\mu (R_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }R)=0$ , we find that the cosmological scalar field satisfies the equation $\\nabla _\\mu \\phi =3\\sigma J_\\mu .$ In particular, $\\phi $ satisfies the wave equation $\\Box \\phi =\\nabla ^\\mu \\nabla _\\mu \\phi =3\\sigma \\nabla ^\\mu J_\\mu =0$ and $J^\\mu \\nabla _\\mu \\phi =3\\sigma J_\\mu J^\\mu <0,$ where for the inequality we have used the fact that $J^\\mu $ is timelike.", "Hence, the cosmological scalar field is decreasing along the matter flow, which can be interpreted as energy being transferred from the scalar field to the particles by diffusion.", "Next, we specialize to spatially homogeneous and isotropic solutions.", "Under these symmetry assumptions, the spacetime metric can be written as $g_{\\mu \\nu }dx^\\mu dx^\\nu =-dt^2+\\frac{a(t)^2}{\\mathcal {K}(r)^{2}}\\delta _{ij}dx^idx^j,$ where $\\mathcal {K}(r)=1+\\frac{k}{4}r^2,\\quad r=|x|,\\quad k\\in \\lbrace -1,0,1\\rbrace .$ Here $x=(x^1,x^2,x^3)$ is a system of spatial isotropic coordinates on the hypersurfaces $t=const.$ , with $t$ denoting the proper time along the normal geodesics, and $|\\cdot |$ denoting the standard Euclidean norm.", "The hypersurfaces of constant proper time have zero/positive/negative constant curvature according to the values $k=0,+1,-1$ of the curvature parameter $k$ .", "Equation (REF ) gives $p^0=\\Bigg [1+\\frac{a(t)^2}{\\mathcal {K}(r)^{2}}|p|^2\\Bigg ]^{1/2},\\quad p=(p^1,p^2,p^3)$ and $p_0=-p^0$ .", "The non-zero Christoffel symbols $\\Gamma ^i_{\\ \\mu \\nu }$ of the metric (REF ) are given by $\\Gamma ^{i}_{\\ i0}=\\frac{\\dot{a(t)}}{a(t)},\\quad \\Gamma ^i_{\\ ij}=-\\frac{k x^j}{2\\mathcal {K}(r)},\\quad \\Gamma ^j_{\\ ii}=\\frac{k x^j}{2\\mathcal {K}(r)} \\quad (i\\ne j)$ and by their symmetric symbols on the lower indexes; the index $i$ is not summed in the previous equations.", "It follows that the Liouville operator in the left hand side of (REF ) is given by $Lf=p^0\\partial _t f+p^i\\partial _{x^i}f-2\\frac{\\dot{a(t)}}{a(t)}p^0p\\cdot \\nabla _p f-\\frac{k}{\\mathcal {K}(r)}(\\tfrac{1}{2}|p|^2x-(x\\cdot p)p)\\cdot \\nabla _p f,$ with $w\\cdot z$ denoting the standard Euclidean scalar product of the vectors $w,z\\in \\mathbb {R}^3$ .", "The metric (REF ) now takes the form $h_{ij}=\\left(\\frac{a(t)^2}{\\mathcal {K}(r)^2}\\delta _{ij}-\\frac{p_ip_j}{(p_0)^2}\\right).$ It follows that $&\\det h=\\frac{a(t)^6}{\\mathcal {K}(r)^6(p_0)^2}\\\\&(h^{-1})^{ij}=g^{ij}+p^ip^j=\\frac{\\mathcal {K}(r)^2}{a(t)^2}\\delta ^{ij}+p^ip^j.$ Hence, the diffusion term in the right hand side of (REF ) ultimately takes the form $\\mathcal {D}_pf=p_0\\partial _{p^i}\\left[\\left(\\frac{g^{ij}+p^ip^j}{p_0}\\right)\\partial _{p^j}f\\right]=p^0\\partial _{p^i}\\left[\\left(\\frac{\\frac{\\mathcal {K}(r)^2}{a(t)^2}\\delta ^{ij}+p^ip^j}{p^0}\\right)\\partial _{p^j}f\\right].$ Definition 2.1 A particle distribution $f$ is said to be spatially homogeneous and isotropic if there exists a function $\\widetilde{F}:\\mathbb {R}\\times [0,\\infty )\\rightarrow [0,\\infty )$ such that $f(t,x,p)=\\widetilde{F}(t,|v|)\\biggr \\vert _{{v=\\frac{a(t)^2}{\\mathcal {K}(r)^2}p}}.$ It can be shown that this definition is equivalent to require that $f$ is invariant by the six-dimensional group of isometries of the metric (REF ), see [10].", "It is now convenient to define the function $F:\\mathbb {R}\\times \\mathbb {R}^3\\rightarrow [0,\\infty )$ by $F(t,v)=\\widetilde{F}(t,|v|),$ in terms of which the Fokker-Planck equation for spatially homogeneous and isotropic distribution functions takes the final form $\\partial _t F=\\sigma a(t)\\partial _{v^i}(D^{ij}\\partial _{v^j}F),$ where the diffusion matrix $D$ is $D^{ij}=\\frac{a(t)^2\\delta ^{ij}+v^iv^j}{\\sqrt{a(t)^2+|v|^2}}.$ In terms of the function $F$ , and recalling (REF ), the energy-momentum tensor and current density (REF ) read $&T_{00}(t)=\\frac{1}{a(t)^4}\\int _{\\mathbb {R}^3}F(t,v)\\sqrt{a(t)^2+|v|^2}\\,dv,\\\\&T_{11}(t)=T_{22}(t)=T_{33}(t)=\\frac{1}{3a(t)^2\\mathcal {K}(r)^2}\\int _{\\mathbb {R}^3}F(t,v)\\frac{|v|^2}{\\sqrt{a(t)^2+|v|^2}}\\\\&J^0(t)=\\frac{1}{a(t)^3}\\int _{\\mathbb {R}^3}F(t,v)\\,dv,\\quad J^i=0,$ and $T_{0i}=T_{ij}=0$ , for $i\\ne j$ .", "Thus, equation (REF ) for the cosmological scalar field $\\phi =\\phi (t)$ becomes $\\dot{\\phi }=-3\\frac{\\sigma }{a^3}N,$ where $N=\\int _{\\mathbb {R}^3}F\\,dv$ is the total number of particles, which is conserved along solutions of (REF ).", "Finally the non-zero components of the Einstein tensor $G_{\\mu \\nu }=R_{\\mu \\nu }-\\tfrac{1}{2}g_{\\mu \\nu }R$ are $G_{00}=\\frac{3(k+\\dot{a}^2)}{a^2},\\quad G_{11}=G_{22}=G_{33}=-\\frac{k+\\dot{a}^2+2a\\ddot{a}}{\\mathcal {K}(r)^2}.$ It follows that the Einstein equations (REF ) are $\\frac{3(k+\\dot{a}^2)}{a^2}-\\phi =\\frac{1}{a^4}\\int _{\\mathbb {R}^3}F\\,\\sqrt{a^2+|v|^2}\\,dv,\\\\-2\\frac{\\ddot{a}}{a}-\\left(\\frac{\\dot{a}}{a}\\right)^2-\\frac{k}{a^2}+\\phi =\\frac{1}{3a^4}\\int _{\\mathbb {R}^3}F\\,\\frac{|v|^2}{\\sqrt{a^2+|v|^2}}\\,dv.$ By introducing the Hubble function $H(t)=\\frac{\\dot{a}(t)}{a(t)},$ as well as the energy density $\\rho (t)$ and the pressure $\\mathcal {P}(t)$ by $\\rho (t)=\\frac{1}{a(t)^4}\\int _{\\mathbb {R}^3}F\\,\\sqrt{a(t)^2+|v|^2}\\,dv,\\quad \\mathcal {P}(t)=\\frac{1}{3a(t)^4}\\int _{\\mathbb {R}^3}F\\,\\frac{|v|^2}{\\sqrt{a(t)^2+|v|^2}}\\,dv,$ we can rewrite the Einstein equations in the form $&H^2=\\frac{1}{3}(\\rho +\\phi )-\\frac{k}{a^2},\\\\&\\dot{H}=-\\frac{\\rho +\\mathcal {P}}{2}+\\frac{k}{a(t)^2}.$ We also have the following auxiliary equations, obtained by combining (): $\\dot{H} = -\\frac{1}{6} ( \\rho + 3\\mathcal {P} ) - H^2 + \\frac{1}{3}\\phi .$ $\\dot{\\rho }=-3H(\\rho +\\mathcal {P})-\\dot{\\phi }.$ Note also the estimates $\\rho \\ge \\frac{N}{a^3},\\qquad \\frac{\\rho }{3}-\\frac{N}{3a^3}\\le \\mathcal {P}\\le \\frac{\\rho }{3},$ which follow straightforwardly by the definitions (REF ) and (REF ).", "The deceleration parameter $Q$ is defined as $Q=-\\frac{a\\ddot{a}}{\\dot{a}^2}=-1-\\frac{\\dot{H}}{H^2}.$ The solution is said to be expanding with acceleration if $Q<0$ and $H>0$ and expanding with deceleration if $Q>0$ and $H>0$ .", "From (REF ), this occurs at time $t$ if and only if $H(t)>0$ and $q(t)=6H^2(t)Q(t)=\\rho (t)+3\\mathcal {P}(t)-2\\phi (t)$ is negative, respectively positive." ], [ "Existence of regular solutions", "In this section we study the existence of regular solutions to the initial value problem for the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field.", "The system, derived in the previous section, is given by $&\\partial _t F=\\sigma a(t)\\partial _{v^i}\\left(\\frac{a(t)^2\\delta ^{ij}+v^iv^j}{\\sqrt{a(t)^2+|v|^2}}\\partial _{v^j}F\\right),\\\\&\\dot{a}=Ha\\\\&\\dot{H}=-\\frac{\\rho +\\mathcal {P}}{2}+\\frac{k}{a^2}\\\\&\\dot{\\phi }=-\\frac{3\\sigma N}{a^3} ,\\\\ &H^2=\\frac{1}{3}(\\rho +\\phi )-\\frac{k}{a^2},$ where $\\rho (t),\\mathcal {P}(t)$ are given by (REF ) and $k$ is either $0,+1$ , or $-1$ .", "Initial data for the system () consist of a quadruple $(F_0,a_0,H_0,\\phi _0)$ , where $F_0:\\mathbb {R}^3\\rightarrow [0,\\infty )$ such that $F_0(v)=\\widetilde{F}_0(|v|)$ , for some $\\widetilde{F}_0:\\mathbb {R}\\rightarrow [0,\\infty )$ , and $a_0>0, \\quad H_0>0,\\quad \\phi _0>0.$ We assume that $F_0$ is not identically zero and belongs to the space $X=\\mathcal {L}_1\\cap H^1$ , where, for $\\gamma >0$ , $\\mathcal {L}_\\gamma =\\lbrace g:\\mathbb {R}^3\\rightarrow \\mathbb {R}:g\\in L^1\\times L^2,\\ \\text{and}\\ v\\rightarrow |v|^\\gamma g\\in L^1\\rbrace .$ Moreover, the initial data are assumed to satisfy () at time $t=0$ , namely $H_0^2=\\frac{1}{3}(\\rho _0+\\phi _0)-\\frac{k}{a_0^2}.$ Given $T>0$ , a quadruple $(F(t,v),a(t),H(t),\\phi (t))$ will be referred to as a regular solution of the system () on the interval $[0,T)$ and with initial data $(F_0,a_0,H_0,\\phi _0)$ if $F\\in C^0((0,T);X),\\quad 0<a\\in C^1((0,T)), \\quad H\\in C^1((0,T)),\\quad \\phi \\in C^1((0,T)),$ $F\\ge 0 \\text{ a.e, }\\quad (a_0,H_0,\\phi _0)=\\lim _{t\\rightarrow 0^+}(a(t),H(t),\\phi (t)),\\quad \\lim _{t\\rightarrow 0^+}\\Vert F(t,\\cdot )-F_0\\Vert _{\\mathcal {L}_1}= 0,$ and () is satisfied in the domain $(t,v)\\in (0,T)\\times \\mathbb {R}^3$ .", "Note that for regular solutions the functions $\\rho ,\\mathcal {P}$ are continuous and hence the Einstein equations are satisfied in the pointwise, classical sense.", "However the Fokker-Planck equation need only be verified in the weak sense.", "Proving existence and uniqueness of local regular solutions is a simple generalization of the argument presented in [2].", "The main tool is the following set of a priori estimates on the solution of the Fokker-Planck equation.", "Proposition 3.1 Let $T_\\mathrm {max}>0$ be the maximal time of existence of a regular solution.", "The following estimates on $F$ hold for all $t\\in [0,T_\\mathrm {max})$ : (i) $N = \\Vert F(t)\\Vert _{L^1}=\\Vert F_0\\Vert _{L^1}$ (conservation of the total number of particles) and $\\Vert F(t)\\Vert _{L^2}\\le \\Vert F_0\\Vert _{L^2}$ (dissipation estimate).", "(ii) Propagation of moments: if $F_0\\in \\mathcal {L}_\\gamma $ , then $F(t,\\cdot )\\in \\mathcal {L}_\\gamma $ and $\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2} F(t,v)\\,dv\\le e^{C\\Big (\\alpha t+\\int _0^t (H)_+(s)\\,ds\\Big )}\\int _{\\mathbb {R}^3}(a_0^2+|v|^2)^{\\gamma /2} F_0(v)\\,dv,$ where $\\alpha =\\alpha (\\sigma )>0$ , with $\\alpha (0)=0$ , and $C>0$ are constants (depending on $\\gamma $ ) and $(y)_+$ denotes the positive part of $y$ .", "(iii) Propagation of derivative moments: if $|v|^{\\gamma /2}\\nabla _v F_0\\in L^2$ then $\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2} |\\nabla _vF(t,v)|^2\\,dv\\le Ce^{C\\Big (\\alpha t+\\int _0^t (H)_+(s)\\,ds\\Big )}.$ The proof is very similar to the proof of Prop.", "2.2 in [2], hence we limit ourself to formally derive the estimate in (ii).", "We compute $\\frac{d}{dt}\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2} F\\,dv&=\\gamma a(t)\\dot{a}(t)\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2-1}F\\,dv\\\\&\\quad +\\sigma a(t)\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2} \\partial _{v^i}(D^{ij}\\partial _{v^j}F)\\,dv\\\\&=\\gamma H(t) a(t)^2\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2-1}F\\,dv\\\\&\\quad -\\sigma \\gamma a(t)\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2-1}v_iD^{ij}\\partial _{v^j}F\\,dv\\\\&\\le \\gamma (H)_+(t)\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2}F\\,dv\\\\&\\quad +\\sigma \\gamma a(t)\\int _{\\mathbb {R}^3}\\partial _{v^j}[(a(t)^2+|v|^2)^{\\gamma /2-1}v_iD^{ij}]F\\,dv.$ In the last integral we use the brief calculation $\\partial _{v^j}[\\dots ]=3(a(t)^2+|v|^2)^{\\gamma /2-1/2}+\\left(\\gamma -1 \\right)(a(t)^2+|v|^2)^{\\gamma /2-3/2}|v|^2\\le C(a(t)^2+|v|^2)^{\\gamma /2-1/2}.$ Hence, we have $\\frac{d}{dt}\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2} F\\,dv\\le C(\\alpha +(H)_+(t))\\int _{\\mathbb {R}^3}(a(t)^2+|v|^2)^{\\gamma /2}F\\,dv$ and Grönwall's Lemma concludes the proof of (ii).", "The following lemma provides a characterization of $\\rho (t)$ that will be useful in subsequent results.", "Lemma 3.2 For any $t\\in [0,T_\\mathrm {max})$ , we have $\\rho (t) = a(t)^{-3} \\left( \\rho _0 a_0^3 + 3\\sigma N t - 3\\int _{0}^t H(s) a(s)^3 \\mathcal {P}(s) \\ ds \\right).$ In particular, if $H(t) \\ge 0$ for all $t \\in [0,T_\\mathrm {max})$ , then $\\rho (t) \\le (\\rho _0 a_0^3 + 3\\sigma N t) a(t)^{-3}$ for $t \\in [0,T_\\mathrm {max})$ .", "Using (REF ) we find $\\dot{\\rho } + 3H\\rho = -3H\\mathcal {P} - \\dot{\\phi },$ and thus using the integrating factor $\\exp (\\int _0^t 3H(s) \\ ds) = \\left( \\frac{a(t)}{a_0} \\right)^3$ , we find $\\frac{d}{dt} \\left( \\rho (t) \\left( \\frac{a(t)}{a_0} \\right)^3 \\right) = (-3H(t)\\mathcal {P}(t) - \\dot{\\phi }(t) ) \\left( \\frac{a(t)}{a_0} \\right)^3.$ Integrating over $[0, t]$ and using (), this becomes $\\rho (t) a(t)^3 - \\rho _0a_0^3 = 3\\int _{0}^t \\left( -H(s)a(s)^3\\mathcal {P}(s) + \\sigma N \\right) \\ ds$ and the conclusion follows.", "In order to determine whether a regular solution exists globally or blows-up in finite time, we shall often apply the following simple result within subsequent sections.", "Lemma 3.3 Let $t _1\\in \\mathbb {R}$ be given and assume there is $t_2 > t_1$ with $I\\in C^1(t_1,t_2)$ satisfying $I_1 := I(t_1) < 0$ and $\\dot{I}(t) \\le -I(t)^2$ for all $t \\in (t_1,t_2)$ .", "If $t_2 \\ge t_1 - \\frac{1}{I_1}$ , then $t_2 = t_1 - \\frac{1}{I_1}$ and $\\lim _{t \\rightarrow t_2^-} I(t) = -\\infty $ with the estimate $I(t) \\le - \\frac{1}{t_2 - t}$ for all $t \\in [t_1, t_2).$ If $t_2 \\le t_1 - \\frac{1}{I_1}$ and $\\lim _{t \\rightarrow t_2^-} I(t) = -\\infty $ , then $I(t) \\ge -\\frac{1}{t_2-t}$ for $t \\in [t_1, t_2)$ .", "Since $\\dot{I}(t) \\le 0$ on $(t_1,t_2)$ and $I_1 < 0$ , it follows that $I(t) < 0$ on the same interval and we divide by $-I(t)^2$ so that the differential inequality becomes $\\frac{d}{dt} \\left( I(t)^{-1}\\right) \\ge 1,$ for all $t \\in (t_1,t_2)$ .", "Integrating over $[t_1, \\tau ]$ for $\\tau \\in (t_1, t_2)$ yields $ \\frac{1}{I(\\tau )} \\ge \\tau - \\left(t_1 - \\frac{1}{I_1} \\right)$ and for $\\tau \\in \\left[t_1, t_1 - \\frac{1}{I_1}\\right)$ , this can be inverted to find $I(\\tau ) \\le \\frac{1}{\\tau - \\left(t_1 - \\frac{1}{I_1} \\right)}.$ Taking the limit as $\\tau \\rightarrow \\left(t_1 - \\frac{1}{I_1} \\right)^-$ implies $I(\\tau ) \\rightarrow -\\infty $ at this time.", "Finally, since the differential inequality cannot be satisfied after the blow-up time $t_1 - \\frac{1}{I_1}$ , it follows that $t_2 = t_1 - \\frac{1}{I_1}$ .", "The resulting estimate then follows by making this replacement in the above inequality.", "In the second case, we first note that the limit condition implies $\\lim _{t \\rightarrow t_2^-} I(t)^{-1} = 0$ .", "Integrating (REF ) over $[\\tau , t_2)$ for $\\tau \\in (t_1, t_2)$ , we find $-\\frac{1}{I(\\tau )} \\ge t_2 - \\tau $ and thus the lower bound $I(\\tau ) \\ge -\\frac{1}{t_2- \\tau }$ for $\\tau \\in [t_1, t_2)$ ." ], [ "Characterization of the maximal time of existence", "Next, we focus on proving a series of criteria for the existence of global regular solutions, or their finite-time blowup.", "We begin by showing that as long as $a(t)$ remains bounded away from zero, the solution remains regular.", "Lemma 3.4 $\\alpha :=\\inf _{t\\in [0,T_\\mathrm {max})}a(t)>0\\Rightarrow T_\\mathrm {max}=+\\infty .$ We limit ourselves to showing that, under the assumption within (REF ), the solution cannot blow-up in finite time.", "Assume $T_\\mathrm {max}<\\infty $ .", "Using $a(t)\\ge \\alpha >0$ in () and () gives $\\phi \\in W^{1,\\infty }([0,T_\\mathrm {max}))$ and $H(t)\\le H_0+Ct$ .", "From the latter inequality and Proposition REF (with $\\gamma =1$ ) we conclude that the integral $\\int _{\\mathbb {R}^3}\\sqrt{a(t)^2+|v|^2}F(t,v)\\,dv$ is bounded.", "In particular $F\\in L^\\infty ([0,T_\\mathrm {max}),X)$ .", "Moreover using $a(t)\\ge \\alpha >0$ in (REF ) we find that $\\rho ,\\mathcal {P}\\in L^\\infty ([0,T_\\mathrm {max}))$ .", "By () and () we obtain $H\\in W^{1,\\infty }([0,T_\\mathrm {max}))$ .", "As $H=\\dot{a}/a$ we also have $a\\in W^{1,\\infty }([0,T_\\mathrm {max}))$ and the proof is complete.", "Theorem 3.5 Let $T_\\mathrm {max}>0$ be the maximal time of existence of a regular solution.", "Then, for $k=0,-1$ the following are equivalent: (a) $\\displaystyle \\lim _{t\\rightarrow T_\\mathrm {max}^-}\\phi (t)\\ge 0$ (b) $H(t)>0$ , for all $t\\in [0,T_\\mathrm {max})$ (c) $T_\\mathrm {max}=+\\infty .$ For $k=1$ , we can only prove the weaker statements $\\left.", "{\\begin{array}{c}T_\\mathrm {max}=+\\infty \\Rightarrow \\lim _{t\\rightarrow T_\\mathrm {max}^-}\\phi (t)\\ge 0,\\\\ \\lim _{t\\rightarrow T_\\mathrm {max}^-}\\phi (t)\\ge \\min \\left\\lbrace \\frac{4}{N^2}, \\frac{9}{4\\rho _0a_0^4}\\right\\rbrace \\Rightarrow T_\\mathrm {max}=+\\infty \\text{ and } H(t)>0\\text{ for all $t\\ge 0$}.\\end{array}}\\right\\rbrace $ We first prove the equivalence of the statements (a), (b), (c) in the cases $k = 0, -1$ .", "(a)$\\Rightarrow $ (b): Assuming (a) and using the fact that $\\phi (t)$ is strictly decreasing on $ [0,T_\\mathrm {max})$ , we have $\\phi (t)>0$ for all $t\\in [0,T_\\mathrm {max})$ .", "Hence () gives $H(t)>0$ , for all $t\\in [0,T_\\mathrm {max})$ when $k=0,-1$ .", "(b)$\\Rightarrow $ (c): As $H(t)$ is strictly positive for $t\\in [0,T_\\mathrm {max})$ , then $a(t)$ is increasing and so $a(t)\\ge a_0$ for all $t\\in [0,T_\\mathrm {max})$ .", "Statement (c) then follows immediately by Lemma REF .", "Next, we prove (c)$\\Rightarrow $ (a), as well as the analogous statement for $k=1$ .", "We do so by using the contrapositive $\\sim $ (a) $\\Rightarrow \\sim $ (c), i.e.", "by showing that negative values of $\\phi $ imply finite-time blow-up of solutions for $k = -1, 0, 1$ .", "Assume $\\lim _{t \\rightarrow T_{\\max }^-} \\phi (t) < 0$ .", "Then, since $\\dot{\\phi }(t) < 0$ there is $T_0 \\in (0,T_{\\max })$ such that $\\phi (t) < 0$ for all $t \\in [T_0,T_{\\max }).$ With this, (REF ) implies $\\dot{H}(t) \\le - H(t)^2$ for all $t \\in [T_0,T_{\\max })$ .", "Again using (REF ), we have $\\dot{H}(t) \\le \\frac{1}{3} \\phi (t) \\le \\frac{1}{3} \\phi (T_0)$ for all $t \\in [T_0, T_{\\max })$ .", "Integrating over $[T_0, t]$ , we have $H(t) \\le H(T_0) + \\frac{1}{3}\\phi (T_0) (t - T_0)$ for $t \\in [T_0, T_{\\max })$ .", "Finally, because $\\phi (T_0) < 0$ , if $T_{\\max } = \\infty $ we may take $t$ large enough in (REF ) so that $H(T_1) < 0$ where $T_1 > \\max \\left\\lbrace T_0, T_0 - \\frac{3H(T_0)}{\\phi (T_0)} \\right\\rbrace $ .", "Applying Lemma REF at $t = T_1$ , this implies $T_{\\max } < \\infty $ contradicting the assumption $T_{\\max } = \\infty $ .", "Thus, in all cases we find $T_{\\max } < \\infty $ .", "Finally, we complete the proof by showing that either positive lower bound on $\\phi $ in the case $k=1$ implies global existence.", "First, assume $\\lim _{t\\rightarrow T_\\mathrm {max}^-}\\phi (t)\\ge \\frac{4}{N^2}.$ As before, since $\\phi $ is decreasing, we have $\\phi (t) > \\frac{4}{N^2}$ for all $t \\in [0,T_{\\max })$ .", "Using the lower bound $\\rho (t)\\ge N/a(t)^3$ in () we obtain $H(t)^2 > \\frac{1}{3} \\left( \\frac{N}{a(t)^3} + \\frac{4}{N^2}\\right) - \\frac{1}{a(t)^2}.$ Defining the function $g(x) = \\frac{1}{3} \\left( \\frac{N}{x^3} + \\frac{4}{N^2} \\right) - \\frac{1}{x^2}$ for all $x > 0$ , we find $g^{\\prime }(x) = \\frac{-N + 2x}{x^4}.$ Therefore, $g^{\\prime }(x) = 0$ only at $x = \\frac{N}{2}$ , and $g$ is minimized at this point.", "Thus, $H(t)^2 > g(a(t)) \\ge g \\left(\\frac{N}{2} \\right) = 0.$ This implies that $H$ does not change sign and thus $H(t) > 0$ on $[0,T_{\\max })$ .", "Hence, $\\dot{a}(t) > 0$ , which implies $a(t)\\ge a_0$ for all $t\\in [0,T_\\mathrm {max})$ , and $T_{\\max } = \\infty $ follows from Lemma REF .", "Now, if we instead assume $\\lim _{t\\rightarrow T_\\mathrm {max}^-}\\phi (t)\\ge \\frac{9}{4\\rho _0a_0^4},$ then define $T^* = \\sup \\left\\lbrace t \\ge 0 : H(s) > 0 \\text{ for all } s\\in [0,t] \\right\\rbrace ,$ and note that $T^* > 0$ because $H_0 > 0$ .", "For $t \\in [0,T^*)$ we use (REF ) to find $\\dot{\\rho } = -3H(\\rho + \\mathcal {P}) - \\dot{\\phi } \\ge -4H\\rho ,$ and thus $\\frac{d}{dt} \\left[a(t)^4 \\rho (t) \\right] \\ge 0.$ Hence, a lower bound on $\\rho $ follows, namely $\\rho (t) \\ge \\rho _0a_0^4 a(t)^{-4}.$ Thus, on the same time interval, we use () to find $H(t)^2 \\ge \\frac{1}{3} \\phi (t) + \\frac{1}{3} \\rho _0 a_0^4 a(t)^{-4} - a(t)^{-2}.$ Defining the function $h(x) = \\frac{1}{3} \\rho _0 a_0^4 x^{-4} - x^{-2}$ for $x > 0$ , we find its minimum value occurs when $x^2 = \\frac{2}{3} \\rho _0 a_0^4$ , and at this value $h(x) = -\\frac{3}{4\\rho _0a_0^4}$ .", "Thus, for every $t \\in [0,T^*)$ $H(t)^2 \\ge \\frac{1}{3} \\phi (t) + h(a(t)) \\ge \\frac{1}{3} \\phi (t) - \\frac{3}{4\\rho _0a_0^4}.$ Using the above assumption on the strictly decreasing function $\\phi (t)$ , we see that $H(T^*)$ remains positive, and as in the previous argument this implies $T^* = T_{\\max } = \\infty $ ." ], [ "Initial data for global existence and blowup", "Our next goal is to prove that for each $k=0,\\pm 1$ there exist conditions on the initial data for which the solution is global and conditions which imply finite time blow-up.", "There are many ways to express these conditions; one is to write them in the form $\\phi _0<\\dots \\Rightarrow $ blow-up and $\\phi _0>\\dots \\Rightarrow $ global existence, where the right hand side of the inequality depends on $N,H_0,a_0$ and the given constants $k$ and $\\sigma $ (but not on $\\phi _0)$ .", "When expressed in this form it is straightforward to show that the conditions on the initial data derived in the present section are compatible with the constraint equation (REF ).", "We begin with the blow-up results.", "Theorem 3.6 Let $k=0,-1$ .", "Then for $\\phi _0<\\frac{\\sigma N}{H_0 a_0^3},$ the regular solution blows-up in finite time, i.e., $T_\\mathrm {max}<\\infty $ .", "If $k=1$ , then the solution blows-up in finite time when $\\phi _0<\\frac{9\\sigma N^2}{4 a_0^3}\\sqrt{\\frac{\\pi }{2}}\\mathrm {Erfc}(x)e^{x^2},\\quad x=\\frac{9H_0N}{4\\sqrt{2}},$ where $\\mathrm {Erfc}(z)$ denotes the complementary error-function.", "First, suppose that (REF ) holds and assume $T_\\mathrm {max}=+\\infty $ .", "By () we have $H(t)\\le H_0$ , and hence for all $t \\in [0,T_{\\max })$ it follows that $a(t)\\le a_0 e^{H_0t}.$ Integrating () we obtain $\\phi (t)=\\phi _0-3\\sigma N\\int _0^t\\frac{ds}{a(s)^3}\\le \\phi _0+\\frac{\\sigma N}{a_0^3H_0}(e^{-3H_0t}-1)\\rightarrow \\phi _0-\\frac{\\sigma N}{a_0^3 H_0}.$ Hence under condition (REF ) we have $\\lim _{t\\rightarrow +\\infty }\\phi (t)<0$ , and so, by Theorem REF , $T_\\mathrm {max} < \\infty $ .", "This contradicts the original assumption and implies $T_{\\max } < \\infty $ in the cases $k=0,-1$ .", "In the case $k=1$ , the bound on $H$ is much weaker.", "Hence, let us suppose that  (REF ) holds and assume $T_\\mathrm {max}=+\\infty $ .", "We first notice that by () and the bound $\\rho (t) \\ge N/a(t)^3$ , $\\dot{H}\\le \\frac{1}{a(t)^2}-\\frac{N}{2a(t)^3}\\le \\max _{x>0} \\left(x^{-2}-\\frac{N}{2} x^{-3} \\right)=\\frac{16}{27 N^2}.$ Hence, $H\\le H_0+\\frac{16}{27 N^2}t$ , which gives $a(t)\\le a_0\\exp \\left(H_0t+\\frac{8}{27 N^2}t^2 \\right)=a_0e^{-\\frac{27}{32}H_0^2N^2}\\exp \\left(\\frac{1}{N}\\sqrt{\\frac{8}{27}}\\,t+\\sqrt{\\frac{27}{32}}H_0N \\right)^2.$ Integrating () we obtain $\\phi (t)&\\le \\phi _0-\\frac{3\\sigma N}{a_0^3}e^{\\frac{81}{32}H_0^2N^2}\\int _0^t\\exp \\left[-3 \\left(\\frac{1}{N}\\sqrt{\\frac{8}{27}}\\,s+\\sqrt{\\frac{27}{32}}H_0N \\right)^2 \\right]ds\\\\&\\rightarrow \\phi _0-\\frac{9\\sigma N^2}{4 a_0^3}\\sqrt{\\frac{\\pi }{2}}\\mathrm {Erfc}(x)e^{x^2},\\quad x=\\frac{9H_0N}{4\\sqrt{2}},$ as $t\\rightarrow \\infty $ .", "Thus, under condition (REF ) we have $\\lim _{t\\rightarrow +\\infty }\\phi (t)<0$ , and by Theorem REF we reach a contradiction.", "With this, we turn to initial data which launch a global-in-time solution.", "Theorem 3.7 Let $k=-1,0$ .", "Then for $\\phi _0\\ge 3\\frac{\\sigma N}{H_0a_0^3},$ the regular solution is global, i.e., $T_\\mathrm {max}=\\infty $ and $\\lim _{t\\rightarrow \\infty }\\phi (t)\\ge \\phi _0- 3\\frac{\\sigma N}{H_0a_0^3}.$ To show global existence under condition (REF ), we define $T^* = \\sup \\left\\lbrace t \\ge 0 : \\phi (\\tau ) > 0 \\ \\mathrm {for \\ all} \\ \\tau \\in [0,t] \\right\\rbrace .$ Note that $T_\\mathrm {max} \\ge T^*$ by Theorem REF .", "Moreover, () implies that $H(t)>0$ for $t\\in [0,T^*)$ .", "By (REF ), () and the bound $\\mathcal {P}\\le \\rho /3$ we have $\\dot{H}\\ge \\frac{1}{3}\\phi -\\frac{1}{3}\\rho -H^2=\\frac{2}{3}\\phi -\\frac{k}{a^2}-2H^2\\ge -2H^2, \\quad t\\in [0,T^*).$ Integrating and using the fact that $H(t)>0$ on $[0,T^*)$ we obtain $H(t)\\ge \\frac{H_0}{1+2H_0t},\\quad \\text{ for $t\\in [0,T^*)$}$ and so $a(t)\\ge a_0(1+2H_0t)^{1/2},\\quad \\text{for $t\\in [0,T^*$}).$ Using () we get $\\phi (t)\\ge \\phi _0-\\frac{3\\sigma N}{a_0^3}\\int _0^t\\frac{ds}{(1+2H_0s)^{3/2}}>\\phi _0-\\frac{3\\sigma N}{a_0^3 H_0}.$ Hence, under condition (REF ), if $T^*<T_\\mathrm {max}$ we would have $\\phi (T^*)>0$ , which contradicts the maximality of $T^*$ .", "It follows that $T^*=T_\\mathrm {max}=+\\infty $ .", "The lower bound on the limit of $\\phi (t)$ follows by letting $t\\rightarrow \\infty $ in (REF ).", "Notice that the last two results suggest a particular quantity involving initial data that can predict the finite or infinite lifespan of the smooth solution in the cases of $k=-1,0$ .", "More specifically, if $\\frac{\\phi _0H_0 a_0^3}{\\sigma N}$ is large enough then the solution must be global, while if this quantity is made too small, then solutions must blow-up in finite time.", "As before, the situation for $k=1$ is more complicated, and the next result establishes conditions on initial data that guarantee global existence in this case.", "Theorem 3.8 Let $k=1$ .", "Then, the solution is global under the conditions $\\frac{3\\sigma }{H_0} \\le 1 \\qquad \\mathrm {and} \\qquad \\rho _0 a_0^2 < \\frac{3}{2}$ or the conditions $\\frac{3\\sigma }{H_0} > 1 \\qquad \\mathrm {and} \\qquad \\rho _0 a_0^2 \\le \\frac{3}{2} \\left(\\frac{H_0}{3\\sigma } \\right) \\exp \\left(1 - \\frac{H_0}{3\\sigma } \\right).$ First, note that under either condition (REF ) or (REF ), we have $\\rho _0 a_0^2<3/2$ ; hence from (REF ) and (), it follows that $\\dot{H}(0)>0$ .", "Define $T^*= \\sup \\left\\lbrace t \\ge 0 : H(s) \\ \\ge \\ \\frac{NH_0}{\\rho _0 a_0^3} \\ \\ \\mathrm {for \\ all} \\ s \\in [0,t] \\right\\rbrace $ and notice that $T^* > 0$ because $\\rho _0 \\ge \\frac{N}{a_0^3}$ and $\\dot{H}(0)>0$ .", "Then, for $t \\in [0,T^*)$ Lemma REF yields $\\rho (t) \\le (\\rho _0 a_0^3 + 3\\sigma N t) a(t)^{-3}$ and similarly, the lower bound $ a(t) \\ge a_0 \\exp \\left(\\frac{NH_0}{\\rho _0 a_0^3} t \\right)$ follows from the bound on $H(t)$ .", "Additionally, on the same time interval (REF ) and () then imply $\\dot{H}(t) & \\ge & -\\frac{2}{3} \\rho (t) + \\frac{1}{a(t)^2}\\\\& \\ge & a(t)^{-2} \\left[ 1- \\frac{2}{3} (\\rho _0 a_0^3 + 3\\sigma Nt ) a(t)^{-1} \\right]\\\\& \\ge & a(t)^{-2} \\left[ 1- \\frac{2}{3} (\\rho _0 a_0^3 + 3\\sigma Nt ) a_0^{-1} \\exp \\left(-\\frac{NH_0}{\\rho _0 a_0^3} t \\right) \\right].$ Now, the function $g(t) = 1- \\frac{2}{3} (\\rho _0 a_0^3 + 3\\sigma Nt ) a_0^{-1} \\exp \\left(-\\frac{NH_0}{\\rho _0 a_0^3} t \\right)$ is minimized at the point $t^* = \\frac{\\rho _0 a_0^3}{3\\sigma N} \\left( \\frac{3\\sigma }{H_0} - 1 \\right).$ If $\\frac{3\\sigma }{H_0} \\le 1$ , then $g(t)$ is increasing for all $t \\ge 0$ and the minimum for nonnegative times occurs at $t = 0$ .", "Therefore, $\\dot{H}(t) \\ge a(t)^{-2} g(0) = a(t)^{-2} \\left( 1- \\frac{2}{3}\\rho _0a_0^2 \\right) > 0$ assuming that (REF ) holds.", "Instead, if $\\frac{3\\sigma }{H_0} > 1$ then $t^* > 0$ .", "Therefore, $\\dot{H}(t) \\ge a(t)^{-2} g(t^*) = a(t)^{-2} \\left[ 1- \\frac{2\\sigma \\rho _0 a_0^2}{H_0} \\exp \\left(\\frac{H_0}{3\\sigma } - 1 \\right) \\right] > 0$ assuming that (REF ) holds.", "In either case, $\\dot{H}(t) > 0$ for all $t \\in (0,T^*)$ , which implies that $H(t) > H_0$ on the same interval.", "Hence, the above lower bound on $a(t)$ continues on $[0,T^*)$ , yielding a regular solution, and because $H(T^*) > H_0$ we find $T^* = \\infty $ .", "Our final result in this section is meant to unify the treatment for all three cases.", "Theorem 3.9 Let $k=-1,0,1$ and take $\\beta \\in (0,1)$ .", "Then, denoting $k_+ = \\max \\lbrace k,0\\rbrace $ , the solution is global under the condition $\\phi _0 \\ge \\frac{3}{\\beta ^2 a_0^2} \\left[ k_+ + 3\\left(\\frac{\\sigma N}{6} \\right)^{2/3}\\right]$ and furthermore $\\lim _{t\\rightarrow \\infty }\\phi (t)\\ge \\phi _0-\\frac{6}{\\beta ^2a_0^2}\\left(\\frac{\\sigma N}{6}\\right)^{2/3}.$ Define $\\gamma = \\frac{1}{\\beta a_0} \\left(\\frac{\\sigma N}{6} \\right)^{1/3} > 0$ and suppose that (REF ) holds.", "After some algebra, we see that this inequality implies $ \\frac{1}{3}\\phi _0 - \\frac{\\sigma N}{3\\beta ^3 a_0^3 \\gamma } - \\frac{k_+}{\\beta ^2 a_0^2} \\ge \\gamma ^2.$ Then, define $T^* = \\sup \\left\\lbrace t \\in [0,T_{\\max }) : a(\\tau ) \\ge \\beta a_0 e^{\\gamma \\tau } \\ \\mathrm {for \\ all} \\ \\tau \\in [0,t] \\right\\rbrace $ and note that since $\\beta < 1$ , we have $T^* > 0$ .", "Now, integrating () for $t \\in [0,T^*)$ , we find $\\phi (t)=\\phi _0-3\\sigma N\\int _0^t\\frac{ds}{a(s)^3}\\ge \\phi _0+\\frac{\\sigma N}{\\gamma \\beta ^3 a_0^3}(e^{-3\\gamma t}-1)\\ge \\phi _0-\\frac{\\sigma N}{\\gamma \\beta ^3 a_0^3}.$ Hence, () implies $H(t)^2 \\ge \\frac{1}{3} \\phi (t) - \\frac{k_+}{a(t)^2} \\ge \\frac{1}{3}\\phi _0-\\frac{\\sigma N}{3\\beta ^3 a_0^3\\gamma } - \\frac{k_+}{\\beta ^2 a_0^2} \\ge \\gamma ^2$ so that $H(t)$ does not change sign on $[0,T^*)$ and $H(t) \\ge \\gamma $ .", "It follows that $a(t) \\ge a_0 e^{\\gamma t}$ for all $t \\in [0,T^*)$ .", "Since $\\beta < 1$ , this then implies $T^* = T_{\\max }$ .", "Finally, since $\\gamma > 0$ we find $\\inf _{t \\in [0,T_{\\max })} a(t) = a_0 > 0.$ It follows by Lemma REF that $T_{\\max } = \\infty $ , and the proof of the theorem is complete." ], [ "Asymptotic behavior of solutions", "In this section we analyze the asymptotic behavior as $t\\rightarrow \\infty $ of global solutions and as $t\\rightarrow T_\\mathrm {max}^-$ of solutions which blow-up in finite time.", "We begin with the former solutions.", "Theorem 4.1 Consider a global regular solution, i.e., $T_\\mathrm {max}=\\infty $ , and let $\\displaystyle \\phi _{\\infty } := \\lim _{t \\rightarrow \\infty } \\phi (t)\\ge 0 $ .", "Then for $k=0,-1$ the following holds: (i) If $\\phi _{\\infty }> 0$ , then there exist $C_1,C_2> 0$ such that for all $t \\ge 0$ $\\sqrt{\\frac{\\phi _{\\infty } }{3}} \\le & H(t) & \\le \\sqrt{\\frac{\\phi _{\\infty } }{3}} + C_2\\exp \\left({-\\sqrt{\\frac{\\phi _{\\infty } }{3}} t} \\right) , \\\\\\phi _{\\infty } + C_1\\sigma \\exp \\left({-3\\sqrt{\\frac{\\phi _{\\infty }}{3}} t} \\right) < & \\phi (t) & \\le \\phi _{\\infty } + C_2 \\sigma \\exp \\left({-3\\sqrt{\\frac{\\phi _{\\infty }}{3}} t} \\right), \\\\a_0 \\exp \\left({\\sqrt{\\frac{\\phi _{\\infty } }{3}} t} \\right) \\le & a(t) & \\le C_2 \\exp \\left({\\sqrt{\\frac{\\phi _{\\infty } }{3}} t} \\right),\\\\C_1\\sigma \\exp \\left({-3\\sqrt{\\frac{\\phi _{\\infty }}{3}} t} \\right) \\le & \\rho (t) & \\le C_2 (1+\\sigma t) \\exp \\left({-3\\sqrt{\\frac{\\phi _{\\infty }}{3}} t} \\right).$ (ii) If $\\phi _{\\infty }= 0$ , then there exist $C_1, C_2 > 0$ such that for all $t \\ge 0$ $C_1 (1+t)^{-1} \\le & H(t) & \\le C_2(1+t)^{-1/2} \\left[1 + \\sigma (1+t)^{1/4} \\right], \\\\0 < & \\phi (t) & \\le C_2(1 + t)^{-1/2}, \\\\C_1(1+t)^{1/2} \\le & a(t) & \\le C_2\\exp { (t^{1/2}[1 + \\sigma t^{1/4}}]),\\\\C_1\\exp { (-3t^{1/2}[1 + \\sigma t^{1/4}}]) \\le & \\rho (t) & \\le C_2(1 +t)^{-3/2} (1+\\sigma t).$ For the case $k=1$ , if $\\phi _{\\infty }> \\min \\left\\lbrace \\frac{4}{N^2}, \\frac{9}{4\\rho _0 a_0^4} \\right\\rbrace :=\\phi _m$ then the same conclusion as in (i) holds with $\\phi _\\infty $ replaced by $\\phi _*=\\phi _\\infty -\\phi _m$ .", "We first assume $k=0,-1$ .", "If $\\phi _{\\infty } > 0$ , then, by (), $H(t)^2 \\ge \\frac{1}{3} \\phi (t) \\ge \\frac{1}{3} \\phi _{\\infty }$ for all $t \\ge 0$ .", "Thus $H$ remains strictly positive for all time with $H(t) \\ge \\sqrt{\\frac{\\phi _{\\infty }}{3}}.$ It follows that for all $t \\ge 0$ $a(t) \\ge a_0 \\exp \\left( \\sqrt{\\frac{\\phi _{\\infty }}{3}} t \\right).$ Furthermore, due to (REF ) and using the positivity of $H$ in Lemma REF implies $N a(t)^{-3} \\le \\rho (t) \\le (\\rho _0 a_0^3 + 3\\sigma N t) a(t)^{-3}$ which yields the claimed behavior of $\\rho $ once the bounds on $a(t)$ are established.", "To arrive at a bound on $\\phi $ , we integrate () and use the lower bound on $a$ to find $\\phi _{\\infty } - \\phi (t) & = & -3\\sigma N \\int _t^{\\infty } a(s)^{-3} \\ ds\\\\& \\ge & -\\frac{3\\sigma N}{a_0^3} \\int _t^{\\infty } e^{-\\sqrt{3\\phi _{\\infty }}s} \\ ds\\\\& = & -\\sqrt{\\frac{3}{\\phi _{\\infty }}} \\frac{\\sigma N}{a_0^3} \\exp \\left(- 3 \\sqrt{\\frac{\\phi _{\\infty }}{3}} t \\right)$ for all $t \\ge 0$ , which implies an upper bound on the behavior of $\\phi $ .", "Using this in () with the upper bound on $\\rho $ and the lower bound on $a$ implies for all $t \\ge 0$ $ H(t)^2 \\le \\frac{1}{3}\\phi _\\infty + C(1+\\sigma t)\\exp \\left(-3\\sqrt{\\frac{\\phi _{\\infty }}{3}}t \\right) + a_0^{-2} \\exp \\left( -2\\sqrt{\\frac{\\phi _{\\infty }}{3}} t \\right).$ This further implies $H(t) \\le \\sqrt{\\frac{\\phi _\\infty }{3}} + C\\exp \\left(-\\sqrt{\\frac{\\phi _{\\infty }}{3}}t \\right),$ and, in conjunction with (REF ), this inequality implies the claimed behavior of $H$ .", "Finally, this upper bound on $H$ implies an analogous upper bound on $a$ , so that $a(t) = a_0 \\exp \\left(\\int _0^t H(s) \\ ds \\right) \\le C \\exp \\left(\\sqrt{\\frac{\\phi _\\infty }{3}} t\\right).$ Pairing this with (REF ) implies the bounds on $a(t)$ , and the upper bound on $a(t)$ further implies the lower bound on $\\phi (t)$ using ().", "Next, assume $\\phi _{\\infty } = 0$ .", "As in the previous case, we find by () $H(t)^2 \\ge \\frac{1}{3} \\phi (t) > 0$ for all $t \\ge 0$ , and thus $H$ remains strictly positive for all time.", "As in the proof of Theorem REF , we use (), (REF ), and the bound $\\mathcal {P}\\le \\rho /3$ to find $\\dot{H}\\ge \\frac{1}{3}\\phi -\\frac{1}{3}\\rho -H^2=\\frac{2}{3}\\phi -\\frac{k}{a^2}-2H^2\\ge -2H^2$ for $t\\ge 0$ .", "Integrating and using the positivity of $H(t)$ we obtain $H(t)\\ge \\frac{H_0}{1+2H_0t}$ and the lower bound $a(t)\\ge a_0(1+2H_0t)^{1/2}$ follows for $t \\ge 0$ .", "Furthermore, the positivity of $H$ implies the same upper bound on $\\rho $ as in the previous case, and we again find for $t \\ge 0$ $N a(t)^{-3} \\le \\rho (t) \\le C(1+\\sigma t) a(t)^{-3}.$ As before, this yields the claimed asymptotic behavior of $\\rho $ once the full behavior of $a$ is obtained.", "Integrating () and using (REF ), we find $ \\phi (t) = \\phi _{\\infty } + 3\\sigma N \\int _t^\\infty a(s)^{-3} \\ ds \\le \\frac{3\\sigma N}{H_0a_0^3} (1 + 2H_0t)^{-1/2}.$ Next, we use (), the upper bounds on $\\rho $ and $\\phi $ , and the lower bound on $a$ so that $H(t)^2 & \\le & \\frac{\\sigma N}{H_0a_0^3} (1 + 2H_0t)^{-1/2} + C(1+\\sigma t) a(t)^{-3} + a_0^{-2}(1+2H_0t)^{-1}\\\\& \\le & C(1+t)^{-1} \\left[1 + \\sigma (1+t)^{1/2} \\right]$ and thus $H(t) \\le C(1+t)^{-1/2} \\left[1 + \\sigma (1+t)^{1/4} \\right].$ With this, an upper bound on $a(t)$ follows, namely $ a(t) \\le C\\exp { (t^{1/2}[1 + \\sigma t^{1/4}}]).$ The last claim for $k=1$ follows by the same argument used to prove (i) and the lower bounds on $H$ attained for $k=1$ within the proof of Theorem REF .", "For example, assuming $\\phi _\\infty \\ge \\frac{4}{N^2}$ the argument proceeds by using $H(t)^2\\ge \\frac{1}{3}\\Big (\\phi _\\infty -\\frac{4}{N^2}\\Big )+\\frac{1}{3}\\Big (\\frac{N}{a(t)^3}+\\frac{4}{N^2}\\Big )-\\frac{1}{a(t)^2}\\ge \\frac{\\phi _*}{3}$ instead of (REF ).", "Similarly, assuming $\\phi _\\infty \\ge \\frac{9}{4\\rho _0a_0^4}$ the argument proceeds by using $H(t)^2\\ge \\frac{1}{3}\\phi (t)-\\frac{3}{4\\rho _0a_0^4} \\ge \\frac{\\phi _*}{3}$ instead of (REF ).", "We remark that sufficient conditions to have $\\phi _\\infty >0$ for $k=0,-1$ and $\\phi _*>0$ for $k=1$ follow by the lower bounds on $\\lim _{t\\rightarrow \\infty }\\phi (t)$ derived in Theorems REF and REF .", "Next, we provide some information on the behavior of solutions that blow-up in finite time as they approach the singularity.", "Theorem 4.2 Let $k=0,-1$ and consider a regular solution that blows up in finite time, i.e., $T_\\mathrm {max} <\\infty $ .", "Then, we have $\\lim _{t \\rightarrow T_{\\max }^-} H(t) = \\lim _{t \\rightarrow T_{\\max }^-} \\phi (t) = - \\infty , \\quad \\lim _{t \\rightarrow T_{\\max }^-} \\rho (t) = \\infty , \\quad \\ \\text{and} \\ \\lim _{t \\rightarrow T_{\\max }^-} a(t) = 0.$ Moreover, the singularity is a curvature singularity, and there exist $C_1, C_2 > 0$ such that for $t$ sufficiently close to $T_{\\max }$ $\\left.", "{\\begin{array}{c}-\\frac{1}{T_{\\max } - t} \\le H(t) \\le -\\frac{C_2}{\\sqrt{T_{\\max } - t}},\\\\-\\frac{C_1}{(T_{\\max } - t)^2} \\le \\phi (t) \\le -\\frac{C_2}{\\sqrt{T_{\\max } - t}},\\\\C_1(T_{\\max } - t) \\le a(t) \\le C_2 \\sqrt{T_{\\max } - t},\\\\\\frac{C_1}{(T_{\\max } - t)^{3/2}} \\le \\rho (t) \\le \\frac{C_2}{(T_{\\max } - t)^2}.\\end{array}}\\quad \\right\\rbrace $ For the case $k=1$ , if $\\displaystyle \\lim _{t \\rightarrow T_{\\max }^-} \\phi (t) < 0$ then the solution blows up in finite time and the same asymptotic behavior follows with identical bounds.", "By Theorem REF the finite time blow-up of the solution in the cases $k=0, -1$ implies $H(t) \\le 0$ for some $t \\in (0, T_{\\max })$ and $\\phi (t)$ must eventually attain negative values.", "Without loss of generality, we can take $T_1 \\in (0,T_{\\max })$ large enough so that $\\phi (t), H(t) < 0$ for $t \\in [T_1, T_{\\max })$ , as $\\phi $ is decreasing and $\\phi (t) < 0$ implies $\\dot{H}(t) < 0$ by (REF ).", "With this, we see that $a(t)$ is strictly decreasing on $(T_1, T_{\\max })$ , and by Lemma REF we find both $a(t) > 0$ for all $t \\in [0,T_{\\max })$ and $\\inf _{t \\in [0,T_{\\max })} a(t) = 0.$ Thus, it follows that $\\lim _{t \\rightarrow T_{\\max }^-} a(t) = 0.$ The limits of $H$ and $\\rho $ in (REF ) follow from this behavior.", "Indeed, the limiting behavior of $\\rho $ follows directly from the lower bound (REF ).", "Also, because $\\phi (t) < 0$ for $t \\in [T_1,T_{\\max })$ , we use (REF ) to arrive at $\\dot{H}(t) < - H(t)^2$ for $t \\in [T_1, T_{\\max })$ .", "Then, using the definition of $H$ , we compute $\\dot{H}(t) = \\frac{d}{dt} \\left(\\frac{\\dot{a}(t)}{a(t)} \\right) = \\frac{\\ddot{a}(t)}{a(t)} - H(t)^2$ so that $\\ddot{a}(t) = \\left(\\dot{H}(t) + H(t)^2 \\right)a(t) < 0$ for $t \\in [T_1,T_{\\max })$ .", "Therefore, $ \\dot{a}(t) < \\dot{a}(T_1) < 0$ on $[T_1, T_{\\max })$ , and $H(t) = \\frac{\\dot{a}(t)}{a(t)} < \\frac{\\dot{a}(T_1)}{a(t)}$ for $t \\in [T_1, T_{\\max })$ then implies $H(t) \\rightarrow -\\infty $ as $t \\rightarrow T_{\\max }^-$ .", "The limiting behavior of $\\phi $ will be obtained from the upper bound established below.", "Next, we turn to establishing the precise asymptotic behavior of these functions at the blow-up time.", "Using (REF ) we find $\\lim _{t \\rightarrow T_{\\max }^-} H(t)^{-1} = 0$ , and because $T_{\\max }$ is the blow-up time, it follows that $T_{\\max } \\le T_1 - \\frac{1}{H(T_1)}$ .", "Therefore, using (REF ) and evoking Lemma REF with $t_1 = T_1$ and $t_2 = T_{\\max }$ , we find $H(t) \\ge - \\frac{1}{T_{\\max } - t}$ for $t \\in [T_1, T_{\\max })$ .", "With this estimate on $H$ we also find for $\\tau \\in [T_1,T_{\\max })$ $\\dot{a}(\\tau ) \\ge -\\frac{1}{T_{\\max } - \\tau } a(\\tau ),$ and upon dividing by $a(\\tau ) > 0$ and integrating over $[T_1,t]$ , the lower bound $a(t) \\ge \\frac{a(T_1)}{T_{\\max } - T_1} \\left(T_{\\max } - t \\right) \\ge C\\left(T_{\\max } - t \\right)$ follows for $t \\in [T_1, T_{\\max })$ .", "The lower bound on $\\phi (t)$ is obtained by using this estimate within () so that $\\phi (t) \\ge \\phi (T_1) - C\\sigma \\int _{T_1}^t \\frac{ds}{(T_{\\max } - s)^3} = \\phi (T_1) + \\frac{C \\sigma }{(T_{\\max } -T_1)^2} - \\frac{C\\sigma }{(T_{\\max } - t)^2}\\ge - \\frac{C}{(T_{\\max } - t)^2}$ for $t$ sufficiently close to $T_{\\max }$ .", "Similarly, to derive the upper bound on $\\rho $ we first use () to find $ \\rho = 3 \\left( H^2 + \\frac{k}{a^2} \\right) - \\phi \\le 3H^2 + \\frac{3}{a^2} - \\phi .$ Then, the previously-obtained lower bound on $H(t)$ for $t \\in [T_1, T_{\\max })$ implies $\\vert H(t) \\vert \\le \\frac{1}{T_{\\max }- t}$ , and thus $H(t)^2 \\le \\frac{1}{(T_{\\max } - t)^2}$ on the same time interval.", "Using this with the lower bounds on $a$ and $\\phi $ , we have $\\rho (t) \\le \\frac{C}{(T_{\\max } - t)^2}$ for $t$ sufficiently close to $T_{\\max }$ .", "Next, we establish bounds on these functions in the opposite directions.", "Since we know $a \\rightarrow 0$ as $t \\rightarrow T_{\\max }^-$ and $H(T_1), \\phi (T_1) < 0$ with each of these functions decreasing on $(T_1, T_{\\max })$ , we can find $T_2 \\in [T_1, T_{\\max })$ such that $H(t), \\phi (t) < 0$ and $a(t) < 1$ for $t \\in [T_2, T_{\\max })$ .", "Then, using (REF ), (REF ), and () we find $\\dot{\\rho }(t) \\le -4H(t)\\rho (t) + 3\\sigma N a(t)^{-3} \\le -4H(t) \\rho (t) + 3\\sigma N a(t)^{-4}$ for $t \\in [T_2, T_{\\max })$ .", "Multiplying by $a(t)^4$ , we can rewrite the inequality as $\\frac{d}{dt} \\left( \\rho (t) a(t)^4 \\right) \\le 3\\sigma N$ and integrating yields $\\rho (t) \\le a(t)^{-4} \\left( \\rho (T_2) a(T_2)^4 + 3\\sigma N(T_{\\max } - T_2) \\right) \\le Ca(t)^{-4}$ for $t \\in [T_2, T_{\\max })$ .", "Using this estimate within (), we find $H(t)^2 \\le \\frac{1}{3} \\rho + a(t)^{-2} \\le C \\left(a(t)^{-4} + a(t)^{-2} \\right) \\le C a(t)^{-4}$ for $t \\in [T_2, T_{\\max })$ .", "Multiplying by $a(t)^4$ , this becomes $\\vert a(t) \\dot{a}(t) \\vert ^2 \\le C$ which, because $\\dot{a}(t) < 0$ , further implies $ -\\frac{1}{2} \\frac{d}{dt} \\left( a(t)^2 \\right) \\le C.$ Integrating over $[\\tau , T_{\\max })$ and using the limiting behavior of $a$ as $t \\rightarrow T_{\\max }^-$ , we find $a(\\tau ) \\le C \\sqrt{T_{\\max } - \\tau }$ for $\\tau \\in [T_2, T_{\\max })$ .", "With the upper bound on $a$ established, the lower bound on $\\rho $ follows directly from (REF ) so that $ \\rho (t) \\ge N a(t)^{-3} \\ge \\frac{C}{(T_{\\max } -t )^{3/2}}$ for $t \\in [T_2, T_{\\max })$ .", "Additionally, the upper bound on $\\phi $ follows exactly as before.", "In particular, integrating () over $[T_2, t]$ we find $\\phi (t) \\le \\phi (T_2) - C \\int _{T_2}^t \\frac{ds}{(T_{\\max } - s)^{3/2}} \\le - \\frac{C}{\\sqrt{T_{\\max } - t}}$ for $t$ sufficiently close to $T_{\\max }$ .", "Furthermore, this estimate implies the limiting behavior of $\\phi $ within (REF ).", "Finally, the upper bound on $H$ is obtained from (REF ) so that $ \\dot{H}(t) \\le - \\frac{1}{6}\\rho \\le -\\frac{C}{(T_{\\max } - t)^{3/2}}$ for $t \\in [T_2, T_{\\max })$ .", "Integrating over $[T_2, \\tau ]$ then yields the result $H(\\tau ) \\le -\\frac{C}{\\sqrt{T_{\\max } - \\tau }}$ for $\\tau $ sufficiently close to $T_{\\max }$ .", "For the case $k = 1$ we merely repeat this argument under the assumption that $\\phi (t) < 0$ for some $t \\in (0, T_{\\max })$ since this gives rise to negative values of $H(t)$ as in the proof of Theorem REF , as well as, the upper bound $\\dot{H}(t) \\le - H(t)^2$ .", "Finally the singularity at $t\\rightarrow T_\\mathrm {max}^-$ is a curvature singularity, for the Einstein equation (REF ) and the proven bounds imply $R=4\\phi -(\\rho +3\\mathcal {P})\\rightarrow -\\infty .$ With these results at hand we can finally discuss the important question of the existence of a phase of accelerated expansion of the Universe in the future of a phase of decelerated expansion.", "There is overwhelming experimental evidence that the Universe is currently expanding with acceleration.", "Nonetheless, the standard cosmological models require the existence in the past of a phase of decelerated expansion, during which the structures visible today (galaxies, clusters, etc.)", "have formed.", "The following last result shows that the diffusion model studied in this paper is able to reproduce this physical behavior.", "Let us define the dimensionless constants: $\\Sigma _0=\\frac{\\sigma }{H_0},\\quad \\Phi _0=\\frac{\\phi _0}{\\rho _0}.$ Corollary 4.3 Let $k=0$ or $-1$ .", "Then for $3\\Sigma _0<\\Phi _0<\\frac{1}{2}$ the regular solution is global and there exist $0<t_0<t_1$ such that $q(t)>0$ for $t\\in [0,t_0)$ , and $q(t)<0$ for $t>t_1$ .", "As $\\rho _0\\ge N/a_0^3$ , then $\\Phi _0>3\\Sigma _0$ implies $\\phi _0>3\\frac{\\sigma N}{H_0 a_0^3}$ , hence the solution is global by Theorem REF .", "Moreover $\\lim _{t\\rightarrow \\infty }q(t)=\\lim _{t\\rightarrow \\infty } \\left[ \\rho (t)+3\\mathcal {P}(t)-2\\phi (t) \\right] =-2\\phi _\\infty <0,$ and therefore there exists $t_1>0$ such that $q(t)$ is negative for $t>t_1$ .", "Furthermore, we find $q(0)=\\rho _0+3\\mathcal {P}_0-2\\phi _0>\\rho _0-2\\phi _0=\\rho _0(1-2\\Phi _0)>0.$ Hence, $\\Phi _0<1/2$ implies $q(0)>0$ , which further guarantees the existence of $t_0>0$ such that $q(t)>0$ for $t\\in [0,t_0)$ ." ] ]
1606.05101
[ [ "Incentivized Campaigning in Social Networks" ], [ "Abstract Campaigners, advertisers and activists are increasingly turning to social recommendation mechanisms, provided by social media, for promoting their products, services, brands and even ideas.", "However, many times, such social network based campaigns perform poorly in practice because the intensity of the recommendations drastically reduces beyond a few hops from the source.", "A natural strategy for maintaining the intensity is to provide incentives.", "In this paper, we address the problem of minimizing the cost incurred by the campaigner for incentivizing a fraction of individuals in the social network, while ensuring that the campaign message reaches a given expected fraction of individuals.", "We also address the dual problem of maximizing the campaign penetration for a resource constrained campaigner.", "To help us understand and solve the above mentioned problems, we use percolation theory to formally state them as optimization problems.", "These problems are not amenable to traditional approaches because of a fixed point equation that needs to be solved numerically.", "However, we use results from reliability theory to establish some key properties of the fixed point, which in turn enables us to solve these problems using algorithms that are linearithmic in maximum node degree.", "Furthermore, we evaluate the efficacy of the analytical solution by performing simulations on real world networks." ], [ "Introduction", "On-line social networking services have enabled advertisers, campaigners and activists to reach millions of individuals.", "In particular, the ability to recommend or share web articles [1], videos [2], and other media can be harnessed by campaigners for disseminating information to a worldwide audience.", "While such social network based campaigns sound promising, due to the fact that ideas propagate only a few hops from their origins [3], often, they are found to perform poorly in practice.", "Providing incentives for recommendations is a natural solution for increasing the hop count.", "For example, Dropbox, which offers cloud storage services, increased its customer base rapidly by offering incentives for social recommendations [4].", "Although such referrals may increase the short term costs to the company by generating a large number of registrations for the free service, in the long term, the free registrations pay off, since a significant portion of free users eventually migrate to the premium service.", "Also, to keep the cost down, they capped the referral payouts — additional 500 MB only for the first 28 referrals, i.e.", "the referral payout is capped at 14 GB.", "An alternative mechanism to limiting the cost is to cap the number of incentivized individuals, i.e., instead of capping incentive payments, offer large referral rewardsFor example the living social website gives $100\\%$ cash-back on a purchased product if the customer persuades three others to buy the (same) product.", "Another famous example of uncapped referrals is Uber, and also the taxi service Lyft.", "to a preselected set of individualsInstead of advertising the referral scheme, a randomized scheme can be advertised.", "For example, the advertisement could be “those who register for the service will get a chance to win discounts for referring their friends.\".", "This approach allows the campaigner to target individuals who are best suited to use and recommend the service to their co-workers or friends.", "In [5], [6], [7], the authors consider nodes to be either active or inactive.", "Making an inactive node active — not through network effects, but through direct intervention for kick-starting the campaign — is termed as “seeding.” Assuming full knowledge of the network structure, i.e., the adjacency matrix, the question asked in [5], [6], [7] is; Given a constraint on the number of seeds, what is the optimal set of seed nodes that would maximize the reach of the campaign?", "However, in several real-world settings, we may just have access to the statistical properties of the network.", "Also, the authors in [5], [6], [7] do not consider the cost incurred due to incentivization.", "Incentivization is the process of providing incentives to nodes who are already active, to persuade their friends to sign up.", "Incentivization happens throughout the campaign, whereas seeding, as discussed in [5], happens only at the beginning.", "A related problem involving the calculation of an optimal pricing strategy for products sold to individuals in a social network was explored in [8], [9].", "The authors in [8], [9] consider the problem of pricing a product and calculating the amount of cash-back (discount) that is provided to individuals as an incentive to evangelize the product.", "In this paper, we do not focus on optimal pricing, but rather focus on the size and cost of the campaign where individuals do not incur a cost to register for the service.", "The problems studied in this paper are more appropriate in settings where the service is free, or is based on a freemium model.", "Furthermore, we only assume partial knowledge of the network and also incorporate constraints that ensure a given campaign penetration, whereas the algorithms proposed in [8], [9] assume full knowledge of the network structure and do not consider any constraint on the campaign size.", "The problem of computing the optimal referral payment mechanisms that maximize profit was studied in [10], by modelling the referral process as a network game.", "The authors in [10] conclude that a combination of linear payment mechanism (linear in the number of referrals) and threshold payment mechanism (payment only when number of referral exceeds a threshold) approximates the optimal pricing scheme.", "In this paper, we focus on the set of nodes to be incentivized while assuming that a pricing scheme, which can be computed based on the results in [10], is provided by the campaigner.", "Similar problems involving the computation of referral rewards in real time, for maximizing the campaign spread, were studied using the theory of optimal control in [11], [12], [13], [14], [15]." ], [ "Our Contributions", "We consider a scheme where preselected incentivized individuals are presented with a reward when they register for the service, encouraging them to spread the news about the service to their friends.", "The decision of whether to offer an incentive to an individual is precomputed based on the solution of an optimization problem.", "We use a variant of the linear threshold model [5] for modelling the campaign spreading process.", "For a given fraction of such incentivized individuals, we first compute the campaign size (expected fraction of registered individuals) using bootstrap percolation.", "We then use this quantity to formulate the following optimization problems: 1).", "minimize the cost for achieving a given expected fraction of registered individuals, and 2).", "maximize the expected fraction of registered individuals for a given cost budget.", "These optimization problems are not amenable to traditional approaches because of a fixed point equation that needs to be solved numerically.", "However, we use results from reliability theory to establish some key properties of the fixed point, which in turn enables us to solve these problems using algorithms that are linearithmic in the maximum node degree.", "Through extensive simulations, we also study the efficacy our incentivization scheme in real world networks.", "Table: A summary of key notationThe key notation used in this paper is summarized in Table REF .", "The remainder of this paper is organized as follows.", "In Section , we present the campaign model.", "In Section , using results from percolation theory, we compute the campaign size (expected proportion of registered individuals).", "Using this quantity, we formulate and solve two relevant optimization problems in Sections  and .", "In Section , we compare the analytical results with simulation performed on real-world networks.", "Finally, in Section , we conclude the paper.", "Figure: Flow chart denoting the various stages of a node; mm denotes the threshold of node ii, α 1 \\alpha _1 and α 2 \\alpha _2 are the activation probabilities of incentivized and non-incentivized nodes, respectively.Consider a set of $\\mathcal {N} = \\lbrace 1, 2, \\cdots , n\\rbrace $ individuals who are connected to one another through a social network.", "For analytical tractability, as in [16], [17], we consider this network to be arbitrary, connectedThe configuration model random graph is connected with high probability if and only if the minimum degree is greater than or equal to 3., locally tree-like and uncorrelatedUncorrelated networks are networks where the degree of a node is statistically independent of the degree of any other node in the network [18].. We represent this network as an undirected graph $\\mathcal {G}(\\mathcal {N}, \\mathcal {L})$ , where $\\mathcal {N}$ and $\\mathcal {L}$ represent the set of nodes and links of graph $\\mathcal {G}$ , respectively.", "An undirected edge $(a,b) \\in \\mathcal {L}$ if individuals $a$ and $b$ are neighbours in the underlying social network.", "In most scenarios, full knowledge of the network structure may not be available to the campaigner.", "In such cases, the campaigner can obtain statistical properties of the network through data mining.", "One such property is the degree distribution.", "Let $\\lbrace p(k), k \\ge 1 \\rbrace $ be the degree distribution of graph $\\mathcal {G}$ .", "We consider a campaign on the network represented by graph $\\mathcal {G}$ (see Fig.", "REF for the flow chart).", "The nodes in graph $\\mathcal {G}$ can be either in “active”, “registered” or “unregistered” state.", "Once active, a node remains active.", "Active nodes are very spirited and express their strong support for the campaign by encouraging their neighbours to register, whereas nodes in the registered and unregistered states refrain from recommendations.", "For a node to become active, it must first show support for the campaign by registering itself.", "However, all registered nodes need not be active.", "The campaign starts with a set of zealous individuals — those who intrinsically desire the service.", "These nodes do not need any recommendation to register, and are registered for the campaign from the very start.", "If a zealous node becomes active, it will recommend the service to its neighbours.", "On the other hand, nodes that are not zealous, register only if the number of active neighbours exceeds a predefined threshold value (see Fig.", "REF ).", "Let $p_{th}(m|k)$ denote the probability of a degree $k$ node having threshold $m$ .", "This is a generalization of the celebrated linear threshold model [5] — a model of choice for studying the dissemination and adoption of new products, technologies and ideas [19], [20], [21], [22].", "We note that, by definition, $\\sum _{k \\ge 1} p(k) \\cdot p_{th}(0|k)$ denotes the fraction of zealous nodes in the networkCampaigner can estimate the fraction of zealous nodes by conducting surveys on a reasonable sized demographic.", "— fraction of nodes that register without any recommendations.", "A node that receives a reward when it registers for the service is more likely to tell its neighbours about the reward and encourage them to register.", "Therefore, we divide the nodes into two categories: the `non-incentivized' (type 1) and the `incentivized' (type 2).", "The campaigner decides the fraction of incentivized nodes before the start of the campaignAlternatively, whenever an individual registers, the campaigner can dynamically decide whether to incentivize this individual.", "This problem is similar to stochastic control problems, and is beyond the scope of this paper..", "However, these nodes become aware of the fact that they are incentivized only if they register for the serviceIncentives are assigned assuming knowledge of the exact degree of the registered node, because the exact node degree can be obtained after registering.", "For example, one can ask individuals to register through an on-line social network, and compute the exact degree from their contact list..", "Since incentivized nodes are provided with incentives upon registration, they are more likely to be enthused, and will play an active part in the campaign than non-incentivized nodes.", "Let $\\alpha _1 \\in (0,1]$ and $\\alpha _2 \\in (0,1]$ be the probabilities of a non-incentivized (type 1) and incentivized (type 2) node becoming active, respectively.", "Due to the presence of incentives, we assume that $\\alpha _2>\\alpha _1$ .", "Figure: NO_CAPTION" ], [ "Computing the Cascade Size", "Before presenting the problems, we need to first compute cascade size, i.e., the expected fraction of nodes that have registered at the end of the campaign.", "Let $p(k^{\\prime } | k)$ be the probability of encountering a degree $k^{\\prime }$ node by traversing a link from a degree $k$ node.", "From [16], we know that this conditional distribution, for an uncorrelated network, has the following form $p(k^{\\prime } | k) = \\frac{k^{\\prime } \\cdot p(k^{\\prime })}{ \\overline{d}}$ , where $\\overline{d}$ denotes the mean degree of the graph $\\mathcal {G}$ .", "Let $q(k)$ be the probability of encountering a type 2 node by traversing a randomly chosen link from a degree $k$ node.", "Then, we have $q(k) &= \\sum \\nolimits _{k^{^{\\prime }} \\ge 1} \\textrm {Pr}[\\textrm {type~2 neigh.}", "| k^{^{\\prime }}] \\cdot p(k^{^{\\prime }}| k) \\nonumber \\\\&= \\sum \\nolimits _{k^{^{\\prime }} \\ge 1} \\phi (k^{^{\\prime }}) \\cdot p(k^{^{\\prime }}| k) = \\frac{1}{\\overline{d}}\\sum \\nolimits _{k^{^{\\prime }} \\ge 1} k^{^{\\prime }} \\phi (k^{^{\\prime }}) p(k^{^{\\prime }})$ Since $q(k)$ is independent of $k$ , let $q(k) = q \\ \\forall \\ k \\in \\lbrace 1,2, \\cdots ,k_{max}\\rbrace $ .", "Here, $k_{max}$ is the maximum node degree.", "Since the network is locally tree-like, with some effort, it can be shown that the number of type 2 neighbours of a degree $k$ node is a binomial random variable with the probability distribution function $\\hat{p}(k_2 | k) = {k \\atopwithdelims ()k_2} \\cdot q^{k_2} \\cdot (1-q)^{k - k_2}$ .", "Let $p_{ex}(k^{^{\\prime }})$ be the excess degree distribution, i.e., the degree distribution of a node encountered by following a randomly chosen link without counting that link.", "For an uncorrelated network, from [17], we know that $p_{ex}(k^{^{\\prime }}) = \\frac{(k^{^{\\prime }}+1) \\cdot p(k^{^{\\prime }}+1)}{ \\overline{d}}$ .", "Now, let $u$ denote the probability of reaching a registered node by following an arbitrary edge of the graph.", "First, let us consider an arbitrary node $j$ of degree $k$ and threshold $m$ .", "Next, we compute the probability that node $j$ registers due to recommendations from its active neighbours, given that we have arrived at node $j$ by following an arbitrary link in the graph.", "For ease of presentation, we do not explicitly write the condition “following an arbitrary link.” $&\\hspace{0.0pt} P[j \\textrm { registers } | j \\textrm { is of degree } k, j \\textrm { has threshold } m] \\nonumber \\\\&\\overset{(a)}{=} {\\textstyle \\sum ^k_{l = m}} P[j \\textrm { has } l \\textrm { active neighbours} | j \\textrm { is of degree } k] \\nonumber \\\\&= {\\textstyle \\sum ^k_{l = m}} {\\textstyle \\sum \\nolimits ^k_{k_2=0}} P[k_2 \\textrm { type~2 neighbours} | j \\textrm { is of degree } k] \\nonumber \\\\& \\hspace{5.69054pt} \\cdot P[l \\textrm { active neigh.", "}| j \\textrm { is of degree } k \\textrm { and } k_2 \\textrm { type~2 neigh.}", "] \\Big ) \\nonumber \\\\&= {\\textstyle \\sum ^k_{l = m}} \\left( {\\textstyle \\sum ^{k}_{k_2=0}} \\hat{p}(k_2 | k) \\cdot P[X_{k_2} + Y_{k-k_2} = l ]\\right) \\nonumber \\\\&= {\\textstyle \\sum ^{k}_{k_2=0}} \\hat{p}(k_2 | k) \\cdot P[X_{k_2} + Y_{k-k_2} \\ge m ] $ where Equality (a) follows because the events $\\lbrace j$ registers$\\rbrace $ and $\\lbrace j$ has $m$ or more active neighbours given that $j$ has threshold $m\\rbrace $ are equivalent, and the events $\\lbrace j $ has $l$ active neighbours$\\rbrace $ and $\\lbrace j$ has threshold $m\\rbrace $ are independent of each other.", "In Equality (REF ), $X_{k_2}$ and $Y_{k-k_2}$ are random variables that denote the number of type 2 and type 1 active neighbours of a degree $k$ node, given that there are $k_2$ type 2 and $k-k_2$ type 1 neighbours.", "With some effort, it can be shown that random variables $X_{k_2}$ and $Y_{k-k_2}$ are independent and have a binomial distribution with parameters $(k_2,\\alpha _2 u)$ and $(k-k_2,\\alpha _1 u)$ , respectively.", "Here, $\\alpha _1$ and $\\alpha _2$ are the probability of a type 1 and type 2 node becoming active, respectively.", "The independence occurs because the probability of incentivizing a node is independent of the degree of its neighbouring nodes.", "The binomial distribution arises here because the network is locally tree-like.", "From Equation (REF ), it is evident that the probability of a node registering is independent of its typeA node registers either if it is zealous (this event is independent of its type), or through recommendations which depend on its neighbours and type of neighbours but not on its type..", "Therefore, the probability that by following an arbitrary link we can reach a node that registers due to recommendations is given by $&P[\\textrm {node registers due to recommendation}] = \\sum \\nolimits _{k \\ge 1} p_{ex}(k) \\cdot \\nonumber \\\\&\\hspace{14.22636pt} P[\\textrm {node reg.", "due to recommendations} | \\textrm {excess degree } k] \\nonumber $ $&= \\sum _{k \\ge 1} p_{ex}(k) \\hspace{-4.2679pt} \\sum _{m \\ge 1} \\hspace{-2.84526pt} P[\\textrm {node registers, node has threshold } m | \\nonumber \\\\& \\hspace{128.0374pt} \\textrm {node has excess degree } k] \\nonumber \\\\&\\overset{(a)}{=} \\sum _{k \\ge 1} p_{ex}(k) \\hspace{-4.2679pt} \\sum _{m \\ge 1} \\hspace{-2.84526pt} p_{th}(m|k+1) \\cdot P[\\textrm {node registers} | \\nonumber \\\\& \\hspace{34.1433pt} \\textrm {node has excess degree } k,\\textrm { node has threshold } m] \\nonumber \\\\& =\\sum _{k \\ge 1} p_{ex}(k) \\sum _{m \\ge 1} p_{th}(m|k+1) \\cdot \\nonumber \\\\& \\hspace{8.53581pt} \\sum ^{k}_{k_2=0} {k \\atopwithdelims ()k_2} \\cdot q^{k_2} \\cdot (1-q)^{k - k_2} \\cdot P[X_{k_2} + Y_{k-k_2} \\ge m ] $ On the RHS of Equation (REF ), we use the excess degree distribution because we discount the link that we followed to arrive at the node.", "In Equality (a), we use $p_{th}(m|k+1)$ because if we include the link on which we arrived, a node of excess degree $k$ will have $k+1$ links.", "Arguing along the lines of [17], we can conclude that $u$ has to satisfy the following self consistency equation $& u = P[\\textrm {node is zealous}] \\nonumber \\\\& \\hspace{56.9055pt} + P[\\textrm {node reg.", "due to recommendations}] \\nonumber \\\\&= \\sum \\nolimits _{k \\ge 0} p_{ex}(k) p_{th}(0|k+1) \\nonumber \\\\& \\hspace{56.9055pt} + P[\\textrm {node reg.", "due to recommendations}] \\nonumber \\\\&= f(q,u) $ where $f(q,u)$ is as given at the top of this page.", "For any $q$ , Equation (REF ) is a fixed point equation in $u$ .", "However, due to the complex nature of the function $f(q,u)$ , the existence of a $u$ that satisfies Equation (REF ) is not obvious.", "Using results from reliability theory, in Proposition REF , we prove the existence and uniqueness of the fixed point of Equation (REF ).", "Before that, the following proposition establishes the nature of function $f(q,u)$ (w.r.t $u$ ).", "Proposition 3.1 If the fraction of zealous nodes lies in the open interval $(0,1)$ , then $ \\forall \\ q \\in [0,1]$ , $f(q,u)$ is a continuously differentiable, convex, monotonically increasing function of $u$ .", "In Appendix REF .", "Proposition REF gives us some intuition about the existence and uniqueness of the fixed point.", "However, we establish the same rigorously in the following proposition.", "Proposition 3.2 For every $q \\in [0,1]$ , If no node in the network is zealous, then $u=0$ is the only solution of Equation (REF ).", "If every node in the network is zealous, then $u=1$ is the only solution of Equation (REF ).", "If the fraction of zealous nodes lies in the interval $(0,1)$ , then Equation (REF ) has a unique fixed point in $(0,1)$ .", "In Appendix REF .", "For ease of presentation, in the remainder of the paper, we assume that the fraction of zealous nodes lies in the open interval $(0,1)$ .", "For any $q \\in [0,1]$ , let $u_q$ denote the fixed point of Equation (REF ).", "Since the fixed point is unique, we can obtain $u_q$ by iteratively solving Equation (REF ).", "Our next proposition qualitatively describes the variation of the fixed point $u_q$ as function of $q$ .", "Proposition 3.3 If $\\alpha _2 > \\alpha _1$ , then $u_q$ is a strictly increasing continuous function of $q$ , i.e., $\\frac{\\partial u_q}{\\partial q} > 0$ .", "In Appendix .", "Now, given a $q \\in [0,1]$ , let $s_k(q)$ be the expected fraction of degree $k$ nodes that have registered at the end of the campaign.", "The expression for $s_k(q)$ can be obtained as follows $& s_k(q) = P[\\textrm {node registers}| \\textrm {degree } k ] \\nonumber \\\\&= P[\\textrm {node is zealous} | \\textrm {degree } k] \\nonumber \\\\& \\hspace{28.45274pt} + P[\\textrm {node reg.", "due to recommendations} | \\textrm {degree } k] \\nonumber \\\\&= \\sum \\nolimits _{m \\ge 1} \\hspace{-2.84526pt} P[\\textrm {node reg, node has threshold } m | \\nonumber \\\\& \\hspace{56.9055pt} \\textrm {node has degree } k] + p_{th}(0|k) \\nonumber \\\\&= p_{th}(0|k) + \\sum \\nolimits _{m \\ge 1} \\hspace{-2.84526pt} p_{th}(m|k) \\cdot \\nonumber \\\\& \\hspace{17.07164pt} P[\\textrm {node reg} | \\textrm {node has degree } k,\\textrm { node has threshold } m] \\nonumber \\\\&= g_k(q,u) |_{u=u_q} $ where $g_k(q, u)$ is as given at the top of this page and $u_q$ is the fixed point of Equation(REF ).", "Given a $q \\in [0,1]$ , let $s(q)$ be the expected fraction of nodes that have registered at the end of the campaign.", "$s(q)$ is also termed as the size of the epidemic.", "Then, using arguments similar to the ones used to derive Equation (REF ), it can be shown that $s(q) &=\\sum \\nolimits _{k \\ge 1} p(k) \\cdot g_k(q,u) \\Big |_{u=u_q} \\hspace{-5.69054pt} = \\sum \\nolimits _{k \\ge 1} p(k) \\cdot s_k(q) $ Once again using results from reliability theory, in Proposition REF , we establish the increasing and non-decreasing nature of functions $g(q,u)$ and $g_k(q,u)$ (with respect to $u$ and $q$ ).", "Proposition 3.4 If $\\alpha _2 > \\alpha _1$ , then $\\frac{\\partial g(q,u)}{\\partial q} > 0$ and $\\frac{\\partial g(q,u)}{\\partial u} > 0$ $\\frac{\\partial g_k(q,u)}{\\partial q} \\ge 0$ and $\\frac{\\partial g_k(q,u)}{\\partial u} \\ge 0$ .", "Similar to that of Propositions REF and REF ." ], [ "Cost Minimization Under Cascade Size Constraint", "Providing incentives is a costly affair.", "The campaigner may either be interested in minimizing the cost while guaranteeing that a given proportion of population registers, or in maximizing the registrations for a given cost budget.", "In this section, we look at the former problem." ], [ "Cost of incentivization", "Incentives provided by the campaigner is usually a function of its degree because the number of potential recommendations depend on the degree.", "Let $c_k$ be the cost of incentivizing a degree $k$ node.", "Incentivized nodes obtain incentives only if they registers.", "Therefore, the expected cost per incentivized degree $k$ node is given by $c_k \\cdot s_k(q)$ , where $s_k(q)$ (see Equation REF ) is the probability that an incentivized degree $k$ node registers.", "Let $\\phi (k)$ denote the probability of incentivizing a degree $k$ node.", "Then, the average cost per node for incentive-policy $(\\phi =\\lbrace \\phi (k), k \\ge 1\\rbrace )$ is given by $\\sum _{k \\ge 1} p(k) \\phi (k) \\times (\\textrm {cost of node of deg. }", "k) = \\sum _{k \\ge 1} p(k) \\cdot c_k \\cdot \\phi (k) \\cdot s_k(q)$ , where $q = \\frac{1}{\\overline{d}} \\sum _{k\\ge 1} k \\cdot p(k) \\cdot \\phi (k)$ ." ], [ "Problem formulation", "Minimizing the cost while providing guarantees on the number of expected registrations is appropriate for campaigns where the campaigners are mandated to achieve a given target.", "This problem can be mathematically formulated as follows $& \\hspace{28.45274pt} \\underset{0 \\le \\phi \\le 1}{\\text{min}} \\ \\ \\ \\ \\sum \\nolimits _{k \\ge 1} c_k \\cdot p(k) \\cdot \\phi (k) \\cdot s_k(q) \\\\& \\hspace{0.0pt} \\textrm {Subject to:} \\quad s(q) \\ge \\gamma \\textrm { and } q = \\frac{1}{\\overline{d}} \\sum \\nolimits _{k\\ge 1} k \\cdot p(k) \\cdot \\phi (k) \\nonumber $ where $\\gamma \\in [0,\\gamma _{max}]$ is the minimum expected fraction of registered individuals that must be achieved and $\\overline{d}$ is the mean degree of the network.", "Here, $\\gamma _{max}$ is the expected fraction of registered individuals obtained by incentivizing everyone.", "The expression for $s_k(q)$ (Equation (REF )) and $s(q)$ (Equation (REF )) involve $u_q$ , which is the solution to the fixed point Equation (REF ).", "Thus, it is not possible to apply traditional analytical techniques such as the Karush-Kuhn-Tucker (KKT) conditions to solve the above problem.", "Furthermore, the problem may be non convex, and applying numerical techniques such as genetic algorithms, or Markov Chain Monte Carlo methods may not yield a globally optimal solution." ], [ "Solution approach", "In this section, we present transformations that allow us to efficiently compute the global optima of the cost minimization problem presented in the previous section.", "To help us with this, we state and prove the following proposition.", "The intuition behind this proposition is that, as $\\alpha _2 > \\alpha _1$ , increasing $q$ increases the proportion of type 2 nodes, which results in a higher $s(q)$ .", "Proposition 4.1 If $\\alpha _2 > \\alpha _1$ , then function $s:[0,1] \\rightarrow [\\gamma _{min},\\gamma _{max}]$ is a monotonically increasing bijection, where $\\gamma _{max}=s(0)$ and $\\gamma _{max}=s(1)$ are the expected fraction of registered individuals obtained by incentivizing nobody and everybody, respectively.", "We have $\\frac{\\partial s(q)}{\\partial q} &= \\frac{\\partial \\left(g(q,u) |_{u=u_q} \\right)}{\\partial q} \\nonumber \\\\& \\overset{(a)}{=} \\frac{\\partial g(q,u)}{\\partial q} \\Bigg |_{u=u_q} + \\frac{\\partial g(q,u)}{\\partial u} \\Bigg |_{u=u_q} \\cdot \\frac{\\partial u_q}{\\partial q} \\nonumber \\\\& = \\underset{\\textrm {Proposition \\ref {prop:g_increasing}}}{(>0)} + \\underset{\\textrm {Proposition \\ref {prop:g_increasing}}}{(> 0)} \\cdot \\underset{\\textrm {Proposition \\ref {prop:fp_increasing}}}{(>0)} \\nonumber \\\\& \\hspace{-42.67912pt} \\Rightarrow \\frac{\\partial s(q)}{\\partial q} > 0 $ where Equality (a) follows from an application of the chain rule.", "From Inequality (REF ), we can see that $s(q)$ is a monotonically increasing function defined on the closed interval $[0,1]$ .", "Hence, $s:[0,1]\\rightarrow [\\gamma _{min},\\gamma _{max}]$ is a bijection.", "Proposition REF establishes that $s(q)$ is a monotonically increasing function of $q$ and is a bijection.", "Therefore, we can replace the constraint $s(q) \\ge \\gamma $ with the constraint $q \\ge q_{\\gamma }$ , where $q_{\\gamma }$ is such that $s(q_\\gamma ) = \\gamma $ .", "Since $s(q)$ is a monotonically increasing function of $q$ , we can compute $q_{\\gamma }$ by performing a line search over the closed interval $[0,1]$ .", "The following algorithm does this.", "Algorithm to compute $q_{\\gamma }$ [1] Set $q_1 :=0$ and $q_2 :=1$ $|s(q_1) - s(q_2)| > \\epsilon $ $q_{mp} := (q_1 + q_2)/2$ Solve equation $u = f(q_{mp},u)$ to obtain $u_{q_{mp}}$ .", "$s(q_{mp}) < \\gamma $ $q_1 := q_{mp}$ $q_2 := q_{mp}$ return $q_{\\gamma } := (q_1+q_2)/2$ Proposition REF does not tell us how the expected fraction of registered degree $k$ nodes ($s_k(q)$ ) responds to changes in $q$ .", "While, the function $s_k(q)$ may not be a bijection for all values of $k$ , we show that it is a non-decreasing function of $q$ , for all values of $k$ .", "Proposition 4.2 If $\\alpha _2 > \\alpha _1$ , then for all $k \\ge 1$ , $\\frac{\\partial s_k(q)}{\\partial q} \\ge 0$ By first writing the total derivative in terms of the partial derivatives (see Proposition REF ), and then by using the non-negativity of these partial derivatives (from Propositions REF and REF ).", "Now, if $\\gamma \\le \\gamma _{min}=s(0)$ , then the cascade constraint is met without incentivizing anyone.", "In such cases, the optimum incentive policy and the optimum cost is given by $\\lbrace \\phi (k)=0, \\forall k \\ge 1\\rbrace $ and 0, respectively.", "On the other hand, if $\\gamma = \\gamma _{max}=s(1)$ , then the cascade constraint is met only if everyone is incentivized, i.e., $\\lbrace \\phi (k)=1, \\forall k \\ge 1\\rbrace $ , and the optimum cost is given by $\\sum _{k \\ge 1} c_k \\cdot p(k) \\cdot s_k(1)$ .", "For the cases when $\\gamma \\in (\\gamma _{min}, \\gamma _{max})$ , due to Propositions REF , the cost minimization problem can be re-written as follows $& \\hspace{0.0pt} P_1: \\underset{0 \\le \\phi \\le 1}{\\text{min}} \\ \\ \\ \\ \\sum \\nolimits _{k \\ge 1} c_k \\cdot p(k) \\cdot \\phi (k) \\cdot s_k(q) \\\\& \\hspace{0.0pt} \\textrm {Subject to:} \\quad q \\ge q_{\\gamma } \\textrm { and } q = \\frac{1}{\\overline{d}} \\cdot \\sum \\nolimits _{k\\ge 1} k \\cdot p(k) \\cdot \\phi (k)\\nonumber $ where $q_{\\gamma }$ is chosen such that $s(q_\\gamma ) = \\gamma $ .", "Since $\\gamma \\in (\\gamma _{min}, \\gamma _{max})$ , we have $q_{\\gamma } \\in (0,1)$ (see Proposition REF ).", "The next proposition establishes the existence of an optimal policy, of problem $P_1$ , that satisfies all its the constraints, with equality.", "Proposition 4.3 If $\\alpha _2 > \\alpha _1$ , then problem $P_1$ has an optimal solution $\\phi ^{opt} = \\lbrace \\phi ^{opt}_k , k \\ge 1\\rbrace $ such that $\\sum _{k\\ge 1} k \\cdot p(k) \\cdot \\phi ^{opt}(k) = \\overline{d} \\cdot q_{\\gamma }$ In Appendix REF .", "From Proposition REF , we can see that problem $P_1$ has an optimal solution that satisfies all its constraints with equality.", "Therefore, to obtain a solution of problem $P_1$ , we restrict our search to the set of policies that satisfy the constraints of problem $P_1$ with equality.", "Subsequently, a change of variable gives us the following linear program $& \\hspace{28.45274pt} P_2: \\underset{\\lbrace \\nu _k , k\\ge 1\\rbrace }{\\text{min}} \\ \\ \\ \\ \\overline{d} \\cdot q_{\\gamma } \\cdot \\sum \\nolimits _{k \\ge 1} \\nu _k \\cdot \\mu _k(q_{\\gamma }) \\\\& \\hspace{0.0pt} \\textrm {Subject to:} \\quad \\sum \\nolimits _{k\\ge 1} \\nu _k = 1 \\textrm { and } 0 \\le \\nu _k \\le \\frac{k \\cdot p(k)}{\\overline{d} \\cdot q_{\\gamma }} \\, \\, \\forall k \\ge 1\\nonumber $ where $\\mu _k(q_{\\gamma }) = \\frac{s_k(q_\\gamma ) \\cdot c_k}{k}$ .", "The following proposition establishes a crucial relation between the optimal solutions of problems $P_1$ and $P_2$ .", "Proposition 4.4 Let $\\lbrace \\nu ^{opt}_k , k \\ge 1\\rbrace $ be an optimal solution of problem $P_2$ .", "If $\\alpha _2 > \\alpha _1$ , then we can obtain an optimal solution of problem $P_1$ as follows $\\phi ^{opt}(k) = {\\left\\lbrace \\begin{array}{ll}0 & \\textrm {if } p(k) = 0 \\\\\\frac{\\nu ^{opt}_k \\cdot \\overline{d} \\cdot q_{\\gamma }}{k \\cdot p(k)} & \\textrm {otherwise}\\end{array}\\right.}", "$ In Appendix REF .", "Therefore, to obtain an optimal solution of problem $P_1$ , we just need to solve $P_2$ and apply Proposition REF .", "Problem $P_2$ is a linear program and can be solved by any well-known LP-solver in polynomial time ($O((k_{max})^{3.5})$ time, where $k_{max}$ is the maximum node degree in the network)[23].", "However, we exploit the rich structure of this problem and solve it in linearithmic time $O(k_{max} \\cdot log(k_{\\max }))$ Let $\\mathcal {S} = \\lbrace 1, 2, \\cdots , k_{max}\\rbrace $ .", "Here, $k_{max}$ is the maximum degree of the network.", "Consider a permutation $\\sigma :\\mathcal {S} \\rightarrow \\mathcal {S}$ such that if $k_1 < k_2$ , then $\\mu _{\\sigma (k_1)}(q_{\\gamma }) \\le \\mu _{\\sigma (k_2)}(q_{\\gamma })$ , i.e., $\\mu _{\\sigma (1)}(q_{\\gamma }) \\le \\mu _{\\sigma (2)}(q_{\\gamma }) \\le \\cdots \\le \\mu _{\\sigma (k_{max})}(q_{\\gamma })$ .", "The permutation $\\sigma $ can be obtained in $O(k_{max} \\cdot log(k_{\\max }))$ time by sorting the set $\\lbrace \\mu _k(q_{\\gamma }), k \\ge 1\\rbrace $ in ascending order.", "Using this order, we can obtain an optimal solution $\\nu ^a$ of problem $P_2$ in $\\Theta (k_{max})$ time by following the steps in Algorithm REF .", "[h] Algorithm to compute optimal solution of $P_2$ [1] $res := 1$ and $\\nu ^a := [0, 0, \\cdots , 0]$ — a $1 \\times k_{max}$ vector $j \\in [1, 2, \\cdots , k_{max}]$ $res > 0$ $\\nu ^a_{\\sigma (j)} = \\min \\left\\lbrace res, \\frac{\\sigma (j) \\cdot p(\\sigma (j))}{\\overline{d} \\cdot q_{\\gamma }} \\right\\rbrace $ $res := res - \\nu ^a_{\\sigma (j)}$ return $\\nu ^a$ In Algorithm REF , $\\nu ^a_k$ denotes the $k^{th}$ component of vector $\\nu ^a$ .", "We illustrate the principle behind Algorithm REF using a jar filling analogy.", "Variable $\\nu ^a_{\\sigma (j)}$ can be interpreted as a jar with capacity $\\frac{\\sigma (j) \\cdot p(\\sigma (j))}{\\overline{d} \\cdot q_{\\gamma }}$ .", "The for loop in the algorithm is equivalent to filling these jars with water from another jar of unit capacity, in an ascending order, as given by the permutation $\\sigma $ .", "From steps 1, and $4-5$ of Algorithm REF , it is easy to see that $\\nu ^a$ is a feasible solution of problem $P_2$ .", "The optimality of $\\nu ^a$ with respect to problem $P_2$ is established by the following proposition.", "Proposition 4.5 The vector $\\nu ^a$ returned by Algorithm REF is an optimal solution of problem $P_2$ .", "In Appendix ." ], [ "Cascade Size Maximization Under Budget Constraint", "In this section, we consider the challenge faced by a resource-constrained campaigner.", "The campaigner desires to maximize the campaign size, but is constrained by limited resources.", "Let $\\overline{c}$ denote the limit on the expected incentivization cost.", "Then, we can formulate the cost-constrained cascade maximization problem as follows $& \\hspace{42.67912pt} P_3: \\underset{0 \\le \\phi \\le 1}{\\text{max}}\\ \\ \\ \\ s(q) \\\\& \\hspace{0.0pt} \\textrm {Subject to:} \\ \\ \\sum \\nolimits _{k \\ge 1} c_k \\cdot p(k) \\phi (k) s_k(q) \\le \\overline{c} \\nonumber \\\\& \\hspace{14.22636pt} \\textrm { and } q = \\frac{1}{\\overline{d}} \\sum \\nolimits _{k\\ge 1} k \\cdot p(k) \\cdot \\phi (k)$" ], [ "Solution approach", "Let $s^{opt}$ be the optimum value of problem $P_3$ .", "From Proposition REF , we know that there exists a unique $q^{opt} \\in (0,1)$ such that $s(q^{opt}) = s^{opt}$ .", "From Proposition REF , we know that $\\frac{\\partial s(q)}{\\partial q}> 0$ , i.e., $s(q)$ is a monotonically increasing bijection.", "Therefore, it can be argued that $q^{opt}$ can be obtained by solving the following optimization problem $& \\hspace{42.67912pt} P_4: \\underset{0 \\le \\phi \\le 1}{\\text{max}}\\ \\ \\ \\ q \\\\& \\hspace{0.0pt} \\textrm {Subject to:} \\sum \\nolimits _{k \\ge 1} c_k \\cdot p(k) \\phi (k) s_k(q) \\le \\overline{c} \\nonumber \\\\& \\hspace{14.22636pt} \\textrm { and } q = \\frac{1}{\\overline{d}} \\sum \\nolimits _{k\\ge 1} k \\cdot p(k) \\cdot \\phi (k)$ Let us define $\\mu _k(q) = \\frac{s_k(q) \\cdot c_k}{k}$ and $\\nu _k(q) = \\frac{k \\cdot p(k) \\cdot \\phi (k)}{\\overline{d} \\cdot q}$ .", "Then, problem $P_4$ can be re-written as follows $& \\hspace{85.35826pt} P_5: \\underset{\\nu , q \\in (0,1)}{\\text{max}} \\ \\ \\ \\ q \\\\& \\hspace{42.67912pt} \\textrm {Subject to:} \\quad \\overline{d} \\cdot q \\cdot \\sum \\nolimits _{k \\ge 1} \\nu _k(q) \\cdot \\mu _k(q) \\le \\overline{c} \\nonumber \\\\& \\hspace{0.0pt} \\sum \\nolimits _{k\\ge 1} \\nu _k(q) = 1 \\quad \\textrm {and} \\quad 0 \\le \\nu _k(q) \\le \\frac{k \\cdot p(k)}{\\overline{d} \\cdot q} \\quad \\forall \\ k \\ge 1\\nonumber $ Now, given a $q \\in (0,1)$ , consider the following criteria $\\overline{d} \\cdot q \\cdot \\sum _{k \\ge 1} \\nu _k(q) \\cdot \\mu _k(q) \\le \\overline{c} $ $ \\sum _{k\\ge 1} \\nu _k(q) = 1$ $0 \\le \\nu _k(q) \\le \\frac{k \\cdot p(k)}{\\overline{d} \\cdot q} \\quad \\forall k \\ge 1$ If we can find a vector $\\nu $ that satisfies the above criteria, then the 2-tuple $(\\nu ,q)$ is a feasible solution of problem $P_5$ .", "Let $\\nu ^a(q)$ be the vector output by Algorithm REF when $q_{\\gamma }$ is set to $q$ .", "From Proposition REF , we know that this vector minimizes the LHS of criterion (1) subject to criteria (2) and (3).", "Therefore, if we have $\\overline{d} \\cdot q \\cdot \\sum _{k \\ge 1} \\nu ^a_k(q) \\cdot \\mu _k(q) > \\overline{c} $ , then for the given value of $q$ , no vector $\\nu $ can satisfy criterion (1).", "Now, let $\\nu ^a(q+ \\Delta q)$ be the vector output by Algorithm REF when $q_{\\gamma }$ is set to $q+\\Delta q$ .", "Then, for any $\\Delta q > 0$ , if $\\overline{d} \\cdot q \\cdot \\sum _{k \\ge 1} \\nu ^a_k(q) \\cdot \\mu _k(q) > \\overline{c}$ , then we have $\\overline{c} & < \\overline{d} \\cdot q \\sum _{k \\ge 1} \\nu ^a_k(q) \\cdot \\mu _k(q) \\overset{(a)}{\\le } \\overline{d} \\cdot q \\cdot \\sum _{k \\ge 1} \\nu ^a_k(q+\\Delta q) \\cdot \\mu _k(q) \\nonumber \\\\& \\overset{(b)}{\\le } \\overline{d} \\cdot (q + \\Delta q) \\cdot \\sum \\nolimits _{k \\ge 1} \\nu ^a_k(q+\\Delta q) \\cdot \\mu _k(q+\\Delta q) $ where Inequality $(a)$ follows because the vector $\\nu ^a(q)$ is an optimal solution and $\\nu ^a(q+\\Delta q)$ is just a feasible point at $q$ , and Inequality $(b)$ follows because $\\frac{\\partial \\mu _k(q)}{\\partial q} = \\frac{c_k}{k} \\cdot \\frac{\\partial s_k(q)}{\\partial q} \\ge 0$ (see Proposition REF ).", "From Inequality REF we can conclude that if there does not exist a vector $\\nu $ that satisfies criteria (1)-(3) for some $q \\in (0,1)$ , then no vector satisfies these criteria for any $q^{^{\\prime }} \\in (q,1)$ .", "Therefore, the optimal value $q^{opt}$ is given by the largest value of $q$ in $ [0,1]$ such that vector $\\nu ^a(q)$ satisfies criteria (1), and can be computed using the following algorithm Algorithm to compute optimum of $P_3$ [1] Set $q_1 :=0$ and $q_2 :=1$ $|s(q_1) - s(q_2)| > \\epsilon $ $q_{mp} := (q_1 + q_2)/2$ Set $q_{\\gamma } = q_{mp}$ in Algorithm REF and obtain $\\nu ^a(q_{mp})$ $\\overline{d} \\cdot q_{mp} \\cdot \\sum _{k \\ge 1} \\nu ^a_k(q_{mp}) \\cdot \\mu _k(q_{mp}) > \\overline{c}$ $q_2 := q_{mp}$ $q_1 := q_{mp}$ Return $q^{opt} := (q_1+q_2)/2$ Algorithm REF does a line search to find the largest values of $q \\in [0,1]$ such that the tuple $(\\nu ^a(q),q)$ meets criteria (1)-(3)." ], [ "Simulation and Numerical Evaluations", "We formulated and solved the optimization problems assuming an uncorrelated, locally tree-like and connected network.", "However, real world social networks may contain triads, loops, multiple connected components, and need not be uncorrelated.", "Despite such differences, we show that, in most cases, our analytical results closely match the simulations on real world social networks.", "We evaluate our analytical results on two very different networks: Gnutella [24] — a p2p file sharing network, and Hamsterster [25] — a social network for people with pet hamsters.", "Table  REF presents a few statistical measures of these networks.", "The degree distributions of these networks are displayed in Figures  REF and  REF .", "All simulations are averaged over 10000 runs.", "To enhance legibility, error bars are suppressed in all the plots.", "Figure: Degree distribution of network B (Hamsterster network); the thick black line denotes the best fit of the power law to the degree distribution.Table: Simple Parameters of the two real-world networks used for simulations.Figure: Analytical and simulated values of average cost vs. γ\\gamma for network A (Gnutella network).We first evaluate the correctness of the analytical calculation of the campaign size, i.e., $s(q)$ .", "The simulation plots were generated by considering a linear threshold model where nodes register if at least $50 \\%$ of their neighbours are active.", "We assume that $30 \\%$ of the nodes in the network were zealous.", "For a different values of $q$ , we studied three schemes: $\\phi ^{all}$ — incentivizing all nodes equally with probability $q$ , $\\phi ^{high}$ — incentivizing nodes starting from the highest degree till degree $k^{\\prime }$ such that $q=\\frac{1}{\\overline{d}} \\sum _{k \\ge k^{\\prime }} p(k) \\phi ^{high}(k)$ , and $\\phi ^{low}$ — incentivizing nodes starting from the lowest degree till degree $k^{\\prime }$ such that $q=\\frac{1}{\\overline{d}} \\sum _{1 \\le k \\le k^{\\prime }} p(k) \\phi ^{high}(k)$ .", "The analytical results were obtained by extracting the degree distribution from the networks.", "The number of triangles in the p2p Gnutella network is very small, which is also reflected in the clustering coefficient.", "Therefore, the Gnutella network is very similar to a locally tree-like network.", "As consequence of this, for the Gnutella network, the simulation and analytical results are in excellent agreement with each other (see Fig.", "REF ).", "On the other hand, one can observe a deviation of the simulation results from the analytical on Hamsterster for large values of $q$ (see Fig.", "REF ).", "This behaviour may in part be due to the presence of significant number of triangles and loops in the network.", "Figure: Analytical and simulated values of cascade size vs. c ¯\\overline{c} for network A (Gnutella network).Figure: Analytical and simulated values of cascade size vs. c ¯\\overline{c} for network B (Hamsterster network).Next, we consider a linear incentive structure, where an incentivized node obtains a unit reward per neighbour, i.e., $c_k = k, \\ \\forall \\ k \\ge 1$ .", "For the cost minimization problem, for a given $\\gamma $ , using the degree distributions, we analytically compute the solution $\\phi $ and the average cost.", "We then simulate the linear threshold process (nodes register if at least $50 \\%$ of their neighbours are active) on the respective real world networks using the analytical solution $\\phi $ , and compare the average cost obtained from analysis and simulations (see Fig.", "REF and Fig.", "REF ).", "Similarly, for the campaign size maximization problem we analytically compute $\\phi $ and size $s$ for a given cost budget $\\overline{c}$ .", "The analytical solution is then used in the simulation of cascade maximization problem (see Fig.", "REF and Fig.", "REF ).", "As observed in Fig.", "REF and  REF , the analytical average cost is in excellent agreement with the simulated one for different values of $\\alpha _1, \\alpha _2$ and $\\gamma $ , on both the networks.", "Similarly Fig.", "REF and  REF show an excellent match between simulation and analytical computation of campaign size in almost all the plots.", "For large values of $\\alpha _1$ and $\\alpha _2$ , the simulated campaign size in the Hamsterster network is larger than the analytically computed campaign size.", "This may in part be due to the large number of triangles in the network.", "This suggests that in social networks containing large number of triangles and loops, the analytically computed size represents a lower bound." ], [ "Conclusion", "In this paper, we have studied the problem of campaigning in social networks by offering incentives for referrals.", "We used ideas from percolation theory to compute the campaign size, which was then used to formulate two optimization problems.", "These problems were not amenable to traditional solutions since they involved a fixed point equation whose solution was analytically intractable.", "We used results from reliability theory to establish some key properties of the fixed point that enabled us to solve these problems with simple algorithms having linearithmic time complexity.", "Although we assumed an uncorrelated and locally tree-like network in the analysis, through extensive simulations on real world social networks, we showed that our analytical results are applicable in real world networks." ], [ "Some Results from Reliability Theory", "In this paper, we prove several propositions using the theory of stochastic order.. Let $X$ and $Y$ be two random variables.", "Then, $X$ is said to be smaller than $Y$ in the usual stochastic order (denoted by $X \\le _{st} Y $ ) [26] if and only if $ P[ X > x ] \\le P[ Y > x ] \\quad \\forall x \\in (-\\infty , \\infty )$ Next, we present two theorem from [26] without proof, and a lemma.", "We will use these theorems and lemma in several of our proofs.", "Theorem 8.1 $X \\le _{st} Y$ if and only if $\\mathbb {E}[\\psi (X)] \\le \\mathbb {E}[\\psi (Y)]$ holds for all non-decreasing functions $\\psi $ for which the expectation exists.", "Refer Chapter 1 of [26].", "Theorem 8.2 The usual stochastic order is closed under convolutions, i.e., If $X \\le _{st} Y$ and $X^{^{\\prime }} \\le _{st} Y^{^{\\prime }}$ , then $X^{^{\\prime }} + X \\le _{st} Y^{^{\\prime }} + Y$ .", "Refer Chapter 1 of [26].", "Lemma 8.1 Let $I(p_1)$ and $I(p_2)$ be two Bernoulli random variable with parameters $p_1$ and $p_2$ , then $I(p_1) \\le _{st} I(p_2)$ if and only if $p_1 \\le p_2$ .", "Let $Z(n, p_1)$ and $Z(n, p_2)$ be two binomial random variable with parameters $(n,p_1)$ and $(n,p_2)$ , then for any $n \\in \\mathbb {N}$ , $Z(n, p_1) \\le _{st} Z(n, p_2)$ if $p_1 \\le p_2$ .", "Part (i): By comparing the complementary cumulative distribution function of the two random variable, and by applying the definition of usual stochastic order.", "Part (ii): A binomial random variable with parameter $(n,p)$ is the sum of $n$ i.i.d.", "Bernoulli random variable with parameter $p$ .", "Now, the result follows from an application of Part (i), and due to the fact that stochastic order is closed under convolution (Theorem REF ).", "We also use some results from the theory of stochastic convexity in our proofs.", "For the sake of completeness, we reproduce some definitions and theorems from [26].", "Let $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace $ be a parametrized collection of random variables.", "We say $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace $ is stochastically increasing (SI) if $\\mathbb {E}[\\psi (X(\\theta ))]$ is non-decreasing in $\\theta $ for all non-decreasing functions $\\psi $ .", "$\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace $ is stochastically increasing and convex (SICX) if $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace \\in SI$ and $\\mathbb {E}[\\psi (X(\\theta ))]$ is non-decreasing and convex in $\\theta $ for all non-decreasing convex functions $\\psi $ .", "$\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace $ is stochastically increasing and linear (SIL) if $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace \\in SICX$ and $\\mathbb {E}[\\psi (X(\\theta ))]$ is non-decreasing and concave in $\\theta $ for all non-decreasing concave functions $\\psi $ .", "Note that, by definition, we have $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace \\in SIL \\Rightarrow \\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace \\in SICX$ .", "Lemma 8.2 Let $X(n,p)$ be a binomial random variable with parameters $(n,p)$ .", "Then, $\\lbrace X(n,p), p \\in (0,1) \\rbrace \\in SICX$ .", "By combining example Example 8.B.3 and Theorem 8.B.9 of [26], it can shown that $\\lbrace X(n,p), p \\in (0,1)\\rbrace \\in SIL$ .", "Then, due to the fact that $SIL \\Rightarrow SICX$ , the lemma follows.", "Theorem 8.3 Suppose $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace $ and $\\lbrace Y(\\theta ), \\theta \\in \\Theta \\rbrace $ are two collection of random variables such that $X(\\theta )$ and $Y(\\theta )$ are independent for each $\\theta $ .", "If $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace \\in SICX$ and $\\lbrace Y(\\theta ), \\theta \\in \\Theta \\rbrace \\in SICX$ , then $\\lbrace X(\\theta )+Y(\\theta ), \\theta \\in \\Theta \\rbrace \\in SICX$ .", "Refer the proof of Theorem 8.A.15 in [26].", "Theorem 8.4 Suppose that for each $\\theta \\in \\Theta $ , the support of $X(\\theta )$ is in $\\mathbb {N}$ .", "Then, $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace \\in SICX$ if and only if $\\lbrace X(\\theta ), \\theta \\in \\Theta \\rbrace \\in SI$ and $P[X(\\mathbf {\\theta }) > m]$ is non-decreasing and convex in $\\theta $ for all $m \\in \\mathbb {N}$ .", "Refer the proof of Theorem 8.A.10 in [26]." ], [ "Existence and Uniqueness of the Fixed Point", "In this section, we show that for any $q \\in [0,1]$ , the fixed point Equation REF has a unique solution.", "The proof in this section is split into two parts.", "In the first part, we show that function $f(q,u)$ is monotonically increasing and convex in $u$ .", "In the section part, we use the result of the first part to establish the existence and uniqueness of the fixed point." ], [ "Proof of Proposition ", "The differentiability of $f(q,u)$ follows from its definition.", "In this section, we show that $f(q,u)$ is monotonically increasing and convex in $u \\in (0,1)$ .", "To show this, we use the theory of stochastic convexity.", "Let $\\lbrace X(k_2, \\alpha _2 u), u \\in (0,1) \\rbrace $ and $\\lbrace Y(k-k_2,\\alpha _1u), u \\in (0,1) \\rbrace $ be two collection of independent binomial random variables with parameters $(k_2,\\alpha _2u)$ and $(k-k_2,\\alpha _1u)$ , respectively.", "Then, from Lemma REF , we have $\\lbrace X(k_2, \\alpha _2 u), u \\in (0,1) \\rbrace \\in SICX$ and $\\lbrace Y(k-k_2,\\alpha _1u), u \\in (0,1) \\rbrace \\in SICX$ .", "Since the random variables are independent, an application of Theorem REF gives us $\\lbrace X(k_2, \\alpha _2 u) + Y(k-k_2,\\alpha _1u), u \\in (0,1) \\rbrace \\in SICX$ .", "Now, by applying Theorem REF we can conclude that $\\lbrace X(k_2, \\alpha _2 u) + Y(k-k_2,\\alpha _1u), u \\in (0,1) \\rbrace \\in SI$ .", "$P[X(k_2, \\alpha _2 u) + Y(k-k_2,\\alpha _1u) > m]$ is a non-decreasing convex function of $u \\in (0,1)$ for all $m \\in \\mathbb {N}$ .", "Let $h(u,k,k_2,m) = P[X(k_2, \\alpha _2 u) + Y(k-k_2,\\alpha _1u) > m]$ .", "We note that function $h(u,k,k_2,m)$ is non-decreasing and convex in $u$ .", "Since the network is connected and the fraction of zealous nodes lies in the open interval $(0,1)$ , there exists positive integers $k_0,m_0$ such that $m_{0} \\in (0, k_0]$ and $p_{ex}(k_0) p_{th}(m_0 | k_0+1) > 0$ .", "Therefore, we have $& \\frac{\\partial f(q, u)}{\\partial u} = \\sum _{k \\ge 1} p_{ex}(k) \\sum _{m \\ge 1} p_{th}(m|k+1) \\cdot \\sum ^{k}_{k_2=0} \\hat{p}(k_2|k) \\cdot \\\\& \\hspace{28.45274pt} \\frac{\\partial h(u,k,k_2,m)}{\\partial u} \\cdot (1 - \\mathbb {I}_{\\lbrace k=k_0,m=m_0,k_2=k\\rbrace }) \\\\& \\hspace{14.22636pt} + p_{ex}(k_0) p_{th}(m_0|k_0+1) \\frac{\\partial P[X(k_0, u\\alpha _2) \\ge m_0]}{\\partial u} \\overset{(a)}{>} 0$ where Inequality (a) follows because $h(u,k,k_2,m)$ and $P[X(k_0, u\\alpha _2) \\ge m_0], m_0 \\in (0, k_0]$ are non-decreasing and monotonically increasing functions of $u$ , respectively.", "Similarly, we have $&\\frac{ \\partial ^2 f(q, u)}{\\partial u^2} = \\sum _{k \\ge 1} p_{ex}(k) \\sum _{m \\ge 0} p_{th}(m|k+1) \\sum ^{k}_{k_2=0} \\hat{p}(k_2|k) \\cdot \\\\&\\hspace{85.35826pt} \\frac{\\partial ^2 h(u,k,k_2,m)}{\\partial u^2} \\overset{(b)}{\\ge } 0$ where Inequality (a) follows because $h(u,k,k_2,m)$ is convex in $u$ ." ], [ "Proof of Proposition ", "We recollect that $u$ denotes the probability of finding a registered node by following an arbitrary link of the network.", "Nodes with zero threshold do not need any recommendations, and are registered from the start of the campaign.", "Thus, if every node has threshold value zero, then the entire network is registered and we have $u=1$ .", "On the other hand, nodes with non-zero threshold have to be recommended by active nodes to register.", "Therefore, if all nodes in the network have non-zero threshold, then there are no node from which the campaign can start.", "Hence, in such scenarios, we have $u=0$ .", "Now, let us consider the case when the fraction of zealous nodes lies in the interval $(0,1)$ .", "Let us define $f^0(u) = f(q,u) - u$ .", "It is easy to see that $u_0$ is a fixed point of Equation REF if and only if $f^{0}(u_0)=0$ .", "By differentiating $f^0(u)$ with respect to $u$ twice, we get $\\frac{\\partial ^2 f^0(u)}{\\partial u^2} = \\frac{\\partial ^2 f(q,u)}{\\partial u^2} \\ge 0$ , i.e., $f^0$ is a convex function of $u$ .", "We also have $f^0(0) &= f(q,0) = \\sum \\nolimits _{k \\ge 0} p_{ex}(k) \\cdot p_{th}(0|k+1) \\overset{(a)}{>} 0 \\\\f^0(1) &= f(q,1) - 1 \\overset{(b)}{<} 0$ where Inequality (a) follows because the network is connected and the fraction of zealous nodes lies in the open interval $(0,1)$ , and Inequality (b) follows because the $f(q,1)$ is a convex combination of non-negative terms less or equal to 1 (some terms are strictly less that 1).", "Since $f^0(0) > 0$ and $f^0(1) < 0$ , due to the continuity of function $f^{0}$ , there exist a point $u_0 \\in (0,1)$ such that $f^{0}(u_0)=0$ i.e., $f(q,u_0) = u_0$ .", "We prove the uniqueness of the fixed point by contradiction.", "Now, if $u_0, u_1 \\in (0,1)$ are two fixed points of Equation (REF ), then we should have $f^{0}(u_0) = f^{0}(u_1) = 0$ .", "Without loss of generality, let us assume that $0 < u_0 < u_1 < 1$ .", "Now, choose positive real numbers $u_2 \\in (u_1,1)$ and $\\lambda \\in (0,1)$ such that $f^0(u_2)<0$ and $u_1 = \\lambda u_0 + (1-\\lambda )u_2$ .", "Then, we have $&f^0(\\lambda u_0 + (1-\\lambda )u_2) = f^0(u_1) = 0 \\quad \\textrm {and}\\\\&\\lambda f^0(u_0) + (1-\\lambda ) f^0(u_2) = (1-\\lambda ) f^0(u_2) < 0 \\\\& \\Longrightarrow f^0(\\lambda u_0 + (1-\\lambda )u_2) \\overset{(c)}{>} \\lambda f^0(u_0) + (1-\\lambda ) f^0(u_2)$ Inequality (c) contradicts the convex nature of function $f^{0}$ .", "Thus, in turn, establishes the uniqueness of the fixed point." ], [ "Increasing Nature of the Fixed Point", "First we introduce some notation and present some preliminary observations.", "We will use these observations in the proof of Proposition REF .", "Let $h_{m,k}(q) = \\sum ^{k}_{k_2=0} {k \\atopwithdelims ()k_2} q^{k_2} (1-q)^{k - k_2} P[X_{k_2} + Y_{k-k_2} \\ge m ]$ where random variables $X_{k_2}$ and $Y_{k-k_2}$ are independent and have a binomial distribution with parameters $(k_2,\\alpha _2 u)$ and $(k-k_2,\\alpha _1 u)$ , respectively.", "The first step in the proof of Proposition REF is to show that for any $m \\in \\mathbb {Z}^{+}$ , $P[X_{k_2} + Y_{k-k_2} \\ge m ]$ is a non-decreasing function of $k_2 \\in [0,k]$ .", "Let $\\varphi _{m,k}(k_2) = P[X_{k_2} + Y_{k-k_2} \\ge m] \\quad \\forall k_2 \\in [0,k]$ We note that if $u \\in (0,1]$ then $\\varphi _{m,k}(k_2) = 0$ only if $m > k$ , i.e., if the number of active neighbours is greater than the node degree.", "Lemma 10.1 For any $m,k \\in \\mathbb {Z}^{+}$ , $\\varphi _{m,k}(k_2)$ is non-decreasing function of $k_2 \\in [0, k]$ .", "Consider two integers $k_2,k^{^{\\prime }}_2 \\in [0,k]$ .", "Without loss of generality, let $k^{^{\\prime }}_2 > k_2$ and define the following random variables $W_{k_2} = X_{k_2} + Y_{k-k_2} \\quad \\textrm {and} \\quad W_{k^{^{\\prime }}_2} = X_{k^{^{\\prime }}_2} + Y_{k-k^{^{\\prime }}_2}$ Then, we have $W_{k_2} &=_{st} X_{k_2} + Y_{k-k_2} \\\\&=_{st} \\sum \\nolimits ^{k_2}_{i=1} I_i(\\alpha _2 u) + \\sum \\nolimits ^{k-k_2}_{i=1} I_i(\\alpha _1 u)$ where $I_i(p)$ are independent Bernoulli random variable with parameter $p$ and $=_{st}$ implies equality in distribution [26].", "Therefore, for any $k^{^{\\prime }}_2 > k_2$ and $k_2,k^{^{\\prime }}_2 \\in [0,k]$ , we have $& W_{k_2} =_{st} \\sum \\nolimits ^{k_2}_{i=1} I_i(\\alpha _2 u) + \\sum \\nolimits ^{(k-k^{^{\\prime }}_2) + (k^{^{\\prime }}_2-k_2)}_{i=1} I_i(\\alpha _1 u) \\\\&=_{st} \\sum \\nolimits ^{k_2}_{i=1} I_i(\\alpha _2 u) + \\sum \\nolimits ^{k-k^{^{\\prime }}_2}_{i=1} I_i(\\alpha _1 u) + \\sum \\nolimits ^{k^{^{\\prime }}_2-k_2}_{i=1} I_i(\\alpha _1 u) \\\\&\\overset{(a)}{\\le }_{st} \\sum \\nolimits ^{k_2}_{i=1} I_i(\\alpha _2 u) + \\sum \\nolimits ^{k-k^{^{\\prime }}_2}_{i=1} I_i(\\alpha _1 u) + \\sum \\nolimits ^{k^{^{\\prime }}_2-k_2}_{i=1} I_i(\\alpha _2 u) \\\\& =_{st} \\sum \\nolimits ^{k^{^{\\prime }}_2}_{i=1} I_i(\\alpha _2 u) + \\sum \\nolimits ^{k-k^{^{\\prime }}_2}_{i=1} I_i(\\alpha _1 u) =_{st} W_{k^{^{\\prime }}_2}$ where Inequality $(a)$ follows because $\\alpha _1 < \\alpha _2 \\Rightarrow \\alpha _1 u \\le \\alpha _2 u$ , and due to Lemma REF .", "Hence, we have established that for $k_2,k^{^{\\prime }}_2 \\in [0,k]$ and $k^{^{\\prime }}_2 > k_2$ , we have $W_{k_2}\\le _{st} W_{k^{^{\\prime }}_2}$ .", "Thus, $\\forall m \\in \\mathbb {Z}^{+}$ and $k^{^{\\prime }}_2 > k_2$ , we have $\\varphi _{m,k}(k_2) = P[X_{k_2} + Y_{k-k_2} \\ge m] &= P[W_{k_2} \\ge m] \\\\& \\hspace{-128.0374pt} \\overset{(b)}{\\le } P[W_{k^{^{\\prime }}_2} \\ge m] = P[X_{k^{^{\\prime }}_2} + Y_{k-k^{^{\\prime }}_2} \\ge m] = \\varphi _{m,k}(k^{^{\\prime }}_2)$ where Inequality (b) follows because $W_{k_2}\\le _{st} W_{k^{^{\\prime }}_2}$ .", "Thus, we have shown that $\\varphi _{m,k}(k_2)$ is a non-decreasing function of $k_2 \\in [0, k]$ .", "Now, let us define $\\psi _{m,k}(k_2) = \\frac{k \\cdot \\varphi _{m,k}(k_2)}{(k-k_2)} \\quad \\forall k_2 \\in [0,k-1]$ Due to Lemma REF , we can conclude that for any $m,k \\in \\mathbb {Z}^{+}$ , $\\psi _{m,k}(k_2)$ is also a non-decreasing function of $k_2 \\in [0, k-1]$ .", "Next, establish that the function $f(q,u)$ is monotonically increasing in $q$ .", "Proposition 10.1 If $\\alpha _2 > \\alpha _1$ , then $\\frac{\\partial f(q,u)}{\\partial q} > 0$ .", "$& h_{m,k}(q) = \\sum ^{k}_{k_2=0} {k \\atopwithdelims ()k_2} q^{k_2} (1-q)^{k - k_2} P[X_{k_2} + Y_{k-k_2} \\ge m ] \\\\&= \\sum \\nolimits ^{k-1}_{k_2=0} {k \\atopwithdelims ()k_2} q^{k_2} (1-q)^{k - k_2} P[X_{k_2} + Y_{k-k_2} \\ge m ] \\\\& \\hspace{113.81102pt} + q^m \\cdot P[X_{k} \\ge m ] \\\\&= (1-q) \\sum \\nolimits ^{k-1}_{k_2=0} {k-1 \\atopwithdelims ()k_2} q^{k_2} (1-q)^{k - k_2} \\cdot \\psi _{m,k}(k_2) \\\\& \\hspace{113.81102pt} + q^m \\cdot P[X_{k} \\ge m ] \\\\&= (1-q) \\cdot \\mathbb {E}[\\psi _{m,k}(Z(k-1,q))] + q^m \\cdot P[X_{k} \\ge m ]$ where $Z(k-1,q)$ is a binomial random variable with parameters $(k-1,q)$ .", "Now, consider two real numbers $q_1,q_2 \\in [0,1]$ such that $q_1 < q_2$ .", "Then, we have $& h_{m,k}(q_1) = (1-q_1) \\mathbb {E}[\\psi _{m,k}(Z(k-1,q_1))] \\nonumber \\\\& \\hspace{142.26378pt} + (q_1)^m P[X_{k} \\ge m ] \\nonumber \\\\& \\overset{(a)}{\\le } (1-q_1) \\mathbb {E}[\\psi _{m,k}(Z(k-1,q_1))] + (q_2)^m P[X_{k} \\ge m ] \\nonumber \\\\& \\overset{(b)}{\\le } (1-q_1) \\mathbb {E}[\\psi _{m,k}(Z(k-1,q_2))] + (q_2)^m P[X_{k} \\ge m ] \\nonumber \\\\& \\overset{}{=} (1-q_2) \\mathbb {E}[\\psi _{m,k}(Z(k-1,q_2))] + (q_2)^m P[X_{k} \\ge m ] \\nonumber \\\\& \\hspace{56.9055pt} + (q_2-q_1) \\mathbb {E}[\\psi _{m,k}(Z(k-1,q_2))] \\nonumber \\\\& \\overset{}{=} h_{m,k}(q_2) + (q_2-q_1) \\mathbb {E}[\\psi _{m,k}(Z(k-1,q_2))] $ where follows because $q_1 < q_2$ .", "since $q_1 < q_2$ , we have $Z(k-1,q_1) \\le _{st} Z(k-1,q_2)$ (Lemma REF ).", "Therefore, due to the non-decreasing nature of function $\\psi _{m,k}$ , we have $\\mathbb {E}[\\psi _{m,k}(Z(k-1,q_1))] \\le \\mathbb {E}[\\psi _{m,k}(Z(k-1,q_2))]$ (Theorem REF ).", "Also, from the definition, we also have $h_{m,k}(q_1) & = \\mathbb {E}[\\varphi _{m,k}(Z(k,q_1))] \\nonumber \\\\&\\overset{(c)}{\\le } \\mathbb {E}[\\varphi _{m,k}(Z(k,q_2))] = h_{m,k}(q_2) $ where Inequality $(c)$ follows because of Lemma REF , Lemma REF and Theorem REF .", "From Equations (REF ) and (REF ), we can see that for any $q_1,q_2 \\in [0,1]$ such that $q_1 < q_2$ , we have $h_{m,k}(q_1)= h_{m,k}(q_2)$ if and only if $\\mathbb {E}[\\psi _{m,k}(Z(k-1,q_2))]=0$ .", "From the definition of function $\\psi _{m,k}$ , we can see that $\\mathbb {E}[\\psi _{m,k}(Z(k-1,q_2))] = 0$ if and only if $m > k$ .", "Thus, if $m \\in [0,k]$ , then we can conclude that $h_{m,k}(q_1) < h_{m,k}(q_2)$ .", "Since the network is connected and the fraction of zealous nodes lies in the open interval $(0,1)$ , there exists positive integers $k_0,m_0$ such that $m_{0} \\in (0, k_0]$ and $p_{ex}(k_0) p_{th}(m_0 | k_0+1) > 0$ .", "Therefore, for $q_1,q_2 \\in [0,1]$ such that $q_1 < q_2$ , we have $h_{m_0,k_0}(q_1) < h_{m_0,k_0}(q_2) $ .", "Let $z_0 = \\sum \\nolimits _{k \\ge 1} p_{ex}(k) p_{th}(0|k+1) $ — fraction of zealous nodes.", "Thus, we have $&f(q_1,u) = z_0 + \\sum _{k \\ge 1} \\sum _{m \\ge 1} p_{ex}(k) p_{th}(m|k+1) h_{m,k}(q_1) \\\\& = \\sum _{k \\ge 1} \\sum _{m \\ge 1} p_{ex}(k) p_{th}(m|k+1) h_{m,k}(q_1) (1-\\mathbb {I}_{\\left\\lbrace \\underset{k=k_0}{m=m_0}\\right\\rbrace }) \\\\& \\hspace{56.9055pt} + p_{ex}(k_0) p_{th}(m_0|k_0+1) \\cdot h_{m_0,k_0}(q_1) + z_0\\\\& < \\sum _{k \\ge 1} \\sum _{m \\ge 1} p_{ex}(k) p_{th}(m|k+1) h_{m,k}(q_1) (1-\\mathbb {I}_{\\left\\lbrace \\underset{k=k_0}{m=m_0}\\right\\rbrace }) \\\\& \\hspace{56.9055pt} + p_{ex}(k_0) p_{th}(m_0|k_0+1) \\cdot h_{m_0,k_0}(q_2) + z_0 \\\\& \\le \\sum _{k \\ge 1} \\sum _{m \\ge 1} p_{ex}(k) p_{th}(m|k+1) h_{m,k}(q_2) (1-\\mathbb {I}_{\\left\\lbrace \\underset{k=k_0}{m=m_0}\\right\\rbrace }) \\\\& \\hspace{56.9055pt} + p_{ex}(k_0) p_{th}(m_0|k_0+1) \\cdot h_{m_0,k_0}(q_2) + z_0\\\\& = f(q_2,u)$ i.e., for $q_1,q_2 \\in [0,1]$ such that $q_1 < q_2$ , we have $f(q_1,u) < f(q_2,u)$ ." ], [ "Proof of Proposition  ", "The continuity of the fixed point follows from the fact that function $f(\\cdot ,u)$ is a continuous map for all $u \\in [0,1]$ .", "To prove the increasing nature of the fixed point, we use the implicit function theorem.", "Let $f^{0}(q,u) = f(q,u) - u$ .", "According to the implicit function theorem, when $f^0(q,u)=0$ , we have $ \\frac{du}{d q} = - \\frac{\\frac{\\partial f^0(q,u)}{\\partial q}}{\\frac{\\partial f^0(q,u)}{\\partial u}} = \\frac{\\frac{\\partial f(q,u)}{\\partial q}}{ \\left(1 - \\frac{\\partial f(q,u)}{\\partial u} \\right)} $ From Proposition REF , we know that for every $q \\in [0,1]$ , there is a unique fixed point $u_q \\in (0,1)$ that satisfies $f(q,u_q) = u_q$ i.e., $f^0(q,u_q)=0$ .", "Therefore, we have $ \\frac{du_q}{d q} = \\frac{\\frac{\\partial f(q,u_q)}{\\partial q}}{ \\left(1 - \\frac{\\partial f(q,u)}{\\partial u} \\Big |_{u = u_q} \\right)} $ From Proposition REF , we know that $\\frac{\\partial f(q,u)}{\\partial q} > 0 , \\forall u \\in (0,1)$ .", "Since $u_q \\in (0,1)$ , we have $\\frac{\\partial f(q,u_q)}{\\partial q} > 0$ .", "Now, to prove Proposition  REF , we just need to show that $\\frac{\\partial f(q,u)}{\\partial u} \\Big |_{u = u_q} < 1$ .", "We show this by contradiction.", "We note that $f(q,1) \\overset{(a)}{<} 1$ , where Inequality (a) follows because the $f(q,1)$ is a convex combination of non-negative terms less than or equal to 1 (some terms are strictly less that 1).", "Since the function $f(q,u)$ is continuous and non-decreasing in $u$ (see Proposition REF ), there exists a point $u^{^{\\prime }}_q \\in (u_q,1)$ such that $f(q,u^{^{\\prime }}_q) < u^{^{\\prime }}_q$ .", "We note that the function $f(q,u)$ is convex in $u \\in (0,1)$ (see Proposition REF ).", "Now, let us assume that $\\frac{\\partial f(q,u)}{\\partial u} \\Big |_{u = u_q} \\ge 1$ .", "Then, from the definition of convex functions, we have $f(q,u^{^{\\prime }}_q) &\\ge f(q,u_q) + (u^{^{\\prime }}_q - u_q) \\cdot \\frac{\\partial f(q,u)}{\\partial u} \\Big |_{u = u_q} \\\\&\\overset{}{\\ge } f(q,u_q) + (u^{^{\\prime }}_q - u_q) \\\\&\\overset{(b)}{=} u_q + (u^{^{\\prime }}_q - u_q) = u^{^{\\prime }}_q \\\\& \\hspace{-56.9055pt} \\Rightarrow f(q,u^{^{\\prime }}_q) \\overset{(c)}{\\ge } u^{^{\\prime }}_q$ where Inequality (b) follows because we have $f(q,u_q) = u_q$ .", "Inequality (c) contradicts the fact that $f(q,u^{^{\\prime }}_q) < u^{^{\\prime }}_q$ .", "Since this contradiction was due to the assumption that $\\frac{\\partial f(q,u)}{\\partial u} \\Big |_{u = u_q} \\ge 1$ , we should have $\\frac{\\partial f(q,u)}{\\partial u} \\Big |_{u = u_q} < 1$ .", "This, in turn, establishes that $\\frac{du_q}{d q} > 0$ ." ], [ "Equivalence to the Linear Program", "In this section, we prove the equivalence between problems $P_1$ and $P_2$ presented in Section" ], [ "Proof of Proposition ", "Let the value of the objective of problem $P_1$ for incentive-policy $\\phi $ be denoted as $c(\\phi )$ .", "Let $\\phi ^{o} = \\lbrace \\phi ^{o}_k , k \\ge 1\\rbrace $ be an optimal solution of problem $P_1$ .", "If we have $\\sum _{k\\ge 1} k \\cdot p(k) \\cdot \\phi ^{o}(k) = \\overline{d} \\cdot q_{\\gamma }$ , then we can choose $\\phi ^{opt} = \\phi ^{o}$ .", "So, let us assume that $\\frac{1}{\\overline{d}} \\sum _{k\\ge 1} k \\cdot p(k) \\cdot \\phi ^{o}(k) = q^o > q_{\\gamma }$ .", "Our goal to obtain a policy $\\phi ^{opt}$ from $\\phi ^{o}$ without increasing the cost.", "Let $\\phi ^{opt} = \\frac{q_{\\gamma }}{q^o} \\phi ^{o} $ .", "Then, we have $&\\frac{1}{\\overline{d}} \\sum _{k\\ge 1} k p(k) \\cdot \\phi ^{opt}(k) = \\frac{q_{\\gamma }}{q^o \\overline{d}} \\sum _{k\\ge 1} k \\cdot p(k) \\cdot \\phi ^{o}(k) = q_{\\gamma } \\\\& \\hspace{56.9055pt} \\textrm {and} \\quad 0 \\le \\phi ^{opt} \\le \\phi ^{o} \\le 1 $ From Equality (REF ) and Inequality () we can see that $\\phi ^{opt}$ is a feasible solution of problem $P_1$ .", "Further, we also have $c(\\phi ^{o}) &= \\sum \\nolimits _{k \\ge 1} c_k \\cdot p(k) \\cdot \\phi ^o(k) \\cdot s_k(q^o) \\nonumber \\\\& \\overset{(a)}{\\ge } \\sum \\nolimits _{k \\ge 1} c_k \\cdot p(k) \\cdot \\phi ^{opt}(k) \\cdot s_k(q^o) \\nonumber \\\\& \\overset{(b)}{\\ge } \\sum \\nolimits _{k \\ge 1} c_k \\cdot p(k) \\cdot \\phi ^{opt}(k) \\cdot s_k(q_{\\gamma }) \\nonumber \\\\& = c(\\phi ^{opt}) \\overset{(c)}{\\ge } c(\\phi ^{o}) \\Rightarrow c(\\phi ^{o}) = c(\\phi ^{opt}) \\nonumber $ where Inequality (a) follows because $\\phi ^{opt} \\le \\phi ^{o}$ , Inequality (b) follows because $q^0 > q_{\\gamma }$ and $s_k(q)$ is non-decreasing in $q$ (Proposition REF ), and Inequality (c) follows because $\\phi ^{o}$ is an optimal solution of problem $P_1$ .", "Since the cost for policies $\\phi ^{o}$ and $\\phi ^{opt}$ are equal, and $\\phi ^{o}$ is an optimal policy, we can conclude that policy $\\phi ^{opt}$ is also an optimal solution of problem $P_1$ ." ], [ "Proof of Proposition ", "Let $\\nu ^{opt} = \\lbrace \\nu ^{opt}_k , k \\ge 1\\rbrace $ be an optimal solution of problem $P_2$ .", "Let us define $\\phi ^{f}(k) = {\\left\\lbrace \\begin{array}{ll}0 & \\textrm {if } p(k) = 0 \\\\\\frac{\\nu ^{opt}_k \\cdot \\overline{d} \\cdot q_{\\gamma }}{k \\cdot p(k)} & \\textrm {otherwise}\\end{array}\\right.}", "$ Then, we have $\\frac{1}{\\overline{d}} \\sum \\nolimits _{k \\ge 1} k \\cdot p(k) \\cdot \\phi ^{f}(k) = q_{\\gamma } \\sum \\nolimits _{k \\ge 1} \\nu ^{opt}_k \\overset{(a)}{=} q_{\\gamma } \\\\\\phi ^{f}(k) = {\\left\\lbrace \\begin{array}{ll}0 < 1 & \\textrm {if } p(k) = 0 \\\\\\frac{\\nu ^{opt}_k \\cdot \\overline{d} \\cdot q_{\\gamma }}{k \\cdot p(k)} \\overset{(b)}{\\le } 1 & \\textrm {otherwise}\\end{array}\\right.}", "$ where Equality (a) and Inequality (b) follows because $\\lbrace \\nu ^{opt}_k , k \\ge 1\\rbrace $ is a feasible solution of problem $P_2$ .", "From Equality (REF ) and Inequality (), we can conclude that $\\phi ^f$ is a feasible solution of problem $P_1$ .", "Now, let $\\phi ^{opt} = \\lbrace \\phi ^{opt}_k , k \\ge 1\\rbrace $ be an optimal solution of problem $P_1$ such that $\\sum _{k\\ge 1} k \\cdot p(k) \\cdot \\phi ^{opt}(k) = \\overline{d} \\cdot q_{\\gamma }$ .", "The existence of such an optimal policy is guaranteed by Proposition REF .", "Let us define $\\nu ^{f}(k) = \\frac{k \\cdot p(k) \\cdot \\phi ^{opt}_k }{\\overline{d} \\cdot q_{\\gamma }} $ Then, we have $\\sum _{k \\ge 1} \\nu ^{f}(k) = \\sum _{k \\ge 1} \\frac{k \\cdot p(k) \\cdot \\phi ^{opt}_k }{\\overline{d} \\cdot q_{\\gamma }} = 1$ and $0 \\le \\nu _k = \\frac{k \\cdot p(k) \\cdot \\phi ^{opt}_k }{\\overline{d} \\cdot q_{\\gamma }} \\le \\frac{k \\cdot p(k)}{\\overline{d} \\cdot q_{\\gamma }}$ .", "Hence, $\\nu ^f$ is a feasible solution of problem $P_2$ .", "Further, we also have $& \\sum \\nolimits _{k \\ge 1} c_k p(k) \\phi ^{opt}(k) s_k(q_{\\gamma }) \\le \\sum \\nolimits _{k \\ge 1} c_k p(k) \\phi ^{f}(k) s_k(q_{\\gamma })\\\\& = \\overline{d} \\cdot q_{\\gamma } \\sum \\nolimits _{k \\ge 1} \\nu ^{opt}_k \\cdot \\mu _k(q_{\\gamma }) \\le \\overline{d} \\cdot q_{\\gamma } \\sum \\nolimits _{k \\ge 1} \\nu ^{f}_k \\cdot \\mu _k(q_{\\gamma }) \\\\& = \\sum _{k \\ge 1} c_k p(k) \\phi ^{opt}(k) s_k(q_{\\gamma }) \\\\& \\Rightarrow \\sum _{k \\ge 1} c_k p(k) \\phi ^{opt}(k) s_k(q_{\\gamma }) = \\sum _{k \\ge 1} c_k p(k) \\phi ^{f}(k) s_k(q_{\\gamma }) \\\\& \\Rightarrow \\phi ^f \\textrm { is an optimal solution of problem P_1.", "}$" ], [ "Optimality of Algorithm ", "In this section, we prove that the vector $\\nu ^a$ returned by Algorithm REF is an optimal point of problem $P_2$ .", "However, as a first step, we prove the following lemma.", "Let $\\mathcal {S} = \\lbrace 1, 2, \\cdots , k_{max}\\rbrace $ and $k_{max}$ is the maximum degree in the network.", "Lemma 12.1 Let $\\nu $ , a $1 \\times k_{max}$ vector, be a feasible solution of problem $P_2$ .", "Then, we have $\\sum \\nolimits ^{j_0}_{j = 1} \\nu ^a_{\\sigma (j)} \\ge \\sum \\nolimits ^{j_0}_{j = 1} \\nu _{\\sigma (j)} \\quad \\forall j_0 \\in \\mathcal {S}$ We prove this lemma by induction.", "For $j_0=1$ , we have $\\nu ^a_{\\sigma (1)} = \\min \\left\\lbrace 1, \\frac{\\sigma (1) \\cdot p(\\sigma (1))}{\\overline{d} \\cdot q_{\\gamma }} \\right\\rbrace \\overset{(a)}{\\ge } \\nu _{\\sigma (1)}$ where Inequality (a) follows because $\\nu $ is a feasible solution of problem $P_2$ .", "Now, let us assume that $\\sum ^{j_1}_{j = 1} \\nu ^a_{\\sigma (j)} \\ge \\sum ^{j_1}_{j = 1} \\nu _{\\sigma (j)}$ for some $j_1 \\in \\mathcal {S}$ .", "We need to show that $\\sum ^{j_1+1}_{j = 1} \\nu ^a_{\\sigma (j)} \\ge \\sum ^{j_1+1}_{j = 1} \\nu _{\\sigma (j)}$ .", "If we have $\\nu ^a_{\\sigma (j_1+1)}\\ge \\nu _{\\sigma (j_1+1)}$ , then the result is trivial.", "Let us therefore consider the case where $\\nu ^a_{\\sigma (j_1+1)} < \\nu _{\\sigma (j_1+1)}$ .", "This implies that $&\\nu ^a_{\\sigma (j_1+1)} < \\nu _{\\sigma (j_1+1)} \\le \\frac{\\sigma (j_1+1) \\cdot p(\\sigma (j_1+1))}{\\overline{d} \\cdot q_{\\gamma }} \\\\&\\Rightarrow \\nu ^a_{\\sigma (j_1+1)} \\overset{(b)}{=} 1 - \\sum \\nolimits ^{j_1}_{j=1} \\nu ^a_{\\sigma (j)}$ where Equality (b) follows because of steps 4 and 5 of Algorithm REF .", "Therefore, if $\\nu ^a_{\\sigma (j_1+1)} < \\nu _{\\sigma (j_1+1)}$ , then we have $\\sum \\nolimits ^{j_1+1}_{j=1} \\nu ^a_{\\sigma (j)} = 1 = \\sum \\nolimits _{j \\in \\mathcal {S}} \\nu _{\\sigma (j)} \\ge \\sum \\nolimits ^{j_1+1}_{j = 1} \\nu _{\\sigma (j)}$" ], [ "Proof of Proposition ", "Let $\\nu $ be a feasible solution of problem $P_2$ .", "Since $\\nu ^a$ and $\\nu $ are feasible solutions of problem $P_2$ , we have $\\sum _{j \\ge 1} \\nu ^a({\\sigma (j)}) = \\sum _{j \\ge 1} \\nu ({\\sigma (j)}) = 1$ .", "Let $X^a$ and $X$ two independent random variable taking values from the set $\\mathcal {S}$ such that $P[X^a = j] = \\nu ^a_{\\sigma (j)} \\quad \\textrm {and} \\quad P[X = j] = \\nu _{\\sigma (j)}$ .", "Then, from Lemma REF , we have $& P[X^a \\le j_0] = \\sum _{j=1}^{j_0} \\nu ^a_{\\sigma (j)} \\ge \\sum _{j=1}^{j_0} \\nu _{\\sigma (j)} = P[X \\le j_0] $ From Equation REF and the definition of usual stochastic orders we can see that $X^a \\le _{st} X$ .", "Therefore, we have $& \\sum \\nolimits _{k \\ge 1} \\nu ^a_k \\cdot \\mu _{k}(q_{\\gamma }) = \\sum \\nolimits _{j \\in \\mathcal {S}} \\nu ^a_{\\sigma (j)} \\cdot \\mu _{\\sigma (j)}(q_{\\gamma }) \\nonumber \\\\&= \\sum \\nolimits _{j \\in \\mathcal {S}} P[X^a = j] \\cdot \\mu _{\\sigma (j)}(q_{\\gamma }) = \\mathbb {E}[\\mu _{\\sigma {(X^a)}}(q_{\\gamma })] \\nonumber \\\\& \\overset{(a)}{\\le } \\mathbb {E}[\\mu _{\\sigma {(X)}}(q_{\\gamma })] = \\sum \\nolimits _{j \\in \\mathcal {S}} P[X = j] \\cdot \\mu _{\\sigma (j)}(q_{\\gamma }) \\nonumber \\\\&= \\sum \\nolimits _{j \\in \\mathcal {S}} \\nu _{\\sigma (j)} \\cdot \\mu _{\\sigma (j)}(q_{\\gamma }) = \\sum \\nolimits _{k \\ge 1} \\nu _k \\cdot \\mu _{k}(q_{\\gamma }) $ where Inequality (a) follows from an application of Theorem REF ($X^a \\le _{st} X$ and $\\mu _{\\sigma (j)}(q_{\\gamma })$ is a non-decreasing function of $j$ ).", "Let $\\nu ^{opt}$ be an optimal solution of $P_2$ .", "Then, we have the following inequalities $& \\sum _{k \\ge 1} \\nu ^{opt}_k \\cdot \\mu _{k}(q_{\\gamma }) \\le \\sum _{k \\ge 1} \\nu ^a_k \\cdot \\mu _{k}(q_{\\gamma }) \\overset{(b)}{\\le } \\sum _{k \\ge 1} \\nu ^{opt}_k \\cdot \\mu _{k}(q_{\\gamma }) \\\\& \\Longrightarrow \\sum _{k \\ge 1} \\nu ^{opt}_k \\cdot \\mu _k (q_{\\gamma }) = \\sum _{k \\ge 1} \\nu ^{a}_k \\cdot \\mu _{k}(q_{\\gamma }) \\\\& \\Longrightarrow \\nu ^{a} \\textrm { is an optimal solution of } P_2$ where Inequality (b) follows by setting $\\nu $ to $\\nu ^{opt}$ in Inequality (REF )." ] ]
1606.05179
[ [ "Fermi Large Area Telescope Detection of Extended Gamma-Ray Emission from\n the Radio Galaxy Fornax A" ], [ "Abstract We report the Fermi Large Area Telescope detection of extended gamma-ray emission from the lobes of the radio galaxy Fornax A using 6.1 years of Pass 8 data.", "After Centaurus A, this is now the second example of an extended gamma-ray source attributed to a radio galaxy.", "Both an extended flat disk morphology and a morphology following the extended radio lobes were preferred over a point-source description, and the core contribution was constrained to be < 14% of the total gamma-ray flux.", "A preferred alignment of the gamma-ray elongation with the radio lobes was demonstrated by rotating the radio lobes template.", "We found no significant evidence for variability on ~0.5 year timescales.", "Taken together, these results strongly suggest a lobe origin for the gamma rays.", "With the extended nature of the > 100 MeV gamma-ray emission established, we model the source broadband emission considering currently available total lobe radio and millimeter flux measurements, as well as X-ray detections attributed to inverse Compton (IC) emission off the cosmic microwave background (CMB).", "Unlike the Centaurus A case, we find that a leptonic model involving IC scattering of CMB and extragalactic background light (EBL) photons underpredicts the gamma-ray fluxes by factors of about ~ 2 - 3, depending on the EBL model adopted.", "An additional gamma-ray spectral component is thus required, and could be due to hadronic emission arising from proton-proton collisions of cosmic rays with thermal plasma within the radio lobes." ], [ "Introduction", "The radio galaxy Fornax A, well known for its radio lobes spanning $\\sim 50\\hbox{$^{\\prime }$}$ , with a lobe-to-lobe separation of $\\sim 33\\hbox{$^{\\prime }$}$ , is one of the closest and brightest radio galaxies, located at a distance of only $18.6$  Mpc [34].", "Hosted by the elliptical galaxy NGC 1316, the radio source contains a low-ionization nuclear emission-line region nucleus, which has been imaged to arcsecond-scale resolution and features a flat spectrum ($\\alpha =0.4$ ; $S_{\\nu } \\propto \\nu ^{-\\alpha }$ ) core with dual-opposing “s”-shaped jets that are detected out to $\\sim 5$  kpc from the core [19].", "The radio lobes are characterized by a complex polarized filamentary structure with no observable hotspots [18].", "Fornax A was the first radio galaxy reported to emit diffuse, non-thermal X-ray emission from within its radio lobes from observations with ROSAT [15] and ASCA [26], which were later confirmed through dedicated observations of the east lobe with XMM-Newton [23] and the west lobe with Suzaku [45].", "The non-thermal X-rays have been widely attributed to inverse Compton (IC) emission of relativistic electrons scattering on cosmic microwave background (CMB) photons, with the same population of relativistic electrons producing both synchrotron and IC emission [21].", "To date, similar leptonic IC/CMB emission has been detected in X-rays from tens of extended lobes in radio galaxies and quasars.", "In general, such detections imply that the ratio of relativistic electron pressure to magnetic field pressure within the lobes is $\\sim 1$ –100 [13], [27], [24].", "More recently, [40] reported a detection of thermal emission from the western lobe of Fornax A using combined Suzaku and XMM-Newton data.", "Thermal emission in the lobes of a radio galaxy is typically not seen, although evidence for this has also been reported in the giant lobes of the nearby radio galaxy Centaurus A [42], [39].", "Motivated by the observed (and presumed IC/CMB) X-ray emission from the lobes of Fornax A, [11] predicted that the high-energy tail of the IC/CMB would be detected by the Fermi Large Area Telescope [8] at $>100$  MeV.", "Following this, [20] predicted that the lobes would also be detected in $\\gamma $ rays at higher energies by the LAT due to IC upscattering of the infrared and optical extragalactic background light (EBL) photons, analogous to the CMB photons upscattered to X-ray energies.", "The association of Fornax A with the Fermi-LAT second year catalog [38] source 2FGL J0322.4$-$ 3717 thus raised an important question regarding the origin of the $\\gamma $ -ray source, which at the time had no evidence presented for significant extension.", "In particular, a distinction between emission arising from the lobes and possible contamination from the central core region could not be established from the $\\gamma $ -ray data alone, although X-ray and radio observations [29] suggested that the contribution from the core was likely to be minimal.", "In a recent study by [36], the spectrum of the Fornax A lobes was modeled in multiple wavelengths using both leptonic and hadronic production scenarios without knowledge of $\\gamma $ -ray spatial extent or $\\gamma $ -ray contamination from the galaxy core.", "They concluded the most likely source of $\\gamma $ -ray production is hadronic processes within filamentary structures of the lobes.", "Our study follows the successful $\\gamma $ -ray detection of the extended lobes from Centaurus A [1], and LAT studies of the lobes of NGC 6251 [44] and Centaurus B [28].", "Gamma-ray upper limits using H.E.S.S.", "and Fermi-LAT observations have been used to constrain the hadronic cosmic-ray population within the radio lobes of Hydra A [2].", "Fornax A is not associated with a $\\gamma $ -ray source in the most recent, third Fermi-LAT catalog based on four years of LAT data [3], [6].", "However, the centroid of the source 3FGL J0322.5$-$ 3721 is offset by 015 from the core of Fornax A.", "This offset is greater than the 95% position uncertainty of the 3FGL source.", "In the following we discuss possible reasons for this offset, detail a significant $\\gamma $ -ray detection of extended emission from Fornax A using $6.1$ years of Fermi-LAT data, and present modeling under leptonic and hadronic scenarios.", "Detecting extended emission from Fornax A with the LAT is challenging because the 68% containment point-spread function (PSF) radius is $\\sim 08$ at 1 GeV, which is larger than the Fornax A lobe-to-lobe separation.", "The LAT PSF is energy dependent going from $5\\hbox{$^\\circ $}$ at 100 MeV to 01 at 100 GeV with 68% confidencehttp://www.slac.stanford.edu/exp/glast/groups/canda/latPerformance.htm.", "Unlike all studies mentioned in Section , we used $6.1$ years (from 2008 August 4 to 2014 September 4) of Pass 8 LAT data.", "Compared to previous iterations of the LAT event-level analysis, Pass 8 provides greater acceptance and a PSF [41], as well as event type partitions according to PSFhttp://fermi.gsfc.nasa.gov/ssc/data/analysis/LATessentials.html, which we used in this analysis.", "All of these factors allowed for a firm detection of extension of Fornax A.", "We selected from all-sky survey data at energies from 0.1 to 300 GeV extracted from a region of interest (ROI) with $10\\hbox{$^\\circ $}$ radius centered at the J2000.0 radio position of Fornax A =50673$, Decl.\\,$ =-37208$,\\endcsname {gel84}.We used the ``source^{\\prime \\prime } event class, recommended for individual sourceanalysis, a zenith angle limit of $ 100$^\\circ $$ to greatly reduce contamination fromthe Earth limb, and a rocking angle limit of $ 52$^\\circ $$.\\textit {Fermi} Science Tools \\texttt {v10r01p00} andinstrument response functions (IRFs) \\texttt {P8R2\\_SOURCE\\_V6} wereused for this analysis\\footnote {{http://fermi.gsfc.nasa.gov/ssc/data/analysis/software}}.$ To model the LAT data, we included all sources from the 3FGL within $10\\hbox{$^\\circ $}$ of the radio core position of Fornax A.", "The diffuse background was modeled using preliminary versions of Galactic diffuse and isotropic spectral templates recommended by the Fermi-LAT collaboration, of which the finalized versions have been released to the publichttp://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html.", "Several tests were performed, and we determined that the results presented here with the preliminary diffuse models are compatible with those obtained with the finalized models.", "We used the same spectral models as in the 3FGL catalog for all background sources, and the normalization and spectral shape parameters of all point sources were left free during optimization.", "For the diffuse models, only the normalization parameters were left free.", "We initially modeled Fornax A as a point source located at the position of the radio core, removing 3FGL J0322.5$-$ 3721 from the model since it is offset from the radio core by 015.", "We optimized the localization using the gtfindsrc tool provided in the Science Tools in unbinned mode.", "The best-fit localization is R.A. $=5073$ , Decl.", "$=-3728$ with a 95% confidence error circle radius of 014, slightly southeast of the position of the Fornax A core and consistent with the reported 3FGL localization.", "Figure REF shows the best-fit localization (point B) and the core (point A) as well as the 2FGL and 3FGL error contours plotted on top of the relative residual counts map.", "Optimizing the model with the single point source at point B, we detect $\\gamma $ -ray emission at a Test Statistic (TS)$TS$ is defined as twice the difference between the logarithmic likelihood of the null hypothesis $\\mathcal {L}_0$ and the alternative hypothesis being tested $\\mathcal {L}_1$ [35]: $TS = 2 (\\log {\\mathcal {L}_1 } - \\log {\\mathcal {L}_0})$ .", "= 121.", "The spectrum was modeled as a single power law with a resulting maximum-likelihood photon index $\\Gamma =2.08 \\pm 0.08$ and a full band energy flux of $(5.34 \\pm 0.78_{\\text{stat}} \\begin{array}{c}+0.03 \\\\ -0.05\\end{array} \\ _{\\text{sys}} )\\times 10^{-12}$  erg cm$^{-2}$ s$^{-1}$ (see Table REF ).", "Systematic errors are due to the systematic uncertainty in the LAT effective areahttp://fermi.gsfc.nasa.gov/ssc/data/analysis/LATcaveats.html.", "These fluxes and indices are consistent with those reported in the catalogs for sources 2FGL J0322.4$-$ 3717 and 3FGL J0322.5$-$ 3721." ], [ "Extension and Morphology", "In the following, we describe several tests performed to determine the morphology of the observed $\\gamma $ -ray emission from the direction of Fornax A, as summarized in Tables REF  & REF .", "All tests on extension and morphology made full use of the additional spatial information brought about by the new Pass 8 PSF event type partitions.", "The broadband flux and spectrum optimizations in addition to these tests were performed using all PSF types in composite likelihood.", "Figure: Relative smoothed (4.7 px, 024 Gaussian FWHM) residual γ\\gamma -ray counts((counts - model)/model) in the 15 square region around thecore of Fornax A between 1 and 300 GeV.Overlaid are the radio contours (gray lines) from the VLA observations of with the radio core (position indicated by A) subtracted.The γ\\gamma -ray residual counts are elongated in a similar way tothe radio lobes.", "Also shown are the 95% confidence error ellipses for 2FGL J0322.4--3717and 3FGL J0322.5--3721; the 3FGL source has the smaller ellipse.", "The dashedcircle shows the 95% confidence error circle from ourmaximum-likelihood localization of the region as a singlepoint source centered at position B.", "The other points(labeled C, D, E) are the locations of various sources and testsources, as detailed in the text and in Table ." ], [ "Spatial Extension", "To determine if the $\\gamma $ -ray emission is extended beyond that of a point source we modeled Fornax A as a flat circular disk of various sizes (003 to 075 in steps of 003) by producing several disk templates centered at the best-fit location of the LAT source described in Section REF (point B in Figure REF ).", "The uniform disk is the simplest spatial model, and the use of a Gaussian profile has typically been shown to produce comparatively little difference in the overall likelihood and best-fit spectral parameters [32].", "Using gtlike in binned mode (with bin size 005), we determined the overall likelihood $\\mathcal {L}$ as a function of the disk radius $r$ .", "As shown in Figure REF , $\\mathcal {L}$ is peaked at $r=033\\pm 005$ , which is roughly compatible with the extent of the lobes as observed in radio Hz at $14\\hbox{$^{\\prime \\prime }$}$ resolution,fom89.", "By comparing the likelihood of the peak radius with the near-zero radius of 003 (effectively a point source), we found that the $\\gamma $ -ray emission is spatial extended with $5.9\\sigma $ confidence L$ = 17.3, 1 degree offreedom,\\endcsname {wilks}.", "See Tables~\\ref {tab:loglike} and \\ref {tab:ts} for more information.$ Figure: (a) Δlogℒ\\Delta \\log \\mathcal {L} between a flat disk of radius 003 (i.e., point-like)and flat disks of various radii.", "A cubic fit gives a maximum likelihood radius of 033±005033\\pm 005.The increase in ℒ\\mathcal {L} from 003 to the maximum likelihood radius indicates the emissionis not point-like.", "(b) Δlogℒ\\Delta \\log \\mathcal {L} between the non-rotated radio lobe templateand the template rotated around the central core indicating that the unrotated radiomorphology is preferred." ], [ "Blind Tests for Morphology", "With the aim of making no prior assumptions about the morphology of the $\\gamma $ -ray emission in this region, we used the ROI fit with our position-optimized point source as described in Section REF and removed that point source from the model, thereby leaving only the background sources.", "Using this background model, we created a map of residual counts above 1 GeV in the ROI by subtracting the model's predicted counts from the observed counts in each bin.", "Apart from the emission near Fornax A, the map of residuals is flat in significance and the distribution of significance values for the bins is Gaussian, indicating there are no significant systematic deviations from the ROI model.", "The map of residual counts shown in Figure REF is cropped to a $15\\times 15$ region centered around the Fornax A core.", "The shape of the residuals suggests a non-circular morphology that resembles the known extended radio morphology.", "The contours of radio emission from a VLA $1.5$  GHz image at $14\\hbox{$^{\\prime \\prime }$}$ resolution [18], with the radio core subtracted, are overlaid in Figure REF .", "We also tested a model which included two separate point sources whose initial locations were chosen by eye based upon the residual counts.", "The locations of these two point sources were then optimized using gtfindsrc.", "The best-fit location of the western point source (labeled D in Figure REF ) matches well the western lobe's centroid while the eastern point (labeled E in Figure REF ) is offset from the centroid of the eastern lobe.", "The double point-source model is preferred over the single point-source model at a confidence level of $4.8\\sigma $ .", "Table REF details these results and Table REF lists the fit parameters.", "Note that the spectral index is statistically compatible with the single point-source model and for both point sources in the double point-source model." ], [ "Radio-Motivated Tests for Morphology", "Assuming that the same electron energy distribution determined from the radio emission scatters optical EBL photons to produce $\\gamma $ rays, then a reasonable guess for the $\\gamma $ -ray morphology should be the observed radio structure.", "The lobes of Fornax A were found to emit non-thermal IC scattered X-rays, with excellent spatial coincidence to the radio structure produced by synchrotron-emitting relativistic electrons [15].", "Diffuse hard X-ray emission associated with the east lobe has been confirmed with XMM-Newton [23], implying the magnetic field is reasonably uniform, further supporting the expectation that the $\\gamma $ rays should match the radio morphology.", "Under these circumstances, as was observed in Centaurus A [1], the relativistic electrons will trace the $\\gamma $ -ray emission [20].", "These assumptions could be incorrect; however, for the purpose of constraining the EBL with the $\\gamma $ -ray flux, using the radio structure is the best choice for the spatial distribution.", "Physically, Fornax A images from the Wilkinson Microwave Anisotropy Probe (WMAP) might be a better choice of template for the $\\gamma $ -ray emission because the synchrotron radiation within $\\sim 23$  – 94 GHz should be originating from the same band of relativistic electrons that IC scatter into $\\sim 0.1$  – 10 GeV $\\gamma $ rays.", "We revisit this topic later in the section.", "We created a spatial template of the lobe emission using VLA $1.5$  GHz radio data [18], which offers a more-than-adequate spatial resolution of $14\\hbox{$^{\\prime \\prime }$}$ .", "Both the central core of Fornax A and various radio point sources inside the lobe structure were manually removed from the spatial template, the former by reduction to zero and the latter by interpolation of the adjacent lobe-dominated flux.", "The central core was reduced to zero because the lobes do not overlap with the central core [14].", "We found this template in addition to a point-source model of the core is preferred over just the point source at the core with a confidence of $6.0\\sigma $ ($\\Delta \\log \\mathcal {L}$ = 19.8, 2 degrees of freedom).", "However, this combined model is preferred over the lobes template alone by only $0.7\\sigma $ .", "Moreover, in the combined point-source-and-lobes template fit only 14% of the energy flux is assigned to the core point source.", "See Tables REF and REF for more information.", "We consider this as evidence that the $\\gamma $ -ray emission from the core of Fornax A is insignificant.", "Minimal $\\gamma $ -ray flux from the central core is expected, as it is assumed to be synchrotron self-Compton, and the radio flux from the core has been reported to be relatively weak compared to the luminous galaxy lobes [19].", "In order to test the uniqueness of the radio template morphology and its rotational symmetry, we rotated the template around the central core in increments of 92 and computed the likelihood at each one.", "The results from this test are shown in Figure REF and indicate that the template in its original orientation is preferred.", "In particular, we see the original orientation is a better model than the one rotated $90^{\\circ }$ clockwise with $5.1\\sigma $ confidence ($\\Delta \\log \\mathcal {L}$ =12.9, 1 degree of freedom) and $90^{\\circ }$ counterclockwise with $6.1\\sigma $ confidence ($\\Delta \\log \\mathcal {L}$ =18.9, 1 degree of freedom).", "The maximum likelihood rotation was $-20^{\\circ } \\pm 10$ Additionally, our plot of $\\log \\mathcal {L}$ over template rotation angle shows a sinusoidal profile with a similar peak in likelihood around $180^{\\circ }$ as around $0^{\\circ }$ .", "This degeneracy indicates that the data are not constraining enough to statistically differentiate the lobes.", "Modeling the lobes as separate point sources (see Section REF ) results in the measurement of similar fluxes and spectral shapes for the two regions, but this is at odds with radio observations of the lobes.", "According to the $1.5$  GHz VLA observation [18], the west lobe has about twice the total flux than the east.", "In fact, the two point-source $\\gamma $ -ray analysis indicated the east lobe may be slightly brighter (but is within the statistical errors).", "This disagreement may be a symptom of our use of $1.5$  GHz VLA data instead of the very similar but more physically motivated $\\sim 23$  – 94 GHz WMAP data as a spatial template for the $\\gamma $ -rays.", "Indeed, the flux ratio of west to east is $\\sim 1.3$ in the WMAP 41 and 61 GHz maps [20], closer to the $\\gamma $ -ray result.", "However, Fornax A is at the resolving power of the LAT in this analysis and the PSF is broad enough ($\\sim 08$ radius at 1 GeV) that fine scale changes would be smoothed out and indistinguishable from the original, and small changes in the flux ratio between the lobes should leave the average flux nearly the same.", "This is highlighted by the similarity in overall likelihood we observe between $0^{\\circ }$ and $180^{\\circ }$ rotations of the $1.5$  GHz VLA template, where the rotation of $180^{\\circ }$ is identical to a flux ratio of $\\sim 0.5$ .", "In addition, even when using the spatial morphology of a single point source, the flux is nearly consistent with the radio template flux (see Section REF ).", "With the aim of testing the region for the possibility of a contaminating background $\\gamma $ -ray source, we added a point source to the lobes template model and optimized its position using gtfindsrc.", "The optimized position of this point source (labeled C in Figure REF ) was R.A. = 5067, Decl.", "= $-3746$ with a 95% confidence error circle radius of 042 (large enough to encompass the whole Fornax A emission region).", "The resulting fit was marginally preferred over the lobes template alone with $2.7\\sigma $ significance.", "Therefore, we do not consider any contribution from a background point source to be significant.", "llrrrr 5 0pc Fornax A spatial model comparisons Null hypothesis Alternative hypothesis DOFa $\\Delta \\log \\mathcal {L}$ $\\sigma $ b Sect.", "Point source (core location A) Point source (best-fit location B) 2 0.6 0.6 REF Disk (best-fit location B, 003 radius) Disk (best-fit location B, 033 radius) 1 17.3 5.9 REF Point source (best-fit location B) Two point sources (locations D and E) 4 16.2 4.8 REF Point source (core location A) Radio lobes template and point source (core location A) 2 19.8 6.0 REF Radio lobes template Radio lobes template and point source (core location A) 2 0.7 0.7 REF Radio lobes template, rotated $90^{\\circ }$ CW Radio lobes template 1 12.9 5.1 REF Radio lobes template, rotated $90^{\\circ }$ CCW Radio lobes template 1 18.9 6.1 REF Radio lobes template Radio lobes template and point source (location C) 4 7.0 2.7 REF Calculated using the likelihood ratio computation as described in Section .", "aThe difference in the number of degrees of freedom between the two hypotheses.", "bThe alternative hypotheses are preferred over the corresponding null hypotheses by the significances $\\sigma $ .", "lrcc 4 0pc Fornax A LAT spectral fit results Model TS Energy flux ($\\times 10^{-12}$  erg cm$^{-2}$ s$^{-1}$ ) Photon index Point (best-fit location B) 121 $5.34 \\pm 0.78 \\begin{array}{c}+0.03 \\\\ -0.05\\end{array} \\ _{\\text{sys}}$ $2.08 \\pm 0.08 \\pm 0.03_{\\text{sys}}$ Disk (best-fit location B, 033 radius) 158 $7 \\pm 1$ $1.99 \\pm 0.07$ Radio lobes template 158 $7.57 \\pm 1.05 \\begin{array}{c}+0.06 \\\\ -0.08\\end{array} \\ _{\\text{sys}}$ $1.99 \\pm 0.07 \\begin{array}{c}+0.03 \\\\ -0.04\\end{array} \\ _{\\text{sys}}$ Two point sources, summed $6.6 \\pm 0.8$         West (location D) 37 $2.9 \\pm 0.7$ $2.02\\pm 0.13$         East (location E) 51 $3.7 \\pm 0.8$ $2.05\\pm 0.11$ Radio lobes template and point source (location C), summed $8 \\pm 1$         Radio lobes template 66 $6 \\pm 1$ $1.97 \\pm 0.08$         Point source (location C) 14 $1.6 \\pm 0.6$ $2.01 \\pm 0.20$ Each fit was performed leaving normalization and spectral shape parameters of all sources free, except for the models for the diffuse background $\\gamma $ rays which were fit with only normalization.", "In the following we assume the radio morphology template (without any core contribution) is the best description of the Fornax A $\\gamma $ -ray emission.", "The likelihood ratio technique cannot quantify whether the radio template is statistically preferred with respect to the best-fit disk model because these models are not nested.", "However, the radio template is the physically motivated model based on the leptonic scenario of $\\gamma $ -ray production.", "Further, the rotation study presented above indicates a preferred axis of the $\\gamma $ -ray emission which mimics the elongated emission observed at radio frequencies, in turn supporting the similarity between the $\\gamma $ -ray and radio emission morphology.", "We tested several broadband spectral models (log-parabola, broken power law, and broken power law with an exponential cutoff), and found none were significantly preferred over the single power law.", "Then, we measured spectral points by fitting each of 6 equal logarithmically spaced energy bins from 0.1 to 300 GeV to a power law and optimized the flux normalizations by maximizing the likelihood function.", "In each bin, the normalization parameters for all sources were free, and all other parameters were fixed to the values obtained from the broadband fit.", "These spectral data points are shown in Figure REF and Table REF .", "The source is detected in 4 of the 6 spectral bins with TS $>5$ , and 95% confidence upper limits were calculated for the two lower-significance bins (at the highest energies).", "Figure: Broadband SED of the lobes of Fornax A (bottom panel) and detailedview of the high-energy part of the SED (top panel).", "As in ,we used archival measurements of the total lobe radio flux densities (shown in red)from , , , , , and ,as collected by , replacing an extrapolated 100 MHz data point from with an 86 MHz measurement .The 3-year integrated WMAP data are shown in magenta ,and X-ray data from ROSAT are shownin green.", "The LAT data points from this study are shown in blue.", "The black upper limit point is from EGRET .The black solid lines from left to right show the synchrotron (<10 13 <10^{13} Hz), IC/CMB (>10 13 >10^{13} Hz), andIC/EBL flux models (>10 15 >10^{15} Hz).", "The dashed black lineshows the IC/EBL flux models assuming the fast evolution model of .The dotted red and cyan lines show the IC upscatteredhost galaxy photon flux from infrared and optical, respectively.The solid green line shows the hadronic model flux, while the solid magenta line showsthe combined hadronic and leptonic model flux.ccc 3 0pc Fornax A total lobe LAT spectral flux Bin energy range (GeV) Energy flux ($\\times 10^{-12}$  erg cm$^{-2}$ s$^{-1}$ ) TS $0.10$ – $0.38$ $0.7 \\pm 0.4$ 6.1 $0.38$ – $1.4$ $1.0 \\pm 0.2$ 49 $1.4$ – $5.5$ $1.1 \\pm 0.2$ 87 $5.5$ – 21 $0.8 \\pm 0.2$ 22 *21 – $79^*$ $<1.7$ 3.5 *79 – $300^*$ $<2.6$ 4.3 Assuming radio lobes template spatial model *95% confidence upper limits To test the $\\gamma $ -ray variability over the $6.1$ year period, we made a $0.1$ –300 GeV light curve in time bins of 185 days, which was found to be the smallest possible time scale while maintaining a reasonable significance of detection in the majority of bins.", "For each time interval, the emission associated with Fornax A was fit to a single point source (positioned at the best-fit location B in Figure REF ), as we expected any potential variable emission to be associated with a point source at the core and not the lobes.", "All sources included in the $6.1$ year analysis were fit with all spectral shape parameters fixed to their optimized values from the full fit, while all normalizations were left free.", "Upper limits were calculated for time bins within which the TS fell below 4 ($<2 \\sigma $ ).", "The significance of variability was determined following the method described in [38].", "Our analysis yielded a $1.3\\sigma $ confidence that the emission is variable, and so we conclude that we do not observe significant variability.", "Our Fermi-LAT study of the region around Fornax A consistently shows that, under all tests performed, the $\\gamma $ -ray emission region is significantly extended and the most likely spatial distribution is delineated by the radio lobes.", "Using a size-optimized flat disk model, extension beyond a point source was found to be significant at $5.9\\sigma $ confidence, with a preferred radius of $033 \\pm 005$ .", "Modeling the emission as two point sources results in a western point source well matched to the radio lobe centroid and an eastern point source offset from the eastern lobe.", "This model is preferred over a single point source at the $4.8\\sigma $ level.", "Furthermore, using the $1.5$  GHz VLA radio morphology [18] as a template in combination with a central core point source results in a significantly greater likelihood than the point source alone with $6.0\\sigma $ confidence.", "Contamination from the core is determined to be at most 14% based on a likelihood fit with the radio lobes template and a point source at the core location.", "While it is difficult to determine the exact morphology of the $\\gamma $ -ray emission, our study shows that it cannot be fully described as a point-like source.", "A few scenarios could explain the offset $\\gamma $ -ray point-source localization seen in the 3FGL and in the single point-source analysis presented here.", "Firstly, since we now know the emission is extended (or at least not point-like), to use a point-source model to localize the emission is to start with a false assumption.", "The distribution of the $\\gamma $ -ray emitting regions may not be uniform across the lobe structure and thus would not result in a symmetric distribution of $\\gamma $ -ray emission.", "Second, based upon the offset eastern lobe point-source localization (point E in Figure REF ), the existence of a background $\\gamma $ -ray source is not ruled out.", "However, adding a point source to the lobes template model and localizing with gtfindsrc yields only a slightly better fit at the $2.7\\sigma $ level (see point C in Figure REF ).", "We also find no evidence for variability in this source over $\\sim 6$ years of observations.", "Variability might support the presence of a common background source such as a blazar.", "We note that another potential source that has been investigated in this region is the Fornax cluster [7], [5] whose center lies 36 northeast of the Fornax A core, and may be contributing contaminating $\\gamma $ -ray flux from various cluster constituents.", "However, no galaxy cluster has been detected in $\\gamma $ rays so far.", "Fermi-LAT data have been previously used by [36] to study Fornax A.", "They reported a photon flux above 100 MeV of $6.7\\times 10^{-9}$  ph cm$^{-2}$  s$^{-1}$ using a point source spatial model, and our finding using the lobes template was close at $(5.7\\pm 0.9)\\times 10^{-9}$  ph cm$^{-2}$  s$^{-1}$ .", "Our study establishes for the first time spatial extension of Fornax A in $\\gamma $ rays, and distinguishes between $\\gamma $ -ray contributions from the core and lobes.", "This result was enabled thanks to the improvements brought about by the new Pass 8 event reconstruction, rather than the marginal increase in exposure time (i.e., 6 years of data in our study instead of 5 years in theirs)." ], [ "Leptonic Modeling", "We model the $\\gamma $ -ray emission following [20], in which the relativistic electrons in the lobes of the radio galaxy are IC scattered off of CMB and EBL photons.", "For a given electron energy distribution (EED) of the lobes, the resulting IC emission will consist of a lower-energy component due to CMB photons, as well as two components at higher energy due to the cosmic infrared and optical backgrounds (CIB and COB, respectively).", "While the EBL energy density is only a few percent that of the CMB, the resulting IC spectrum due to CIB and COB photons will be shifted in frequency by $\\gamma ^{2}_{\\rm max}$ , where $\\gamma _{\\rm max}$ is the maximum Lorentz factor of the EED.", "The EED used in this model is a broken power law that breaks at $\\gamma _{break}=1.3\\times 10^5$ from an electron index of $2.3$ to a much larger value to mimic a cutoff.", "This break was chosen so as to not overproduce the emission in the lowest-energy LAT band.", "With peak wavelengths of $\\lambda \\sim 100 \\ $ m and $\\sim 1 \\ $ m for the CIB and COB, the resulting IC spectrum will be shifted in frequency by factors of $\\sim 10$ and 1000, respectively from that of the upscattered CMB.", "This model is shown along with the $\\gamma $ -ray spectral energy distribution and the radio-to-sub-mm measurements of the total emission from the lobes presented in [20] in Figure REF .", "Note that because Fornax A is an extended source in other wavelengths as well, care must be taken in defining the spatial structure in all wavelengths in order to draw meaningful comparisons.", "This model makes use of currently available total lobe fluxes.", "WMAP and Planck fluxes reported by [36] were obtained using resolution-dependent apertures that did not fully enclose the extent of the synchrotron lobe emission.", "We assumed a lobe magnetic field strength of 1.65 $$ G, constrained to the X-ray flux data point, and we include photon contributions from the host galaxy following [20].", "The extracted spectral data points do not appear to match the predicted model shape based on IC/EBL emission alone.", "Fully accounting for the Fermi-LAT observed fluxes under the IC/EBL hypothesis alone would imply an EBL level that is even higher than the Stecker model [43], which was ruled out by [4].", "Consequently, the applied leptonic model cannot completely explain the observed emission.", "The model relies upon the assumption that all of the X-ray flux observed from the lobes is created by IC/CMB scattering to obtain the magnetic field strength.", "If some amount of the X-ray flux is thermal emission , our expected IC/EBL level would decrease, creating further discrepancy between model and data.", "As discussed in Section REF , a more physically motivated choice of spatial template would be the higher-frequency WMAP data.", "However, the resulting changes in flux and spectral shape should be within the statistical errors of our current results, and therefore should not alter our result that the flux exceeds the leptonic model of the Fornax A lobes.", "Note that the intensity of the IC contribution from the host galaxy photons of Fornax A is comparable to that of the EBL photons in the lobes, and it actually dominates at higher energies ($>1$  GeV, see Figure REF ).", "This differs from the case of Centaurus A, wherein the predicted EBL photon intensity is roughly five times that of the starlight [1].", "Were it true that host galaxy photons dominate in Fornax A, the expected spatial distribution of $\\gamma $ rays from the lobes would not be uniform, with brighter emission nearer the center and less away from the core.", "Testing for this feature requires spatial resolution that is beyond the capabilities of the LAT with current statistics." ], [ "Hadronic Modeling", "The problem of the model not fitting the $\\gamma $ -ray spectrum in Fornax A may be solved by an additional contribution from hadronic cosmic rays interacting within the lobes, as found by [36].", "We created a model of hadronic emission (proton-proton interactions) assuming a total emitting volume of $7 \\times 10^{70}$  cm$^3$ and a uniform distribution of thermal gas with number density $3 \\times 10^{-4}$  cm$^{-3}$ following [40], and a power-law cosmic-ray spectrum with energy index 2.3 extending from $\\sim 3$  GeV up to more than 10 TeV.", "Modeling the $\\gamma $ rays as entirely hadronic in origin requires a large total cosmic-ray energy of $\\sim 1 \\times 10^{61}$  erg, which is twice the observed energy of $\\sim 5\\times 10^{60}$  erg in the lobes of comparable radio galaxy Hydra A [2] and very high compared to an estimate of $\\sim 5\\times 10^{58}$  erg in the outburst that is assumed to have created the lobes of Fornax A [33].", "We then subtracted the lowest IC/EBL model [16] from our LAT spectral points and fit the residual flux as hadronic emission, and found we could achieve a reasonable fit, shown in Figure REF .", "The resulting cosmic-ray pressure fitted from this residual flux is $\\sim 2 \\times 10^{-11}$  dyn cm$^{-2}$ and the total energy stored in cosmic rays is $\\sim 5 \\times 10^{60}$  erg.", "This total energy is similar to that of Hydra A [2] and closer to an estimate of the total energy in the Fornax A lobes [33].", "This result agrees with analogous calculations by [36], in which the discrepancy is explained by suggesting the emission is primarily hadronic and localized to relatively denser sub-structures within the lobes, thereby decreasing the effective emitting volume." ], [ "Conclusions", "We report the first Fermi-LAT detection of extended $\\gamma $ -ray emission from the radio galaxy Fornax A using $6.1$ years of Fermi-LAT data.", "We conclude that a point-source spatial model is insufficient to describe the $\\gamma $ -ray emission, and our analysis indicates it is likely the emission originates in the lobes.", "We investigated the origin of the extended emission by assuming leptonic emission that arises due to IC scattering of EBL photons off of relativistic electrons in the radio lobes.", "This leptonic modeling underestimates the observed $\\gamma $ -ray emission for any current EBL estimate, consistent with the recent findings by [36], even after accounting for the additional contribution of IC emission off of the host galaxy light.", "A hadronic-only model (proton-proton interactions) requires implausibly large total cosmic-ray energy when compared to an estimate of the Fornax A outburst assumed to have created the lobes [33], and this problem can be alleviated by invoking denser sub-structures in the lobes [36].", "When we assume the lowest EBL model [16] and fit the residual $\\gamma $ -ray flux as hadronic production, our fit yields a total cosmic-ray energy of $\\sim 5 \\times 10^{60}$  erg, matching well with the Hydra A energy [2], but still at least 100 times greater than the estimated total energy in the Fornax A lobes [33].", "Thus, even the combined leptonic and hadronic scenario may not be able to explain the $\\gamma $ -ray lobe emission.", "Given our current understanding of the content of the radio lobes and the EBL, some contribution from leptonic processes must exist.", "If it is true that there is a $\\gamma $ -ray component other than leptonic in Fornax A, we should expect to observe such a component in other nearby radio galaxies as well.", "Our spatial analysis hinted at the existence of a background $\\gamma $ -ray source (C in Figure REF ).", "We re-evaluated the SED of Fornax A including this source in the model, but we found this background source addition is only marginally preferred and cannot fully make up the difference between the data and the IC/EBL model.", "In any case, $\\gamma $ -ray contamination from an unresolved background source or the Fornax cluster could be present.", "Our modeling was done using previously published multiwavelength data.", "Further analysis in other wavelengths in the future will yield a more definitive picture of the SED.", "More detailed observations, such as with the hard X-ray telescope NuSTAR, would help determine if the X-ray emission is contaminated by thermal processes [40], which would test our initial assumptions that the radio data traces the X-rays and $\\gamma $ rays, and that the X-rays could be used to constrain the lobe magnetic field.", "Fermi-LAT analysis at even lower energies (below 100 MeV) may provide more information about the IC/CMB component [11].", "Potentially with greater statistics, the Fermi-LAT could extend the lobes' detection to higher energies and additionally observe the effect of the host-galaxy photons on the spatial distribution of $\\gamma $ rays in the Fornax A lobes.", "The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis.", "These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.", "Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France.", "Facility: Fermi" ] ]
1606.04905
[ [ "Stellar populations in the outskirts of M31: the mid-infrared view" ], [ "Abstract The mid-infrared provides a unique view of galaxy stellar populations, sensitive to both the integrated light of old, low-mass stars and to individual dusty mass-losing stars.", "We present results from an extended Spitzer/IRAC survey of M31 with total lengths of 6.6 and 4.4 degrees along the major and minor axes, respectively.", "The integrated surface brightness profile proves to be surprisingly diffcult to trace in the outskirts of the galaxy, but we can also investigate the disk/halo transition via a star count profile, with careful correction for foreground and background contamination.", "Our point-source catalog allows us to report on mid-infrared properties of individual objects in the outskirts of M31, via cross-correlation with PAndAS, WISE, and other catalogs." ], [ "Introduction", "Getting the full picture of the stellar populations in the nearest large galaxy, M31, poses challenges because of its large angular extent on the sky.", "The PAndAS project [7] covers a huge area around M31 in visible light, but observations at other wavelengths are needed to tell the whole story.", "As seen by the S$^4$ G survey, the mid-infrared has the advantages of tracing the integrated light of low-mass stars while being relatively unaffected by interstellar extinction [8].", "Mid-infrared point-source observations can be used to identify mass-losing stars with circumstellar dust [4] and red supergiants [3].", "Here we present preliminary results on the stellar populations observed in a wide-field mid-infrared survey of M31." ], [ "Observations", "With about 40 hours of Spitzer observations in the post-cryogenic mission, we extended the IRAC 3.6 and 4.5 $\\mu $ m mapping of M31 presented by [1] to larger distances along the major and minor axes of the galaxy (Fig.", "REF ).", "[9] describes the observations, the challenging process of background subtraction, and construction of a point-source catalog containing over $4\\times 10^5$ individual sources.", "The catalog reaches (Vega) magnitude of $\\approx 19$ at both 3.6 and 4.5 $\\mu $ m, about 1 magnitude fainter than the tip of the red giant branch (TRGB) luminosity at the distance of M31.", "Using wide-field, shallow surveys to model Galactic foreground and narrow-field, deep surveys to model unresolved background galaxies, we developed a scheme to estimate the probability that a given mid-infrared source belonged to M31 from its magnitude and colour.", "Probable M31 AGB stars are found at $[4.5]\\lesssim 15$ and $[3.6]-[4.5] \\ge 0.4$ .", "Figure: Sky positions of point sources in the outskirts of M31 matched to SIMBAD sources with a tolerance of 1 arcsec.", "North is up and East to the left.Blank regions along the major (NE-SW) and minor (NW-SE) axes are locations where crowding was too severe for accurate point-source extraction.The concentration of galaxies in the northeast part of the disc is due to identification in a study of M31 star clusters by the PHAT project(Johnson 2012)while the carbon stars to the northwest were found in a study of the satellite galaxy NGC 205 by Demers, Battinelli & Letarte (2003)." ], [ "Analysis: mid-infrared point sources in M31", "Our extensive point-source catalog allowed for detailed comparisons between the stellar populations as seen in the IRAC 3.6 and 4.5 $\\mu $ m bands and those observed at other wavelengths.", "Comparing with 2MASS and WISE [9], we find that sources detected only by IRAC and WISE detections are redder and fainter than those detected by all three of IRAC, WISE, and 2MASS.", "This is consistent with the greater depth and redder response of WISE compared to 2MASS.", "Comparing with PAndAS [9], we find that IRAC+PAndAS detections are optically-redder and more likely to be Galactic foreground dwarfs compared to the PAndAS-only detections; however, the IRAC observations do detect some probable M31 red giants as expected from the detection limit below the TRGB.", "Both the SIMBAD and NED databases include individual components of nearby galaxies (e.g., stars, star clusters, planetary nebulae, Hii regions), although these are not the main focus of either database.", "We matched the IRAC sources against SIMBAD objects with a distance tolerance of 1 arcsec, with the results shown in Figures 1 and 2.", "Figure 1 shows that, as expected, the sources categorized as stars or star-related (PNe, Hii regions) are strongly concentrated toward the disk of the galaxy, while background galaxies and active galactic nuclei are more uniformly distributed.", "Globular clusters fall in between these two distributions and are easily detectible in this dataset.", "Figure: Spitzer/IRAC colour-magnitude diagram showing point sources in the outskirts of M31 matched to SIMBADsources with a tolerance of 1 arcsec.", "All magnitudes are in the Vega system; the red colours of objects brighter than[4.5]≈11[4.5]\\approx 11 are due to saturation in the 3.6 μ\\mu m band.", "Some of the SIMBAD sub-classes (e.g.", "of variable stars)have been combined.Figure 2 shows the colour-magnitude diagram of matched sources.", "As expected from deep-field extragalactic surveys, the active galactic nuclei and quasars have very red $[3.6]-[4.5]$ colours; background galaxies are mostly found at $[3.6]-[4.5]>0.1$ .", "Globular clusters have Vega-magnitude colours $[3.6]-[4.5]\\approx 0$ ; see also [2].", "Individual stars in M31 show a wide range of mid-infrared colours; interestingly, (optically) red supergiants are mostly found at $[3.6]-[4.5]<0$ while the handful of blue supergiants have $[3.6]-[4.5]>0.2$ , so `red' and `blue' are reversed in the mid-infrared.", "[3] found similar results for supergiants in Local Group dwarf irregulars.", "Combining mid-infrared surveys with those from other wavelengths should provide a rich variety of targets for spectroscopic follow-up with the James Webb Space Telescope." ] ]
1606.05215
[ [ "Learning Optimal Interventions" ], [ "Abstract Our goal is to identify beneficial interventions from observational data.", "We consider interventions that are narrowly focused (impacting few covariates) and may be tailored to each individual or globally enacted over a population.", "For applications where harmful intervention is drastically worse than proposing no change, we propose a conservative definition of the optimal intervention.", "Assuming the underlying relationship remains invariant under intervention, we develop efficient algorithms to identify the optimal intervention policy from limited data and provide theoretical guarantees for our approach in a Gaussian Process setting.", "Although our methods assume covariates can be precisely adjusted, they remain capable of improving outcomes in misspecified settings where interventions incur unintentional downstream effects.", "Empirically, our approach identifies good interventions in two practical applications: gene perturbation and writing improvement." ], [ "Introduction", "In many data-driven applications, including medicine, the primary interest is identifying interventions that produce a desired change in some associated outcome.", "Due to experimental limitations, learning in such domains is commonly restricted to an observational dataset $\\mathcal {D}_n := \\left\\lbrace \\left(x^{(i)}, y^{(i)}\\right) \\right\\rbrace _{i=1}^n$ which consists of IID samples from a population with joint distribution $\\mathbb {P}_{XY}$ over covariates (features) $X \\in \\mathbb {R}^d$ and outcomes $Y \\in \\mathbb {R}$ .", "Typically, such data is analyzed using models which facilitate understanding of the relations between variables (eg.", "assuming linearity/additivity).", "Based on conclusions drawn from this analysis, the analyst decides how to intervene in a manner they confidently believe will improve outcomes.", "Formalizing such beliefs via Bayesian inference, we develop a framework that identifies beneficial interventions directly from the data.", "In our setup, an intervention on an individual with pre-treatment covariates $X$ produces post-treatment covariate values $\\widetilde{X}$ that determine the resulting outcome $Y$ (depicted as the graphical model: $X \\rightarrow \\widetilde{X} \\rightarrow Y)$ .", "Each possible intervention results in a diffferent $\\widetilde{X}$ .", "More concretely, we make the following simplifying assumption: $Y = f(\\widetilde{X}) + \\varepsilon \\ \\text{ with } \\mathbb {E}[\\varepsilon ] = 0, \\varepsilon \\protect \\mathchoice{\\protect \\mathrel {\\unknown.", "{\\displaystyle \\perp }\\hspace{1.111pt}{\\displaystyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\textstyle \\perp }\\hspace{1.111pt}{\\textstyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\scriptstyle \\perp }\\hspace{1.111pt}{\\scriptstyle \\perp }}}{\\protect \\mathrel {\\unknown.", "{\\scriptscriptstyle \\perp }\\hspace{1.111pt}{\\scriptscriptstyle \\perp }}}\\widetilde{X}, X$ for some underlying function $f$ that encodes the effects of causal mechanisms (ie.", "$\\widetilde{X}$ represents a fair description of the system state, and some covariates in $\\widetilde{X}$ causally affect $Y$ , not vice-versa).", "The observed data is comprised of naturally occurring covariate values where we presume $\\widetilde{x}^{(i)} = x^{(i)}$ for $i=1,\\dots ,n$ (ie.", "the system state remains static without intervention, so the observed covariate values directly influence the observed outcomes).", "Moreover, we assume the relationship between these covariate values and the outcomes remains invariant, following the same (unknown) function $f$ for any $\\widetilde{X}$ arising from one of our feasible interventions (or no intervention at all).", "Note that this assumption precludes the presence of hidden confounding.", "have also relied on this invariance assumption, verifying it as a reasonable property of causal mechanisms in nature.", "Given this data, we aim to learn an intervention policy defined by a covariate transformation $T : \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ , applied to each individual in the population.", "Here, $T(x)$ presents a desired setting of the covariates that should be reflected by subsequent intervention to actually influence outcomes.", "When $T$ only specifies changes to a subset of the covariates, an intervention seeking to realize $T$ may have unintended side-effects on covariates outside of this subset.", "We ignore such “fat hand” settings until §.", "Instead, our methods assume interventions can always be carried out with great precision to ensure the desired transformation $T$ is exactly reflected in the post-treatment values: $\\widetilde{x} = T(x)$ .", "Our goal is to identify the transformation $T$ which produces the largest corresponding post-treatment improvement with high certainty.", "$T(x)$ can either represent a single mapping to be performed on all individuals (global policy) or encode a personalized policy where the intervened upon variables and their values may change with $x$ .", "Our strong assumptions are made to ensure that statistical modeling alone suffices to identify beneficial interventions.", "While many real-world tasks violate these conditions, there exist important domains in which violations are sufficiently minor that our methods can discover effective interventions (cf.", ", ).", "We use two applications to illustrate our framework.", "One is a writing improvement task where the data consists of documents labeled with associated outcomes (eg.", "grades or popularity) and the goal is to suggest beneficial changes to the author.", "Our second example is a gene perturbation task where the expression of some regulatory genes can be up/down-regulated in a population (eg.", "cells or bacteria) with the goal of inducing a particular phenotype or activation/repression of a downstream gene.", "In these examples, covariates are known to cause outcomes and our other assumptions may hold to some degree, depending on the type of external intervention used to alter covariate values.", "The contributions of this work include: (1) a formal definition of the optimal intervention that exhibits desirable characteristics under uncertainty due to limited data, (2) widely applicable types of (sparse) intervention policy that are easily enacted across a whole population, (3) algorithms to find the optimal intervention under practical constraints, (4) theoretical insight regarding our methods' properties in Gaussian Process settings as well as certain misspecified applications." ], [ "Related Work", "The same invariance assumption has been exploited by and for causal variable selection in regression models.", "Recently, researchers such as and have supported a greater role for predictive modeling in various decision-making settings.", "use gradient boosting to predict glycemic response based on diet (and personal/microbiome covariates), and found they can naively leverage their regressor to select personalized diets which result in superior glucose levels than the meals proposed by a clinical dietitian.", "As treatment-selection in high-impact applications (eg.", "healthcare) grows increasingly reliant on supervised learning methods, it is imperative to properly handle uncertainty.", "Nonlinear Bayesian predictive models have been employed by , , and for quantifying the effects of a given treatment from observations of individuals who have been treated and those who have not.", "Rather than considering a single given intervention, we introduce the notion of an optimal intervention under various practical constraints, and how to identify such a policy from a limited dataset (in which no individuals have necessarily received any interventions).", "Although our goals appear similar to Bayesian optimization and bandit problems , , additional data is not collected in our setup.", "Since we consider settings where interventions are proposed based on all available data, acquisition functions for sequential exploration of the response-surface are not appropriate.", "As most existing data is not generated through sequential experimentation, our methods are more broadly applicable than iterative approaches like Bayesian optimization.", "A greater distinction is our work's focus on the pre vs. post-intervention change in outcome for each particular individual, whereas Bayesian optimization seeks a single globally optimal configuration of covariates.", "In practice, feasible covariate transformations are constrained based on an individual's naturally occurring covariate-values, which stem from some underlying population beyond our control.", "For example in the writing improvement task, the goal is not to identify a globally optimal configuration of covariates that all texts should strive to achieve, but rather to inform a particular author of simple modifications likely to improve the outcome of his/her existing article.", "Appropriately treating such constraints is particularly important when we wish to prescribe a global policy corresponding to a single intervention applied to all individuals from the population (there is no notion of an underlying population in Bayesian optimization)." ], [ "Methods", "Our strategy is to first fit a Bayesian model for $Y \\mid X$ whose posterior encodes our beliefs about the underlying function $f$ given the observed data.", "Subsequently, the posterior for $f \\mid \\mathcal {D}_n$ is used to identify a transformation of the covariates $T : \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ which is likely to improve expected post-intervention outcomes according to our current beliefs.", "The posterior for $f \\mid \\mathcal {D}_n$ may be summarized at any points $x, x^{\\prime } \\in \\mathbb {R}^d$ by mean function $\\mathbb {E}[f(x) \\mid \\mathcal {D}_n]$ and covariance function $\\text{Cov}(f(x) , f(x^{\\prime }) \\mid \\mathcal {D}_n)$ ." ], [ "Intervening at the Individual Level", "For $x \\in \\mathbb {R}^d$ representing the covariate-measurements from an individual, we are given a set $\\mathcal {C}_x \\subset \\mathbb {R}^d$ that denotes constraints of possible transformations of $x$ .", "Let $T(x) = \\widetilde{x} \\in \\mathcal {C}_x$ denote the new covariate-measurements of this individual after a particular intervention on $x$ which alters covariates as specified by transformation $T: \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ .", "Recall that we assume an intervention can be conducted to produce post-treatment covariate-values that exactly match any feasible transformation: $\\widetilde{x} = T(x)$ , and we thus write $f(T(x))$ in place of $\\mathbb {E}_{\\varepsilon }[Y \\mid \\widetilde{X} = T(x)]$ .", "We first consider personalized interventions in which $T$ may be tailored to a particular $x$ .", "Under the Bayesian perspective, $f \\mid \\mathcal {D}_n$ is randomly distributed according to our posterior beliefs, and we define the individual expected gain function: $ G_x(T) := f(T(x)) - f(x) \\mid \\mathcal {D}_n$ Since $f(x) = \\mathbb {E}_\\varepsilon [Y \\mid \\widetilde{X} = x]$ , random function $G_x$ evaluates the expected outcome-difference at the post vs. pre-intervention setting of the covariates (this expectation is over the noise $\\varepsilon $ , not our posterior).", "To infer the best personalized intervention (assuming higher outcomes are desired), we use optimization over vectors $T(x) \\in \\mathbb {R}^d$ to find: $T^*(x) = \\operatornamewithlimits{argmax}_{T(x) \\in \\mathcal {C}_x} \\ F^{-1}_{G_x(T)}(\\alpha )$ where $F^{-1}_{G(\\cdot )}(\\alpha )$ denotes the $\\alpha ^{\\text{th}}$ quantile of our posterior distribution over $G(\\cdot )$ .", "We choose $0 < \\alpha < 0.5$ , which implies the intervention that produces $T^*(x)$ should improve the expected outcome with probability $\\ge 1-\\alpha $ under our posterior beliefs.", "Defined based on known constraints of feasible interventions, the set $\\mathcal {C}_x \\subset \\mathbb {R}^d$ enumerates possible transformations that can be applied to an individual with covariate values $x$ .", "If the set of possible interventions is independent of $x$ (ie.", "$\\mathcal {C}_x = \\mathcal {C} \\ \\forall x$ ), then our goal is similar to the optimal covariate-configuration problem studied in Bayesian optimization.", "However, in many practical applications, $x$ -independent transformations are not realizable through intervention.", "Consider gene perturbation, a scenario where it is impractical to simultaneously target more than a few genes due to technological limitations.", "If alternatively intervening on a quantity like caloric intake, it is only realistic to change an individual's current value by at most a small amount.", "The choice $\\mathcal {C}_x := \\lbrace z \\in \\mathbb {R}^d : ||x - z||_0 \\le k \\rbrace $ reflects the constraint that at most $k$ covariates can be intervened upon.", "We can denote limits on the amount that the $s^{\\text{th}}$ covariate may be altered by $\\mathcal {C}_x := \\lbrace z \\in \\mathbb {R}^d : |x_s - z_s| \\le \\gamma _s \\rbrace $ for $s \\in \\lbrace 1,\\dots ,d\\rbrace $ .", "In realistic settings, $\\mathcal {C}_x$ may be the intersection of many such sets reflecting other possible constraints such as boundedness, impossible joint configurations of multiple covariates, etc.", "For any $x, T(x) \\in \\mathbb {R}^d$ : the posterior distribution for $G_x(T)$ has: $ & \\text{mean} = \\mathbb {E}[f(T(x) \\mid \\mathcal {D}_n ] - \\mathbb {E}[f(x) \\mid \\mathcal {D}_n \\\\& \\text{variance} = \\text{Var}(f(T(x)) \\mid \\mathcal {D}_n) + \\text{Var}(f(x) \\mid \\mathcal {D}_n) \\\\& \\hspace*{51.21495pt} - 2\\text{Cov}(f(T(x)), f(x) \\mid \\mathcal {D}_n) $ which is easily computed using the corresponding mean/covariance functions of the posterior $f \\mid \\mathcal {D}_n$ .", "When $T(x) = x$ , the objective in (REF ) takes value 0, so any superior optimum corresponds to an intervention we are confident will lead to expected improvement.", "If there is no good intervention in $\\mathcal {C}_x$ (corresponding to a large increase in the posterior mean) or too much uncertainty about $f(x)$ given limited data, then our method simply returns $T^*(x) = x$ indicating no intervention should be performed.", "Our objective exhibits these desirable characteristics because it relies on the posterior beliefs regarding both $f(T(x))$ and $f(x)$ , which are tied via the covariance function.", "In contrast, a similarly-conservative lower confidence bound objective (ie.", "the UCB acquisition function with lower rather than upper quantiles) would only consider $f(T(x))$ , and could propose unsatisfactory transformations where ${{\\mathbb {E}[f(x) \\mid \\mathcal {D}_n ]} > \\mathbb {E}[f(T(x)) \\mid \\mathcal {D}_n ]}$ ." ], [ "Intervening on Entire Populations", "The above discussion focused on personalized interventions tailored on an individual basis.", "In certain applications, policy-makers are interested in designing a single intervention which will be applied to all individuals from the same underlying population as the data.", "Relying on such a global policy is the only option in cases where we no longer observe covariate-measurements of new individuals outside the data.", "In our gene perturbation example, gene expression may no longer be individually profiled in future specimens that receive the decided-upon intervention to save costs/labor.", "Here, the covariates $X$ are assumed distributed according to some underlying (pre-intervention) population, and we define the population expected gain function: $ G_X(T) := \\mathbb {E}_X [ G_x(T) ] = \\mathbb {E}_X \\big [ f(T(x)) - f(x) \\mid \\mathcal {D}_n \\big ]$ which is also randomly distributed based on our posterior ($\\mathbb {E}_X$ is expectation with respect to the covariate-distribution $X$ which is not modeled by $f \\mid \\mathcal {D}_n$ ).", "Our goal is now to find a single transformation $T : \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ corresponding to a population intervention which will (with high certainty under our posterior beliefs) lead to large outcome improvements on average across the population: $T^* = \\operatornamewithlimits{argmax}_{T \\in \\mathcal {T}} \\ F^{-1}_{G_X(T)}(\\alpha )$ Here, the family of possible transformations $\\mathcal {T}$ is constrained such that $T(x) \\in \\mathcal {C}_x$ for all $T \\in \\mathcal {T}, x \\in \\mathbb {R}^d$ .", "As a good model of our multivariate features may be unknown, we instead work with the empirical estimate: $&& T^* & = \\operatornamewithlimits{argmax}_{T \\in \\mathcal {T}} \\ F^{-1}_{G_n(T)}(\\alpha ) && \\\\\\text{where } && G_n(T) & := \\frac{1}{n} \\hspace*{-0.28453pt} \\sum _{i=1}^n \\big [ f(T(x^{(i)})) - f(x^{(i)}) \\big ] \\ \\mid \\mathcal {D}_n &&$ is the empirical population expected gain, whose posterior distribution has: $& \\hspace*{-5.69054pt} \\text{mean} = \\frac{1}{n} \\hspace*{-1.42262pt} \\sum _{i=1}^n \\mathbb {E}[f(T(x^{(i)})) \\mid \\mathcal {D}_n] - \\mathbb {E}[f(x^{(i)}) \\mid \\mathcal {D}_n] \\\\& \\hspace*{-5.69054pt} \\text{variance} = \\frac{1}{n^2} \\sum _{i=1}^n \\sum _{j=1}^n \\Big [ \\text{Cov}\\left( f(x^{(i)}), f(x^{(j)}) \\mid \\mathcal {D}_n \\right) \\\\& \\hspace*{44.10176pt} - \\text{Cov}\\big ( f(T(x^{(i)})), f(x^{(j)}) \\mid \\mathcal {D}_n \\big ) \\\\& \\hspace*{44.10176pt} - \\text{Cov}\\big ( f(x^{(i)}), f(T(x^{(j)})) \\mid \\mathcal {D}_n \\big ) \\\\[-5pt]& \\hspace*{44.10176pt} + \\text{Cov}\\big (f(T(x^{(i)})), f(T(x^{(j)})) \\mid \\mathcal {D}_n \\big ) \\Big ] $ The population intervention objective in (REF ) is again 0 for the identity mapping $T(x) = x$ .", "Under excessive uncertainty or a dearth of beneficial transformations in $\\mathcal {T}$ , the policy produced by this method will again simply be to perform no intervention.", "In this population intervention setting, $T$ is designed assuming future individuals will stem from the same underlying distribution as the samples in $\\mathcal {D}_n$ .", "Although $T$ is a function of $x$ , the form of the transformation must be agnostic to the specific values of $x$ (so the intervention can be applied to new individuals without measuring their covariates).", "We consider two types of transformations that we find widely applicable.", "Shift interventions involve transformations of the form: $T(x) = x + \\Delta $ where $\\Delta \\in \\mathbb {R}^d$ represents a (sparse) shift that the policy applies to each individuals' covariates (eg.", "always adding 3 to the value of the second covariate corresponds to $T(x) = [x_1, x_2 + 3, \\dots , x_d]$ ).", "Covariate-fixing interventions are policies which set certain covariates to a constant value for all individuals, and involve transformations $T_{\\mathcal {I}\\rightarrow z}(x) = [z_1, \\dots , z_d]$ such that for some covariate-subset $\\mathcal {I} \\subseteq \\lbrace 1,\\dots ,d\\rbrace : z_j = x_j \\ \\forall j \\notin \\mathcal {I}$ and for $j \\in \\mathcal {I}$ : $z_j \\in \\mathbb {R}$ is fixed across all $x$ (eg.", "always setting the first covariate to 0, for example in gene knockout, corresponds to $T(x) = [0, x_2, \\dots , x_d]$ $\\forall x$ ).", "Figure REF depicts examples of these different interventions.", "Under a sparsity constraint, we must carefully model the underlying population in order to identify the best covariate-fixing intervention (here, setting $X_1$ to a large value is superior to intervening on $X_2$ ).", "Figure: Contour plot of expected outcomes over feature space [X 1 ,X 2 ] [X_1, X_2] for relationship Y=X 1 ·X 2 +ε{Y = X_1 \\cdot X_2 +~\\varepsilon }.", "Black points: the underlying population.", "Gold diamond: optimal covariate-setting if any transformation in the box were feasible.", "Red points: same population after shift intervention Δ=[-3,0]\\Delta = [-3, 0].", "Light (or dark) green points (along border): best covariate-fixing intervention which can only set X 2 X_2 (or only X 1 X_1) to a fixed value.", "Blue, purple, light blue points: individuals who receive a single-variable personalized intervention (arrows indicate direction of optimal transformation)." ], [ "Algorithms", "Throughout this work, we use Gaussian Process (GP) regression to model $Y \\mid X$ as described in § (`S' indicates references in the Supplementary Material).", "This nonparametric method has been favored in many applications as it produces both accurate predictions and effective measures of uncertainty (with closed-form estimators available in the standard case).", "Furthermore, a variety of GP models exist for different settings including: non-Gaussian response variables , non-stationary relationships , deep representations , measurement error , and heteroscedastic noise .", "While these variants are not employed in this work, our methodology can be directly used in conjunction with such extensions (or more generally, any model which produces a useful posterior for $f \\mid \\mathcal {D}_n$ ).", "Under the standard GP model, $G_x(T)$ follows a Gaussian distribution and the $\\alpha ^{\\text{th}}$ quantile of our personalized gain is simply given by: $F^{-1}_{G_x(T)} = & \\mathbb {E}[ G_x(T) ] + \\Phi ^{-1}(\\alpha ) \\cdot \\text{Var} [ G_x(T)] $ where $\\Phi ^{-1}$ denotes the $N(0,1)$ quantile function.", "The quantiles of the empirical population gain may be similarly obtained.", "When a smooth smooth covariance kernel $k(\\cdot , \\cdot )$ is adopted in the GP prior, derivatives of our intervention-objectives are easily computed with respect to $T$ .", "In many practical settings, an intervention that only affects a small subset of variables is desired.", "Software to improve text, for example, should not overwhelm authors with a multitude of desired changes, but rather present a concise list of the most beneficial revisions in order to retain underlying semantics.", "Note that identifying a sparse transformation of the covariates is different from feature selection in supervised learning (where the goal is to identify dimensions along which $f$ varies most).", "In contrast, we seek the dimensions $\\mathcal {I} \\subset \\lbrace 1,\\dots ,d \\rbrace $ along which one of our feasible covariate-transformations can produce the largest high-probability increase in $f$ , assuming the other covariates remain fixed at their initial pre-treatment values (in the case of personalized intervention) or follow the same distribution as the pre-intervention population (in the case of a global policy).", "For a shift intervention $T(x) = x+ \\Delta $ , we introduce the convenient notation $G_n(\\Delta ) := G_n(T)$ .", "In applications where shifting $x_s$ (the $s^{\\text{th}}$ covariate for $s \\in \\lbrace 1,\\dots , d\\rbrace $ ) by one unit incurs cost $\\gamma _s$ , we account for these costs by considering the following regularized intervention-objective: $ J_\\lambda (\\Delta ) := F^{-1}_{G_n(\\Delta )}(\\alpha ) - \\lambda \\sum _{s=1}^d \\gamma _s | \\Delta _s|$ By maximizing this objective over feasible set $\\mathcal {C}_{\\Delta } := \\lbrace \\Delta \\in \\mathbb {R}^d : x + \\Delta \\in \\mathcal {C}_{x} \\text{ for all } x \\in \\mathbb {R}^d \\rbrace $ , policy-makers can decide which variables to intervene upon (and how much to shift them), depending on the relative value of outcome-improvements (specified by $\\lambda $ ).", "This optimization is performed using the proximal gradient method , where at each iterate: a step in the gradient direction is followed by a soft-thresholding operation as well as a projection back onto the feasible set $\\mathcal {C}_{\\Delta }$ .", "However, a simple gradient method may suffer from local optima.", "To avoid severely suboptimal solutions, we develop a continuation technique that performs a series of gradient-based optimizations over variants of this objective with tapering levels of added smoothness (details in §).", "In some settings, one may want to ensure at most $k < d$ covariates are intervened upon.", "We identify the optimal $k$ -sparse shift intervention via the Sparse Shift Algorithm below, which relies on $\\ell _1$ -relaxation and the regularization path of our penalized objective in (REF ).", "[leftmargin=0cm,rightmargin=0cm,skipbelow=0cm,skipabove=0cm, innertopmargin=3pt, innerbottommargin=3pt, innerrightmargin=0.8pt, innerleftmargin=0.8pt] Sparse Shift Algorithm: Finds best $k$ -sparse shift intervention.", "Set $\\gamma _s = 1$ for $s = 1,\\dots , d$ Perform binary search over $\\lambda $ to find: $\\lambda ^* \\leftarrow \\operatornamewithlimits{argmin}\\Big \\lbrace \\lambda \\ge 0 & \\text{ s.t.}", "\\ \\Delta ^* := \\operatornamewithlimits{argmax}_{\\Delta \\in \\mathcal {C}_{\\Delta }} J_\\lambda (\\Delta ) \\\\[-0.75em]& \\text{ has $\\le k$ nonzero entries} \\Big \\rbrace $ Define $\\displaystyle \\mathcal {I} \\leftarrow \\text{support}(\\Delta ^*_{\\lambda ^*}) \\subseteq \\lbrace 1,\\dots , d\\rbrace $ where $\\displaystyle \\Delta ^*_{\\lambda ^*} := \\operatornamewithlimits{argmax}_{\\Delta \\in \\mathcal {C}_{\\Delta }} J_{\\lambda ^*}(\\Delta )$ Return: $\\displaystyle \\Delta ^* \\in \\mathbb {R}^d \\leftarrow \\operatornamewithlimits{argmax}_{\\Delta \\in B} J_{\\hspace*{0.28453pt} 0}(\\Delta )$ where $B := \\mathcal {C}_\\Delta \\bigcap \\big \\lbrace \\Delta \\in \\mathbb {R}^d : \\Delta _s = 0 \\text{ if } s \\notin \\mathcal {I} \\big \\rbrace $ Recall that in the case of personalized intervention, we simply optimize over vectors $T(x) \\in \\mathcal {C}_x$ .", "Any personalized transformation can therefore be equivalently expressed as a shift in terms of $\\Delta _x \\in \\mathbb {R}^d$ such that $T(x) = x + \\Delta _x$ .", "After substituting the individual gain $G_x(\\Delta _x)$ in place of the population gain $G_n(\\Delta )$ within our definition of $J_\\lambda $ in (REF ), we can thus employ the same algorithms to identify sparse/cost-sensitive personalized interventions.", "To find a covariate-fixing intervention which sets $k$ of the covariates to particular fixed constants across all individuals from the population, we instead employ a forward step-wise selection algorithm (detailed in §REF ), as the form of the optimization is not amenable to $\\ell _1$ -relaxation in this case." ], [ "Theoretical Results", "Consider the following basic conditions: (A1) all data lies in $\\mathcal {C} := [0,1]^d$ , (A2) ${0 < \\alpha \\le 0.5}$ .", "Throughout this section, we assume (A), (A), and the conditions laid out in § hold.", "For clarity, we rewrite the true underlying relationship as $f^*$ , letting $f$ now denote arbitrary functions.", "Our results are with respect to the true improvement of an intervention ${G^*_x(T) := f^*(T(x)) - f^*(x)}$ , $G^*_X(T) := \\mathbb {E}_X [G^*_x(T)]$ (note that $G^*_x, G^*_X$ are no longer random).", "Our theory relies on Gaussian Process results derived by , , and we relegate proofs and technical definitions to §.", "Theorem 1 Suppose we adopt a GP$\\big (0, k(x,x^{\\prime })\\big )$ prior and the following conditions hold: (A3) noise variables $\\varepsilon ^{(i)} \\overset{iid}{\\sim } N(0, \\sigma ^2)$ (A4) there exist $\\rho > 0$ such that the Hölder space $C^\\rho [0,1]^d$ has probability one under our prior (see ).", "(A5) $f^*$ and any $f$ supported by the prior are Lipschitz continuous over $\\mathcal {C}$ with constant $L$ (A6) the density of our input covariates $p_X \\in [a,b]$ is bounded above and below over domain $\\mathcal {C}$ .", "Then, for all $x, T(x) \\in \\mathcal {C}$ : $ \\mathbb {E}_{\\mathcal {D}_n} \\Big | F^{-1}_{G_x(T)}(\\alpha ) - G^*_x(T) \\Big | \\le \\frac{C}{\\alpha } \\Big (L + \\frac{1}{a} \\Big ) \\cdot \\Psi _{\\hspace*{-1.13809pt}f^*}\\hspace*{-0.56905pt}(n)^{1 / [2(d+1)]}$ where constant $C$ depends on the prior and density $p_X$ and we define: $ \\Psi _{f}(n) := {\\left\\lbrace \\begin{array}{ll}\\big [\\psi _{f}^{-1}(n)\\big ]^2 & \\hspace*{-22.76219pt} \\text{ if } \\psi _{f}^{-1}(n) \\le n^{-d/(4\\rho + 2d)} \\\\n \\cdot [\\psi _{f^*}^{-1}(n)]^{(4\\rho + 4d)/d} & \\text{ otherwise } \\\\\\end{array}\\right.", "}$ $\\psi _{f^*}^{-1}(n)$ is the (generalized) inverse of $\\psi _{f^*}(\\epsilon ) := \\frac{\\phi _{f^*}(\\epsilon )}{\\epsilon ^2}$ which depends on the concentration function $\\displaystyle \\phi _{f^*}(\\epsilon ) = \\inf _{h \\in \\mathcal {H}_k : || h - f^*||_\\infty < \\epsilon } ||h||^2_k - \\log \\Pi \\big ( f: ||f ||_\\infty < \\epsilon \\big )$ .", "$\\phi _{f^*}$ measures how well the RKHS of our GP prior $\\mathcal {H}_k$ approximates $f^*$ (see for more details).", "The expectation $ \\mathbb {E}_{\\mathcal {D}_n}$ is over the distribution of the data $\\lbrace (X^{(i)}, Y^{(i)})\\rbrace _{i=1}^n$ .", "Importantly, Theorem REF does not assume anything about the true relationship $f^*$ , and the bound depends on the distance between $f^*$ and our prior.", "When $f^*$ is a $\\rho $ -smooth function, a typical bound is given by $\\psi _{f^*}^{-1}(n) = \\mathcal {O}( n^{- \\min \\lbrace \\nu , \\rho \\rbrace /(2\\nu + d)})$ if $k$ is the Matérn kernel with smoothness parameter $\\nu $ .", "When $k$ is the squared exponential kernel and $f^*$ is $\\beta $ -regular (in Sobolev sense), $\\psi _{f^*}^{-1}(n) = \\mathcal {O}((1 / \\log n)^{\\beta /2 - d/4})$ .", "Theorem 2 Under the assumptions of Theorem REF , for any $T$ such that $\\Pr (T(X) \\in \\mathcal {C}) = 1$ : $& \\mathbb {E}_{\\mathcal {D}_n} \\Big | F^{-1}_{G_n(T)}(\\alpha ) - G_X^*(T) \\Big | \\\\& \\le \\frac{C}{\\alpha } \\Big [L\\sqrt{ \\frac{d}{n} } + \\Big (L + \\frac{1}{a} \\Big ) \\Psi _{f^*}(n)^{\\frac{1}{2(d+1)}} \\Big ]$ Theorems REF and REF characterize the rate at which our personalized/population-intervention objectives are expected to converge to the true improvement (due to contraction of the posterior as $n$ grows).", "Since these results hold for all $T$ , this implies the maximizer of our intervention-objectives will converge to the true optimal transformation as $n \\rightarrow \\infty $ (under a reasonable prior).", "Complementing these results, Theorem REF in § ensures that for any $n$ : optimizing our personalized intervention objective corresponds to improving a lower bound on the true improvement with high probability, when $\\alpha $ is small and $f^*$ belongs to the RKHS of our prior.", "In this case, the optimal transformation inferred by our approach only worsens the actual expected outcome with low probability." ], [ "Results", "§ contains an analysis of our approach on simulated data from simple covariate-outcome relationships.", "The average improvement produced by our chosen interventions rapidly converges to the best possible value with increasing $n$ .", "In these experiments, sparse-interventions consistently alter the correct feature subset, and proposed transformations under our conservative $\\alpha = 0.05$ criterion are much more rarely harmful than those suggested by optimizing the posterior mean function (which ignores uncertainty)." ], [ "Gene Perturbation", "Next, we applied our method to search for population interventions in observational yeast gene expression data from .", "We evaluated the effects of proposed interventions (restricted to single gene knockouts) over a set $X$ of 10 transcription factors ($n=161$ ) with the goal of down-regulating each of a set of 16 small molecule metabolism target genes, $Y$ .", "Results for all methods are compared to the actual expression change of the target gene found experimentally under individual knockouts of each transcription factor in $X$ .", "Compared to marginal linear regressions and multivariate linear regression, our method's uncertainty prevents it from proposing harmful interventions, and the interventions it proposes are optimal or near optimal (Figure REF ).", "Insets (a) and (b) in Figure REF show empirical marginal distributions between target gene TSL1 and members of $X$ identified for knockout by our method (CIN5) and marginal regression (GAT2).", "From the linear perspective, these relationships are fairly indistinguishable, but only CIN5 displays a strong inhibitory effect in the knockout experiments.", "Inset (c) shows the empirical marginal for a harmful intervention proposed by multivariate regression for down-regulating GPH1, where the overall correlation is significantly positive, but the few lowest expression values (which influence our GP intervention objective the most) do not provide strong evidence of a large knockdown effect.", "Figure: Actual effects of proposed interventions (single gene knockout) over a set transcription factors on down-regulation of each of a set of 16 small molecule metabolism target genes." ], [ "Writing Improvement", "Finally, we apply our personalized intervention methodology to the task of transforming a given news article into one which will be more widely-shared on social media.", "We use a dataset from containing various features about individual Mashable articles along with their subsequent popularity in social networks (detailed description/results for this analysis in §).", "We train a GP regressor on 5,000 articles labeled with popularity-annotations and evaluate sparse interventions on a held-out set of 300 articles based on changes they induce in article benchmark popularity (defined in §).", "When $\\alpha = 0.05$ , the average benchmark popularity increase produced by our personalized intervention methodology is 0.59, whereas it statistically significantly decreases to 0.55 if $\\alpha = 0.5$ is chosen.", "Thus, even given this large sample size, ignoring uncertainty appears detrimental for this application, and $\\alpha = 0.5$ results in 4 articles whose benchmark popularity worsens post-intervention (compared to only 2 for $\\alpha = 0.05$ ).", "Nonetheless, both methods generally produce very beneficial improvements in this analysis, as seen in Figure REF .", "As an example of the personalization of proposed interventions, our method ($\\alpha = 0.05$ ) generally proposes different sparse interventions for articles in the Business category vs. the Entertainment category.", "On average, the sparse transformation for business articles uniquely advocates decreasing global sentiment polarity and increasing word count (which are not commonly altered in the personalized interventions found for entertainment articles), whereas interventions to decrease title subjectivity are uniquely prevalent throughout the entertainment category.", "These findings appear intuitive (eg.", "critical business articles likely receive more discussion, and titles of popular entertainment articles often contain startling statements written non-subjectively as fact).", "Interestingly, the model also tends to advise shorter titles for business articles, but increasing the length for entertainment articles.", "Articles across all categories are universally encouraged to include more references to other articles and keywords that were historically popular." ], [ "Misspecified Interventions", "Our methodology heavily relies on the assumption that the outcome-determining covariate values $\\widetilde{x}$ produced through intervention exactly match the desired covariate transformation $T(x)$ .", "When transformations are only allowed to alter at most $k < d$ covariates, this requires that we can intervene to alter only this subset without affecting the values of other covariates.", "If $T$ specifies a sparse change affecting only a subset of the covariates $\\mathcal {I} \\subset \\lbrace 1,\\dots ,d\\rbrace $ , our methods assume the post-treatment value of any non-intervened-upon covariate remains at its initial value (ie.", "$\\widetilde{x}_s = x_s \\ \\forall s \\notin \\mathcal {I}$ ).", "In some domains, the covariate-transformation induced via sparse external intervention can only be roughly controlled (eg.", "our gene perturbation example when the profiled genes belong to a common regulatory network).", "Let $T_{\\mathcal {I}\\rightarrow z}$ denote a covariate-fixing transformation which sets a subset of covariates in $\\mathcal {I} \\subset \\lbrace 1,\\dots ,d\\rbrace $ to constant values $z_\\mathcal {I} \\in \\mathbb {R}^{|\\mathcal {I}|}$ across all individuals in the population.", "In this section, we consider an alternative assumption under which the intervention applied in hopes of achieving $T_{\\mathcal {I}\\rightarrow z}$ propagates downstream to affect other covariates outside $\\mathcal {I}$ (so there may exist $s \\notin \\mathcal {I}$ : $\\widetilde{x}_s \\ne x_s$ ), which we formalize as the $\\emph {do}$ -operation in the causal calculus of .", "Here, we suppose the underlying population of $X, Y$ follows a structural equation model (SEM) .", "The outcome $Y$ is restricted to be a sink node of the causal DAG, so we can still write $Y = f^*(\\widetilde{X}) + \\varepsilon $ and maintain the other conditions from §.", "Rather than exhibiting covariate-distribution $T_{\\mathcal {I}\\rightarrow z}(X)$ with $Y = f^*(T_{\\mathcal {I}\\rightarrow z}(X)) + \\varepsilon $ (as presumed in our methods), the post-treatment population which arises from an intervention seeking to enact transformation $T_{\\mathcal {I}\\rightarrow z}$ is now assumed to follow the distribution specified by $p(X, Y \\mid do(X_{\\mathcal {I}} = z_{\\mathcal {I}}))$ .", "Note that the $\\emph {do}$ -operation here is only applied to some nodes in the DAG (variables in subset $\\mathcal {I}$ ) as discussed by , but its effects can alter the distributions of non-intervened-upon covariates outside of $\\mathcal {I}$ which lie downstream in the DAG.", "Theorem 3 For some $\\mathcal {I} \\subseteq \\lbrace 1, \\dots ,d\\rbrace $ , suppose the condition: (A7) $pa(Y) \\subseteq \\mathcal {I} \\ \\bigcup \\ \\text{desc}(\\mathcal {I})^C$ holds.", "Then, for any covariate-fixing transformation $T_{\\mathcal {I}\\rightarrow z}$ : $\\displaystyle \\mathbb {E}_X \\big [ f^*(T_{\\mathcal {I}\\rightarrow z}(x)) - f^*(x) \\big ]$ and $\\mathbb {E}_{\\widetilde{x} \\sim \\text{do}(X_{\\mathcal {I}} = z_{\\mathcal {I}})} \\big [ f^*(\\widetilde{x}) \\big ] - \\mathbb {E}_{X}\\big [f^*(x) \\big ]$ are equal.", "Here, $\\text{pa}(Y)$ denotes the variables which are parents of outcome $Y$ in the underlying causal DAG, and $\\text{desc}(\\mathcal {I})^C$ is the set of variables which are not descendants of variables in subset $\\mathcal {I}$ .", "For the next result, we define: $\\displaystyle \\mathcal {I^*} := \\operatornamewithlimits{argmin}\\Big \\lbrace |\\mathcal {I}^{\\prime }| \\text{ s.t. }", "\\exists \\ T_{\\mathcal {I}^{\\prime } \\rightarrow z} \\in \\operatornamewithlimits{argmax}_{T_{\\mathcal {I}\\rightarrow z} : |\\mathcal {I}| \\le k} \\hspace*{-0.28453pt} \\mathbb {E}_X \\big [ f^*(T_{\\mathcal {I}\\rightarrow z}(x)) - f^*(x) \\big ] \\Big \\rbrace $ as the intervention set corresponding to the optimal $k$ -sparse covariate-fixing transformation (where in the case of ties, the set of smallest cardinality is chosen), if transformations were exactly realized by our interventions (which is not necessarily the case in this section).", "Theorem 4 Suppose the underlying DAG satisfies: (A8) No variable in $\\text{pa}(Y)$ is a descendant of other parents, ie.", "$\\nexists \\ j \\in \\text{pa}(Y)$ s.t.", "$j \\in \\text{desc}(\\text{pa}(Y) \\setminus \\lbrace j\\rbrace )$ .", "Then, $\\mathcal {I^*}$ satisfies (AREF ).", "In the absence of extremely strong interactions between variables in $\\text{pa}(Y)$ , the equality of Theorem REF will also hold for $\\mathcal {I^*}$ if $| \\text{pa}(Y) | \\le k$ .", "For settings where sparse interventions elicit unintentional $do$ -effects and the causal DAG meets condition (AREF ), Theorems REF and REF imply that, under complete certainty about $f^*$ , the (minimum cardinality) maximizer of our covariate-fixing intervention objective corresponds to an transformation that produces an equally good outcome change when the corresponding intervention is actually realized as a $do$ -operation in the underlying population.", "Combined with Theorem REF , our results ensure that, even in this misspecified setting, the empirical maximizer of our sparse covariate-fixing intervention objective (REF ) produces (in expectation as $n \\rightarrow \\infty $ ) beneficial interventions for populations whose underlying causal relationships satisfy certain conditions.", "Next, we empirically investigate how effective our methods are in this misspecified SEM setting, where a proposed sparse population transformation is actually realized as a do-operation and can therefore unintentionally affect other covariates in the post-intervention population.", "We generate data from an underlying linear non-Gaussian SEM, and where $Y$ is a sink node in the corresponding causal DAG (see §REF for details).", "Our approach to identify a beneficial sparse population intervention is compared with inferring the complete SEM using the LinGAM estimator of and subsequently identifying the optimal single-node do-operation in the inferred SEM.", "Note that LinGAM is explicitly designed for this setting, while both our method and the relied-upon Gaussian Process model are severely misspecified.", "Figures REF A and REF B demonstrate that the inferred best single-variable shift population intervention (under constraints on the magnitude of the shift) matches the performance the interventions suggested by LinGAM (except for in rare cases with tiny sample size) when the proposed interventions are evaluated as do-operations in the true underlying SEM.", "Thus, we believe a supervised learning approach like ours is preferable in practical applications where interpreting the underlying causal structure is not as important as producing good outcomes (especially for higher dimensional data where estimation of the causal structure becomes difficult ).", "Figure: The average (solid) and 0.05 th 0.05^{\\text{th}} quantile (dashed) expected outcome change produced by our method (red) vs LinGAM (blue) over 100 datasets drawn from two underlying SEMs chosen by .", "The black dashed line indicates the best possible improvement in each case.The assumption of sparse interventions realized as a $do$ -operation (as defined by ) may also be an inappropriate in many domains, particularly if off-target effects of interventions are explicitly mitigated via external controls.", "To appreciate the intricate nature of assumptions regarding non-intervened-upon variables, consider our example of modeling text documents represented using two features: polarity and word count.", "A desired transformation to increase the text's polarity can be accomplished by inserting additional positive adjectives, but such an intervention also increases articles' word count.", "Alternatively, polarity may be identically increased by replacing words with more positive alternatives, an external intervention which would not affect the word count (and thus follows the assumptions of our framework)." ], [ "Discussion", "This work introduces methods for directly learning beneficial interventions from purely observational data without treatments.", "While this objective is, strictly speaking, only possible under stringent assumptions, our approach performs well in both intentionally-misspecified and complex real-world settings.", "As supervised learning algorithms grow ever more popular, we expect intervention-decisions in many domains will increasingly rely on predictive models.", "Our conservative definition of the optimal intervention provides a principled approach to handle the inherent uncertainty in these settings due to finite data.", "Able to employ any Bayesian regressor, our ideas are widely applicable, considering practical types of interventions that can either be personalized or enacted uniformly over a population.", "Acknowledgements: We thank David Gifford for helpful comments.", "DR was funded by an award from IARPA (under research contract 2015-15061000003)." ], [ "References", "Supplementary Material toc" ], [ "Gaussian Process Regression", "Gaussian Process regression Rasmussen2006si adopts a prior under which $f(x^{(1)}), \\dots , f(x^{(n)})$ follow multivariate Gaussian distribution N$(\\mathbf {m}_{n}, \\mathbf {K}_{n,n})$ for any collection $\\lbrace x^{(i)}\\rbrace _{i=1}^n$ .", "The model is specified by a prior mean function $m : \\mathbb {R}^d \\rightarrow \\mathbb {R}$ and positive-definite covariance function $k : \\mathbb {R}^d \\times \\mathbb {R}^d \\rightarrow \\mathbb {R}$ which encodes our prior belief regarding properties of the underlying relationship between $X$ and $Y$ (such as smoothness or periodicity).", "Here, the vector $\\mathbf {m}_{n} \\in \\mathbb {R}^n$ denotes the evaluation of function $m$ at each point $\\lbrace x^{(i)}\\rbrace _{i=1}^n$ , and $\\mathbf {K}_{n,n}$ denotes the matrix whose $i,j^{\\text{th}}$ component is $k(x^{(i)}, x^{(j)})$ .", "Given test input points $x_*^{(1)},\\dots ,x_*^{(n_*)} \\in \\mathbb {R}^d$ in addition to training data $\\mathcal {D}_n$ , we additionally define: $\\mathbf {f}_* := [f(x_*^{(1)}), \\dots , f(x_*^{(n_*)})]$ , $\\mathbf {y}_n = [y^{(1)}, \\dots , y^{(n)}]$ , matrix $\\mathbf {K}_{n,*}$ with $i,j^{\\text{th}}$ entry $k(x^{(i)}, x_*^{(j)})$ (where $x^{(i)}$ is the $i^{\\text{th}}$ training input), and matrix $\\mathbf {K}_{*,*}$ which contains pairwise covariances between test inputs.", "Assuming the noise $\\varepsilon \\sim \\text{N}(0, \\sigma ^2$ ) is independently sampled for each observation, the posterior for $f$ at the test inputs, $\\mathbf {f}_* \\mid \\mathcal {D}_n$ , follows $\\text{N}(\\mathbf {\\mu _n}_*, \\mathbf {\\Sigma _n}_* )$ distribution with the following mean vector and covariance matrix: $\\mathbf {\\mu _n}_* = \\mathbf {m}_{*} + (\\mathbf {K}_{n,n} + \\sigma ^2 \\mathbf {I})^{-1} (\\mathbf {y}_n - \\mathbf {m}_{n}), \\ \\mathbf {\\Sigma _n}_* = \\mathbf {K}_{*,*} - \\mathbf {K}_{*,n} (\\mathbf {K}_{n,n} + \\sigma ^2 \\mathbf {I})^{-1} \\mathbf {K}_{n,*}$ Note that our intervention-optimization framework is not specific to this GP model, but can be combined with any algorithm that learns a reasonable posterior for $f$ .", "While employing a more powerful model should improve the results of our approach, comparing various regressors is not our focus.", "Thus, all practical results of our methodology are presented using only the standard GP regression model, under which the posterior distribution over $f$ is given by the above expressions.", "In each application presented here, our GP uses the Automatic-Relevance-Determination (ARD) covariance function, a popular choice for multi-dimensional data Rasmussen2006si: $k(x, x^{\\prime }) = \\sigma _0^2 \\cdot \\exp \\left[-\\frac{1}{2} \\sum _{s=1}^d \\left( \\frac{x_s - x_s^{\\prime }}{l_s} \\right)^2 \\right]$ The ARD kernel relies on length-scale hyperparameters $l_1, \\dots , l_d$ which determine how much $f$ can depend on each dimension of the feature-space.", "All hyperparameters of our GP regression model (covariance-kernel parameters $l_1 \\dots , l_d$ and $\\sigma _0$ (the output variance) as well as the variance of the noise $\\sigma ^2$ ) are empirically selected via marginal-likelihood maximization Rasmussen2006si.", "In each application, we employ the $0.05^\\text{th}$ posterior-quantile ($\\alpha = 0.05$ ) in our method to ensure that with high probability, the intervention it infers to be optimal induces a nonnegative change in expected outcomes." ], [ "Algorithmic Details", "To find an optimal transformation of our regularized objective $J_\\lambda $ in (REF ), we employ the proximal gradient method described in §.", "When $\\lambda = 0$ and there is no penalty, we instead use Sequential Least Squares Programming Kraft1988si.", "However, the intervention objective $J_\\lambda $ may be highly nonconcave.", "To deal with local optima in acquisition functions, Bayesian optimization methods employ heuristics like combining the results of many local optimizers or operating over a fine partitioning of the feature space Shahriari2016si, Lizotte2008.", "We instead propose a continuation technique that solves a series of optimization problems, each of which operates on our objective under a smoothed posterior (and the amount of additional smoothing is gradually decreased to zero).", "Excessive smoothing of the posterior is achieved by simply considering GP models whose kernels are given overly large length-scale parameters.", "Each time the amount of smoothing is tapered, we initialize our local optimizer using the solution found at the previously greater smoothing level.", "Intuitively, the highly smoothed GP model is primarily influenced by the global structure in the data, and thus our optimization with respect to the posterior of this model is far less susceptible to low-quality local maxima.", "Analysis of a similar homotopy strategy under radial basis kernels has been conducted by Mohabi2012si." ], [ "Sparse Shift Intervention", "Here, we provide an explanatory description of the Sparse Shift Algorithm from §.", "To find the best $k$ -sparse population shift intervention, we resort to $\\ell _1$ relaxation.", "As the $\\ell _1$ -norm provides the closest convex relaxation to the $\\ell _0$ norm, this is a a commonly adopted strategy to avoid combinatorial search in feature selection Bach2012si.", "First, we compute the regularization path over different settings of the penalty $\\lambda > 0$ for the following regularized objective: $ J_\\lambda (\\Delta ) := F^{-1}_{G_n(\\Delta )}(\\alpha ) - \\lambda ||\\Delta ||_1$ which is maximized over the feasible set $\\mathcal {C}_{\\Delta } := \\lbrace \\Delta \\in \\mathbb {R}^d : x + \\Delta \\in \\mathcal {C}_{x} \\text{ for all } x \\in \\mathbb {R}^d \\rbrace $ (recall we write $G_n(\\Delta ) := G_n(T)$ when $T(x) = x+ \\Delta $ ).", "Subsequently, we identify the regularization penalty which produces a shift of desired cardinality and select our intervention set $\\mathcal {I}$ as the covariates which receive nonzero shift.", "Finally, we optimize the original unregularized objective ($\\lambda = 0$ ) with respect to only the selected covariates in $\\mathcal {I}$ to remove bias induced by the regularizer.", "Each inner maximization in both the Sparse Shift/Covariate-fixing algorithms is performed via the proximal gradient methods combined with our continuation approach introduced in §." ], [ "Sparse Covariate-fixing Intervention", "Another goal is to identify the optimal covariate-fixing intervention which sets $k$ of the covariates to particular fixed constants uniformly across all individuals from the population.", "We employ the forward step-wise selection algorithm described below, as the form of the optimization in this case is not amenable to $\\ell _1$ -relaxation.", "Recall $\\mathcal {I} \\subseteq \\lbrace 1,\\dots ,d\\rbrace $ denotes the subset of covariates which are intervened upon, and the covariate-fixing intervention produces vector $T_{\\mathcal {I} \\rightarrow z}(x) \\in \\mathbb {R}^d$ such that $T_{\\mathcal {I} \\rightarrow z}(x)_s = x_s$ if $s \\notin \\mathcal {I}$ , otherwise $T_{\\mathcal {I} \\rightarrow z}(x)_s = z_s$ which is a constant chosen by the policy-maker.", "This same transformation is applied to each individual in the population, creating a more homogeneous group which share the same value for the covariates in $\\mathcal {I}$ .", "For a given $\\mathcal {I}$ , the objective function to find the best constants is: $& J^{\\text{unif}}_\\mathcal {I}\\big (\\lbrace z_s\\rbrace _{s \\in \\mathcal {I}} \\big ) := F^{-1}_{G_n(T_{\\mathcal {I}\\rightarrow z})} (\\alpha ) \\\\\\text{ with } \\ & G_n(T_{\\mathcal {I}\\rightarrow z}) = \\frac{1}{n} \\sum _{i=1}^n \\big [ f(z^{(i)}) - f(x^{(i)}) \\big ] \\mid \\mathcal {D}_n \\ \\text{ where } z_s^{(i)} = {\\left\\lbrace \\begin{array}{ll}x^{(i)} & \\text{ if } s \\notin \\mathcal {I} \\\\z_s & \\text{ otherwise } \\\\\\end{array}\\right.", "}$ which is maximized over the constraints: $z_s \\in \\mathcal {C}_s \\subseteq \\mathbb {R}$ for $s \\in \\mathcal {I}$ .", "Sparse Covariate-fixing Algorithm: Identifies best $k$ -sparse covariate-fixing intervention.", "Input: Dataset $\\mathcal {D}_n = \\lbrace (x^{(i)}, y^{(i)}) \\rbrace _{i=1}^n$ , Posterior $f \\mid \\mathcal {D}_n$ Parameters: $k \\in \\lbrace 1, \\dots , d\\rbrace $ specifies the maximal number of covariates which may be set by the covariate-fixing intervention, $\\mathcal {C}_1,\\dots , \\mathcal {C}_d \\subseteq \\mathbb {R}$ are sets of feasible settings for each covariate.", "Initialize $\\mathcal {I} \\leftarrow \\varnothing $ , $\\mathcal {U} \\leftarrow \\lbrace 1,\\dots , d\\rbrace $ , $J^* \\leftarrow 0$ While $|\\mathcal {I}| < k$ :      Set $\\displaystyle J^*_s \\leftarrow \\max _{\\mathcal {C}_r : r \\in \\mathcal {I} \\cup \\lbrace s\\rbrace } J^{\\text{unif}}_{\\mathcal {I} \\cup \\lbrace s\\rbrace } \\big (\\lbrace z_r\\rbrace _{r \\in \\mathcal {I} \\cup \\lbrace s\\rbrace } \\big )$      for each $s \\in \\mathcal {U}$      Find $ s^* \\leftarrow \\operatornamewithlimits{argmax}_{s \\in \\mathcal {U}} \\big \\lbrace J^*_s \\big \\rbrace $      If $J^*_{s^*} > J^*$ :      $J^* \\leftarrow J^*_{s^*}$ , $\\mathcal {I} \\leftarrow \\mathcal {I} \\cup \\lbrace s^*\\rbrace $ , $\\mathcal {U} \\leftarrow \\mathcal {U} \\setminus s^*$      Else:     break Return: $\\lbrace z_s^*\\rbrace _{s \\in \\mathcal {I}} \\leftarrow \\operatornamewithlimits{argmax}_{\\mathcal {C}_s : s \\in \\mathcal {I}} \\ J^{\\text{unif}}_{\\mathcal {I}} \\big (\\lbrace z_s\\rbrace _{s \\in \\mathcal {I}} \\big )$" ], [ "Simulations", "Over the simulated data summarized in Figure REF , we apply our basic personalized intervention method ($\\alpha = 0.05$ ) with purely local optimization (standard) and our continuation technique (smoothed), which significantly improves results.", "For each of the 100 datasets, we randomly sampled a new point (from the same underlying distribution) to receive a personalized intervention.", "The magnitude of each intervention is bounded by 1, except for in data from the quadratic relationship.", "We also infer sparse interventions (with a cardinality constraint of 2 for the linear and quadratic relationships, 1 for the product relationship).", "When $Y = X_1 \\cdot X_2 + \\varepsilon $ , the optimal (constrained) intervention may drastically vary depending upon the individual's covariate-values, and our algorithm is able to correctly infer this behavior (Simulation C).", "Finally, we also apply a variant of our method which entirely ignores uncertainty ($\\alpha = 0.5$ ).", "While this approach is on average better for larger sample sizes, highly harmful interventions are occasionally proposed, whereas our uncertainty-adverse method ($\\alpha =0.05$ ) is much less prone to producing damaging interventions (preferring to abstain by returning $T(x) = x$ instead).", "This is an invaluable characteristic since interventions generally require effort and are only worth conducting when they are likely to produce a substantial benefit.", "Figure REF displays the behavior of both the population shift intervention in the linear setting, and the population covariate-fixing intervention under the quadratic relationship.", "The population intervention is notably safer than the individually tailored variants, producing no negative changes in our experiments." ], [ "Linear SEM Analysis", "tocsubsectionsupsec:SEMdetails Here, we suppose that a desired transformation upon variable $s \\in \\lbrace 1,\\dots , d\\rbrace $ cannot be enacted exactly and the $Y$ which arises post-treatment is distributed according to $do(X_s = \\mathbb {E}[X_s] + \\Delta )$ , where $\\mathbb {E}[X_s]$ is the mean of the pre-treatment marginal distribution of the $s$ th covariate.", "In this case, do-effects can propagate to other covariates which are descendants of $s$ in the DAG because the values of descendant variables are redrawn from the do-distribution which arises as a result of shifting $\\mathbb {E}[X_s]$ .", "Because all relationships are linear in our SEMs, the actual expected outcome change resulting from a particular shift (resulting from the corresponding do-operation) is easily obtained in closed form.", "Our GP framework is applied to the data to infer an optimal 1-sparse shift population intervention (only interventions on a single variable are allowed).", "The maximal allowed magnitude of the shift is constrained to ensure the optimum is well-defined (to $\\pm 1$ times the standard deviation of each variable in the underlying SEM distribution).", "An alternative approach to improve outcomes in contrast to our black-box approach is to apply a causal inference method like LinGAM Shimizu2006si to estimate the SEM from the data, and then identify the optimal single-variable shift $\\Delta _s^*$ in the LinGAM-inferred SEM (since all inferred relationships are also linear, the optimal single-variable shift will be either 0 or the lower/upper allowed shift and we simply search over these possibilities).", "We compare our approach against LinGAM by evaluating the actual expected outcome change produced by the shift $\\Delta ^*_s$ proposed by each method (where the actual expected outcome change is found by analytically performing the $do(X_s = x_s + \\Delta ^*_s)$ operation in the true underlying SEM) .", "In our experiment, two underlying SEM models are considered which were used by Shimizu2006si to demonstrate the utility of their LinGAM method (albeit with impractically large sample size = 10,000).", "SEM$_A$ is used to refer to the model depicted in Figure 3 of Shimizu2006si, where we define $Y$ as x6 (a sink node in the causal DAG).", "SEM$_B$ denotes the underlying model of Figure 4 in the same paper ($Y$ is defined as sink node x7).", "The remainder of the variables in each SEM are adopted as our observed covariates $X$ .", "This experiment represents an application of our method in a highly misspecified setting.", "The true data-generating mechanism differs significantly from assumptions of our GP regressor (output noise is now fairly non-Gaussian, the underlying relationships are all linear while we use an ARD kernel).", "Furthermore, an intervention to transform a single covariate incurs a multitude of unintentional off-target effects resulting from the do-effects propagating to downstream covariates in the SEM, whereas our method believes only the chosen covariate is changed.", "In contrast, this data exactly follows the special assumptions required by LinGAM, and we properly account for inferred downstream do-operation effects when identifying the best inferred intervention under LinGAM.", "The only disadvantage of the LinGAM method is that it does not know the direction of the causal relationship $X \\rightarrow Y$ (although we found it always estimated this direction correctly except on rare occasions with tiny sample sizes of $n = 20$ ).", "Since LinGAM only estimates linear relations, the best inferred shift-intervention found by this approach will always be 0 or the minimal/maximal shift allowed for a particular covariate.", "Searching over these three values for each covariate ensures the actual optimal shift will be recovered if the LinGAM SEM-estimate were correct.", "However, under our approach, identifying the optimal population shift-intervention requires solving an optimization problem.", "Even if the GP regression posterior were to exactly reflect the true data-generating mechanism, our approach might get stuck in a suboptimal local maximum or avoid the minimal/maximal allowed shift due to too much uncertainty about $f$ in the resulting region of feature-space.", "In practice, these potential difficulties do not pose much of an issue for our approach." ], [ "Gene Knockout Interventions", "The data set used for this analysis contains gene expression levels for a set of wild type (ie.", "`observational') samples, $\\mathcal {D}_{obs} \\ (n=161)$ , as well as for a set of `interventional' samples, $\\mathcal {D}_{int}$ , in which each individual gene was serially knocked out.", "In our analysis, we search for potential interventions for affecting the expression of a desired target gene by training our GP regressor on $\\mathcal {D}_{obs}$ and determining which knockout produces the best value of our empirical covariate-fixing population intervention objective (for down-regulating the target).", "Subsequently, we use $\\mathcal {D}_{int}$ to evaluate the actual effectiveness of proposed interventions in the knockout experiments.", "We only search for interventions present in $\\mathcal {D}_{int}$ (single gene knockouts) rather than optimizing to infer optimal covariate transformations.", "As candidate genes for this analysis we used only the 700 genes that Kemmeren2014largesi classified as responsive mutants (at least four transcripts show robust changes in response to the knockout).", "Furthermore, we omitted genes whose expression over the 161 observational samples had standard deviation $< 0.1$ .", "Out of the transcription factors present in the remaining set of genes, we defined the top 10 factors as our feature set $X$ , after ranking the transcription factors by the difference between their expression when they were knocked out in the interventional data and their $0.1^{\\text{th}}$ quantile expression level in the observational data.", "This was to ensure that our model would be trained on data that at least resembled the experimental data $\\mathcal {D}_{int}$ .", "The set of genes to down-regulate was simply chosen to be those classified by Kemmeren2014largesi as small molecule metabolism genes that met the minimum standard deviation requirement in their observational expression marginal distribution.", "The resulting set was 16 target genes, and the (negative) expression of each of was treated as an outcome $Y$ in our analyses.", "Each method evaluated in this analysis was to propose an intervention (single gene knockout) to down-regulate the expression of each target gene (separately).", "Once a gene to knock out was proposed, this intervention was evaluated by comparing the resulting expression of the target when the proposed knockout was actually performed in the experimental data $\\mathcal {D}_{int}$ .", "This expression level could then be compared to the `optimal' choice of gene from $X$ to intervene upon (the gene in $X$ whose knockout produced the largest down-regulation of the target in $\\mathcal {D}_{int}$ ).", "We compared our approach against two methods popularly used to draw conclusions about affecting outcomes in the sciences.", "First, we applied a multivariate regression analysis in which a linear regression model was fit to the observations of $(X, Y)$ in $\\mathcal {D}_{obs}$ .", "The best gene to knockout was inferred on the basis of the regression coefficients and expression values (if no beneficial regression coefficient was found significant at the 0.05 level under the standard $t$ -test, then no intervention was proposed).", "Second, we performed a marginal analysis in which separate univariate linear regression models were fit to $(X_1, Y), \\dots , (X_d, Y)$ , and the best knockout was again inferred on the basis of the regression coefficients and expression values (again, no intervention was recommended if there was no statistically significant beneficial regression coefficient at the 0.05 level, after correcting for multiple testing via the False Discovery Rate).", "Figure REF compares the results produced by these methods to the optimal intervention over $X$ for down-regulating each $Y$ , as found in the experimental data $\\mathcal {D}_{int}$ .", "Of the 16 small molecule metabolism target genes tested, in three cases our method proposed an intervention which was found to be optimal or near optimal in $\\mathcal {D}_{int}$ , while in the remaining cases, the model uncertainty causes the method not to recommend any intervention (except for one very minorly harmful intervention for target SAM3).", "On the other hand, neither form of linear regression proposed effective interventions for any target other than FKS1, and in some cases, the linear regressors proposed counterproductive interventions that up-regulated the target.", "This highlights the importance of a model that properly accounts uncertainty when evaluating potential interventions." ], [ "Interventions to Improve Article Popularity", "We demonstrate our personalized intervention methodology in a setting with rich nonlinear underlying relationships.", "The data consist of 39,000 news articles published by Mashable around 2013-15 Fernandes2015si.", "Each article is annotated with the number of shares it received in social networks (which we use as our outcome variable after log-transform and rescaling).", "A multitude of features have been extracted from each article (eg.", "word count, the category such as “tech” or “lifestyle”, keyword properties), many of which Fernandes2015si produced using natural language processing algorithms (eg.", "subjectivity, polarity, alignment with topics found by Latent Dirichlet Allocation).", "After removing many highly redundant covariates, we center and rescale all variables to unit-variance (see Table REF for a complete description of the 29 covariates used in this analysis).", "We randomly partition the articles into 3 disjoint groups: a training set (5,000 articles on which scaling-factors are computed and our GP regressor is trained), an improvement set (300 articles we find interventions for), and a held-out set (over 34,000 articles used for evaluation).", "A large group is left out for validation to ensure there are many near-neighbors for any given article, so we can reasonably estimate the true expected popularity given any setting of the article-covariates.", "Subsequently, a basic GP regression model is fitted to the training set.", "As the predictive power of our GP regressor did not measurably benefit from ARD feature-weighting, we simply use the squared exponential kernel.", "Over the held-out articles, the Pearson correlation between the observed popularity and the GP (posterior mean) predictions is 0.35.", "Furthermore, there is a highly significant ($p < 8 \\cdot 10^{-41}$ ) positive correlation of 0.07 between the model's predictive variance and the actual squared errors of GP predictions over this held-out set.", "Our model is thus able to make reasonable predictions of popularity based on the available covariates, and its uncertainty estimates tend to be larger in areas of the feature-space where the posterior mean lies further from actual popularity values.", "In this analysis, we compare our personalized intervention methodology which rejects uncertainty (using $\\alpha = 0.05$ ) with a variant of the this approach that ignores uncertainty (using the same objective function with $\\alpha = 0.5$ ).", "Both methods share the same GP regressor, optimization procedure, and set of constraints.", "For the 300 articles in the intervention set (not part of the training data) we allow intervening upon all covariates except for the article category which presumably is fixed from an author's perspective.", "All covariate-transformations are constrained to lie within [-2,2] of the original (rescaled) covariate value, and we impose a sparsity constraint that at most 10 covariates can be intervened upon for a given article.", "Unfortunately, no pre-and-post-intervention articles are available for us to ascertain a ground truth evaluation.", "To crudely measure performance, we estimate the underlying expected popularity of a given covariate-setting using benchmark popularity: the (weighted) average observed popularity amongst 100 nearest neighbors (in the feature-space) from the set of held-out articles (with weights based on inverse Euclidean distance).", "Over our improvement set, the Pearson correlation between articles' observed popularity and benchmark popularity is 0.28 (highly significant: $p \\le 2\\cdot 10^{-10}$ ).", "This approach thus appears to be, on average, a reasonable way to benchmark performance (even though nearest-neighbor held-out articles can individually differ from the text of a particular pre/post-intervention article despite sharing similar values of our 29 measured covariates).", "Figure REF depicts the results of our personalized intervention for each article in our intervention set.", "The expected improvement produced by a particular intervention is estimated as the difference between the benchmark popularity of the post-intervention covariate-settings and the original covariate-settings of the article receiving the personalized intervention.", "Table REF summarizes these results.", "A paired-sample $t$ -test suggests our method is significantly superior on average ($p < 2 \\cdot 10^{-6}$ ).", "Figure: Benchmark popularity changes produced by the personalized interventions for 300 articles suggested by our method with α=0.05\\alpha = 0.05 (Rejecting Uncertainty) vs. α=0.5\\alpha = 0.5 (Ignoring Uncertainty).", "The points (ie.", "articles) are colored according to the value of our personalized intervention objective with α=0.05\\alpha = 0.05.", "Using α=0.05\\alpha =0.05 outperforms α=0.5\\alpha =0.5 in this analysis in 177/300 articles in the improvement set.", "Table: Summary statistics for the benchmark popularity change produced by each method over the 300 articles of the intervention set.", "The last column counts the number of harmful interventions (with change <0< 0).To provide concrete examples, we present some articles of the Business and Entertainment categories (taken from our improvement set).", "For this business article: http://mashable.com/2014/07/30/how-to-beat-the-heat/, our method proposes shifting the following 10 covariates (see Table REF for feature descriptions): num_hrefs: +2, num_self_hrefs: -1.25, average_token_length: -1.771, kw_avg_min: +1.71, kw_avg_avg: +2, self_reference_min_shares: +2, self_reference_max_shares: +1.68, self_reference_avg_sharess: +2, global_subjectivity: +1.57, global_sentiment_polarity: -2 For this entertainment article: http://mashable.com/2014/07/30/how-to-beat-the-heat/, our method proposes shifting the following 10 covariates: average_token_length: -1.55, kw_avg_min: + 1.63, kw_avg_avg: +2, self_reference_min_shares: +2 self_reference_max_shares: +1.85, self_reference_avg_shares: +2.0, LDA_00: +1.63, LDA_01: -2, LDA_04: +0.82, global_subjectivity: +1.62 Indifferent to uncertainty, the method with $\\alpha = 0.5$ advocates shifting all these covariates by the $\\pm 2$ maximal allowed amounts, which leads to a 0.04 worse improvement in benchmark popularity compared with the covariate-changes specified above for this article.", "Table: The 29 covariates of each article (dimensions of XX in this analysis).", "Features involving the share-counts of other articles and LDA were based only on data known before the publication date." ], [ "Notation and Definitions", "tocsubsubsectionnotation All points $x \\in \\mathbb {R}^d$ lie in convex and compact domain $\\mathcal {C} \\subset \\mathbb {R}^d$ .", "$C$ denotes constants whose value may change from line to line.", "All occurrences of $f$ are implicitly referring to $f \\mid \\mathcal {D}_n$ .", "$\\mu _n(\\cdot )$ , $\\sigma ^2_n(\\cdot )$ , and $\\sigma _n(\\cdot , \\cdot )$ respectively denote the mean, variance, and covariance function of our posterior for $f \\mid \\mathcal {D}_n$ under the GP$\\big (0, k(x,x^{\\prime })\\big )$ prior.", "$F^{-1}_Z(\\alpha )$ denotes the $\\alpha ^{\\text{th}}$ quantile of random variable $Z$ .", "$\\Phi ^{-1}(\\cdot )$ denotes the $N(0,1)$ quantile function.", "$||\\cdot ||_k$ denotes the norm of reproducing kernel Hilbert space $\\mathcal {H}_k$ .", "$\\mathcal {B}_\\delta (x) \\subset \\mathbb {R}^d$ denotes the ball of radius $\\delta $ centered at $x \\in \\mathcal {C}$ .", "$\\mathcal {I} \\subseteq \\lbrace 1,\\dots ,d\\rbrace $ represents the set of variables which are intervened upon in sparse settings.", "$\\text{pa}(Y)$ denotes the set of variables which are parents of $Y$ in a causal directed acyclic graph (DAG) Pearl2000si $\\text{desc}(\\mathcal {I})$ is the set of variables which are descendants of at least one variable in $\\mathcal {I}$ according to the causal DAG.", "$A^C$ denotes the complement of set $A$ .", "The squared exponential kernel (with length-scale parameter $l > 0$ ) is defined: $k(x,x^{\\prime }) = \\exp \\Big ( -\\frac{1}{2 l^2} ||x - x^{\\prime } ||^2 \\Big ) $ The Matérn kernel (with another parameter $\\nu > 0$ controlling smoothness of sample paths) is defined: $k(x,x^{\\prime }) = \\frac{2^{1-\\nu }}{\\Gamma (\\nu )} r^\\nu B_\\nu (r) \\ \\ \\ \\text{ where } \\ r = \\frac{\\sqrt{2\\nu }}{l} || x - x^{\\prime }||, B_\\nu \\text{ is a modified Bessel function}$ Random variables $\\varepsilon ^{(1)}, \\dots , \\varepsilon ^{(n)}$ form a martingale difference sequence which is uniformly bounded by $\\sigma $ if $ \\ \\mathbb {E}[\\varepsilon ^{(i)} \\mid \\varepsilon ^{(i-1)},\\dots , \\varepsilon ^{(1)}] = 0 $ and $\\varepsilon ^{(i)} \\le \\sigma \\ \\ \\forall i \\in \\mathbb {N}$ .", "A function $f$ is Lipshitz continuous with constant $L$ if: $|f(x) - f(x^{\\prime })| \\le L |x - x^{\\prime }|$ for every $x, x^{\\prime } \\in \\mathcal {C}$ .", "Suppose $\\rho > 0$ is expressed as $\\rho = m + \\eta $ for nonnegative integer $m$ and $0 < \\eta \\le 1$ .", "The Hölder space $C^\\rho [0,1]^d$ is the space of functions with existing partial derivatives of orders $(k_1, \\dots , k_d)$ for all integers $k_1, \\dots , k_d \\ge 0$ satisfying $k_1 + \\dots + k_d \\le m$ .", "Additionally, each function's highest order partial derivative must form a function $h$ that satisfies: $|h(x) - h(y)| \\le C |x-y|^\\eta $ for any $x,y$ .", "Theorem 5 (VanderVaart2011si) Under the assumptions of Theorem REF : $\\mathbb {E}_{\\mathcal {D}_n} \\int \\int _\\mathcal {C} [f(x) - f^*(x)]^2 p_X(x) \\mathrm {d}x \\ \\mathrm {d} \\Pi _n ( f \\mid \\mathcal {D}_n) \\ \\le \\ C \\cdot \\Psi _{\\hspace*{-0.56905pt}f^*}(n)$ where $\\Psi _{\\hspace*{-0.56905pt}f^*}^{-1}(n)$ is defined as in §.", "See VanderVaart2011si for a detailed discussion about this function.", "tocsubsubsectionproof:close Recall $G_{x}(T) := f(T(x)) - f(x) \\mid \\mathcal {D}_n$ depends on $f$ .", "We fix $x_0, T(x_0) \\in \\mathcal {C}$ and adapt the bound provided by Theorem REF to show our result.", "Let $\\mathcal {B}_\\delta (x) \\subset \\mathcal {C}$ denote the ball of radius $0 < \\delta < \\frac{1}{2}$ centered at $x \\in \\mathcal {C}$ .", "We first establish the bound: $& \\int _\\mathcal {C} \\big |f(x) - f^*(x) \\big | p_X(x) \\ \\mathrm {d}x\\\\\\ge & \\int _{\\mathcal {B}_\\delta (x_0)} \\big | f(x) - f^*(x) \\big | p_X(x) \\ \\mathrm {d}x + \\int _{\\mathcal {B}_\\delta (T(x_0))} \\big | f(x) - f^*(x) \\big | p_X(x) \\ \\mathrm {d}x\\\\\\ge & a \\cdot \\text{Vol}(\\mathcal {B}_\\delta ) \\Big [ \\min _{x \\in \\mathcal {B}_\\delta (x_0)} \\big | f(x) - f^*(x) \\big | + \\min _{x \\in \\mathcal {B}_\\delta (T(x_0))} \\big | f(x) - f^*(x) \\big | \\Big ]\\\\\\ge & a \\cdot \\text{Vol}(\\mathcal {B}_\\delta ) \\cdot \\Big [ \\Big | f(T(x_0)) - f(x_0) - \\big [ f^*(T(x_0)) - f^*(x_0) \\big ] \\Big | - 8 \\delta L \\Big ] \\\\\\ge & a \\cdot \\text{Vol}(\\mathcal {B}_\\delta ) \\cdot \\Big [ \\Big | G_{x_0}(T) - G^*_{x_0}(T) \\Big | - 8 \\delta L \\Big ] $ where Vol$(\\mathcal {B}_\\delta ) = \\mathcal {O}( \\delta ^d)$ .", "Theorem REF implies the following inequality (ignoring constant factors): $[C \\cdot & \\Psi _{\\hspace*{-0.56905pt}f^*}(n) ]^{1/2}\\\\\\ge & \\Bigg [ \\mathbb {E}_{\\mathcal {D}_n} \\int \\int _\\mathcal {C} [f(x) - f^*(x)]^2 p_X(x) \\ \\mathrm {d}x \\ \\mathrm {d} \\Pi _n ( f \\mid \\mathcal {D}_n) \\Bigg ]^{1/2}\\\\\\ge & \\mathbb {E}_{\\mathcal {D}_n} \\int \\int _\\mathcal {C} \\big | f(x) - f^*(x) \\big | p_X(x) \\ \\mathrm {d}x \\ \\mathrm {d} \\Pi _n ( f \\mid \\mathcal {D}_n) \\\\\\ge & a \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\int \\big | G_{x_0}(T) - G^*_{x_0}(T) \\big | - \\delta L \\ \\ \\mathrm {d} \\Pi _n ( f \\mid \\mathcal {D}_n)\\\\= & - a L \\delta ^{d+1} + a \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\int _0^\\infty \\Pr \\Big ( \\big | G_{x_0}(T) - G^*_{x_0}(T) \\big | \\ge r \\Big ) \\ \\mathrm {d}r\\\\= & - a L \\delta ^{d+1} + a \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\int _0^1 F^{-1}_{ | G_{x_0}(T) - G^*_{x_0}(T) | }(\\widetilde{\\alpha }) \\ \\ \\mathrm {d}\\widetilde{\\alpha }\\\\\\ge & - a L \\delta ^{d+1} + a \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\int _\\alpha ^1 F^{-1}_{G_{x_0}(T)}(\\widetilde{\\alpha }) - G^*_{x_0}(T) \\ \\ \\mathrm {d}\\widetilde{\\alpha }\\\\\\ge & - a L \\delta ^{d+1} + a (1-\\alpha ) \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\Big [ F^{-1}_{G_{x_0}(T)}(\\alpha ) - G^*_{x_0}(T) \\Big ]$ We can similarly bound $G^*_{x_0}(T) - F^{-1}_{G_{x_0}(T)}(\\alpha )$ : $& - a L \\delta ^{d+1} + a \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\int _0^1 F^{-1}_{ | G^*_{x_0}(T) - G_{x_0}(T) |}(\\widetilde{\\alpha }) \\ \\mathrm {d}\\widetilde{\\alpha }\\\\\\ge & - a L \\delta ^{d+1} + a \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\int _0^\\alpha G^*_{x_0}(T) - F^{-1}_{G_{x_0}(T) }(\\widetilde{\\alpha }) \\ \\ \\mathrm {d}\\widetilde{\\alpha }\\\\\\ge & - a L \\delta ^{d+1} + a \\alpha \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\Big [ G^*_{x_0}(T) - F^{-1}_{G_{x_0}(T) }(\\alpha ) \\Big ]$ Choosing $\\delta := [\\Psi _{\\hspace*{-0.56905pt}f^*}(n)]^{\\frac{1}{2(d+1)}}$ and combining (REF ) and (REF ) produces the desired result, since assuming $\\alpha < 0.5$ implies $\\alpha < 1 - \\alpha $ .", "tocsubsubsectionproof:popclose Combining the results of Lemmas REF and REF below, we obtain the desired upper bound through a straightforward application of the triangle inequality.", "Note that we've simplified the bound using the identity $-\\log (1-\\alpha ) < 1/\\alpha $ for $\\alpha < 0.5$ .", "Lemma 1 Under the assumptions of Theorem REF , for any $x, T(x) \\in \\mathcal {C}$ : $ \\mathbb {E}_{\\mathcal {D}_n} \\ \\Big |F^{-1}_{G_n(T)}(\\alpha ) - F^{-1}_{G_X(T)}(\\alpha ) \\Big |\\le C \\cdot \\Big [ \\frac{-L^2d}{n} \\log (1-\\alpha ) \\Big ]^{1/2}$ [Proof of Lemma REF ] Define random variables $Z_i := f(T(x^{(i)}) - f(x^{(i)}) \\mid \\mathcal {D}_n$ for $i=1,\\dots , n$ .", "Note that these variables all share the same expectation: $\\mathbb {E}_X[Z] := \\mathbb {E}_X[Z_i] = G_X(T)$ and $G_n(T) = \\frac{1}{n}\\sum _{i=1}^n Z_i$ .", "The Lipschitz continuity of $f$ combined with the fact that $\\mathcal {C} = [0,1]^d$ implies: $Z_i \\in [-L \\sqrt{d}, L \\sqrt{d}]$ for all $i$ .", "Thus, Hoeffding's inequality ensures: $& \\Pr \\Bigg (\\Bigg | G_n(T) - G_X(T) \\Bigg | \\ge t \\Bigg ) \\le 2 \\exp \\Bigg ( \\frac{-n t^2}{2 L^2 d} \\Bigg )\\\\\\Rightarrow \\ & F^{-1}_{\\big | G_n(T) - G_X(T) \\big |}(\\alpha )\\le C \\cdot \\Big [ \\frac{-L^2d}{n} \\log (1-\\alpha ) \\Big ]^{1/2}\\\\$ Because posteriors $ G_n(T), G_X(T)$ follow a Gaussian distribution: $& F^{-1}_{G_n(T)}(\\alpha ) - F^{-1}_{G_X(T)}(\\alpha )\\le F^{-1}_{\\big | G_n(T) - G_X(T) \\big |}(\\alpha )\\\\\\text{ and } & F^{-1}_{G_X(T)}(\\alpha ) - F^{-1}_{G_n(T)}(\\alpha )\\le F^{-1}_{\\big | G_n(T) - G_X(T) \\big |}(\\alpha )$ Lemma 2 Under the assumptions of Theorem REF , for any $x, T(x) \\in \\mathcal {C}$ : $ \\mathbb {E}_{\\mathcal {D}_n} \\ \\Big |F^{-1}_{G_X(T)}(\\alpha ) - G_X^*(T) \\Big |\\le \\frac{C}{\\alpha } \\cdot \\Big (L + \\frac{1}{a} \\Big ) \\cdot [\\Psi _{f^*}(n)]^{1 / [2(d+1)]}$ [Proof of Lemma REF ] A similar argument as the proof of Theorem REF applies here.", "We again first bound: $& \\int _\\mathcal {C} \\big |f(x) - f^*(x) \\big | p_X(x) \\ \\mathrm {d}x\\\\\\ge & a \\cdot \\text{Vol}(\\mathcal {B}_\\delta ) \\cdot \\Bigg [ \\int _\\mathcal {C} \\big |f(x) - f^*(x) \\big | p_X(x) \\ \\mathrm {d}x + \\int _\\mathcal {C} \\big | f(T(x))- f^*(T(x)) \\big | p_X(x) \\ \\mathrm {d}x - 8 \\delta L \\Bigg ]\\\\\\ge & a \\cdot \\text{Vol}(\\mathcal {B}_\\delta ) \\cdot \\Bigg [ \\Big | \\mathbb {E}_X [f(x) - f^*(x)] + \\mathbb {E}_X [f(T(x)) - f^*(T(x))] \\Big | - 8 \\delta L \\Bigg ]$ Following the same reasoning as in the proof of Theorem REF , we obtain (up to constant factors): $- a L \\delta ^{d+1} + a \\alpha \\delta ^d \\cdot \\mathbb {E}_{\\mathcal {D}_n} \\Big [ G_X^*(T) - F^{-1}_{G_X(T)}(\\alpha ) \\Big ] \\le [C \\cdot \\Psi _{\\hspace*{-0.56905pt}f^*}(n) ]^{1/2}$ and we can use the same argument to similarly bound $\\mathbb {E}_{\\mathcal {D}_n} \\Big [ F^{-1}_{G_X(T)}(\\alpha ) - G_X^*(T) \\Big ]$ tocsubsubsectionproof:dooperation Here, we employ subscripts to index particular covariates of $X$ .", "The notation $[a_{R}, a_{S}] = a \\in \\mathbb {R}^d$ is used to denote a vector assembled from disjoint subsets of dimensions $R,S \\subseteq \\lbrace 1,\\dots , d\\rbrace $ .", "Regardless of the ordering of these partitions in our notation, we assume they are correctly arranged in the assembled vector based on their subscript-indices (ie.", "$a = [a_{R}, a_{S}] = [a_{S}, a_{R}]$ ).", "$& \\mathbb {E}_{\\text{do}(X_{\\mathcal {I}} = z_{\\mathcal {I}})} \\big [ f^*(x) \\big ]\\\\= & \\int f^*\\big ([x_{\\mathcal {I}^C} , z_{\\mathcal {I}}]\\big ) \\ p\\big (x_{\\mathcal {I}^C} \\mid do(X_\\mathcal {I} = z_\\mathcal {I}) \\big ) \\ \\mathrm {d}x_{\\mathcal {I}^C}\\\\= & \\int \\int f^*\\big ([x_{\\text{pa}(Y) \\setminus \\mathcal {I}} , z_{\\mathcal {I} \\cap \\text{pa}(Y)} , a_{\\mathcal {I}^C \\setminus \\text{pa}(Y)}]\\big ) \\cdot p\\big (x_{\\mathcal {I}^C \\setminus \\text{pa}(Y)}\\mid x_{\\text{pa}(Y) \\setminus \\mathcal {I}} ,do(X_\\mathcal {I} = z_\\mathcal {I})\\big )\\\\& \\hspace*{42.67912pt}\\cdot p\\big (x_{\\text{pa}(Y) \\setminus \\mathcal {I}} \\mid do(X_\\mathcal {I} = z_\\mathcal {I})\\big ) \\ \\mathrm {d}x_{\\mathcal {I}^C \\setminus \\text{pa}(Y)} \\ \\mathrm {d}x_{\\text{pa}(Y) \\setminus \\mathcal {I}}\\\\& \\text{ where covariate-subset } a_{\\mathcal {I}^C \\setminus \\text{pa}(Y)} \\text{ can take arbitrary values since $f^*$ is constant along covariates $\\notin \\text{pa}(Y)$}\\\\= & \\int f^*\\big ([x_{\\text{pa}(Y) \\setminus \\mathcal {I}} , z_{\\mathcal {I} \\cap \\text{pa}(Y)} , a_{\\mathcal {I}^C \\setminus \\text{pa}(Y)}]\\big ) \\ p\\big (x_{\\text{pa}(Y) \\setminus \\mathcal {I}} \\mid do(X_\\mathcal {I} = z_\\mathcal {I})\\big ) \\ \\mathrm {d}x_{\\text{pa}(Y) \\setminus \\mathcal {I}}\\\\= & \\int f^*\\big ([x_{\\text{pa}(Y) \\setminus \\mathcal {I}} , z_{\\mathcal {I} \\cap \\text{pa}(Y)} , a_{\\mathcal {I}^C \\setminus \\text{pa}(Y)}]\\big ) \\ p\\big (x_{\\text{pa}(Y) \\setminus \\mathcal {I}} \\big )\\ \\mathrm {d}x_{\\text{pa}(Y) \\setminus \\mathcal {I}}\\\\& \\text{ since the marginal distribution over } X_{\\text{pa}(Y) \\setminus \\mathcal {I}} \\text{ equals the \\emph {do}-distribution by assumption (A\\ref {as:bestdo})}\\\\= & \\int \\int f^*\\big ([x_{\\text{pa}(Y) \\setminus \\mathcal {I}} , z_{\\mathcal {I} \\cap \\text{pa}(Y)} , x_{\\mathcal {I}^C \\setminus \\text{pa}(Y)}]\\big ) \\ p\\big (x_{\\mathcal {I}^C \\setminus \\text{pa}(Y)}\\mid x_{\\text{pa}(Y) \\setminus \\mathcal {I}}\\big )\\ p\\big ( x_{\\text{pa}(Y) \\setminus \\mathcal {I}} \\big ) \\ \\mathrm {d}x_{\\mathcal {I}^C \\setminus \\text{pa}(Y)} \\ \\mathrm {d}x_{\\text{pa}(Y) \\setminus \\mathcal {I}}\\\\= & \\mathbb {E}_X \\Big [ f^*(T_{\\mathcal {I}\\rightarrow z}(x)) \\Big ]$ tocsubsubsectionproof:bestdofound Recall we defined: $\\mathcal {I^*} := \\operatornamewithlimits{argmin}\\Big \\lbrace |\\mathcal {I}^{\\prime }| \\ \\text{ s.t. }", "\\ \\exists \\ T_{\\mathcal {I}^{\\prime }\\rightarrow z} \\in \\operatornamewithlimits{argmax}_{T_{\\mathcal {I}\\rightarrow z} : |\\mathcal {I}| \\le k} \\mathbb {E}_X \\big [ f^*(T_{\\mathcal {I}\\rightarrow z}(x)) - f^*(x) \\big ] \\Big \\rbrace $ as the intervention set corresponding to the optimal sparse covariate-fixing transformation (taken to be the set of minimal cardinality in cases with multiple maxima).", "Since $\\mathbb {E}_X [f^*(T_{\\mathcal {I}\\rightarrow z}(x))]$ does not change when $z_j := [T_{\\mathcal {I}\\rightarrow z}(x)]_j$ is altered for any $j \\notin \\text{pa}(Y)$ , including variables outside of the parent set in $\\mathcal {I}$ does not improve this quantity.", "Thus, either $ \\text{pa}(Y) \\subseteq \\mathcal {I}^*$ , or $\\mathcal {I}^* \\subset \\text{pa}(Y)$ .", "The first case immediately implies (AREF ).", "When $\\mathcal {I}^* \\subset \\text{pa}(Y)$ : our assumption that no variable in $\\text{pa}(Y)$ is a descendant of other parents implies the other parents must belong the complement of $\\text{desc}(\\mathcal {I}^*)$ , since this is a subset of $\\text{desc}\\big (\\text{pa}(Y)\\big )$ .", "tocsubsubsectionsupsec:thmnotbad Theorem 6 Suppose we adopt a GP$\\big (0, k(x,x^{\\prime })\\big )$ prior and, in addition to the assumptions outlined in §, the following conditions hold: (A9) $f^* \\in \\mathcal {H}_k(\\mathcal {C})$ which is the RKHS induced by our covariance function $k$ with norm $||\\cdot ||_k$ (cf.", "Rasmussen2006si §6.1), (A10) noise variables $\\varepsilon ^{(i)}$ form a uniformly bounded martingale difference sequence $\\varepsilon ^{(i)} \\le \\sigma $ for $i= 1,\\dots , n$ .", "Then, for any $\\displaystyle x, T(x) \\in \\mathcal {C} : \\ \\ F^{-1}_{G_x(T)}(\\alpha ) \\le G^*_x(T) $ $\\text{with probability (over the noise) greater than } \\ 1 - C(n+1) \\cdot \\exp \\left( -\\frac{[\\Phi ^{-1}(\\alpha )]^2 - 2 ||f^*||^2_k}{\\gamma _n} \\right)$ In Theorem REF , $\\displaystyle \\gamma _n := \\max _{A \\subset \\mathcal {C} : |A| = n} \\frac{1}{2}\\log \\big | \\mathbf {I} + \\sigma ^{-2} \\mathbf {K}_A \\big |$ is a kernel-dependent quantity ($\\mathbf {K}_A := [k(x,x^{\\prime })]_{x,x^{\\prime } \\in A}$ ) which, in the Gaussian setting, is the mutual information between $f$ and observations of $Y$ at the most informative choice of $n$ points.", "When the kernel satisfies $k(x, x^{\\prime }) \\le 1$ , the following bounds are known Srinivas2010si: $\\gamma _n = \\mathcal {O}(d \\log n)$ for the linear kernel, $\\gamma _n = \\mathcal {O}((\\log n)^{d+1})$ for the squared exponential kernel, and $\\gamma _n = \\mathcal {O}(n^{d(d+1)/(2\\nu + d(d+1))} (\\log n))$ for the Matérn kernel with smoothness parameter $\\nu $ .", "Note that while $f^*$ is not required to be drawn from our prior and $\\varepsilon $ may be non-Gaussian, this result assumes the kernel $k$ and noise-level $\\sigma $ are correctly set.", "Our proof relies on the following statement: Theorem 7 (Srinivas2010si) Assume conditions (AREF ) - (AREF ), fix $\\delta \\in (0,1)$ , and define: $\\displaystyle \\beta _n := 2 || f^* ||^2_k + 300 \\gamma _n [\\log (n / \\delta )]^3 $ Then: $\\begin{aligned}[t]\\Pr \\Big [ \\forall x \\in \\mathcal {C} : \\ | \\mu _n (x) - f^*(x) | \\le \\sqrt{\\beta _{n+1}} \\sigma _n(x) \\Big ] \\ge 1- \\delta \\end{aligned}$ [Proof of Theorem REF ] Fix $x, T(x) \\in \\mathcal {C}$ , and define $\\displaystyle \\delta := (n+1) \\cdot \\exp \\left( -\\frac{[\\Phi ^{-1}(\\alpha )]^2 - 2 ||f^*||^2_k}{300 \\gamma _n} \\right)$ .", "In this case, $-\\sqrt{\\beta _{n+1}} = \\Phi ^{-1}(\\alpha )$ (see definition in previous theorem).", "Theorem REF implies that with probability $\\ge 1 - \\delta $ : $ | \\mu _n (x) - f^*(x) | \\le -\\Phi ^{-1}(\\alpha ) \\cdot \\sigma _n(x) $ and $| \\mu _n (T(x)) - f^*(T(x)) | \\le -\\Phi ^{-1}(\\alpha ) \\cdot \\sigma _n(T(x)) $ Since our posterior is Gaussian: $ F^{-1}_{G_x(T)} (\\alpha ) = \\mu _n(T(x)) - \\mu _n(x) + \\Phi ^{-1}(\\alpha ) \\bigg [\\sigma ^2_n(T(x)) + \\sigma ^2_n(x) - 2 \\sigma _n(x, T(x)) \\bigg ]^{1/2}$ Therefore: $& f^*(T(x)) - f^*(x) - F^{-1}_{G_x(T)} (\\alpha )\\\\= & f^*(T(x)) - \\mu _n(T(x)) + \\mu _n(x) - f^*(x) - \\Phi ^{-1}(\\alpha ) \\bigg [\\sigma ^2_n(T(x)) + \\sigma ^2_n(x) - 2 \\sigma _n(x, T(x)) \\bigg ]^{1/2} \\\\\\le & f^*(T(x)) - \\mu _n(T(x)) + \\mu _n(x) - f^*(x) - \\Phi ^{-1}(\\alpha ) \\bigg [\\sigma ^2_n(T(x)) + \\sigma ^2_n(x) + 2 \\sqrt{ \\sigma ^2_n(x) \\sigma ^2_n(T(x))} \\bigg ]^{1/2}\\\\& \\\\= & f^*(T(x)) - \\mu _n(T(x)) + \\mu _n(x) - f^*(x) - \\Phi ^{-1}(\\alpha ) \\bigg [\\sigma _n(T(x)) + \\sigma _n(x) \\bigg ]\\\\= & \\big [f^*(T(x)) - \\mu _n(T(x)) - \\Phi ^{-1}(\\alpha ) \\sigma _n(T(x)) \\big ] + \\big [ \\mu _n(x) - f^*(x) - \\Phi ^{-1}(\\alpha ) \\sigma _n(x) \\big ]$ which is less than 0 with probability at most $\\delta $ .", "agsm InterventionOptBibliography" ] ]
1606.05027
[ [ "Investigation of potential fluctuating intra-unit cell magnetic order in\n cuprates by muon spin relaxation" ], [ "Abstract We report low temperature muon spin relaxation (muSR) measurements of the high-transition-temperature (Tc) cuprate superconductors Bi{2+x}Sr{2-x}CaCu2O{8+\\delta} and YBa2Cu3O6.57, aimed at detecting the mysterious intra-unit cell (IUC) magnetic order that has been observed by spin polarized neutron scattering in the pseudogap phase of four different cuprate families.", "A lack of confirmation by local magnetic probe methods has raised the possibility that the magnetic order fluctuates slowly enough to appear static on the time scale of neutron scattering, but too fast to affect $\\mu$SR or nuclear magnetic resonance (NMR) signals.", "The IUC magnetic order has been linked to a theoretical model for the cuprates, which predicts a long-range ordered phase of electron-current loop order that terminates at a quantum crictical point (QCP).", "Our study suggests that lowering the temperature to T ~ 25 mK and moving far below the purported QCP does not cause enough of a slowing down of fluctuations for the IUC magnetic order to become detectable on the time scale of muSR.", "Our measurements place narrow limits on the fluctuation rate of this unidentified magnetic order." ], [ " Investigation of potential fluctuating intra-unit cell magnetic order in cuprates by $\\mu $ SR A. Pal,$^1$ K. Akintola,$^1$ M. Potma,$^1$ M. Ishikado,$^2$ H. Eisaki,$^3$ W.N.", "Hardy,$^{4,5}$ D.A.", "Bonn,$^{4,5}$ R. Liang,$^{4,5}$ and J.E.", "Sonier,$^{1,5}$ $^1$ Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada $^2$ Research Center for Neutron Science and Technology, Tokai, Naka, Ibaraki, Japan 319-1106 $^3$ National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki, Japan 305-8568 $^4$ Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada $^5$ Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada We report low temperature muon spin relaxation ($\\mu $ SR) measurements of the high-transition-temperature ($T_c$ ) cuprate superconductors Bi$_{2+x}$ Sr$_{2-x}$ CaCu$_2$ O$_{8+\\delta }$ and YBa$_2$ Cu$_3$ O$_{6.57}$ , aimed at detecting the mysterious intra-unit cell (IUC) magnetic order that has been observed by spin polarized neutron scattering in the pseudogap phase of four different cuprate families.", "A lack of confirmation by local magnetic probe methods has raised the possibility that the magnetic order fluctuates slowly enough to appear static on the time scale of neutron scattering, but too fast to affect $\\mu $ SR or nuclear magnetic resonance (NMR) signals.", "The IUC magnetic order has been linked to a theoretical model for the cuprates, which predicts a long-range ordered phase of electron-current loop order that terminates at a quantum crictical point (QCP).", "Our study suggests that lowering the temperature to $T \\!", "\\sim \\!", "25$  mK and moving far below the purported QCP does not cause enough of a slowing down of fluctuations for the IUC magnetic order to become detectable on the time scale of $\\mu $ SR. Our measurements place narrow limits on the fluctuation rate of this unidentified magnetic order.", "74.72.-h, 74.25.Ha, 76.75.+i An enduring and central open question concerning cuprate superconductors is the nature of the mysterious pseudogap regime above $T_c$ .", "Achieving an understanding of the pseudogap (PG) has long been viewed as key to understanding high-$T_c$ superconductivity.", "A clue to the origin of the PG has come from spin-polarized neutron diffraction studies that have detected the onset of an unusual three-dimensional (3-D), long-range IUC magnetic order at a temperature concomitant with the PG onset temperature $T^*$ in YBa$_2$ Cu$_3$ O$_{6+x}$ (Y123), HgBa$_2$ CuO$_{4+\\delta }$ (Hg1201) and Bi$_2$ Sr$_2$ CaCu$_2$ O$_{8+\\delta }$ (Bi2212).", "[1], [2], [3], [4], [5], [6], [7] This finding provides evidence for a change in symmetry at $T^*$ associated with the onset of a novel type of order, which is supported by other kinds of measurements that indicate the the PG is related to a true phase transition.", "[9], [10], [11], [12] The magnetic order observed by polarized neutron diffraction is described by staggered out-of-plane magnetic moments that diminish in magnitude from the underdoped to optimally-doped regime.", "[7], [8] A similar mysterious magnetic order is also observed in $x \\!", "= \\!", "0.085$ La$_{2-x}$ Sr$_x$ CuO$_4$ (LSCO),[13] although it is short-range, two-dimensional, and onsets at a temperature far below $T^*$ .", "The latter is also the case in underdoped YBa$_2$ Cu$_3$ O$_{6.45}$ — suggesting a potential competition with Cu spin density wave order at low doping.", "The magnetic structure and the hole-doping dependence of the onset temperature of the IUC magnetic order are somewhat compatible with a model derived from a three-band Hubbard model, which attributes the PG to a time-reversal symmetry breaking phase consisting of a pattern of circulating electron currents that preserve translational symmetry.", "[14] With increased hole doping the transition temperature of the circulating-current (CC) ordered phase is reduced towards zero, terminating at a QCP within the superconducting phase near or above optimal doping.", "Yet zero-field (ZF) $\\mu $ SR experiments have found no evidence for such a magnetically ordered phase.", "[15], [16], [17], [18] While it has been suggested that charge screening of the positively charged muon ($\\mu ^+$ ) causes severe underdoping of its local environment, resulting in the loss of CC order over a distance of several lattice constants,[19] such severe perturbation of the local environment is inconsistent with $\\mu ^+$ -Knight shift measurements that show a linear scaling with the bulk magnetic susceptibility.", "[20] Moreover, non-perturbative NMR and nuclear quadupole resonance (NQR) experiments also find no evidence of IUC magnetic order.", "[22], [23], [24], [25] It has been argued from calculations in a multi-orbital Hubbard model and for parameters relevant to cuprate superconductors, that the CC phase proposed in Ref.", "Varma:97 or variations of it are unlikely to be stabilized as the ground state.", "[26] A staggered ordering of Ising-like oxygen orbital magnetic moments has been offered as an alternative explanation of the IUC magnetic order.", "[27] Since the original CC phase proposal, the model has been extended to include quantum critical fluctuations of the CC order parameter.", "[28], [29] The extended model attributes the anomalous normal-state properties of cuprates to a funnel-shaped quantum critical region in the $T$ -versus-$p$ phase diagram that extends to temperatures well above the QCP at $p \\!", "= \\!", "p_c$ , $T \\!", "= \\!", "0$ .", "In the quantum-critical region the CC order spatially and temporally fluctuates between four possible ground-state configurations characterized by different directions of the CC order parameter.", "Local disorder is argued to couple to the CC order, leading to four distinct domains consisting of one of the four possible CC order configurations.", "The fluctuation rate between the different CC order configurations has been estimated to be slow enough to appear static on the time scale of neutron scattering, but too fast to cause relaxation of $\\mu $ SR or NMR spectra.", "[30] One exception to the null local-probe results is a ZF-$\\mu $ SR study of a large YBa$_2$ Cu$_3$ O$_{6.6}$ single crystal in which the unusual 3-D IUC magnetic order has been detected by polarized neutron scattering.", "[17] Static magnetic order with an onset temperature and local magnetic field consistent with the neutron findings was observed, but only in $\\sim \\!", "3$  % of the sample.", "This raises the possibility of fluctuating IUC magnetic order (that is not necessarily CC order) being locally pinned in a static configuration by disorder.", "The impurity/disorder type must be fairly specific though, since it has been shown that Zn substitution of Cu in YBa$_2$ Cu$_3$ O$_{6.6}$ does not affect the magnetic-onset temperature, but does reduce the magnetic Bragg scattering intensity.", "[4] In other words, the Zn impurity apparently reduces the volume of the sample containing the IUC magnetic order.", "Here we investigate whether there is fluctuating IUC magnetic order that slows down enough near $T \\!", "= \\!", "0$ , where thermal fluctuations vanish, to become detectable by ZF-$\\mu $ SR.", "If the mysterious magnetic order is associated with a QCP, then near $T \\!", "= \\!", "0$ we expect quantum fluctuations to dominate close to $p_c$ , but in the absence of significant disorder to have a diminishing effect as the hole concentration is lowered.", "The neutron experiments on Y123 and Hg1201 suggest $p_c \\!", "\\sim \\!", "0.19$ , and previous ZF-$\\mu $ SR measurements on Y-doped Bi2212, pure LSCO, and Zn-doped LSCO, extending down to 40 mK show a vanishing of low-frequency spin fluctuations above this critical doping.", "[31] However, a similar ZF-$\\mu $ SR study down to such low temperatures has not been performed on the other cuprates in which IUC magnetic order has been detected by neutrons.", "An exception are ZF-$\\mu $ SR measurements on a $p \\!", "\\sim \\!", "0.167$ Bi2212 powdered sample, which indicate the onset of spin fluctuations below $T \\!", "\\sim \\!", "5$  K, but no spin freezing down to 40 mK.", "[31] ZF-$\\mu $ SR measurements with the initital muon spin polarization P(0) parallel to the $\\hat{c}$ -axis were performed on underdoped ($p \\!", "= \\!", "0.094$ , $T_c \\!", "= \\!", "58$  K) and optimally-doped ($p \\!", "= \\!", "0.16$ , $T_c \\!", "= \\!", "90$  K) Bi2212 single crystals, and single crystals of underdoped ($p \\!", "= \\!", "0.11$ , $T_c \\!", "= \\!", "62.5$  K) YBa$_2$ Cu$_3$ O$_{6.57}$ .", "The samples were prepared as described elsewhere.", "[32], [33] Spectra were collected down to as low as $T \\!", "= \\!", "24$  mK using a dilution refrigerator on the M15 surface muon beam line at the TRIUMF subatomic physics laboratory in Vancouver, Canada.", "The single crystals were mounted on a silver (Ag) sample holder, covering a 8 mm $\\times $  5 mm area.", "A scintillation detector placed downstream was used to reject muons that missed the sample.", "A fraction ($\\le \\!", "40$  %) of the incoming muons stopped in the uncovered portion of the Ag sample holder, and a fraction ($\\sim \\!", "20$  %) of the muons stopped in the copper (Cu) heat shields of the dilution refrigerator.", "Since the nuclear dipole fields in Ag are negligible, there is no appreciable time or temperature dependence to the background component from the sample holder.", "While the relaxation rate of the ZF-$\\mu $ SR signal from Cu does have a temperature dependence caused by muon diffusion,[34] the Cu shields are at constant temperature.", "We also performed longitudinal-field (LF) $\\mu $ SR measurements on $p \\!", "= \\!", "0.11$ Y123 single crystals at a fixed temperature far below $T_c$ using a helium-gas flow cryostat and low-background sample holder, for the purpose of determining whether the internal magnetic fields are static or dynamic.", "In this setup there is no Cu component and the background contribution to the LF-$\\mu $ SR signal is less than 20 %.", "Figure: (Color online) Representative normalized ZF-μ\\mu SR asymmetry spectra for underdoped YBa 2 _2Cu 3 _3O 6.57 _{6.57} single crystals.These spectra were recorded with the initial muon spin polarization P(0) parallel to the c ^\\hat{c}-axis.The solid curves through the data points are fits to Eq.", "(), assuming Eq.", "()for the relaxation function G s (t)G_{\\rm s}(t).", "The inset shows the ZF-μ\\mu SR spectrum for T=27T \\!", "= \\!", "27 mK at early times.The solid green curve simulates the presence of a 3 % damped-oscillating contribution to the sample componentassuming a mean local field of 141 G.The ZF-$\\mu $ SR asymmetry spectra were fit to the sum of sample and backgrounds terms as follows $A(t)=a_{\\rm s}G_{\\rm s}(t) + a_{\\rm b}G_{\\rm b}(t) \\, ,$ where $a_{\\rm s}$ and $G_{\\rm s}(t)$ [$a_{\\rm b}$ and $G_{\\rm b}(t)$ ] are the amplitude and ZF relaxation function for the sample (background) contribution.", "The background term originating from muons stopping outside of the sample was assumed to be independent of temperature and approximately described by the following relaxation function $G_{\\rm b}(t) = G_{z}^{\\rm KT}(\\Delta _{\\rm b}, t) \\, ,$ where $G_{z}^{\\rm KT}(\\Delta _{\\rm b}, t)$ is a static Gaussian Kubo-Toyabe function.", "In particular, $G_{z}^{\\rm KT}(\\Delta _{\\rm b}, t) = \\dfrac{1}{3} +\\dfrac{2}{3}(1-\\Delta _{\\rm b}^2 t^2) \\exp \\left(-\\dfrac{1}{2}\\Delta _{\\rm b}^2 t^2 \\right) \\, ,$ where $\\gamma _\\mu $ is the muon gyromagnetic ratio and ${\\Delta _{\\rm b}}/{\\gamma _\\mu }$ is the width of the Gaussian distribution in field sensed by the implanted muon ensemble.", "The sample contribution was assumed to be the product of two relaxation functions $G_{\\rm s}(t) = G_{z}^{\\rm KT}(\\Delta _{\\rm s}, t) \\exp (-\\lambda t) \\, ,$ which assumes that muons stopping in the sample sense the vector sum of static nuclear dipolar fields and fields of some other origin that generate a weak exponential relaxation rate $\\lambda $ .", "An exception is Bi2212 at $p \\!", "= \\!", "0.094$ , where the ZF-$\\mu $ SR asymmetry spectra below $T \\!", "= \\!", "1$  K were better described by $G_{\\rm s}(t) = [f \\exp (-\\eta t) + (1-f)]G_{z}^{\\rm KT}(\\Delta _{\\rm s}, t) \\exp (-\\lambda t) \\, .$ This function assumes an enhanced exponential relaxation rate $\\lambda \\!", "+ \\!", "\\eta $ due to a fraction $f$ of the muons experiencing additional fields in some parts of the sample.", "In contrast to the relaxation rates $\\Delta _{\\rm s}$ and $\\Delta _{\\rm b}$ , the exponential relaxation rates $\\lambda $ and $\\eta $ were allowed to vary with temperature in the fitting of the ZF-$\\mu $ SR spectra.", "In addition, $f$ was assumed to be independent of temperature.", "Figure: (Color online) Representative normalized ZF-μ\\mu SR asymmetry spectra for (a) underdoped and (b) optimally-dopedBi2212 single crystals.", "The insets show the ZF-μ\\mu SR spectra for the lowest temperature at early times.These spectra were recorded with the initial muon spin polarization parallel to the c ^\\hat{c}-axis.The solid curves through the data points are fits to Eq.", "(), assuming Eq.", "()for the relaxation function G s (t)G_{\\rm s}(t).", "An exception is the solid curve for the p=0.094p \\!", "= \\!", "0.094 sample atT=24T \\!", "= \\!", "24 mK, which is a fit assuming Eq.", "() for G s (t)G_{\\rm s}(t).The insets show the ZF-μ\\mu SR spectra for T=24T \\!", "= \\!", "24 mK at early times.The solid green curves simulate the presence of a 3 % damped-oscillating contribution to the sample componentassuming a mean local field of 141 G.Figure: (Color online) Temperature dependence of the ZF exponential relaxation rate λ\\lambda (open symbols).Also shown is the enhanced exponential relaxation rate λ+η\\lambda \\!", "+ \\!", "\\eta (solid circles)for p=0.094p \\!", "= \\!", "0.094 Bi-2212 below T=1T \\!", "= \\!", "1 K, which is due to a fraction (34.6 %) of the implantedmuons experiencing additional internal magnetic fields.Figures REF and REF show representative ZF-$\\mu $ SR asymmetry spectra for the Y123 and Bi2212 samples.", "The fits described above yielded $\\Delta _{\\rm s} \\!", "= \\!", "0.156(1)$  $\\mu $ s$^{-1}$ and $\\Delta _{\\rm b} \\!", "= \\!", "0.405(6)$  $\\mu $ s$^{-1}$ for the $p \\!", "= \\!", "0.11$ Y123 sample, and $\\Delta _{\\rm s} \\!", "= \\!", "0.134(6)$  $\\mu $ s$^{-1}$ and $\\Delta _{\\rm b} \\!", "= \\!", "0.395(1)$  $\\mu $ s$^{-1}$ ($\\Delta _{\\rm s} \\!", "= \\!", "0.134(6)$  $\\mu $ s$^{-1}$ and $\\Delta _{\\rm b} \\!", "= \\!", "0.393(6)$  $\\mu $ s$^{-1}$ ) for the $p \\!", "= \\!", "0.094$ ($p \\!", "= \\!", "0.16$ ) Bi2212 sample.", "The values of $\\Delta _{\\rm b}$ are consistent with the background relaxation being dominated by the Cu heat shields.", "Below $T \\!", "= \\!", "1$  K the fits of the $p \\!", "= \\!", "0.094$ Bi2212 ZF-$\\mu $ SR signals yielded $f \\!", "= \\!", "0.346$ , indicating that about one third of the muons implanted in the sample sense local magnetic fields in addition to the host magnetic nuclear dipole moments.", "The ZF-$\\mu $ SR spectra do not exhibit an oscillatory component indicative of long-range magnetic order.", "The presence of a magnetically-ordered state with a broad distribution of local magnetic fields or a small magnetically-ordered volume fraction would result in a rapidly damped oscillatory signal.", "The insets of Figs.", "REF and REF show the ZF-$\\mu $ SR signal for $T \\!", "\\le \\!", "27$  mK plotted over the first 1.5 $\\mu $ s. While there is no apparent oscillatory component, simulations of a 3 % magnetically-ordered phase of the kind observed in the large YBa$_2$ Cu$_3$ O$_{6.60}$ single crystal in Ref.", "Sonier:09 superimposed on the early-time ZF-$\\mu $ SR spectra (green curves in the insets of Figs.", "REF and REF ) show that a small $0.4 \\!", "\\times \\ 3$  % $\\!", "= \\!", "1.2$  % contribution to the total signal cannot be ruled out.", "However, it is worth mentioning that no such minority phase was previously observed in low-background measurements of the $p \\!", "= \\!", "0.11$ Y123 sample above $T \\!", "= \\!", "2.3$  K.[17] Figure: (Color online) (a) Normalized LF-μ\\mu SR asymmetry spectra for underdoped YBa 2 _2Cu 3 _3O 6.57 _{6.57} singlecrystals at T=5T \\!", "= \\!", "5 K, recorded with the initial muon spin polarization P(0) parallel to the c ^\\hat{c}-axis.", "The circles, nablas,and triangles correspond to data recorded in external longitudinal magnetic fields of H=0H \\!", "= \\!", "0, 0.5 and 1.5 kOe, respectively.The green curve is the sum of a pure exponential function and a constant, 0.8exp(-λ LF t)+0.20.8\\exp (-\\lambda _{\\rm LF} t) \\!", "+ \\!", "0.2, whereλ LF =0.01\\lambda _{\\rm LF} \\!", "= \\!", "0.01 μ\\mu s -1 ^{-1}.", "(b) Calculated relaxation rate λ LF \\lambda _{\\rm LF} from Eq.", "() for a longitudinal magnetic field of 0.50.5 kOe anddifferent values of the local-field fluctuation amplitude Δ/γ μ \\Delta /\\gamma _{\\mu }.The dashed green line indicates the maximum value of λ LF \\lambda _{\\rm LF} inferred from the corresponding LF-μ\\mu SR spectrum in (a).Figure REF shows the temperature dependence of the exponential relaxation rate $\\lambda $ for all three samples, along with $\\lambda \\!", "+ \\!", "\\eta $ for $p \\!", "= \\!", "0.094$ Bi2212 below $T \\!", "= \\!", "1$  K. While there is an increase in the relaxation rate for the $p \\!", "= \\!", "0.094$ Bi2212 sample below $T \\!", "= \\!", "1$  K, this is most likely due to low-energy spin fluctuations in the CuO$_2$ planes, as spin freezing is observed in Y-doped Bi2212 below $p \\!", "\\sim \\!", "0.10$ .", "[31] The lack of any increase of $\\lambda $ at low temperatures for the $p \\!", "= \\!", "0.11$ Y123 and the $p \\!", "= \\!", "0.16$ Bi2212 samples rules out the onset of quasi-static magnetism below $T \\!", "= \\!", "5$  K. However, these ZF-$\\mu $ SR results do not rule out the possibility that even at these low temperatures and at a hole doping far below $p_c$ , the IUC magnetic order fluctuates too fast to be detectable on the time scale of ZF-$\\mu $ SR.", "Assuming the local magnetic field due to IUC magnetic order is 141 G (as estimated in Ref.", "Sonier:09), the ZF-$\\mu $ SR results for $p \\!", "= \\!", "0.11$ Y123 and $p \\!", "= \\!", "0.16$ Bi2212 imply a lower limit of $1.9 \\!", "\\times \\!", "10^6$  Hz for the fluctuation rate.", "This is far below the upper limit of $10^{11}$  Hz imposed by the energy resolution of the polarized neutron experiments.", "Our LF-$\\mu $ SR measurements in a different experimental setup greatly increase the lower limit of the fluctuation rate.", "Figure REF (a) shows LF-$\\mu $ SR spectra recorded for $p \\!", "= \\!", "0.11$ Y123 well below $T_c$ .", "Below $T_c$ weak applied fields are completely or partially screened from the bulk, and hence external fields well in excess of the lower critical field $H_{c1}$ were applied.", "A longitudinal field of $B_{\\rm LF} \\!", "= \\!", "0.5$  kOe completely decouples the muon spin from the nuclear dipoles of the background and the internal magnetic fields of the sample.", "If the muons sense a rapidly fluctuating nearly Gaussian distribution of field, the ZF-$\\mu $ SR signal will decay with a pure exponential relaxation $G_{\\rm s}(t) \\!", "= \\!", "\\exp (- \\lambda t)$ .", "In this case the dependence of the dynamic relaxation rate on the LF is given by the Redfield equation[35] $\\lambda _{\\rm LF} = \\frac{2 \\Delta ^2/\\nu }{1 + (\\gamma _{\\mu } B_{\\rm LF}/\\nu )^2} \\, ,$ where $\\Delta /\\gamma _{\\mu }$ is the width of the field distribution and $\\nu $ is the local-field fluctuation frequency.", "In the previous $\\mu $ SR study of a large single crystal of YBa$_2$ Cu$_3$ O$_{6.6}$ in which static magnetic order was detected in 3 % of the sample,[17] the mean local field detected was $\\sim \\!", "141$  G — which was shown to be in good agreement with the magnitude and direction of the ordered moment determined by polarized neutron diffraction.", "Figure REF (b) shows a simulation of the dependence of $\\lambda _{\\rm LF}$ on $\\nu $ for a LF of $B_{\\rm LF} \\!", "= \\!", "500$  G and different values of the local-field fluctuation amplitude $\\Delta /\\gamma _{\\mu }$ .", "The values $\\Delta /\\gamma _{\\mu } \\!", "= \\!", "141$  G and 24 G assume the polarized neutron measurements of $p \\!", "= \\!", "0.11$ Y123 (Ref.", "Mook:08) detect IUC magnetic order within the CuO$_2$ planes in 3 % and 100 % of the sample, respectively.", "Also shown is the upper limit $\\lambda _{\\rm LF} \\!", "\\le \\!", "0.01$  $\\mu $ s$^{-1}$ inferred from the corresponding LF-$\\mu $ SR spectrum in Fig.", "REF (a) under the assumption that fluctuating magnetism occurs throughout the sample.", "The simulation of $\\lambda _{\\rm LF}$ for $\\Delta /\\gamma _{\\mu } \\!", "= \\!", "141$  G exceeds the upper limit of the relaxation rate observed in $p \\!", "= \\!", "0.11$ Y123 below $\\nu \\!", "\\sim \\!", "3 \\!", "\\times \\!", "10^{10}$  Hz.", "On the other hand, the simulation for $\\Delta /\\gamma _{\\mu } \\!", "= \\!", "24$  G only rules out a fluctuation rate below $\\nu \\!", "\\sim \\!", "10^{9}$  Hz.", "If the IUC magnetic order is due to loop currents flowing out of the CuO$_2$ plane through the apical oxygen as proposed in Ref.", "Weber:09, $\\Delta /\\gamma _{\\mu } \\!", "= \\!", "22$  G and the lower limit of the fluctuation rate is slightly reduced.", "Regardless, the combined LF-$\\mu $ SR results and the polarized neutron measurements place narrow limits of $10^{9}$ to $10^{11}$  Hz on the fluctuation rate of the IUC magnetic order.", "If the IUC magnetic order is associated with fluctuations between different orientations of a CC-ordered state in finite size domains, rather than spatially-uniform long range magnetic order, quantum fluctuations will not diminish away from the QCP.", "[30] The lowest quantum fluctuation frequency between the distinct CC configurations is estimated to be less than 10$^{10}$  Hz — a scenario not completely ruled out by our LF-$\\mu $ SR results.", "As for other possible origins of the IUC magnetic order, while our estimated lower limit of the fluctuation frequency assumes fluctuating magnetic order throughout the sample volume, the current measurements do not rule out the possibility that there is slower fluctuating IUC magnetic order contained in a small volume fraction.", "We thank the staff of TRIUMF's Centre for Molecular and Materials Science for technical assistance.", "JES, WNH, DAB, and RL acknowledge support from CIFAR and NSERC of Canada." ] ]
1606.04865
[ [ "Atom-light interactions in quasi-1D nanostructures: a Green's function\n perspective" ], [ "Abstract Based on a formalism that describes atom-light interactions in terms of the classical electromagnetic Green's function, we study the optical response of atoms and other quantum emitters coupled to one-dimensional photonic structures, such as cavities, waveguides, and photonic crystals.", "We demonstrate a clear mapping between the transmission spectra and the local Green's function that allows to identify signatures of dispersive and dissipative interactions between atoms.", "We also demonstrate the applicability of our analysis to problems involving three-level atoms, such as electromagnetically induced transparency.", "Finally we examine recent experiments, and anticipate future observations of atom-atom interactions in photonic bandgaps." ], [ "Introduction", "As already noticed by Purcell in the first half of the past century, the decay rate of an atom can be either diminished or enhanced by tailoring its dielectric environment [1], [2], [3].", "Likewise, by placing more than one atom in the vicinity of photonic nanostructures, one can curtail or accelerate their collective decay.", "In addition to modifying the radiative decay, nanophotonic structures can be employed to spatially and spectrally engineer atom-light interactions, thus obtaining fundamentally different atom dynamics to those observed in free-space [4].", "In the past decade, atoms and other quantum emitters have been interfaced with the electromagnetic fields of a plethora of quasi-1D nanostructured reservoirs, ranging from high-quality optical [5], [6], [7], [8], [9], [10] and microwave [11], [12] cavities to dielectric [13], [14], [15], [16], [17], [18], [19], [20], metallic [21], [22], [23], [24], and superconducting [25], [26] waveguides.", "Photonic crystal waveguides, periodic dielectric structures that display a bandgap where light propagation is forbidden [27], [28], have been proposed as promising candidates to study long- and tunable-range coherent interactions between quantum emitters [29], [30], [31], [32].", "Due to the different character of the guided modes at various frequencies within the band structure of the photonic crystal, the interaction of the quantum emitters with the nanostructure can be remarkably distinct depending on the emitter resonance frequency.", "Far away from the bandgap, where light propagates, the guided modes resemble those of a conventional waveguide.", "Close to the bandgap, but still in the propagating region, the fields are similar to those of a quasi-1D cavity, whereas inside the bandgap the fields become evanescent, decaying exponentially.", "All these regimes have been recently explored in the lab, where atoms [33], [34], [35] and quantum dots [36], [37], [4] have been interfaced with photonic crystal waveguides.", "Most of these experiments have been performed in conditions where the resonance frequency of the emitter lies outside the bandgap.", "However, very recently, the first experiments of atoms [38] and superconducting qubits [39] interacting with evanescent modes in the bandgap of photonic crystal waveguides have been reported.", "Within this context, it has become a necessity to understand the rich spectral signatures of atom-like emitters interacting through the guided modes of quasi one-dimensional nanophotonic structures within a unified framework that extends beyond those of cavity [40] or waveguide QED [41].", "In this work, we employ a formalism based on the classical electromagnetic Green's function [42], [43], [44], [45], [46] to characterize the response of atoms that interact by emitting and absorbing photons through the guided mode of the nanostructure.", "Since the fields in the vicinity of the structure might have complex spatial and polarization patterns, the full Green's function is only known analytically for a handful of systems (such as planar multilayer stacks [47], infinite nanofibers [48], [49], [50], and a few more [44]) and beyond that one has to resort to numerical solvers of Maxwell's equations.", "However, in quasi-1D nanostructures, one can isolate the most relevant guided mode and build a simple prescription for the 1D Green's function that accounts for the behavior of this mode, greatly simplifying the problem.", "In the first part of the article, we summarize the procedure to obtain an effective atom-atom Hamiltonian, in which the guided-mode fields are effectively eliminated and the atom interactions are written in terms of Green's functions [42], [43], [44].", "We then apply this formalism to a collection of atoms in different quasi one-dimensional dielectric environments, and analyze the atomic transmission and reflection spectra in terms of the eigenvalues of the matrix consisting of the Green's functions between every pair of atoms.", "We show that, in the linear (low-saturation) regime, asymmetry in the transmission spectra and frequency shifts are signatures of coherent atom-light interactions, whereas symmetric lineshapes reveal dissipation.", "We also tackle the problem of three-level atoms coupled to quasi-1D nanostructures under electromagnetically induced transparency (EIT) conditions, and derive analytical expressions for the polaritonic band structure of such systems in terms of the eigenvalues of the Green's function matrix.", "Finally, based on the rapid technical advances in fabrication of both optical and microwave structures, we project observable signatures that can be made in the next generation of experiments of atoms and superconducting qubits interacting in the bandgap of photonic crystal waveguides." ], [ "Atom-light interactions in terms of Green's functions", "Much effort has gone into developing a quantum formalism to describe atoms coupled to radiation.", "A conventional technique is to express the field in terms of a set of eigenmodes of the system, with corresponding creation and annihilation operators $a^{\\dagger }$ and $a$ [41].", "This canonical quantization technique is well suited for approximately closed systems such as high-Q cavities and homogeneous structures such as waveguides, both of which have simple eigenmode decompositions.", "However, the application of this quantization scheme to more involved nanostructures is not straight forward.", "Further, the formalism is not suited for dispersive and absorbing media as the commutation relations for the field operators are not conserved [51].", "Instead, here we describe atom-light interactions using a quantization scheme based on the classical electromagnetic Green's function, valid for any medium characterized by a linear and isotropic dielectric function $\\epsilon ({\\bf r},\\omega )$ , closely following the work of Welsch and colleagues [42], [43], [44], [46].", "In the following, we employ this formalism to derive an atom-atom Hamiltonian in which the field is effectively eliminated, yielding an expression that only depends on atomic operators.", "Moreover, once the dynamics of the atoms is solved, the electric field at every point along the quasi one-dimensional structure can be recovered through an expression that relates the field to the atomic operators.", "Classically, the field ${\\bf E}({\\bf r},\\omega )$ at a point ${\\bf r}$ due to a source current $\\textbf {j}({\\bf r}^{\\prime },\\omega )$ at ${\\bf r}^{\\prime }$ is obtained by means of the propagator of the electromagnetic field, the dyadic Green's function (or Green's tensor), as ${\\bf E}({\\bf r},\\omega )={\\rm i}\\mu _0\\omega \\int d{\\bf r}^{\\prime } \\,{\\bf G}({\\bf r},{\\bf r}^{\\prime },\\omega )\\cdot \\textbf {j}({\\bf r}^{\\prime },\\omega )$ .", "In particular, for a dipole source ${\\bf p}$ located at ${\\bf r}_0$ , the current is $\\textbf {j}({\\bf r},\\omega )=-{\\rm i}\\omega {\\bf p}\\,\\delta ({\\bf r}-{\\bf r}_0)$ , and the field reads ${\\bf E}({\\bf r},\\omega )=\\mu _0\\omega ^2\\,{\\bf G}({\\bf r},{\\bf r}_0,\\omega )\\cdot {\\bf p}$ .", "The tensorial structure of the Green's function accounts for the vectorial nature of the electromagnetic field, as a dipole directed along the $\\hat{x}$ -direction can create a field polarized not only along $\\hat{x}$ , but also along $\\hat{y}$ and $\\hat{z}$ .", "Throughout this manuscript, the Green's tensor will be also denoted as Green's function.", "The Green's function ${\\bf G}({\\bf r},{\\bf r}^{\\prime },\\omega )$ is the fundamental solution of the electromagnetic wave equation, and obeys [52]: $\\mathbf {\\nabla }\\times \\mathbf {\\nabla }\\times {\\bf G}({\\bf r},{\\bf r}^{\\prime },\\omega )-\\frac{\\omega ^2}{c^2}\\epsilon ({\\bf r},\\omega )\\, {\\bf G}({\\bf r},{\\bf r}^{\\prime },\\omega )=\\delta ({\\bf r}-{\\bf r}^{\\prime })\\mathbb {1},$ where $\\epsilon ({\\bf r},\\omega )$ is the medium relative permittivity.", "For a scalar permittivity, Lorentz reciprocity holds and, then, ${\\bf G}^{\\rm T}({\\bf r},{\\bf r}^{\\prime },\\omega )={\\bf G}({\\bf r}^{\\prime },{\\bf r},\\omega )$ , where T stands for transpose (and operates on the polarization indexes).", "This formalism ignores the possibility of systems made non-reciprocal by the material response [53] (e.g., non-symmetric permittivity tensors), although it does in principle cover the interesting case of chiral atom-light interactions [54], [55], where the time-reversal symmetry breaking is due to the atomic states.", "In analogy to its classical counterpart, the electric field operator at frequency $\\omega $ can be written in terms of bosonic annihilation (creation) operators $\\hat{\\textbf {f}}$ ($\\hat{\\textbf {f}}^{\\dagger }$ ) as [42] $\\hat{{\\bf E}}({\\bf r},\\omega )&={\\rm i}\\mu _0\\,\\omega ^2 \\sqrt{\\frac{\\hbar \\epsilon _0}{\\pi }}\\int d{\\bf r}^{\\prime }\\,\\sqrt{\\text{Im}\\lbrace \\epsilon ({\\bf r}^{\\prime },\\omega )\\rbrace }\\, \\text{{\\bf G}}({\\bf r},{\\bf r}^{\\prime }\\omega )\\cdot \\hat{{\\bf f}} ({\\bf r}^{\\prime },\\omega )+ \\text{h.c.}=\\hat{{\\bf E}}^+({\\bf r},\\omega )+\\hat{{\\bf E}}^-({\\bf r},\\omega ),$ where $\\hat{{\\bf E}}^{+(-)}({\\bf r},\\omega )$ is the positive (negative) frequency component of the field operator, h.c. stands for Hermitian conjugate, and the total field operator reads $\\hat{{\\bf E}}({\\bf r})=\\int d\\omega \\,\\hat{{\\bf E}}({\\bf r},\\omega )$ .", "Within this quantization framework, $\\hat{\\textbf {f}}({\\bf r},\\omega )$ is associated with the degrees of freedom of local material polarization noise, which accompanies the material dissipation $\\text{Im}\\lbrace \\epsilon ({\\bf r},\\omega )\\rbrace $ as required by the fluctuation-dissipation theorem [46].", "This expression guarantees the fulfillment of the canonical field commutation relations, even in the presence of material loss.", "The appearance of the Green's function reveals that the quantumness of the system is encoded in either the correlations of the noise operators $\\hat{\\textbf {f}}$ or in any other quantum sources (such as atoms), but the field propagation obeys the wave equation and as such the spatial profile of the photons is determined by the classical propagator.", "We now want to investigate the evolution of $N$ identical two-level atoms of resonance frequency $\\omega _{\\rm A}$ that interact through a guided mode probe field of frequency $\\omega _{\\rm p}$ .", "Within the Born-Markov approximation, we trace out the photonic degrees of freedom, obtaining an effective atom-atom Hamiltonian [43], [56], [57].", "This approximation is valid when the atomic correlations decay much slower than the photon bath correlations, or, in other words, when the Green's function is characterized by a broad spectrum, which can be considered to be flat over the atomic linewidth.", "Then, the atomic density matrix $\\hat{\\rho }_{\\rm A}$ evolves according to $\\dot{\\hat{\\rho }}_{\\rm A}=-({\\rm i}/\\hbar )\\,[\\mathcal {H},\\hat{\\rho }_{\\rm A}]+\\mathcal {L}[\\hat{\\rho }_{\\rm A}]$ [40].", "Within the rotating wave approximation, and in the frame rotating with the probe field frequency, the Hamiltonian and Lindblad operators read $\\mathcal {H}=&-\\hbar \\Delta _{\\rm A}\\sum _{i=1}^N\\hat{\\sigma }_{ee}^i-\\hbar \\sum _{i,j=1}^N J^{ij}\\hat{\\sigma }_{eg}^i\\hat{\\sigma }_{ge}^j\\\\\\nonumber &-\\sum _{i=1}^N\\left(\\textbf {d}\\cdot \\hat{{\\bf E}}_{\\rm p}^- ({\\bf r}_i)\\,\\hat{\\sigma }_{ge}^i+\\textbf {d}^*\\cdot \\hat{{\\bf E}}_{\\rm p}^+({\\bf r}_i)\\,\\hat{\\sigma }_{eg}^i\\right),$ $\\mathcal {L}[\\hat{\\rho }_{\\rm A}]=\\sum _{i,j=1}^N\\frac{\\Gamma ^{ij}}{2}\\,\\left(2\\hat{\\sigma }_{ge}^i\\hat{\\rho }_{\\rm A}\\hat{\\sigma }_{eg}^j-\\hat{\\sigma }_{eg}^i\\hat{\\sigma }_{ge}^j\\hat{\\rho }_{\\rm A}-\\hat{\\rho }_{\\rm A}\\hat{\\sigma }_{eg}^i\\hat{\\sigma }_{ge}^j\\right),$ where $\\hat{{\\bf E}}_{\\rm p}$ is the guided mode probe field, and $\\Delta _{\\rm A}=\\omega _{\\rm p}-\\omega _{\\rm A}$ is the detuning between the guided mode probe field and the atom.", "The dipole moment operator is expressed in terms of the dipole matrix elements as $\\hat{{\\bf p}}_j = \\textbf {d}^* \\, \\hat{\\sigma }^j_{eg} + \\textbf {d}\\, \\hat{\\sigma }^j_{ge} $ , where $\\hat{\\sigma }_{eg}^j=\\mathinner {|{e}\\rangle }\\mathinner {\\langle {g}|}$ is the atomic coherence operator between the ground and excited states of atom $j$ , and $\\textbf {d}=\\mathinner {\\langle {g|\\hat{{\\bf p}}_j |e}\\rangle }$ is the dipole matrix element associated with that transition.", "The spin-exchange and decay rates are $J^{ij}=(\\mu _0\\omega _{\\rm p}^2/\\hbar )\\,\\textbf {d}^*\\cdot \\text{Re}\\,\\mathbf {G}({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})\\cdot \\textbf {d},$ $\\Gamma ^{ij} =(2\\mu _0\\,\\omega _{\\rm p}^2/\\hbar )\\,\\textbf {d}^*\\cdot \\text{Im}\\,\\mathbf {G}({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})\\cdot \\textbf {d}.$ Note that the dispersive and dissipative atom-atom couplings are given in terms of the total Green's function of the medium.", "For a given dielectric geometry, $\\mathbf {G}({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})$ can be calculated either numerically or analytically to obtain quantitative predictions for the spin exchange and decay matrix elements.", "On the other hand, given some basic assumptions about the regime of interest, one can construct a simple effective model for $\\mathbf {G}({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})$ .", "This enables one to broadly capture a number of physical systems and gain general insight, and we take this approach here.", "In particular, we assume that there is a single 1D guided band to which the atoms predominantly couple, and explicitly separate its contribution from the Green's function, ${\\bf G}({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})={\\bf G}_{\\rm 1D}({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})+{\\bf G}^{\\prime }({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})$ .", "The second term in the sum contains the atom-atom interactions mediated by all other free space and guided modes.", "We further assume that only collective interactions via the explicitly separated 1D channel are important, while ${\\bf G}^{\\prime }({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})$ only provides independent single-atom decay and energy shifts.", "Then, we can write $J^{ij}=J^{ij}_{\\rm 1D}+J^{\\prime }\\delta _{ij}$ and $\\Gamma ^{ij}=\\Gamma ^{ij}_{\\rm 1D}+\\Gamma ^{\\prime }\\delta _{ij}$ , where $\\delta _{ij}$ is the Kronecker delta.", "In particular, in free-space, $\\Gamma ^{\\prime }$ is simply $\\Gamma _0=(2\\mu _0\\,\\omega _{\\rm p}^2/\\hbar )\\,\\textbf {d}^*\\cdot \\text{Im}\\,\\mathbf {G}_0({\\bf r}_i,{\\bf r}_i,\\omega _{\\rm p})\\cdot \\textbf {d}=\\omega _{\\rm p}^3|\\textbf {d}|^2/3\\pi \\hbar \\epsilon _0 c^3$ , where ${\\bf G}_0$ is the vacuum's Green's function [i.e.", "the solution to Eq.", "(REF ) when $\\epsilon ({\\bf r},\\omega )=1$ ].", "Depending on the geometry and dielectric response of the nanostructure, and on the atom position, $\\Gamma ^{\\prime }$ can be larger or smaller than $\\Gamma _0$ .", "$J^{\\prime }$ accounts for frequency shifts due to other guided and non-guided modes, and is in general spatially dependent.", "For an atom placed in free-space, $J^{\\prime }$ is the Lamb shift, which renormalizes the atomic resonance frequency, and is considered to be included in the definition of the free-space $\\omega _{\\rm A}$ .", "The presence of a nanostructure shifts the atomic resonance frequency.", "We will for simplicity consider this shift identical for every atom and assume its value to be zero.", "Once the dynamics of the atomic coherences are solved for, one can reconstruct the field at any point in space.", "Generalizing Eq.", "(6.16) of Ref.", "[43] for more than a single atom, the evolution of the bosonic field operator is given by $\\dot{\\hat{\\textbf {f}}}\\,({\\bf r},\\omega )&=-{\\rm i}\\omega \\, \\hat{\\textbf {f}}({\\bf r},\\omega )\\\\\\nonumber &+\\frac{\\omega ^2}{c^2}\\sqrt{\\frac{1}{\\pi \\hbar \\epsilon _0}\\text{Im}\\lbrace \\epsilon ({\\bf r},\\omega )}\\rbrace \\sum _{j=1}^N {\\bf G}^*({\\bf r},{\\bf r}_j,\\omega )\\cdot \\textbf {d}\\,\\hat{\\sigma }_{ge}^j,$ where the atoms act as sources for the bosonic fields.", "We can formally integrate this expression and plug it into the equation for the field [Eq.", "(REF )].", "After some algebra, and performing Markov's approximation, we arrive at the final expression for the field operator, which is simply $\\hat{{\\bf E}}^+({\\bf r})&=\\hat{{\\bf E}}_{\\rm p}^+({\\bf r})+\\mu _0 \\omega ^2_{\\rm p} \\sum _{j=1}^N {\\bf G}({\\bf r},{\\bf r}_j,\\omega _{\\rm p})\\cdot \\textbf {d}\\,\\hat{\\sigma }_{ge}^j.$ This expression can be understood as a generalized input-output equation, where the total guided mode field is the sum of the probe, i.e.", "free, field $\\hat{{\\bf E}}_{\\rm p}^+({\\bf r})$ and the field re-scattered by the atoms.", "The quantum nature of these equations has been treated before when deriving a generalized input-output formalism for unstructured waveguides [58], [59]." ], [ "Atomic coherences in the low saturation regime", "We now explore the behavior of the atoms under a coherent, continuous-wave probe field.", "In the single-excitation manifold and low saturation (linear) regime ($\\mathinner {\\langle {\\hat{\\sigma }_{ee}}\\rangle }=0$ ), the atoms behave as classical dipoles.", "Then, the Heisenberg equations for the expectation value of the atomic coherences ($\\mathinner {\\langle {\\hat{\\sigma }_{eg}}\\rangle }=\\sigma _{eg}$ ) are linear on the atomic operators, and read $\\dot{\\sigma }_{ge}^i={\\rm i}\\left(\\Delta _{\\rm A}+{\\rm i}\\frac{\\Gamma ^{\\prime }}{2}\\right)\\sigma _{ge}^i+{\\rm i}\\Omega _i+{\\rm i}\\sum _{j=1}^N g_{ij}\\,\\sigma _{ge}^j,$ where $\\Omega _i=\\textbf {d}^*\\cdot {\\bf E}_{\\rm p}^+({\\bf r}_i)/\\hbar $ is the guided mode Rabi frequency (with ${\\bf E}_{\\rm p}=\\mathinner {\\langle {\\hat{{\\bf E}}_{\\rm p}}\\rangle }$ ), and $g_{ij}=J_{\\text{1D}}^{ij}+{\\rm i}\\Gamma _{\\text{1D}}^{ij}/2\\,=(\\mu _0\\omega _{\\rm p}^2/\\hbar )\\,\\textbf {d}^*\\cdot \\mathbf {G}_{\\rm 1D}({\\bf r}_i,{\\bf r}_j,\\omega _{\\rm p})\\cdot \\textbf {d}$ depends only on the Green's function of the guided mode.", "For long times, the coherences will damp out to a steady state ($\\dot{\\sigma }_{ge}^i=0$ ).", "The solution for the atomic coherences is then $\\vec{\\sigma }_{ge}=-\\mathcal {M}^{-1}{\\bf \\Omega }\\;\\text{ with }\\;\\mathcal {M}=\\left(\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2\\right)\\mathbb {1}+\\mathfrak {g}.$ In the above equation, $\\vec{\\sigma }_{ge}=(\\sigma _{ge}^1,\\,\\hdots \\,,\\sigma _{ge}^N)$ and ${\\bf \\Omega }=(\\Omega _1,\\,\\hdots \\,,\\Omega _N)$ are vectors of $N$ components, and $\\mathcal {M}$ is a $N\\times N$ matrix that includes the dipole-projected matrix $\\mathfrak {g}$ of elements $g_{ij}$ .", "Significantly, the matrix is not Hermitian, as there is radiation loss.", "However, due to reciprocity, the Green's function matrix is complex symmetric [${\\bf G}^{\\rm T}({\\bf r},{\\bf r}^{\\prime },\\omega )={\\bf G}({\\bf r}^{\\prime },{\\bf r},\\omega )$ ], and $\\mathfrak {g}$ inherits this property if the dipole matrix elements are real, which will be a condition enforced from now on.", "Complex symmetric matrices can be diagonalized, $\\mathfrak {g}\\textbf {v}_{\\xi }=\\lambda _{\\xi }\\textbf {v}_{\\xi }$ with $\\xi =1\\ldots N$ , where $\\lambda _{\\xi }$ and $\\textbf {v}_{\\xi }$ are the eigenvalues and eigenvectors of $\\mathfrak {g}$ , respectively.", "Since the first term of $\\mathcal {M}$ is proportional to the identity, $\\mathcal {M}$ and $\\mathfrak {g}$ share the same set of eigenvectors.", "The eigenmodes represent the spatial profile of the collective atomic excitation, i.e., the dipole amplitude and phase at each atom.", "However, as the matrix $\\mathfrak {g}$ is non-Hermitian, the eigenmodes are not orthonormal in the regular sense, but instead follow different orthogonality and completeness prescriptions, namely $\\textbf {v}_{\\xi }^T\\cdot \\textbf {v}_{\\xi ^{\\prime }}=\\delta _{\\xi ,\\xi ^{\\prime }}$ and $\\sum _{\\xi =1}^N \\textbf {v}_{\\xi }\\otimes \\textbf {v}_{\\xi }^T=\\mathbb {1}$ , where $T$ indicates transpose instead of the customary conjugate transpose [60].", "After inserting the completeness relation into Eq.", "(REF ), we find that the expected value of the atomic coherences in the steady state in terms of the eigenvalues and eigenvectors of the quasi-1D Green's function is $\\vec{\\sigma }_{ge}=-\\sum _{\\xi \\in \\text{mode}}\\frac{(\\textbf {v}^T_{\\xi }\\cdot {\\bf \\Omega })}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2}\\textbf {v}_{\\xi },$ where $J_{\\xi ,\\rm 1D}=\\text{Re}\\,\\lambda _{\\xi }$ and $\\Gamma _{\\xi ,\\rm 1D}=2\\,\\text{Im}\\,\\lambda _{\\xi }$ are the frequency shifts and decay rates corresponding to mode $\\xi $ , and the sum is performed over mode number from 1 to $N$ .", "The scalar product in the numerator $\\textbf {v}^T_{\\xi }\\cdot \\vec{\\Omega }=\\sum _{j=1}^N v_{\\xi ,j}\\,\\Omega _j$ describes the coupling between the probe field and a particular collective atomic mode.", "Both the frequency shifts and decay rates, as well as the spatial profile eigenstates of $\\mathfrak {g}$ , are frequency dependent.", "The dynamics of the atoms can be understood in terms of the eigenmodes of $\\mathfrak {g}$ , where the real and imaginary parts of the eigenvalues correspond to cooperative frequency shifts and decay rates of the collective atomic modes $\\lbrace \\xi \\rbrace $ .", "As the modes are non-normal, the observables cannot be expressed as the sum over all different mode contributions but, instead, any measurable quantity will show signatures of interference between different modes.", "Although it could be considered a mathematical detail, the fact that the modes of a system are non-normal has deep physical consequences.", "For instance, non-normal dynamics is responsible of phenomena as different as the Petermann excess-noise factor observed in lasers [61], [62], [63] or the transient growth of the shaking of a building after an earthquake [64]." ], [ "Transmission and reflection coefficients", "Having previously calculated the linear response of an ensemble of atoms to an input field, we now relate the response to observable outputs, i.e.", "the reflected and transmitted fields.", "One can calculate the total field from Eq.", "(REF ), by substituting in the solution of Eq.", "(REF ) for the atomic coherences $\\sigma _{ge}$ .", "For a dipole moment directed along $\\alpha $ (i.e.", "$\\textbf {d}=d\\hat{\\alpha }$ ), the $\\beta -$ polarization component of the field reads $E^+_\\beta ({\\bf r})&=E^{+}_{\\rm p,\\beta }({\\bf r})-\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}^T_{\\alpha \\beta }({\\bf r})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf E}_{\\rm p,\\alpha }^+\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2},$ where the $j$ -component of the electric field vector ${\\bf E}_{\\rm p,\\alpha }^+$ , which reads ${\\bf E}_{\\rm p,\\alpha ,j}^+=\\hat{\\alpha }\\cdot {\\bf E}_{\\rm p}^+({\\bf r}_j)$ , no longer represents different polarization components, but the dipole-projected field evaluated at the atoms' positions ${\\bf r}_j$ .", "The $j$ -component of vector ${\\bf g}_{\\alpha \\beta } ({\\bf r})$ is $g_{\\alpha \\beta ,j}({\\bf r})=g_{\\alpha \\beta }({\\bf r},{\\bf r}_j)=(\\mu _0\\omega _{\\rm p}^2d^2/\\hbar )\\hat{\\alpha }\\cdot {\\bf G}_{{\\rm 1D}} ({\\bf r},{\\bf r}_j,\\omega _p)\\cdot \\hat{\\beta }$ , where $j$ runs over the atom number.", "In particular, the scalar product ${\\bf g}_{\\alpha \\beta }^T({\\bf r})\\cdot \\textbf {v}_{\\xi }=\\sum _{i=j}^N\\,g_{\\beta \\alpha ,j}({\\bf r})v_{\\xi ,j}$ represents how much the mode $\\xi $ contributes to the field emitted by the atoms.", "Figure: (a) Sketch of an atom interacting with the guided mode of a structured 1D nanostructure.", "The single-atom decay rate is Γ 1D \\Gamma _{\\rm 1D}, and the decay into non-guided modes is characterized by Γ ' \\Gamma ^{\\prime }.", "(b) Normalized transmission spectra (|t/t 0 | 2 |t/t_0|^2) for a single atom for different values of the ratio between the real and imaginary parts of the guided mode Green's function, following Eq. ().", "The decay rate into the guided modes is taken to be Γ 1D =Γ ' \\Gamma _{\\rm 1D}=\\Gamma ^{\\prime } for all cases.We now assume that the atomic chain and the main axis of the nanostructure are oriented along $\\hat{x}$ .", "In order to connect the above expression to the transmission and reflection coefficients, we evaluate the field $E_\\beta ^+({\\bf r})$ at the positions ${\\bf r}={\\bf r}_{\\rm right}$ and ${\\bf r}={\\bf r}_{\\rm left}$ , which are considered to be immediately outside the atomic chain, and only differ in the $x$ -component.", "Following Appendix , the normalized transmission and reflection coefficients are $\\frac{t(\\Delta _{\\rm A})}{t_0(\\Delta _{\\rm A})}=1-\\frac{1}{g_{\\beta \\beta }({\\bf r}_{\\rm right},{\\bf r}_{\\rm left})}\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}^T_{\\alpha \\beta }({\\bf r}_{\\rm right})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf g}_{\\alpha \\beta }({\\bf r}_{\\rm left})\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2},$ $r(\\Delta _{\\rm A})=r_0(\\Delta _{\\rm A})-\\frac{1}{g_{\\beta \\beta }({\\bf r}_{\\rm left},{\\bf r}_{\\rm left})}\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}^T_{\\alpha \\beta }({\\bf r}_{\\rm left})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf g}_{\\alpha \\beta }({\\bf r}_{\\rm left})\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2},$ where $t_0(\\Delta _{\\rm A})$ and $r_0(\\Delta _{\\rm A})$ are the transmission and reflection coefficients for the 1D photonic structure when no atoms are present." ], [ "Simplified expression for the transmission", "For linearly-polarized, transverse guided modes with an approximately uniform transverse field distribution, one can further simplify the expression for the transmission coefficient.", "We find a product equation that only depends on the eigenvalues of the Green's function matrix $\\mathfrak {g}$ , and not on their spatial structure (i.e., the eigenfunctions).", "Following Appendix , we obtain $\\nonumber \\frac{t(\\Delta _{\\rm A})}{t_0(\\Delta _{\\rm A})}&=\\prod _{\\xi =1}^N\\,\\frac{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2}\\\\&\\equiv \\prod _{\\xi =1}^N\\,t_{\\xi }(\\Delta _{\\rm A}).$ The total transmission coefficient can thus be written as the product of the transmission coefficients of each of the collective atomic modes.", "Noticeably, when looking at the transmission spectrum of atoms that interact through the guided mode of a quasi-1D nanostructure, there is a redundancy between the eigenfunctions and eigenvalues, and one is able to obtain an expression that does not depend on the former (i.e., all the relevant information about the geometry is contained in the collective frequency shifts and decay rates).", "In particular, for a single atom located at $x_j$ with $J^{jj}_{\\rm 1D}\\equiv J_{\\rm 1D}$ and $\\Gamma ^{jj}_{\\rm 1D}\\equiv \\Gamma _{\\rm 1D}$ , the eigenvalues are directly proportional to the local Green's function, and $\\frac{t(\\Delta _{\\rm A})}{t_0(\\Delta _{\\rm A})}=\\frac{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2}{(\\Delta _{\\rm A}+J_{\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\rm 1D})/2}.$ The transmittance $T=|t|^2$ can be recast into a Fano-like lineshape [65] as $\\frac{T}{T_0}=\\frac{(q+\\chi )^2}{1+\\chi ^2}+\\left(\\frac{\\Gamma ^{\\prime }}{\\Gamma ^{\\prime }+\\Gamma _{\\rm 1D}}\\right)^2\\frac{1}{1+\\chi ^2},$ where $\\chi =2(\\Delta _{\\rm A}+J_{\\rm 1D})/(\\Gamma _{\\rm 1D}+\\Gamma ^{\\prime })$ and $q=-2J_{\\rm 1D}/(\\Gamma _{\\rm 1D}+\\Gamma ^{\\prime })$ is the so-called asymmetry parameter.", "For $\\Gamma ^{\\prime }\\ll \\Gamma _{\\rm 1D}$ , the second term is negligible and the normalized transmittance is a pure Fano resonance, with $q=-\\text{Re}\\lbrace \\text{G}_{\\rm 1D}({\\bf r}_j,{\\bf r}_j,\\omega _{\\rm p})\\rbrace /\\text{Im}\\lbrace \\text{G}_{\\rm 1D}({\\bf r}_j,{\\bf r}_j,\\omega _{\\rm p})\\rbrace $ .", "Fano resonances arise whenever there is interference between two different transport channels.", "For instance, in a cavity far from resonance, there is interference arising from all the possible optical paths that contribute to the transmission signal due to reflections at the mirrors, whereas in an unstructured waveguide there is no such interference and thus the lineshape is Lorentzian.", "For a single atom, there is a clear mapping between the spectrum lineshape and the local 1D Green's function.", "For a nanostructure with a purely imaginary self Green's function ${\\bf G}(x_i,x_i)$ (such as a wave-guide or a cavity at resonance), the spectrum is Lorentzian, and centered around the atomic frequency.", "However, if the real part is finite, one would observe a frequency shift of the spectrum, which becomes asymmetric.", "Figure REF (b) shows how the normalized transmission spectrum for a single atom becomes more and more asymmetric for higher ratios $J_{\\rm 1D}/\\Gamma _{\\rm 1D}$ .", "Also, there is an appreciable blueshift of the spectral features.", "We would like to remark that the Markov approximation has thus far been employed in our analysis, as every Green's function is considered to be a complex constant over frequency ranges larger than the linewidth of the atoms.", "However, the expressions for the transmission and reflection coefficients are valid also in the non-Markovian regime.", "We analyze this issue in more detail in Appendix ." ], [ "Application to several one-dimensional photonic structures", "In this section, we analyze the transmission spectra of atoms placed along common quasi-1D nanostructures, such as cavities, waveguides, and photonic crystals." ], [ "Standing-wave cavities", "To begin with, we want to illustrate the connection between the Green's function formalism and the well-known Jaynes Cummings (JC) model [40], [66].", "For $N$ atoms in a driven cavity of length $L$ and effective area $A$ , the JC Hamiltonian, and its corresponding Lindblad operator read $\\nonumber \\mathcal {H}&=-\\hbar \\Delta _{\\rm c}\\hat{a}^\\dagger \\hat{a}-\\hbar \\Delta _{\\rm A}\\sum _{i=1}^N \\hat{\\sigma }_{ee}^i+\\hbar \\sum _{i=1}^N \\mathcal {q}_i\\left(\\hat{a}^\\dagger \\hat{\\sigma }_{ge}^i+\\hat{\\sigma }_{eg}^i\\hat{a}\\right)\\\\&+\\hbar \\eta \\,(\\hat{a}+\\hat{a}^\\dagger ),$ $\\nonumber \\mathcal {L}[\\hat{\\rho }]&=\\frac{\\Gamma ^{\\prime }}{2}\\,\\sum _{i,j=1}^N\\left(2\\hat{\\sigma }_{ge}^i\\hat{\\rho }\\hat{\\sigma }_{eg}^j-\\hat{\\sigma }_{eg}^i\\hat{\\sigma }_{ge}^j\\hat{\\rho }-\\hat{\\rho }\\hat{\\sigma }_{eg}^i\\hat{\\sigma }_{ge}^j\\right)\\\\&+\\frac{\\kappa _c}{2}\\,\\left(2\\hat{a}\\hat{\\rho }\\hat{a}^\\dagger -\\hat{a}^\\dagger \\hat{a}\\hat{\\rho }-\\hat{\\rho }\\hat{a}^\\dagger \\hat{a}\\right),$ where $\\hat{a}$ is the cavity-field annihilation operator, $\\hat{\\rho }$ is the density matrix for the atoms and the cavity field, $\\eta $ is a frequency that represents the amplitude of the classical driving field, $\\Delta _{\\rm c}=\\omega _{\\rm p}-\\omega _{\\rm c}$ is the detuning between the driving (probe) and the cavity fields, and $\\kappa _c$ is the cavity-field decay.", "The atom cavity coupling is $\\mathcal {q}_i=\\mathcal {q}\\cos (k_{\\rm c} x_i)$ , where $\\mathcal {q}=d\\sqrt{\\omega _{\\rm c}/(\\hbar \\epsilon _0 LA)}$ is modulated by a function that depends on the atoms' positions and the cavity wave-vector $k_{\\rm c}$ .", "The Heisenberg equations of motion for the field and atomic operators are $&\\dot{\\hat{a}}=\\left({\\rm i}\\Delta _{\\rm c}-\\frac{\\kappa _{\\rm c}}{2}\\right)\\hat{a}-{\\rm i}\\sum _{i=1}^N \\mathcal {q}_i\\hat{\\sigma }_{ge}^i-{\\rm i}\\eta ,\\\\&\\dot{\\hat{\\sigma }}_{ge}^i=\\left({\\rm i}\\Delta _{\\rm A}-\\frac{\\Gamma ^{\\prime }}{2}\\right)\\hat{\\sigma }_{ge}^i+{\\rm i}\\mathcal {q}_i\\left(\\hat{\\sigma }_{ee}^i-\\hat{\\sigma }_{gg}^i\\right)\\hat{a}.$ When $ \\Gamma ^{\\prime }\\ll \\kappa _{\\rm c}$ and $\\mathcal {q}<\\text{min}\\lbrace \\Delta _{\\rm c},\\kappa _{\\rm c}\\rbrace $ , the cavity field can be adiabatically eliminated, and the field operator re-expressed in terms of the atomic ones, i.e., $\\nonumber \\dot{\\hat{a}}=0\\quad \\rightarrow \\quad \\hat{a}=\\frac{1}{\\left(\\Delta _{\\rm c}+{\\rm i}\\frac{\\kappa _{\\rm c}}{2}\\right)}\\left(\\eta +\\sum _{i=1}^N \\mathcal {q}_i\\hat{\\sigma }_{ge}^i\\right).$ Introducing this expression back into the equation for the atomic operator, one can deduce a master equation for the atomic density matrix $\\hat{\\rho }_{\\rm A}$ .", "The new Hamiltonian and Lindblad operators read just as those of Eqs.", "(REF ) and (REF ), but for a classical driving field, and with spin exchange and decay rates into the cavity mode given by [67] $J^{ij}_{\\rm 1D}&=-\\frac{\\mathcal {q}^2\\Delta _{\\rm c}}{(\\Delta _{\\rm c}^2+\\kappa _{\\rm c}^2/4)}\\cos (k_{\\rm c} x_i)\\cos (k_{\\rm c} x_j),\\\\\\Gamma ^{ij}_{\\rm 1D}&=\\frac{\\mathcal {q}^2\\kappa _{\\rm c}}{(\\Delta _{\\rm c}^2+\\kappa _{\\rm c}^2/4)}\\cos (k_{\\rm c} x_i)\\cos (k_{\\rm c} x_j).$ It can thus be seen that the Markovian approximation to arrive at these equations is equivalent to the absence of strong coupling effects within the JC model.", "The last step for connecting this simple model with our formalism is to calculate the Green's function of a cavity and confirm that $J^{ij}_{\\rm 1D}$ and $\\Gamma ^{ij}_{\\rm 1D}$ are precisely those obtained within the JC framework.", "The Green's function of a quasi-1D cavity formed by partially transmitting mirrors of reflection coefficient $r$ (chosen to be real) is [68] $\\nonumber &G_{\\rm 1D}(x_i,x_j,\\omega _{\\rm p})\\simeq \\frac{{\\rm i}c^2}{2v_g\\omega _p A (1-r^2 e^{2{\\rm i}k_{\\rm p} L})}\\left[e^{{\\rm i}k_{\\rm p} |x_i-x_j|}\\right.\\\\&\\left.+re^{{\\rm i}k_{\\rm p}(L+x_i+x_j)}+re^{{\\rm i}k_{\\rm p}[L-(x_i+x_j)]}+r^2e^{{\\rm i}k_{\\rm p}(2L-|x_i-x_j|)}\\right],$ where $v_{\\rm g}$ is the group velocity.", "For high-Q standing-wave cavities, i.e.", "with $r\\simeq 1$ , and choosing $v_{\\rm g}=c$ , the Green's function can be approximated as $\\nonumber G_{\\rm 1D}(x_i,x_j,\\omega _{\\rm p})\\simeq &\\left(\\frac{2{\\rm i}c}{\\omega _p A}\\right)\\frac{1}{1-r^2e^{2{\\rm i}k_{\\rm p} L}}\\\\&\\times \\cos (k_{\\rm p} x_i) \\cos (k_{\\rm p} x_j).$ The cavity is resonant at a frequency $\\omega _{\\rm c}$ with corresponding wave-vector $k_{\\rm c}$ , chosen to be such that $k_{\\rm c} L=2\\pi m$ , with $m$ being an integer.", "Close to resonance, one can write $k_{\\rm p}=k_{\\rm c}+\\delta k$ , and assume that $\\delta k L\\ll 1$ .", "Then $1-r^2e^{2{\\rm i}k_{\\rm p} L}\\simeq 1-r^2-2{\\rm i}r^2\\delta k L$ , and the Green's function is simply $G_{\\rm 1D}(x_i,x_j,\\omega _{\\rm p})\\simeq -\\left(\\frac{c^2}{\\omega _{\\rm p} L A}\\right)\\frac{\\cos (k_{\\rm c} x_i) \\cos (k_{\\rm c} x_j)}{\\Delta _c+{\\rm i}\\kappa _c/2},$ where $\\kappa _c=(1-r^2) c/L$ is the cavity linewidth.", "Therefore, the atoms' spin-exchange and decay rates are given by $\\nonumber J^{ij}_{\\rm 1D}&=\\frac{\\mu _0\\omega _{\\rm p}^2 d^2}{\\hbar }\\,\\text{Re}\\,G_{\\rm 1D}(x_i,x_j,\\omega _{\\rm p})=-\\mathcal {q}_i\\mathcal {q}_j\\frac{\\Delta _{\\rm c}}{(\\Delta _{\\rm c}^2+\\kappa _{\\rm c}^2/4)},$ $\\nonumber \\Gamma ^{ij}_{\\rm 1D}&=\\frac{2\\mu _0\\omega _{\\rm p}^2 d^2}{\\hbar }\\,\\text{Im}\\,G_{\\rm 1D}(x_i,x_j,\\omega _{\\rm p})=\\mathcal {q}_i\\mathcal {q}_j\\frac{\\kappa _c}{(\\Delta _{\\rm c}^2+\\kappa _{\\rm c}^2/4)},$ which is precisely what is obtained within the Jaynes Cummings model.", "Figure: (a) Frequency shifts and (b) decay rates of the collective modes of a regular chain of 5 atoms placed along a waveguide normalized to the single-atom decay rate into the guided mode Γ 1D \\Gamma _{\\rm 1D}, as a function of the distance dd between the atoms in units of the probe wavelength.Let's now look at the transmission spectrum of $N$ atoms in a cavity.", "As we have just demonstrated, coefficients of the dipole-projected Green's function matrix $\\mathfrak {g}$ read $g_{ij}=g(\\omega _{\\rm p})\\cos (k_{\\rm c} x_i) \\cos (k_{\\rm c} x_j),$ where $g(\\omega _{\\rm p})=J^{\\rm max}_{\\rm 1D}+{\\rm i}\\Gamma ^{\\rm max}_{\\rm 1D}/2$ , where $J^{\\rm max}_{\\rm 1D}$ and $\\Gamma ^{\\rm max}_{\\rm 1D}$ are the spin-exchange and decay rates at the antinode of the cavity field.", "Depending on the detuning between the probe field and the cavity resonance, $g(\\omega _{\\rm p})$ can be purely imaginary, yielding dissipative atom-atom interactions, or can have both real and imaginary parts, resulting in both dissipative and dispersive couplings.", "The matrix $\\mathfrak {g}$ is separable (has rank one) as it can be written as the tensor product of just one vector by itself.", "The matrix has one eigenstate describing a superposition of atomic coherences that couples to the cavity (a \"bright mode\"), with eigenvalue $\\lambda _{\\rm B}=\\sum _{i=1}^Ng^{ii}= \\left(J^{\\rm max}_{\\text{1D}}+{\\rm i}\\Gamma ^{\\rm max}_{\\text{1D}}/2\\right)\\sum _{i=1}^N \\cos ^2(k_{\\rm c}x_i)$ .", "This atomic collective excitation follows spatially the mode profile of the cavity, i.e.", "$\\sigma _{ge}^i\\propto \\cos (k_{\\rm c} x_i)$ .", "The matrix $\\mathfrak {g}$ has also $N-1$ decoupled (\"dark\") modes of eigenvalue 0.", "Because these dark modes have a zero decay rate into the cavity mode, it is also impossible to excite them employing the cavity field.", "The optical response is thus entirely controlled by the bright mode, and the transmission is simply $\\frac{t(\\Delta _{\\rm A})}{t_0(\\Delta _{\\rm A})}=\\frac{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2}{(\\Delta _{\\rm A}+\\sum _{i=1}^N J^{ii}_{\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\sum _{i=1}^N \\Gamma ^{ii}_{\\rm 1D})/2}.$ Remarkably, this expression is valid no matter the separation between the atoms or whether they form an ordered or disordered chain.", "The transmission spectrum corresponds to that of a `super-atom', where the decay rates and the frequency shifts are enhanced (N-fold if all the diagonal components of $\\mathfrak {g}$ are equal) compared to those of a single atom.", "This result replicates the well-known expressions for conventional cavity QED." ], [ "Unstructured Waveguides", "Another paradigm that has been investigated frequently is that of \"waveguide QED\" [41].", "The simple model of such a system consists of a single guided mode with translational invariance, and where the dispersion relation is well-approximated as linear around the atomic resonance frequency.", "In a 1D translationally invariant system, a source simply emits a plane wave whose phase at the detection point is proportional to the distance of separation.", "Therefore, the elements of the Green's function matrix $\\mathfrak {g}$ depend on the distance between the atoms, and read $g_{ij}={\\rm i}\\frac{\\Gamma _{\\rm 1D}}{2}e^{{\\rm i}k_p |x_i-x_j|}.$ Remarkably, the self Green's function in a waveguide is purely imaginary.", "The coherent interactions between atom $i$ and atom $j$ are dictated by the Hamiltonian [given by Eq.", "(REF )], and are proportional to $\\text{Re}\\lbrace g_{ij}\\rbrace =-(\\Gamma _{\\rm 1D}/2)\\sin k_{\\rm p}|x_i-x_j|$ , whereas the dissipation is given by the Lindblad operator [given by Eq.", "(REF )], which is proportional to $\\text{Im}\\lbrace g_{ij}\\rbrace =(\\Gamma _{\\rm 1D}/2)\\cos k_{\\rm p}(x_i-x_j)$ [24], [56].", "It is thus clear that by carefully tuning the distance between the emitters, one can engineer fully dissipative interactions.", "If the atoms form a regular chain and are spaced by a distance $d$ such that $k_pd=n\\pi $ , where $n$ is an integer number, the matrix $\\mathfrak {g}$ has only one non-zero eigenvalue $\\lambda _{\\rm B}={\\rm i}N\\Gamma _{\\rm 1D}/2$ associated with the bright atomic mode.", "This situation is analogous to the case of atoms interacting in an on-resonance cavity.", "Therefore, there will not be any collective frequency shift, and the lineshape will be a Lorentzian of width $N\\Gamma _{\\rm 1D}+\\Gamma ^{\\prime }$ .", "For $n$ even, the phases of the dipole moments of the atoms are all identical, whereas for odd $n$ the dipole moments of adjacent atoms are $\\pi $ out of phase.", "For a regular chain with lattice constant different from $k_{\\rm p}d=n\\pi $ , or for atoms placed randomly along the waveguide, the coefficients of matrix $\\mathfrak {g}$ have both a real and imaginary part, and, to the best of our knowledge, there is no analytic expression for the eigenvalues of $\\mathfrak {g}$ .", "Figure REF shows the frequency shifts and decay rates of the collective modes of a $N=5$ atom chain as a function of the separation between the atoms.", "For separations where $k_{\\rm p}d=n\\pi $ , the real part of the Green's function is zero and the imaginary part of all modes but one goes to zero, whereas for other spacings one generically gets a zoo of coherent and dissipative couplings of comparable strength.", "This occurs because the real and imaginary parts of $g_{ij}$ are generically of similar magnitude.", "Figure REF shows the transmission and reflection spectra for $N=20$ atoms separated by $k_{\\rm p}d=\\pi $ (blue dashed curve), and for several random realizations where each atomic position is chosen randomly from a distribution $k_{\\rm p} x_{i} \\in [0,2\\pi ]$ (orange curves).", "The thick black line in Fig.", "REF (a) represents the non-interacting case, which is obtained by setting the non-diagonal terms of $\\mathfrak {g}$ to zero, yielding a transmission spectrum $\\frac{t(\\Delta _{\\rm A})}{t_0(\\Delta _{\\rm A})}=\\left(\\frac{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2}{\\Delta _{\\rm A}+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\rm 1D})/2}\\right)^N,$ where the transmission coefficient is a product of the transmission coefficient of each single atom, and the frequency shifts and decay rates are not collective quantities but, instead, single-atom parameters.", "Figure: (a) Transmission spectra for 20 atoms interacting through the guided modes of an unstructured waveguide.", "The dashed blue line represents a regular separation between the atoms of d=λ p /2d=\\lambda _{\\rm p}/2.", "The orange curves show 10 different spectra obtained by randomly placing the atoms along the nanostructure.", "The thick black curve represents the \"non-interacting\" case of Eq. ().", "(b) Reflection spectra for the same situations as in (a).", "We have chosen Γ 1D =Γ ' \\Gamma _{\\rm 1D}=\\Gamma ^{\\prime }, T 0 =1T_0=1, and R 0 =0R_0=0.Figure: (a) When the atomic resonance frequency lies within the band-gap region of a photonic crystal, an excited atom becomes dressed by a photonic “cloud” of tunable size.", "A second atom sitting at a distance dd interacts with this cloud, giving rise to an effective atomic interaction of spatial range κ x -1 \\kappa _x^{-1}.", "(b) Collective frequency shifts of the modes of a regular chain of NN=10 atoms in the bandgap of an infinite photonic crystal as a function of κ x d\\kappa _xd.", "The atoms are placed at even antinodes of the Bloch modes.Figure REF (a) also shows that, for random filling, although the atoms interact with each other ($\\mathfrak {g}_{ij\\ne i}\\ne 0$ ), the transmission spectra follow closely that of a non-interacting system, for which all the off-diagonal elements are zero ($\\mathfrak {g}_{ij\\ne i}=0$ ), and the eigenvalues of matrix $\\mathfrak {g}$ are proportional to the self Green's functions [${\\bf G}(x_i,x_i)$ ] at the atoms' positions.", "In this case, the behavior of the emitters cannot be understood in terms of the `super-atom' picture, as the transmission spectrum of the system is significantly different from a Lorentzian.", "In particular, for the non-interacting scenario, one can recast Eq.", "(REF ) into an exponential, and the transmittance recovers the well-known form of a Beer-Lambert law, reading $\\nonumber \\frac{T(\\Delta _{\\rm A})}{T_0(\\Delta _{\\rm A})}&=\\text{exp}\\left[-N \\ln \\frac{\\Delta _{\\rm A}^2+(\\Gamma ^{\\prime }+\\Gamma _{\\rm 1D})^2/4}{\\Delta _{\\rm A}^2+\\Gamma ^{\\prime 2}/4}\\right]\\\\&\\simeq \\text{exp}\\left[-\\frac{\\text{OD}}{1+(2\\Delta _{\\rm A}/\\Gamma ^{\\prime })^2}\\right],$ where $\\text{OD}\\equiv 2N\\Gamma _{\\rm 1D}/\\Gamma ^{\\prime }$ is the optical depth and the last equality holds for $\\Gamma _{\\rm 1D}\\ll \\Gamma ^{\\prime }$ .", "This is exactly the same behavior that an atomic ensemble in free space would exhibit.", "This occurs only for non-negligible $\\Gamma ^{\\prime }$ , which suppresses multiple reflections.", "Otherwise one would see huge fluctuations associated with Anderson localization in the spectra.", "The reflectance spectrum, on the other hand, is more complex and carries more information than the transmittance, as shown in Fig.", "REF (b).", "In contrast to the case of the transmission coefficient, the reflection does not admit a simple formula in terms of the eigenvalues of the system.", "This is only possible when the Green's function is separable, namely, when the distance between the atoms is $d=n\\lambda _{p}/2$ ." ], [ "Photonic crystal bandgaps", "The band-gap region of a photonic crystal waveguide (PCW) is a very appealing scenario to explore coherent atom-atom interactions, as light cannot propagate, and atoms interact with each other through evanescent fields [32], as depicted in Fig.", "REF (a).", "Band-gap interactions facilitate long range dispersive dynamics, enabling the engineering of many-body states for atoms and photons with very low dissipation.", "For a photonic crystal waveguide of lattice constant $a$ the elements of matrix $\\mathfrak {g}$ are well approximated by $g_{ij}=J^{\\rm max}_{\\rm 1D}\\cos (\\pi x_i/a) \\cos (\\pi x_j/a) e^{-\\kappa _x|x_i-x_j|},$ where the cosine terms account for the spatial profile of the Bloch modes, $J^{\\rm max}_{\\rm 1D}$ is the value of the spin-exchange rate for an atom located at the maximum of the Bloch mode, and $\\kappa _x^{-1}$ is the finite range of interaction due to the evanescent decay of the guided mode field in the bandgap, which is controlled by detuning the band-edge frequency from the atomic resonance.", "The value of $J^{\\rm max}_{\\rm 1D}$ is also determined by this detuning.", "It should be noted that in this idealized picture, $g_{ij}$ is purely real, indicating the absence of collective emission into the PCW.", "This is naturally expected, due to the absence of guided modes at the atomic frequency.", "In practice, residual decay might still exist to the extent that the mediating photon has a decay channel.", "This could be either due to the finite length of the PCW, which can cause the photon to leak out the ends and is suppressed when $\\kappa _{x}L \\gg 1$ , or through scattering and absorption losses of the PCW.", "Given that these photonic decay processes can be made small, for conceptual simplicity here we treat the idealized case.", "For a chain of periodically spaced atoms placed in even antinodes of the Bloch modes, the dipole-projected Green's function matrix reads $\\mathfrak {g}=J^{\\rm max}_{\\rm 1D}\\begin{pmatrix}1 & \\chi & \\chi ^2 &\\cdots & \\chi ^{N-1} \\\\\\chi & 1 & \\chi &\\cdots & \\chi ^{N-2}\\\\\\vdots & \\vdots & \\vdots &\\ddots & \\vdots \\\\\\chi ^{N-1} & \\chi ^{N-2} & \\chi ^{N-3} &\\cdots & 1\\end{pmatrix},$ where we have defined $\\chi \\equiv e^{-\\kappa _xd}$ , with $d$ being the distance between nearest-neighbor atoms.", "The matrix $\\mathfrak {g}$ is a real symmetric Toeplitz matrix (or bisymmetric matrix).", "Neglecting higher order contributions besides first-neighbor, an approximation valid for $\\kappa _xd\\gg 1$ , $\\mathfrak {g}$ becomes a tridiagonal Toeplitz matrix whose eigenvalues and eigenvectors are [69]: $\\lambda _{\\xi }\\equiv J_{{\\rm 1D},\\xi }=J^{\\rm max}_{\\rm 1D}+2J^{\\rm max}_{\\rm 1D}e^{-\\kappa _xd}\\cos \\left(\\frac{\\xi \\pi }{N+1}\\right),$ $v_{\\xi ,j}=\\sqrt{\\frac{2}{N+1}}\\sin \\left(\\frac{\\xi j \\pi }{N+1}\\right).$ In this simple tight binding model, the frequency shifts of the collective atomic modes are distributed around $J^{\\rm max}_{\\rm 1D}$ with a frequency spread controlled by $\\kappa _{x}$ (i.e., for larger $\\kappa _x$ , the modes are closer in frequency).", "This can be observed in Fig.", "REF (b), which shows how the collective frequency shifts coalesce towards $J^{\\rm max}_{\\rm 1D}$ for large $\\kappa _x d$ .", "However, if the interaction length is very large compared to the distance between the atoms, the approximation of neglecting higher order neighbors falls apart, and the eigenvalues start to show a different behavior.", "Eventually, when the interaction length becomes infinite (or much larger than the length of the atomic distribution), there is only one bright mode, of eigenvalue $\\lambda _{\\rm B}=NJ^{\\rm max}_{\\rm 1D}$ .", "This is analogous to the cavity case, where the interaction range is also infinite, except now the eigenvalue is purely real.", "The band-edge of a photonic crystal is thus a cross-over region in which the single bright mode approximation holds and then transitions to another regime where it breaks down, as the guided mode becomes evanescent and decays substantially within the length of the PCW.", "Importantly, the bandgap of a photonic crystal provides a tunable interaction range, a feature which is unique to this kind of nanostructure, and makes PCWs remarkably different reservoirs from either cavities or unstructured waveguides.", "In the final section, we present some predictions for the transmission spectrum of two atoms coupled to a PCW for $\\Gamma _{\\rm 1D}$ and $J_{\\rm 1D}$ values that can be achieved experimentally in the coming years.", "We hope that the foreseen large coherent couplings between the atoms combined with low dissipation through the guided mode help to stimulate a new generation of experiments that go beyond the current state of the art." ], [ "Electromagnetically induced transparency", "Throughout this work, we have dealt with two-level atoms, but our formalism is not restricted to these kind of emitters and simple expressions for light propagation can also be derived in the case of multilevel atoms.", "In this section we apply the derived expression for the normalized transmission in terms of the collective energy shifts and decay rates [i.e.", "Eq.", "(REF )] to one of the paradigmatic problems within quantum optics: electromagnetically induced transparency (EIT) [70], [71], [72].", "We consider a chain of $\\Lambda $ -atoms whose $\\mathinner {|{e}\\rangle }$ to $\\mathinner {|{g}\\rangle }$ transition is coupled to the guided mode of a quasi-1D nanostructure, as shown in Fig.", "REF (a).", "The excited state can also decay to a metastable state $\\mathinner {|{s}\\rangle }$ of frequency $\\omega _s$ .", "This transition, whose polarization is chosen to be orthogonal to the guided mode, is accessed via an external, classical, and uniform control field of Rabi frequency $\\Omega _c$ .", "Under the action of the control field, the guided mode photons promote the atoms from the ground state to the so-called dark state, a superposition of both $\\mathinner {|{g}\\rangle }$ and $\\mathinner {|{s}\\rangle }$ .", "The creation of these dark states has two main consequences: first, it prevents photon loss due to the the long lifetime of the $\\mathinner {|{s}\\rangle }$ state; second, the group velocity is greatly reduced as the probe field is converted into a polaritonic excitation made of spins and photons moving at a diminished speed.", "In order to describe EIT, we have to add another term to the Hamiltonian of Eq.", "(REF ).", "The new term accounts for the presence of the $\\mathinner {|{s}\\rangle }$ state and the driving of the $\\mathinner {|{e}\\rangle }$ to $\\mathinner {|{s}\\rangle }$ transition, and reads $\\mathcal {H}_{\\rm c}=-\\hbar \\sum ^N_{i=1}\\Delta _{\\rm s}\\sigma _{ss}^i+\\Omega _c\\left(\\sigma _{es}^i+\\sigma _{se}^i\\right)$ , where $\\Delta _{\\rm s}=\\omega _{\\rm p}-\\omega _c-\\omega _{\\rm s}$ , and $\\omega _c$ is the control field frequency.", "For the sake of simplicity, we take $\\Omega _c$ to be real.", "In the single-excitation manifold, and for low saturation ($\\sigma _{gg}\\simeq 1$ , and $\\sigma _{ss}\\simeq \\sigma _{ee}\\simeq \\sigma _{es}\\simeq 0$ ), the equations of motion for the atomic coherences are given by $&\\dot{\\sigma }_{ge}^i={\\rm i}\\Delta _{\\rm A}\\sigma _{ge}^i+{\\rm i}\\Omega _i+{\\rm i}\\Omega _c\\sigma _{gs}^i+{\\rm i}\\sum _{j=1}^N g_{ij}\\,\\sigma _{ge}^j,\\\\&\\dot{\\sigma }_{gs}^i={\\rm i}\\Delta _{\\rm s}\\sigma _{gs}^i+{\\rm i}\\Omega _c\\sigma _{ge}^i,$ where $\\Delta _{\\rm A}=\\omega _{\\rm p}-\\omega _{\\rm A}$ is the detuning between the guided mode probe field and the atomic $\\mathinner {|{e}\\rangle }-\\mathinner {|{g}\\rangle }$ transition.", "Following the steps described in Section II, we solve these equations in the steady state.", "Then, $\\sigma _{gs}^i=-(\\Omega _c/\\Delta _{\\rm s})\\,\\sigma _{ge}^i$ , and $\\vec{\\sigma }_{ge}=-\\Delta _{\\rm s}\\sum _{\\xi \\in \\text{mode}}\\frac{(\\textbf {v}^T_{\\xi }\\cdot {\\bf \\Omega })}{\\Delta _{\\rm s}[\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2+\\lambda _\\xi ]-|\\Omega _c|^2}\\textbf {v}_{\\xi }.$ Since this equation has the same mathematical structure of Eq.", "(REF ), one can proceed as before and find the product expression for the normalized transmission, which reads $\\frac{t_{\\rm EIT}}{t_0}=\\prod _{\\xi =1}^N\\,\\frac{\\Delta _{\\rm A}(\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)-\\Omega _c^2}{\\Delta _{\\rm A}[\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2+\\lambda _\\xi ]-\\Omega _c^2}=\\prod _{\\xi =1}^N t_{\\xi },$ where we have chosen $\\Delta _{\\rm s}=\\Delta _{\\rm A}$ .", "When the probe field is on resonance with the $\\mathinner {|{e}\\rangle }-\\mathinner {|{g}\\rangle }$ transition, the two-photon resonance condition ($\\Delta _{\\rm s}=0$ ) is met, guaranteeing maximal transparency precisely when $\\Delta _\\text{A}=0$ .", "Figure REF (b) depicts the transmittance spectrum for a chain of $N=5$ atoms separated by a distance $d=\\lambda _\\text{p}/4$ and coupled to an unstructured waveguide.", "Close to resonance, the probe field is fully transmitted and the spectrum displays the characteristic transparency window.", "The limiting result of Eq.", "(REF ) when applied to one atom agrees with what has been previously found [73].", "From the above equation it is easy to find a general expression for the effective wave-vector of the polaritonic excitation.", "After a propagation distance spanning $N$ atoms, the transmission becomes $t_{\\rm EIT}/t_0=e^{{\\rm i}k_{\\rm eff} Nd}$ , where $k_{\\rm eff}$ is a complex number that accounts for both dispersion and absorption.", "Up to third order in the atom-probe detuning $\\Delta _{\\rm A}$ , we find $\\nonumber k_{\\rm eff}&=-\\frac{{\\rm i}}{Nd}\\sum _{\\xi =1}^N \\lambda _\\xi \\left\\lbrace \\frac{\\Delta _{\\rm A}}{\\Omega _c^2}+\\frac{\\Delta _{\\rm A}^2}{2\\Omega _c^4}(\\lambda _\\xi +{\\rm i}\\Gamma ^{\\prime })\\right.\\\\&\\left.+\\frac{\\Delta _{\\rm A}^3}{12\\Omega _c^6}\\left[12\\Omega _c^2-3\\Gamma ^{\\prime 2}+6{\\rm i}\\Gamma ^{\\prime } \\lambda _\\xi +4\\lambda _\\xi ^2\\right]\\right\\rbrace .$ The term quadratic in the detuning accounts for absorption, whereas the cubic one describes group velocity dispersion [74].", "The above expression is valid for any linear and isotropic quasi 1D-nanostructure.", "We would like now to focus on the case of an unstructured waveguide.", "As we have shown in the previous section, if the atoms are arranged in the mirror configuration (i.e.", "$k_{\\rm p}d=n\\pi $ , with $n\\in \\mathbb {N}$ ) there is only a single eigenstate, of eigenvalue $\\lambda _\\xi ={\\rm i}N\\Gamma _{\\rm 1D}/2$ , and thus the effective wave-vector is $k_{\\rm eff}&=\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}}{2d\\Omega _c^2}+{\\rm i}\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}^2}{8d\\Omega _c^4}(2\\Gamma ^{\\prime }+N\\Gamma _{\\rm 1D})\\\\\\nonumber &+\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}^3}{24d\\Omega _c^6}\\left[12\\Omega _c^2-3N\\Gamma _{\\rm 1D}\\Gamma ^{\\prime }-N^2\\Gamma _{\\rm 1D}^2-3\\Gamma ^{\\prime 2}\\right].$ For any other configuration, there are $N$ eigenstates, and the calculation of $k_{\\rm eff}$ is not as simple.", "However, realizing that $\\sum _{\\xi } \\lambda _\\xi ^\\beta =\\text{Tr} \\,\\lbrace \\mathfrak {g}^\\beta \\rbrace $ , it is easy to evaluate Eq.", "(REF ) for any inter-atomic separation.", "In particular, when $k_{\\rm p}d=n\\pi /2$ , with $n$ being an odd integer, we find another closed-form solution, namely $k_{\\rm eff}&=\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}}{2d\\Omega _c^2}+{\\rm i}\\frac{\\Gamma _{\\rm 1D}\\Gamma ^{\\prime }\\Delta _{\\rm A}^2}{4d\\Omega _c^4}+\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}^3}{24 d\\Omega _c^6}[12\\Omega _c^2+2\\Gamma _{\\rm 1D}^2-3\\Gamma ^{\\prime 2}]\\quad \\text{if $N$ is even},$ $k_{\\rm eff}&=\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}}{2d\\Omega _c^2}+{\\rm i}\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}^2}{8d\\Omega _c^4}(2\\Gamma ^{\\prime }+\\Gamma _{\\rm 1D}/N)+\\frac{\\Gamma _{\\rm 1D}\\Delta _{\\rm A}^3}{24 d\\Omega _c^6}[12\\Omega _c^2-\\Gamma _{\\rm 1D}^2-3\\Gamma ^{\\prime 2}-3\\Gamma _{\\rm 1D}\\Gamma ^{\\prime }/N]\\quad \\text{if $N$ is odd}.$ Therefore, for a chain of atoms in a waveguide, the effective polaritonic wavevector scales very differently with the number of atoms depending on the distance between them.", "For some specific separations, even the parity of the number of emitters modifies the dispersion relation.", "However, the group velocity at zero detuning, $v_g(\\Delta _{\\rm A}=0)=2\\Omega _c^2d/\\Gamma _{\\rm 1D}$ is not affected by the specific value of $k_{\\rm p}d$ , as it is calculated from the trace of the Green's function matrix (i.e., it only depends on the local Green's function).", "We have analyzed the problem of EIT as a possible example where important properties of the system, such as the band structure, can easily be obtained by applying the product expression for the normalized transmission.", "In principle, any other process that respects the mathematical structure of Eq.", "(REF ) could be subject to such an analysis." ], [ "Experimental perspectives", "In a recent experiment [38], the authors have observed signatures of collective atom-light interactions in the transmission spectra of atoms coupled to an alligator photonic crystal waveguide.", "They have recorded these spectra for various frequencies around the band edge of the PCW, exploring different physical regimes.", "Outside the bandgap, due to the finite size of the PCW, they observe the formation of a low-finesse cavity mode [as shown in Fig.", "3(a) of Ref.", "[38], at a frequency $\\nu _1$ ].", "At resonance with this cavity mode, the dissipative single-atom coupling to the structure for an atom at the peak of the Bloch function is $\\Gamma ^{\\rm max}_{\\rm 1D} (\\nu _1)\\simeq 1.5\\Gamma _0$ , as obtained from steady-state transmission lineshape measurements.", "The decay rate into leaky modes is $\\Gamma ^{\\prime }/\\Gamma _0\\simeq 1.1$ , estimated from finite-difference time-domain (FDTD) numerical calculations.", "After tuning the spectral features of the PCW so that the resonance frequency of the atoms moves into the bandgap, they observe asymmetric lineshapes, revealing significant coherent coupling.", "Specifically at $\\nu _{\\rm BG}=60$  GHz inside the bandgap, the spin exchange and decay rates are $J^{\\rm max}_{\\rm 1D}(\\nu _{\\rm BG})/\\Gamma _0\\simeq -0.2$ and $\\Gamma ^{\\rm max}_{\\rm 1D}(\\nu _{\\rm BG})/\\Gamma _0\\simeq 0.01$ , respectively.", "Due to the evanescent character of the field in the bandgap, the interaction range is finite, and at $\\nu _{\\rm BG}$ its value is $\\kappa _x^{-1}\\simeq 80a$ , being $a=370$  nm the lattice constant of the alligator PCW.", "While this experiment constitutes the first observation of more than one emitter interacting through the guided modes around the band edge of a PCW, the values of $J_{\\rm 1D}$ and $\\Gamma _{\\rm 1D}$ are not yet good enough to observe further signatures of atom-atom interactions such as time-dependent spin exchange.", "Nevertheless, we expect that near-term advances of the current set up will yield dramatic improvements on these rates, opening the door to exploring exciting collective atomic phenomena.", "In particular, instead of using an alligator PCW, one can employ a slot photonic crystal waveguide [4], [75], i.e.", "a quasi-1D waveguide embedded in a 2D photonic crystal.", "This structure would be advantageous due to several reasons.", "First of all, it inhibits atomic emission into non-guided modes due to the surrounding 2D photonic bandgap that reduces the modes into which the atom can radiate.", "Absent inhomogeneous broadening, early simulations demonstrate that it is possible to achieve a very small non-guided decay rate, i.e.", "$\\Gamma ^{\\prime }\\simeq 0.5\\Gamma _0$ .", "Moreover, one can engineer flatter bands, which leads to an increase of the group index of $n_{\\rm g}\\simeq 30$ near the band-edge (three times larger than that of the current alligator), according to FDTD simulations.", "Then, both $J_{\\rm 1D}$ and $\\Gamma _{\\rm 1D}$ would experience a three-fold increase.", "Finally, by trapping the atoms at the center of the nanostructure, in between the two slots and not above as it is currently done, we have estimated that $J_{\\rm 1D}$ and $\\Gamma _{\\rm 1D}$ would be five times larger.", "Summarizing, we project $\\Gamma ^{\\rm max}_{\\rm 1D}(\\nu _1)/\\Gamma ^{\\prime }\\simeq 44$ at the first cavity resonance.", "This yields the values of $J^{\\rm max}_{\\rm 1D}(\\nu _{\\rm BG})/\\Gamma ^{\\prime }\\simeq -6$ , $\\Gamma ^{\\rm max}_{\\rm 1D}(\\nu _{\\rm BG})/\\Gamma ^{\\prime }\\simeq 0.3$ , and $\\Gamma ^{\\prime }/\\Gamma _0\\simeq 0.5$ for a detuning from the band edge $\\nu _{\\rm BG}=20$  GHz, where the range of interaction is $\\kappa _x^{-1}\\simeq 80a$ .", "Figure REF (a) compares the ratio $|J_{\\rm 1D}/\\Gamma _{\\rm 1D}|$ between the coherent and dissipative guided-mode rates for the current alligator PCW (dashed line) and the described slot PCW (continuous line).", "The improved ratio for the later structure can already be observed at frequencies just beyond the band-edge, and becomes $|J_{\\rm 1D}/\\Gamma _{\\rm 1D}|\\simeq 10^4$ at a detuning of $0.5$  THz from the band-edge.", "Another signature of collective behavior is represented in Fig.", "REF (b), which shows the evolution of the excited state populations of two atoms placed at successive even antinodes (continuous and dashed curves), after initially inverting one of them.", "The atoms interact through the guided modes of the already described slot PCW, and their resonance frequency lies inside the bandgap, at the frequency for which the interaction range is $\\kappa _x^{-1}\\simeq 80a$ .", "The dotted lines show the expected result for non interacting atoms, where the off-diagonal terms of $\\mathfrak {g}$ are zero, a situation that occurs when the atoms are separated by a distance $d\\gg \\kappa _x^{-1}$ .", "To summarize, we believe that there is a bright future for experiments involving not only atoms, but also superconducting qubits interacting through the guided modes of a microwave photonic crystal.", "In a recent experiment, a ratio of $\\Gamma _{\\rm 1D}/\\Gamma ^{\\prime }=50$ has already been achieved for transmon qubits connected to a 1D coplanar microwave transmission line [25].", "Combined with the exciting recent advances in microwave photonic crystal fabrication [39], we expect a next generation of experiments where many qubits interact with each other in a mostly coherent manner." ], [ "Conclusion", "We have analyzed the optical response of a chain of atoms placed along a quasi-1D nanophotonic structure in terms of the classical electromagnetic Green's function.", "This formalism is valid in the presence of absorptive and dispersive media.", "We find that the linear response of the atoms can be understood in terms of collective atomic eigenstates of the Green's function matrix $\\mathfrak {g}(x_i,x_j)$ for all pairs of atoms.", "In particular, we have derived a closed expression for the transmission spectra that only depends on the cooperative frequency shifts and decay rates of these modes.", "We have shown that the transmission coefficient is a direct probe of the Green's function of the nanostructure, enabling us to determine whether the atom-light interactions are fundamentally dispersive or dissipative in character as well as to quantify the degree of cooperative interaction.", "We have gained insight into the interactions between atoms and quasi-1D cavities, waveguides, and photonic crystals, structures of relevance in recent experiments, as well as provided estimations of what can be observed in the near future.", "The Green's function formalism provides a natural language that unifies nanophotonics and quantum optics, and our results apply not only to atoms [38], but to many other quantum emitters, such as superconducting qubits [39], NV centers [76], rare earth ions [77] or quantum dots [4], interacting with any kind of quasi-1D photonic structures or circuits.", "Acknowledgments – We thank S.-P. Yu for sharing his insight on slot photonic waveguides.", "We are also grateful to O. J. Painter, A. Keller, M. Fang, and P. Dieterle for stimulating discussions about superconducting qubits.", "Funding is provided by the AFOSR QuMPASS MURI, NSF Grant PHY-1205729, the Office of Naval Research (ONR) Award N00014-16-1-2399; the ONR QOMAND MURI; the DOD NSSEFF program, and the IQIM, an NSF Physics Frontiers Center with support of the Moore Foundation.", "A. A.- G. was supported by the IQIM Postdoctoral Fellowship and the Global Marie Curie Fellowship LANTERN (655701).", "DEC acknowledges support from Fundacio Privada Cellex Barcelona, Marie Curie CIG ATOMNANO, MINECO Severo Ochoa Grant SEV-2015-0522, ERC Starting Grant FoQAL, and CERCA Programme / Generalitat de Catalunya." ], [ "Transmission and reflection coefficients in terms of Green's functions", "We begin by recalling Eq.", "(REF ), $E^+_\\beta ({\\bf r})&=E^{+}_{\\rm p,\\beta }({\\bf r})-\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}^T_{\\alpha \\beta }({\\bf r})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf E}_{\\rm p,\\alpha }^+\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2},$ which relates the $\\beta $ -polarization component of the field along any point of the structure with the collective atomic modes.", "In order to calculate the transmission spectra, we need an expression that connects the output and the input fields.", "To do so, let's consider that we have a dipole $p_{\\rm left}$ placed to the left of the first atom of the chain, at position ${\\bf r}_{\\rm left}$ , which is the source of the probe field $E_{\\rm p,\\beta }^+$ .", "The dipole $p_{\\rm left}$ is polarized along $\\hat{\\beta }$ , the same polarization of the guided mode field.", "To obtain the transmission coefficient, we evaluate the field at position ${\\bf r}_{\\rm right}$ , immediately to the right of the last atom of the chain.", "When the atoms are not present, the probe field at the left and right positions of the quasi-1D nanostructure are $E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left})=\\mu _0\\omega ^2_{\\rm p}\\, G_{\\rm 1D,\\beta \\beta }({\\bf r}_{\\rm left},{\\bf r}_{\\rm left})\\, p_{\\rm left},$ $E_{\\rm p,\\beta }^+({\\bf r}_{\\rm right})=\\mu _0\\omega ^2_{\\rm p}\\, G_{\\rm 1D,\\beta \\beta }({\\bf r}_{\\rm right},{\\bf r}_{\\rm left}) \\,p_{\\rm left}.$ Then, the transmission for the system without the atoms is simply $t_0(\\Delta _{\\rm A})=\\frac{E_{\\rm p,\\beta }^+({\\bf r}_{\\rm right})}{E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left})}=\\frac{G_{\\rm 1D,\\beta \\beta }({\\bf r}_{\\rm right},{\\bf r}_{\\rm left})}{G_{\\rm 1D,\\beta \\beta }({\\bf r}_{\\rm left},{\\bf r}_{\\rm left})}.$ When $N$ atoms are placed in the vicinity of the nanostructure, the field at position ${\\bf r}_{\\rm right}$ is $\\nonumber E^+_\\beta ({\\bf r}_{\\rm right})&=E_{\\rm p,\\beta }^+({\\bf r}_{\\rm right})-\\frac{1}{g_{\\beta \\beta }({\\bf r}_{\\rm left},{\\bf r}_{\\rm left})}\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}^T_{\\alpha \\beta }({\\bf r}_{\\rm right})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf g}_{\\alpha \\beta }({\\bf r}_{\\rm left})\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2}E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left})\\\\&=\\left(t_0(\\Delta _{\\rm A})-\\frac{1}{g_{\\beta \\beta }({\\bf r}_{\\rm left},{\\bf r}_{\\rm left})}\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}_{\\alpha \\beta }^T({\\bf r}_{\\rm right})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf g}_{\\alpha \\beta }({\\bf r}_{\\rm left})\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2}\\right)E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left}),$ where we have employed that the probe field at atom ${\\bf r}_j$ can be related to $E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left})$ as $\\nonumber E_{\\rm p,\\beta }^+({\\bf r}_j)&=\\mu _0\\omega ^2_{\\rm p} \\,G_{\\rm 1D,\\beta \\beta }({\\bf r}_j,{\\bf r}_{\\rm left})\\, p_{\\rm left}\\\\&=\\frac{G_{\\rm 1D,\\beta \\beta }({\\bf r}_j,{\\bf r}_{\\rm left})}{G_{\\rm 1D,\\beta \\beta }({\\bf r}_{\\rm left},{\\bf r}_{\\rm left})}E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left}).$ Then, the normalized transmission acquires the form given by Eq.", "(REF a) in the main text.", "It is important to notice that if there is only one guided mode (or if the coupling of the atoms to the other ones can be neglected), the transmission coefficient is an scalar, i.e., by calculating the coefficient for a component of the field, we calculate it for the full guided mode.", "Let's now calculate the reflection coefficient.", "Without the atoms, the field at ${\\bf r}_{\\rm left}$ is $E^+_\\beta ({\\bf r}_{\\rm left})=[1+r_0(\\Delta _{\\rm A})]E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left})$ .", "When the atoms are present, the field reads $&E_\\beta ^+({\\bf r}_{\\rm left})=[1+r_0(\\Delta _{\\rm A})]E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left})\\\\\\nonumber &-\\frac{E_{\\rm p,\\beta }^+({\\bf r}_{\\rm left})}{g_{\\beta \\beta }({\\bf r}_{\\rm left},{\\bf r}_{\\rm left})}\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}_{\\alpha \\beta }^T({\\bf r}_{\\rm left})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf g}_{\\alpha \\beta }({\\bf r}_{\\rm left})\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2}.$ Following similar steps as those above, we find Eq.", "(REF b) of the main text." ], [ "Derivation of Equation (", "We can exploit some properties of 1D systems to arrive to the closed expression for the transmission shown in Eq.", "(REF ), which only depends on the decay rates and frequency shifts of the modes, not on their spatial structure (i.e.", "the eigenfunctions).", "We first show how to derive the 1D Green's function wave equation, and how the solution is related to the full quasi-1D solution.", "We start with the 3D Green's function ${\\bf G}_{\\rm 1D}$ for the guided mode, which follows Eq.", "(REF ).", "We assume that the guided modes are transverse waves that travel in the $\\pm \\hat{x}$ direction and are polarized along $\\hat{y}$ , and that the field is approximately uniform in the transverse directions.", "From 3D, one can in principle construct the guided modes and their dispersion relations $\\omega (k)$ , from which one can identify an effective dielectric constant $\\epsilon _{\\rm eff}(x,\\omega )$ which produces the same behavior (at least within some bandwidth).", "The final answer that we are trying to achieve does not depend on explicit construction of $\\epsilon _{\\rm eff} (x,\\omega )$ .", "The result is a Helmoltz equation for the Green's function that reads $\\left[ \\frac{d^2}{d x^2} + \\frac{\\omega ^2}{c^2} \\epsilon _{\\rm eff}(x,\\omega ) \\right] \\tilde{G}_\\text{1D}(x,x^{\\prime },\\omega ) = - \\delta (x-x^{\\prime }),$ where $\\tilde{G}_{\\rm 1D}=AG_{\\rm 1D}$ , being $A$ the effective mode area.", "The solution for this second order linear ordinary differential equation can be expressed as the sum of the two homogeneous solutions.", "The Green's function can then be written in terms of the auxiliary fields $\\tilde{\\phi }_{L,R}(x)$ , which are solutions of the homogeneous equation, as $\\nonumber \\tilde{G}_\\text{1D}(x,x^{\\prime }) =\\frac{1}{W}&\\left[\\Theta (x^{\\prime }-x) \\tilde{\\phi }_L(x^{\\prime }) \\tilde{\\phi }_R(x)\\right.\\\\+&\\left.\\Theta (x-x^{\\prime }) \\tilde{\\phi }_R(x^{\\prime }) \\tilde{\\phi }_L(x)\\right],$ where $W$ is the Wronskian, which does not depend on the position, and is given by $W = \\tilde{\\phi }_R(x^{\\prime }) \\frac{\\textrm {d} \\, \\tilde{\\phi }_L(x^{\\prime }) }{\\textrm {d}x^{\\prime }} - \\frac{\\textrm {d} \\,\\tilde{\\phi }_R(x^{\\prime })}{\\textrm {d}x^{\\prime }}\\tilde{\\phi }_L(x^{\\prime }).$ We can then recover the full Green's function between atom $i$ and atom $j$ as $&G_{\\rm 1D} (x_i,x_j,\\omega )=\\frac{1}{A} \\tilde{G}_{\\rm 1D} (x_i,x_j,\\omega )\\\\\\nonumber &= \\left[\\Theta (x_j-x_i)\\phi _L(x_j) \\phi _R(x_i) +\\Theta (x_i-x_j) \\phi _L(x_i) \\phi _R(x_j)\\right],$ where $\\phi _{R,L}\\equiv \\tilde{\\phi }_{R,L}/\\sqrt{AW}$ .", "Then, the dipole-projected Green's function is $g_{ij}=\\Theta (x_j-x_i)s_{ji}+\\Theta (x_i-x_j)s_{ij},$ where $s_{ij}=\\varphi _{L}(x_i) \\varphi _{R}(x_j)$ , with $\\varphi _{L,i}=\\sqrt{\\mu _0\\omega _{\\rm p}^2d^2/\\hbar }\\,\\,\\phi _L(x_i)$ and $\\varphi _{R,j}=\\sqrt{\\mu _0\\omega _{\\rm p}^2d^2/\\hbar }\\,\\,\\phi _R(x_j)$ .", "It is convenient to define the rank-one matrix $\\mathfrak {s}={\\mathbf {\\varphi }}_L\\otimes {\\mathbf {\\varphi }}_R^T$ , where ${\\mathbf {\\varphi }}_{\\lbrace R,L\\rbrace }=(\\varphi _{\\lbrace R,L\\rbrace }(x_1), ..., \\varphi _{\\lbrace R,L\\rbrace }(x_N))$ is a vector of $N$ components.", "Let's now proceed to demonstrate Eq.", "(REF ).", "In terms of the eigenfunctions of $\\mathfrak {g}$ , the transmission is $\\nonumber &t(\\Delta _{\\rm A})/t_0(\\Delta _{\\rm A})=\\\\\\nonumber &1-\\frac{1}{g(x_{\\rm right},x_{\\rm left})}\\sum _{\\xi =1}^N \\frac{\\left({\\bf g}^T(x_{\\rm right})\\cdot \\textbf {v}_{\\xi }\\right) \\left(\\textbf {v}_{\\xi }^T \\cdot {\\bf g}(x_{\\rm left})\\right)}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2}\\\\&=1-\\frac{1}{g(x_{\\rm right},x_{\\rm left})}\\left({\\bf g}^T(x_{\\rm right})\\cdot \\mathcal {M}^{-1}\\cdot {\\bf g}(x_{\\rm left})\\right),$ where $\\mathcal {M}$ is given in Eq.", "(REF ).", "Since ${\\bf g}\\propto G_{\\rm 1D}$ , and using the expression for the Green's function in terms of the right-going and left-going field solutions [Eq.", "(REF )], we find $\\nonumber t(\\Delta _{\\rm A})/t_0(\\Delta _{\\rm A})&=1-{\\mathbf {\\varphi }}^T_R\\cdot \\frac{1}{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2+\\mathfrak {g}}\\cdot {\\mathbf {\\varphi }}_L\\\\&= 1-\\textbf {w}^T\\cdot \\frac{1}{\\mathbb {1}+\\tilde{\\mathfrak {g}}}\\cdot \\textbf {u},$ where we have defined $\\textbf {w}\\equiv {\\mathbf {\\varphi }}_R/\\sqrt{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2}$ , $\\textbf {u}\\equiv {\\mathbf {\\varphi }}_L/\\sqrt{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2}$ , and $\\tilde{\\mathfrak {g}}\\equiv \\mathfrak {g}/(\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)$ .", "By the matrix determinant lemma [60], we know that for a invertible matrix $\\textbf {A}$ and a pair of vectors $\\textbf {u},\\textbf {w}$ , we can write $\\text{det}(\\textbf {A}+\\textbf {u}\\otimes \\textbf {w}^T)=\\text{det}(\\textbf {A})\\left(1+\\textbf {w}^T\\cdot \\textbf {A}^{-1}\\cdot \\textbf {u}\\right)$ .", "Choosing $\\textbf {A}=-(\\mathbb {1}+\\tilde{\\mathfrak {g}})$ , we find $\\nonumber t(\\Delta _{\\rm A})/t_0(\\Delta _{\\rm A})&=\\frac{\\text{det}(\\mathbb {1}+\\tilde{\\mathfrak {g}}-\\textbf {u}\\otimes \\textbf {w}^T)}{\\text{det}(\\mathbb {1}+\\tilde{\\mathfrak {g}})}\\\\&=\\frac{\\text{det}((\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)\\mathbb {1}+\\mathfrak {g}-\\mathfrak {s})}{\\text{det}((\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)\\mathbb {1}+\\mathfrak {g})}.$ Since $(\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)\\mathbb {1}+\\mathfrak {g}-\\mathfrak {s}$ is a triangular matrix with $(\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)$ in the diagonal entries, and the determinant of a triangular matrix is the product of the diagonal entries, we find $\\text{det}((\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)\\mathbb {1}+\\mathfrak {g}-\\mathfrak {s})=(\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)^N$ , which yields $t(\\Delta _{\\rm A})/t_0(\\Delta _{\\rm A})&=\\frac{(\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)^N}{\\text{det}((\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2)\\mathbb {1}+\\mathfrak {g})}\\\\\\nonumber &=\\prod _{\\xi =1}^N\\,\\frac{\\Delta _{\\rm A}+{\\rm i}\\Gamma ^{\\prime }/2}{(\\Delta _{\\rm A}+J_{\\xi ,\\rm 1D})+{\\rm i}(\\Gamma ^{\\prime }+\\Gamma _{\\xi ,\\rm 1D})/2},$ as the determinant of a matrix is the product of its eigenvalues.", "The above expression is precisely Eq.", "(REF ).", "To the best of our knowledge, it is not possible to obtain a simplified expression for the reflection coefficient." ], [ "Non-Markovian effects: colored reservoirs", "The Markov approximation has been employed in the analysis carried out in the main text, as the Green's function of the nanostructures are considered to be constant over frequency ranges larger than the linewidth of the atoms.", "If that is not the case, it is not possible to find simple expressions for the Hamiltonian and Lindblad terms for the atomic density matrix.", "However, the expressions for the transmission and reflection coefficients are valid even when the spectral variation of the Green's function occurs within frequency intervals comparable to and smaller than the atomic linewidth.", "This fact might not be surprising, as in the low saturation limit, atoms behave as classical dipoles, and an equation for the transmission and reflection coefficients identical to those in Eqs.", "(REF a-b) can be found for classical emitters, without resorting to Markov's approximation.", "Following Refs.", "[78], [79], one finds the following equations for the field and atomic coherence operators in the frequency domain $&\\hat{{\\bf E}}^+({\\bf r},\\omega )=\\hat{{\\bf E}}_{\\rm p}^+({\\bf r},\\omega )+\\mu _0 \\omega ^2 \\sum _{j=1}^N {\\bf G}({\\bf r},{\\bf r}_j,\\omega )\\cdot \\textbf {d}\\,\\hat{\\sigma }_{ge}^j(\\omega ),\\\\&\\hat{\\sigma }_{ge}^j(\\omega )=-\\frac{1}{\\hbar \\Delta _{\\rm A}}\\textbf {d}^*\\cdot \\hat{{\\bf E}}^+({\\bf r},\\omega ).$ By plugging the expression for the field into the equation for the atomic coherence operators, one obtains Eq.", "(REF ), but for the operators in the frequency domain and without any Markovian impositions on the Green's function.", "Then, one finds the expressions for the transmission and reflection coefficients of Eqs.", "(REF a-b), but where now the eigenvalues are frequency-dependent and not constant quantities.", "Therefore, these expressions could be employed to understand the spectrum of a very recent experiment of a superconducting qubit in the bandgap of a photonic crystal waveguide, where the Green's function of the structure varies significantly within the linewidth of the qubit [39].", "Figure REF shows Markovian (continuous lines) and non-Markovian transmission spectra (dashed lines) in the dissipative and dispersive regimes of a cavity in the strong coupling limit.", "The non-Markovian curves show the typically encountered Rabi splitting, which results in fractional decay of the population and oscillations in time domain, in contrast with the exponential decay of the Markovian case." ] ]
1606.04977
[ [ "Detailed study of the microwave emission of the supernova remnant 3C 396" ], [ "Abstract We have observed the supernova remnant 3C~396 in the microwave region using the Parkes 64-m telescope.", "Observations have been made at 8.4 GHz, 13.5 GHz, and 18.6 GHz and in polarisation at 21.5 GHz.", "We have used data from several other observatories, including previously unpublished observations performed by the Green Bank Telescope at 31.2 GHz, to investigate the nature of the microwave emission of 3C 396.", "Results show a spectral energy distribution dominated by a single component power law emission with $\\alpha=(-0.364 \\pm 0.017)$.", "Data do not favour the presence of anomalous microwave emission coming from the source.", "Polarised emission at 21.5 GHz is consistent with synchrotron-dominated emission.", "We present microwave maps and correlate them with infrared (IR) maps in order to characterise the interplay between thermal dust and microwave emission.", "IR vs. microwave TT plots reveal poor correlation between mid-infrared and microwave emission from the core of the source.", "On the other hand, a correlation is detected in the tail emission of the outer shell of 3C 396, which could be ascribed to Galactic contamination." ], [ "Introduction", "The nature of the emission of supernova remnants (SNRs) and their interaction with the surrounding medium can be studied with multifrequency observations ranging from the radio to the X-ray regions of the spectrum.", "In the microwave region, a detailed multifrequency analysis of SNR emission can shed light on the interplay between synchrotron emission, which typically dominates SNR radio emission, and the possible presence of other mechanisms such as anomalous microwave emission (AME, [27]).", "Of key importance is the ability to distinguish between different components within the emitting source as well as to disentangle local and foreground contaminants.", "To this end, high angular resolution (i.e.", "arcminute-level) observations are crucial and allow one to distinguish different contributions within the same Galactic region (e.g.", "[10]).", "3C 396 is a shell-like SNR with a mean angular diameter of $7^{\\prime }.8$ [57].", "Its distance, estimated from Chandra data [55] and updated by means of CO observations, is about 6.2 kpc, and its age is about 3000 yr [84].", "[57] performed a complete overview of the numerous radio observations of the source before 1990.", "Using high angular resolution observations at 1.4 GHz obtained with the Very Large Array (VLA), [57] were able to identify the presence of two separate components in the source: the core, dominated by non-thermal synchrotron emission, and the tail, representing about $10\\%$ of the total flux at 1.4 GHz with a less steep spectral index compatible with free–free emission.", "[8] discussed the spatial spectral index variations of the spectral energy distribution (SED) in this SNR and concluded that these variations do not coincide with features in total intensity, but also found that the region associated with the brightest feature in the SNR has a somewhat flatter spectral index with respect to the average SNR spectral index.", "The presence of a small synchrotron pulsar wind nebula (PWN) within 3C 396 has been reported, although there is no spatial correspondence with the radio feature in high-resolution 20-cm VLA images of the remnant.", "This component could [would?]", "have a contribution of about $4\\%$ of the total radio flux density at 1.4 GHz [55], not enough to explain the spatial spectral index variations of the SED in this SNR.", "[82] observed 3C 396 with the Very Small Array (VSA) experiment [81] and tentatively reported finding anomalously high emission at 33 GHz, which suggested for the first time the possible presence of AME due to spinning dust in an SNR.", "As reported by [60], this evidence could not be confirmed by the Planck experiment owing to the lack of angular resolution and its vicinity to the Galactic plane.", "Besides AME, the excess seen by the VSA could also be explained by the presence of a significant level of thermal emission (about $50~\\%$ of the total emission at 1.4 GHz), causing the flattening of the spectrum at frequencies greater than 10 GHz [56].", "[82] emphasize the need for further measurements in the 10–20 GHz range, which is among the goals of the present paper.", "An attempt to evaluate the amount of free–free emission was performed, using the radio recombination line (RRL) survey of the Galactic plane from the H I Parkes All-sky Survey [4], [5]; the presence of diffuse free–free emission in the 3C 396 region is quite evident, although the angular resolution is poor (beamwidth of 14.4 arcmin FWHM).", "A rough estimate of the total flux gives an upper limit of 3 Jy at 1.4 GHz.", "Polarisation measurements of the SNR have been made, mainly at 5 GHz, and show the presence of a mean polarisation fraction of about $3\\%$ [75], with some regions being polarised at the 5–10% level and a small outer region being up to $50\\%$ polarised [57].", "In this paper we present observations at arcminute-level resolutions of 3C 396 performed with the Parkes single-dish 64 m telescope at 8.4 GHz, 13.5 GHz, 18.6 GHz, and 21.5 GHz.", "In Sections REF –REF we present the observations performed with the Parkes telescope, in Section REF we introduce the previously-unpublished 3C 396 observations performed by the 100-m Green Bank Telescope (GBT) and in Section we briefly introduce the remaining ancillary data used to fit the SED of the source.", "In Section we study the morphology of the SNR and in Section we present the SED, concentrating on the microwave band.", "The polarisation data, based on measurements at 21.5 GHz, are presented in Section .", "In Section  we show the correlation between IR and microwave data." ], [ "Observations and data reduction", "Observations were made in four different frequency bands with the 64-m Parkes Radio Telescope, NSW Australia, operated by ATNF-CASS, CSIRO: photometric observations were conducted at three frequencies (8.4 GHz, 13.5 GHz, and 18.6 GHz) and polarisation observations at 21.5 GHz.", "A further unpublished observation performed by GBT at 31.2 GHz is presented in the last subsection." ], [ "8.4 GHz observations", "The 8.4 GHz observations were conducted with the MARS receiver of the Parkes telescope on 26 July 2011 for 4 h. The receiver is a circular polarisation system with $T_{\\rm sys} \\sim 30$  K, a resolution of 2.4 arcmin, and a bandwidth of 400 MHz centred at 8.4 GHz.", "To measure the whole useful bandwidth the backend Digital Filter Banks Mark 3 (DFB3) was used with a configuration of 512 frequency channels of 2 MHz resolution for a total bandwidth of 1024 MHz.", "The correlator has full Stokes parameter capability, recording the two autocorrelation products $RR^*$ , $LL^*$ and the complex cross-product $RL^*$ whose real and imaginary parts are the two Stokes parameters $Q$ and $U$ .", "The gain is 1.18 Jy K$^{-1}$ .", "The source PKS B1934-638 was used for flux density scale calibration with an accuracy of 5% [69].", "The channels spanning the 400 MHz band were then binned into twenty 20-MHz sub-bands.", "A standard basket-weaving technique with two orthogonal scan sets along R.A. and Dec. spaced by 45 arcsec was used to observe an area of $20^{\\prime }\\times 20^{\\prime }$ centred on the source.", "The scan speed was $0^{\\circ }.5$ /min, with a sampling time of 0.25 s. For each scan, a linear fit performed off-source is removed.", "Map-making software based on the [30] Fourier algorithm was applied to make the maps [16].", "This technique effectively reduces 1/$f$ noise and removes stripes and features that differ between the two orthogonal sets of scans.", "The twenty sub-band maps were binned together in one map for the analysis.", "The final rms on the map is 22 mJy/beam on a beam-size scale.", "This is higher than the expected sensitivity ($\\sim $ 0.7 mJy/beam); however, 3C 396 is locate at ($l$ , $b$ ) = (39$^\\circ $ .2, -0$^\\circ $ .3), close to the Galactic plane, and the excess signal could be due to diffuse Galactic emission.", "To investigate this possibility we analysed the 1.4 GHz CHIPASS map [14], which has a resolution of 14.5 arcmin, and found an rms of 650 mK in the area around 3C 396.", "Following [11] we estimate an rms from the Galactic signal at our frequency and resolution of 23 mJy/beam, quite consistent with the rms signal directly measured around the source in our map.", "Figure: 18.6 GHz map of 3C 396 obtained with the Parkes Radio Telescope using the 13-mm receiver.The angular resolution (after smearing to Ku-band angular resolution) is 1.7 arcmin (FWHM) and the final rms of the map is 32 mJy/beam (0.11 MJy/sr)." ], [ "13.5 GHz observations", "The 13.5 GHz observations were conducted with the Ku-band receiver of the Parkes telescope on 31 August 2011 and 1 September 2011, for a total of 3 h. The receiver is a linear polarisation package with $T_{\\rm sys} \\sim 150$  K, a resolution of 1.7 arcmin, and a bandwidth of 700 MHz centred at 13.55 GHz.", "To detect the whole useful bandwidth the backend Digital Filter Banks Mark 3 (DFB3) was used with a configuration of 512 frequency channels of 2 MHz resolution for a total bandwidth of 1024 MHz.", "Only the two autocorrelation products were used ($XX^*$ , $YY^*$ ) for Stokes I measurements.", "The gain is 1.55 Jy K$^{-1}$ .", "The source PKS B1934-638 was used for flux density scale calibration with an accuracy of 5% [69].", "The atmospheric opacity when observing the calibrator was 0.085 dB at the observing elevation (EL); we correct for this attenuation in all our data.", "During the observations the opacity ranged from 0.074 to 0.120 dB (including EL effects) for a maximum variation compared to the constant opacity correction of 0.035 dB (0.8%), with marginal effects on the flux densitiy scale accuracy.", "The opacity at zenith was computed from atmospheric parameters (temperature, pressure, and relative humidity) and used to compute the opacity at the observing elevation correcting for EL effects (1/cos(EL)).", "The channels spanning the 700 MHz band were binned into seven 100-MHz sub-bands for the subsequent map-making processing.", "An area of $20^{\\prime }\\times 20^{\\prime }$ centred at the source was observed with a standard basket-weaving pattern, in which we performed two orthogonal scan sets along R.A. and Dec.", "The scans were spaced by 30 arcsec, the scan speed was 0$^\\circ $ .5/min, and the sampling time 0.25 s. After running the map-making software, the seven sub-band maps were co-added.", "The final rms signal on the map is 18 mJy/beam on beam-sized scales, larger than the expected sensitivity (4 mJy/beam).", "Following the same argument presented in Section 2.1 REF , we estimate a Galactic emission contribution of rms$_{13.5\\ \\rm GHz}^{1.7^{\\prime }} = $ 12 mJy/beam at the frequency and resolution of our observations, consistent with the measured value." ], [ "18.6 GHz observations", "The 18.6 GHz observations were conducted with the 13-mm receiver of the Parkes telescope on 29 August 2011 for a total of 3 h. This receiver can be set up either with a linear polarisation feed covering the frequency range 16–26 GHz or with a circular polariser covering 21.0–22.3 GHz.", "We used the linear polarisation configuration with a 800 MHz IF band centred at 18.6 GHz.", "The system temperature was $T_{\\rm sys} \\sim 70$  K, and the resolution was 76 arcsec.", "The backend DFB3 was used with with a configuration of 512 frequency channels of 2 MHz resolution for a total bandwidth of 1024 MHz.", "Only the two autocorrelation products were used ($XX^*$ , $YY^*$ ) for Stokes $I$ measurements.", "The gain is 1.56 Jy K$^{-1}$ .", "The source PKS B1921-293 was used to calibrate the flux density scale with an assumed flux of 17.2 Jy, and accuracy of 10%.", "This is a variable source on a time scale of a few weeks and its flux density was measured with the Australia Telescope Compact Array the day after the Parkes observations (31 August 2011).", "The opacity when observing the calibrator was 0.19 dB at the observing elevation, and we corrected all our data for this attentuation.", "During the observations the opacity ranged from 0.18 to 0.28 dB (including EL effects) for a maximum variation compared to the constant opacity correction of 0.09 dB (2%), with marginal effects on the flux density scale accuracy.", "The frequency channels spanning the 800 MHz band were binned into four 200-MHz sub-bands for the subsequent map-making step.", "An area of $20^{\\prime }\\times 20^{\\prime }$ centred at the source was observed with a standard basket-weaving pattern, in which we performed two orthogonal scan sets along R.A. and Dec.", "The scans were spaced by 24 arcsec, the scan speed was 0$^\\circ $ .5/min, and the sampling time 0.25 s. The map-making software was run and the four sub-band maps were co-added into a single map.", "We reached an rms sensitivity of 32 mJy/beam on beam-sized scales, after smearing the map to the angular resolution to the Ku-band observations, larger than the expected sensitivity (2 mJy/beam).", "As described in Section REF , we estimate a Galactic emission contribution of rms$_{18.6\\ \\rm GHz}^{1.7^{\\prime }} = $ 12 mJy/beam at the frequency and final resolution of our map.", "This can only partly explain the rms signal excess we measure.", "Possible sources of the additional noise are atmospheric emission fluctuations or a 1/$f$ -noise contribution from the receiver amplifiers.", "We do not have enough data to determine which of the two effects might have the greater contribution." ], [ "21.5 GHz observations", "The 21.5 GHz observations were conducted with the 13 mm receiver of the Parkes telescope on 30 August 2011 for a total of 3 h. We used the circular polarisation set-up with a 900 MHz band centred at 21.55 GHz.", "The system temperature was $T_{\\rm sys} \\sim 95$  K, and the resolution was 67 arcsec.", "The backend DFB3 was used with a configuration of 512 frequency channels of 2 MHz resolution for a total bandwidth of 1024 MHz.", "All autocorrelation and complex cross-products of the two circular polarisations were recorded ($RR^*$ , $LL^*$ , $RL^*$ ).", "The gain was 1.70 Jy K$^{-1}$ .", "Observations consisted of 111 repeated scans of one strip through the source at fixed DEC=5$^{\\circ }$ .43 from RA=285$^{\\circ }$ .90 to RA=286$^{\\circ }$ .20 and back.", "The source PKS B1921-293 was used to calibrate the flux density scale with an assumed flux of 16.5 Jy, and accuracy of 10%.", "The opacity when observing the calibrator was 0.47 dB at the observing elevation, and all our data are corrected for this attenuation.", "During the observations the opacity ranged from 0.39 to 0.65 dB (including EL effects) for a maximum variation compared to the constant opacity correction of 0.19 dB (4%).", "Combined with the flux scale accuracy (10%) this gives a final accuracy of 11%.", "Off-axis instrumental polarisation can be as high as $0.6\\%$ in single observations.", "Cancellation effects [15], [54] and the rotation of the parallactic angle during the execution of the scans reduces this effect when averaging scans taken at different parallactic angles.", "Following Battistelli et al (2015), we estimated a level of $0.2\\%$ after averaging all scans.", "Combining with the on-axis term, the overall systematic residual instrumental polarisation is estimated at $0.3\\%$ .", "The frequency channels over the 900 MHz band were binned into 90 sub-bands for flux and instrumental polarisation calibration.", "All the sub-bands were then combined for the subsequent analysis.", "We reached a sensitivity per beam-sized pixel of $\\sigma _{Q,U}^{21.5\\ \\rm {GHz}}$  = 0.2 mJy/beam in polarisation, consistent with the expected value.", "The fluctuations in Stokes $I$ are larger with an rms of 6.0 mJy/beam, consistent with the rms of the Galactic signal that, following the procedure of Section REF , is estimated to be rms$_{21.5\\ GHz}^{67\"^{\\prime }} = $ 5.2 mJy/beam at the frequency and resolution of the observations.", "The longer exposure time allowed the 1/$f$ and atmospheric noise terms to be reduced below the Galactic diffuse emission confusion, making the latter the leading term of the noise budget." ], [ "Ka-band - Green Bank Telescope", "A $\\sim 9^{\\prime } \\times 9^{\\prime }$ map of 3C 396 was made at 31.2 GHz under project AGBT10A-001 using the GBT Ka-band receiver [43] and Caltech Continuum Backend (CCB).http://www.astro.caltech.edu/ tjp/GBT/ .", "The Ka-band receiver is a beam-switched pseudo-correlation receiver which, when used with the CCB, provides excellent continuum sensitivity; technical details of these systems are given in [47].", "The GBT beam at 31 GHz is 24 arcsec (FWHM), and the two (differenced) beams read out by the CCB are separated by 78 arcsec in the cross-elevation direction on the sky.", "Data are calibrated using 3C 286, referenced to the WMAP absolute measurement of Jupiter [88] There is a $15\\%$ calibration uncertainty, dominated by uncertainties in the overall beam area of the GBT.", "The sky map (see Figure REF ) was estimated from the beam-switched data using a maximum entropy method (MEM) deconvolution (e.g., [19] and [53]).", "Figure: 31.2 GHz map of the core of 3C 396 obtained with the GBT Ka-band receiver.The angular resolution is 0.4 arcmin (FWHM)." ], [ "Ancillary data", "Radio emission from 3C 396 has been investigated in depth in the past.", "We selected a number of previous observations, taking into consideration the quality, accuracy and angular resolution of the data.", "In Table REF we show a summary of the data we used.", "When multiple observations are available, the flux at a given frequency is the weighted average of all the different measurements.", "[57] compiled a detailed list and analysis of the observations made before 1990.", "We include all these data.", "We were also able to retrieve the original map performed with the VLA at 1.468 GHz (with a beam FWHM of 54.4 arcsec and a noise rms of 16 mJy/beam) and at 4.860 GHz (with a beam FWHM of 16 arcsec and a noise rms of 2 mJy/beam), which will be used in the morphology section.", "In addition, we include more recent observations made with the Effelsberg telescope [67] as in [71].", "Recently, the SNR was observed at 5 GHz (in intensity and polarisation) by the Sino–German survey with the Urumqi telescope [75].", "The results are consistent with other observations in the literature at the same frequency.", "We analysed further observations at 8, 10, and 14 GHz ([45], [38]), which, however, were excluded owing to the low S/N ratio.", "We also include the 33 GHz measurement presented in [82].", "Any comparison of data from Parkes and the Very Small Array (VSA) interferometer needs to take into account the different experiment architecture and their different sensitivities as a function of angular scale.", "A linear fit was subtracted from each linear scan in the Parkes observations to reduce atmospheric contamination and 1/f detector noise.", "Due to this procedure these maps are sensitive to angular scales roughly between the beam size and the map size (20 arcmin).", "The VSA observation was made in the extended configuration with a primary beam FWHM of $\\approx 72$  arcmin and a synthesized beam FWHM of $\\approx 6$  arcmin.", "(More details of the experimental set-up and data reduction can be found in [24].)", "The VSA observation is more sensitive than the Parkes observation for scales between 20 and 72 arcmin, though Parkes has a finer angular resolution.", "In order to compare the same angular scales of in our two observations, each of which has a different window function, we reanalysed the VSA data.", "First, we filtered the VSA visibilities using the Parkes window function.", "Then, using standard aips [37] routines, we produced a sky map which we cleaned down to about $1.5$ times the map noise sensitivity.", "In order to derive the flux densities of 3C 396 and the nearby HII region NRAO 591 simultaneousely, we applied a double Gaussian fit.", "The resulting 3C 396 flux density is $5.20\\pm 0.33$  Jy.", "This value is lower than that derived in Scaife et al.", "(2007), $6.64\\pm 0.33$  Jy, as a consequence of the filtering process.", "We also included in our analysis Planck-HFI observations between 100 and 853 GHz and Herschel Space Observatory data between 600 and 4300 GHz in order to properly characterise the contribution from thermal dust in the microwave region.", "Planck-LFI data of 3C 396 are not included owing to the lack of angular resolution and the vicinity to the Galactic plane [60].", "Planck DR2 maps [59], downloaded from the Planck Legacy Archive,http://pla.esac.esa.int/pla/ were used to derive fluxes in the six frequency bands (100, 143, 217, 353, 545, and 857 GHz) of the High Frequency Instrument (HFI).", "These maps are supplied in the HEALPix pixelization [36] at a resolution of $N_{\\rm side}=2048$ .", "The fluxes shown in Table 1 were obtained within an aperture of radius 0$^\\circ $ .14 centred at $(l,b)=(39^{\\circ }.22,-0^{\\circ }.31)$ , after subtracting a median background level calculated in an external annulus between radii $0^{\\circ }.14$ and $0^{\\circ }.18$ .", "Owing to the proximity of the Galactic plane, the uncertainties of these fluxes are dominated by background fluctuations rather than by instrumental noise.", "The flux uncertainties are estimated from the background fluctuations in the ring, taking into account the number of independent samples and the number of pixels in the aperture.", "3C 396 was also observed by the Herschel Space Observatory during the project Herschel infrared Galactic Plane Survey (Hi-GAL) [52].", "We derive flux in five bands (600, 857, 1200, 1870, and 4300 GHz), processing the data as in [85].", "The fluxes were calculated using aperture photometry in the same manner as the Planck HFI case described above.", "In order to calculate the uncertainty, which is dominated by background fluctuations, we split the annulus used to subtract the background in ten radial sub-rings and calculated the dispersion of the recovered fluxes using this ensemble of sub-rings.", "Planck and Herschel flux estimates are strongly affected by the foreground emission evident in their maps.", "This makes them only tentative detections and could be interpreted as upper limits of the mm and sub-mm emission from 3C 396 SNR.", "Table: Integrated fluxes used for the 3C 396 SED fit." ], [ "Morphology", "3C 396 has a complex morphology that makes it difficult to determine the nature of its observed emission.", "The SNR is a shell-type object with highly non-uniform emission, and [57] observed the presence of a tail at $RA \\simeq 286^{\\circ }.07$ , $DEC \\simeq 5^{\\circ }.47$ that accounts for about $10\\%$ of total integrated flux at 1.4 GHz.", "The tail emanates from the eastern side of the remnant and curves more than 120$^{\\circ }$ before falling below detectability.", "Parkes measurements have been able – for the first time in the microwave – to partially resolve the structure of the SNR and its tail.", "A rough estimate of its contribution at the observed frequencies (8.4, 13.5 and 18.6 GHz) yields 10–20% of the total integrated flux.", "Using a TT-plot with the VLA 1.4 GHz VLA map and Parkes 13.5 GHz Parkes map and together with the definition of tail shown in Fig.", "REF , we find a spectral index for the tail of $(-0.16 \\pm 0.02 (stat) \\pm 0.05 (cal))$ .", "This is compatible with free-free emission, as previuosly suggested by [57].", "In order to investigate the nature and the spatial properties of the emission, we made a spectral index map (Figure REF ) from the two maps with highest angular resolution (VLA at 4.8 GHZ and GBT at 31.2 GHz).", "The two maps where degraded to the same angular resolution and the spectral index was derived pixel-by-pixel using the relation $f(\\nu )=k\\nu ^{-\\alpha }$ .", "The data at both frequencies have a high S/N ratio, and so the error in the fitting procedure is dominated by systematic errors (e.g., calibration error).", "This map is useful for investigating variations in the spectral index within the source itself: it exhibits a steeper spectral index ($-0.6< \\alpha <-0.4$ ) in the outer part of the core than in the inner part, where $\\alpha $ is between $-0.4$ and $-0.3$ .", "This confirms claims by [8] from data at 1.4 and 5 GHz.", "The observation of such spectral variations in shell remnants can be associated with spatially-dependent particle acceleration or bends in the relativistic electron energy spectrum.", "We cannot exclude the possibility, however, that these variations are caused by free–free presence within the source, as suggested by [56], or along the line of sight.", "Analysis of the IR and sub-mm maps (see Fig.", "REF ) retrieved from the Herschel and SPITZER experiments and ranging from 500 $\\mu $ m to 24 $\\mu $ m shows a lack of emission towards the core of 3C 396, while there is evidence of emission in the surrounding area, especially around the tail where the emission peaks at around 160 $\\mu $ m. It should be stressed, however, that the closeness of the source to the Galactic plane makes its emission at all wavelengths fairly contaminated by the Galactic plane itself.", "It is therefore difficult to disentangle local and diffuse emission.", "A cross-correlation analysis between IR and microwave emission is performed in section .", "Figure: Spectral index map of 3C 396 calculated using the 4.8 GHz VLA map (beam FWHM of 16 arcsec)and the 31.2 GHz GBT map (beam FWHM of 25 arcsec).", "Contour levels are from the 31.2 GHz map and are at 0.23 , 0.46, 0.68, 0.91, 1.14, 1.37, 1.59, 1.82, 2.05 and 2.28 MJy/sr." ], [ "Spectral Energy Distribution", "In order to retrieve information about the overall emission of 3C 396 we have studied the SED of the SNR.", "To compare fluxes from all our available data, we reduced the maps in the same way: aperture photometry was performed with a radius of 8.5 arcmin after subtracting a background measured in an annulus with inner radius of 8.5 arcmin and outer radius of 10 arcmin.", "Reported errors are the combination of calibration error and map noise, estimated from background fluctuations.", "Our baseline model of the SED has one power law component, $Sy$ , and a thermal dust emission.", "Millimetre, sub-millimetre and FIR emission is clearly dominated by thermal dust emission, which we modelled as modified blackbody emission with fixed dust emissivity spectral index $\\beta =1.6$ [58] and free parameters for the temperature $T_{\\rm d}$ and the optical depth at 100 $\\mu $ m, $\\tau _{100}$ .", "The resulting model, which we fit with a routine based on the IDL MPFIT one [48], is therefore: $S(\\nu )= Sy \\cdot \\nu ^{\\alpha _{1}}+ \\tau _{100} \\left( \\frac{\\nu }{3 \\rm THz} \\right)^{\\beta }\\cdot BB_{\\nu }(T_{\\rm d}) \\Omega $ where $BB_{\\nu }$ is the Blackbody brightness and $\\Omega $ is the total solid angle of the source.", "Results are shown in Figure REF .", "The SED of 3C 396 is well-described in the 1-30 GHz range Crab-like synchrotron component with $\\alpha =(-0.364 \\pm 0.017)$ .", "The dust temperature is $(25.1 \\pm 1.2)$  K and $\\tau _{100}$ is $(3.6 \\pm 1.2) 10^{-4}$ .", "The fit has $\\chi $ -square = 24.6 (DOF=21).", "We also tested more complex models by adding AME, using a fiducial model derived from [27], or including contaminations from free-free or a second power law component.", "To select the best model, we used the Akaike information criterion (AIC, [1], [13]), in which AIC=$2k-\\chi ^{2}$ , where k is the number of estimated parameters.", "These three modified models (AIC $\\approx $ 34.5) are all rejected by AIC with respect to the baseline model (AIC $\\approx $ 32.5).", "We used the high angular resolution maps of VLA, Parkes and GBT to make a further SED fit restricted only to the core of 3C 396.", "Its flux was estimated from aperture photometry with a radius of 4.5 arcmin and background subtraction in an annulus with inner radius of 4.5 arcmin and outer radius of 6 arcmin (excluding the tail region).", "We obtain $\\alpha =(-0.466 \\pm 0.024)$ and find no anomalous emission.", "Infrared and mm-wave measurements can constrain the contribution from dust at the frequencies where AME should be dominant (i.e., below 100 GHz).", "However, tt is important to stress that 3C 396 is not unambiguously detected in these maps.", "Consequently the estimated fluxes can only be taken as upper limits on the SNR emission.", "Figure: SED of 3C 396 including Parkes (in red) and the reprocessed VSA flux at 33 GHz.", "The fitindicates that one synchrotron component is the best explaination of the SED behaviour in the microwave range.The fit does not favour the presence of any spinning dust component or further power law components (e.g., free–free)." ], [ "Polarisation", "Polarisation-sensitive observations at the fixed declination of $5 ^{\\circ } 26^{\\prime } 6^{\\prime \\prime }$ were conducted at 21.5 GHz with the 13 mm receiver of the Parkes telescope.", "In Figure REF we show the percentage polarised emission obtained from the combination of our one-dimensional scans.", "Only the polarisation fraction is presented since the measurements are affected by gain fluctuations on long time scales ($>$ 6 hrs) and cannot be reliably calibrated.", "As previously described, spurious polarisation and depolarisation are verified to be less than $0.3\\%$ .", "Any estimate of polarised intensity ($PI$ ), and consequently of polarisation fraction, is intrinsically overestimated unless a debasing procedure is applied ([80]).", "For this reason, we adopted a debiasing procedure described in [11], which is based on the Bayesian approach of [78].", "For high signal-to-noise data ($PI/\\sigma > 5$ ), we set a debiased polarised intensity of $\\sqrt{(PI^{2}-\\sigma ^{2})}$ as suggested by [80].", "Some of our polarised measurements are in the low signal-to-noise regime in which the debiasing is particularly complex.", "For $PI / \\sigma < 2$ we set upper limits, while in the intermediate range we integrate the posterior probability density function over the parameter space of the true polarisation.", "We detect a polarisation of about $5~\\%$ in the outer regions of the source, while in the core values are lower ($1-4~\\%$ ).", "These results are in good agreement with previous measurements at 5 GHz ([57], [75]).", "[75] observed a mean polarisation in the same one-dimensional scan of the source of $(2.4 \\pm 0.4) \\%$ , which is compatible with our average measurement of $(2.4 \\pm 0.1$ (rand)$\\pm 0.3$ (sys))$~\\%$ .", "This result seems to discount the possibility of a large amount of free–free emission at 1 GHz (of about $40 \\%$ ) as suggested by [56], since one that would require the polarisation at 21.5 GHz to be about half of that at 5 GHz.", "Further, we can assume that the nature of 3C 396 emission does not dramatically change from radio to microwave frequencies.", "Figure: Observation of the fraction of polarised intensity over the total intensity at 21.5 GHz(fixed declination DEC=5 ∘ .435^{\\circ }.43 ), observed with Parkes.", "Error bars are quoted at the 2 σ\\sigma level.", "The dot–dashed line is the profile of the total intensity emission of the source at 21.5 GHz in arbitrary units." ], [ "Infrared Correlation", "Studying the possible spatial correlations between our microwave data and IR data is of interest since the correlation between AME and thermal dust emission is well known (e.g.", "[20], [79], [25], [77], [34], [76], [23]).", "In the case of 3C 396 any correlation between our microwave data and IR data would hint at the presence of AME below the detection limit.", "The region emissivity is usually defined as the ratio between the source emission at a frequency near the spinning dust peak ($\\sim $ 20–30 GHz) and the FIR signals at 100 $\\mu $ m. [23] reports HII region emissivity in units of $\\mu K/(MJy/sr)$ for microwave emission in the range 10 to 70 GHz relative to IRAS 100 $\\mu $ m. These show emissivities of the same order of magnitude as high latitude diffuse cirrus (i.e., $\\sim 10 \\mu $ K/(MJy/sr)), though it is significantly lower in some regions.", "Pixel-by-pixel correlations measured by [79] in $\\rho $ Ophiuchi and LDN1780 show a clear correlation with a trend favouring the short wavelengths IR bands (i.e., 60 $\\mu $ m and 25 $\\mu $ m).", "Recently, the Planck experiment studied all-sky correlations, showing an excellent correlation (Pearson $r=0.98$ ) with the map at 545 GHz and an average emissivity of about 70 $\\mu $ K/(MJy/sr) [61].", "The analysis of selected known AME sources has significant correlation (Pearson $r>0.6$ ) and similar values of emissivity.", "In order to compare our microwave measurements with IR emission, we make use of the SPIRE, PACS and MIPS IR maps from the Herschel [52] and SPITZER [70] experiments, ranging from 500 $\\mu $ m to 24 $\\mu $ m. In Figure REF we compare the Parkes Ku 13.5 GHz map with those extracted from the SPIRE instrument (500 $\\mu $ m) and the PACS instrument (160 $\\mu $ m and 70 $\\mu $ m) after smoothing them to the angular resolution of the Parkes 13.5 GHz band.", "3C 396 has been divided into two regions (tail and core), as shown in Figure REF .", "A lack of FIR emission is evident towards the core region of 3C 396, whereas there are bright FIR structures towards the northeast, where the tail identified by [57] and also hinted at in our Parkes data is located.", "These FIR structures have been studied in different publications ([62], [46], [7]).", "In the three right panels of Figure REF we show TT correlation plots between the FIR and the radio emission at 13.5 GHz from Parkes.", "A significant correlation is evident between the data sets for the tail region, whereas for the core region we see possible anti-correlation or no correlation at all.", "Pearson correlation coefficients for the tail emission are reported in Table REF .", "The statistical significance is at the 0.2% level and we make significant measurements of all the coefficients.", "The slope of a linear fit to the pixels defining the tail-region gives an estimate of the emissivity in the range 70–3500 $\\mu $ K/(MJy sr$^{-1}$ ).", "These values are considerably higher than what was found for typical AME and HII regions.", "The correlation coefficient $R$ between 13.5 GHz Parkes map and the 500 $\\mu $ m map is lower than typical values estimated by the Planck experiment.", "Moreover, a close comparison between the IR maps and the microwave maps does not allow us to exclude the possibility that the IR emission is unassociated with the SNR and its microwave emission.", "Figure: Parkes 13.5 GHz map of 3C 396 with superimposed indications of ourregion selection and the polarisation scan path.", "In order to perform ameaningful correlation analysis and to obtain the TT-plots shown inFigure , we have divided 3C 396 source into two regions, onecovering the core (central elliptical contour) and one covering the tail(lateral semicircular contour).", "The white dashed line illustrates thepath of the polarisation scan reported in Figure .", "Contour levels, from the 13.5 GHz map, are at 0.2, 0.70, 1.2 , 1.70, and 2.2 MJy/sr.Figure: Left column from top to bottom: 13.5 GHz Parkes map (contours) vs. Pacs 70 μ\\mu m, Pacs 160 μ\\mu m,and SPIRE 500 μ\\mu m maps.", "The right column has the corresponding scatter plots, i.e., microwave map vs IR map.White diamonds denote emission from the core of 3C 396 while black circles denote that of the tail, with the definition of these two regions shown in Figure .", "Contour levels, from the 13.5 GHz map, are at 0.2, 0.70, 1.2, 1.70, and 2.2 MJy/sr.Table: Results of the correlation analysis of tail emission between 13.5 GHz and IR experiments" ], [ "Conclusions", "We have performed new intensity observations of the SNR 3C 396 with the Parkes telescope at 8.4, 13.5, and 18.6 GHz and analysed them together with unpublished 31.2 GHz data from the GBT telescope.", "We have calculated the SED of the core of the source and and its surroundings.", "Our observations argue against the presence of an AME component from the core of the source.", "When accounting for the surrounding region, the SED is well-described by a power law with a Crab-like spectral index.", "We find a spatial spectral index variation between 5 GHz and 31.2 GHz that is consistent with the variation reported by [8] between 1.4 GHz and 5 GHz.", "We have also performed new polarisation observations at 21.5 GHz that seem to confirm that 3C 396 emission is dominated by synchrotron, without any important contamination due to free–free as suggested by [56].", "The absence of correlation between IR and 13.5 GHz emission in the core of 3C 396 is a further finding, confirming that the emission of 3C 396 is not anomalous.", "On the other hand, we find a significant correlation in the tail-region at short wavelengths (i.e., $ < 160~\\mu $ m), as in other AME sources; however, this is probably due to diffuse emission from the Galactic plane." ], [ "Acknowledgments", "We acknowledge the logistic support provided by Parkes operators.", "The Parkes radio telescope is part of the Australia Telescope National Facility, which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.", "The American National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We would like to thank A.D. Hincks, M. Hobson, A. Scaife for useful suggestions and comments.", "We thank the referee for providing constructive comments and help in improving the contents of this paper." ] ]
1606.05193
[ [ "Predictivity of Neutrino Mass Sum Rules" ], [ "Abstract Correlations between light neutrino observables are arguably the strongest predictions of lepton avour models based on (discrete) symmetries, except for the very few cases which unambiguously predict the full set of leptonic mixing angles.", "A subclass of these correlations are neutrino mass sum rules, which connect the three (complex) light neutrino mass eigenvalues among each other.", "This connection constrains both the light neutrino mass scale and the Majorana phases, so that mass sum rules generically lead to a nonzero value of the lightest neutrino mass and to distinct predictions for the e ective mass probed in neutrinoless double beta decay.", "However, in nearly all cases known, the neutrino mass sum rules are not exact and receive corrections from various sources.", "We introduce a formalism to handle these corrections perturbatively in a model-independent manner, which overcomes issues present in earlier approaches.", "Our ansatz allows us to quantify the modi cation of the predictions derived from neutrino mass sum rules.", "We show that, in most cases, the predictions are fairly stable: while small quantitative changes can appear, they are generally rather mild.", "We therefore establish the predictivity of neutrino mass sum rules on a level far more general than previously known." ], [ "Introduction", "Neutrinos keep on surprising us when it comes to experiments.", "We are still puzzled by their curious properties which are experimentally established but which we still do not understand on a fundamental level.", "For example, the neutrino mass is extremely small compared to any other fermionic mass we know, namely below 1 eV (or possibly even $0.1$  eV).", "This we know from several experiments and observations, either in the lab by kinematical determinations of the neutrino mass [1] and from the hunt for neutrinoless double beta decay [2], or in space by the time-of-flight measurements of supernova neutrinos [3] or from cosmological considerations [4].", "Similarly, the leptonic mixing angles are a mystery to us, for we have measured their values to be fairly large (see nu-fit.org [5]), but yet we have absolutely no theoretical understanding of these numbers.", "Although far from perfect, still the best idea we have to explain leptonic mixing is to motivate the values of the mixing angles by relating them to discrete flavour symmetries, see [6] for recent reviews.For alternative ideas, one could for example consider anarchy [7] or radiative transmission [8].", "However, flavour models based on discrete symmetries generically face one big problem: they often do not give us any testable prediction beyond fitting the known mixing angles within their experimental ranges (note that, if a model did not, it would in any case be discarded).", "On the other hand, at least some groups of flavour models are more powerful than that, in the sense that they predict certain correlations between different observable quantities.", "Among these correlations, the most popular ones discussed in the literature are mixing angle sum rules [9]; but, when looking at the total mass matrix of neutrinos, a second class of correlations arises, neutrino mass sum rules.", "It is these mass sum rules (in the following referred to by “SRs”) that we will investigate in this text.", "Basically, what they do is to connect the three complex neutrino mass eigenvalues $\\tilde{m}_i$ in a simple relation, where all three contributions sum to zero.", "For example, $\\tilde{m}_1 + \\tilde{m}_2 - \\tilde{m}_3 = 0$ would be a valid SR, as well as $ \\tilde{m}_1^{-1} + \\tilde{m}_2^{-1} - \\tilde{m}_3^{-1} = 0$ .", "Several studies of SRs have been presented earlier.", "Among the first works investigating SRs were Refs.", "[10], [11], [12].", "However, these actually did not mention the name “sum rules” at all.", "That term came up only later, in Refs.", "[13], [14], [15], [16].", "The probably most comprehensive study of SRs, which includes all known cases we are aware of, had been presented in Ref. [17].", "Based on this study, Ref.", "[18] has shown that, indeed, with realistic assumptions on the experimental side one could truly distinguish at least some classes of models with near-future data – as long as our current understanding of the nuclear physics aspects of neutrinoless double beta decay is not totally flawed.", "However, it is clear in nearly all cases that sum rules, even though predicted, are in fact not exact.See Ref.", "[19] for a notable exception, where the SR at least holds to next-to-leading order.", "These can arise, e.g., from higher-order correction terms arising from flavour symmetry breaking, or from corrections transmitted from the charged lepton sector.", "A further correction that we had investigated earlier are those arising from renormalisation group running, see Ref. [20].", "We have indeed been able to show that at least these corrections do not change the predictions from SRs significantly, but we cannot claim that this particular type of corrections would be the most general one.", "In this manuscript, we will try to close this gap by computing the effect of nonexact SRs on the predictions in a very general framework, based on a perturbative approach.", "The only attempt to investigate approximate sum rules that we are aware of had been presented in Ref.", "[15], however, it had been pointed out [17], [20] that the approach in that reference is likely to be insufficient because the corrections have been attributed to one particular mass eigenvalue only, namely $m_3$ .", "However, this may create problems both due to measure dependence of the perturbations and due to the fact that $m_3$ plays a very different role for the two mass orderings.", "This is another shortcoming which we will at least partially overcome with our approach, by describing a procedure by which the dependence on a choice of mass eigenvalue can be minimised.", "Using the techniques developed, we will investigate the predictions of all sum rules we found in the literature.", "As we will see, while the predictions are of course changed for approximate SRs, in most cases these changes are comparatively mild, thereby keeping the predictivity of the SRs alive.", "Only in one case, namely SR 10, a qualitative change does happen which could in fact be visible in an experiment.", "This work is organised as follows.", "We start in Sec.", "by reintroducing the parametrisation of SRs used by us, and we will also visualise how to interpret neutrino mass sum rules, no matter if exact or not, in a geometrical manner.", "Sec.", "is dedicated to a discussion of the possible origins of the various corrections, to clarify which cases are covered by our formalism.", "Our numerical results, along with detailed plots for each sum rule, are presented in Sec. .", "We conclude in Sec. .", "Technical details on how to derive the physical leptonic mixing parameters from the charged lepton and neutrino sectors are summarised in Appendix ." ], [ "Parametrisation and geometrical interpretation", "To start off, let us define our conventions.", "First of all, we parametrise the leptonic mixing matrix, the so-called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, by using the standard parametrisation suggested by the Particle Data Group (PDG) [21] $U_\\text{PMNS}&=R_{23}U_{13}R_{12}P_{0}\\nonumber \\\\&=\\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}\\mathrm {e}^{-\\mathrm {i}\\delta }\\\\-s_{12}c_{23}-c_{12}s_{23}s_{13}\\mathrm {e}^{\\mathrm {i}\\delta }&c_{12}c_{23}-s_{12}s_{23}s_{13}\\mathrm {e}^{\\mathrm {i}\\delta }&s_{23}c_{13}\\\\s_{12}s_{23}-c_{12}c_{23}s_{13}\\mathrm {e}^{\\mathrm {i}\\delta }&-c_{12}s_{23}-s_{12}c_{23}s_{13}\\mathrm {e}^{\\mathrm {i}\\delta }&c_{23}c_{13}\\\\\\end{pmatrix}P_{0},$ where $\\delta $ is the Dirac CP-phase and $P_{0}$ =diag$(\\mathrm {e}^{-\\mathrm {i}\\phi _1/2},\\mathrm {e}^{-\\mathrm {i}\\phi _2/2},1)$ is a diagonal matrix containing the two Majorana phases $\\phi _{1,2}$ .", "However, note that our definition of the Majorana phases is different compared to what the PDG uses [21] in their Eq.", "(14.78): $\\phi _1 = -\\alpha _{31}$ and $\\phi _2 = \\alpha _{21}-\\alpha _{31}$ .", "As for the mixing parameters, we have used the v2.0 version from nu-fit.org, as reported in Tab.", "REF .Note that, just while this work was in its final stages, a new version of mixing parameters v2.1 was released.", "We have verified that our results are not significantly changed for some example cases, however, given the time-consuming numerics behind this manuscript we have decided against rerunning all of our code and have instead decided to consistently present the results obtained for v2.0.", "With that said, let us next introduce our formalism to treat neutrino mass sum rules (SRs).", "In [20] we have investigated the effect of renormalisation group corrections to neutrino mass SRs.", "In this paper we want to examine the impact of next-to-leading-order (NLO) corrections to neutrino mass matrices on mass SRs.", "A general exact SR can be parametrised according to [20]: $s(m_1,m_2,&m_3,\\phi _1,\\phi _2;c_1,c_2,d,\\Delta \\chi _{13},\\Delta \\chi _{23}) \\equiv \\nonumber \\\\&c_1 \\left(m_1 \\text{e}^{-\\text{i}\\phi _{1}}\\right)^d\\text{e}^{\\text{i}\\Delta \\chi _{13}}+c_2 \\left(m_2 \\text{e}^{-\\text{i}\\phi _{2}}\\right)^d\\text{e}^{\\text{i}\\Delta \\chi _{23}}+m_3^d~ \\stackrel{!", "}{=} 0 \\;,$ where $\\phi _i$ , $i=1,~2$ are the Majorana phases.", "The quantities $c_1,~c_2,~ d,~\\Delta \\chi _{13}$ , and $\\Delta \\chi _{23}$ are parameters which characterise the SR, e.g., SR 1, $\\tilde{m}_1 + \\tilde{m}_2 = \\tilde{m}_3$ , is characterised by $(c_1, c_2, d, \\Delta \\chi _{13}, \\Delta \\chi _{23}) = (1, 1, 1, \\pi , \\pi )$ , while SR 7, $\\tilde{m}_1^{-1} = 2 \\tilde{m}_2^{-1} + \\tilde{m}_3^{-1}$ , is characterised by $(c_1, c_2, d, \\Delta \\chi _{13}, \\Delta \\chi _{23}) = (1, 2, -1, \\pi , 0)$ , see Tab.", "REF for a summary.", "Note that, in this notation, $\\tilde{m}_i$ are the complex mass eigenvalues, i.e., with the phase information included.", "In Tab.", "REF we have collected all the SRs we found in the literature with their parameters $c_1,~c_2,~ d,~\\Delta \\chi _{13}$ , and $\\Delta \\chi _{23}$ .", "Table: Summary table of the SRs we will analyse in the following.", "The parameters c 1 ,c 2 ,d,Δχ 13 c_1, c_2, d, \\Delta \\chi _{13}, and Δχ 23 \\Delta \\chi _{23} that characterise them are defined in Eq. ().", "In SRs 9 and 10, two possible signs appear which lead to two possible values of Δχ i3 \\Delta \\chi _{i3}.A complex perturbation governed by the complex parameter $\\delta m_i~\\text{e}^{\\text{i}\\delta \\phi _i}$ to the neutrino mass matrix shifts its complex eigenvalues to $\\tilde{m}_i=m_i \\text{e}^{-\\text{i}\\phi _i}=m_i^{(0)} \\text{e}^{-\\text{i}\\phi _i^ {(0)}}+ \\delta m_i~\\text{e}^{\\text{i}\\delta \\phi _i}~.$ Thus, explicitly, the corrections are connected to the physical parameters as follows: $m_i&\\approx m_i^{(0)} \\left( 1 + \\frac{\\delta m_i}{m_i^{(0)}} \\cos (\\delta \\phi _i-\\phi _i^{(0)}) \\right) ~,\\\\\\phi _i&\\approx -\\arctan \\left(\\frac{-m_i^{(0)}\\sin (\\phi _i^{(0)})+\\delta m_i \\sin (\\delta \\phi _i)}{m_i^{(0)}\\cos (\\phi _i^{(0)})+\\delta m_i \\cos (\\delta \\phi _i)}\\right) ~.$ Thus, e.g.", "$\\delta \\phi _i$ is not the correction to the phase of the complex mass.", "We assume that, in general, the correction to each mass can have a different phase $\\delta \\phi _i$ than that of the zeroth-order SR, i.e., $\\delta \\phi _i$ need not be a small number.", "We furthermore assume that $\\delta m_i/m_i^ {(0)}\\ll 1$ , with $\\delta m_i > 0$ without loss of generality.", "So we can expand Eq.", "(REF ) in the small parameters $\\delta m_i$ to obtain the deviation from the exact SR.", "This results in $0 \\ne s \\approx s^{(0)}+\\delta s \\;,$ where $s^{(0)}= c_1 \\left(m_1^{(0)} \\text{e}^{-\\text{i}\\phi _{1}^{(0)}}\\right)^d\\text{e}^{\\text{i}\\Delta \\chi _{13}}+c_2 \\left(m_2^{(0)} \\text{e}^{-\\text{i}\\phi _{2}^{(0)}}\\right)^d\\text{e}^{\\text{i}\\Delta \\chi _{23}}+\\left(m_3^{(0)}\\right)^d~$ and $\\delta s &= d \\Big [ c_1 \\left(m_1^{(0)} \\text{e}^{-\\text{i}\\phi _1^{(0)}}\\right)^d \\text{e}^{\\text{i}(\\Delta \\chi _{13}+\\phi _1^ {(0)})} \\frac{\\delta m_1 \\text{e}^{\\text{i}\\delta \\phi _1} }{m_1^ {(0)}} \\nonumber \\\\&+c_2 \\left(m_2^{(0)} \\text{e}^{-\\text{i}\\phi _2^{(0)}}\\right)^d \\text{e}^{\\text{i}(\\Delta \\chi _{23}+\\phi _2^{(0)})} \\frac{\\delta m_2 \\text{e}^{\\text{i}\\delta \\phi _2} }{m_2^ {(0)}} + \\left(m_3^{(0)}\\right)^{d} \\frac{\\delta m_3 \\text{e}^{\\text{i}\\delta \\phi _3} }{m_3^ {(0)}} \\Big ]~.$ With the leading-order (LO) expressions for the masses the SR is exactly fulfilled (i.e., $s^{(0)} = 0$ ), in case the SR does allow for the mass ordering under consideration.", "Note that the SR is complex and hence the correction $\\delta s$ will in general be complex as well.", "Graphically the deviation from a SR corresponds to an “incomplete” triangle in the complex plane, as illustrated in Fig.", "REF .", "Note that the parameters which appear in the triangle are now the corrected masses and phases, which are complicated functions of the leading-order parameters and of the corrections.", "We will use the parameters $\\delta s_r$ and $\\delta s_i$ to measure the effect of the perturbation, where $\\delta s_r$ ($\\delta s_i$ ) corresponds to the real (imaginary) part of the deviation.", "At this point we also want to note that $\\delta s$ is a dimensionful quantity, and hence not well suited to express corrections since the notion of small and big is not meaningful.", "We therefore introduce the normalised hatted quantities $\\hat{s} \\equiv \\frac{ s}{m_n^d} \\text{ and } \\delta \\hat{s} \\equiv \\frac{\\delta s}{m_n^d} \\;, ~$ where $m_n$ is chosen in such a way that the coefficients in front of $\\delta m_i/m_i^{(0)}$ in Eq.", "(REF ) are not artificially enhanced by $m_i/m_j \\gg 1$ .", "Explicitly, that is: $m_n = {\\left\\lbrace \\begin{array}{ll}m_3 & \\text{for } d>0 \\text{ and NO,} \\\\m_2 & \\text{for } d>0 \\text{ and IO,} \\\\m_1 & \\text{for } d<0 \\text{ and NO,} \\\\m_3 & \\text{for } d<0 \\text{ and IO,} \\\\\\end{array}\\right.}", "$ where NO (IO) stands for normal (inverted) ordering.", "With this choice, $|\\delta \\hat{s}|$ should be much smaller than one – if we want to talk about small corrections and one of the sides of the triangle in Fig.", "REF has a length of $\\mathcal {O}(1)$ .", "Before going on, it is important to realise that Eq.", "(REF ) implies that the quantity $\\delta \\hat{s} < b$ can in fact have a slightly different meaning even for one and the same bound $b$ , depending on the mass ordering and on the sign of $d$ .", "This is, however, not so much an inconsistency than simply a convenient approach to use the same formalism for all cases considered.", "The decisive point is that, in any case, the two mass orderings are physically different, and so are cases with a different sign of $d$ , so that it is simply impossible to put all SRs on the same footing for all cases.", "Thus, a bound such as, e.g., $\\delta \\hat{s} < 0.1$ may be more or less restrictive, depending on the actual case under consideration.", "However, as we will see, the difference induced by this subtlety is not really decisive and will thus not be a major concern for the remainder of this manuscript.", "Figure: Definition of the parameters δs r \\delta s_r and δs i \\delta s_i used to measure the corrections to the SRs.", "Note that these parameters are dimensionful.", "We normalise δs\\delta s according to Eq.", "() such that we obtain dimensionless quantities.", "The parameters m i m_i and φ i \\phi _i are already corrected." ], [ "Main origin of corrections", "In this section, we will give examples for a possible origin of deviations from exact SRs.", "Note that, in principle, no matter where the corrections arise from, they will always be covered by our formalism, cf.", "section .", "However, the important restriction is that our Eq.", "(REF ) relies on the assumption that one can expand the full SR $s$ in the small quantities $\\delta m_i/m_i^ {(0)}$ .", "If this is not possible for some reason, our formalism will not apply.", "Keeping this in mind, we will now discuss three possible origins for deviations to SRs.", "The first origin is higher-dimensional operators, Sec.", "REF , which typically arise from including suppressed terms that ultimately arise from the flavour symmetry being broken.", "The next possibility to modify SRs is to have corrections from the charged-lepton mass matrix.", "These can arise if the mass matrix in the charged lepton sector is not diagonal, but has to be diagonalised to arrive at the standard definition of leptonic mixing angles; this is discussed in Sec.", "REF .", "Finally, as discussed in our earlier reference [20], renormalisation group evolution (RGE) corrections can also lead to modifications.", "This possibility, to be introduced in Sec.", "REF , is discussed in some more detail in what follows, for the simple reason that we can give a detailed comparison to the previous results." ], [ "Higher-dimensional operators", "In many models, the mass matrices have a leading-order structure which is supposed here to give one of the mass sum rules.", "But then due to some higher-dimensional operators this leading structure gets disturbed.", "As an example for a possible higher order correction we want to study the $A_{5}\\times SU(5)$ model proposed in [22], [45], [46] where a correction to the leading Yukawa matrix is introduced to account for the observed baryon asymmetry of the universe via the leptogenesis mechanism.", "In order to simplify the notation, we introduce the dimensionless matrices $\\hat{y}&=\\begin{pmatrix}1 & 0 & 0 \\\\0 & 0 & 1 \\\\0 & 1 & 0\\end{pmatrix} \\text{ and }\\delta \\hat{y}= \\begin{pmatrix}0&1&0\\\\-1&0&0\\\\0&0&0\\end{pmatrix} ~.$ The neutrino Yukawa coupling in this model is then proportional to $\\hat{y} + c \\text{ e}^{\\text{i } \\gamma }\\delta \\hat{y}$ , with $c \\ll 1$ and $\\gamma $ being an arbitrary phase coming from a higher-dimensional operator.", "The mass matrix for the right-handed neutrinos is proportional to (see also [19]): $\\hat{m}_{\\text{RR}} &= \\begin{pmatrix}\\frac{X}{\\sqrt{6}}+\\frac{Y \\text{e}^{\\text{i}\\chi }}{\\sqrt{30}}&-\\frac{Y \\text{e}^{\\text{i}\\chi }}{\\sqrt{15}}& -\\frac{Y \\text{e}^{\\text{i}\\chi }}{\\sqrt{15}}\\\\-\\frac{Y \\text{e}^{\\text{i}\\chi }}{\\sqrt{15}}&\\frac{15 X-\\sqrt{5}Y\\text{e}^{\\text{i}\\chi }}{10\\sqrt{6}}&\\frac{-5 X-\\sqrt{5}Y\\text{e}^{\\text{i}\\chi }}{10\\sqrt{6}}\\\\-\\frac{Y \\text{e}^{\\text{i}\\chi }}{\\sqrt{15}}&\\frac{-5 X-\\sqrt{5}Y\\text{e}^{\\text{i}\\chi }}{10\\sqrt{6}}&\\frac{15 X-\\sqrt{5}Y\\text{e}^{\\text{i}\\chi }}{10\\sqrt{6}}\\\\\\end{pmatrix} \\; .$ The light neutrino mass matrix is then generated via the type I seesaw mechanism [23].", "It is up to $\\mathcal {O}(c)$ given by $m_\\nu =-\\hat{y}~ \\hat{m}_{\\text{RR}} ^{-1}~ \\hat{y}-c~\\text{e}^ {\\text{i}\\gamma } \\left(\\delta \\hat{y}~ \\hat{m}_{\\text{RR}}^{-1}~ \\hat{y}+\\hat{y}~ \\hat{m}_{\\text{RR}} ^{-1}~ \\delta \\hat{y}\\right) \\;,$ where we have absorbed all coefficients into $c$ and $\\hat{m}_{\\text{RR}}^{-1}$ , and we thus have only five effective parameters.", "These are the dimensionful $X,~Y$ ; the phases $\\chi $ and $\\gamma $ , and the small parameter $c$ .", "One can easily map our simplified notation here to the original notation used in [45], [46], by rewriting these parameters with the respective prefactors.", "Since the leading-order neutrino mass matrix ($c\\equiv 0$ ) depends only on two (complex) parameters, we find a mass SR which corresponds to SR 6 from Tab.", "REF : $\\frac{\\text{e}^{\\text{i}\\phi _1}}{m^{(0)}_1}+\\frac{\\text{e}^{\\text{i}\\phi _2}}{m^{(0)}_2}-\\frac{1}{m_3^{(0)}}=0~.$ The corrections to the complex masses from Eq.", "(REF ) are up to order $c$ $\\delta m_1 &= \\sqrt{6 (3 + \\sqrt{5})}\\frac{Y \\sin \\gamma }{X \\cos \\chi } \\sqrt{\\frac{1 }{ (X^2+Y^2+2 XY \\cos \\chi )}} \\, c~,\\\\\\delta m_2&= \\sqrt{6 (3 - \\sqrt{5})}\\frac{Y \\sin \\gamma }{X \\cos \\chi } \\sqrt{\\frac{1 }{ (X^2+Y^2-2 XY \\cos \\chi )}} \\, c~\\\\\\delta m_3&=0~,\\\\\\delta \\phi _1&=\\arctan \\left(\\frac{1}{\\tan \\chi }+\\frac{X}{Y}\\frac{1}{\\sin \\chi }\\right)~,\\\\\\delta \\phi _2&=\\arctan \\left(\\frac{1}{\\tan \\chi }-\\frac{X}{Y}\\frac{1}{\\sin \\chi }\\right)~,\\\\\\delta \\phi _3&=0~.$ Note that for $\\sin \\gamma = 0$ , $\\delta m_1$ and $\\delta m_2$ get corrected only at $\\mathcal {O}(c^2)$ and $\\tilde{m}_3$ does not get a correction at $\\mathcal {O}(c)$ .", "The correction is enhanced by the ratio $Y/X$ and $\\chi \\approx n \\pi /2$ ($n \\in \\mathbb {Z}$ ).", "As an example parameter point we consider normal ordering with $X=24.0,~Y=185.3,~\\chi =0.4$ so that, for $c=0.05$ and $\\gamma =0.2$ , we obtain $\\delta \\hat{s}_r\\approx -0.013 $ and $\\delta \\hat{s}_i\\approx 0.13$ , which corresponds to $\\delta \\hat{s}\\approx 0.13$ ." ], [ "Charged lepton sector", "Despite the fact that the mass SR is a feature of the neutrino sector of a given model, it can be influenced by a nondiagonal charged lepton mass matrix.", "The phases which appear in the PMNS matrix (i.e., the Dirac CP-phase and the two Majorana phases) depend on the phases from the neutrino and the charged lepton mixing matrices, since $U_{\\text{PMNS}}=U_e^\\dagger U_{\\nu }$ .", "This leads to relations between the leptonic mixing angles and phases.", "In the derivation of the formulas, however, some subtleties can arise, since all unphysical phases have to be correctly extracted.", "In App.", "we present a comprehensive derivation of relations between the parameters in the PMNS matrix and the neutrino and charged lepton mixing parameters.", "Here we will quote the results for the approximate expressions for the PMNS parameters in terms of SRs of neutrino mixing angles and the charged lepton mixing angles.", "For $\\theta _{13}^\\nu \\approx 0$ and $\\theta _{ 23}^e\\approx \\theta _{13}^e\\approx 0$ , they read $c_{13}^{\\text{PMNS}} c_{12}^{\\text{PMNS}} \\text{e}^{\\text{i}(\\eta _1-\\phi _1/2)}&=c_{12}^e c_{12}^\\nu \\text{e}^{\\text{i}\\omega _1^\\nu }+c_{23}^\\nu s_{12}^e s_{12}^\\nu \\text{e}^{\\text{i}(\\omega _2^\\nu +\\delta _{12}^\\nu -\\delta _{12}^e)}~,\\\\s_{12}^{\\text{PMNS}} c_{13}^{\\text{PMNS}}\\text{e}^{\\text{i} (\\eta _1-\\phi _2/2)} &= s_{12}^{\\nu }c_{12}^e \\text{e}^{-\\text{i}\\delta _{12}^{\\nu } }\\text{e}^{\\text{i}\\omega _1^\\nu }-s_{12}^{e}c_{23}^{\\nu }c_{12}^{\\nu }\\text{e}^{-\\text{i}\\delta _{12}^e }\\text{e}^{\\text{i}\\omega _2^\\nu } \\;,\\\\s_{13}^{\\text{PMNS}}\\text{e}^{\\text{i}( \\eta _1-\\delta )} &=-s_{12}^e s_{23}^{\\nu } \\text{e}^{-\\text{i}(\\delta _{12}^{e}+\\delta _{23}^\\nu )}\\text{e}^{\\text{i}\\omega _2^\\nu }~.", "$ The superscript $\\nu $ denotes the neutrino mixing parameters, while the superscript $e$ signifies the charged lepton mixing parameters.", "The phases $\\eta _i$ and $\\omega _i^\\nu $ are unphysical, but they nevertheless have to be treated with care in order to obtain the correct results for the phases.", "Together with $c_{12}^{\\text{PMNS}} \\left(c_{13}^{\\text{PMNS}}\\right)^2 c_{23}^{\\text{PMNS}} s_{13}^{\\text{PMNS}} \\left(s_{12}^{\\text{PMNS}}s_{23}^{\\text{PMNS}}\\text{e}^{-\\text{i}\\delta }-c_{12}^{\\text{PMNS}}c_{23}^{\\text{PMNS}}s_{13}^{\\text{PMNS}}\\right) = \\\\\\left(U_{11}^{\\text{PMNS}}\\right)^*U_{13}^{\\text{PMNS}}U_{31}^{\\text{PMNS}} \\left(U_{33}^{\\text{PMNS}}\\right)^*~,$ which we get if we exploit the structure of the PMNS matrix, we can close the system to solve the four equations (REF ) to (REF ) to determine the Majorana phases.", "In [48], [47], [49] these expressions were also derived but their formulas apply directly to the case where the unphysical phases are taken correctly into account and then subsequently absorbed.", "From Eqs.", "(REF , , ) we see that the Majorana phases indeed depend on the charged lepton phases.", "As a concrete example, we consider again the $A_5\\times SU(5)$ model proposed in [22], [45], [46], which features a nondiagonal charged lepton mass matrix, a vanishing reactor angle, and a maximal atmospheric neutrino mixing angle.", "We introduced only a small 1-2 mixing in the charged lepton sector.", "The parameters in the neutrino mass matrix are complex, and hence the $\\delta _{ij}^{\\nu }$ depend on these parameters.", "But, for simplicity, we take the neutrino mass matrix to be real ($\\chi = c = 0$ ).", "Taking into account all phases in the PMNS matrix, we obtain for the physical Majorana phases $\\phi _1=\\sqrt{3+\\sqrt{5}}\\theta _{12}^e \\sin \\delta _{12}^e~\\text{ and }~\\phi _2=\\pi -\\frac{\\sqrt{5}-1}{\\sqrt{2}}\\theta _{12}^e\\sin \\delta _{12}^e~.$ This result tells us that $\\delta _{12}^e\\approx \\pi /2$ leads to the maximal correction to the Majorana phases and $\\phi _1$ gets more affected by the charged lepton phases than $\\phi _2$ .", "We see that the physical Majorana phases have a dependence on the charged lepton phases, and thus the Majorana phases which appear in the SR are in general not equal to the physical Majorana phases one obtains in a model with a nondiagonal charged lepton mass matrix.", "For this reason, a SR can get destroyed in this type of models.", "To demonstrate how powerful our formalism is, we will now express these corrections in terms of $\\delta \\hat{s}$ for the given example.", "On leading order we find that $\\phi _1^{(0)}=0,&~\\phi _2^{(0)}=\\pi \\;,~\\\\m_1 = m_1^{(0)}+\\mathcal {O}(\\delta m_1^2),~m_2 &= m_2^{(0)}+\\mathcal {O}(\\delta m_2^2),~m_3 = m_3^{(0)} + \\mathcal {O}(\\delta m_3^2)\\;~.$ Note that the physical neutrino masses will only get corrected at the order $\\mathcal {O}(\\delta m_i^2)$ .", "With $\\delta m_1&=-\\sqrt{3 + \\sqrt{5}} \\, \\theta _{12}^e \\, m_1^{(0)} \\sin \\delta _{12}^e~,\\\\\\delta \\phi _1&=-\\frac{1}{2} \\left(\\pi + \\sqrt{3 + \\sqrt{5}} \\, \\theta _{12}^e \\, \\sin \\delta _{12}^e \\right)~,\\\\\\delta m_2&= -\\sqrt{3-\\sqrt{5}} \\, \\theta _{12}^e m_2^{(0)} \\, \\sin \\delta _{12}^e~,\\\\\\delta \\phi _2&=-\\frac{1}{2}\\left(\\pi -\\theta _{12}^e \\sqrt{3-\\sqrt{5}} \\sin \\delta _{12}^e \\right)~,\\\\\\delta m_3 &= 0~,$ we get from eq.", "(REF ) normalised to $m_1^d$ in leading order in $\\theta _{12}^e$ $\\delta \\hat{s}\\approx - \\text{i} \\frac{ \\sqrt{3 - \\sqrt{5}} \\, m_1 - \\sqrt{3 + \\sqrt{5}} \\, m_2 }{m_2} \\theta _{12}^e \\sin \\delta _{12}^e~,$ where we have used that $m_i \\approx m_i^{(0)}$ in this approximation.", "With $\\theta _{12}^e\\approx 12^{\\circ }$ and $\\delta _{12}^e=\\pi /2$ we obtain as the maximal correction $\\delta \\hat{s}\\approx 0.36$ and the correction vanishes for $\\delta _{12}^e=0$ ." ], [ "RGE corrections", "Finally, a very generic correction to the mass SR are renormalisation group effects on the masses and Majorana phases.", "In [20], we have already investigated the effect of such most generic corrections on the predictions of SRs.", "Now we want to match the corrections to the complex masses from Eq.", "(REF ) to the corrections coming from the RGEs, as far as possible.", "We therefore extract the RGE corrections from the absolute values of the masses and from the phases, and we rewrite the corrected complex masses as $m_i \\text{e}^{-\\text{i}\\phi _i} = \\left(m_i^{(0)}+m_i^{\\text{RGE}}\\right)\\text{e}^{\\text{i} (-\\phi _i^{(0)}+\\phi _i^{\\text{RGE}})}~,$ where the superscript $(0)$ denotes the LO masses and Majorana phases, and the superscript RGE labels the corrections from the renormalisation group running.", "The connection to Eq.", "(REF ) is $\\delta m_i&=\\sqrt{\\left(m_i^{\\text{RGE}}\\right)^2 + \\left(m_i^{(0)}\\right)^2 \\left(\\phi _i^{\\text{RGE}}\\right)^2}~,\\\\\\delta \\phi _i&=-\\phi _i^{(0)}+\\text{arctan}\\left( \\frac{m_i^{(0)}\\phi _i^{\\text{RGE}}}{m_i^{\\text{RGE}}}\\right)~.$ Note that the index $i$ in Eq.", "(REF ) runs from 1 to 3, whereas $i$ in Eq.", "() is either 1 or 2.", "The explicit formulas for $\\phi _i^{\\text{RGE}}$ and $\\delta m_i^{\\text{RGE}}$ can be found in [43].", "The RGE corrections to the phases and masses have a dependence on the mass scale and in the minimal supersymmetric Standard Model (MSSM) additionally on $\\tan \\beta $ .", "For large mass scales and large $\\tan \\beta $ , the effect of the corrections is enhanced.", "The $\\beta $ functions of the Majorana phases also depend on the phases themselves which leads to a dependence of $\\delta m_i$ and $\\delta \\phi _i$ on both leading-order Majorana phases.", "The RGE correction to the masses is always positive in the MSSM and negative in the Standard Model (SM), and hence the sign of the contribution to $\\phi _i^{(0)}$ in Eq.", "() is fixed.", "Except for $\\phi _1^{(0)}\\approx \\phi _2^{(0)}$ , where the running of the phases is suppressed, the running of the Majorana phases is stronger than the running of the masses, since the $\\beta $ -functions of the masses depend on the values of the masses themselves (see also the discussion in [20]).", "For $\\phi _1^{(0)}\\ne \\phi _2^{(0)}$ , the main contribution to $\\delta m_i$ comes from the term $m_i^{(0)}\\phi _i^{\\text{RGE}}$ , and the correction to $\\phi _i^{(0)}$ is close to maximal (i.e.", "$\\pi /2$ ) in Eq. ().", "In the next section we will discuss the effect of RGE corrections to various sum rules in more detail." ], [ "Numerical results", "In this section, we will present our numerical results.", "We will answer the nontrivial question of whether it is possible to reconstitute forbidden mass orderings by corrections, since some SRs allow for only one of the two mass orderings.", "Furthermore, we will present our numerical results for the allowed ranges for the effective mass $|m_{ee}|$ and for the lightest neutrino mass eigenvalues $m$ , which we obtain from the corrected SRs." ], [ "Reconstituting forbidden orderings", "Some SRs only allow for one particular mass ordering [15], [16], [17], i.e., normal ($m_1<m_2<m_3$ ) or inverted ($m_3<m_1<m_2$ ).", "The question might arise if we can reconstitute those forbidden mass orderings by large enough corrections.", "In [20], we did a similar study concerning the generic corrections arising from renormalisation group effects, where we concluded that this was not possible.", "For SRs 2, 3, 4, 5, 10 and 12 from Tab.", "REF , only one mass ordering is allowed.", "Hence, we have $s\\ne 0$ for the forbidden orderings.", "In order to obtain $s^{(0)}= 0$ we need the leading-order masses to respect the allowed ordering in these SRs, and due to corrections, the observed masses will obey the forbidden ordering.", "In other words, the corrections to the masses have to alter the mass ordering.", "In principle the ordering of the leading-order masses is not restricted to be normal or inverted, and one can also imagine having $m_1^{(0)}>m_3^{(0)}>m_2^{(0)}$ or other variations as long as they fulfill the leading-order SR.", "However, one would usually discard such cases, as they apparently do not correspond to reality.", "To be more precise, we want to ask the question of how large $\\delta \\hat{s}$ has to be at least, such that $s^{(0)} = 0$ is fulfilled and the $m_i$ follow the “forbidden” mass ordering.", "This can be done most easily by considering again the geometrical interpretation of the SRs, as done in Fig.", "REF .", "We see that a SR without corrections is fulfilled if the values of $\\delta s_r = \\delta s_i =0$ (i.e., the triangle closes).", "Now, on the other hand, it can happen that – for experimentally allowed values of the neutrino masses – the triangle never closes.", "For this case, we want to determine $\\delta \\hat{s}$ with $|\\delta \\hat{s}|$ being minimal.", "Obviously, $|\\delta \\hat{s}|$ is minimal for $\\delta \\hat{s}_i &= 0 \\;, \\\\\\Delta \\chi _{23} - d \\phi _2 &= \\pi \\;, \\\\d \\phi _1- \\Delta \\chi _{13} &= -\\pi \\;.$ Then $|\\delta \\hat{s}| = |\\delta \\hat{s}_r|$ is minimal.", "Plugging these expressions into the general formulas results in a rather long expressions, and hence we prefer to discuss what happens for the concrete example of SR 2, where $(c_1, c_2, d, \\Delta \\chi _{13}, \\Delta \\chi _{23}) = (1, 2, 1, \\pi , \\pi )$ , and we want to focus on inverted mass ordering for the observed (corrected) masses.", "Since we have $\\delta \\hat{s}_i=0$ , and the minimal value for $\\delta \\hat{s}_r$ corresponds to the situation where the sides of the triangle lie on the baseline, which is $\\delta \\hat{s}_r&= 2 - \\frac{m_3}{m_2} - \\frac{m_1}{m_2}.$ For three different exemplary mass scales ($m_3 = 0.0,\\ 0.001,\\ 0.05$  eV), we obtain: $\\delta \\hat{s}_r(m_3=0~\\text{eV})&=1.02 \\;,\\\\\\delta \\hat{s}_r(m_3=0.001~\\text{eV})&=1.00 \\;,\\\\\\delta \\hat{s}_r(m_3=0.05~\\text{eV})&=0.30 \\;.$ The corrections thus have to have at least this size to reconstitute inverted mass ordering.", "Table: Summary table of the SRs which allow only one mass ordering and the minimal value of the correction δs ^ r \\delta \\hat{s}_r that is needed to reconstitute the forbidden mass orderings for different mass scales.In Tab.", "REF we have calculated the minimal values of $\\delta \\hat{s}_r$ for different mass scales, for a large mass scale, for a small mass scale and for a vanishing value of $m$ for the SRs which allow for only one mass ordering (SRs 2, 3, 4, 5, 10 and 12).", "We see that, in principle, we can reconstitute all forbidden orderings although we might need quite sizeable corrections.", "Especially for the case of very small neutrino masses, the corrections are so large that a perturbative approach is not suitable anymore, and it is simply not appropriate to talk of a neutrino mass SR at all.", "Furthermore, the attentive reader might be surprised that we can suddenly reconstitute all forbidden orderings which was in no case possible for the RGE corrections only.", "We will show now that this is due to the fact that the correction to the SR had in nearly all cases a fixed sign pointing in the “wrong” direction, i.e., the RGE corrections – although potentially sizeable – tend to make the forbidden mass orderings even less likely than the LO SRs.", "In other words, instead of making the deviation of the SR smaller, RGE corrections increased it.", "Only for one case, the sign mentioned above was in principle suitable, but for that particular case, the corrections were way too small.", "Thus, our previous conclusions remain perfectly valid for the RGE-corrected case; however, as we will see, the general case actually can make the otherwise forbidden orderings possible – as to be expected.", "As an example we will consider SR 2 in Tab.", "REF , where we try to reconstitute inverted mass ordering by RGE corrections.", "In order to do so, we will analyse the corrections to the lengths of the sides of the triangle.", "If these corrections are positive and large enough, they can close the open triangle.", "The nontrivial lengths of the triangle in case of SR 2 are $m_1/m_2$ and $m_3/m_2$ where we set $m_i=m_i^{(0)}+m_i^{\\text{RGE}}~.$ The RGE corrections for the masses are of the form $m_i^{\\text{RGE}}=\\frac{1}{16 \\pi ^2}\\left[\\alpha _{\\text{RGE}} \\, m_i^{(0)}+F_i m_i^{(0)}\\right] \\log \\left(\\frac{\\mu }{M_Z} \\right)~,$ where $\\alpha _{\\text{RGE}}\\approx 3$ is a function which depends on the gauge and Yukawa couplings, $\\mu >M_Z$ is the high energy scale and $F_i$ is a function which depends on the angles and on $\\tan \\beta $ in the MSSM.", "In the MSSM we expect the largest effect for $F_i>0$ , whereas $F_i<0$ holds in the SM.", "We will focus on the MSSM case first.", "If we plug Eq.", "(REF ) into Eq.", "(REF ) and expand in $m_i^{\\text{RGE}}$ , we see that the terms $\\propto \\alpha _{\\text{RGE}}$ drop out since $\\alpha _{\\text{RGE}}$ is the same for all masses.", "We obtain for the corrections of the length of the sides: $\\delta \\left( \\frac{m_1}{m_2}\\right)&=\\frac{m_1^ {(0)}}{m_2^ {(0)}}(F_1-F_2)~,\\\\\\delta \\left( \\frac{m_3}{m_2}\\right)&=\\frac{m_3^ {(0)}}{m_2^ {(0)}}(F_3-F_2)~.$ In the $3 \\sigma $ ranges for the mixing angles from Tab.", "REF , we have $F_1-F_2<0$ and $F_3-F_2>0$ .", "Hence the length $m_3/m_2$ increases whereas the length $m_1/m_2$ decreases.", "If $m_3/m_2$ increases more strongly than $m_1/m_2$ decreases we can hope to close the triangle.", "But for SR 2 this is not the case since $m_1/m_2$ decreases more strongly than $m_3/m_2$ increases.", "We conclude that the RGE corrections make the deviation from the SR even larger.", "Hence the inverted ordering cannot be reconstituted.", "This statement can be transferred to SRs 3 and 10, where we also have to normalise the sides of the triangle by $m_2$ .", "Also in these cases the corrections to the lengths of the sides are $\\propto (F_1-F_2) $ and $\\propto (F_3-F_2)$ .", "In case of SRs 4 and 5, the sides of the triangle have to be normalised to $m_3$ .", "The corrections to the length of the sides are $\\propto (F_1-F_2)$ and $\\propto (F_2-F_3)$ .", "For the $3 \\sigma $ ranges of the mixing angles, $F_1-F_2<0$ and $F_2-F_3<0$ .", "Hence, the sides of the triangle both decrease due to RGE corrections.", "Also in these cases we cannot reconstitute the forbidden orderings.", "Only in the case of SR 12 both sides of the triangle increase.", "In SR 12, $d<0$ , and we have to normalise the sides by $m_3^d$ to reconstitute normal ordering.", "This leads to terms $\\propto (F_3-F_1)$ and $\\propto (F_3-F_2)$ in the corrections to the lengths of the sides, which are both positive.", "However, numerically we would need a mass scale larger than 1 eV for $\\tan \\beta =200$ to fulfil the sum rule, which is simply unrealistic.", "In fact, for any SR that is not fulfilled with the low energy masses, the RGE effects in the MSSM have the wrong sign if $d>0$ for both orderings and if $d<0$ for normal ordering.", "In the SM, $\\delta \\hat{s}$ has the right sign to decrease the deviation from the SR but nevertheless the effects are too small to fulfil the sum rule.", "In conclusion the fixed sign of the RGE corrections makes it barely possible to reconstitute forbidden orderings.", "Only in the case of $d<0$ for inverted ordering or in the SM case the sign was suitable, but the effects are nevertheless too small.", "Since, however, any corrections to the masses beyond those from RGEs do not have a fixed sign, it is nevertheless possible to reconstitute forbidden orderings in the general case.", "In the following section we want to confirm these estimates also numerically." ], [ "Effects on the lower bound of $m$", "One major prediction of the SRs is the lower bound on the smallest neutrino mass eigenvalue $m$ .", "The question arises as to how this bound changes in the light of corrections to the SR. To answer this question, we will consider – similarly as in the previous section – the case where the sides of the triangle lie on the baseline, i.e., the case where the SR is just fulfilled.", "This clearly leads to the lower bound on the lightest mass $m$ .", "In Tab.", "REF we have summarised the results for the relative change of the mass scale in leading order in $\\zeta \\equiv \\frac{\\Delta m_{21}^2}{|\\Delta m_ {32}^2|}\\approx 0.03$ .", "The results are presented in the form $\\frac{m}{m^ {(0)}}=1-\\kappa ~\\delta \\hat{s}~,$ where $\\kappa $ is a parameter that depends on the SR and on the mass ordering.", "Table: Estimates for the relative change of the lower bound of the lightest mass, where ζ≡Δm 21 2 |Δm 32 2 |\\zeta \\equiv \\frac{\\Delta m_{21}^2}{|\\Delta m_ {32}^2|}.", "Please see the main text for more details.For SRs 1 and 4 in IO, the effect of the correction is enhanced because of the small $\\zeta $ in the denominator.", "For the other SRs we find that the relative change is between $20\\%$ and $80\\%$ for $\\delta \\hat{s}=0.1$ and $0.3$ .", "In the next section, we will verify these estimates numerically.", "The enhancement of the effect of the corrections for the mass scale for SR 1 and SR 4, for IO in both cases, is not visible when considering RGE corrections only, because $\\delta \\hat{s}$ for a small mass scale is very small.", "In the case of SR 7 for NO, we additionally get a correction to the upper mass bound: $\\frac{m}{m^ {(0)}}=1+\\frac{4}{3} ~\\delta \\hat{s}~,$ which is $1.4$ for $\\delta \\hat{s}=0.3$ .", "Furthermore we obtain for $\\delta \\hat{s}\\ne 0$ a new allowed mass region in NO.", "To open up this region we need for the lightest neutrino mass $m=(0.01, 0.05, 0.1)$  eV a $\\delta \\hat{s}= (-0.31, -0.27, -0.10)$ .", "These values are well within our scan regions.", "In the case of RGE effects only we encountered a parametric enhancement for the RGE effects for SR 1 and 4 in IO for a medium mass scale.", "This enhancement is not present now because we parametrised the corrections differently, to avoid any artificial parametric enhancement." ], [ "Neutrinoless double beta decay", "Finally, our main predictions are those for the effective mass $|m_{ee}| = \\left|m_{1} U_{e1}^{2}+m_{2} U_{e2}^{2}+m_{3} U_{e3}^{2}\\right|=\\left| m_{1}c_{12}^{2}c_{13}^{2}\\text{e}^{-\\text{i}\\phi _{1}}+m_{2}s_{12}^{2}c_{13}^{2}\\text{e}^{-\\text{i}\\phi _{2}}+m_{3}s_{13}^{2}\\text{e}^{-2 \\text{i} \\delta }\\right|$ as probed in neutrinoless double beta decay, see, e.g., Refs.", "[44] for detailed discussions on this quantity.", "We have, for each SR, numerically scanned the parameter space and we have derived the allowed regions for both normal and inverted mass orderings, see Figs.", "REF to REF .", "For each SR, we have investigated the following three cases depending on the size of $ \\hat{s}$ , as defined in Eq.", "(REF ): $ \\hat{s} < 10^{-4}$ (left panels): This case basically means that the SR is taken to be exact.", "All plots perfectly match the unperturbed SR predictions, as presented in Refs.", "[17], [20].", "$ \\hat{s} < 0.1$ (middle panels): This corresponds to a considerable perturbation of each SR.", "In particular, a correction of this size will reveal that, in cases where one mass ordering is forbidden for an exact SR, this ordering will open up due to the corrections.", "$ \\hat{s} < 0.3$ (right panels): Here, the SRs are even less exact.", "This case is more or less the limiting case of what can be described by the approach followed in this work, given that we ultimately rely on a perturbative expansion.", "This case should in particular include the RGE corrections, as discussed in Ref.", "[20], as long as they can be described accurately as small perturbations.", "Let us discuss the results for the different SRs one by one, with a particular focus on how the allowed regions for the perturbed SRs compare to those derived from the RGE-corrected SRs (cf.", "Ref.", "[20], in which reference Sec.", "4.X always contains the plots and the discussion on SR X).", "The parameters for the respective SRs can be read off from Tab.", "REF .", "Starting with SR 1 in Fig.", "REF , it is immediately visible from the left panel that, indeed, a very small deviation from an exact SR, $ \\hat{s} < 10^{-4}$ , practically does not change the prediction compared to that of the exact SR.", "This observation will hold true for all SRs, as to be expected.", "If we increase $ \\hat{s}$ , it is visible that, for NO, the allowed regions visibly increase.", "In particular, the lowest allowed value for $m$ evolves as $(0.027, 0.023, 0.017)$  eV for $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ , while at the same time the lowest possible value for the effective mass $|m_{ee}|$ changes as $(0.025, 0.022, 0.016)$  eV.", "In particular, with increasing $ \\hat{s}$ , the allowed range for the effective mass becomes even slightly larger than if only RGE corrections were included (cf.", "Sec.", "4.1 in Ref.", "[20]), where the minimal value for the effective mass would be at $0.026$  eV.", "For IO, in turn, while the allowed region also increased with increasing $ \\hat{s}$ , the main difference is that the area does not grow sufficiently large as to cover the complete allowed region obtained when taking into account RGE-corrections.", "As already hinted, this is a reflection of the RGE corrections not always staying within the perturbative range for $ \\hat{s}$ , which is what we are considering in this work.", "As shown in [20], we encounter a parametric enhancement for the RGE effects for a small mass scale, because the correction is proportional to the inverse mass scale.", "Since we avoid parametric enhancements of the corrections in our parametrisation from Eqs.", "(REF ) to (REF ) we do not obtain the same result as in [20].", "Nevertheless, given that even with RGE-corrections the change in the prediction was less than dramatic, in particular when taking into account the nuclear uncertainties, we can again conclude that the predictions of the SRs are relatively robust compared to corrections.", "Figure: Effective mass with SR 1 for s ^=10 -4 \\hat{ s}=10^{-4}, 0.10.1, 0.30.3.Let us press on and jump to SR 2, see Fig.", "REF .", "Starting with NO, the qualitative change with increasing $ \\hat{s}$ is similar to that with increasingly strong RGE-corrections, however, the allowed regions are different (and possibly slightly larger for the RGE-corrections).", "For $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ , the lowest $m$ evolves as $(0.015, 0.013, 0.009)$  eV and the lower bound on the effective mass $|m_{ee}|$ as $(0.014, 0.012, 0.008)$  eV (compared to $0.016$  eV and $0.015$  eV for the RGE corrections).", "For IO, something interesting happens.", "As is visible from the left panel of Fig.", "REF , this mass ordering is not allowed for an exact SR. As noted in Sec.", "REF , this behaviour remained true for the RGE corrections.", "However, with a more general correction to the SR, as implied by an approximate SR, the IO starts opening up.", "While it is hardly visible for $ \\hat{s} < 0.1$ , it is easy to spot for $\\hat{s} < 0.3$ .", "However, while the otherwise forbidden ordering does in principle open up, the resulting predicted region unfortunately lies in the part of the plot that is in any case strongly disfavoured by cosmology, cf.", "grey rectangle on the right in the plots.", "Thus, in essence, the prediction of IO being forbidden does not change.", "Figure: Effective mass with SR 2 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Similarly to the previous case, for SR 3 the allowed area for NO also broadens, while for IO a small regions opens up which is, however, strongly disfavoured, cf.", "Fig.", "REF .", "However, the difference is that, for large enough $ \\hat{s}$ , the allowed region for NO may enter the “tube” in which cancellations inside $|m_{ee}|$ to practically zero are possible.", "This implies that, for $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ , while the smallest neutrino mass eigenvalue $m$ is only reduced as $(0.015, 0.013, 0.009)$  eV, the minimum value of the effective mass $|m_{ee}|$ changes more dramatically, $(0.0030, 0.0020, 0.00029)$  eV, compared to a considerably larger lowest value of $0.0036$  eV for RGE corrections only.", "Figure: Effective mass with SR 3 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.For SR 4, cf.", "Fig.", "REF , no dramatic changes are visible.", "The allowed region is a bit broader for strong RGE corrections, but the minimal value of the effective mass is in any case $0.015$  eV, the lowest value at all possible for IO.", "The minimal value for the smallest neutrino mass eigenvalue $m$ decreases more strongly than for RGE corrections only, however, such small values are in any case not accessible by experiments.", "For NO, a small area opens up which is allowed by the SR as such, but again it is located in the disfavoured region of the plot.", "Figure: Effective mass with SR 4 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Sum rule 5 exhibits once more the characteristic behaviour, see Fig.", "REF .", "For IO, the allowed region broadens even more strongly than for RGE corrections only, leading to a smallest neutrino mass $m$ of $(0.024, 0.020, 0.012)$  eV and a minimum effective mass $|m_{ee}|$ of $(0.051, 0.050, 0.047)$  eV for $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ .", "For NO, the small region opening up is again disfavoured.", "Figure: Effective mass with SR 5 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Sum rule 6 – see Fig.", "REF – looks similar to SR 5 for IO and to SR 3 for NO, in both cases fully including the regions predicted by the RGE corrections.", "For $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ , the smallest neutrino mass $m$ and the minimum effective mass $|m_{ee}|$ change as $(0.028, 0.025, 0.020)$  eV and $(0.053, 0.051, 0.045)$  eV, respectively, for IO.", "The corresponding values for NO are, in turn, $(0.011, 0.0090, 0.0061)$  eV and $(0.0011, 0.00031, 3.5\\cdot 10^{-12})$  eV, where the latter value simply indicates that a full cancellation of the effective mass can happen.", "Figure: Effective mass with SR 6 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Figs.", "REF and REF look nearly identical and quite interesting, too, as several changes happen.", "Starting with IO, the allowed region again broadens, however, while a new edgy corner had appeared for the RGE-corrections, see Sec.", "4.7 for SR 7 in Ref.", "[20], in the case of approximate SRs the band simply broadens.", "The lowest value possible for the effective mass quickly reaches its absolute minimum and then cannot change anymore, $(0.017, 0.015, 0.015)$  eV for $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ , while the smallest neutrino mass $m$ varies as $(0.017, 0.015, 0.012)$  eV.", "For NO, in turn, the changes are more dramatic.", "First of all, the allowed patch considerably grows for $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ , such that the minimum [maximum] mass eigenvalue $m$ varies as $(0.0043, 0.0040, 0.0040)$  eV [$(0.0060, 0.0071, 0.010)$  eV], while the minimum [maximum] effective mass changes as $(0.0043, 0.0040, 0.0039)$  eV [$(0.0087, 0.0073, 0.012)$  eV].", "Furthermore, for large enough mass $m$ , a second (disjoint) allowed region opens up for NO in the quasidegenerate mass region for $ \\hat{s} < (0.1, 0.3)$ .", "However, again this new addition is located in that part of the parameter space that is strongly disfavoured by cosmology.", "Figure: Effective mass with SR 7 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Figure: Effective mass with SR 8 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Sum rule 9, cf.", "Fig.", "REF , hardly change at all for RGE corrections only.", "And also for approximate SRs, hardly any significant change is visible for IO.", "Already for the exact SR, the entire range is allowed for $|m_{ee}|$ , while the smallest mass $m$ changes slightly, $(0.017, 0.015, 0.012)$  eV, for $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ .", "However, for NO, the ranges for $m$ change as $(0.0043 ... 0.0060, 0.0040 ... 0.0071, 0.0040 ... 0.010)$  eV, while the ones for $|m_{ee}|$ evolve as $(0.0027 ... 0.0067, 0.0022 ... 0.0073, 0.0017 ... 0.0095)$  eV.", "Again, for NO, a small region opens up for large masses which are, however, disfavoured.", "Figure: Effective mass with SR 9 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Coming to SR 10, the change in NO looks more dramatic than it actually is, given that in all cases the full range for the effective mass is allowed.", "Also the smallest neutrino mass $m$ is always below detectability.", "However, for IO – which looks rather innocent at first sight – a small region opens up that can actually be detectable!", "While IO is completely forbidden for $ \\hat{s}=10^{-4}$ and only opens up in the disfavoured region for $\\hat{s}=0.1$ , for the largest value of $ \\hat{s}=0.3$ the IO points start to penetrate the allowed region, predicting smallest values of $(m, |m_{ee}|) = (0.031, 0.054)$  eV.", "Figure: Effective mass with SR 10 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.For SR 11 both the NO and IO allowed band broaden to some extent for increasing $ \\hat{s}$ , see Fig.", "REF , and in particular they broaden more than if only the RGE corrections were taken into account.", "For IO, the smallest value of $m$ [of $|m_{ee}|$ ] varies as $(0.024, 0.021, 0.016)$  eV [as $(0.044, 0.039, 0.031)$  eV] for $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ .", "For NO, the smallest value of $m$ varies as $(0.031, 0.029, 0.023)$  eV, while that of $|m_{ee}|$ varies as $(0.022, 0.017, 0.013)$  eV.", "As a comparison, the minimum values for $(m, |m_{ee}|)$ have been determined to be $(0.024, 0.042)$  eV [$(0.032, 0.021)$  eV] for IO [NO], cf.", "Sec.", "4.11 in Ref. [20].", "Figure: Effective mass with SR 11 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3.Finally, for SR 12, the broadening for NO appears to be much stronger than in the case where only RGE corrections are taken into account.", "For $ \\hat{s} < (10^{-4}, 0.1, 0.3)$ , the smallest value for $m$ [of $|m_{ee}|$ ] varies as $(0.0026, 0.0021, 0.0012)$  eV [as $(0.0029, 0.0025, 0.0014)$  eV] in the case of NO.", "For IO, again a small region opens up for larger $ \\hat{s}$ , but only in the disfavoured region.", "Figure: Effective mass with SR 12 for s ^=10 -4 \\hat{s}=10^{-4}, 0.10.1, 0.30.3." ], [ "Summary and conclusions", "In this paper, we have used a perturbative approach to investigate how the predictions of neutrino mass sum rules change if the sum rules are not exact relations, but rather approximate to a given degree.", "After establishing a formalism to treat corrections to an exact sum rule, we show that the perturbations can be linked to the geometrical image of a “nonperfect” triangle.", "This illustration makes it relatively easy to understand the generality of our approach, as long as the corrections are small enough to be covered by a perturbative computation.", "We then discuss several scenarios in which corrections to an exact sum rule can potentially arise.", "The three most generic frameworks are higher-order terms resulting from flavour symmetry breaking, corrections to the light neutrino masses arising from the charged lepton sector, and modifications of sum rules due to renormalisation group running.", "We have for each approach presented several analytic approximations, which serve as analytic estimates to be compared to the numerical computations.", "The latter are our main results.", "We have, for each of the known mass sum rules, investigated the effect of perturbations to the exact formulas.", "As to be expected, as long as the correction terms are very small, we basically recover the result obtained from an exact sum rule.", "However, as we turn on the perturbations, we can see that the allowed regions are modified.", "The first type of modification is a simple increase of the allowed area in the parameter space.", "This is to be expected, since suddenly more parameter combinations are allowed, but in fact these broadenings are rather mild for most of the cases (at least so long as we stay in the perturbative regime).", "On the other hand, in several cases, qualitatively new predictions arise: depending on the sum rule, the exact formulas may forbid one of the neutrino mass orderings, which can be restored in the perturbed case.", "This could strongly alter the predictions, however, it turns out that in most cases (except for SR 10) the newly allowed regions are not very big and are practically excluded by the cosmological bounds on the neutrino mass.", "Finally, an interesting result is that the predictions for approximate sum rules do in many cases not cover the regions which had been derived by us in an earlier work on the corrections from renormalisation group running.", "While this result may seem to come as a surprise at first sight, it is in fact easy to understand, since in some cases the corrections induced by running due to some parametric enhancement are larger than a $30\\%$ correction, so that they are not covered by our formalism.", "Summing up, we have treated the topic of neutrino mass sum rules in unprecedented generality, to the point that we may have delivered the final step to what can at all be said about neutrino mass sum rules from a phenomenological point of view.", "Using this work as well as previous ones, any model predicting a new sum rule can be analysed to the point that it can clearly be matched to the experimental results.", "We thus pass the ball to the experimentalists, who will hopefully be able to deliver further new bounds which allow us to constrain whole groups of flavour models in a reliable manner.", "This can push our understanding of the leptonic flavour sector to a new level." ], [ "Acknowledgements", "AM acknowledges partial support by the Micron Technology Foundation, Inc. AM furthermore acknowledges partial support by the European Union through the FP7 Marie Curie Actions ITN INVISIBLES (PITN-GA-2011-289442) and by the Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreements No.", "690575 (InvisiblesPlus RISE) and No.", "674896 (Elusives ITN).", "MS would like to thank Stefan Antusch for useful discussions on parametrisations and acknowledges support by BMBF under contract no.", "05H12VKF." ], [ " Notes on parametrisations and phases", "In this appendix we want to give a comprehensive derivation of the relations between the mixing parameters in the PMNS matrix and the neutrino and charged lepton mixing parameters.", "First of all we parametrise the relevant matrices as unitary $3\\times 3$ matrices by 3 angles and 6 phases.", "One possible parametrisation is [48]: $U=P_1 U_{23} U_{13}U_{12},$ where the $U_{ij}$ are $U_{23}=\\begin{pmatrix}1&0&0\\\\0&c_{23}&s_{23}\\text{e}^{-\\text{i}\\delta _{23}}\\\\0&-s_{23}\\text{e}^{\\text{i}\\delta _{23}}&c_{23}\\\\\\end{pmatrix},$ and analogous expressions for $U_{12}$ and $U_{13}$ .", "We use the usual abbreviations $c_{ij}\\equiv \\cos \\theta _{ij}$ and $s_{ij}\\equiv \\sin \\theta _{ij}$ .", "The matrix $P_1$ is a diagonal matrix which only contains phases: $P_1&=\\begin{pmatrix}\\text{e}^{\\text{i}\\omega _1}&0&0\\\\0&\\text{e}^{\\text{i}\\omega _2}&0\\\\0&0&\\text{e}^{\\text{i}\\omega _3}\\\\\\end{pmatrix}.$ The phase matrix $P_1$ can be removed by an additional charged lepton phase rotation to make the charged lepton masses real [48].", "Hence we are left with only three phases $\\delta _{ij}$ and $U= U_{23} U_{13}U_{12}~.$ In the standard parametrisation of the PMNS matrix which contains the Dirac phases and the two Majorana phases we will also introduce an unphysical phase matrix $P_2$ , which can be removed by a charged lepton phase rotation as $P_2=\\begin{pmatrix}\\text{e}^{\\text{i}\\eta _1}&0&0\\\\0&\\text{e}^{\\text{i}\\eta _2}&0\\\\0&0&\\text{e}^{\\text{i}\\eta _3}\\\\\\end{pmatrix}~.$ in the combination $P_2 R_{23}U_{13}R_{12}P_0~.$ Where $\\delta _{13}$ in $U_{13}$ is replaced with the Dirac phase $\\delta $ and $P_0$ contains the Majorana phases: $P_0=\\begin{pmatrix}\\text{e}^{-\\text{i}\\phi _1/2}&0&0\\\\0&\\text{e}^{-\\text{i}\\phi _2/2}&0\\\\0&0&1\\\\\\end{pmatrix}~.$ $R_{ij}$ are the Euler matrices which are of the form of Eq.", "(REF ) but without any phases.", "The relation between the $\\delta _{ij}$ in Eq.", "(REF ) and the phases in the matrix $P_0$ is [48]: $\\phi _1&=-2(\\delta _{12}+\\delta _{23})~,\\\\\\phi _2&=-2\\delta _{23}~,\\\\\\delta &=\\delta _{13}-\\delta _{23}-\\delta _{12}~.$ Replacing the $\\delta _{ij}$ in Eq.", "(REF ) with Eqs.", "(REF , , ) leads to $P_1 U_{23} U_{13}U_{12} \\underbrace{=}_{\\text{Eqs.~(\\ref {eq:maj1},\\ref {eq:maj2},\\ref {eq:delta})}}P_2 R_{23}U_{13}R_{12}P_0~.$ By comparing both sides of the equation we see that the phases in $P_1$ are related to the phases in $P_2$ as $\\omega _1&=\\eta _1-\\frac{\\phi _1}{2}~,\\\\\\omega _2&=\\eta _2-\\frac{\\phi _2}{2}~,\\\\\\omega _3&=\\eta _3~.$ In the following we will use the parametrisation in Eq.", "(REF ) for the mixing matrix of the neutrinos and the charged leptons.", "For the PMNS matrix we will replace the $\\omega _i$ in Eq.", "(REF ) by Eqs.", "(REF , , ).", "Our aim is to obtain expressions for the physical phases in the PMNS matrix (the Majorana phases and the Dirac phase) in terms of the neutrino and charged lepton mixing parameters.", "We therefore use the relation $U_{\\text{PMNS}}=U_e^\\dagger U_{\\nu }$ .", "From the elements in the first row the PMNS matrix we obtain: $c_{13}^{\\text{PMNS}} c_{12}^{\\text{PMNS}} \\text{e}^{\\text{i}(\\eta _1-\\phi _1/2)}&=c_{12}^e c_{12}^\\nu c_{13}^e c_{13}^\\nu \\text{e}^{-\\text{i}(\\omega _1^e-\\omega _1^\\nu )}+ \\text{e}^{-\\text{i}(\\delta _{12}^e+\\delta _{13}^e+\\delta _{23}^e+\\omega _3^e-\\omega _3^\\nu )} \\nonumber \\\\& (\\text{e}^{\\text{i}(\\delta _{12}^e+\\delta _{23}^e)}\\theta _{13}^e c_{12}^e-\\text{e}^{\\text{i}\\delta _{13}^e} \\theta _{23}^e s_{12}^e)(\\text{e}^{\\text{i}\\delta _{13}^\\nu }c_{12}^\\nu c_{23}^\\nu s_{13}^\\nu -\\text{e}^{\\text{i}(\\delta _{12}^e+\\delta _{23}^\\nu )} s_{12}^\\nu s_{23}^\\nu ) \\nonumber \\\\& +\\text{e}^{-\\text{i}(\\delta _{12}^e+\\delta _{23}^\\nu +\\omega _2^e-\\omega _2^\\nu )} s_{12}^e(\\text{e}^{\\text{i}(\\delta _{12}^\\nu +\\delta _{23}^\\nu )} c_{23}^\\nu s_{12}^\\nu +\\text{e}^{\\text{i}\\delta _{13}^\\nu }c_{12}^\\nu s_{13}^\\nu s_{23}^\\nu , \\\\s_{12}^{\\text{PMNS}} c_{13}^{\\text{PMNS}}\\text{e}^{\\text{i} (\\eta _1-\\phi _2/2)} &= \\text{e}^{-\\text{i}(\\delta _{12}^e+\\delta _{12}^\\nu +\\delta _{13}^e+\\delta _{23}^e+\\delta _{23}^\\nu +\\omega _1^e+\\omega _2^e+\\omega _3^e)}(\\text{e}^{\\text{i}(\\delta _{12}^\\nu +\\delta _{23}^\\nu +\\omega _1^e)}c_{12}^\\nu \\nonumber \\\\& (-\\text{e}^{\\text{i}(\\delta _{13}^e+\\delta _{23}^e+\\omega _2^\\nu +\\omega _3^e)}c_{23}^\\nu s_{12}^e +\\text{e}^{\\text{i}(\\delta _{12}^e+\\delta _{23}^e+\\delta _{23}^\\nu +\\omega _2^e+\\omega _3^\\nu )}\\theta _{13}^e c_{12}^e s_{23}^\\nu \\nonumber \\\\& -\\text{e}^{\\text{i}(\\delta _{13}^e+\\delta _{23}^\\nu +\\omega _2^e+\\omega _3^\\nu )}\\theta _{23}^e s_{12}^e s_{23}^\\nu +s_{12}^\\nu (\\text{e}^{\\text{i}(\\delta _{12}^e+\\delta _{13}^e+\\delta _{23}^e+\\delta _{23}^\\nu +\\omega _1^\\nu +\\omega _2^e \\omega _3^e)}c_{12}^e c_{13}^\\nu \\nonumber \\\\&+\\text{e}^{\\text{i}(\\delta _{13}^e+\\delta _{13}^\\nu +\\delta _{23}^\\nu +\\omega _1^e+\\omega _2^e+\\omega _3^\\nu )} \\theta _{13}^e c_{12}^e c_{23}^\\nu s_{13}^\\nu \\nonumber \\\\& -\\text{e}^{\\text{i}(\\delta _{13}^e+\\delta _{13}^\\nu +\\delta _{23}^\\nu +\\omega _1^e+\\omega _2^e+\\omega _3^\\nu )}\\theta _{23}^e c_{23}^\\nu s_{12}^e s_{13}^\\nu \\nonumber \\\\& +\\text{e}^{\\text{i}(\\delta _{13}^e+\\delta _{13}^\\nu +\\delta _{23}^e+\\omega _1^e+\\omega _2^\\nu +\\omega _3^e)}s_{12}^e s_{13}^\\nu s_{23}^\\nu )), \\\\s_{13}^{\\text{PMNS}}\\text{e}^{\\text{i}( \\eta _1-\\delta )} &= \\text{e}^{-\\text{i}(\\delta _{12}^e+\\delta _ {13}^e+\\delta _{13}^\\nu +\\delta _{23}^e+\\delta _{23}^\\nu +\\omega _1^e+\\omega _2^e+\\omega _3^e)} \\nonumber \\\\& (-\\text{e}^{\\text{i}(\\delta _{12}^e+\\delta _{13}^\\nu +\\delta _{23}^e +\\delta _{23}^\\nu +\\omega _1^e+\\omega _2^e+\\omega _3^\\nu )}\\theta _{13}^e c_{12}^e c_{13}^\\nu c_{23}^\\nu \\nonumber \\\\& +\\text{e}^{\\text{i}(\\delta _{13}^e+\\delta _{13}^\\nu +\\delta _{23}^\\nu +\\omega _1^e+\\omega _2^e+\\omega _3^\\nu )}\\theta _{23}^e c_{13}^\\nu c_{23}^\\nu s_{12}^e \\nonumber \\\\& +\\text{e}^{\\text{i}(\\delta _{12}^e+\\delta _{ 13}^e+\\delta _{23}^e+\\delta _{23}^\\nu +\\omega _1^\\nu +\\omega _2^e+\\omega _3^e)}c_{12}^e s_{13}^\\nu \\nonumber \\\\& -\\text{e}^{\\text{i}(\\delta _{13}^e +\\delta _{13}^\\nu +\\delta _{23}^e+\\omega _1^e+\\omega _2^\\nu +\\omega _3^e)} c_{13}^\\nu s_{12}^e s_{23}^\\nu )~.", "$ These expressions are exact to leading order in $\\theta _{13}^e$ and $\\theta _{23}^e$ .", "If we exploit the structure of the PMNS matrix we furthermore obtain: $c_{12}^{\\text{PMNS}} \\left(c_{13}^{\\text{PMNS}}\\right)^2 c_{23}^{\\text{PMNS}} s_{13}^{\\text{PMNS}} \\left(s_{12}^{\\text{PMNS}}s_{23}^{\\text{PMNS}}\\text{e}^{-\\text{i}\\delta }-c_{12}^{\\text{PMNS}}c_{23}^{\\text{PMNS}}s_{13}^{\\text{PMNS}}\\right)=\\\\\\left(U_{11}^{\\text{PMNS}}\\right)^*U_{13}^{\\text{PMNS}}U_{31}^{\\text{PMNS}} \\left(U_{33}^{\\text{PMNS}}\\right)^*~.$ Together with Eqs.", "(REF , , , REF ) we obtain equations for $\\delta $ , $\\eta _1$ , and for the Majorana phases.", "The mixing angles in Eq.", "(REF ) can be expressed in terms of the leptonic mixing parameters using Eqs.", "(REF , , ).", "As a concrete example to employ the formalism to derive the expressions for the Majorana phases, we consider the $A_5\\times SU(5)$ model proposed in [22], [45], [46].", "We have $~\\theta _{13}^e\\approx 0,~\\theta _{23}^e\\approx 0,~\\theta _{12}^e\\ne 0$ and in the neutrino sector we have Golden Ratio mixing with $\\theta _{13}^\\nu =0,~\\theta _{23}^\\nu =45^\\circ $ and $\\theta _{12}^\\nu =\\text{arctan}\\left(\\frac{2}{1+\\sqrt{5}}\\right)$ .", "Since $\\theta _{13}^e $ and $\\theta _{23}^e$ are negligibly small, we will set their values to zero in the following.", "The phase $\\delta _{12}^e $ in the charged lepton sector will be treated as a free parameter.", "For simplicity, we take the neutrino mass matrix to be real.", "The phases in the neutrino sector which lead to positive eigenvalues are then: $\\omega _1^\\nu =\\pi /2, ~\\omega _2^\\nu =\\pi ,~\\omega _3^\\nu =\\pi /2,~\\delta _{12}^{\\nu }=3\\pi /2 ,~\\delta _{23}^\\nu =3\\pi /2.$ With these parameters we obtain for Eqs.", "(REF , , , REF ,) to first order in $\\theta _{12}^e$ : $c_{12}^{\\text{PMNS}} \\text{e}^{\\text{i}(\\eta _1-\\phi _1/2)}&\\approx \\frac{\\text{e}^{\\text{i}\\pi /2} (\\sqrt{3+\\sqrt{5}} + \\theta _{12}^e \\text{e}^{-\\text{i}\\delta _{12}^e})}{\\sqrt{5+\\sqrt{5}}}~,\\\\s_{12}^{\\text{PMNS}} \\text{e}^{\\text{i}(\\eta _1-\\phi _2/2)}&\\approx -\\frac{2}{\\sqrt{10+2\\sqrt{5}}}+ \\frac{\\theta _{12}^{e}\\text{e}^{-\\text{i}\\delta _{12}^e }}{\\sqrt{5-\\sqrt{5}}}~,\\\\\\theta _{13}^{\\text{PMNS}}\\text{e}^{\\text{i}( \\eta _1-\\delta )} &\\approx \\frac{\\text{e}^{\\text{i}\\pi /2}\\theta _{12}^e \\text{e}^{-\\text{i}\\delta _{12}^{e}}}{\\sqrt{2}}~,\\\\s_{12}^{\\text{PMNS}}s_{23}^{\\text{PMNS}}\\text{e}^{-\\text{i}\\delta }&\\approx \\frac{-\\sqrt{2}(15+7\\sqrt{5})\\text{e}^{-\\text{i}\\delta _{12}^e}+(5+2\\sqrt{5})\\theta _{12}^e \\text{e}^{-2\\text{i}\\delta _{12}^e}+(20+9\\sqrt{5})\\theta _{12}^e}{4(5+2\\sqrt{5})^{3/2}}~.$ One might wonder if it is possible to take the limit $\\theta _{12}^e\\rightarrow 0$ in Eq.", "(REF ) and to obtain a sensible result.", "This is not possible since a diagonal charged lepton mass matrix corresponds to $\\theta _{13}^{\\text{PMNS}}=0$ where $\\delta $ is unphysical and in the derivation of Eq.", "(REF ) from Eq.", "() we have to divide by $\\theta _{13}^{\\text{PMNS}}$ .", "For a nondiagonal charged lepton mass matrix, we obtain for $\\delta $ $\\delta \\approx \\pi +\\delta _{12}^e+\\frac{\\theta _{12}^e \\sin (\\delta _{12}^e)}{\\sqrt{2}}~.$ We can easily obtain that $\\eta _1$ is $\\eta _1\\approx -\\frac{\\pi }{2}+\\frac{1}{\\sqrt{2}}\\theta _{12}^e \\sin (\\delta _{12}^e)~.$ Finally we get $\\phi _1&\\approx \\sqrt{3+\\sqrt{5}}\\theta _{12}^e \\sin (\\delta _{12}^e)~,\\\\\\phi _2&\\approx \\pi -\\frac{\\sqrt{5}-1}{\\sqrt{2}}\\theta _{12}^e\\sin (\\delta _{12}^e) ~.", "$ One might wonder if it is necessary to include the unphysical phases in order to derive the expressions for Majorana phases.", "Indeed, the correct consideration of the unphysical phases is essential since taking the $\\omega _i$ in Eq.", "(REF ) to zero which corresponds to $\\eta _i=\\phi _i/2$ for $i=1,~2$ changes the matrix element from which we extract the Majorana phases.", "For example, to obtain information about $\\phi _2$ , we consider the 1-2 element if we include the unphysical phases, but if we neglect the unphysical phases, we would have to consider the 2-3 element of the PMNS matrix.", "The dependence of these elements on the charged lepton mixing parameters differs in general.", "Even in the case of only a 1-2 mixing in the charged lepton we would miss the correct $\\theta _{12}^e$ dependence of $\\phi _2$ without the unphysical phases.", "The formulas derived in [48], [47], [49] assume that the unphysical phases have been already correctly taken into account.", "The reader has to be aware that these formulas therefore strictly apply to this case only.", "A phase matrix on the left side of the PMNS matrix can always be absorbed by an additional charged lepton phase rotation.", "For this reason, the phases $\\eta _i$ do not appear in physical observables." ] ]
1606.04965
[ [ "On goodness-of-fit tests for parametric hypotheses in perturbed\n dynamical systems using a minimum distance estimator" ], [ "Abstract We consider the problem of the construction of the Goodness-of-Fit test in the case of continuous time observations of a diffusion process with small noise.", "The null hypothesis is parametric and we use a minimum distance estimator of the unknown parameter.", "We propose an asymptotically distribution free test for this model." ], [ "Introduction", "We consider the problem of the construction of the asymptotically distribution free (ADF) test in the case of continuous time observations of a diffusion process.", "We suppose that under the null hypothesis the trend coefficient depends on some unknown one-dimensional parameter.", "Therefore the basic (null) hypothesis is parametric.", "The goodness-of-fit (GoF) test plays an important role in mathematical statistics because its application allows to check if the observations correspond well to the proposed mathematical model.", "Remind that in the case of i.i.d.", "observations the problem of the construction of GoF tests attracts attention of the statisticians since 1928 due to the works by Cramér (1928), von Mises (1931) and Smirnov (1937) (see, e.g., [2] and [3]).", "Let us recall the well-known basic results for the i.i.d.", "model.", "Suppose that the null hypothesis is simple.", "Denoting the continuous distribution function under the null hypothesis by $F_0\\left(x\\right)$ , we have to check if the i.i.d.", "observations $X^n=\\left(X_1,\\ldots ,X_n\\right)$ have this continuous distribution function.", "Many GoF tests are based on the following property: the normalized empirical distribution function $\\sqrt{n} \\left( \\hat{F}_n\\left(x\\right)-F_0\\left(x\\right)\\right), \\qquad \\hat{F}_n\\left(x\\right)=\\frac{1}{n}\\sum _{j=1}^n \\mbox{1\\hspace{-2.5pt}I}_{\\lbrace X_j < x\\rbrace }$ converges in distribution, under the null hypothesis, to the Brownian bridge $B\\left(F_0\\left(x\\right)\\right)$ .", "In particular, for the Cramér-von Mises statistic we have (with the change of variable $s=F_0(x)$ ) $\\Delta _n&={n}\\int _{-\\infty }^{\\infty }\\left( \\hat{F}_n\\left(x\\right)-F_0\\left(x\\right)\\right)^2{\\rm d}F_0\\left(x\\right) \\\\&\\qquad \\Longrightarrow \\int _{-\\infty }^{\\infty }B\\left(F_0\\left(x\\right)\\right)^2{\\rm d} F_0\\left(x\\right)=\\int _{0}^{1}B\\left(s\\right)^2{\\rm d}s\\equiv \\Delta .$ Therefore the limit $\\Delta $ does not depend on $F_0\\left(\\cdot \\right)$ (distribution free) and the Cramér-von Mises test $\\psi _n=\\mbox{1\\hspace{-2.5pt}I}_{\\left\\lbrace \\Delta _n>c_\\alpha \\right\\rbrace },\\qquad \\mathbf {P}\\left\\lbrace \\Delta >c_\\alpha \\right\\rbrace =\\alpha $ has the asymptotic $(n\\rightarrow \\infty )$ size $\\alpha \\in \\left(0,1\\right)$ (see, e.g., [14]).", "In the case of parametric null hypothesis $\\mathcal {H}_0 \\quad : \\qquad F\\left(x\\right)=F_0\\left(\\vartheta ,x\\right),\\qquad \\vartheta \\in \\Theta \\subseteq \\mathbb {R},$ where $\\vartheta $ is some unknown one-dimensional parameter, the similar limit $U_n\\left(x\\right) = \\sqrt{n}\\left(\\hat{F}_n\\left(x\\right)-F_0(\\hat{\\vartheta }_n,x)\\right)\\Longrightarrow B\\left(F_0\\left(\\vartheta ,x\\right)\\right)-\\zeta \\dot{F}_0\\left(\\vartheta ,x\\right)\\equiv U\\left(x\\right)$ is no more distribution free and the choice of the threshold $c_\\alpha $ is much more complicated.", "Here $\\hat{\\vartheta }_n$ is the maximum likelihood estimator (MLE), $\\dot{F}_0\\left(\\vartheta ,\\cdot \\right)$ is the derivative of $F_0\\left(\\vartheta ,\\cdot \\right)$ w.r.t.", "$\\vartheta $ and $\\zeta $ is a Gaussian variable.", "We have for $U\\left(x\\right)$ , by the change of variables $t=F_0\\left(\\vartheta ,x\\right)$ and $s=F_0\\left(\\vartheta ,y\\right)$ , the representation (see, e.g., Darling [2]) $u\\left(t\\right)=B\\left(t\\right)-\\int _{0}^{1}h\\left(\\vartheta ,s\\right){\\rm d}B\\left(s\\right)\\int _{0}^{t}h\\left(\\vartheta ,s\\right){\\rm d}s,$ where $h\\left(\\vartheta ,s\\right)=\\frac{\\dot{l}\\left(\\vartheta ,F_\\vartheta ^{-1}(s)\\right)}{\\sqrt{I\\left(\\vartheta \\right)}},\\qquad \\int _{0}^{1}h\\left(\\vartheta ,s\\right)^2{\\rm d}s=1.$ Here $l\\left(\\vartheta ,x\\right)=\\ln f\\left(\\vartheta ,x\\right),$ $\\dot{l}\\left(\\vartheta ,x\\right)$ is the derivative of $l\\left(\\vartheta ,x\\right)$ w.r.t.", "$\\vartheta $ , $f\\left(\\vartheta ,x\\right)$ is the density function, $I\\left(\\vartheta \\right)$ is the Fisher information and $y=F_\\vartheta ^{-1}(s)$ is the inverse function to $F_0\\left(\\vartheta ,y\\right)$ , i.e., $F_0(\\vartheta ,y)=s$ .", "Note that the function $h\\left(\\cdot ,\\cdot \\right)$ appears here twice because we used the MLE.", "For the minimum distance estimator (MDE) using the same arguments as in the case of the MLE we obtain the limit $u\\left(t\\right)=B\\left(t\\right)-\\int _{0}^{1}g\\left(\\vartheta ,s\\right)\\;{\\rm d}B\\left(s\\right)\\int _{0}^{t}h\\left(\\vartheta ,s\\right)\\;{\\rm d}s,\\qquad \\int _{0}^{1}g\\left(\\vartheta ,s\\right)^2{\\rm d}s=1$ with two different functions.", "In both cases, the corresponding tests are no more asymptotically distribution free and the choice of the threshold from the equation $\\mathbf {P}\\left\\lbrace \\int _{0}^{1}u\\left(s\\right)^2{\\rm d}s>c_\\alpha \\right\\rbrace =\\alpha $ can be a difficult problem.", "There are several possibilities to solve this problem in the case of the limit (REF ).", "One of them is to find a linear transformation $L\\left[\\cdot \\right]$ of the random function $u\\left(\\cdot \\right)$ , such that $L\\left[u\\right](t)=W\\left(t\\right)$ , where $W\\left(t \\right),0\\le t\\le 1$ is some Wiener process.", "Then the statistic $\\hat{\\Delta }_n&=\\int _{-\\infty }^\\infty L\\left[ U_n\\right]\\left(x\\right)^2{\\rm d}F_0(\\hat{\\vartheta }_n,x) \\Longrightarrow \\int _{-\\infty }^\\infty W\\left(F_0\\left(\\vartheta ,x\\right)\\right)^2{\\rm d} F_0\\left(\\vartheta ,x\\right)\\\\&\\qquad =\\int _{0}^{1}W\\left(t\\right)^2{\\rm d}t \\equiv \\hat{\\Delta }$ and we obtain once more the distribution free limit $\\hat{\\Delta }$ .", "Therefore the test $\\hat{\\psi }_n=\\mbox{1\\hspace{-2.5pt}I}_{\\left\\lbrace \\hat{\\Delta }_n>d_\\alpha \\right\\rbrace },\\qquad \\qquad \\mathbf {P}\\left\\lbrace \\hat{\\Delta } >d_\\alpha \\right\\rbrace =\\alpha $ is ADF.", "Such transformation was proposed by Khmaladze [6].", "We have to emphasize that in [6] and in many other works (see, e.g., the paper Maglapheridze et al.", "[16]) the estimator used was always the MLE and this is important for the construction of this linear transformation.", "Many authors wrote that similar transformation can be obtained in the case of other estimators with limit (REF ), but as we know this work (construction of the linear transformation with other estimators) was not done.", "The problem of GoF testing for the model of continuous time observations of diffusion process, with a simple null hypothesis $\\Theta =\\lbrace \\vartheta _{0}\\rbrace $ , was studied in [1] and [10].", "Suppose that the observed diffusion process under hypothesis is $\\textrm {d}X_t=S_0(X_t)\\;\\textrm {d}t+\\varepsilon \\sigma \\left(X_t\\right)\\;\\textrm {d}W_t,\\qquad X_0=x_0, \\quad 0\\le t \\le T,$ with deterministic initial value $x_0$ , known diffusion coefficient $\\varepsilon ^2 \\sigma \\left(\\cdot \\right)^2>0$ , some known smooth function $S_0(\\cdot )$ and $W_t, 0 \\le t \\le 1$ is a Wiener process.", "Note that if the functions $S_0\\left(x\\right)$ and $\\sigma \\left(x\\right)$ are Lipschitz w.r.t.", "$x$ , then we have with probability 1 $\\sup _{0\\le t \\le T}\\left|X_t-x_t \\right| \\le C\\varepsilon \\sup _{0\\le t \\le T}\\left|W_t\\right|,$ where $C$ is some constant.", "Therefore the process $X_t$ converges to $x_t=x_t(\\vartheta _0)$ (solution of the equation (REF ) as $\\varepsilon =0$ ) uniformly w.r.t.", "$t\\in \\left[0,T\\right]$ with probability 1.", "We have as well $\\sup _{0\\le t \\le T}\\mathbf {E}\\left|X_t-x_t \\right|^2 \\le C\\varepsilon ^2.$ For the proof see [9], Lemma 1.13.", "The GoF test was constructed on the basis of the normalized difference $\\varepsilon ^{-1}\\left(X_t-x_t\\right)$ and the limit of this statistic is a Gaussian process.", "This process can be transformed into a Wiener process $w(s), 0 \\le s \\le 1$ as follows.", "Introduce the statistic $\\delta _\\varepsilon = \\left[\\int _0^T\\left(\\frac{\\sigma \\left(x_t\\right)}{S_0(x_t)}\\right)^2\\textrm {d}t\\right]^{-2}\\int _0^T\\left(\\frac{X_t-x_t}{\\varepsilon S_0(x_t)^2}\\right)^2\\sigma \\left(x_t\\right)^2\\textrm {d}t.$ The following convergence $\\delta _\\varepsilon \\Longrightarrow \\delta \\equiv \\int _0^1w(s)^2\\;\\textrm {d}s$ was proved and therefore the test $\\phi _\\varepsilon = \\mbox{1\\hspace{-2.5pt}I}_{\\lbrace \\delta _\\varepsilon > c_\\varepsilon \\rbrace }$ with $\\mathbf {P}\\left(\\delta >c_\\varepsilon \\right)=\\varepsilon $ is ADF.", "The case of parametric basic hypothesis and ADF tests for “small noise\" diffusion processes was studied, for example, in [5], [11]$-$[13] and the estimator used, for the construction of the linear transformation and tests, was always the MLE (see Kutoyants [8]).", "There are several ADF GoF tests for the ergodic diffusion processes proposed, for example, in the works [17]$-$[18] and [7].", "Note that in some problems it is not possible to have an explicit expression for the MLE and therefore sometimes it is better to use other estimators, which can be easily calculated.", "For example, this can be the minimum distance estimator, a method of moments estimator or a trajectory fitting estimator and so on.", "In all such cases, the limit expression for the underlying statistics will be like (REF ) but with two different functions $h\\left(\\cdot ,\\cdot \\right)$ and $g\\left(\\cdot ,\\cdot \\right)$ (see below).", "In this work, we observe continuous time process $X^{\\varepsilon }$ , which is the solution of the stochastic differential equation $\\textrm {d}X_t=S(X_t)\\;\\textrm {d}t+\\varepsilon \\;\\textrm {d}W_t,\\qquad X_0=x_0, \\quad 0\\le t \\le T,$ where $W_t, \\;0\\le t\\le T$ is a Wiener process and $S(x)$ is some unknown smooth function.", "Based on a minimum distance estimator, we construct an ADF test in the case of parametric null hypothesis $\\mathcal {H}_0: \\quad S(x) = S(\\vartheta ,x),\\qquad \\vartheta \\in \\Theta =(a,b).$ The main results are presented in Theorem REF and Theorem REF .", "In particular, the ADF test is given in Theorem REF below.", "We realize the following program.", "First, we show that the basic statistic $u_\\varepsilon \\left(t\\right)=\\frac{X_t-x_t\\left(\\vartheta _\\varepsilon ^*\\right)}{\\varepsilon \\;S\\left(\\vartheta _\\varepsilon ^*,X_t \\right)}, \\qquad \\qquad 0\\le t \\le T$ ($\\vartheta _\\varepsilon ^*$ is the MDE) converges to the random process $u\\left(t\\right), 0\\le t \\le T$ (see (REF ) below).", "Then we transform $u\\left(t\\right)$ in $U\\left(\\frac{t}{T} \\right), 0\\le t \\le T$ (see (REF ) below).", "We obtain for $U\\left(\\cdot \\right)$ the following representation $U\\left(\\nu \\right)=W\\left(\\nu \\right)-\\int _{0}^{1}g\\left(\\vartheta ,r\\right){\\rm d}W\\left(r\\right)\\int _{0}^{\\nu }h\\left(\\vartheta ,r\\right){\\rm d}r,\\quad \\int _{0}^{1}g\\left(\\vartheta ,r\\right)^2{\\rm d}r=1.$ The last step is to apply the special linear transformation $L\\left[U\\right]\\left(\\nu \\right)=w_\\nu , 0\\le \\nu \\le 1$ , where $w_\\nu , 0\\le \\nu \\le 1$ is a Wiener process (see Theorem REF ).", "This allows us to construct an ADF test (see Theorem REF ).", "The main contribution of this work is the form of this linear transformation.", "Then we realize similar transformations with the “empirical” process $u_\\varepsilon \\left(\\cdot \\right)$ , apply the linear transformation $L\\left[\\cdot \\right]$ and obtain the convergence $\\Delta _\\varepsilon =\\frac{1}{T}\\int _{0}^{T}L\\left[U_\\varepsilon \\right]\\left(t\\right)^2{\\rm d}t \\Longrightarrow \\int _{0}^{1}w_\\nu ^2\\;{\\rm d}\\nu .$ Here the process $U_\\varepsilon (\\cdot )$ will be defined in Section 5 by (REF ).", "Therefore the test $\\psi _\\varepsilon =\\mbox{1\\hspace{-2.5pt}I}_{\\left\\lbrace \\Delta _\\varepsilon >c_\\alpha \\right\\rbrace }$ will be ADF, because the limit distribution of $\\Delta _\\varepsilon $ does not depend on $S\\left(\\cdot \\right)$ and $\\vartheta $ and the test is of asymptotic $(\\varepsilon \\rightarrow 0)$ size $\\alpha \\in (0,1)$ .", "Note that if in our problem we use the MLE of the unknown parameter, then the limit representation is $U\\left(\\nu \\right)=W\\left(\\nu \\right)-\\int _{0}^{1}h\\left(\\vartheta ,s\\right){\\rm d}W\\left(s\\right)\\int _{0}^{\\nu }h\\left(\\vartheta ,s\\right){\\rm d}s,\\qquad \\int _{0}^{1}h\\left(\\vartheta ,s\\right)^2{\\rm d}s=1$ and our transformation coincides with the one proposed in [6].", "We discuss this case in Section 6.", "We have to note that this linear transformation is rather cumbersome and the realization of the test can be a computationally difficult problem too.", "We suppose that the presented result is of theoretical interest and allows to “close the gap” in this field.", "At the same time, we understand that this result is in some sense “negative” and says that if we have no MLE it is better to seek another GoF test, which is ADF.", "Note as well that in i.i.d.", "case, even if the estimated parameter is one-dimensional, the reduction of the equation with Brownian bridge (REF ) to the equation (REF ), using the relation $(B\\left(t\\right)=W(t) -t \\;W(1))$ , leads to the corresponding Fredholm equation (see (REF ) below) with two-dimensional $g\\left(\\cdot \\right)$ and $h\\left(\\cdot \\right)$ .", "The expression for the solution of this equation and the form of the linear transformation becomes much more complicated.", "This is probably the reason why this problem was not considered till now.", "Our results thus should be understood as a constructive existence result for ADF tests based on the MDE." ], [ "Minimum distance estimator", "Suppose that the continuous time observed process $X^{\\varepsilon } = (X_t,0\\le t \\le T)$ is the solution of the stochastic differential equation $\\textrm {d}X_t=S(X_t)\\;\\textrm {d}t+\\varepsilon \\;\\textrm {d}W_t,\\qquad X_0=x_0, \\quad 0\\le t \\le T,$ where $W_t, \\;0\\le t\\le T$ is a Wiener process, the initial value $x_0$ is deterministic and the trend coefficient $S(x)$ is some unknown smooth function.", "We consider the composite basic hypothesis $\\mathcal {H}_0: \\quad S(x) = S(\\vartheta ,x),\\qquad \\vartheta \\in \\Theta =(a,b),$ where $\\vartheta $ is the one-dimensional (unknown) parameter, against alternative $\\mathcal {H}_1:$ not $\\mathcal {H}_0$ , i.e., the trend coefficient $S\\left(x\\right)$ in the observed diffusion process (REF ) does not belong to the parametric family $\\left\\lbrace S\\left(\\vartheta ,x\\right),\\vartheta \\in \\Theta \\right\\rbrace $ .", "Therefore the process, under hypothesis $\\mathcal {H}_0$ , has the stochastic differential $\\textrm {d}X_t=S(\\vartheta ,X_t)\\;\\textrm {d}t+\\varepsilon \\;\\textrm {d}W_t,\\qquad X_0=x_0, \\quad 0\\le t \\le T.$ We are interested in the properties of the test in the asymptotics of small noise, i.e., as $\\varepsilon \\rightarrow 0$ .", "Below and in the sequel the dot means derivation w.r.t.", "$\\vartheta $ .", "Let us introduce the regularity conditions.", "${\\cal R}.$ The function $S(\\vartheta ,x)$ is strictly positive and has two continuous bounded derivatives with respect to $\\vartheta $ and $x$ .", "In the presentation below we suppose that these conditions and the basic hypothesis $\\mathcal {H}_0$ are always fulfilled.", "It is known that the solution $X_t$ converges uniformly in $t\\in [0,T]$ to the solution $x_t =x_t(\\vartheta )$ of the ordinary differential equation $\\frac{\\textrm {d}x_t}{\\textrm {d}t} = S(\\vartheta ,x_t),\\qquad x_0 ,\\quad 0\\le t \\le T,$ where $x_0$ is the same as in (REF ) (see [4] or [9], Lemma 1.13 for the proof).", "Recall the properties of maximum likelihood and minimum distance estimators of the parameter $\\vartheta $ .", "The likelihood ratio function in the case of observations (REF ) is $\\ell \\left(\\vartheta ,X^\\varepsilon \\right)=\\frac{\\textrm {d}P_\\vartheta }{\\textrm {d}P_0}=\\exp \\left\\lbrace \\int _0^T\\frac{S\\left(\\vartheta ,X_t\\right)}{\\varepsilon ^2}\\;\\textrm {d}X_t-\\int _0^T\\frac{S\\left(\\vartheta ,X_t\\right)^2}{2\\;\\varepsilon ^2}\\;\\textrm {d}t\\right\\rbrace , \\quad \\vartheta \\in \\Theta ,$ where $P_\\vartheta $ and $P_0$ are the measures induced respectively by the processes (REF ) and $\\textrm {d}X_t=\\varepsilon \\;\\textrm {d}W_t, \\qquad X_0=x_0, \\qquad 0 \\le t \\le T,$ (see [15] for more details).", "The MLE $\\hat{\\vartheta }_\\varepsilon $ is solution of the equation $\\sup _{\\vartheta \\in \\Theta }\\ell \\left(\\vartheta ,X^\\varepsilon \\right)=\\ell (\\hat{\\vartheta }_\\varepsilon ,X^\\varepsilon ) .$ This estimator is consistent and asymptotically normal (as $\\varepsilon \\rightarrow 0$ ) $\\varepsilon ^{-1}(\\hat{\\vartheta }_\\varepsilon -\\vartheta )\\Longrightarrow {\\cal N}\\left(0,{\\rm I}\\left(\\vartheta \\right)^{-1}\\right),\\qquad {\\rm I}\\left(\\vartheta \\right)=\\int _{0}^{T}\\dot{S}\\left(\\vartheta ,x_t\\right)^2{\\rm d}t.$ Here and in the sequel $x_t=x_t\\left(\\vartheta \\right)$ .", "For the proof see, Kutoyants [9].", "Introduce the minimum distance estimator $\\vartheta ^{*}_{\\varepsilon } =\\operatornamewithlimits{arg\\,min}\\limits _{\\vartheta \\in \\Theta } \\left\\Vert X-x(\\vartheta )\\right\\Vert ,$ where $\\Vert \\cdot \\Vert $ is $L^2[0,T]$ norm defined by $\\left\\Vert X-x(\\vartheta )\\right\\Vert ^2 = \\int _0^T\\left(X_t-x_t(\\vartheta )\\right)^2\\textrm {d}t.$ The properties of the MDE for this model were studied in [9].", "The MDE satisfies the minimum distance equation (MDEq) $\\int _0^T \\left(X_t-x_t(\\vartheta ^{*}_{\\varepsilon })\\right)\\dot{x}_t(\\vartheta ^{*}_{\\varepsilon })\\;\\textrm {d}t=0.$ Let us put $u_{\\varepsilon }^{*}=\\varepsilon ^{-1}\\left(\\vartheta ^{*}_{\\varepsilon }-\\vartheta \\right)$ .", "Therefore the MDEq is as follows: $\\int _0^T \\left(X_t-x_t\\left(\\vartheta + \\varepsilon u_{\\varepsilon }^{*}\\right)\\right)\\dot{x}_t(\\vartheta ^{*}_{\\varepsilon })\\;\\textrm {d}t=0.$ Then, by the Taylor formula, we can write $\\int _0^T \\left(X_t- x_t(\\vartheta ) - \\varepsilon \\;u_{\\varepsilon }^{*} \\;\\dot{x}_t(\\tilde{\\vartheta })\\right)\\dot{x}_t(\\vartheta ^{*}_{\\varepsilon })\\;\\textrm {d}t=0,$ where $ |\\tilde{\\vartheta }-\\vartheta |\\le \\left| \\vartheta ^{*}_{\\varepsilon }-\\vartheta \\right|$ and for $u_\\varepsilon ^*$ we obtain the following representation $u_{\\varepsilon }^{*} =J(\\vartheta )^{-1}\\int _0^T\\varepsilon ^{-1}\\left(X_t-x_t(\\vartheta )\\right)\\dot{x}_t(\\vartheta )\\;\\textrm {d}t+o(1),$ where $J(\\vartheta ) =\\int _0^T\\dot{x}_t(\\vartheta )^2\\textrm {d}t.$ Let us consider the random process $x_t^{(1)}$ defined as the derivative of $X_t$ w.r.t.", "$\\varepsilon $ at $\\varepsilon =0$ , i.e., we have $\\varepsilon ^{-1}\\left(X_t-x_t\\left(\\vartheta \\right)\\right)\\rightarrow x_t^{(1)}$ .", "Here the random process $x_t^{(1)}$ satisfies the linear equation $\\textrm {d}x_t^{(1)} = S^{\\prime }(\\vartheta ,x_t)\\;x_t^{(1)}\\textrm {d}t +\\textrm {d}W_t,\\qquad x_0^{(1)}=0,\\quad 0\\le t \\le T,$ where $S^{\\prime }(\\vartheta ,x) = \\displaystyle \\frac{\\partial S(\\vartheta ,x)}{\\partial x}.$ The process $x_t^{(1)}$ has the representation $x_t^{(1)}=S(\\vartheta ,x_t)\\int _0^t \\frac{1}{S(\\vartheta ,x_s)}\\;\\textrm {d}W_s$ (see, e.g., Section $3.3$ in [9]).", "Due to the above representation of $x_t^{(1)},$ the equation (REF ) becomes $u_{\\varepsilon }^{*}&= & J(\\vartheta )^{-1}\\int _0^Tx_t^{(1)}\\dot{x}_t(\\vartheta )\\;\\textrm {d}t+o(1)\\\\ & = &J(\\vartheta )^{-1}\\int _0^TS(\\vartheta ,x_t)\\int _0^t\\frac{1}{S(\\vartheta ,x_s)}\\;\\textrm {d}W_s\\;\\dot{x}_t(\\vartheta )\\;\\textrm {d}t+o(1).$ Here $o(1)$ means the convergence in probability, i.e., for any $\\nu >0$ , we have $\\lim _{\\varepsilon \\rightarrow 0}\\mathbf {P}_\\vartheta \\left(\\left|u_{\\varepsilon }^{*}-\\rho (\\vartheta )\\right|>\\nu \\right)=0,$ where $\\rho (\\vartheta )=J(\\vartheta )^{-1}\\int _0^TS(\\vartheta ,x_t)\\int _0^t\\frac{1}{S(\\vartheta ,x_s)}\\;\\textrm {d}W_s\\;\\dot{x}_t(\\vartheta )\\;\\textrm {d}t.$ Therefore $\\vartheta ^{*}_{\\varepsilon }$ admits the following representation, by Fubini's Theorem, $\\varepsilon ^{-1}\\left(\\vartheta ^{*}_\\varepsilon -\\vartheta \\right)=J(\\vartheta )^{-1}\\int _0^T\\frac{1}{S(\\vartheta ,x_v)}\\int _v^T S(\\vartheta ,x_s)\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s\\;\\textrm {d}W_v+o(1).$ Moreover, under conditions of regularity, the estimator $\\vartheta ^{*}_{\\varepsilon }$ is consistent and asymptotically normal (see Chapter 7 in [9]) $\\mathcal {L}_{\\vartheta }\\lbrace \\varepsilon ^{-1}(\\vartheta ^{*}_{\\varepsilon }-\\vartheta )\\rbrace \\Longrightarrow \\mathcal {L}\\lbrace \\xi \\rbrace =\\mathcal {N}(0,\\sigma ^2\\left(\\vartheta \\right)),$ where $\\sigma ^2\\left(\\vartheta \\right) = J(\\vartheta )^{-2}\\int _0^T\\frac{1}{S(\\vartheta ,x_v)^2}\\left(\\int _v^TS(\\vartheta ,x_s)\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s\\right)^2\\textrm {d}v.$" ], [ "Basic statistic", "Our goal is to find the GoF test, which is ADF, i.e., we seek the test statistic whose limit distribution, under hypothesis, does not depend on the underlying model given by the function $S(\\vartheta ,x)$ and parameter $\\vartheta $ .", "Introduce the statistic $\\delta ^*_{\\varepsilon }=\\int _0^T\\left[\\frac{X_t-x_t(\\vartheta ^{*}_{\\varepsilon })}{\\varepsilon \\;S\\left(\\vartheta ^{*}_{\\varepsilon },X_t\\right)}\\right]^2\\textrm {d}t.$ To study it we need the behavior of the difference $X_t-x_t(\\vartheta ^{*}_{\\varepsilon })$ , which can be described as follows: $&&\\varepsilon ^{-1}\\left(X_t-x_t(\\vartheta ^{*}_{\\varepsilon })\\right) =\\varepsilon ^{-1}\\left(X_t-x_t(\\vartheta )\\right)-\\varepsilon ^{-1}\\left(\\vartheta ^{*}_{\\varepsilon }-\\vartheta \\right)\\dot{x}_t(\\vartheta )+o(1)\\\\ &&\\qquad \\qquad =x_t^{(1)}(\\vartheta )-J(\\vartheta )^{-1}\\int _0^Tx_s^{(1)}\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s\\;\\dot{x}_t(\\vartheta )+o(1)\\\\ &&\\qquad \\qquad = S(\\vartheta ,x_t)\\int _0^t\\frac{\\textrm {d}W_v}{S(\\vartheta ,x_v)}\\\\ && \\qquad \\quad \\quad -J(\\vartheta )^{-1}\\int _0^T\\frac{1}{S(\\vartheta ,x_v)}\\int _v^TS(\\vartheta ,x_s)\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s\\;\\textrm {d}W_v\\;\\dot{x}_t(\\vartheta )+o(1),$ where the process $x_t^{(1)}$ was defined by (REF ) and the derivative $\\dot{x}_t(\\vartheta )$ w.r.t.", "$\\vartheta $ satisfies the equation $\\frac{\\textrm {d}\\dot{x}_t(\\vartheta )}{\\textrm {d}t} =S^{\\prime }(\\vartheta ,x_t) \\;\\dot{x}_t(\\vartheta )+\\dot{S}(\\vartheta ,x_t),\\qquad \\dot{x}_0(\\vartheta )=0.$ Its solution is the function (it can be found in [11]) $\\dot{x}_t(\\vartheta ) = S(\\vartheta ,x_t)\\int _0^t\\frac{\\dot{S}(\\vartheta ,x_v)}{S(\\vartheta ,x_v)}\\;\\textrm {d}v.$ Here $o(1)$ is the uniform convergence in probability w.r.t.", "$t\\in [0,T]$ , i.e., for any $\\nu >0$ , we have $\\lim _{\\varepsilon \\rightarrow 0}\\mathbf {P}_\\vartheta \\left(\\sup _{t\\in [0,T]} \\left|\\varepsilon ^{-1}\\left(X_t-x_t(\\vartheta ^{*}_{\\varepsilon })\\right)-\\mu _t(\\vartheta )\\right|>\\nu \\right)=0,$ where $&\\mu _t(\\vartheta )=S(\\vartheta ,x_t)\\int _0^t\\frac{\\textrm {d}W_v}{S(\\vartheta ,x_v)}\\\\&\\qquad \\qquad -J(\\vartheta )^{-1}\\int _0^T\\frac{1}{S(\\vartheta ,x_v)}\\int _v^TS(\\vartheta ,x_s)\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s\\;\\textrm {d}W_v\\;\\dot{x}_t(\\vartheta ).$ For the details see [9], Chapter 7.", "Hence we have the uniform convergence w.r.t.", "$t\\in [0,T]$ (in probability) $\\begin{split}&u_{\\varepsilon }(t) =\\frac{X_t-x_t(\\vartheta ^{*}_{\\varepsilon })}{\\varepsilon \\;S\\left(\\vartheta ^{*}_{\\varepsilon },X_t\\right)}\\longrightarrow u(t) = \\int _0^t\\frac{\\textrm {d}W_v}{S(\\vartheta ,x_v)}\\\\ &\\qquad -J(\\vartheta )^{-1}\\int _0^T\\frac{1}{S(\\vartheta ,x_v)}\\int _v^TS(\\vartheta ,x_s)\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s\\;\\textrm {d}W_v\\int _0^t\\frac{\\dot{S}(\\vartheta ,x_v)}{S(\\vartheta ,x_v)}\\;\\textrm {d}v\\end{split}$ and it can be shown (see in [11] the details in the similar problem where the MLE was used) that $\\delta ^*_{\\varepsilon }\\Longrightarrow \\int _0^Tu(t)^2\\;\\textrm {d}t.$ Therefore the test based on this statistic is not ADF.", "Hence to obtain an ADF GoF test we introduce the Gaussian process $U\\left(\\frac{t}{T}\\right) = \\frac{1}{\\sqrt{T}}\\int _0^{t}S(\\vartheta ,x_s)\\;\\textrm {d}u(s),\\qquad 0\\le t\\le T,$ where $u\\left(\\cdot \\right)$ was given by (REF ).", "Then, by Itô formula, $\\textrm {d}\\left(S(\\vartheta ,x_s)u(s)\\right)=\\textrm {d}S(\\vartheta ,x_s)\\;u(s)+\\textrm {d}u(s)\\;S(\\vartheta ,x_s)$ and using the equality $\\textrm {d}S(\\vartheta ,x_s)=S^{\\prime }(\\vartheta ,x_s)\\;S(\\vartheta ,x_s)\\;\\textrm {d}s,$ we have $S(\\vartheta ,x_t)\\;u(t)=\\int _0^tS^{\\prime }(\\vartheta ,x_s)\\;S(\\vartheta ,x_s)\\;u(s)\\;\\textrm {d}s+\\int _0^tS(\\vartheta ,x_s)\\;\\textrm {d}u(s).$ Here and in the sequel we denoted by prime the derivative w.r.t.", "$x$ .", "Therefore the process $U(\\cdot )$ defined by (REF ) admits the following representation $(0\\le t \\le T)$ $&U\\left(\\frac{t}{T}\\right) = \\frac{1}{\\sqrt{T}}\\;S(\\vartheta ,x_t)\\;u(t)- \\frac{1}{\\sqrt{T}}\\int _0^tS^{\\prime }(\\vartheta ,x_s)\\;S(\\vartheta ,x_s)\\;u(s)\\;\\textrm {d}s.$ Further we define two functions $\\begin{split}h(\\vartheta ,r)&=T\\tilde{J}(\\vartheta )^{-1}\\dot{S}(\\vartheta ,x_{rT}(\\vartheta ))\\;C\\left(\\vartheta \\right)^{1/2}\\end{split}$ and $\\begin{split}g(\\vartheta ,r) &= S(\\vartheta ,x_{rT}(\\vartheta ))^{-1}\\displaystyle \\int _{r}^1S(\\vartheta ,x_{zT}(\\vartheta ))\\;\\dot{x}_{zT}(\\vartheta )\\;\\textrm {d}z\\;C\\left(\\vartheta \\right)^{-1/2}.\\end{split}$ Here $\\tilde{J}(\\vartheta ) = \\displaystyle \\int _0^1 \\dot{x}_{vT}(\\vartheta )^2 \\textrm {d}v$ and $C\\left(\\vartheta \\right)=\\int _0^1S(\\vartheta ,x_{vT}(\\vartheta ))^{-2}\\left(\\int _{v}^1S(\\vartheta ,x_{zT}(\\vartheta ))\\;\\dot{x}_{zT}(\\vartheta )\\;\\textrm {d}z\\right)^2\\textrm {d}v.$ Observe that $\\int _0^1 g(\\vartheta ,r)^2\\textrm {d}r=1.$ Lemma 1 We have the equality $U(\\nu ) = W(\\nu ) - \\int _0^1g(\\vartheta ,r)\\;\\textrm {d}W(r)\\;\\int _0^\\nu h(\\vartheta ,r)\\;\\textrm {d}r, \\quad 0\\le \\nu \\le 1,$ where $W(\\nu ), 0\\le \\nu \\le 1$ is a Wiener process.", "Proof.", "We have by Itô formula, for $0\\le t \\le T$ , $&\\textrm {d}U\\left(\\frac{t}{T}\\right)=\\frac{1}{\\sqrt{T}}\\;\\textrm {d}S(\\vartheta ,x_t)\\;u(t)\\\\&\\qquad \\qquad \\quad +\\frac{1}{\\sqrt{T}}\\;S(\\vartheta ,x_t)\\;\\textrm {d}u(t)- \\frac{1}{\\sqrt{T}}\\;S^{\\prime }(\\vartheta ,x_t)\\;S(\\vartheta ,x_t)\\;u(t)\\;\\textrm {d}t.$ Then by (REF ) and using the representation (REF ) of the process $u(\\cdot )$ , we obtain $&U\\left(\\frac{t}{T}\\right)=\\frac{1}{\\sqrt{T}}\\int _0^tS(\\vartheta ,x_s)\\;\\textrm {d}u(s)\\\\&=\\frac{W_t}{\\sqrt{T}}-\\frac{J(\\vartheta )^{-1}}{\\sqrt{T}} \\int _0^T\\frac{1}{S(\\vartheta ,x_v)}\\int _v^TS(\\vartheta ,x_s)\\;\\dot{x}_{s}(\\vartheta )\\;\\textrm {d}s\\;\\textrm {d}W_v\\int _0^t\\dot{S}(\\vartheta ,x_v)\\;\\textrm {d}v.$ Hence, by the change of variables $\\nu =\\displaystyle \\frac{t}{T}$ and $W(\\nu ) = T^{-1/2}W_{\\nu T}, 0\\le \\nu \\le 1$ , we have $&U(\\nu ) =W\\left(\\nu \\right) \\\\&\\qquad \\qquad -\\frac{J(\\vartheta )^{-1}}{\\sqrt{T}} \\int _0^T\\frac{1}{S(\\vartheta ,x_v)}\\int _v^TS(\\vartheta ,x_s)\\;\\dot{x}_{s}(\\vartheta )\\;\\textrm {d}s\\;\\textrm {d}W_v\\int _0^{\\nu T}\\dot{S}(\\vartheta ,x_v)\\;\\textrm {d}v.$ Let us change the variables $r=\\displaystyle \\frac{v}{T}$ , $z=\\displaystyle \\frac{s}{T}$ , $W(r) = T^{-1/2}W_{rT}, 0\\le r \\le 1$ .", "Then we can write $U(\\nu )& = W(\\nu ) \\\\& \\quad - \\tilde{J}(\\vartheta )^{-1} \\int _0^1\\frac{1}{S(\\vartheta ,x_{rT})} \\int _{rT}^TS(\\vartheta ,x_s)\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s\\;\\textrm {d}W(r)\\int _0^{\\nu }\\dot{S}(\\vartheta ,x_{rT})\\;\\textrm {d}r\\\\& = W(\\nu ) \\\\& \\quad -\\frac{T}{\\tilde{J}(\\vartheta )}\\int _0^1\\frac{1}{S(\\vartheta ,x_{rT})} \\int _{r}^1S(\\vartheta ,x_{zT})\\;\\dot{x}_{zT}(\\vartheta )\\;\\textrm {d}z\\;\\textrm {d}W(r)\\int _0^{\\nu }\\dot{S}(\\vartheta ,x_{rT})\\;\\textrm {d}r\\\\& = W(\\nu ) -\\int _0^1g(\\vartheta ,r)\\;\\textrm {d}W(r)\\;\\int _0^{\\nu }h(\\vartheta ,r)\\;\\textrm {d}r.$ Therefore we obtain the representation (REF ) and this proves the Lemma REF .", "It can be shown using the convergence of the empirical version $U_\\varepsilon \\left(\\cdot \\right)$ to $U\\left(\\cdot \\right)$ (see proof of Theorem REF below) and due to the continuous mapping Theorem, that we have the convergence $\\widetilde{\\Delta }_{\\varepsilon } = \\frac{1}{T^2}\\int _0^T\\left( \\int _0^{t}S(\\vartheta ^*_{\\varepsilon },X_s)\\;\\textrm {d}u_{\\varepsilon }(s)\\right)^2\\textrm {d}t\\Longrightarrow \\int _0^1 U(\\nu )^2\\textrm {d}\\nu .$ We remark that the test based on this statistic is not ADF.", "Hence we have to find the transformation $L[U](\\cdot )$ into the Wiener process such that $\\int _0^1 L[U](\\nu )^2\\textrm {d}\\nu = \\int _0^1w_{\\nu }^2\\;\\textrm {d}\\nu ,\\qquad 0\\le \\nu \\le 1,$ where $w_{\\nu }, 0\\le \\nu \\le 1$ is a Wiener process.", "This property allows us to present the “empirical version” of the test statistic with the same limit.", "Therefore the test based on this statistic is ADF." ], [ "Linear transformation", "Now the problem is to find such transformation $L\\left[\\cdot \\right]$ of $U(\\cdot )$ (see (REF )) using the MDE that $L[U](\\nu )=w_{\\nu }.$ Recall that for the limit process (REF ) such linear transformation and the corresponding ADF test were proposed by Khmaladze [6].", "Another (direct) proof of this result was recently obtained by Kleptsyna and Kutoyants [7].", "Note that in these works the estimator used was always the MLE and in our work it is the MDE.", "The limit processes in these two cases are quite different.", "That is why we have to present here a special modification of the proof given in [7].", "Our proof follows the main steps of the work [7].", "Specifically, we have to solve Fredholm equation of the second kind with degenerated kernel.", "The solution of it gives us the desired linear transformation.", "Denote $\\begin{split}&I_1 = \\int _0^r g(\\vartheta ,q)^2\\textrm {d}q,\\; I_2 = \\int _0^r h(\\vartheta ,q)\\;g(\\vartheta ,q)\\;\\textrm {d}q,\\; I_3 =\\int _0^rh(\\vartheta ,q)\\;\\textrm {d}q,\\\\ &\\qquad \\qquad \\quad I_4 =\\int _0^rh(\\vartheta ,q)^2\\textrm {d}q,\\; I_5 =\\int _0^rg(\\vartheta ,q)\\;\\textrm {d}q.", "\\end{split}$ Below we omit $\\vartheta $ and $r$ for simplicity and put $g=g(\\vartheta ,r)$ and $h=h(\\vartheta ,r)$ .", "Introduce the functions $\\varphi _1(r) & = g-h-3I_2g+I_5hg+ I_3g^2+2I_2h-2I_2I_3g^2+I_1I_2^2h+I_4I_5g^2-I_2^3g \\nonumber \\\\& \\quad - I_2I_4g+3I_2^2g+ I_2I_5h^2-2I_2I_5hg-2I_1I_2h + I_2^2I_5hg+I_1^2I_3h^2-I_4h\\nonumber \\\\& \\quad + 2I_1I_4h-I_1I_4g + I_1I_2I_4g + I_1I_4I_5hg-I_1^2I_4h+I_1h+2I_2I_3hg\\nonumber \\\\& \\quad - I_2I_4I_5g^2-I_5h^2 + 2I_1I_3hg - 2I_1I_2I_3hg-2I_1I_3h^2-I_2^2h+I_3h^2\\nonumber \\\\& \\quad - 2I_3hg-I_4I_5hg-I_1I_2I_5h^2+I_2^2I_3g^2+I_1I_5h^2+I_4g,$ $\\varphi _2(r) & = 1 + I_5h-3I_2I_5h+I_1I_3h+I_3g-3I_2I_3g+I_4I_5g-I_3h-I_1I_4^2I_5g\\nonumber \\\\& \\quad + 3I_2^2I_3g- 2I_2I_4I_5g+2I_2I_3h-I_2^3I_5h+I_1I_2^2I_3h-I_2^3I_3g+3I_2^2I_5h\\nonumber \\\\& \\quad + I_4I_5h-I_2I_4I_5h+2I_1I_3I_4h+I_3I_4g-I_2I_3I_4g+I_4^2+ 2I_2^2I_4+I_2^4 \\nonumber \\\\& \\quad + I_1I_2I_4I_5h-I_1^2I_3I_4h-I_1I_3I_4g + I_1I_2I_3I_4g- I_2^2I_3h -2I_1I_2^2I_4\\nonumber \\\\& \\quad - 2I_1I_4^2+I_1^2I_4^2+2I_4- 2I_1I_4-4I_2I_4+4I_1I_2I_4-4I_2 + I_4^2I_5g \\nonumber \\\\& \\quad + I_2^2I_4I_5g - 2I_1I_2I_3h-I_3I_4h+6I_2^2- 4I_2^3- I_1I_4I_5h$ and $\\psi _2(r) & = h+I_3hg-2I_2I_4g+I_5h^2-3I_2h-2I_2I_3hg-I_1I_2I_3h^2+I_4g+ 3I_2^2h \\nonumber \\\\& \\quad + I_2^2I_4g+I_2^2I_5h^2-I_2^3h-I_3I_4hg+I_4^2g+I_4h - I_2I_4h-2I_2I_5h^2\\nonumber \\\\& \\quad - I_1I_4h+I_1I_2I_4h+I_1I_3h^2 + I_3I_4g^2+ I_2I_3h^2-I_3h^2+I_1I_3I_4hg\\nonumber \\\\& \\quad - I_2I_3I_4g^2-2I_2I_4I_5hg +2I_4I_5hg +I_2^2I_3hg-I_1I_4^2g+I_4^2I_5g^2.$ The following Theorem is the main result of this work.", "Theorem 1 Suppose that $h(q)$ and $g(q)$ are continuous functions such that $\\displaystyle \\int _0^1 g(q)^2 \\textrm {d}q = 1 $ and $\\varphi _2(r)$ is a strictly positive function on $[0,1)$ .", "Then the equality $L[U](\\nu ) = U(\\nu )+\\displaystyle \\int _0^{\\nu } \\int _0^r\\frac{\\varphi _1(r)\\;h(q)+\\psi _2(r)\\;g(q)}{\\varphi _2(r)}\\;\\textrm {d}U(q)\\;\\textrm {d}r=w_{\\nu }$ holds.", "Here $w_{\\nu }, \\;0\\le \\nu \\le 1$ is a Wiener process.", "Proof.", "The proof will be done in several steps.", "Step 1: Introduce a Gaussian process $ M_t = \\int _0^t q(t,s) \\;\\textrm {d}U(s),\\qquad 0\\le t \\le 1,$ where the function $q(t,s)$ is chosen as solution of Fredholm equation described in the next step.", "Observe that $M_t = \\int _0^t q(t,u) \\;\\textrm {d}W(u) - \\int _0^1 g(u) \\;\\textrm {d}W(u)\\int _0^t q(t,u) \\; h(u)\\;\\textrm {d}u.$ Step 2: For the correlation function of $M_t$ $R(t,s) = \\textbf {\\textrm {E}}\\left[M_t M_s\\right] ,\\qquad t>s,$ we have $\\textbf {\\textrm {E}}\\left[M_t M_s\\right] & = \\textbf {\\textrm {E}}\\left[ \\int _0^t q(t,u) \\;\\textrm {d}W(u) - \\int _0^1 g(u) \\;\\textrm {d}W(u)\\int _0^t q(t,u)\\;h(u)\\;\\textrm {d}u\\right]\\\\& \\qquad \\;\\left[\\int _0^s q(s,v) \\;\\textrm {d}W(v) - \\int _0^1 g(v) \\;\\textrm {d}W(v)\\int _0^s q(s,v) \\;h(v)\\;\\textrm {d}v\\right]\\\\& = \\int _0^s q(t,u)\\;q(s,u)\\;\\textrm {d}u - \\int _0^t q(t,u)\\; g(u)\\;\\textrm {d}u \\int _0^s q(s,v)\\; h(v)\\;\\textrm {d}v \\\\& \\quad \\quad - \\; \\int _0^s q(s,v) \\;g(v)\\;\\textrm {d}v \\int _0^t q(t,u) \\;h(u)\\;\\textrm {d}u \\\\& \\quad \\quad + \\underbrace{\\int _0^1 g(v)^2 \\textrm {d}v}_{=1} \\int _0^t q(t,u)\\;h(u)\\;\\textrm {d}u \\int _0^s q(s,v) \\;h(v)\\;\\textrm {d}v\\\\& = \\int _0^s q(s,u)\\biggl [q(t,u) - \\int _0^t q(t,v)\\; g(v)\\;\\textrm {d}v\\;h(u) \\\\& \\quad \\quad - \\int _0^t q(t,v)\\; h(v)\\;\\textrm {d}v\\; g(u)+ \\int _0^t q(t,v)\\; h(v)\\;\\textrm {d}v\\; h(u)\\biggl ]\\textrm {d}u\\\\& = \\int _0^s q(s,u)\\biggl [q(t,u) - \\int _0^t q(t,v)\\;\\bigl [ g(v)\\;h(u)\\\\& \\qquad \\qquad \\qquad \\qquad + \\;h(v)\\;g(u)-h(v)\\;h(u)\\bigl ]\\;\\textrm {d}v \\biggl ]\\textrm {d}u.$ Denote the kernel $K(u,v) = g(v)\\;h(u)+h(v)\\;g(u)-h(v)\\;h(u).$ Then $\\textbf {\\textrm {E}}\\left[M_t M_s\\right] = \\int _0^s q(s,u)\\biggl [q(t,u) - \\int _0^t q(t,v)\\;K(u,v)\\;\\textrm {d}v \\biggl ]\\textrm {d}u.$ Therefore if we take $q(t,s)$ such that it solves the Fredholm equation of the second kind $(t$ is fixed$)$ $ q(t,s) - \\int _0^t q(t,v)\\;K(s,v)\\;\\textrm {d}v = 1,\\qquad s \\in [0,t],$ then $(\\ref {ch115})$ becomes $ \\textbf {\\textrm {E}}\\left[M_t M_s\\right] = \\textbf {\\textrm {E}}\\left[M_s^2\\right] = \\int _0^sq(s,u)\\;\\textrm {d}u.$ Step 3: The solution $q(t,s)$ can be found as follows.", "We have $q(t,s) = 1+ \\int _0^t q(t,v)\\;K(s,v)\\;\\textrm {d}v.$ Denote $A(t) = \\int _0^t q(t,v)\\; h(v)\\;\\textrm {d}v$ and $B(t) = \\int _0^t q(t,v)\\; g(v)\\;\\textrm {d}v.$ Then $q(t,s)$ has the representation $q(t,s) = 1+ B(t) \\; h(s) + A(t) \\left( g(s)-h(s)\\right),$ where the function $A(t)$ itself is solution of the following equation (after multiplying (REF ) by $h(s)$ and integrating) $\\begin{split}\\int _0^t h(s) \\; \\textrm {d}s &= A(t) - B(t) \\int _0^t h(s)^2\\textrm {d}s \\\\& - A(t) \\int _0^t h(s)\\; g(s)\\;\\textrm {d}s + A(t) \\int _0^t h(s)^2\\textrm {d}s.\\end{split}$ The function $B(t)$ is solution of the following equation (after multiplying (REF ) by $g(s)$ and integrating) $\\begin{split}\\int _0^t g(s) \\; \\textrm {d}s & = B(t) - B(t) \\int _0^t h(s)\\; g(s)\\;\\textrm {d}s \\\\& - \\;A(t) \\int _0^t g(s)^2\\textrm {d}s + A(t) \\int _0^t h(s)\\; g(s)\\;\\textrm {d}s.\\end{split}$ Using the notation (REF ), we can write (REF )$-$ (REF ) as follows: $ A(t) - B(t)\\; I_4 - A(t) \\;I_2 + A(t)\\; I_4 = I_3$ and $ B(t) - B(t)\\; I_2 - A(t)\\; I_1 + A(t) \\; I_2 = I_5.$ Further, we have to find the expressions of $A(t)$ and $B(t).$ Therefore we obtain from (REF ) $B(t) = \\displaystyle \\frac{I_5 + A(t) \\left(I_1 -I_2 \\right)}{\\left(1-I_2\\right)}.$ Then we insert (REF ) in (REF ) and obtain $ A(t) = \\displaystyle \\frac{I_3 \\left(1- I_2\\right)+I_4\\;I_5}{\\left(1-I_2\\right)^2+I_4\\;I_6}$ and $ B(t) = \\displaystyle \\frac{ I_5\\left(1+I_4-I_2\\right)+ I_3\\left(I_1 - I_2\\right)}{\\left(1-I_2\\right)^2+I_4\\;I_6},$ where $I_6 =\\displaystyle \\int _t^1 g(s)^2\\;\\textrm {d}s.$ Therefore the solution $q(t,s)$ of (REF ) is $q(t,s)&= 1+\\displaystyle \\frac{ I_5\\left(1+I_4-I_2\\right)+ I_3\\left(I_1 - I_2\\right)}{\\left(1-I_2\\right)^2+I_4\\;I_6}\\; h(s)\\\\& \\qquad + \\displaystyle \\frac{I_3 \\left(1- I_2\\right)+I_4\\;I_5}{\\left(1-I_2\\right)^2+I_4\\;I_6}\\;\\left(g(s) - h(s)\\right).$ The final expression of $q(t,s)$ is $&q(t,s)= 1+\\displaystyle \\frac{\\displaystyle \\int _0^tg(s)\\textrm {d}s\\left(1+\\displaystyle \\int _0^th(s)^2\\textrm {d}s-\\displaystyle \\int _0^th(s)g(s)\\;\\textrm {d}s\\right)h(s)}{\\left(1-\\displaystyle \\int _0^th(s)g(s)\\;\\textrm {d}s\\right)^2+\\displaystyle \\int _0^th(s)^2\\textrm {d}s\\displaystyle \\int _t^1g(s)^2\\textrm {d}s} \\\\& \\qquad \\qquad \\;\\;+ \\displaystyle \\frac{\\displaystyle \\int _0^th(s)\\textrm {d}s\\left(\\displaystyle \\int _0^tg(s)^2\\textrm {d}s -\\displaystyle \\int _0^th(s)g(s)\\;\\textrm {d}s\\right)h(s)}{\\left(1-\\displaystyle \\int _0^th(s)g(s)\\;\\textrm {d}s\\right)^2+\\displaystyle \\int _0^th(s)^2\\textrm {d}s\\displaystyle \\int _t^1g(s)^2\\textrm {d}s} \\\\& \\quad + \\displaystyle \\frac{\\displaystyle \\int _0^th(s)\\textrm {d}s \\left(1- \\displaystyle \\int _0^th(s)g(s)\\textrm {d}s\\right)+\\displaystyle \\int _0^th(s)^2\\textrm {d}s\\int _0^tg(s)\\textrm {d}s}{\\left(1-\\displaystyle \\int _0^th(s)g(s)\\textrm {d}s\\right)^2+\\displaystyle \\int _0^th(s)^2\\textrm {d}s\\displaystyle \\int _t^1g(s)^2\\textrm {d}s}(g(s) -h(s))$ Step 4: To show that $M_t$ is martingale we need the following Lemma.", "Lemma 2 We have the following equality $\\displaystyle \\int _0^t q(t,s)\\;\\textrm {d}s = \\displaystyle \\int _0^t q(s,s)^2\\textrm {d}s, \\qquad 0\\le t \\le 1.$ Proof.", "Show that $\\frac{\\textrm {d}}{\\textrm {d}t}\\int _0^t q(t,s)\\;\\textrm {d}s = \\frac{\\textrm {d}}{\\textrm {d}t}\\int _0^t q(s,s)^2\\textrm {d}s = q(t,t)^2.$ Denote $C(t) & = 1 - \\displaystyle \\int _0^t h(s)\\; g(s)\\;\\textrm {d}s,\\\\D(t) & =\\displaystyle \\int _0^tg(s)\\;\\textrm {d}s\\left(1+\\displaystyle \\int _0^th(s)^2\\textrm {d}s-\\displaystyle \\int _0^th(s)\\;g(s)\\;\\textrm {d}s\\right)\\\\& \\quad + \\displaystyle \\int _0^th(s)\\;\\textrm {d}s\\left(\\displaystyle \\int _0^tg(s)^2\\textrm {d}s -\\displaystyle \\int _0^th(s)\\;g(s)\\;\\textrm {d}s\\right),\\\\K(t) & = C(t)^2 + \\displaystyle \\int _0^th(s)^2\\textrm {d}s\\displaystyle \\int _t^1 g(s)^2\\textrm {d}s \\\\\\textrm {and}\\qquad \\qquad \\qquad \\;\\\\N(t) & = C(t)\\; \\int _0^t h(s)\\; \\textrm {d}s + \\displaystyle \\int _0^th(s)^2\\textrm {d}s\\displaystyle \\int _0^t g(s)\\;\\textrm {d}s.$ Then $q(t,s)$ has the following expression $q(t,s) = 1+\\frac{D(t)}{K(t)}\\;h(s) + \\frac{N(t)}{K(t)}\\;(g(s) - h(s)).$ Hence $\\begin{split}\\frac{\\textrm {d}}{\\textrm {d}t}\\displaystyle \\int _0^t q(t,s)\\;\\textrm {d}s & = 1+\\frac{D(t)\\;h(t) +N(t)\\;(g(t) - h(t))}{K(t)} \\\\& \\quad +\\frac{\\Big (D^{\\prime }(t)\\;K(t) -D(t)\\;K^{\\prime }(t)\\Big )}{K(t)^2}\\int _0^th(s)\\;\\textrm {d}s\\\\& \\quad + \\frac{\\big (N^{\\prime }(t)K(t) -N(t)K^{\\prime }(t)\\big )}{K(t)^2}\\int _0^t(g(s) - h(s))\\;\\textrm {d}s.\\end{split}$ Then we obtain the equalities $C(t) & =& 1 - I_2,\\\\D(t) & =& I_5\\left(1+I_4-I_2\\right)+ I_3\\left(I_1 - I_2\\right),\\\\K(t) & =& \\left(1 - I_2\\right)^2 + I_4 \\left(1-I_1\\right),\\\\N(t) & =& I_3 \\left(1 - I_2\\right)+ I_4 \\;I_5.$ The derivatives of these functions w.r.t.", "$t$ have such expressions $C^{\\prime }(t) & = & - h(t)\\; g(t),\\\\D^{\\prime }(t) & = &g(t) + I_4 \\; g(t) - I_2\\; g(t) + I_5 \\;h(t)^2 - I_5 \\;h(t)\\;g(t) \\\\& & \\qquad + \\; I_1 \\;h(t) - I_2 \\;h(t)+I_3 \\;g(t)^2 - I_3\\;h(t)\\;g(t),\\\\K^{\\prime }(t) & = & -2 h(t) \\;g(t) + 2I_2 \\;h(t) g(t) + h(t)^2 - I_1 \\;h(t)^2 - I_4 \\;g(t)^2,\\\\N^{\\prime }(t) & = & -I_3\\;h(t)\\; g(t) + I_4\\;g(t) + I_5 \\;h(t)^2 + h(t) - I_2 \\;h(t).$ Thus, (REF ) has the following representation $\\frac{\\textrm {d}}{\\textrm {d}t}\\displaystyle \\int _0^t q(t,s)\\;\\textrm {d}s = 1+\\frac{\\Phi _1(t)+\\Phi _2(t)+\\Phi _3(t)}{K(t)^2},$ where $\\Phi _1(t) &= \\big (D(t)\\;h(t) +N(t)\\;(g(t) - h(t))\\big )K(t),\\\\\\Phi _2(t) & = \\big (D^{\\prime }(t)\\;K(t) -D(t)\\;K^{\\prime }(t)\\big )I_3$ and $\\Phi _3(t) = \\big (N^{\\prime }(t)\\;K(t) - N(t)\\;K^{\\prime }(t)\\big )\\left(I_5- I_3\\right).$ Then returning to initial notation we obtain the expressions $\\Phi _1(t) & = 3I_2^2I_5h-3I_2I_5h+I_1I_3h - 3I_2I_3g+ I_1I_2^2I_3h-I_2^3I_5h + I_3g - I_2^3I_3g\\\\& \\; + I_4I_5g-I_1I_3I_4g-I_3h-I_1^2I_3I_4h +2I_2I_3h+ I_5h- I_1I_4^2I_5g- I_2I_3I_4g \\\\& \\; - 2I_1I_2I_3h+3I_2^2I_3g- 2I_2I_4I_5g + I_4I_5h-I_2I_4I_5h+I_1I_2I_3I_4g+I_4^2I_5g\\\\& \\; + I_2^2I_4I_5g - I_2^2I_3h+ 2I_1I_3I_4h- I_3I_4h - I_1I_4I_5h + I_3I_4g+ I_1I_2I_4I_5h, \\\\\\Phi _2(t) & = I_2I_3^2h^2+ 2I_3I_4g-3I_2I_3g + I_3I_5hg+I_3I_4^2I_5g^2+I_3^2I_4g^2+ I_1I_3h\\\\& \\; - 3I_2I_3I_4g+3I_2^2I_3g-I_2I_3I_5h^2-2I_2I_3I_5hg+I_3^2g^2-I_1I_3^2h^2\\\\& \\; + I_2^2I_3I_4g-I_2^3I_3g+I_2^2I_3I_5h^2+I_2^2I_3I_5hg+I_1I_2^2I_3h-2I_1I_2I_3h\\\\& \\; - I_3^2hg-2I_2I_3^2g^2+I_2^2I_3^2g^2+2I_2^2I_3h-2I_2I_3I_4I_5hg-I_1^2I_3I_4h\\\\& \\;+ I_2^2I_3^2hg+I_3I_4^2g+I_3I_4I_5hg-2I_1I_2I_3^2hg+I_1^2I_3^2h^2- I_1I_2I_3^2h^2\\\\& \\; - I_1I_3I_4g-I_1I_3I_4^2g+I_1I_2I_3I_4g+I_1I_3I_4I_5hg-I_1I_2I_3I_5h^2+I_3g\\\\& \\; + I_1I_3^2I_4hg + I_1I_3I_5h^2+I_3I_4I_5g^2+I_1I_3I_4h-I_2I_3h-I_2I_3^2I_4g^2 \\\\& \\; - I_2I_3I_4I_5g^2 + 2I_1I_3^2hg-I_3^2I_4hg+I_1I_2I_3I_4h-I_2I_3I_4h-I_2^3I_3h$ and $\\Phi _3(t) & = I_3I_5hg+ I_2I_3I_4h-I_3I_5h^2 -3I_2I_5h-I_3I_4^2g-I_2^2I_3^2hg\\\\& \\;- 2 I_2I_5^2h^2 + 3I_2^2I_5h+I_2^2I_3I_5hg- I_2^2I_3I_5h^2+I_2^2I_5^2h^2-I_3I_4^2I_5g^2\\\\& \\; + I_4^2I_5g-3I_2^2I_3h - I_2I_4I_5h+I_1I_3I_4I_5hg- I_3^2hg -I_1I_4^2I_5g\\\\& \\; +I_2I_3^2I_4g^2 +I_1I_3I_5h^2+I_3I_4I_5g^2 + I_2I_3I_5h^2+ I_5h-2I_2I_3I_5hg \\\\& \\; + 2I_4I_5^2hg - 2I_2I_4I_5^2hg+I_4^2I_5^2g^2+I_2^2I_4I_5g-I_3h+ I_1I_2I_4I_5h \\\\& \\; + 3I_2I_3h+2I_2I_3^2hg+2I_2I_3I_4g + 2I_2I_3I_5h^2+I_5^2h^2- I_1I_4I_5h \\\\& \\; - I_2^2I_3I_4g +I_2^3I_3h + I_3^2I_4hg+I_4I_5h-I_2^3I_5h+I_3^2h^2+I_4I_5g\\\\& \\; - I_1I_3^2I_4hg+ I_1I_2I_3^2h^2 + I_1I_3I_4^2g+I_1I_3I_4h-I_3I_4h-I_3I_4g \\\\& \\; - I_3^2I_4g^2 - I_2I_3^2h^2 -I_3I_5h^2-I_1I_2I_3I_4h -I_2I_3I_4I_5g^2- I_1I_3^2h^2\\\\& \\; - 2I_3I_4I_5hg + 2I_2I_3I_4I_5hg-2I_2I_4I_5g- I_1I_2I_3I_5h^2- I_3I_4I_5hg .$ Denote $\\Phi (t) = \\Phi _1(t)+\\Phi _2(t)+\\Phi _3(t),$ then we can write $\\Phi (t) & = h(t)\\;\\Big (2I_5-6I_2I_5+2I_1I_3-2I_3+6I_2^2I_5-4I_1I_2I_3+4I_2I_3+ 4I_1I_3I_4\\\\& \\;-2I_2^3I_5+2I_1I_2^2I_3+ 2I_4I_5 - 2I_2I_4I_5 - 2I_3I_4+2I_1I_2I_4I_5-2I_1^2I_3I_4\\\\& \\;-2I_1I_4I_5-2I_2^2I_3\\Big )+\\;g(t)\\;\\Big (2I_3- 2I_1I_3I_4+2I_1I_2I_3I_4- 6I_2I_3+2I_4I_5\\\\& \\;-2I_1I_4^2I_5 + 6I_2^2I_3 - 4I_2I_4I_5- 2I_2^3I_3+2I_2^2I_4I_5+2I_3I_4+2I_4^2I_5 \\\\& \\;- 2I_2I_3I_4\\Big )+ h(t)g(t)\\Big (2I_3I_5 - 2I_3^2-4I_2I_3I_5- 2I_3I_4I_5 + 2I_1I_3I_4I_5\\\\& \\;+2I_2I_3^2+2I_4I_5^2+2I_1I_3^2-2I_1I_2I_3^2 - 2I_2I_4I_5^2+2I_2^2I_3I_5\\Big )+ h(t)^2\\\\&\\;\\Big (2I_1I_3I_5- 2I_1I_2I_3I_5 - 2I_2I_5^2+I_2^2I_5^2-2I_3I_5 + 2I_2I_3I_5-2I_1I_3^2+I_3^2\\\\& \\;+I_5^2+ I_1^2I_3^2\\Big )+ g(t)^2\\Big (I_3^2-2I_2I_3^2+2I_3I_4I_5-2I_2I_3I_4I_5+I_4^2I_5^2+I_2^2I_3^2\\Big ).$ Finally, (REF ) can be written as follows: $\\frac{\\textrm {d}}{\\textrm {d}t}\\displaystyle \\int _0^t q(t,s)\\;\\textrm {d}s = 1+\\frac{\\Phi (t)}{K(t)^2}.$ Now the expression of $q(t,t)^2$ is $q(t,t)^2& = &\\left(1+\\frac{D(t)}{K(t)}\\;h(t) +\\frac{N(t)}{K(t)}\\;(g(t) - h(t))\\right)^2\\\\& = & 1+ \\frac{2\\;h(t)\\;K(t)\\big (D(t) - N(t)\\big )\\; + 2 \\;g(t)\\; K(t)\\;N(t)}{K(t)^2}\\\\& & \\quad +\\; \\frac{h(t)^2\\big (D(t) -N(t)\\big )^2+ g(t)^2 N(t)^2}{K(t)^2}\\\\& & \\quad +\\;\\frac{2\\;h(t)\\;g(t) N(t) \\big (D(t) - N(t)\\big )}{K(t)^2}.$ Denote $M(t) & = & 2\\;K(t)\\left(D(t) - N(t)\\right), \\\\Q(t) & = & 2\\; K(t)\\;N(t),\\\\L(t) & = & \\big (D(t) -N(t)\\big )^2, \\\\E(t) & = & N(t)^2,\\\\H(t) & = & 2\\;N(t) \\big (D(t)-N(t)\\big ).$ Therefore the final expression for $q(t,t)^2$ is $\\begin{split}q(t,t)^2 & = 1+ \\frac{h(t)\\;M(t) + g(t)\\; Q(t)+h(t)^2L(t) }{K(t)^2}\\\\& \\quad +\\;\\frac{g(t)^2 E(t)+ h(t)\\;g(t) \\;H(t)}{K(t)^2},\\end{split}$ where $M(t) & = 2I_1I_3-6I_2I_5-2I_3+6I_2^2I_5-4I_1I_2I_3+4I_2I_3-2I_2^3I_5+ 2I_4I_5\\\\& \\; +2I_1I_2^2I_3-2I_1^2I_3I_4 - 2I_2^2I_3 -2I_2I_4I_5+2I_5-2I_3I_4-2I_1I_4I_5\\\\& \\; + 2I_1I_2I_4I_5+4I_1I_3I_4,\\\\Q(t) & = 2I_3I_4 -6I_2I_3 + 6I_2^2I_3 -2I_2^3I_3 + 2I_4I_5 - 4I_2I_4I_5 + 2I_2^2I_4I_5\\\\& \\; + 2I_3-2I_2I_3I_4- 2I_1I_3I_4+2I_1I_2I_3I_4+2I_4^2I_5-2I_1I_4^2I_5,\\\\L(t) & = I_5^2+I_2^2I_5^2-2I_2I_5^2+I_1^2I_3^2+I_3^2-2I_1I_3^2+2I_1I_3I_5-2I_3I_5\\\\& \\; + 2I_2I_3I_5-2I_1I_2I_3I_5, \\\\E(t) & = I_3^2-2I_2I_3^2+I_2^2I_3^2+I_4^2I_5^2+2I_3I_4I_5-2I_2I_3I_4I_5,\\\\H(t) & = 2I_3I_5-4I_2I_3I_5-2I_3I_4I_5+2I_1I_3^2+2I_2I_3^2+2I_2^2I_3I_5\\\\& \\; - 2I_2I_4I_5^2 + 2I_1I_3I_4I_5-2I_3^2-2I_1I_2I_3^2+2I_4I_5^2.$ The comparison of these expressions with (REF )$-$ (REF ) shows that the Lemma is proved.", "Step 5: In the next step, we need the following Lemma to show that the linear transformation is a Wiener process.", "Lemma 3 If the Gaussian process $M_s$ satisfies (REF ) and we have relation (REF ), then $z(t) = \\displaystyle \\int _0^t q(s,s)^{-1}\\;dM_s$ is a Wiener process.", "Proof.", "The proof can be found, e.g., in [7], Lemma 2.", "Hence $M_t = \\int _0^t q(s,s) \\;\\textrm {d}w_s,\\qquad 0\\le t\\le 1$ is a Gaussian martingale, where $w_s,0\\le s \\le 1$ is a Wiener process.", "Therefore we have the equality $\\textit {w}_t = \\displaystyle \\int _0^t q(s,s)^{-1}\\;\\textrm {d}M_s =U(t)+\\displaystyle \\int _0^t q(s,s)^{-1}\\int _0^s q_s^{\\prime }(s,v)\\;\\textrm {d}U(v)\\;\\textrm {d}s,$ where $w_t, 0 \\le t \\le 1$ is a Wiener process (by Lemma 3).", "Now we have to calculate the right side of the above expression.", "The derivative $q_t^{\\prime }(t,s)$ w.r.t.", "$t$ can be written as follows: $q_t^{\\prime }(t,s) = \\frac{\\left(\\psi _1(t) - \\psi _2(t)\\right)h(s)+ \\psi _2(t)\\; g(s)}{K(t)^2},$ where $\\psi _1(t) = D^{\\prime }(t)\\;K(t) -D(t)\\;K^{\\prime }(t)$ and $\\psi _2(t) = N^{\\prime }(t)\\;K(t) - N(t)\\;K^{\\prime }(t).$ Returning to initial notation we obtain the following expression: $\\psi _1(t) & = g+2I_4g-3I_2g+I_5hg+I_1h-I_2h+I_1I_4I_5hg+I_4^2g+3I_2^2g\\\\& \\quad -\\; I_2I_5h^2 -2I_2I_5hg-2I_1I_2h+2I_2^2h+I_2^2I_3g^2-I_1I_2I_3h^2-I_2^3g\\\\& \\quad +\\; I_2^2I_5hg+I_1I_2^2h-I_2^3h-2I_2I_4I_5hg+I_1^2I_3h^2-I_1I_2I_5h^2-I_3hg\\\\& \\quad -\\; I_2I_4h+I_3I_4g^2-I_3I_4hg-I_1I_4g-I_1I_4^2g+I_1I_2I_4g-2I_2I_3g^2\\\\& \\quad +\\; I_1I_2I_4h+I_1I_5h^2+I_4I_5g^2+I_4^2g+I_2^2I_3hg-I_1I_3h^2+I_4^2I_5g^2\\\\& \\quad -\\; I_2I_4I_5g^2+2I_1I_3hg-2I_1I_2I_3hg+I_2^2I_4g-I_1^2I_4h+I_2I_3h^2\\\\& \\quad -\\; I_2I_3I_4g^2-3I_2I_4g+I_2^2I_5h^2+I_1I_4h+I_1I_3I_4hg+I_4I_5hg+I_3g^2.$ The function $\\psi _2(t)$ is defined by (REF ).", "Hence we obtain $\\frac{q_t^{\\prime }(t,s)}{q(t,t)} = \\frac{\\left(\\psi _1(t) - \\psi _2(t)\\right)\\;h(s)+ \\psi _2(t)\\; g(s)}{K(t)^2+ \\Phi _1(t)}.$ Then if we put $\\varphi _1(t) = \\psi _1(t) - \\psi _2(t)$ and $\\varphi _2(t) = K(t)^2+ \\Phi _1(t),$ then this implies that $\\frac{q_t^{\\prime }(t,s)}{q(t,t)} = \\frac{\\varphi _1(t)\\;h(s)+ \\psi _2(t)\\; g(s)}{\\varphi _2(t)},$ with $\\varphi _1(t)$ and $\\varphi _2(t)$ were defined by (REF )$-$ (REF ).", "Finally, we obtain the expression $w_t = \\displaystyle \\int _0^t q(s,s)^{-1}\\textrm {d}M_s =U(t)+\\displaystyle \\int _0^t \\int _0^s \\frac{\\varphi _1(s)\\;h(v)+ \\psi _2(s)\\;g(v)}{\\varphi _2(s)}\\;\\textrm {d}U(v)\\;\\textrm {d}s.$ This is the linear transformation $w_t = L[U](t)$ of the process $U(\\cdot )$ into the Wiener process $w_t$ and this proves the Theorem REF .", "Remark.", "Let us present a sufficient condition for $\\varphi _2(t)>0$ .", "$\\mathcal {R}_0.$ Suppose that $h(t)$ and $g(t)$ are continuous strictly positive functions such that $g(t)>h(t)$ and $\\int _0^th(s)\\;g(s)\\;\\textrm {d}s <1,\\qquad \\int _0^1 g(s)^2 \\textrm {d}s = 1, \\qquad 0\\le t <1,$ then $\\varphi _2(t)$ defined by (REF ) is strictly positive function on $[0,1)$ .", "Now we will verify that if the condition $\\mathcal {R}_0$ is satisfied, then $\\varphi _2(t)$ is strictly positive function.", "Remind that $\\varphi _2(t)$ has the following expression: $\\varphi _2(t) = K(t)^2+ \\Phi _1(t),\\qquad \\qquad 0\\le t <1,$ where $\\Phi _1(t) = \\big (D(t)\\;h(t) +N(t)\\;(g(t) - h(t))\\big )K(t),$ with $D(t) & =\\displaystyle \\int _0^tg(s)\\;\\textrm {d}s\\left(1+\\displaystyle \\int _0^th(s)^2\\textrm {d}s-\\displaystyle \\int _0^th(s)\\;g(s)\\;\\textrm {d}s\\right)\\\\& \\quad + \\displaystyle \\int _0^th(s)\\;\\textrm {d}s\\left(\\displaystyle \\int _0^tg(s)^2\\textrm {d}s -\\displaystyle \\int _0^th(s)\\;g(s)\\;\\textrm {d}s\\right),$ $N(t) = C(t)\\; \\int _0^t h(s)\\; \\textrm {d}s + \\displaystyle \\int _0^th(s)^2\\textrm {d}s\\displaystyle \\int _0^t g(s)\\;\\textrm {d}s$ and $K(t) = C(t)^2 + \\displaystyle \\int _0^th(s)^2\\textrm {d}s\\displaystyle \\int _t^1 g(s)^2\\textrm {d}s.$ Here $C(t)= 1 - \\displaystyle \\int _0^t h(s)\\; g(s)\\;\\textrm {d}s.$ Note that it is sufficient to check that $\\Phi _1(t) >0$ to obtain $\\varphi _2(t)>0$ .", "Recall that $K(t)$ is strictly positive function.", "Then due to the following condition $\\displaystyle \\int _0^th(s)\\;g(s)\\;\\textrm {d}s <1,$ we have $C(t)>0$ .", "Consequently, we obtain $N(t)>0$ by the conditions $h(t)>0 \\qquad \\textrm {and} \\qquad g(t)>0.$ Finally, we see that $D(t)>0$ by the conditions $\\displaystyle \\int _0^1 g(s)^2 \\textrm {d}s = 1$ , (REF ) and (REF ).", "We conclude that we have $\\varphi _2(t)>0$ if we suppose that $g(t)>h(t)$ ." ], [ "Test", "Our objective is to test the composite parametric hypothesis ${\\cal H}_0$ and to do this we will propose a statistic based on the MDE $\\vartheta _\\varepsilon ^*$ .", "Recall that the starting statistic $u_\\varepsilon \\left(t\\right)=\\frac{X_t-x_t\\left(\\vartheta _\\varepsilon ^*\\right)}{\\varepsilon \\;S\\left(\\vartheta _\\varepsilon ^*,X_t \\right)}$ converges to the random function $u\\left(t\\right)=\\int _{0}^{t}\\frac{{\\rm d}W_s}{S(\\vartheta ,x_s)}-\\int _0^T\\int _v^T\\frac{S(\\vartheta ,x_s)\\;\\dot{x}_s(\\vartheta )}{ J(\\vartheta ) S\\left(\\vartheta ,x_v\\right) }\\;\\textrm {d}s\\;\\textrm {d}W_v\\int _0^t\\frac{\\dot{S}(\\vartheta ,x_s)}{S(\\vartheta ,x_s)}\\;\\textrm {d}s .$ Then the linear transformation $U\\left(\\frac{t}{T}\\right)=\\frac{1}{\\sqrt{T}}\\int _{0}^{t}S\\left(\\vartheta ,x_s\\right){\\rm d}u\\left(s\\right)$ has the following representation, by Itô formula, $&U\\left(\\frac{t}{T}\\right) = \\frac{1}{\\sqrt{T}}\\;S(\\vartheta ,x_t)\\;u(t)- \\frac{1}{\\sqrt{T}}\\int _0^tS^{\\prime }(\\vartheta ,x_s)\\;S(\\vartheta ,x_s)\\;u(s)\\;\\textrm {d}s$ which leads to the random function $U\\left(\\nu \\right)=W\\left(\\nu \\right)-\\int _{0}^{1}g\\left(\\vartheta ,r\\right){\\rm d}W\\left(r\\right)\\int _{0}^{\\nu }h\\left(\\vartheta ,r\\right){\\rm d}r.$ The last step is to apply the transformation $L\\left[U\\right](\\cdot )$ from Theorem REF and to obtain the Wiener process $L\\left[U\\right]\\left(\\nu \\right)=w_\\nu ,\\qquad 0\\le \\nu \\le 1.$ Now we have to realize the similar transformations with the “empirical” process $u_\\varepsilon \\left(\\cdot \\right)$ defined by (REF ), i.e., we (formally) calculate $U_\\varepsilon \\left(\\frac{t}{T}\\right)=\\frac{1}{\\sqrt{T}}\\int _{0}^{t}S\\left(\\vartheta _\\varepsilon ^*,X_s \\right)\\;{\\rm d}u_\\varepsilon \\left(s\\right).$ Then we apply the transformation $L\\left[\\cdot \\right]$ to the process $U_\\varepsilon \\left(\\cdot \\right) $ and we show that this statistic converges in distribution to the Wiener process $w_\\nu , 0\\le \\nu \\le 1$ .", "Therefore the test $\\psi _\\varepsilon =\\mbox{1\\hspace{-2.5pt}I}_{\\left\\lbrace \\Delta _\\varepsilon >c_\\alpha \\right\\rbrace }$ with $\\Delta _\\varepsilon =\\frac{1}{T}\\int _{0}^{T}L\\left[U_\\varepsilon \\right]\\left(t \\right)^2{\\rm d}t \\Longrightarrow \\int _{0}^{1}w_\\nu ^2\\;{\\rm d}\\nu $ will be ADF because the limit distribution of $\\Delta _\\varepsilon $ does not depend on $S\\left(\\cdot ,\\cdot \\right)$ and $\\vartheta $ .", "Let us realize this program.", "We have the following representation for the process $U_{\\varepsilon }(\\cdot )$ $\\begin{split}&U_{\\varepsilon }\\left(\\frac{t}{T}\\right) = \\frac{1}{\\sqrt{T}}\\;S\\left(\\vartheta ^{*}_{\\varepsilon },X_t\\right)\\;u_\\varepsilon \\left(t\\right)\\\\&\\qquad \\qquad \\qquad - \\frac{1}{\\sqrt{T}}\\int _0^tS^{\\prime }\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)\\;S\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)\\;u_\\varepsilon \\left(s\\right)\\;\\textrm {d}s.\\end{split}$ Introduce the functions $\\hat{h}\\left(\\vartheta ,v\\right)=\\frac{1}{\\sqrt{T}}\\;J(\\vartheta )^{-1}\\dot{S}(\\vartheta ,x_v), \\qquad J(\\vartheta )=\\int _0^T\\dot{x}_s(\\vartheta )^2\\textrm {d}s$ and $\\hat{g}\\left(\\vartheta ,v\\right)=S(\\vartheta ,x_v)^{-1} I\\left(\\vartheta ,v\\right), \\qquad I\\left(\\vartheta ,v\\right)=\\int _v^TS(\\vartheta ,x_s)\\;\\dot{x}_{s}(\\vartheta )\\;\\textrm {d}s$ and their “empirical versions”, respectively $& h_{\\varepsilon }\\left(\\vartheta ^{*}_{\\varepsilon },v\\right) =\\frac{1}{\\sqrt{T}}\\;J_\\varepsilon (\\vartheta ^{*}_{\\varepsilon })^{-1}\\;\\dot{S}(\\vartheta ^*_{\\varepsilon },X_v),\\qquad g_{\\varepsilon }\\left(\\vartheta ^{*}_{\\varepsilon },v\\right)=S(\\vartheta ^{*}_{\\varepsilon },X_v)^{-1}I_{\\varepsilon }(\\vartheta ^{*}_{\\varepsilon },v).$ Here $&J_\\varepsilon (\\vartheta ^{*}_{\\varepsilon })=\\int _0^T\\dot{x}_s(\\vartheta ^{*}_{\\varepsilon })^2\\textrm {d}s,\\qquad I_{\\varepsilon }(\\vartheta ^{*}_{\\varepsilon },v)=\\int _v^TS(\\vartheta ^{*}_{\\varepsilon },X_s)\\;\\dot{x}_s(\\vartheta ^{*}_{\\varepsilon })\\;\\textrm {d}s.$ Note that in the functions $\\hat{h}\\left(\\cdot ,\\cdot \\right)$ and $\\hat{g}\\left(\\cdot ,\\cdot \\right)$ we omit the normalizing constants in the expressions of the functions $h\\left(\\cdot ,\\cdot \\right)$ and $g\\left(\\cdot ,\\cdot \\right)$ defined by (REF )$-$ (REF ), for simplicity of exposition, because the structure of the used statistic is such that we can do this without changing the limit distribution of the statistic.", "Then denote the “empirical versions” $I_{1,\\varepsilon } =C_1\\left(T\\right) \\int _0^sg_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },v\\right)^2\\textrm {d}v,\\quad I_{2,\\varepsilon } =C_2\\left(T\\right) \\int _0^sh_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },v\\right)g_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },v\\right)\\;\\textrm {d}v,$ $I_{3,\\varepsilon } =C_3\\left(T\\right)\\int _0^sh_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },v\\right)\\;\\textrm {d}v,\\quad I_{4,\\varepsilon } =C_4\\left(T\\right)\\int _0^sh_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },v\\right)^2\\textrm {d}v$ and $I_{5,\\varepsilon } =C_5\\left(T\\right)\\int _0^sg_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },v\\right)\\;\\textrm {d}v$ of the integrals $\\hat{I}_1 = C_1\\left(T\\right)\\int _0^s\\hat{g}\\left(\\vartheta ,v\\right)^2\\textrm {d}v,\\quad \\hat{I}_2 = C_2\\left(T\\right)\\int _0^s\\hat{h}\\left(\\vartheta ,v\\right)\\hat{g}\\left(\\vartheta ,v\\right)\\;\\textrm {d}v,$ $\\hat{I}_3 =C_3\\left(T\\right)\\int _0^s\\hat{h}\\left(\\vartheta ,v\\right)\\;\\textrm {d}v,\\quad \\hat{I}_4 =C_4\\left(T\\right)\\int _0^s\\hat{h}\\left(\\vartheta ,v\\right)^2\\textrm {d}v$ and $\\hat{I}_5 =C_5\\left(T\\right)\\int _0^s\\hat{g}\\left(\\vartheta ,v\\right)\\;\\textrm {d}v,$ where $C_1\\left(T\\right)=\\displaystyle \\frac{1}{T^3}$ , $C_2\\left(T\\right)=\\sqrt{T}$ , $C_3\\left(T\\right)=T\\sqrt{T}$ , $C_4\\left(T\\right)=T^4$ and $C_5\\left(T\\right)=\\displaystyle \\frac{1}{T^2}$ .", "This allows us to introduce the “empirical versions” $\\varphi _{1,\\varepsilon }(\\cdot ), \\varphi _{2,\\varepsilon }(\\cdot )$ and $\\psi _{2,\\varepsilon }(\\cdot )$ of $\\hat{\\varphi }_{1}(\\cdot ), \\hat{\\varphi }_{2}(\\cdot )$ and $\\hat{\\psi }_2(\\cdot )$ defined respectively by (REF )$-$ (REF ), where we replace the functions $g$ by $\\hat{g}=\\displaystyle \\;\\frac{1}{T}\\;\\hat{g}\\left(\\vartheta ,s\\right)$ and $h$ by $\\hat{h}=T^2\\sqrt{T}\\;\\hat{h}\\left(\\vartheta ,s\\right)$ , $\\varphi _{1,\\varepsilon }(s) & =g_{\\varepsilon }-h_{\\varepsilon }+I_{4,\\varepsilon }g_{\\varepsilon }-3I_{2,\\varepsilon }g_{\\varepsilon }+I_{5,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }+I_{1,\\varepsilon }h_{\\varepsilon }+2I_{2,\\varepsilon }h_{\\varepsilon }+I_{2,\\varepsilon }^2I_{3,\\varepsilon }g_{\\varepsilon }^2 \\nonumber \\\\ &\\quad -2I_{2,\\varepsilon }I_{3,\\varepsilon }g_{\\varepsilon }^2-I_{5,\\varepsilon }h_{\\varepsilon }^2 + I_{3,\\varepsilon }g_{\\varepsilon }^2-I_{2,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }+3I_{2,\\varepsilon }^2g_{\\varepsilon }-2I_{3,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }\\nonumber \\\\ & \\quad -2I_{2,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }-I_{2,\\varepsilon }^3g_{\\varepsilon }+I_{2,\\varepsilon }^2I_{5,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }+I_{1,\\varepsilon }I_{2,\\varepsilon }^2h_{\\varepsilon }-I_{2,\\varepsilon }^2h_{\\varepsilon }\\nonumber \\\\ &\\quad +2I_{1,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }-I_{1,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }+ I_{1,\\varepsilon }^2I_{3,\\varepsilon }h_{\\varepsilon }^2+I_{1,\\varepsilon }I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }+I_{2,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }^2 \\nonumber \\\\ & \\quad +I_{1,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }^2+I_{4,\\varepsilon }I_{5,\\varepsilon }g_{\\varepsilon }^2-I_{1,\\varepsilon }I_{2,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }^2-I_{4,\\varepsilon }h_{\\varepsilon }-I_{2,\\varepsilon }I_{4,\\varepsilon }I_{5,\\varepsilon }g_{\\varepsilon }^2\\nonumber \\\\ & \\quad +2I_{1,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon } -2I_{1,\\varepsilon }I_{2,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }-2I_{1,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }^2-2I_{1,\\varepsilon }I_{2,\\varepsilon }h_{\\varepsilon }\\nonumber \\\\ &\\quad +I_{1,\\varepsilon }I_{2,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }+2I_{2,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }-I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon } g_{\\varepsilon }\\nonumber +I_{3,\\varepsilon }h_{\\varepsilon }^2-I_{1,\\varepsilon }^2I_{4,\\varepsilon }h_{\\varepsilon },\\\\\\varphi _{2,\\varepsilon }(s) & = 1 -2I_{1,\\varepsilon }I_{4,\\varepsilon }^2-3I_{2,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }+I_{1,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }+I_{3,\\varepsilon }g_{\\varepsilon }-3I_{2,\\varepsilon }I_{3,\\varepsilon }g_{\\varepsilon }\\nonumber \\\\ &\\quad +I_{4,\\varepsilon }I_{5,\\varepsilon }g_{\\varepsilon } -I_{3,\\varepsilon }h_{\\varepsilon }+3I_{2,\\varepsilon }^2I_{5,\\varepsilon }h_{\\varepsilon }-2I_{1,\\varepsilon }I_{2,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }-2I_{1,\\varepsilon }I_{2,\\varepsilon }^2I_{4,\\varepsilon }\\nonumber \\\\ & \\quad -2I_{2,\\varepsilon }I_{4,\\varepsilon }I_{5,\\varepsilon }g_{\\varepsilon }+2I_{2,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }-I_{2,\\varepsilon }^3I_{5,\\varepsilon }h_{\\varepsilon }+I_{1,\\varepsilon }I_{2,\\varepsilon }^2I_{3,\\varepsilon }h_{\\varepsilon }-2I_{1,\\varepsilon }I_{4,\\varepsilon }\\nonumber \\\\ &\\quad -I_{2,\\varepsilon }^3I_{3,\\varepsilon }g_{\\varepsilon } +I_{4,\\varepsilon }^2 +I_{2,\\varepsilon }^2I_{4,\\varepsilon }I_{5,\\varepsilon }g_{\\varepsilon }+I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }+I_{4,\\varepsilon }^2I_{5,\\varepsilon }g_{\\varepsilon }\\nonumber \\\\ &\\quad -I_{2,\\varepsilon }I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }+2I_{1,\\varepsilon }I_{3,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }+I_{3,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }+3I_{2,\\varepsilon }^2I_{3,\\varepsilon }g_{\\varepsilon }\\nonumber \\\\ & \\quad +2I_{2,\\varepsilon }^2I_{4,\\varepsilon } + 2I_{4,\\varepsilon }-I_{3,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }+6I_{2,\\varepsilon }^2+I_{1,\\varepsilon }I_{2,\\varepsilon }I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }\\nonumber \\\\ & \\quad -I_{1,\\varepsilon }^2I_{3,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }-I_{1,\\varepsilon }I_{3,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }+I_{5,\\varepsilon }h_{\\varepsilon }+I_{1,\\varepsilon }I_{2,\\varepsilon }I_{3,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }\\nonumber \\\\ & \\quad -I_{1,\\varepsilon }I_{4,\\varepsilon }^2I_{5,\\varepsilon }g_{\\varepsilon }-I_{1,\\varepsilon }I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }+4I_{1,\\varepsilon }I_{2,\\varepsilon }I_{4,\\varepsilon }-I_{2,\\varepsilon }I_{3,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }\\nonumber \\\\&\\quad -I_{2,\\varepsilon }^2I_{3,\\varepsilon }h_{\\varepsilon }+I_{2,\\varepsilon }^4-4I_{2,\\varepsilon }^3-4I_{2,\\varepsilon }-4I_{2,\\varepsilon }I_{4,\\varepsilon }+ I_{1,\\varepsilon }^2I_{4,\\varepsilon }^2$ and $\\psi _{2,\\varepsilon }(s) & = h_{\\varepsilon }+I_{3,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }+I_{4,\\varepsilon }g_{\\varepsilon }+I_{5,\\varepsilon }h_{\\varepsilon }^2-3I_{2,\\varepsilon }h_{\\varepsilon }-I_{1,\\varepsilon }I_{2,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }^2\\nonumber \\\\& \\quad -2I_{2,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }+I_{4,\\varepsilon }h_{\\varepsilon } -2I_{2,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }^2 + 3I_{2,\\varepsilon }^2h_{\\varepsilon }+ I_{2,\\varepsilon }^2I_{4,\\varepsilon }g_{\\varepsilon }\\nonumber \\\\& \\quad +I_{2,\\varepsilon }^2I_{5,\\varepsilon }h_{\\varepsilon }^2-I_{2,\\varepsilon }^3h_{\\varepsilon }-I_{3,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }+I_{4,\\varepsilon }^2g_{\\varepsilon } - I_{2,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }\\nonumber \\\\& \\quad +I_{1,\\varepsilon }I_{3,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }-I_{1,\\varepsilon }I_{4,\\varepsilon }^2g_{\\varepsilon } - I_{1,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }+I_{1,\\varepsilon }I_{2,\\varepsilon }I_{4,\\varepsilon }h_{\\varepsilon }\\nonumber \\\\& \\quad -I_{3,\\varepsilon }h_{\\varepsilon }^2 + I_{1,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }^2 + I_{3,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }^2+ I_{2,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }^2-2I_{2,\\varepsilon }I_{3,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }\\nonumber \\\\& \\quad - 2I_{2,\\varepsilon }I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }+2I_{4,\\varepsilon }I_{5,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }-I_{2,\\varepsilon }I_{3,\\varepsilon }I_{4,\\varepsilon }g_{\\varepsilon }^2+I_{2,\\varepsilon }^2I_{3,\\varepsilon }h_{\\varepsilon }g_{\\varepsilon }\\\\& \\quad + I_{4,\\varepsilon }^2I_{5,\\varepsilon }g_{\\varepsilon }^2.$ Here $g_{\\varepsilon }=g_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right),$ $h_{\\varepsilon }=h_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)$ and $I_{1,\\varepsilon }$ , $I_{2,\\varepsilon }$ , $I_{3,\\varepsilon }$ , $I_{4,\\varepsilon }$ , $I_{5,\\varepsilon }$ are the “empirical versions” of $\\hat{I}_1$ , $\\hat{I}_2$ , $\\hat{I}_3$ , $\\hat{I}_4$ , $\\hat{I}_5$ , respectively.", "In the construction of the test we introduce one condition else.", "$\\mathcal {R}_1.$ We suppose that $\\varphi _2\\left(r\\right),$ $r\\in [0,1)$ defined by (REF ) is strictly positive function.", "We have the uniform convergence in probability w.r.t.", "$s\\in [0,T]$ $(\\varepsilon \\rightarrow 0)$ $\\sup _{s\\in [0,T]}\\left|\\varphi _{2,\\varepsilon }(s)- \\hat{\\varphi }_2(s)\\right|\\longrightarrow 0.$ This convergence we obtain due to the consistency of the estimator and the smoothness of the functions $g_\\varepsilon \\left(\\cdot ,\\cdot \\right)$ and $h_\\varepsilon \\left(\\cdot ,\\cdot \\right)$ .", "Therefore we can introduce the function $\\varphi _{2,\\varepsilon }^+\\left(s\\right)=\\left\\lbrace \\begin{array}{ll}\\varphi _{2,\\varepsilon }\\left(s\\right)^{-1}, \\quad & \\textrm {if} \\;\\varphi _{2,\\varepsilon }\\left(s\\right)>0,\\\\0, & \\textrm {else},\\end{array}\\right.$ which asymptotically coincides with $\\hat{\\varphi }_2(s)^{-1}$ and therefore the limit distribution does not change.", "Hence we consider (formally) the statistic $&W_{\\varepsilon }(t) = U_{\\varepsilon }\\left(\\frac{t}{T}\\right)\\\\& +\\frac{1}{T}\\int _0^{t} \\int _{0}^s\\varphi _{2,\\varepsilon }^+(s)[\\lambda _1(T)\\varphi _{1,\\varepsilon }(s)h_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },q)+\\lambda _2(T)\\psi _{2,\\varepsilon }(s)g_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },q)]\\textrm {d}U_{\\varepsilon }\\left(\\frac{q}{T}\\right)\\textrm {d}s\\nonumber $ where $\\lambda _1(T)=T^2\\sqrt{T}, \\lambda _2(T)=\\displaystyle \\frac{1}{T}$ and $U_{\\varepsilon }(\\cdot )$ was defined by (REF ).", "If we prove that $W_{\\varepsilon }(t) \\longrightarrow L\\left[U\\right](\\nu )=w_\\nu ,$ then the test based on this statistic will be ADF.", "The main technical problem in carrying out this program is to define the stochastic integrals $K_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s) = \\int _0^sh_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },q\\right)\\;\\textrm {d}U_{\\varepsilon }\\left(\\frac{q}{T}\\right)$ and $L_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s) = \\int _0^sg_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },q\\right)\\;\\textrm {d}U_{\\varepsilon }\\left(\\frac{q}{T}\\right).$ Unfortunately we can not calculate them as they are written now, because the integrand contains the MDE $\\vartheta ^{*}_{\\varepsilon }$ and this estimator depends on the whole trajectory $X^\\varepsilon =\\left(X_t,0\\le t\\le T\\right)$ .", "Therefore the corresponding stochastic integrals $K_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)$ and $L_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)$ are not well defined.", "To avoid this problem we use an approach which is based on the application of the Itô formula, i.e., we replace the corresponding stochastic integrals by the ordinary ones.", "Note that this approach was applied in the similar problem in [12].", "Introduce the statistic $&K\\left(\\vartheta ,s\\right)=\\int _0^s\\hat{h}\\left(\\vartheta ,q\\right)\\;\\textrm {d}U\\left(\\frac{q}{T}\\right)=\\frac{1}{\\sqrt{T}}\\int _0^s\\hat{h}\\left(\\vartheta ,q\\right)\\;S\\left(\\vartheta ,x_q\\right)\\;\\textrm {d}u(q),$ where the process $U\\left(\\cdot \\right)$ and the function $\\hat{h}\\left(\\cdot ,\\cdot \\right)$ were defined by (REF ) and (REF ), respectively.", "Indeed the Itô formula gives us the following representation $&\\textrm {d}(\\hat{h}(\\vartheta ,q)S(\\vartheta ,x_q)u(q))=\\left(\\hat{h}^{\\prime }(\\vartheta ,q)S(\\vartheta ,x_q)+\\hat{h}(\\vartheta ,q)S^{\\prime }(\\vartheta ,x_q)S(\\vartheta ,x_q)\\right)u(q)\\textrm {d}q\\\\&\\qquad \\qquad \\qquad \\qquad \\quad \\qquad +\\hat{h}\\left(\\vartheta ,q\\right)S\\left(\\vartheta ,x_q\\right)\\textrm {d}u(q).$ Here $\\hat{h}^{\\prime }\\left(\\vartheta ,q\\right)$ is the derivative of $\\hat{h}\\left(\\vartheta ,q\\right)$ w.r.t.", "$q$ , given by the following expression $\\begin{split}\\hat{h}^{\\prime }\\left(\\vartheta ,q\\right)=\\frac{1}{\\sqrt{T}}\\;J(\\vartheta )^{-1}\\dot{S}^{\\prime }(\\vartheta ,x_q(\\vartheta ))\\;S\\left(\\vartheta ,x_q(\\vartheta )\\right).\\end{split}$ Therefore the statistic $K\\left(\\vartheta ,\\cdot \\right)$ defined by (REF ) can be written as follows: $&K\\left(\\vartheta ,s\\right)=\\frac{1}{\\sqrt{T}}\\;\\hat{h}\\left(\\vartheta ,s\\right)S\\left(\\vartheta ,x_s\\right)u(s)\\\\&\\qquad \\qquad -\\frac{1}{\\sqrt{T}}\\int _0^s\\left(\\hat{h}^{\\prime }\\left(\\vartheta ,q\\right)S\\left(\\vartheta ,x_q\\right)+\\hat{h}\\left(\\vartheta ,q\\right)S^{\\prime }\\left(\\vartheta ,x_q\\right)S\\left(\\vartheta ,x_q\\right)\\right)u(q)\\;\\textrm {d}q.$ Hence we obtain for the process (REF ) the following representation $\\begin{split}&K_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)=\\frac{1}{\\sqrt{T}}\\;h_\\varepsilon \\left(\\vartheta ^*_{\\varepsilon },s\\right)S\\left(\\vartheta ^*_{\\varepsilon },X_s\\right)u_\\varepsilon (s)\\\\&-\\frac{1}{\\sqrt{T}}\\int _0^s\\left(h_\\varepsilon ^{\\prime }(\\vartheta ^*_{\\varepsilon },q)S(\\vartheta ^*_{\\varepsilon },X_q)+h_\\varepsilon (\\vartheta ^*_{\\varepsilon },q)S^{\\prime }(\\vartheta ^*_{\\varepsilon },X_q)S(\\vartheta ^*_{\\varepsilon },X_q)\\right)u_\\varepsilon (q)\\;\\textrm {d}q\\end{split}$ and the integral is now well defined.", "Here $h^{\\prime }_\\varepsilon \\left(\\vartheta ^*_{\\varepsilon },q\\right)=\\frac{1}{\\sqrt{T}}\\;J_\\varepsilon (\\vartheta ^*_{\\varepsilon })^{-1}\\dot{S}^{\\prime }(\\vartheta ^*_{\\varepsilon },X_q)\\;S\\left(\\vartheta ^*_{\\varepsilon },X_q\\right).$ Similarly, the process (REF ) can be written as follows: $\\begin{split}&L_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)=\\frac{1}{\\sqrt{T}}\\;g_\\varepsilon \\left(\\vartheta ^*_{\\varepsilon },s\\right)S\\left(\\vartheta ^*_{\\varepsilon },X_s\\right)u_\\varepsilon (s)\\\\&-\\frac{1}{\\sqrt{T}}\\int _0^s\\left(g_\\varepsilon ^{\\prime }(\\vartheta ^*_{\\varepsilon },q)S(\\vartheta ^*_{\\varepsilon },X_q)+g_\\varepsilon (\\vartheta ^*_{\\varepsilon },q)S^{\\prime }(\\vartheta ^*_{\\varepsilon },X_q)S(\\vartheta ^*_{\\varepsilon },X_q)\\right)u_\\varepsilon (q)\\textrm {d}q,\\end{split}$ where $g^{\\prime }_\\varepsilon (\\vartheta ^*_{\\varepsilon },q)=-S(\\vartheta ^*_{\\varepsilon },X_q)^{-1}S^{\\prime }(\\vartheta ^*_{\\varepsilon },X_q)\\int _{q}^TS(\\vartheta ^*_{\\varepsilon },X_s)\\;\\dot{x}_s(\\vartheta ^*_{\\varepsilon })\\;\\textrm {d}s-\\dot{x}_q(\\vartheta ^*_{\\varepsilon })$ is the “empirical version” of $\\begin{split}&\\hat{g}^{\\prime }(\\vartheta ,q)=-S(\\vartheta ,x_q(\\vartheta ))^{-1}S^{\\prime }(\\vartheta ,x_q(\\vartheta ))\\int _{q}^TS(\\vartheta ,x_s(\\vartheta ))\\;\\dot{x}_s(\\vartheta )\\;\\textrm {d}s-\\dot{x}_q(\\vartheta ).\\end{split}$ Then the formal expression (REF ) for $W_\\varepsilon (t)$ can be replaced by $\\begin{split}&\\tilde{W}_{\\varepsilon }(t) = U_{\\varepsilon }\\left(\\frac{t}{T}\\right)\\\\& +\\frac{1}{T}\\displaystyle \\int _0^{t} \\varphi _{2,\\varepsilon }^+\\left(s\\right)\\left[\\lambda _1(T)\\;\\varphi _{1,\\varepsilon }(s)\\;K_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)+\\lambda _2(T)\\;\\psi _{2,\\varepsilon }(s)\\;L_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)\\right]\\;\\textrm {d}s,\\end{split}$ where $\\lambda _1(T)=T^2\\sqrt{T}, \\lambda _2(T)=\\displaystyle \\frac{1}{T}$ and the processes $U_{\\varepsilon }(\\cdot )$ , $K_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },\\cdot \\right)$ and $L_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },\\cdot \\right)$ admit the representations (REF ), (REF ) and (REF ), respectively.", "The test is given in the following Theorem.", "Theorem 2 Suppose that the conditions of regularity $\\mathcal {R}$ and $\\mathcal {R}_1$ are fulfilled, then the test $\\psi _{\\varepsilon } = \\mbox{1\\hspace{-2.5pt}I}_{\\left\\lbrace \\Delta _\\varepsilon > c_\\alpha \\right\\rbrace },\\qquad \\mathbf {P}\\left(\\tilde{\\Delta }> c_{\\alpha }\\right)=\\alpha ,$ with $\\Delta _{\\varepsilon } = \\frac{1}{T}\\int _0^T\\tilde{W}_{\\varepsilon }(t)^2 \\textrm {d}t,\\quad \\tilde{\\Delta } \\equiv \\int _0^1 w_{\\nu }^2\\;\\textrm {d}\\nu $ is ADF and of asymptotic size $\\alpha \\in (0,1)$ .", "Proof.", "We have to show that, under hypothesis $\\mathcal {H}_0$ , the convergence $\\Delta _{\\varepsilon } \\Longrightarrow \\tilde{\\Delta }$ holds.", "Recall that the process $U_{\\varepsilon }(\\cdot )$ has the following representation $&U_{\\varepsilon }\\left(\\frac{t}{T}\\right) = \\frac{1}{\\sqrt{T}}\\;S\\left(\\vartheta ^{*}_{\\varepsilon },X_t\\right)\\;u_\\varepsilon \\left(t\\right)\\\\&\\qquad \\qquad \\qquad - \\frac{1}{\\sqrt{T}}\\int _0^tS^{\\prime }\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)\\;S\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)\\;u_\\varepsilon \\left(s\\right)\\;\\textrm {d}s.$ Note that we have already the convergence in probability uniformly w.r.t.", "$s\\in [0,T]$ (as $\\varepsilon \\rightarrow 0$ ) $\\sup _{s\\in [0,T]}\\left|X_s- x_s(\\vartheta )\\right|\\longrightarrow 0, \\qquad \\sup _{s\\in [0,T]}\\left|u_\\varepsilon \\left(s\\right)- u\\left(s\\right)\\right|\\longrightarrow 0.$ Further, we can write $&\\left|S\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)-S\\left(\\vartheta ,x_s\\right)\\right|\\le \\left|S\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)-S\\left(\\vartheta ,X_s\\right)\\right|+ \\left|S\\left(\\vartheta ,X_s\\right)-S\\left(\\vartheta ,x_s\\right)\\right|\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\le \\left|\\vartheta ^{*}_{\\varepsilon }-\\vartheta \\right|\\left|\\dot{S}(\\tilde{\\vartheta },X_s)\\right|+\\left|X_s-x_s\\right|\\left|S^{\\prime }(\\vartheta ,\\tilde{X}_s)\\right|.$ Here $|\\tilde{\\vartheta } - \\vartheta | \\le |\\vartheta ^{*}_{\\varepsilon }- \\vartheta |$ and $\\left|\\tilde{X}_s - X_s\\right|\\le \\left| x_s(\\vartheta ^*_{\\varepsilon }) - X_s\\right| \\le \\left| x_s(\\vartheta ^*_{\\varepsilon })-x_s(\\vartheta )\\right|+\\left|x_s(\\vartheta )-X_s\\right|\\rightarrow 0.$ This convergence is uniform w.r.t.", "$s\\in [0,T]$ .", "Then we know that the functions $\\dot{S}(\\vartheta ,x)$ and $S^{\\prime }(\\vartheta ,x)$ are bounded by regularity conditions $\\mathcal {R}$ , the process $X_s$ converges uniformly w.r.t.", "$s\\in [0,T]$ to $x_s\\left(\\vartheta \\right)$ and due to the consistency of the estimator $\\vartheta ^{*}_{\\varepsilon }$ , we obtain the uniform convergence w.r.t.", "$s\\in [0,T]$ (in probability) $\\sup _{s\\in [0,T]}\\left|S\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)-S\\left(\\vartheta ,x_s\\right)\\right|\\longrightarrow 0.$ Further, similar arguments give the uniform convergence w.r.t.", "$s\\in [0,T]$ (in probability) $\\sup _{s\\in [0,T]}\\left|S^{\\prime }\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)-S^{\\prime }\\left(\\vartheta ,x_s\\right)\\right|\\longrightarrow 0.$ Therefore, we obtain the uniform convergence w.r.t.", "$t\\in [0,T]$ (in probability) $&U_{\\varepsilon }\\left(\\frac{t}{T}\\right) \\longrightarrow \\frac{1}{\\sqrt{T}}\\;S(\\vartheta ,x_t)\\;u(t)- \\frac{1}{\\sqrt{T}}\\int _0^tS^{\\prime }(\\vartheta ,x_s)\\;S(\\vartheta ,x_s)\\;u(s)\\;\\textrm {d}s\\\\&\\qquad \\qquad \\quad = \\frac{1}{\\sqrt{T}}\\int _0^tS(\\vartheta ,x_s)\\;\\textrm {d}u(s)= U\\left(\\frac{t}{T}\\right).$ Now we have to show that $K_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)\\longrightarrow K(\\vartheta ,s)$ , where $&K_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)=\\frac{1}{\\sqrt{T}}\\;h_\\varepsilon \\left(\\vartheta ^*_{\\varepsilon },s\\right)S\\left(\\vartheta ^*_{\\varepsilon },X_s\\right)u_\\varepsilon (s)\\\\&-\\frac{1}{\\sqrt{T}}\\int _0^s\\left(h_\\varepsilon ^{\\prime }(\\vartheta ^*_{\\varepsilon },q)S(\\vartheta ^*_{\\varepsilon },X_q)+h_\\varepsilon (\\vartheta ^*_{\\varepsilon },q)S^{\\prime }(\\vartheta ^*_{\\varepsilon },X_q)S(\\vartheta ^*_{\\varepsilon },X_q)\\right)u_\\varepsilon (q)\\;\\textrm {d}q$ and $&K\\left(\\vartheta ,s\\right)=\\frac{1}{\\sqrt{T}}\\;\\hat{h}\\left(\\vartheta ,s\\right)S\\left(\\vartheta ,x_s\\right)u(s)\\\\&\\qquad \\quad \\quad -\\frac{1}{\\sqrt{T}}\\int _0^s\\left(\\hat{h}^{\\prime }\\left(\\vartheta ,q\\right)S\\left(\\vartheta ,x_q\\right)+\\hat{h}\\left(\\vartheta ,q\\right)S^{\\prime }\\left(\\vartheta ,x_q\\right)S\\left(\\vartheta ,x_q\\right)\\right)u(q)\\;\\textrm {d}q,$ where $\\hat{h}\\left(\\cdot ,\\cdot \\right)$ and $\\hat{h}^{\\prime }(\\cdot ,\\cdot )$ were defined by (REF )$-$ (REF ).", "We have $&\\left|\\dot{S}\\left(\\vartheta ^*_{\\varepsilon },X_s\\right)-\\dot{S}\\left(\\vartheta ,x_s\\right)\\right|\\le \\left|\\dot{S}\\left(\\vartheta ^*_{\\varepsilon },X_s\\right)-\\dot{S}\\left(\\vartheta ,X_s\\right)\\right| +\\left|\\dot{S}\\left(\\vartheta ,X_s\\right)-\\dot{S}\\left(\\vartheta ,x_s\\right)\\right|\\\\&\\qquad \\qquad \\qquad \\qquad \\quad \\quad \\;\\le \\left|\\vartheta ^*_{\\varepsilon }-\\vartheta \\right|\\left|\\ddot{S}(\\tilde{\\vartheta },X_s)\\right|+\\left|X_s-x_s\\right|\\left|\\dot{S}^{\\prime }(\\vartheta ,\\tilde{X}_s)\\right|\\\\&\\qquad \\qquad \\qquad \\qquad \\quad \\quad \\;\\le \\tilde{C}_1\\left|\\vartheta ^*_{\\varepsilon }-\\vartheta \\right|+\\tilde{C}_2\\left|X_s-x_s\\right|.$ Here $\\tilde{C}_1$ and $\\tilde{C}_2$ are some constants and $\\ddot{S}(\\vartheta ,x)$ , which means the second derivative w.r.t.", "$\\vartheta $ , and $\\dot{S}^{\\prime }(\\vartheta ,x)$ are bounded functions due to the regularity conditions $\\mathcal {R}$ .", "Therefore we have the convergence in probability $\\sup _{s\\in [0,T]}\\left|\\dot{S}\\left(\\vartheta ^{*}_{\\varepsilon },X_s\\right)-\\dot{S}\\left(\\vartheta ,x_s\\right)\\right|\\longrightarrow 0.$ Further, we have $J_\\varepsilon (\\vartheta ^*_{\\varepsilon })- J(\\vartheta )=\\displaystyle \\int _0^T\\dot{x}_s(\\vartheta ^*_{\\varepsilon })^2\\textrm {d}s -\\displaystyle \\int _0^T\\dot{x}_s(\\vartheta )^2\\textrm {d}s\\longrightarrow 0.$ Hence we obtain the convergence in probability $\\sup _{s\\in [0,T]}\\left|h_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)-\\hat{h}\\left(\\vartheta ,s\\right)\\right|\\longrightarrow 0.$ Similarly, it is shown that we have the convergence in probability $\\sup _{s\\in [0,T]}\\left|h^{\\prime }_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)-\\hat{h}^{\\prime }\\left(\\vartheta ,s\\right)\\right|\\longrightarrow 0.$ Therefore we obtain the convergence in probability, by the uniform convergence of $u_\\varepsilon \\left(s\\right)$ to $u(s)$ w.r.t.", "$s\\in [0,T]$ , $K_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)\\longrightarrow K(\\vartheta ,s).$ Now we have to show that $L_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s) \\longrightarrow L(\\vartheta ,s),$ where $&L_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)=\\frac{1}{\\sqrt{T}}\\;g_\\varepsilon \\left(\\vartheta ^*_{\\varepsilon },s\\right)S\\left(\\vartheta ^*_{\\varepsilon },X_s\\right)u_\\varepsilon (s)\\\\&-\\frac{1}{\\sqrt{T}}\\int _0^s\\left(g_\\varepsilon ^{\\prime }(\\vartheta ^*_{\\varepsilon },q)S(\\vartheta ^*_{\\varepsilon },X_q)+g_\\varepsilon (\\vartheta ^*_{\\varepsilon },q)S^{\\prime }(\\vartheta ^*_{\\varepsilon },X_q)S(\\vartheta ^*_{\\varepsilon },X_q)\\right)u_\\varepsilon (q)\\;\\textrm {d}q$ and $&L\\left(\\vartheta ,s\\right)=\\frac{1}{\\sqrt{T}}\\;\\hat{g}\\left(\\vartheta ,s\\right)S\\left(\\vartheta ,x_s\\right)u(s)\\\\&\\qquad \\qquad -\\frac{1}{\\sqrt{T}}\\int _0^s\\left(\\hat{g}^{\\prime }\\left(\\vartheta ,q\\right)S\\left(\\vartheta ,x_q\\right)+\\hat{g}\\left(\\vartheta ,q\\right)S^{\\prime }\\left(\\vartheta ,x_q\\right)S\\left(\\vartheta ,x_q\\right)\\right)u(q)\\;\\textrm {d}q,$ where $\\hat{g}\\left(\\cdot ,\\cdot \\right)$ and $\\hat{g}^{\\prime }(\\cdot ,\\cdot )$ were defined by (REF ) and (REF ), respectively.", "Observe that for $s\\in [0,T]$ , we have $&\\left|I_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)- I(\\vartheta ,s)\\right|\\le \\int _s^T \\left|S(\\vartheta ^*_{\\varepsilon },X_v)\\left(\\dot{x}_v(\\vartheta ^*_{\\varepsilon })-\\dot{x}_v(\\vartheta )\\right)\\right|\\textrm {d}v \\\\ &\\qquad \\qquad \\qquad \\qquad \\qquad +\\int _s^T \\left|\\dot{x}_v(\\vartheta )\\left(S(\\vartheta ^*_{\\varepsilon },X_v)-S(\\vartheta ,x_v(\\vartheta ))\\right)\\right|\\textrm {d}v\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\le \\int _s^T \\left|S(\\vartheta ^*_{\\varepsilon },X_v)\\right|\\left|\\dot{x}_v(\\vartheta ^*_{\\varepsilon })-\\dot{x}_v(\\vartheta )\\right|\\textrm {d}v \\\\ &\\qquad \\qquad \\qquad \\qquad \\qquad +\\left|\\vartheta ^*_{\\varepsilon }-\\vartheta \\right| \\int _s^T \\left|\\dot{x}_v(\\vartheta )\\right| \\left|\\dot{S}(\\tilde{\\vartheta },X_v)\\right|\\textrm {d}v\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad +\\int _s^T \\left|\\dot{x}_v(\\vartheta )\\right| \\left|X_v-x_v(\\vartheta )\\right| \\left|S^{\\prime }(\\vartheta ,\\tilde{X}_v)\\right|\\textrm {d}v\\\\&\\qquad \\qquad \\le \\left(\\tilde{C}_3\\left|\\vartheta ^*_{\\varepsilon }-\\vartheta \\right|+\\tilde{C}_4\\left|\\vartheta ^*_{\\varepsilon }-\\vartheta \\right|+\\tilde{C}_5\\sup _{s\\in [0,T]}\\left|X_s-x_s(\\vartheta )\\right|\\right)\\left(T-s\\right),$ where $\\tilde{C}_3$ , $\\tilde{C}_4$ and $\\tilde{C}_5$ are constants.", "Therefore we obtain the convergence in probability uniformly w.r.t.", "$s\\in [0,T]$ $\\sup _{s\\in [0,T]}\\left|I_{\\varepsilon }(\\vartheta ^*_{\\varepsilon },s)- I(\\vartheta ,s)\\right|\\longrightarrow 0$ because the estimator $\\vartheta ^*_{\\varepsilon }$ is consistent, the process $X_s$ converges uniformly w.r.t.", "$s\\in [0,T]$ to $x_s(\\vartheta )$ and the derivatives are bounded due to the conditions of regularity $\\mathcal {R}$ .", "Further, we proved already the uniform convergence w.r.t.", "$s\\in [0,T]$ of $u_\\varepsilon (s)$ to $u(s)$ and $S(\\vartheta ^*_{\\varepsilon },X_s)$ to $S(\\vartheta ,x_v)$ .", "Hence, we have the convergence in probability $\\sup _{s\\in [0,T]}\\left|g_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)-\\hat{g}\\left(\\vartheta ,s\\right)\\right|\\longrightarrow 0.$ Similarly, by the regularity conditions $\\mathcal {R}$ , it is shown that (in probability) $\\sup _{s\\in [0,T]}\\left|g^{\\prime }_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)-\\hat{g}^{\\prime }\\left(\\vartheta ,s\\right)\\right|\\longrightarrow 0.$ Therefore we obtain the convergence in probability $L_{\\varepsilon }\\left(\\vartheta ^*_{\\varepsilon },s\\right)\\longrightarrow L\\left(\\vartheta ,s\\right).$ Further, a similar arguments give the convergence in probability uniformly w.r.t.", "$s\\in [0,T]$ due to the regularity conditions $\\mathcal {R}$ and the consistency of the estimator $\\vartheta ^*_{\\varepsilon }$ $\\sup _{s\\in [0,T]}\\left| \\psi _{2,\\varepsilon }(s) -\\hat{\\psi }_2(s)\\right|\\longrightarrow 0$ and $\\sup _{s\\in [0,T]}\\left| \\varphi _{1,\\varepsilon }(s) -\\hat{\\varphi }_1(s)\\right|\\longrightarrow 0, \\sup _{s\\in [0,T]}\\left|\\varphi _{2,\\varepsilon }(s)- \\hat{\\varphi }_2(s)\\right|\\longrightarrow 0.$ Finally, the convergence mentioned in (REF ) is proved and using this result the test $\\psi _{\\varepsilon }$ is ADF and of asymptotic size $\\alpha \\in (0,1)$ ." ], [ "The case of MLE", "This case was studied in [6]$-$[7].", "They proposed a linear transformation, which yields the convergence of the test statistic to the integral of Wiener process.", "Therefore they showed that the test based on this statistic is ADF.", "To obtain the linear transformation mentioned in [7], we put $h(\\vartheta ,r)=g(\\vartheta ,r)$ in (REF ) and we obtain $I_1 = I_2 =I_4= \\int _0^r h(\\vartheta ,q)^2\\textrm {d}q,\\quad I_3 =I_5=\\int _0^r h(\\vartheta ,q)\\;\\textrm {d}q.$ Therefore we can write $\\varphi _1(r) & = h-h+I_1h-3I_1h+I_3h^2+ I_3h^2+2I_1h-I_1^2h-2I_1I_3h^2-I_1^2I_3h^2\\\\& \\quad - I_1h+I_1^2I_3h^2 - I_1^2h+3I_1^2h+ I_1I_3h^2-2I_1I_3h^2-2I_1^2h+ 2I_1I_3h^2\\\\& \\quad + I_1^2I_3h^2+I_1^3h+I_3h^2+I_1h+ 2I_1^2h-I_1^2h + I_1^3h + I_1^2I_3h^2- 2I_1^2I_3h^2\\\\& \\quad -I_1^3h+I_1I_3h^2+I_1I_3h^2+2I_1I_3h^2-I_1^3h- I_1^2I_3h^2-I_3h^2-2I_1I_3h^2\\\\& \\quad +I_1^2I_3h^2-I_1I_3h^2- 2I_3h^2\\\\& = 0,\\\\\\varphi _2(r) & = 1 + I_3h-3I_1I_3h+I_1I_3h+I_3h-3I_1I_3h+I_1I_3h-I_3h+6I_1^2-2I_1^4\\\\& \\quad + 3I_1^2I_3h- 2I_1^2I_3h+2I_1I_3h-I_1^3I_3h+I_1^3I_3h-I_1^3I_3h + I_1^2-4I_1 \\\\& \\quad + I_1I_3h-I_1^2I_3h+2I_1^2I_3h+I_1I_3h-I_1^2I_3h + 2I_1^3 + I_1^2I_3h- 4I_1^3\\\\& \\quad + I_1^3I_3h-I_1^3I_3h-I_1^2I_3h + I_1^3I_3h- I_1^2I_3h+4I_1^3+ 3I_1^2I_3h- I_1^3I_3h\\\\& \\quad + I_1^4-2I_1^3+I_1^4+2I_1- 2I_1^2+I_1^3I_3h - 2I_1^2I_3h-I_1I_3h - I_1^2I_3h-4I_1^2 \\\\& = 1+I_3h-I_1I_3h+I_1^2-2I_1\\\\& = (1-I_1)(1+I_3h-I_1)$ and $\\psi _2(r) & = h+I_3h^2+I_1h+I_3h^2-3I_1h-2I_1I_3h^2-2I_1^2h-2I_1I_3h^2-I_1^3h \\\\& \\quad + I_1^3h+I_1^2I_3h^2-I_1^3h-I_1I_3h^2+I_1^2h+I_1h - I_1^2h-I_3h^2+ 3I_1^2h \\\\& \\quad - I_1^2h+I_1^3h+I_1I_3h^2 + I_1I_3h^2+ I_1I_3h^2-I_1^2I_3h^2+2 I_1^2I_3h^2\\\\& \\quad - I_1^2I_3h^2-2I_1^2I_3h^2 +2I_1I_3h^2 +I_1^2I_3h^2\\\\& = h\\;(1+I_3h-I_1),$ where $h=h(\\vartheta ,r)$ .", "Hence the linear transformation (REF ) will have the following expression: $L[U](\\nu ) = U(\\nu )+\\displaystyle \\int _0^{\\nu } \\int _0^r h(r)\\;\\mathbb {N}(r)^{-1}h(q)\\;\\textrm {d}U(q)\\;\\textrm {d}r=w_{\\nu },$ where $\\mathbb {N}(r) = \\displaystyle \\int _r^1 h(q)^2\\textrm {d}q$ and $U(\\nu ) = W(\\nu ) -\\int _0^1h(r)\\;\\textrm {d}W(r)\\;\\int _0^\\nu h(r)\\;\\textrm {d}r,$ with $W(\\nu )$ and $w_{\\nu }, 0\\le \\nu \\le 1$ are some standard Wiener processes.", "The transformation $L[U](\\cdot )$ of the limit process $U(\\cdot )$ given by (REF ) coincides with one by Khmaladze [6].", "Acknowledgements.", "The author is grateful to Yu.", "A. Kutoyants for the statement of the problem and many helpful suggestions during the preparation of this paper.", "The author would like to thank the two Referees for the comments which allowed to improve essentially the exposition." ] ]
1606.05167
[ [ "Chiral kinetic theory and anomalous hydrodynamics in even spacetime\n dimensions" ], [ "Abstract We study the hydrodynamics of a gas of noninteracting Weyl fermions coupled to the electromagnetic field in $(2N + 1) + 1$ spacetime dimensions using the chiral kinetic theory, which encodes the gauge anomaly in the Chern character of the nonabelian Berry connection over the Fermi surface.", "We derive the anomalous contributions to the relativistic hydrodynamic currents in equilibrium and at a finite temperature, which agree with and provides an approach complementary to the results derived previously using thermodynamic constraints." ], [ "Introduction", "The equations of relativistic hydrodynamics, proposed in the 1940s[1], [2], have been a versatile tool for the study of a wide variety of classical fluids.", "They have also been used to describe the long wavelength behavior of quantum field theories(QFTs), especially the usually intractable strongly interacting cases.", "A fluid description is useful as it can often be systematically constructed as a derivative expansion, knowing only the symmetries (and corresponding conservation laws) of the underlying QFT[3].", "Anomalies, the breakdown of a conservation law of a classical field theory when the theory is quantized[4], are one of the most subtle and interesting aspects of quantum field theory.", "Their importance lies in their topological nature, as they correspond to the density of the topological index of a Dirac operator[5].", "In principle, a theory with an anomalous dynamical gauge field is inconsistent, as the anomaly implies a breakdown of the gauge invariance; however, anomalies in a non-dynamical background gauge field can offer insightful diagnostics into the theory.", "The anomalies can have macroscopic consequences in the hydrodynamic regime of the QFT[6], [7], as seen in the investigation of the so-called “anomalous fluids”.", "In Ref son-surowka, Son and Surówka show that in $3+1$ spacetime dimensions, the presence of an anomalous conservation law for a $U(1)$ current necessitates adding terms to the constitutive relation at the first order in derivative expansion, which can be constrained using the second law of thermodynamics.", "Subsequently, Loganayagam[8] derived the general solutions to the second law constraint in arbitrary even spacetime dimensions.", "More recently, Loganayagam and Surówka[9] have conjectured a very powerful result for the hydrodynamic description of Weyl fermions in even spacetime dimensions.", "They argue that the anomalous contributions to the hydrodynamic equations can be derived from a “Gibbs free energy current” $\\mathcal {G}$ .", "Furthermore, in $d$ spacetime dimensions, $\\mathcal {G}$ can be obtained by the replacements $F \\rightarrow \\mu $ and $\\text{tr}\\left\\lbrace R^{2n} \\right\\rbrace \\rightarrow 2 (2 \\pi T)^{2n} \\; \\forall \\, n \\in \\mathbb {Z}^+$ in the anomaly polynomial in $d+2$ dimensions, where $\\mu $ is the chemical potential and $T$ the temperature, and $F$ and $R$ denote the Maxwell and Riemann curvatures, respectively.", "These replacement rules have been further studied[10] in a holographic setting using the tools of fluid-gravity duality[3].", "An alternative perspective on anomalous QFTs is provided by the semiclassical approach, where one studies the dynamics of wavepackets treated as classical particles.", "The only quantum aspects are the coupling to the Berry connection and the $\\hbar $ occurring in the phase space volume.", "In a Hamiltonian picture, the coupling to the Berry curvature leads to a nontrivial (“anomalous”) symplectic form on the phase space, so that the position and momentum coordinates are no more conjugate[11], [12].", "This view has proved particularly useful in condensed matter physics[13], [14], for instance, in the study of transport in Weyl semimetals[15].", "In Ref mishackt, Stephanov and Yin showed that a kinetic theory based on a semiclassical description of charged noninteracting Weyl fermions in $3+1$ spacetime dimensions reproduces the Adler-Bell-Jackiw anomaly[4] correctly.", "Subsequently, their computation was generalized to nonabelian gauge anomalies[17] in arbitrary even spacetime dimensions[18] by constructing an anomalous symplectic form on an extended phase space, where the anomaly signals a breakdown of the Liouville's theorem.", "The formalism has also been used to describe the transport processes associated with gauge anomalies, for instance, the chiral magnetic effect(CME) and chiral vortical effect(CVE)[16], [15].", "In this paper, we study the hydrodynamics of a gas of charged noninteracting Weyl fermions in arbitrary even spacetime dimensions.", "Starting from a semiclassical microscopic description and assuming the system to be in thermodynamic equilibrium with the comoving frame, we derive the anomalous contributions to the hydrodynamic currents, which depend on the electromagnetic field and vorticity of the fluid.", "At a finite temperature, we include both positive and negative energy sectors to get a closed form expressions for the currents, which are identical to those obtained in Ref logaweylgas using thermodynamic constraints.", "The rest of this paper is organized as follows: in Sec , we review the anomalous symplectic form and the extended phase space proposed in Ref.", "vd-msarbt.", "In Sec , we review the basics of relativistic hydrodynamics, including the differential form notation proposed in Ref.", "logaanomtransport.", "In Sec , we set up the formalism to derive expressions for macroscopic currents using the anomalous symplectic form, using which we derive the anomalous hydrodynamic currents in Sec .", "Finally, we discuss our conclusions in Sec .", "In the Appendices, we review the Fermi-Dirac distribution and associated quantities in Appendix , derive the symplectic form in a noninertial reference frame in Appendix and show that the comoving frame used in our calculation satisfies the no-drag condition[19] in Appendix We follow the general relativity convention for the Minkowski metric, where $\\eta ^{\\mu \\nu } = \\mathrm {diag}\\lbrace -1,1,\\dots 1\\rbrace $ on $\\mathbb {R}^{2N+1,1}$ .", "The Greek indices ($\\mu ,\\nu $ ) run over all the spacetime coordinates and the Latin indices from the middle of the alphabet $(i,j,k)$ run over only the space coordinates, with Einstein summation for repeated indices.", "We set $\\hbar = c = 1$ ." ], [ "Semiclassical description of Weyl fermions", "We briefly review the semiclassical description of Weyl fermions[16], [17], [18] and the symplectic formulation thereof." ], [ "Extended phase space", "Consider a classical system on a $2M$ -dimensional phase space $\\mathcal {M}$ with a set of coordinates ${\\zeta }= \\left( \\zeta ^1, \\dots \\zeta ^{2M} \\right)$ .", "A generic action functional on this phase space is given by $S[{\\zeta }] = \\int dt \\left( \\eta _i \\left( {\\zeta },t \\right) \\dot{\\zeta }^i - \\mathcal {H}({\\zeta }, t) \\right).$ The system is nonautonomous as $\\eta $ is time-dependent, so that the standard symplectic formalism[20] cannot be used.", "Instead, we extend the phase space to $\\mathcal {M}_H = \\mathcal {M}\\times \\mathbb {R}$ , with the time coordinate $t \\in \\mathbb {R}$ .", "The action can be written as a line integral of the so-called Liouville 1-form $\\eta _H$ along the trajectory: $S[{\\zeta }] = \\int \\eta _H, \\quad \\eta _H = \\eta _i \\left( {\\zeta },t \\right) d \\zeta ^i - \\mathcal {H}({\\zeta }, t) dt.$ The equation of motion can be written elegantly in a coordinate independent fashion in terms of a generalized symplectic form $\\rho _H = d\\eta _H$ as $i_V \\rho _H = 0, \\quad \\quad V = \\frac{\\partial }{\\partial t} + \\dot{\\zeta }^i \\frac{\\partial }{\\partial \\zeta ^i}.$ In other words, the “symplectic form” is now degenerate with corank 1, and the null direction gives us the trajectory in the extended phase space.", "This setup has also been studied under the name of contact structure, with the above equation due to Élie Cartan (For instance, see Ref abraham-marsden, Theorem 5.1.13).", "Explicitly, $\\rho _H = \\frac{1}{2} \\left( \\frac{\\partial \\eta _j}{\\partial \\zeta ^i} - \\frac{\\partial \\eta _i }{\\partial \\zeta ^j} \\right) d\\zeta ^i \\wedge d\\zeta ^j- \\left( \\frac{\\partial \\eta _i }{\\partial t} + \\frac{\\partial \\mathcal {H}}{ \\partial \\zeta ^i} \\right) d\\zeta ^i \\wedge dt.$ so that the equation of motion, $i_V \\rho _H = 0$ , demands that $\\dot{\\zeta }^i \\left( \\frac{\\partial \\eta _j}{\\partial \\zeta ^i} - \\frac{\\partial \\eta _i }{\\partial \\zeta ^j} \\right) + \\left( \\frac{\\partial \\eta _j }{\\partial t} + \\frac{\\partial \\mathcal {H}}{ \\partial \\zeta ^j} \\right) = & \\; 0 \\\\\\dot{\\zeta }^i \\left( \\frac{\\partial \\eta _i }{\\partial t} + \\frac{\\partial \\mathcal {H}}{ \\partial \\zeta ^i} \\right) = & \\; 0$ The former is precisely the Euler-Lagrange equation $\\delta S[{\\zeta }] = 0$ , while the latter can be obtained from the former by multiplying with $\\dot{\\zeta }^j$ .", "The extended phase space is also equipped with a symplectic volume form $\\Omega _H= \\frac{1}{M!}", "\\rho _H^{M} dt = \\sqrt{\\rho } \\left( \\bigwedge _{i=1}^{2M} d\\zeta ^i \\right) \\wedge dt.", "$ Liouville's theorem, the statement that the phase space volume is conserved under Hamiltonian flows, is simply $\\mathcal {L}_V \\Omega _H= 0$ , where $\\mathcal {L}_V = i_V d + d i_V$ is the Lie derivative." ], [ "Anomalous symplectic form", "In Ref mishackt, Stephanov and Yin showed that in $3 + 1$ spacetime dimensions, positive chirality Weyl fermions with charge $q$ coupled to a background electromagnetic field can be described by the classical action $S[\\mathbf {x},\\mathbf {p}] = \\int dt \\left( \\mathbf {p}\\cdot \\dot{\\mathbf {x}}- \\varepsilon - q\\phi + q \\mathbf {A}\\cdot \\dot{\\mathbf {x}}- \\mathfrak {a}\\cdot \\dot{\\mathbf {p}}\\right),$ where $\\varepsilon = c|\\mathbf {p}|, \\, c = \\pm 1$ is the energy of the particle.", "The electromagnetic field is minimally coupled, with $\\phi $ and $\\mathbf {A}$ being the electromagnetic scalar and vector potential, respectively.", "The quantum effects at $O(\\hbar )$ are encoded in the Berry connection, $\\mathfrak {a}$ , which can be thought of as a $\\mathrm {U}(1)$ gauge field on the momentum space.", "We use the extended phase space formalism described Sec REF to define the Liouville 1-formFor Weyl fermions, $\\mathfrak {a}$ corresponds to a monopole field, with the monopole located at the band-touching point.", "Thus, $\\eta _H$ is not globally well-defined.", "$\\eta _H = p_i dx^i - c|\\mathbf {p}| dt + q A - \\mathfrak {a},$ where we have defined the 1-forms $A = A_\\mu dx^\\mu = - \\phi dt + \\mathbf {A}_i dx^i$ and $\\mathfrak {a}= \\mathfrak {a}_i dp^i$ .", "The corresponding symplectic form is $\\rho _H \\equiv d \\eta _H = d p_i \\wedge dx^i - c\\, d |\\mathbf {p}| \\wedge dt + q F - \\mathfrak {F},$ where $F = dA = \\frac{1}{2} F_{\\mu \\nu } dx^\\mu \\wedge dx^\\nu $ and $\\mathfrak {F}= d \\mathfrak {a}= \\frac{1}{2} \\mathfrak {F}_{ij} dp^i \\wedge dp^j$ .", "In $3+1$ dimensions, the Berry connection is abelian, since the Weyl fermions have only two components so that the positive energy sector is described by a single component.", "However, in $2N + 2$ spacetime dimensions, the positive energy sector is $2^{N+1}$ -fold degenerate, so that the Berry connection becomes nonabelian with the gauge group $\\mathrm {Spin}(2^{N+1})$ .", "In Ref.", "vd-msarbt, we include this nonabelian Berry connection in the classical description by “dequantizing” it à la Wong[21].", "Given a representation of a compact gauge group $G$ with the highest weight vector $\\Lambda $ , one chooses an element $\\alpha _\\Lambda $ in the Cartan subalgebra of the Lie algebra, $\\mathfrak {g}$ .", "The classical description then involves enlarging the phase space to include the adjoint orbit[22] of $\\alpha _\\Lambda $ , denoted by $\\mathcal {O}_\\Lambda $ .", "Explicitly, we can define coordinates on $\\mathcal {O}_\\Lambda $ as $\\mathfrak {S}= \\sigma \\alpha _\\Lambda \\sigma ^{-1}, \\; \\sigma \\in G$ .", "Clearly, $\\mathfrak {S}$ is invariant under $\\sigma \\rightarrow \\sigma \\cdot \\eta \\; \\forall \\eta \\in H$ , where $H \\subset G$ is the subgroup generated by the elements of Lie algebra that commute with $\\alpha _\\Lambda $ , so that the orbit can be identified with the quotient $G/H$ .", "Choose a basis $\\lbrace \\lambda _a\\rbrace $ of $\\mathfrak {g}$ , which satisfies the orthonormality condition $\\text{tr}_\\Lambda \\lbrace \\lambda _a \\lambda _b \\rbrace = \\delta _{ab}$ .", "Then, $\\mathfrak {S}\\in \\mathfrak {g}$ , being simply an adjoint action on $\\alpha _\\Lambda \\in \\mathfrak {g}$ , can be written as $\\mathfrak {S}= \\mathfrak {S}^a \\lambda _a$ .", "Similarly, $\\mathfrak {a}= \\mathfrak {a}^a \\lambda _a$ and $\\mathfrak {F}= \\mathfrak {F}^a \\lambda _a$ .", "For the “dequantization”, the matrix-valued gauge connection and curvature are then replaced by $\\mathfrak {a}& \\; \\mapsto \\widetilde{\\mathfrak {a}} \\equiv \\text{tr}\\lbrace \\mathfrak {S}\\mathfrak {a}\\rbrace = \\mathfrak {S}^a \\mathfrak {a}_a \\nonumber \\\\\\mathfrak {F}& \\; \\mapsto \\widetilde{\\mathfrak {F}} \\equiv \\text{tr}\\lbrace \\mathfrak {S}\\mathfrak {F}\\rbrace = \\mathfrak {S}^a \\mathfrak {F}_a.$ To make $\\mathfrak {S}$ , dynamical, we add the corresponding right Maurer-Cartan form $\\mathfrak {w}_R = d\\sigma \\sigma ^{-1}$ to the Liouville 1-form.", "Thus, in $2N+2$ spacetime dimensions, our Liouville 1-form on the extended phase space $\\mathcal {M}_H = \\mathbb {R}^{4N+3} \\times \\mathcal {O}_\\Lambda $ becomes $\\eta _H = p_i dx^i - c\\, |\\mathbf {p}| dt + q A - \\text{tr}\\lbrace \\mathfrak {S}\\left( \\mathfrak {a}+ i \\mathfrak {w}_R \\right) \\rbrace .$ The corresponding symplectic form is $\\rho _H \\equiv d \\eta _H = d p_i \\wedge dx^i - c\\, d |\\mathbf {p}| \\wedge dt + q F - \\widetilde{\\mathfrak {F}} - i \\, \\text{tr}\\left\\lbrace \\mathfrak {S}\\left( \\mathfrak {w}_R - i \\mathfrak {a}\\right)^2 \\right\\rbrace .$" ], [ "Anomalous fluids", "The dynamics of an anomalous fluid with a $U(1)$ anomaly is described by $\\partial _\\mu \\mathcal {T}^{\\mu \\nu } = F^{\\nu \\lambda } J_\\lambda , \\quad \\partial _\\mu J^\\mu = \\mathcal {A},$ where $\\mathcal {T}^{\\mu \\nu }$ is the energy-momentum tensor (“energy current”) of the fluid, $J^\\nu $ is the charge current, $F_{\\mu \\nu }$ is the Maxwell gauge field corresponding to the gauge connection $A_\\mu (x)$ and $\\mathcal {A}(F)$ is the anomaly polynomial.", "One also defines an entropy current $S^\\mu $ , which must satisfy the second law of thermodynamics, $\\partial _\\mu S^\\mu \\ge 0$ .", "In order to obtain a closed system of equations, the hydrodynamic currents need to be expressed in terms of the thermodynamic fields, viz, the velocity $u^\\mu (x)$ (satisfying $u_\\mu u^\\mu = -1$ ), the temperature $T(x)$ , the chemical potential $\\mu (x)$ and the gauge connection $A_\\mu (x)$ .", "These are the so-called constitutive relations, which can be constructed systematically in a derivative expansion, where each spacetime derivative of the thermodynamic fields counts as dimension 1 for the bookkeeping[3].", "For anomalous and dissipationless fluids, the most generalIn principle, there can also be tensor corrections to $\\mathcal {T}$ due to the anomaly.", "They are usually ignored in thermodynamic calculations, as they cannot be constrained by the second law of thermodynamics[8].", "constitutive relations can be written as[6], [8], [9] $\\mathcal {T}^{\\mu \\nu } = & \\, (\\varepsilon + p) u^\\mu u^\\nu + p \\eta ^{\\mu \\nu } + (q^\\mu u^\\nu + u^\\mu q^\\nu ), \\nonumber \\\\J^\\mu = & \\, n u^\\mu + \\mathcal {J}^\\mu , \\nonumber \\\\S^\\mu = & \\, s u^\\mu + \\mathcal {S}^\\mu , $ where $q, \\mathcal {J}$ and $\\mathcal {S}$ contain one or more spacetime derivatives of $u_\\mu $ or $A_\\mu $ .", "We also set $u_\\mu q^\\mu = u_\\mu \\mathcal {J}^\\mu = u_\\mu \\mathcal {S}^\\mu = 0, $ so that in the frame where $u^\\mu = (1, 0, \\dots 0)$ , the components $\\mathcal {T}^{00} \\equiv \\varepsilon $ and $J^0 \\equiv n$ represent the actual energy density and charge density, respectively.", "At the first order in a derivative expansion, we can define the “curvatures” $F_{\\mu \\nu } = \\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu , \\quad \\Omega _{\\mu \\nu } = \\partial _\\mu u_\\nu - \\partial _\\nu u_\\mu .$ In a frame specified by $u$ , the electric field is defined as $E_\\mu = F_{\\mu \\nu } u^\\nu $ .", "Since the acceleration is defined as $a_\\mu = u^\\nu \\partial _\\nu u_\\mu $ , we can use $u^\\nu \\partial _\\mu u_\\nu = \\frac{1}{2} \\partial _\\mu \\left( u_\\nu u^\\nu \\right) = 0$ to derive an analogous expression $a_\\mu = u^\\nu \\left( \\partial _\\nu u_\\mu - \\partial _\\mu u_\\nu \\right) = - \\Omega _{\\mu \\nu } u^\\nu .$ It is convenient to rephrase the above expressions in the language of differential forms.", "We define the 1-forms $u = u_\\mu dx^\\mu $ and $A = A_\\mu dx^\\mu $ , and decompose their exterior derivatives into the “magnetic” and “electric” components[8] as $F = & \\; dA = B + u \\wedge E, \\nonumber \\\\\\Omega = & \\; du \\, = \\omega - u \\wedge a,$ where $B$ and $\\omega $ are 2-forms.", "Clearly, $u \\wedge F = u \\wedge B, \\quad \\quad u\\wedge \\Omega = u \\wedge \\omega .", "$ Note that in order to expose the mathematical symmetry between these decompositions, we have followed the traditional fluid mechanics convention[2] in defining the vorticity $\\omega $ , so that $\\vec{\\omega } = \\nabla \\times \\vec{u}$ in 3+1 dimensions.", "This is in contrast to the angular velocity, $\\vec{\\omega }_{A} = \\frac{1}{2} \\nabla \\times \\vec{u} = \\frac{1}{2} \\omega $ , which is sometimes referred to as the “vorticity” in relativistic hydrodynamics (for instance, Refs fluid-gravity, logaanomtransport).", "The decompositions are particularly transparent in a frame where $u^\\mu = \\left( 1, 0, \\dots 0 \\right)$ , so that $u \\equiv u_\\mu dx^\\mu = -dt$ .", "Then, $E_0 = 0, \\; E_i = F_{i 0}$ and $a_0 = 0, \\; a_i = -\\Omega _{i0} = \\partial _t u_i$ , so that $F = & \\; B - F_{i0} \\, dt \\wedge dx^i \\Rightarrow B = \\frac{1}{2} F_{ij} dx^i \\wedge dx^j \\\\\\Omega = & \\; \\omega - \\Omega _{i0} \\, dt \\wedge dx^i \\Rightarrow \\omega = \\frac{1}{2} \\Omega _{ij} dx^i \\wedge dx^j = \\partial _i u_j dx^i \\wedge dx^j.$ To express the conservation laws in the language of differential forms, we also need the Hodge dual[23].", "We follow Loganayagam[8] in denoting the Hodge dual by an overbar as well as the usual $\\star $ .", "For instance, in $3 + 1$ dimensions, $\\bar{u}= u_\\mu (\\star dx^\\mu ) = \\frac{1}{3!}", "u_\\mu \\epsilon ^\\mu _{\\phantom{\\mu }\\nu \\rho \\lambda } dx^\\nu dx^\\lambda dx^\\rho = \\left( \\frac{1}{3!}", "\\epsilon _{\\mu \\nu \\rho \\lambda } u^\\mu \\right) dx^\\nu dx^\\lambda dx^\\rho .$ Given 1-forms $u$ and $v$ , the inner product and gradient can be written as $\\star \\, (v_\\mu u^\\mu ) = v \\wedge \\bar{u}, \\quad \\star \\, (\\partial _\\mu u^\\mu ) = d\\bar{u},$ where $\\star 1 = \\mathbb {V}$ , the Euclidean volume form on $\\mathbb {R}^{2N+1,1}$ .", "This is the dictionary to go between the differential forms and vectors on $\\mathbb {R}^{2N+1, 1}$ .", "Finally, we define the 1-forms corresponding to the anomalous currents as $q = q_\\mu dx^\\mu $ , etc and their Hodge duals as $\\bar{q}= q_\\mu \\left( \\star dx^\\mu \\right)$ , which are $(2N+1)$ -forms.", "Schematically, we can construct such a form by wedging the 1-form $u$ to $N-1$ copies of $F$ or $\\Omega $ , i.e, $\\bar{q}\\sim \\sum _{k=1}^{N-1} \\xi _{q, k} \\, u \\wedge \\underbrace{F \\wedge \\dots \\wedge F}_{k} \\wedge \\underbrace{\\Omega \\wedge \\dots \\wedge \\Omega }_{N-1-k}, $ with similar expressions for $\\bar{\\mathcal {J}}$ and $\\bar{\\mathcal {S}}$ .", "The task of hydrodynamics is then to constrain the transport coefficients ($\\xi $ 's here) using general principles such as those of thermodynamics.", "The authors of Ref.", "logaanomtransport also define a grand potentialIn Ref logaweylgas, $\\mathcal {G}$ is referred to as the Gibbs free energy current.", "However, as the Gibbs free energy (per unit volume) is defined as $G = \\varepsilon + p - Ts = \\mu n$ , the “Gibbs free energy current” would be defined as $\\mu J$ .", "current $\\bar{\\mathcal {G}}$ , which acts as a generating function for $\\bar{q}$ , $\\bar{\\mathcal {J}}$ and $\\bar{\\mathcal {S}}$ : $\\bar{\\mathcal {G}}= \\bar{q}- \\mu \\bar{\\mathcal {J}}- T \\bar{\\mathcal {S}}; \\quad \\quad \\bar{\\mathcal {J}}= -\\frac{\\partial \\bar{\\mathcal {G}}}{\\partial \\mu }, \\quad \\bar{\\mathcal {S}}= -\\frac{\\partial \\bar{\\mathcal {G}}}{\\partial T}.$ We shall derive this current by an explicit semiclassical calculation." ], [ "Symplectic form and currents", "In Sec , we reviewed the semiclassical Hamiltonian description of Weyl fermions in an inertial reference frame.", "However, for hydrodynamics, it is more natural to consider the co-moving frame, defined by the given velocity field $u^\\mu (x)$ .", "As the frame may in general possesses a nonzero acceleration as well as vorticity $(\\Omega = du \\ne 0)$ , we need a way to include the inertial forces in our formalism.", "In Appendix , we derive the generalized symplectic form in a noninertial reference frame, and show that for massless particles, it is reasonable to include the inertial forces in the symplectic form as $\\rho _H \\rightarrow \\rho _H + \\varepsilon \\Omega $ (at the linear order in $\\Omega $ ).", "This is reminiscent of the minimal coupling to the electromagnetic field, with $\\varepsilon = c|\\mathbf {p}|$ serving as the “charge”.", "Thus, the semiclassical dynamics of Weyl fermions in the co-moving frame on $\\mathbb {R}^{2N+1,1}$ is described by the generalized symplectic form $\\rho _H \\equiv d \\eta _H = d p_i \\wedge dx^i - c\\, d |\\mathbf {p}| \\wedge dt + q F + c\\, |\\mathbf {p}|\\Omega - \\widetilde{\\mathfrak {F}} - i \\, \\text{tr}\\left\\lbrace \\mathfrak {S}\\left( \\mathfrak {w}_R - i \\mathfrak {a}\\right)^2 \\right\\rbrace , $ where we have locally set $u^\\mu = (1,0, \\dots 0)$ by suitable Lorentz transforms, so that $-dt = u$ .", "The $(2M+1)$ -dimensional extended phase space is $\\mathcal {M}_H = \\mathbb {R}^{2n} \\times \\mathbb {R}\\times \\mathcal {O}_\\Lambda , \\, n = 2N+1$ , where $M = n + \\frac{1}{2} \\text{dim}\\left( \\mathcal {O}_\\Lambda \\right) \\equiv n + m_\\Lambda $ .", "Consider now the space density of a energy-dependent physical quantity, $Q(\\varepsilon ,x)$ .", "The corresponding current is defined as $J_c^i = \\int _{{P}} \\, \\frac{d^np}{(2\\pi )^n} \\, d\\mu _\\Lambda \\, \\sqrt{\\rho }\\, \\dot{x}^i \\, Q(\\varepsilon ,x),$ where ${P}= \\mathbb {R}^n \\times \\mathcal {O}_\\Lambda $ and $d\\mu _\\Lambda $ is an invariant measure on $\\mathcal {O}_\\Lambda $ .", "In principle, one next needs to solve the equation of motion $(i_V \\rho _H = 0)$ for $\\sqrt{\\rho } \\, \\dot{x}^i$ , which can then be integrated over the momentum space and the co-adjoint orbit.", "In 3+1 dimensions, this is straightforward[16], and one gets $\\sqrt{\\rho } \\, \\dot{\\mathbf {x}}= c\\,\\hat{\\mathbf {p}} + \\mathfrak {b} \\times \\mathbf {E} + \\left( \\hat{\\mathbf {p}} \\cdot \\mathfrak {b} \\right) \\left( q \\mathbf {B} + c|\\mathbf {p}| \\omega \\right), \\quad \\quad \\quad \\mathfrak {b}^i = \\frac{1}{2} \\epsilon ^{ijk} \\mathfrak {F}_{jk}.$ However, the task is much more complicated in spacetime dimensions greater than 4.", "Thus, we follow an alternative approach using the symplectic formulation of classical mechanics, which lets us compute such currents without computing $\\sqrt{\\rho } \\, \\dot{x}^i$ explicitly.", "Define the current 1-form $J_c = J_{c,i} dx^i$ , whose Hodge dual can be explicitly written as $\\bar{J}_c = \\frac{1}{(2\\pi )^n} \\int _{P}Q(\\varepsilon ,x) \\sqrt{\\rho } \\left( \\sum _{i=1}^n (-1)^i dx^1 \\wedge \\dots dx^{i-1} \\wedge \\dot{x}^i dx^{i+1} \\wedge \\dots dx^{n} \\right) \\frac{d^np}{(2\\pi )^n} \\wedge dt \\wedge d\\mu _\\Lambda ,$ The differential form in the parenthesis is simply $i_V \\left( d^{n} x \\right) = i_V \\left( \\bigwedge _{i=1}^n dx^i \\right) = \\sum _{i=1}^n (-1)^i dx^1 \\wedge \\dots dx^{i-1} \\wedge \\dot{x}^i dx^{i+1} \\wedge \\dots dx^{n},$ so that the measure on the right hand side can readily be obtained as an antiderivation of the symplectic volume form, with only one term at $O(d^{n-1}x)$ : $i_V \\Omega _H= & \\; i_V \\left( \\sqrt{\\rho } \\, d^nx \\wedge d^np \\wedge dt \\wedge d\\mu _\\Lambda \\right) \\nonumber \\\\= & \\; \\sqrt{\\rho } \\left[ i_V \\left( d^{n}x \\right) \\wedge d^n p \\wedge dt \\wedge d\\mu _\\Lambda + \\text{terms involving }d^nx \\right].$ Using eq.", "(REF ) and our explicit expression for the symplectic form in eq.", "(REF ), we can explicitly write the symplectic volume form as $\\Omega _H = \\frac{1}{M!}", "\\rho _H^M dt = \\frac{1}{n!}", "\\rho _0^n dt \\, d\\mu _\\Lambda , \\quad d\\mu _\\Lambda = \\frac{1}{m_\\Lambda !}", "\\left[ - i \\text{tr}\\left\\lbrace \\mathfrak {S}\\mathfrak {w}_R^2 \\right\\rbrace \\right]^{m_\\Lambda }$ where we have defined $\\rho _0 = d p_i \\wedge dx^i - c\\, d |\\mathbf {p}| \\wedge dt + qF + c|\\mathbf {p}| \\Omega - \\widetilde{\\mathfrak {F}},$ which contains only the spacetime and momentum differentials.", "Thus, using $i_V dt = 1$ and the equation of motion $i_V \\rho _H = 0$ , we get $i_V \\Omega _H= \\frac{1}{n!}", "i_V \\left[ \\rho _0^{n} \\, dt \\, d\\mu _\\Lambda \\right] = \\frac{1}{n!}", "\\left[ \\rho _0^n d\\mu _\\Lambda + \\rho _0^n \\, dt (i_V d\\mu _\\Lambda ) \\right].$ As we seek the terms at $O(d^{n-1}x)$ , we simply need to read off the coefficient of $\\left( i_V d^nx \\right) d^np \\, dt$ in $\\frac{1}{n!}", "\\rho _0^n$ .", "We obtain terms of the form $\\left( d p_i \\wedge dx^i - c\\, d |\\mathbf {p}| \\wedge dt \\right)^{2(N-k)+1} \\left( qF + c|\\mathbf {p}| \\Omega \\right)^k \\wedge \\left( \\widetilde{\\mathfrak {F}} \\right)^k; \\quad 0 \\le k \\le N.$ To compute $\\bar{J}_c$ , we need to integrate these terms over ${P}= \\mathbb {R}^{2N+1} \\times \\mathcal {O}_\\Lambda \\cong \\mathbb {R}^+ \\times S^{2N} \\times \\mathcal {O}_\\Lambda $ , where $\\mathbb {R}^+$ denotes the radial $|\\mathbf {p}|$ axis.", "We now show that only the term with $k = N$ integrates to a nonzero value over ${P}$ .", "For the Weyl Hamiltonian $\\mathcal {H}= \\mathbf {p}\\cdot \\Gamma $ , the Berry curvature is singular at the band touching point $\\mathbf {p}= 0$ , which acts as a nonabelian monopole in the Berry curvature field[18].", "Mathematically, the states correspond to a complex line bundle $\\mathfrak {C}$ over the unit sphere in momentum space, which can be written as a direct sum of the subbundles corresponding to positive and negative energies.", "The positive energy subbundle carries a Chern number $\\chi $ , equal to the chirality of the node, and as $\\mathfrak {C}$ is trivial, the negative energy subbundle carries a Chern number $-\\chi $ (also see Ref.", "vd-msarbt, eq B9).", "As we are only considering a positive chirality Weyl node, we set $\\chi = +1$ .", "Thus, using the definition of Chern number[5] and $\\widetilde{\\mathfrak {F}} = \\mathfrak {S}^a \\mathfrak {F}_a$ , we get $\\frac{(-1)^N}{N!", "(2\\pi )^N} \\int _{S^{2N} \\times \\mathcal {O}_\\Lambda } \\widetilde{\\mathfrak {F}}^N \\wedge d\\mu _\\Lambda = & \\; \\frac{(-1)^N}{N!", "(2\\pi )^N} \\int _{S^{2N}} \\mathfrak {F}^{a_1} \\dots \\mathfrak {F}^{a_N} \\int _{\\mathcal {O}_\\Lambda } d\\mu _\\Lambda \\mathfrak {S}_{a_1} \\dots \\mathfrak {S}_{a_N} \\nonumber \\\\= & \\; \\frac{(-1)^N}{N!", "(2\\pi )^N} \\int _{S^{2N}} \\mathfrak {F}^{a_1} \\dots \\mathfrak {F}^{a_N} \\text{tr}\\lbrace \\lambda _{a_1} \\dots \\lambda _{a_N} \\rbrace \\nonumber \\\\= & \\; \\frac{1}{N!}", "\\int _{S^{2N}} \\text{tr}\\left\\lbrace \\left( - \\frac{\\mathfrak {F}}{2\\pi } \\right)^N \\right\\rbrace = c, $ where $c= \\pm 1$ for the positive/negative energy subspace.", "In the first line, we have replaced the integral of $\\mathfrak {S}$ 's over $\\mathcal {O}_\\Lambda $ , the classical phase space, with a trace of a product of generators over the quantum representation.", "This is only an approximation, which can be improved by integrating over $\\mathcal {O}_{\\Lambda + \\mathfrak {W}}$ instead of $\\mathcal {O}_\\Lambda $ , where we have shifted the highest weight vector $\\Lambda $ by the Weyl vector $\\mathfrak {W}$ (See Appendix C of Ref.", "vd-msarbt for details).", "Better still, we can “requantize” the co-adjoint orbit to reproduce the quantum traces, using the Borel-Weil-Bott construction (Ref stone-MM, Sec 16.2.3).", "Next, we note that any terms with $0<k<N$ integrate to zero, since in order for $\\text{tr}\\lbrace \\mathfrak {F}^k\\rbrace \\wedge d^{2(N-k)}p$ to integrate to a nonzero value over $S^{2N}$ , we need to integrate $\\text{tr}\\lbrace \\mathfrak {F}^k\\rbrace $ over a nontrivial $2k$ -cycle in $S^{2N}$ .", "However, the only nontrivial cycles in $S^{2N}$ are in dimensions $2N$ and zero[5], [23], so that all such integrals with $0 < k < N$ evaluate to zero.", "Finally, for $k = 0$ , considering the integral for $J^i$ and using $d|\\mathbf {p}| = \\hat{p_j} dp^j$ , the integral over the momentum space $\\mathbb {R}^n, \\; n = 2N+1$ , is $\\int _{\\mathbb {R}^{n}} Q(c|\\mathbf {p}|,x) \\hat{p}_j dp^j \\left(\\bigwedge _{\\ell \\ne i} dp^\\ell \\right) = \\int _{\\mathbb {R}^{n}} \\hat{p}_i Q(c|\\mathbf {p}|,x) \\left(\\bigwedge _{\\ell =1}^n dp^\\ell \\right),$ which vanishes, as the integrand is odd under $p_i \\rightarrow -p_i$ .", "Thus, $J_c$ would involve only the integral of $\\frac{1}{(N!", ")^2} (- c\\, d|\\mathbf {p}| \\wedge dt) \\wedge (qF + c|\\mathbf {p}|\\Omega )^N \\wedge \\left( -\\widetilde{\\mathfrak {F}} \\right)^N$ over ${P}$ .", "Explicitly, $\\bar{J}_c = - \\frac{c}{2\\pi (N!", ")^2} \\int _{P}Q(c|\\mathbf {p}|,x) d|\\mathbf {p}| \\wedge \\left( - \\frac{\\widetilde{\\mathfrak {F}}}{2\\pi } \\right)^N \\wedge d\\mu _\\Lambda \\wedge dt \\wedge \\left( \\frac{qF + c|\\mathbf {p}|\\Omega }{2\\pi } \\right)^N.$ Integrating over $S^{2N} \\times \\mathcal {O}_\\Lambda $ using eq.", "(REF ), we get $\\bar{J}_c = \\frac{c^2}{N!}", "(-dt) \\wedge \\int _0^\\infty \\frac{d|\\mathbf {p}|}{2\\pi } Q(\\varepsilon ,x) \\left( \\frac{qF + c|\\mathbf {p}|\\Omega }{2\\pi } \\right)^N .$ Finally, substituting $u = -dt$ and $c^2 = 1$ and using eq.", "(REF ), $\\bar{J}_c = \\frac{1}{N!}", "\\, u \\wedge \\int _0^\\infty \\frac{d|\\mathbf {p}|}{2\\pi } Q(c|\\mathbf {p}|,x) \\left( \\frac{qB + c|\\mathbf {p}|\\omega }{2\\pi } \\right)^N.", "$ This is an explicit expression for the contribution of one energy sector ($c = \\pm 1$ ) of a positive chirality Weyl node to the current $J$ in arbitrary even spacetime dimensions.", "In the next section, we derive the relevant $Q(\\varepsilon ,x)$ for the grand potential current in relativistic hydrodynamics of anomalous fluids." ], [ "Microscopic derivation of hydrodynamic currents", "Consider a gas of Weyl particles with positive chirality in the phase space in equilibrium with a given frame of reference, so that the phase space distribution is simply the Fermi-Dirac distribution $f(p, x) \\equiv f(\\varepsilon ) = \\frac{1}{1 + e^{\\beta (\\varepsilon - \\mu (x)) }}, \\quad \\beta = \\frac{1}{T},$ Since the particles are fermions, we can define the microscopic entropy density $h(\\varepsilon ) = -\\sum _{\\textrm {states}} p_i \\log p_i = - f \\log f - (1-f) \\log (1-f).$ Given the trajectory of a single particle, $\\dot{x}^i$ , the number current and energy-momentum tensor are defined asWe have suppressed the Dirac delta functions localizing these quantities to the particle trajectory.", "$j^i = \\dot{x}^i, \\quad t^{i\\nu } = \\varepsilon \\dot{x}^i \\dot{x}^\\nu .$ The anomalous hydrodynamic currents of eq.", "(REF ) can then be defined simply by averaging over all particles.", "Following the definitions of Sec.", ", we fefine the anomalous currents for a given energy sector asStrictly speaking, we should be subtracting off the “normal” contribution, i.e, replacing $\\dot{x}^i \\rightarrow \\dot{x}^i - \\dot{x}_0^i$ , where $x_0^i(t)$ solves the equation of motion with $ \\mathfrak {F}= 0$ .", "But as the system is in equilibrium in the given frame, the normal component vanishes.", "$q^i = & \\; \\int _{{P}} \\, \\frac{d^np}{(2\\pi )^n} \\, d\\mu _\\Lambda \\, \\sqrt{\\rho }\\, \\dot{x}^i \\, \\varepsilon \\, f(c|\\mathbf {p}|), \\nonumber \\\\\\mathcal {J}^i = & \\; \\int _{{P}} \\, \\frac{d^np}{(2\\pi )^n} \\, d\\mu _\\Lambda \\, \\sqrt{\\rho }\\, \\dot{x}^i \\, f(c|\\mathbf {p}|), \\nonumber \\\\\\mathcal {S}^i = & \\; \\int _{{P}} \\, \\frac{d^np}{(2\\pi )^n} \\, d\\mu _\\Lambda \\, \\sqrt{\\rho }\\, \\dot{x}^i \\, h(c|\\mathbf {p}|),$ where $q^i = \\mathcal {T}^{i0}$ .", "As $u^\\mu = \\left( 1, 0, \\dots 0 \\right)$ and the anomalous contributions are transverse to $u$ (eq.", "(REF )), we also have $q^0 = \\mathcal {J}^0 = \\mathcal {S}^0 = 0$ .", "The corresponding grand potential current is given by $\\mathcal {G}_c^i = q^i - \\mu \\mathcal {J}^i - T \\mathcal {S}^i = \\int _{{P}} \\, \\frac{d^np}{(2\\pi )^n} \\, d\\mu _\\Lambda \\, \\sqrt{\\rho }\\, \\dot{x}^i g(c|\\mathbf {p}|),$ where (using eq.", "(REF )) $(\\varepsilon - \\mu ) f(\\varepsilon ) - T h (\\varepsilon ) = -\\frac{1}{\\beta } \\ln \\left(1 + e^{-\\beta (\\varepsilon - \\mu )} \\right) \\equiv g (\\varepsilon ).$ In order to derive physically meaningful expressions at a finite temperature, we must include both positive and negative energy sectors.", "Using eq.", "(REF ), we define $\\bar{\\mathcal {G}}= & \\; \\bar{\\mathcal {G}}_+ + \\bar{\\mathcal {G}}_- \\nonumber \\\\= & \\; \\frac{u}{N!}", "\\wedge \\left[ \\int _0^\\infty \\frac{d|\\mathbf {p}|}{2\\pi } g(|\\mathbf {p}|) \\left( \\frac{qB + |\\mathbf {p}|\\omega }{2\\pi } \\right)^N + \\int _0^\\infty \\frac{d|\\mathbf {p}|}{2\\pi } g(-|\\mathbf {p}|) \\left( \\frac{qB - |\\mathbf {p}|\\omega }{2\\pi } \\right)^N \\right].$ Substituting $|\\mathbf {p}| = \\varepsilon $ in the first integral and $|\\mathbf {p}| = -\\varepsilon $ in the second, we get $\\bar{\\mathcal {G}}= \\frac{u}{N!}", "\\wedge \\int _{-\\infty }^\\infty \\frac{d\\varepsilon }{2\\pi } g(\\varepsilon ) \\left( \\frac{qB + \\varepsilon \\omega }{2\\pi } \\right)^N.", "$ This integral is clearly divergent, as $g(\\varepsilon ) \\sim (\\varepsilon - \\mu )$ for $\\varepsilon \\rightarrow -\\infty $ .", "This is expected, as we are integrating over an infinitely deep Dirac sea.", "In order to regularize this integral, we need to subtract off the zero temperature vacuum contribution, where we define the “vacuum” as the many-body state where all 1-particle states with $\\varepsilon < 0$ (i.e, below the Weyl node) are filled up.", "Since at $T = 0$ , $g(\\varepsilon ) = (\\varepsilon - \\mu ) \\Theta (\\mu - \\varepsilon )$ , where $\\mu > 0$ , define the regularized grand potential current as $\\bar{\\mathcal {G}}^{reg} = \\frac{u}{N!}", "\\wedge \\int _{-\\infty }^\\infty \\frac{d\\varepsilon }{2\\pi } \\left[g(\\varepsilon ) - (\\varepsilon - \\mu ) \\Theta (-\\varepsilon ) \\right] \\left( \\frac{qB + \\varepsilon \\omega }{2\\pi } \\right)^N.", "$ Following Ref.", "logaweylgas, we define a generating function $\\bar{\\mathcal {G}}^{reg}_\\tau $ by multiplying the integral by $\\tau ^N$ and formallyi.e, we treat $\\omega $ and $B$ as c-numbers instead of differential forms.", "summing over $N$ to get $\\bar{\\mathcal {G}}_\\tau ^{reg} = u \\wedge e^{\\frac{\\tau q B}{2\\pi }} \\int _{-\\infty }^\\infty \\frac{d\\varepsilon }{2\\pi } \\left[g(\\varepsilon ) - (\\varepsilon - \\mu ) \\Theta (-\\varepsilon ) \\right] e^{\\frac{\\tau \\varepsilon \\omega }{2\\pi }}.$ We evaluate the integral explicitly (Appendix , eq.", "(REF )) to get $\\bar{\\mathcal {G}}_\\tau ^{reg} = -u \\wedge e^{\\frac{\\tau q B}{2\\pi }} \\frac{2\\pi }{(\\omega \\tau )^2} \\left[ \\frac{\\frac{\\omega \\tau }{2\\beta }}{\\sin \\left( \\frac{\\omega \\tau }{2\\beta } \\right)} e^{\\frac{\\mu \\omega \\tau }{2\\pi }} - \\left( 1 + \\frac{\\mu }{2\\pi } \\omega \\tau \\right) \\right],$ which is identical to the expression obtainedAfter replacing $\\omega \\rightarrow 2 \\omega _{A}$ , as they defines their vorticity as the angular velocity, $\\omega _{A}$ .", "in Ref.", "logaweylgas.", "In order to obtain the anomalous currents in $2N+2$ spacetime dimensions, $\\left.", "\\bar{\\mathcal {G}}^{reg} \\right|_{d=2N+2}$ , we expand $\\bar{\\mathcal {G}}^{reg}_\\tau $ in a power series in $\\tau $ and pick out the coefficient of $\\tau ^N$ .", "The generating function $\\bar{\\mathcal {G}}^{reg}_\\tau $ , as pointed out in Ref.", "logaweylgas, is remarkable in its form, which is identical to that of the generating function for gauge and gravitational anomaly polynomials.", "An obvious but nontrivial consequence is that the transport coefficients, which can be derived using eq.", "(REF ), are polynomials in $T$ and $\\mu $ in all spacetime dimensions." ], [ "Conclusion and Discussion", "We have derived the anomalous contributions to the relativistic hydrodynamic currents from a microscopic semiclassical description of Weyl fermions, which do agree with the corresponding expressions derived earlier using thermodynamic constraints[9], [19].", "Starting from a semiclassical theory, our calculation exposes the role the (nonabelian) Berry curvature plays in the semiclassial dynamics of chiral fermions.", "The semiclassical formalism encodes the anomaly via the collisionless Boltzmann equation[18], $\\mathcal {L}_V (f \\Omega _H) \\sim \\mathcal {A}$ .", "Our approach, originally proposed to derive the anomalies from a semiclassical calculation, is complementary to the usual hydrodynamic approach, which takes the anomaly as given and explores its conseqences on the transport properties of the system.", "In order to compute the hydrodynamic currents at a finite temperature, we needed both positive and negative energy sectors of a single positive chirality Weyl node.", "Even though the contribution of negative energy states is minuscule for $\\mu \\gg T$ , it was required to obtain closed form expressions for the currents, where the transport coefficients turn out to be polynomials in $T$ and $\\mu $ .", "This is in contrast to the calculation in Ref logaweylgas, where the contributions from the “particle” and “antiparticle” sectors were needed, which correspond to Weyl nodes of different chiralities in $4N$ spacetime dimensions.", "A careful reader might have noticed that despite our claims to have derived the relativistic currents, our expression for the symplectic form is not manifestly Lorentz invariant, as the definition of a Berry phase explicitly requires us to choose a foliation of the spacetime and to treat space and time on different footings.", "Indeed, a Lorentz invariant description has been attempted[24], [25], leading to a nontrivial implementation of the Lorentz symmetry on our $x$ and $p$ variables as well as a magnetic moment correction to the energy.", "However, such corrections occur at $O(\\hbar ^2)$ , so that our semiclassical symplectic form is accurate to $O(\\hbar )$ , and furthermore, the corrections do not affect the derivation of anomaly.", "Thus, we believe that the anomalous contributions obtained in this paper should be accurate to the lowest nontrivial order in $\\hbar $ in a WKB-like expansion.", "The semiclassical description used in this paper was proposed[18] for Weyl fermions coupled to a nonabelian gauge field with a compact gauge group.", "Hence, the calculation presented in this paper readily generalizes to anomalous currents for hydrodynamics coupled to a nonabelian gauge field, provided the chemical potentials commute.", "However, as the nonabelian anomalies, being a breakdown of a covariant conservation law, cannot be interpreted as a spectral flow in any obvious way, the physical content of such a calculation is open to interpretation.", "The similarity of the generating function for the grand potential current to the generating function for the gauge anomaly, and the corresponding replacement rules that it naturally suggests, is one of the many remarkable and mysterious results concerning anomalies.", "Whether the similarity points to a deeper connection (for instance, in holographic terms), or is a mere mathematical coincidence remains a very intriguing open question.", "This project was supported by the National Science Foundation under grant NSF DMR 13-06011." ], [ "Fermi-Dirac distribution and integrals", "Consider a gas of fermions in the grand canonical ensemble.", "Owing to the Pauli exclusion principle, a given microstate can either be unoccupied or occupied by exactly one fermion.", "Thus, the 1-particle grand canonical partition function $z$ is $z = \\sum _{\\text{states}} e^{-\\beta (\\varepsilon -\\mu )} = 1 + e^{-\\beta (\\varepsilon -\\mu )},$ where $\\varepsilon $ is the energy of the microstate, $\\beta = T^{-1}$ is the inverse temperature and $\\mu $ is the chemical potential.", "The corresponding grand potential $g$ is $g = -\\frac{1}{\\beta } \\ln z = -\\frac{1}{\\beta } \\ln \\left( 1 + e^{-\\beta (\\varepsilon - \\mu )} \\right).$ The grand potential is the generator for a variety of other relevant functions, such as the probability of occupation of a given state $f$ (Fermi-Dirac distribution) or the 1-particle entropy $h$ : $f = - \\frac{\\partial g}{\\partial \\mu } = \\frac{1}{1 + e^{\\beta (\\varepsilon - \\mu )}}, \\quad h = -\\frac{\\partial g}{\\partial T}.$ For fermions, we also have the highly nontrivial relation $g = (\\varepsilon - \\mu )f - T h, $ which can be obtained in a straightforward fashion using a definition of $h$ in terms of the occupation probability $f$ : $h= & \\; - \\sum _{\\text{states}} p_i \\ln p_i = - f \\ln f - (1-f) \\ln (1-f) \\\\= & \\; \\frac{1}{1 + e^{\\beta (\\varepsilon - \\mu )}} \\ln \\left( 1 + e^{\\beta (\\varepsilon - \\mu )} \\right) + \\frac{e^{\\beta (\\varepsilon - \\mu )}}{1 + e^{\\beta (\\varepsilon - \\mu )}} \\ln \\left( \\frac{1 + e^{\\beta (\\varepsilon - \\mu )}}{e^{\\beta (\\varepsilon - \\mu )}} \\right) \\\\= & \\; \\ln \\left( 1 + e^{\\beta (\\varepsilon - \\mu )} \\right) - \\frac{e^{\\beta (\\varepsilon - \\mu )}}{1 + e^{\\beta (\\varepsilon - \\mu )}} \\beta (\\varepsilon - \\mu ) \\\\= & \\; \\ln \\left(1 + e^{-\\beta (\\varepsilon - \\mu )} \\right) + \\beta (\\varepsilon - \\mu ) \\left[1 - \\frac{e^{\\beta (\\varepsilon - \\mu )}}{1 + e^{\\beta (\\varepsilon - \\mu )}} \\right] \\\\= & \\; \\beta \\left[ - g + (\\varepsilon - \\mu ) f \\right].$ Integrals: We seek to evaluate the integral $I(\\sigma ) = \\int _{-\\infty }^\\infty \\frac{d\\varepsilon }{2\\pi } e^{\\sigma \\varepsilon } \\left[ g(\\varepsilon ) - (\\varepsilon - \\mu ) \\Theta (-\\varepsilon ) \\right].$ The Heaviside integral part is easy to evaluate: $\\int _{-\\infty }^0 \\frac{d\\varepsilon }{2\\pi } (\\varepsilon - \\mu ) e^{\\sigma \\varepsilon } = \\left( \\frac{\\partial }{\\partial \\sigma } -\\mu \\right) \\int _{-\\infty }^0 \\frac{d\\varepsilon }{2\\pi } e^{\\sigma \\varepsilon } = \\left( \\frac{\\partial }{\\partial \\sigma } -\\mu \\right) \\frac{1}{2\\pi \\sigma } = - \\frac{1 + \\mu \\sigma }{2\\pi \\sigma ^2}.", "$ For the remaining integral, substitute $s = e^{\\beta (\\varepsilon - \\mu )} \\Rightarrow ds = \\beta s \\, d\\varepsilon $ and integrate by parts to get $\\int _{-\\infty }^\\infty \\frac{d\\varepsilon }{2\\pi } g(\\varepsilon ) e^{\\sigma \\varepsilon }= & \\; - \\frac{1}{\\beta } \\int _0^\\infty \\frac{ds}{2\\pi \\beta \\, s} \\ln \\left( 1 + \\frac{1}{s} \\right) \\left( s\\, e^{\\beta \\mu } \\right)^{\\frac{\\sigma }{\\beta }} \\nonumber \\\\= & \\; - \\frac{e^{\\mu \\sigma }}{2\\pi \\beta ^2} \\int _0^\\infty ds \\, s^{\\frac{\\sigma }{\\beta }-1} \\ln \\left( 1 + \\frac{1}{s} \\right) \\nonumber \\\\= & \\; - \\frac{e^{\\mu \\sigma }}{2\\pi \\beta ^2} \\left[ \\left.", "\\frac{\\beta }{\\sigma }s^{\\frac{\\sigma }{\\beta }} \\ln \\left( 1 + \\frac{1}{s} \\right) \\right|_0^\\infty - \\frac{\\beta }{\\sigma } \\int _0^\\infty ds \\, s^{\\frac{\\sigma }{\\beta }} \\left( -\\frac{1}{s(s+1)} \\right) \\right] \\nonumber \\\\= & \\; - \\frac{e^{\\mu \\sigma }}{2\\pi \\sigma \\beta } \\int _0^\\infty ds \\frac{s^{\\frac{\\sigma }{\\beta }-1}}{s+1} = -\\frac{e^{\\mu \\sigma }}{2\\sigma \\beta \\sin \\left( \\pi \\sigma / \\beta \\right)} $ where in the last line, we have used the integral $\\int _0^\\infty ds \\frac{s^{\\alpha -1}}{1+s} = \\frac{\\pi }{\\sin (\\pi \\alpha )}, \\quad 0 < \\alpha < 1.$ Thus, from eq.", "(REF ) and eq.", "(REF ), we get $I(\\sigma ) = -\\frac{1}{2\\pi \\sigma ^2} \\left[ \\frac{\\frac{\\pi \\sigma }{\\beta }}{\\sin \\left( \\frac{\\pi \\sigma }{\\beta } \\right)} e^{\\mu \\sigma } - (1 + \\mu \\sigma ) \\right], $ We also note that $\\frac{1}{2\\pi \\sigma ^2} \\frac{\\frac{\\pi \\sigma }{\\beta }}{\\sin \\left( \\frac{\\pi \\sigma }{\\beta } \\right)} e^{\\mu \\sigma } = \\frac{1}{2\\pi \\sigma ^2} + \\frac{\\mu }{2\\pi \\sigma } + \\left( \\frac{\\mu ^2}{4\\pi } + \\frac{\\pi }{12 \\beta ^2} \\right)\\sigma + O(\\sigma ^2).$ Thus, the integral over the $\\Theta (-\\varepsilon )$ term precisely subtracts off the singularities of the divergent integral over $g(\\varepsilon )$ ." ], [ "Symplectic forms in noninertial frames", "Consider the generalized Liouville 1-form for the dynamics of a classical particle on $\\mathbb {R}^{n,1}$ with an isotropic momentum-dependent Hamiltonian: $\\eta _H = p_i dx^i - \\mathcal {H}(|\\mathbf {p}|) dt$ We seek a Hamiltonian formulation of this system as seen from a noninertial frame of reference.", "We switch frames by a time-dependent change of coordinate $x^i = O^i_{\\, j}(w^j + \\xi _j)$ , where $w(t)$ corresponds to a Galilean boost and $O(t) \\in \\mathrm {SO}(n)$ to a time-dependent rotation, so that $\\xi _i$ is the position coordinate in the noninertial frame.", "The derivation of a suitable symplectic form describing the dynamics in the noninertial frame then involves a choice of the definition of “momentum”.", "The most straightforward choice is the canonical momentum, defined as $\\pi _i = \\pi _j (O^{-1})^{\\, j}_i$ , which preserves the canonical ($p_idx^i$ ) form of $\\eta _H$ .", "We also define the velocity of the frame as $v^i = \\partial _t w^i$ and its vorticity as $ (O^{-1} \\partial _t O)_{ij} = -\\frac{1}{2} \\omega _{ij}$ , both of which may depend on time.", "The vorticity satisfies $\\omega _{ij} = -\\omega _{ji}$ , which simply follows from the orthogonality of $O$ .", "The Liouville form becomes $\\eta _H = \\pi _i d\\xi ^i - \\left[ \\mathcal {H}- \\pi _i v^i + \\frac{1}{2} \\omega ^{ij} \\pi _i \\xi _j \\right] dt \\equiv \\pi _i d\\xi ^i - \\mathcal {H}^{\\prime } dt,$ where assuming a slowly accelerating and rotating frame, we have only retained the terms linear in $\\omega $ and $v$ .", "Thus, for the canonical momentum, the change of frame keeps the symplectic structure invariant, while changing the Hamiltonian.", "In other words, ${\\xi }$ and $\\pi $ are canonically conjugate.", "An alternative choice of momentum is the kinetic momentum, which intends to keep the equation of motion for $\\dot{\\xi }^i$ invariant.", "To wit, consider the symplectic form in rotating coordinates $\\rho _H \\equiv d\\rho _H = d\\pi _i \\wedge d\\xi ^i - \\left[ \\frac{\\partial \\mathcal {H}}{\\partial \\pi _i} d\\pi _i - v^i d\\pi _i + \\frac{1}{2} \\omega ^{ij} \\left( \\pi _i d\\xi _j + \\xi _j d\\pi _i \\right) \\right] \\wedge dt.$ The equations of motion become $\\dot{\\xi }_i = \\frac{\\partial \\mathcal {H}}{\\partial \\pi _i} - v^i + \\frac{1}{2} \\omega ^{ij} \\xi _j, \\quad \\dot{\\pi }^i = - \\frac{1}{2} \\omega ^{ij} \\pi _j.$ Then, one seeks the kinetic momentum $\\psi ^i$ , in terms of which the equation of motion for $\\xi $ becomes $\\dot{\\xi }^i = \\partial \\mathcal {H}/ \\partial \\psi _i$ .", "We elucidate this by examples in the following.", "Massive case: Consider a massive classical particle, so that $\\mathcal {H}= \\frac{|\\mathbf {p}|^2}{2m} = \\frac{|\\pi |^2}{2m} \\Rightarrow \\frac{\\partial \\mathcal {H}}{\\partial \\psi _i} = \\frac{\\pi _i}{m}.$ Then, the kinetic momentum is defined by setting $\\dot{\\xi }_i = \\frac{\\psi _i}{m} \\Rightarrow \\pi ^i = \\psi _i + m v_i - \\frac{1}{2} m \\, \\omega ^{ij} \\xi ^j,$ so that $\\mathcal {H}^{\\prime }= & \\; \\frac{1}{2 m} \\left| \\psi _i + m v_i - \\frac{1}{2} m \\, \\omega ^{ij} \\xi ^j \\right|^2 - \\left( \\psi _i + m v_i - \\frac{1}{2} m \\, \\omega ^{ij} \\xi ^j \\right) \\left( v^i - \\frac{1}{2} \\omega ^{ik} \\right) \\\\= & \\; \\frac{\\psi ^2}{2 m^2} + \\psi _i v^i - \\frac{1}{2} \\omega ^{ij} \\psi _i \\xi _j - \\psi _i v^i + \\frac{1}{2} \\omega ^{ij} \\psi _i \\xi _j + \\text{second order terms} \\\\= & \\; \\frac{\\psi ^2}{2 m^2} + \\text{second order terms}.$ It is precisely this cancellation that we seek in defining the kinetic momentum.", "Thus to linear order in $v$ and $\\omega $ , defining $a^i = \\partial _t v^i$ and $\\alpha _{ij} = \\partial _t \\omega _{ij}$ , the symplectic form becomes $\\rho _H = d\\psi _i \\wedge d\\xi ^i + \\frac{1}{2} m \\, \\omega _{ij} d\\xi ^i \\wedge d\\xi ^j + m \\left(a_i + \\alpha _{ij} \\xi ^j \\right) dt \\wedge d\\xi ^i - d \\left( \\frac{|\\psi |^2}{2m} \\right) \\wedge dt.$ We combine the inertial terms as $\\Omega = \\frac{1}{2} \\Omega _{\\mu \\nu } dx^\\mu dx^\\nu = \\frac{1}{2} \\omega _{ij} d\\xi ^i \\wedge d\\xi ^j + \\left(a_i + \\alpha _{ij} \\xi ^j \\right) dx^0 \\wedge d\\xi ^i,$ the symplectic form simply becomes $\\rho _H = d\\psi _i \\wedge d\\xi ^i + m \\, \\Omega - d\\mathcal {H}\\wedge dt.$ Here, $m \\omega $ corresponds to the Coriolis force, $m a$ to the inertial force and $m \\alpha _{ij} \\xi ^j dt dx^i$ to the tangential acceleration due to a variable angular velocity.", "This does not capture the centrifugal force, as we have ignored the terms at $O(\\omega ^2)$ .", "Massless case: For massless particles, $\\mathcal {H}= c|\\mathbf {p}| = c|\\pi |; \\; c= \\pm 1$ , so that the equations of motion become $\\dot{\\xi }_i = c\\hat{\\pi }^i - v^i - \\frac{1}{2} \\omega ^{ij} \\xi _j, \\quad \\dot{\\pi }^i = \\frac{1}{2} \\omega ^{ij} \\pi _j.$ Taking a cue from the massive case, consider a definition of kinetic momentum as $\\pi ^i = \\psi _i + c |\\psi | \\left( v_i - \\frac{1}{2} \\omega ^{ij} \\xi ^j \\right); \\quad c^2 = 1.$ Expanding $\\mathcal {H}^{\\prime } = c|\\pi |= c|\\psi | \\left[ 1 + 2 c\\hat{\\psi }^i \\left( v_i - \\frac{1}{2} \\omega ^{ij} \\xi ^j \\right) \\right]^{1/2} = c|\\psi | + \\psi ^i \\left( v_i - \\frac{1}{2} \\omega ^{ij} \\xi ^j \\right) + O(\\Omega ^2),$ we again get a cancellation in $\\mathcal {H}^{\\prime }$ at linear order, so that $\\mathcal {H}^{\\prime } = c|\\psi |$ .", "Thus, the symplectic form becomes $\\rho _H = d\\psi _i \\wedge d\\xi ^i + c|\\psi | \\, \\Omega - c\\, d|\\psi | \\wedge dt.$ Note that in considering the Galilean boosts (instead of Lorentz boosts), we are ignoring the effect of time dilation, including which will lead to corrections at the next order in $\\Omega $ ." ], [ "No-drag frame", "In this appendix, we show that the frame with respect to which our Weyl fluid is in equilibrium satisfies the “no-drag” condition described by Stephanov and Yee[19].", "In $3+1$ dimensions, the grand potential current becomesNote that there are missing factors of $2\\pi $ in the expansions of $\\bar{\\mathcal {G}}_{anom}$ in Ref.", "logaweylgas, eq.", "A.12 - A.15, as the expansion should always have the combination $\\frac{qB}{2\\pi }$ instead of $qB$ .", "$\\bar{\\mathcal {G}}= - \\left( \\frac{\\mu ^3}{24\\pi ^2} + \\frac{\\mu T^2}{24} \\right) u \\wedge \\omega - \\left( \\frac{\\mu ^2}{8\\pi ^2} + \\frac{T^2}{24} \\right) u \\wedge B,$ so that $\\bar{\\mathcal {J}}= & \\; -\\frac{\\partial \\bar{\\mathcal {G}}}{\\partial \\mu } = \\left( \\frac{\\mu ^2}{4\\pi ^2} + \\frac{T^2}{12} \\right) u \\wedge \\omega _{A} + \\frac{\\mu }{4\\pi ^2} u \\wedge B \\nonumber \\\\\\bar{\\mathcal {S}}= & \\; -\\frac{\\partial \\bar{\\mathcal {G}}}{\\partial T} = \\frac{\\mu T}{6} u \\wedge \\omega _{A} + \\frac{T}{12} u \\wedge B \\nonumber \\\\\\bar{q}= & \\; \\bar{\\mathcal {G}}+ \\mu \\bar{\\mathcal {J}}+ T \\bar{\\mathcal {S}}= \\left( \\frac{\\mu ^3}{6\\pi ^2} + \\frac{\\mu T^2}{6} \\right) u \\wedge \\omega _{A} + \\left( \\frac{\\mu ^2}{8\\pi ^2} + \\frac{T^2}{24} \\right) u \\wedge B,$ where $\\omega _{A} = \\frac{1}{2} \\omega $ .", "Thus, we can identify the coefficients $\\begin{array}{lll}\\xi _{J, \\omega } = \\frac{1}{12} T^2 + \\frac{1}{4\\pi ^2}\\mu ^2, \\quad &\\xi _{S, \\omega } = \\frac{1}{6} \\mu T, \\quad &\\xi _{T, \\omega } = \\frac{1}{6}\\mu T^2 + \\frac{1}{6\\pi ^2}\\mu ^3, \\\\\\xi _{J, B} = \\frac{1}{4\\pi ^2}\\mu , &\\xi _{S, B} = \\frac{1}{12}T, &\\xi _{T, B} = \\frac{1}{24} T^2 + \\frac{1}{8\\pi ^2} \\mu ^2.\\end{array}$ Comparing with the expressions obtained in Ref [19] $\\begin{array}{lll}\\xi _{J, \\omega } = X_B T^2 + C \\mu ^2, \\quad &\\xi _{S, \\omega } = X_\\omega T^2 + 2 X_B \\mu T, \\quad &\\xi _{T, \\omega } = \\frac{2}{3} \\left( X_\\omega T^3 + 3 X_B \\mu T^2 + C \\mu ^3 \\right) \\\\\\xi _{J, B} = C \\mu , &\\xi _{S, B} = X_B T, &\\xi _{q, B} = \\frac{1}{2} \\left( X_B T^2 + C \\mu ^2 \\right),\\end{array}$ we can readily identify $C = \\frac{1}{4\\pi ^2}, \\quad X_B = \\frac{1}{12}, \\quad X_\\omega = 0.$ Thus, our transport coefficients are indeed consistent with the no-drag frame." ] ]
1606.04945
[ [ "Ring wormholes via duality rotations" ], [ "Abstract We apply duality rotations and complex transformations to the Schwarzschild metric to obtain wormhole geometries with two asymptotically flat regions connected by a throat.", "In the simplest case these are the well-known wormholes supported by phantom scalar field.", "Further duality rotations remove the scalar field to yield less well known vacuum metrics of the oblate Zipoy-Voorhees-Weyl class, which describe ring wormholes.", "The ring encircles the wormhole throat and can have any radius, whereas its tension is always negative and should be less than $-c^4/4G$.", "If the tension reaches the maximal value, the geometry becomes exactly flat, but the topology remains non-trivial and corresponds to two copies of Minkowski space glued together along the disk encircled by the ring.", "The geodesics are straight lines, and those which traverse the ring get to the other universe.", "The ring therefore literally produces a hole in space.", "Such wormholes could perhaps be created by negative energies concentrated in toroidal volumes, for example by vacuum fluctuations." ], [ "Model", "We consider the theory with a scalar field, ${\\cal L}=[R-2\\epsilon \\, (\\partial \\Phi )^2]\\sqrt{-g}\\,.$ Here the parameter takes two values, either $\\epsilon =+1$ corresponding to the conventional scalar field, or $\\epsilon =-1$ corresponding to the phantom field.", "We shall denote $\\Phi =\\phi $ for $\\epsilon =1$ and $\\Phi =\\psi $ for $\\epsilon =-1$ .", "The phantom field $\\psi $ can formally be viewed as $\\phi $ continued to imaginary values.", "We use units in which $G=c=1$ , unless otherwise stated.", "Assume the spacetime metric to be static, $ds^2=-e^{2U} dt^2+e^{-2U} dl^2,$ where $dl^2=\\gamma _{ik}dx^i dx^k$ and $U$ , $\\gamma _{ik}$ , $\\Phi $ depend on the spatial coordinates $x^k$ .", "Denoting by $U_k\\equiv \\partial _k U$ , the Lagrangian becomes ${\\cal L}=[\\stackrel{(3)}{R}-2 \\gamma ^{ik}(U_i U_k+\\epsilon \\,\\Phi _i\\Phi _k)]\\,\\sqrt{\\gamma }\\,,$ where $\\stackrel{(3)}{R}$ is the Ricci scalar for $\\gamma _{ik}$ .", "The field equations are $\\stackrel{(3)}{R}_{ik}=2(U_i U_k +\\epsilon \\, \\Phi _i \\Phi _k) \\,,~~~\\Delta U=0,~~~\\Delta \\Phi =0.$ We notice that for $\\epsilon =1$ these equations are invariant under $U\\leftrightarrow \\Phi =\\phi ,~~~~~~~~\\gamma _{ik}\\rightarrow \\gamma _{ik}\\,.$ If the 3-metric $\\gamma _{ik}$ is chosen to be of the Weyl form, $dl^2 = e^{2k} \\bigl ( d \\rho ^2 + dz ^2 \\bigr ) + \\rho ^2 d \\varphi ^2,$ where $k$ and also $U,\\Phi $ depend only on $\\rho ,z$ , then the equations reduce toOne can check that this reduction is indeed consistent, which would not be the case if the scalar field had a potential.", "The Weyl formulation is also consistent for an electrostatic vector field, so that it applies, for example, within the electrostatic sector of dilaton gravity.", "$&&U_{\\rho \\rho }+\\frac{1}{\\rho }\\,U_\\rho +U_{zz}=0,~~~~~~~~~~\\Phi _{\\rho \\rho }+\\frac{1}{\\rho }\\,\\Phi _\\rho +\\Phi _{zz}=0,\\nonumber \\\\&&k_z=2\\rho \\,[U_\\rho U_z+\\epsilon \\, \\Phi _\\rho \\Phi _z],~~~~~~~k_\\rho =\\rho \\,[U_\\rho ^2-U_z^2+\\epsilon \\,(\\Phi _\\rho ^2-\\Phi _z^2)].$ These equations admit a scaling symmetry mapping solutions to solutions, $U\\rightarrow \\lambda \\, U,~~~~~\\Phi \\rightarrow \\lambda \\, \\Phi ,~~~~~k\\rightarrow \\lambda ^2\\, k,$ where $\\lambda $ is a constant parameter.", "In addition, they are invariant under $U\\leftrightarrow \\Phi ,~~~~~~k\\rightarrow \\epsilon \\, k\\,.$" ], [ "BE wormhole", "We start from the Schwarzschild metric (here $d\\Omega ^2=d\\vartheta ^2+\\sin ^2\\vartheta d\\varphi ^2$ ) $ds^2=-\\left(1-\\frac{2m}{r}\\right)dt^2+\\frac{dr^2}{1-2m/r}+r^2 d\\Omega ^2.$ Introducing $x=r-m$ it can be put to the form (REF ) with $U=\\frac{1}{2}\\ln \\left(\\frac{x-m}{x+m}\\right),~~~~~~~dl^2=dx^2+(x^2-m^2)\\,d\\Omega ^2,~~~~~\\Phi =0.$ Applying to this the symmetry (REF ) gives a new solution which is ultrastatic but has a non-trivial scalar fieldThis is a member of the Fisher-Janis-Newman-Winicour solution family [21], [22].", "The rest of this family can be similarly obtained from the Schwarzschild metric by applying, instead of the discrete symmetry (REF ), continuous $SO(2)$ rotations in the $U,\\phi $ space.", "$U=0,~~~dl^2=dx^2+(x^2-m^2)\\,d\\Omega ^2,~~~\\phi =\\frac{1}{2}\\ln \\left(\\frac{x-m}{x+m}\\right).$ If we now continue the parameter $m$ to the imaginary region, $m\\rightarrow i\\mu $ , then the metric remains real, while $\\phi \\rightarrow \\frac{1}{2}\\ln \\left(\\frac{x-i\\mu }{x+i\\mu }\\right)=i\\psi ,$ where $\\psi $ is real.", "This gives the solution for the phantom field, $ds^2&=&-dt^2+dx^2+(x^2+\\mu ^2) (d\\vartheta ^2+\\sin ^2\\vartheta d\\varphi ^2), \\nonumber \\\\\\psi &=&\\arctan \\left(\\frac{x}{\\mu }\\right),$ which is precisely the BE wormhole [13], [14].", "Here $x\\in (-\\infty ,+\\infty )$ and the limits $x\\rightarrow \\pm \\infty $ correspond to two asymptotically flat regions, while $x=0$ is the wormhole throat – the 2-sphere of minimal radius $\\mu $ .", "The geodesics can get through the throat from one asymptotic region to the other.", "Passing to the coordinates $z=x\\cos \\vartheta ,~~\\rho =\\sqrt{x^2+\\mu ^2}\\,\\sin \\vartheta ~~\\leftrightarrow ~~x\\pm i\\mu \\cos \\vartheta =\\sqrt{\\rho ^2+(z\\pm i\\mu )^2}\\,,$ the wormhole metric assumes the Weyl form, $ds^2=-dt^2+e^{2k}(d\\rho ^2+dz^2)+\\rho ^2d\\varphi ^2$ with $e^{2k}=\\frac{x^2+\\mu ^2}{x^2+\\mu ^2\\cos ^2\\vartheta }\\,.$ Since $d\\rho ^2+dz^2=\\left.\\left.\\frac{x^2+\\mu ^2\\cos ^2\\vartheta }{x^2+\\mu ^2}\\right[dx^2+(x^2+\\mu ^2) d\\vartheta ^2\\right],$ the metric (REF ) is indeed equivalent to the one in Eq.", "(REF )." ], [ "Ring wormhole", "Let us apply to the solution (REF ) the symmetry (REF ), $U\\leftrightarrow \\psi $ , $k\\rightarrow -k$ , and then act on the result with the scaling symmetry (REF ) with $\\lambda =\\sigma $ .", "This gives the axially symmetric metrics with $\\Phi =0$ , $ds^2&=&-e^{2 U}dt^2+e^{-2 U}\\,dl^2,~~~~~~~~U=\\sigma \\arctan \\left(\\frac{x}{\\mu }\\right), \\\\dl^2&=&\\left.\\left.\\left(\\frac{x^2+\\mu ^2\\cos ^2\\vartheta }{x^2+\\mu ^2}\\right)^{\\sigma ^2+1}\\right[dx^2+(x^2+\\mu ^2) d\\vartheta ^2\\right]+(x^2+\\mu ^2) \\sin ^2\\vartheta \\, d\\varphi ^2.", "\\nonumber $ We note that the 3-metric $dl^2$ is invariant under $x\\rightarrow -x$ while the Newtonian potential $U$ changes sign.", "As we shall see below, these solutions describe ring wormholes.", "We thought initially these solutions were new, but actually they were described before although remain very little known.", "We obtained them from the BE wormhole by rotating away the scalar field.", "However, they can also be obtained from the Schwarzschild metric without introducing any scalar fields at all.", "Passing to the coordinates $z=x\\cos \\vartheta ,~~\\rho =\\sqrt{x^2-m^2}\\sin \\vartheta ~~\\leftrightarrow ~~x\\pm m\\cos \\vartheta =\\sqrt{\\rho ^2+(z\\pm m)^2}\\,,$ the Schwarzschild metric (REF ) assumes the Weyl form with $dl^2=dx^2+(x^2-m^2)\\,d\\Omega ^2=e^{2k}(d\\rho ^2+d z^2)+\\rho ^2d\\varphi ^2,$ where $e^{2k}=\\frac{x^2-m^2}{x^2-m^2 \\cos ^2\\vartheta },$ since $d\\rho ^2+dz^2=\\left.\\left.\\frac{x^2-m^2\\cos ^2\\vartheta }{x^2-m^2}\\right[dx^2+(x^2-m^2) d\\vartheta ^2\\right].$ Acting on this with the scaling symmetry (REF ) with $\\lambda =-\\delta $ gives vacuum metrics of the prolate Zipoy-Voorhees (ZV) class [23], [24] $ds^2&=&-\\left(\\frac{x-m}{x+m}\\right)^{-\\delta } +\\left(\\frac{x-m}{x+m}\\right)^{\\delta }dl^2, \\\\dl^2&=&\\left.\\left.\\left(\\frac{x^2-m^2\\cos ^2\\vartheta }{x^2-m^2}\\right)^{1-\\delta ^2}\\right[dx^2+(x^2-m^2) d\\vartheta ^2\\right]+(x^2-m^2) \\sin ^2\\vartheta \\, d\\varphi ^2\\,.", "\\nonumber $ These metrics have been relatively well studied (see for example [25], [26]) since they can be used to describe deformations of the Schwarzschild metric.", "If we now continue the parameters to the imaginary region, $\\delta \\rightarrow i\\sigma ,~~~~~~m\\rightarrow i\\mu ,$ then metrics (REF ) remain real and reduce to the oblate ZV solutions [23], [24], which are precisely the metrics (REF ).", "Much less is known about these solutions.", "Their wormhole nature has been discussed [27], [28] but no systematic description is currently available and moreover the solutions remain largely unknown.", "We shall therefore describe their essential properties and find new surprising features, notably the existence of a non-trivial flat space limit." ], [ "Properties of the ring wormhole", "Similarly to the BE wormhole (REF ), solution (REF ) has a throat and two asymptotically flat regions.", "This can be most easily seen by noting that at the symmetry axis, where $\\cos ^2\\vartheta =1$ , the line element $dl^2$ reduces precisely to that in (REF ), while $U$ is a bounded function.", "Therefore, the solution indeed describes a wormhole interpolating between two asymptotically flat regions.", "One has for $x\\rightarrow \\pm \\infty $ (assuming that $\\mu >0$ ) $e^{2U}\\rightarrow e^{\\pm \\sigma \\pi }\\left(1- \\frac{2\\sigma \\mu }{x}+\\ldots \\right),$ hence the time coordinates in both limits differ by a factor of $e^{2\\sigma \\pi }$ while the ADM mass $M=\\pm \\sigma \\mu $ is positive when seen from one wormhole side and negative from the other.", "The wormhole is traversable, which can be easily seen by considering geodesics along the symmetry axis.", "A particle of mass ${\\rm m}$ and energy $E$ follows a trajectory $x(s)$ defined by $\\left(\\frac{dx}{ds}\\right)^2+{\\rm m}^2 e^{2U(x)}=E^2\\,,$ hence $x(s)\\in (-\\infty ,+\\infty )$ if $E^2>{\\rm m}^2 e^{\\sigma \\pi }$ .", "Since $U(x)$ grows with $x$ , it follows that the wormhole attracts particles in the $x>0$ region but repels them in the $x<0$ region.", "Therefore, it acts as a “drainhole\" sucking in matter from the $x>0$ region and spitting it out to the $x<0$ region.", "The metric (REF ) degenerates at $x=0$ , $\\vartheta =\\pi /2$ , which corresponds to a circle of radius $\\mu $ .", "In its vicinity one can define $y=\\mu \\cos \\vartheta $ and then $dl^2$ in (REF ) becomes $dl^2&=&\\left(\\frac{x^2+y^2}{\\mu ^2}\\right)^{\\sigma ^2+1}(dx^2+dy^2)+\\mu ^2 d\\varphi ^2 \\\\&=&\\mu ^2\\,r^{2\\sigma ^2+2}(dr^2+r^2d\\alpha ^2)+\\mu ^2 d\\varphi ^2 =\\mu ^2\\,(dR^2+R^2d\\omega ^2)+\\mu ^2 d\\varphi ^2, \\nonumber $ where $x=\\mu r\\cos \\alpha $ , $y=\\mu r\\sin \\alpha $ while $R={r^{\\sigma ^2+2}}/{(\\sigma ^2 +2)}$ and $\\omega =(\\sigma ^2+2)\\,\\alpha $ .", "The metric in the last line in (REF ) looks flat, however, since $\\alpha \\in [0,2\\pi )$ , the angle $\\omega $ ranges from zero to $\\omega _{\\rm max}=2\\pi \\,(\\sigma ^2+2)> 2\\pi .$ Therefore, there is a negative angle deficit $\\Delta \\omega =2\\pi -\\omega _{\\rm max}=-(\\sigma ^2+1)\\,2\\pi $ and hence a conical singularity at $R=0$ .", "Since the singularity stretches in the $\\varphi $ -direction, it sweeps a ring of radius $\\mu $ .", "Such line singularities are known to be generated by singular matter sources distributed along lines – cosmic strings.", "Their energy per unit length (tension) ${T}$ is related to the angle deficit via $\\Delta \\omega =(8\\pi G/c^4)\\,{T}$ (we restore for a moment the correct physical dimensions), hence the string has a negative tension ${T}=-\\frac{(1+\\sigma ^2)\\,c^4}{4G}\\,.$ Therefore, the wormhole solution (REF ) can be viewed as sourced by a ring with negative tension.", "The solution carries the free parameter $\\mu $ , which determines the radius of the ring, and also $\\sigma $ determining the value of the ring tension.", "It is quite surprising that the self-gravitating ring can be in equilibrium and moreover admits an exact solution.", "If the string tension was positive then the string loop would be shrinking and the system would not be static (we are unaware of any exact solutions in this case, although one can construct exact initial data for a string loop [29]).", "For the solution (REF ) the ring has a negative tension and is similar to a strut, hence it should rather tend to expand, which tendency is counterbalanced by the gravitational attraction.", "This must be the reason for which such an equilibrium system exists (we do not know if the equilibrium is stable).", "Even more remarkable is the following.", "Setting $\\sigma =0$ in (REF ) yields $ds^2=-dt^2+\\left.\\left.\\frac{x^2+\\mu ^2\\cos ^2\\vartheta }{x^2+\\mu ^2}\\,\\right[dx^2+(x^2+\\mu ^2) d\\vartheta ^2\\right]+(x^2+\\mu ^2) \\sin ^2\\vartheta \\, d\\varphi ^2.$ This metric is vacuum and ultrastatic hence it must be flat, so one may think this limit is trivial.", "However, close to the symmetry axis, where $\\cos ^2\\vartheta \\approx 1$ , the geometry reduces to that for the BE wormhole (REF ).", "The wormhole throat is located at $x=0$ , but, unlike in the BE case, this is not a sphere but rather a sphere squashed to a disk with the geometry $\\mu ^2(\\cos ^2\\vartheta d\\vartheta ^2+\\sin ^2\\vartheta d\\varphi ^2)=\\mu ^2 (d\\xi ^2+\\xi ^2 d\\varphi ^2),$ where $\\xi =\\sin \\vartheta \\in [0,1]$ .", "The disk is encircled by the ring.", "Figure: Wormhole topology.", "The x,ϑx,\\vartheta coordinates cover the whole of the manifold, the throatbeing at x=0x=0, ϑ∈[0,π]\\vartheta \\in [0,\\pi ] and the ring is at x=0x=0, θ=π/2\\theta =\\pi /2.The Weyl charts D + D_{+} and D - D_{-} cover, respectively, the x>0x>0 and x<0x<0 regions.Lines of constant xx are the (half)-ellipses in the Weyl coordinates,the ring corresponds to the branch points at z=0,ρ=μz=0,\\rho =\\mu , while the throat corresponds to the branch cuts OμO\\mu .The upper edge of the cut in the D + D_{+} patch is identified with the lower edge of the cut in the D - D_{-} patchand vice-versa.", "A winding around the ring in the x,ϑx,\\vartheta coordinates corresponds to two windingsin Weyl coordinates.The coordinates $x,\\vartheta ,\\varphi $ are global, but to study the geodesics it is convenient to pass to the Weyl coordinates (REF ).", "The metric then becomes manifestly flat $ds^2=-dt^2+d\\rho ^2+dz^2+\\rho ^2d\\varphi ^2.$ However, this does not mean the solution describes empty Minkowski space.", "Indeed, the ring is still present and has the tension ${T}=-\\frac{c^4}{4G}\\,.$ As a consequence of this the topology is non-trivial.", "This follows from the fact that the Weyl coordinates $\\rho ,z$ cover either only the $x<0$ part or only the $x>0$ part of the wormhole.", "Indeed, one has $\\frac{z^2}{x^2}+\\frac{\\rho ^2}{x^2+\\mu ^2}=1,$ hence lines of constant $x$ in the $\\rho ,z$ plane are ellipses insensitive to the sign of $x$ , therefore one needs two Weyl charts, $D_\\pm :\\lbrace \\rho \\ge 0,\\, -\\infty <z<\\infty \\rbrace $ , to cover the whole of the manifold – one for $x<0$ and one for $x>0$ (see Fig.REF ) [30].", "The coordinate transformation $(x,\\vartheta )\\rightarrow (\\rho ,z)$ degenerates at the branch point $z=0$ , $\\rho =\\mu $ corresponding to the position of the ring.", "Each chart therefore has a branch cut ending at the branch point, which may be chosen to be $z=0$ , $\\rho \\in [0,\\mu ]$ (see Fig.REF ).", "The upper edge of the $D_{+}$ cut is identified with the lower edge of the $D_{-}$ cut and vice-versa.", "These cuts correspond to the wormhole throat, so that we see once again that the throat is a disk encircled by the ring.", "The geodesics in Weyl coordinates are straight lines.", "Those which miss the ring always stay in the same coordinate chart (geodesic $A$ in Fig.REF ), while those threading the ring (geodesics $B,C$ in Fig.REF ) continue from $D_{+}$ to $D_{-}$ thus traversing the wormhole.", "Therefore, the negative tension ring genuinely creates a hole in space through which one can observe another universe as well as get thereA similar possibility to create a hole in flat space, although without specifying the precise structure of the matter source needed, was discussed in Ref.", "[31], following the discussion in Ref. [9]..", "This reminds one of Alice observing the room behind the looking glass and next jumping there.", "An object falling through the ring can be seen from behind, while viewed from the side it is not seen coming from the other side (see Fig.REF ).", "Figure: Particles entering the ring are not seen coming out from the other sideIt is striking that the line source is enough to create the wormhole.", "Usually the negative energy supporting the wormhole is distributed over the 3-volume, as for the BE solution, or at least over a 2-surface, as for the thin-shell wormholes [32].", "We see however that it is sufficient to distribute the negative energy only along the one-dimensional ring, which is presumably easier to achieve than in other cases since a smaller amount of the NEC violation is needed.", "It is also interesting to note that the arguments based on the Raychaudhuri equation do not apply since the wormhole either does not affect the geodesics at all if they miss it, or it absorbs them if they hit it – its edges are sharp.", "It seems plausible that the wormhole edges could be smoothened without changing the global structure if the singular ring is replaced by a regular hoop-shaped energy distribution of finite thickness and with the same tension.", "Inside the hoop the energy density is finite hence the geometry must be regular, while outside the energy is zero and the geometry should be more or less the same as for the original ring.", "This suggests that wormholes could be created by negative energies concentrated in toroidal volumes, for example by vacuum fluctuations.", "However, the energy density needed to create a ring wormhole is extremely high.", "The absolute value of the negative tension (REF ) coincides with the highest possible value for a positive tension (force), according to the maximum tension principle in General Relativity conjectured in [33], [34].", "This conjecture is supported, for example, by the fact that the angle deficit of a cosmic string cannot exceed $2\\pi $ .", "Numerically, ${T}=-3.0257\\times 10^{43}$  Newtons $\\approx -3\\times 10^{39}$ Tonnes.", "To create a ring of radius $R=1$ metre, say, one needs a negative energy equivalent to the $10^{-3}$ Solar masses, $2\\pi RT/ c^2\\approx -0.001\\times M_{\\odot }$ .", "At the same time, one can imagine that such rings could by quantum fluctuations appear spontaneously from the vacuum and then disappear again.", "Particles crossing the ring during its existence would no longer be accessible from our universe after the ring disappears.", "If true, this would be a potential mechanism for the loss of quantum coherence.", "In summary, by applying generating techniques we (re)-discovered solutions describing ring wormholes, computed the ring tension, and showed that their geometry can be precisely flat: the ring creates a hole in space." ], [ "Acknowledgements", "G.W.G.", "thanks the LMPT for hospitality and acknowledges the support of “Le Studium\" – Institute for Advanced Studies of the Loire Valley.", "M.S.V.", "was partly supported by the Russian Government Program of Competitive Growth of the Kazan Federal University." ] ]
1606.04879
[ [ "Characterization of Parity-Time Symmetry in Photonic Lattices Using\n Heesh-Shubnikov Group Theory" ], [ "Abstract We investigate the properties of parity-time symmetric periodic photonic structures using Heesh-Shubnikov group theory.", "Classical group theory cannot be used to categorize the symmetry of the eigenmodes because the time-inversion operator is antiunitary.", "Fortunately, corepresentations of Heesh-Shubnikov groups have been developed to characterize the effect of antiunitary operators on eigenfunctions.", "Using the example structure of a one-dimensional photonic lattice, we identify the corepresentations of eigenmodes at both low and high symmetry points in the photonic band diagram.", "We find that thresholdless parity-time transitions are associated with particular classes of corepresentations.", "The approach is completely general and can be applied to parity-time symmetric photonic lattices of any dimension.", "The predictive power of this approach provides a powerful design tool for parity-time symmetric photonic device design." ], [ " Supplemental Material For Characterization of Parity-Time Symmetry in Photonic Lattices Using Heesh-Shubnikov Group Theory Adam Mock School of Engineering and Technology, Central Michigan University, ET 100, Mount Pleasant, MI 48859, USA and Science of Advanced Materials Program Central Michigan University, Mount Pleasant, MI 48859, USA 1.", "Symmetry operators at $k=0$ The group multiplication for the Heesh-Shubnikov little group (HSLG) of $k=0$ , $\\mathcal {M}^{k=0} = (e, m, \\xi , \\mu )$ , is provided below.", "Because $k=0$ , the space group representation does not play a role.", "$\\mathcal {M}^{k=0}$ is isomorphic to $C_{2v} (2mm)$ .", "Table: Multiplication table for ℳ k=0 \\mathcal {M}^{k=0}.The unitary subgroup of $\\mathcal {M}^{k=0}$ is isomorphic to $C_{1h} (m)$ .", "Table: Multiplication table for unitary subgroup of ℳ k=0 \\mathcal {M}^{k=0}.Table: Character table of C 1h (m)C_{1h} (m) point group along with results of Dimmock and Wheeler test (α\\alpha ) and corepresentation type (Correp.).2.", "Discussion of modes at $k=0$ To provide some insight into the lack of $\\mathcal {PT}$ degeneracy at $k=0$ , consider the effect of $\\xi $ on an eigenfunction at $k = 0$ : $\\xi H^{0,i}(x) = [H^{0,i}(x-\\Lambda /2)]^*$ where $H^{0,i}(x)$ is the eigenfunction of the $i$ th band at $k = 0$ .", "From Table I in the main text we know that $\\xi H^{0,i}(x) = \\pm H^{0,i}(x)$ , so we conclude $\\pm H^{0,i}(x) = [H^{0,i}(x-\\Lambda /2)]^*$ .", "The implication is that the field intensity $|H^{0,i}(x)| = |[H^{0,i}(x-\\Lambda /2)]^*| = |H^{0,i}(x-\\Lambda /2)|$ is spatially periodic with a period of $\\Lambda / 2$ .", "With this periodicity, spatially shifting an eigenfunction by $\\Lambda / 2$ reproduces the same field intensity instead of transforming into a field intensity with a different preferential overlap with the gain or loss regions.", "Therefore, these modes overlap the gain and loss regions equally which results in real eigenfrequencies.", "The $\\Lambda /2$ periodicity of $|H^{0,i}(x)|$ is apparent in the fields shown in Fig.", "2 in the main text.", "3.", "Symmetry operators at $k=\\pi / \\Lambda $ At $k=\\pi / \\Lambda $ , the space group representation plays a role in the analysis.", "Space group operations are described by the Seitz operator $\\lbrace R|t \\rbrace $ which consists of a point symmetry operation $R$ followed by a translation $t$ .", "The multiplication operation between Seitz operators is given by $ \\lbrace R|t \\rbrace \\lbrace S|t^{\\prime } \\rbrace = \\lbrace RS|Rt^{\\prime }+t \\rbrace $ As example consider $\\overline{e}m = \\lbrace E|2m\\Lambda +\\Lambda \\rbrace \\lbrace \\sigma | 2n\\Lambda \\rbrace = \\lbrace \\sigma | 2(m+n)\\Lambda + \\Lambda \\rbrace \\equiv \\lbrace \\sigma | 2n\\Lambda + \\Lambda \\rbrace =\\overline{m} $ where we have noted that a translation of $2(m+n)\\Lambda $ is equivalent to $2n\\Lambda $ for integer $m$ and $n$ at $k = \\frac{\\pi }{\\Lambda }$ .", "Consider now $m \\overline{e} = \\lbrace \\sigma | 2n\\Lambda \\rbrace \\lbrace E|2m\\Lambda + \\Lambda \\rbrace = \\lbrace \\sigma | \\sigma (2m\\Lambda +\\Lambda ) + 2n\\Lambda \\rbrace = \\lbrace \\sigma | -(2m\\Lambda + \\Lambda ) + 2n\\Lambda \\rbrace = \\lbrace \\sigma | 2(n-m)\\Lambda - \\Lambda \\rbrace \\equiv \\lbrace \\sigma | 2n\\Lambda +\\Lambda \\rbrace = \\overline{m} $ .", "When the Seitz operator contains the time inversion operation $\\mathcal {T}$ , it applies only to other time inversion operators and not to the point and spatial symmetry operations.", "As example consider $m \\mu = \\lbrace \\sigma | 2m\\Lambda \\rbrace \\lbrace \\mathcal {T}\\sigma |2n\\Lambda + \\Lambda /2 \\rbrace = \\mathcal {T} \\lbrace \\sigma \\sigma | 2m\\Lambda + \\sigma (2n\\Lambda + \\Lambda /2) \\rbrace =\\mathcal {T} \\lbrace E | 2m\\Lambda - (2n\\Lambda + \\Lambda /2) \\rbrace = \\lbrace \\mathcal {T} | 2(m-n) \\Lambda - \\Lambda /2) \\rbrace =\\lbrace \\mathcal {T} | 2(m-n-1) \\Lambda + \\Lambda + \\Lambda /2) \\rbrace \\equiv \\lbrace \\mathcal {T} | 2n \\Lambda + \\Lambda + \\Lambda /2) \\rbrace = \\overline{\\xi }$ .", "Table REF shows the group multiplication table of the HSLG of $k = \\frac{\\pi }{\\Lambda }$ , $\\mathcal {M}^{k = \\pi / \\Lambda }$ .", "$\\mathcal {M}^{k = \\pi / \\Lambda }$ is isomorphic to $C_{4v} (4mm)$ .", "Table: Multiplication table for ℳ k=π/Λ \\mathcal {M}^{k=\\pi / \\Lambda }.Table: Multiplication table for unitary subgroup of ℳ k=π/Λ \\mathcal {M}^{k=\\pi / \\Lambda }.", "It is isomorphic to C 2v (2mm)C_{2v} (2mm).4.", "Construction of corepresentation table for $\\mathcal {M}^{k=\\pi / \\Lambda }$ Here we provide calculation details for the construction of the corepresentation table for the group $\\mathcal {M}^{k=\\pi / \\Lambda }$ .", "For Type (a) corepresentations, the elements of the unitary subgroup retain their classical representations $\\mathbf {\\Gamma }_i(R) = \\mathbf {\\Delta }_i(R)$ for $R \\in \\mathcal {N}$ .", "Corepresentations of the antiunitary elements are given in terms of the classical representations of the unitary subgroup according to $\\mathbf {\\Gamma }_i(RA) = \\mathbf {\\Delta }_i(R)\\beta $ with $\\beta \\beta ^* = \\mathbf {\\Delta }_i(A^2)$ .", "Here we choose $A = \\xi $ .", "Using a different operator for $A$ will give the same results.", "$\\mathbf {\\Gamma }_i(\\xi e) = \\mathbf {\\Gamma }_i(\\xi ) = \\mathbf {\\Delta }_i(e)\\beta $ $\\mathbf {\\Gamma }_i(\\xi \\overline{e}) = \\mathbf {\\Gamma }_i(\\overline{\\xi }) = \\mathbf {\\Delta }_i(\\overline{e})\\beta $ $\\mathbf {\\Gamma }_i(\\xi m) = \\mathbf {\\Gamma }_i(\\mu ) = \\mathbf {\\Delta }_i(m)\\beta $ $\\mathbf {\\Gamma }_i(\\xi \\overline{m}) = \\mathbf {\\Gamma }_i(\\overline{\\mu }) = \\mathbf {\\Delta }_i(\\overline{m})\\beta $ The classical representations of the group $C_{2v}$ are provided in Table II in the main text.", "The corepresentations resulting from this calculation make up the first two rows of Table III in the main text.", "For Type (c) corepresentations, the corepresentations of the elements of the unitary subgroup are given by Eq.", "5 in the main text.", "With $A = \\xi = S\\mathcal {T}$ , we identify $S = \\lbrace E|2n\\Lambda + \\Lambda /2 \\rbrace $ and $S^{-1} = \\lbrace E | -2n\\Lambda - \\Lambda /2 \\rbrace $ .", "Determination of the matrix elements is shown in Table REF .", "Table: Determination of the matrix elements for the Type (c) corepresentations of the unitary operators in ℳ k=π/Λ \\mathcal {M}^{k=\\pi / \\Lambda }For Type (c) corepresentations, the corepresentations of the antiunitary elements are given by Eq.", "6 in the main text.", "With $A = \\xi $ , we identify $A^{-1} = \\overline{\\xi }$ .", "Determination of the matrix elements is shown in Table REF .", "Table: Determination of the matrix elements for the Type (c) corepresentations of the antiunitary operators in ℳ k=π/Λ \\mathcal {M}^{k=\\pi / \\Lambda }.The corepresentations resulting from this calculation make up the last two rows of Table III in the main text.", "5.", "Unitary matrix that transforms $\\mathbf {\\Gamma _3}$ into $\\mathbf {\\Gamma _4}$ at $k=\\pi / \\Lambda $ .", "Corepresentations $\\mathbf {\\Gamma }_3$ and $\\mathbf {\\Gamma }_4$ are equivalent because they can be transformed into each other via $U \\mathbf {\\Gamma }_3 U^{-1} = \\mathbf {\\Gamma }_4$ for the unitary elements and $U \\mathbf {\\Gamma }_3 (U^*)^{-1} = \\mathbf {\\Gamma }_4$ for the antiunitary elements.", "Any matrix of the form $U = \\left( \\begin{array}{cc}0 & -e^{-i \\theta } \\\\e^{i \\theta } & 0 \\\\\\end{array} \\right)$ accomplishes this transformation for real $\\theta $ .", "6.", "Discussion of modes at $k=\\pi / \\Lambda $ .", "Using the gain and loss modes ($H^{k,i}_g(x)$ and $H^{k,i}_l(x)$ , respectively) as a basis of the corepresentation at wave number $k=\\pi / \\Lambda $ and empty-lattice band crossing point $i$ , Table III in the main text shows that when the unitary symmetry operators ($R$ ) are applied, the eigenfunctions transform according to diagonal matrices $R \\left( \\begin{array}{c}H^{ k,i}_g(x) \\\\H^{ k,i}_l(x) \\\\\\end{array} \\right) =\\left( \\begin{array}{cc}r_{11} & 0 \\\\0 & r_{22} \\\\\\end{array} \\right)\\left( \\begin{array}{c}H^{k,i}_g(x) \\\\H^{k,i}_l(x) \\\\\\end{array} \\right) =\\left( \\begin{array}{c}r_{11} H^{ k,i}_g(x) \\\\r_{22} H^{ k,i}_l(x) \\\\\\end{array} \\right)$ showing that application of $R$ to the 2D basis does not mix or exchange the eigenfunctions.", "This is to be contrasted to the transformation of the eigenfunctions upon application of the antiunitary operators ($A$ ) which are represented by anti-symmetric matrices: $A \\left( \\begin{array}{c}H^{k,i}_g(x) \\\\H^{k,i}_l(x) \\\\\\end{array} \\right) =\\left( \\begin{array}{cc}0 & a_{12} \\\\a_{21} & 0 \\\\\\end{array} \\right)\\left( \\begin{array}{c}H^{k,i}_g(x) \\\\H^{k,i}_l(x) \\\\\\end{array} \\right) =\\left( \\begin{array}{c}a_{12} H^{ k,i}_l(x) \\\\a_{21} H^{ k,i}_g(x) \\\\\\end{array} \\right).$ Here it is seen that application of the antiunitary operator transforms a loss mode into a gain mode and vice versa.", "As discussed previously [40] gain and loss modes have similar symmetry, and if only the field intensity is visualized, the difference between them is a spatial shift such that the gain modes preferentially overlap the gain regions, and the loss modes preferentially overlap the loss regions.", "Because the antiunitary operators include this spatial shift, the mode transformation properties of the matrix corepresentations make physical sense.", "7.", "Corepresentation for $\\mathcal {M}^{k= 0.8 (\\pi / \\Lambda )}$ Corepresentation for $\\mathcal {M}^{k= 0.8 (\\pi / \\Lambda )}$ are Type (a).", "Using $A = \\mu $ yields $\\beta \\beta ^* = \\mathbf {\\Delta }_i(\\mu ^2) = 1$ , so use $\\beta = \\pm 1$ .", "$\\mathbf {\\Gamma }_i(R\\mu ) = \\mathbf {\\Gamma }_i(e \\mu ) = \\mathbf {\\Delta }_i(e)\\beta = \\beta = \\pm 1$ .", "Inspection of the fields in Fig.", "REF indicates that $H_a$ has corepresentation $\\mathbf {\\Gamma }^+$ , and $H_b$ has corepresentation $\\mathbf {\\Gamma }^-$ .", "Table: Corepresentations of ℳ k=0.8(π/Λ) \\mathcal {M}^{k=0.8 (\\pi / \\Lambda )}.Figure: Magnetic field (H z (x)H_z(x)) spatial distribution in 𝒫𝒯\\mathcal {PT} symmetric 1D lattice at k=0.8(π/Λ)k = 0.8 (\\pi / \\Lambda ) (points a and b in Fig.", "2(b) in the main text).Transformed fields are shown to verify the characters in Table .8.", "Broken $\\mathcal {PT}$ symmetry Fig.", "REF shows a band diagram for the 1D photonic lattice shown in Fig.", "1(a) in the main text but with $n = 2 \\pm i0.7$ .", "This larger value for $n_i$ results in $\\mathcal {PT}$ transition points closer to $k=0$ than in Fig.", "1(b).", "In fact for the fifth and sixth bands, the $\\mathcal {PT}$ transition point has reached $k=0$ , and these bands exhibit broken $\\mathcal {PT}$ symmetry for $0 \\le k < \\pi / \\Lambda $ .", "Fig.", "REF shows the spatial field distribution for the modes marked by a circle at $k=0.8(\\pi / \\Lambda )$ in Fig.", "REF (a) and labeled g and l in Fig.", "REF (b).", "When $n_i = 0.25$ , the fields at $k=0.8(\\pi / \\Lambda )$ are shown in Fig.", "REF , and the characters in Table REF accurately describe the symmetry of the fields.", "When $n_i = 0.70$ , the field symmetry is no longer of Type (a), and the mode is in the broken $\\mathcal {PT}$ symmetry regime.", "As shown in Fig.", "REF , the fields have Type (c) corepresentations where the antiunitary operator has a matrix corepresentation of the form $\\mu = \\left( \\begin{array}{cc}0 & 1 \\\\1 & 0 \\\\\\end{array} \\right)$ This matrix is determined from observation of the field and does not follow from symmetry analysis.", "Figure: Photonic band diagram for the 1D photonic lattice shown in Fig.", "1(a) in the main text with n=2±i0.7n = 2 \\pm i0.7.", "(a) Real part of the frequencies.", "(b) Imaginary part of the frequencies.Figure: Magnetic field (H z (x)H_z(x)) spatial distribution in 𝒫𝒯\\mathcal {PT} symmetric 1D lattice with n i =0.70n_i = 0.70 at k=0.8(π/Λ)k = 0.8 (\\pi / \\Lambda ) (point marked by a circle in Fig.", ").Transformed fields are shown to illustrate that mode symmetry differs from that predicted by group theory." ] ]
1606.05044
[ [ "Global existence of weak solutions for generalized quantum MHD equation" ], [ "Abstract We prove the existence of a weak solution to a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data.", "The existence of a global weak solution is established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent \\gamma>1." ], [ "Introduction", "   The evolution of quantum MHD equations in $\\Omega = T^{2}$ is described by the following system $&\\partial _{t}n +{\\rm div}(n u) = 0, \\\\&\\partial _{t}(n u)+{\\rm div}(n u\\otimes u)+\\nabla (P(n)+P_{c}(n))- 2 {\\rm div} (\\mu (n) D(u)) \\\\ \\nonumber &- \\nabla (\\lambda (n) {\\rm div} u) - \\frac{\\hbar ^{2}}{2} n \\nabla (\\varphi ^{\\prime }(n) \\Delta \\varphi (n))- (\\nabla \\times B)\\times B=0, \\\\&\\partial _{t} B - \\nabla \\times (u\\times B) + \\nabla \\times (\\nu _{b}(\\rho )B \\times B)=0,\\\\& n(x,0)= n_{0}(x), ~nu(x,0)=m_{0},\\\\& B(x,0)= B_{0}(x),~{\\rm div} B_{0}=0,$ where the functions $n, u $ and $B$ represent the mass density,the velocity field, the magnetic field.", "$P(n)=n^{\\gamma }$ stands for the pressure, $P_{c}$ is a singular continuous function and called cold pressure.", "$\\mu (n), \\lambda (n)$ denote the fluid viscosity coefficient.", "$\\hbar >0$ quantum plank constant, $\\nu _{b}$ the magnetic viscosity coefficient.", "Our analysis is based on the following physically grounded assumptions: [A1]The viscosity coefficient is determined by the Newton's rheological law $\\mu (n)=\\mu _{0} n^{\\alpha },~0<\\alpha \\le 1, \\lambda (n)=2(n\\mu ^{\\prime }(n)- \\mu (n)).$ where $\\mu $ and $\\lambda $ are respectively the shear and bulk constant viscosity coefficients, the dispersion term $\\varphi $ satisfying $\\varphi (n)=n^{\\alpha },$ [A2]The cold pressure $P_{c}$ obeys the following growth assumption: $\\lim _{n\\rightarrow 0}P_{c}(n)= + \\infty ,$ More precisely, we assume $P_{c}^{\\prime }(n)=\\left\\lbrace \\begin{aligned}&c_{1} n^{-\\gamma ^{-}-1}~~n\\le 1,\\\\&c_{2} n^{\\gamma -1} ~~n >1,\\end{aligned}\\right.$ where constants $\\gamma ^{-}, \\gamma ^{+}\\ge 1, c_{1},c_{2}>0$ .", "[A3]The positive coefficient $\\nu _{b}$ is supposed to be a continuous function of the density, bounded from above and taking large values for small and large densities.", "More precisely, we assume that there exists $B>0$ , positive constants $d_{0},d_{0}^{\\prime }, d_{1},d_{1}^{\\prime }$ large enough, $2\\le a<a^{\\prime }<3$ and $b\\in [0,\\infty ]$ such that $\\forall s<B,~~\\frac{d_{0}}{s^{a}}\\le \\nu _{b}(s)\\le \\frac{d_{0}^{\\prime }}{s^{a^{\\prime }}}~~and ~~\\forall s\\ge B, d_{1}\\le \\nu _{b}(s) \\le d_{1}^{\\prime }s^{b}.$ Define function $H(n)$ and $\\xi (n)$ as follow : $\\left\\lbrace \\begin{aligned}&nH^{\\prime }(n)-H(n)=P(n),~~nH^{\\prime }_{c}(n)-H_{c}(n)=P_{c}(n)\\\\&n\\xi ^{\\prime }(n)=\\mu ^{\\prime }(n),\\end{aligned}\\right.$ Now, we give the definition of a weak solution to ().", "Definition 1.1 We call $(n, u, B)$ is as a weak solution to the problem (), if the following is satisfied.", "(1)the density $n$ is a non-negative function satisfying the internal identity $\\int _{0}^{T} \\int _{\\Omega } n \\partial _{t} \\phi + n u\\cdot \\nabla \\phi dx dt+ \\int _{\\Omega } n_{0} \\phi (0)dx=0,$ for any test function $\\phi \\in C^{\\infty }([0,T]\\times \\overline{\\Omega }), \\phi (T)=0$ .", "(2) the momentum equation in () holds in $D^{\\prime }((0,T)\\times \\Omega )$ (in the sense of distributions), that means, $\\begin{aligned}&\\int _{\\Omega }m_{0} \\phi (0) dx+\\int _{0}^{T} \\int _{\\Omega } n u \\cdot \\partial _{t} \\phi + n (u\\otimes u): \\nabla \\phi + P {\\rm div} \\phi dx dt\\\\&=\\frac{\\hbar ^{2}}{2} \\int _{0}^{T} \\int _{\\Omega } \\varphi ^{\\prime }(n) \\Delta \\varphi (n) \\nabla n \\phi + n \\phi ^{\\prime } \\Delta \\varphi (n) {\\rm div} \\phi dx dt + 2\\int _{0}^{T} \\int _{\\Omega } \\mu (n) D(u) \\nabla \\phi dx dt\\\\& + \\int _{0}^{T} \\int _{\\Omega }\\lambda (n) {\\rm div} u {\\rm div} \\phi dx dt- \\nu _{b} \\int _{0}^{T} \\int _{\\Omega }(\\nabla \\times B) \\times B \\cdot \\phi dx dt ,\\end{aligned}$ for any test function $\\phi \\in C^{\\infty }([0,T]\\times \\overline{\\Omega }), \\phi (T)=0$ .", "(3) the magnetic field $B$ is a non-negative function satisfying $\\begin{aligned}&\\int _{\\Omega } B_{0} \\phi (0)dx =\\int _{0}^{T} \\int _{\\Omega } (B \\cdot \\partial _{t}\\phi + (u \\times B) \\cdot (\\nabla \\times \\phi )- \\nu _{b} \\nabla B :\\nabla \\phi ) dx dt ,\\end{aligned}$ for any test function $\\phi \\in C^{\\infty }([0,T]\\times \\overline{\\Omega }), \\phi (T)=0$ .", "Remark 1.1 If $\\mu (n)=0, \\lambda (n)=0, \\alpha =\\frac{1}{2}, B=0, P_{c}(n)=0, $ , then quantum hydrodynamic equation () becomes $&\\partial _{t}n +{\\rm div}(n u) = 0, \\\\&\\partial _{t}(n u)+{\\rm div}(n u\\otimes u)+\\nabla P(n)- \\frac{\\hbar ^{2}}{2} n \\nabla (\\frac{\\Delta \\sqrt{n}}{\\sqrt{n}})=0, $ If $\\mu (n)=0, \\lambda (n)=0, \\alpha =1, B=0, P_{c}(n)=0, \\nu =0$ , then quantum hydrodynamic equation () becomes $&\\partial _{t}n +{\\rm div}(n u) = 0, \\\\&\\partial _{t}(n u)+{\\rm div}(n u\\otimes u)+\\nabla P(n)- \\sigma n \\nabla \\Delta n=0, $ Now, we are ready to formulate the main result of this paper.", "Theorem 1.2 (global existence for the quantum Euler model) Let $\\Omega =T^{2}$ be a periodic box.", "Assume $ T>0$ .", "Let the initial data satisfy $\\left\\lbrace \\begin{aligned}&\\frac{1}{2}\\int _{T^{2}} (\\frac{|m|^{2}}{2n_{0}} dx + [H(n_{0})+ H_{c}(n_{0})]+ \\frac{\\hbar ^{2}}{2} |\\nabla \\varphi (n_{0})|^{2} +\\int _{T^{d}} |B_{0}|^{2})dx <+\\infty ,\\\\&\\frac{\\nabla \\mu (n_{0})}{\\sqrt{n_{0}}} \\in L^{2}(\\Omega ), \\ \\end{aligned}\\right.$ Then problem ()-(REF ) posses at least one global weak solution $n,u, B$ .", "This paper is organized as follows.", "In section 2, we establish the global existence of solutions to the Faedo-Galerkin approximation to ().", "In section 3 we deduce the B-D entropy energy estimates, which is a key part in the analysis process.", "In section 4 and 5, we use the uniform estimates to recover the original system by vanishing the artificial viscosity and artificial pressure respectively, therefore the main theorem is proved by using the weak convergence method." ], [ "Faedo-Galerkin approximation", "   In this section, we prove the existence of solutions to approximate viscous quantum Euler equations.", "We proceed similarly as [16.", "Chap.", "7]." ], [ "Local existence of solutions.", "Let $T>0$ , and let $(e_{k})$ be an orthonormal basis of $L^{2}(T^{d})$ which is also an orthogonal basis of $H^{1}(T^{d})$ .", "Introduce the finite-dimensional space $X_{N}=span \\lbrace e_{1}, e_{2}, ..., e_{N} \\rbrace $ , $N\\in \\mathbb {N}$ .", "Let $(n_{0},u_{0}, B_{0})\\in C^{\\infty }(T^{d})^{3}$ be some initial data satisfying $n_{0}\\ge \\delta >0$ for $x\\in T^{d}$ for some $\\delta >0$ , and let the velocity $u \\in C^{0}([0,T]; X_{n})$ be given.", "We notice that $u$ can be written as $u(x,t)= \\sum _{j=1}^{N} \\lambda _{j} (t) e_{j}(x), ~~(x,t)\\in T^{d} \\times [0,T]$ for some function $\\lambda _{i}(t),$ and the norm of $u$ in $C^{0}([0,T];X_{N})$ can be formulated as $\\Vert u\\Vert _{C^{0}([0,T];X_{N})} = \\max _{t\\in [0,T]} |\\sum _{j=1}^{N} \\lambda _{j}(t)|,$ As a consequence, $u$ can be bounded in $C^{0}([0,T]; C^{k}(T^{d}))$ for any $k\\in \\mathbb {N}$ , and there exists a constant $C>0$ depending on $k$ such that $\\Vert u\\Vert _{C^{0}([0,T];C^{k}(T^{d}))} \\le C \\Vert u\\Vert _{C^{0}([0,T];L^{2}(T^{d}))}.$ Therefore there exists solution operator $F: C^{0}([0,T];X_{N})\\rightarrow C^{0}([0,T];C^{3}(T^{d}))$ such that $n=F(u)$ be that the classical solution to $n_{t} + {\\rm div} (n u)= \\varepsilon \\Delta n,~n(x,0)=n_{0}~T^{d}\\times (0,T)$ The maximum principle provides the lower and upper bounds $\\begin{aligned}&\\inf _{x\\in T^{d}} n_{0}(x) \\exp (-\\int _{0}^{t}\\Vert {\\rm div} u\\Vert _{L^{\\infty }(T^{d})}ds) \\le n(x,t)\\\\&\\le \\sup _{x\\in T^{d}} n_{0}(x) \\exp (\\int _{0}^{t}\\Vert {\\rm div} u\\Vert _{L^{\\infty }(T^{d})}ds),~for~(x,t)\\in T^{d}\\times [0,T].\\end{aligned}$ Since the equation is linear with respect to $n$ , the operator $F$ is Lipschitz continuous in the following sense: $\\Vert F(v_{1})-F(v_{2})\\Vert _{C^{0}([0,T];C^{k}(T^{d}))} \\le C \\Vert v_{1}-v_{2}\\Vert _{C^{0}([0,T];L^{2}(T^{d}))}.$ Since we assumed that $n_{0}(x)\\ge \\delta >0$ , $n(t,x)$ is strictly positive.", "In view of (2.1), for $\\Vert v\\Vert _{C^{0}([0,T];L^{2}(T^{d}))}\\le c$ , there exist constants $\\underline{n}(c)$ and $\\overline{n}(c)$ such that $0 < \\underline{n}(c, \\varepsilon )\\le n(x,t) \\le \\overline{n}(c, \\varepsilon ).$ Next, we wish to obtain the solvability of the magnetic field on the space $X_{N}$ .", "To this end, for given $u$ above, we are looking for a unique function $B$ satisfying $&\\partial _{t} B - \\nabla \\times (u\\times B) + \\nabla \\times (\\nu _{b} \\times B)=0,\\\\& {\\rm div} B=0,\\\\& B(x,0)= B_{0}(x),$ which is a linear parabolic-type problem in $B$ .", "Therefore, by the standard Faedo-Galerkin methods, there exists a solution $B\\in L^{2}([0,T]; H^{1}(T^{3}))\\cap L^{\\infty }([0,T];L^{2}(T^{3})).$ to Eqs.(2.7).", "Further, there exists a continuous solution operator $G: C^{0}([0,T]; X_{N})\\rightarrow L^{2}([0,T]; H^{1}(T^{3})) \\cap L^{\\infty }([0,T]; L^{2}(T^{3}))$ by $G(v)=B$ .", "Now, for all test function $\\psi \\in C([0,T];X_{N}) $ satisfying $\\psi (\\cdot ,T)=0$ , we wish to solve the momentum equation on the space $X_{N}$ .", "To this end, for given $n=F(u), B=G(u)$ , we are looking for a function $u_{N}\\in C^{0}([0,T]; X_{N})$ such that $\\begin{aligned}&-\\int _{\\Omega } n_{0}u_{0}\\psi (\\cdot ,0)dx= \\int _{0}^{T}\\int _{T^{d}} ( nu_{N} \\cdot \\psi _{t} + (nu\\otimes u_{N}): \\nabla \\psi + P(n) {\\rm div} \\psi dx dt \\\\&- \\lambda \\int _{0}^{T}\\int _{T^{d}} \\Delta ^{s+1} (nu) : \\Delta ^{s}(n\\psi ) dx dt -\\frac{\\hbar ^{2}}{2} \\int _{0}^{T}\\int _{T^{d}} (\\varphi ^{\\prime }(n) \\Delta \\varphi (n) \\nabla n \\psi + n \\varphi ^{\\prime }(n) \\Delta \\varphi (n) {\\rm div} \\psi ) dx dt\\\\&-\\lambda \\int _{0}^{T} \\int _{T^{d}} \\Delta ^{s} ({\\rm div} (n \\psi )): \\Delta ^{s+1} n dx dt- 2\\int _{0}^{T} \\int _{T^{d}} \\mu (n) D(u_{N}) \\cdot \\nabla \\psi dx dt- \\varepsilon \\int _{0}^{T} \\int _{T^{d}} (\\nabla n \\nabla \\cdot ) u \\psi dx dt\\\\& - \\int _{0}^{T} \\int _{T^{d}} \\lambda (n) {\\rm div} u_{N} \\cdot {\\rm div} \\psi dx dt+ \\mu _{b} \\int _{0}^{T}\\int _{T^{d}} (\\nabla \\times B)\\times B \\cdot \\psi dxdt\\end{aligned}$ we will apply Banach's fixed point theorem to prove the local-in-time existence of solutions in the above equation.", "The regularization yields the $H^{1}$ regularity of $u_{N}$ which is needed to conclude the global existence of solutions.", "To solve (2.9), we follow Ref.6 and consider a family of linear operators, given a function $\\rho \\in L^{1}(T^{d})$ with $\\rho \\ge \\underline{\\rho }>0$ $M[\\rho ]: X_{N}\\rightarrow X_{N}^{\\star },~~<M[n]v,u>=\\int _{T^{d}} nv\\cdot u dx, ~~v,u\\in X_{N}.$ where the symbol $X_{N}^{\\star }$ stands for the dual space of $X_{N}$ .", "It is easy to see that the operator $M$ is invertible provided $n$ is strictly positive on $T^{d}$ , and $\\Vert M^{-1}[n]\\Vert _{L(X_{N}^{\\star }, X_{N})} \\le \\underline{\\rho }^{-1},$ where $L(X_{N}^{\\star }, X_{N})$ is the set of bounded linear mappings from $X_{N}^{\\star }$ to $X_{N}$ .", "Moreover, the identity $M^{-1}[n_{1}]-M^{-1}[n_{2}]= M^{-1}[n_{2}](M[n_{1}]-M[n_{2}])M^{-1}[n_{1}],$ can be used to get $\\Vert M^{-1}[n_{1}]-M^{-1}[n_{2}]\\Vert _{L(X_{N}^{\\star }, X_{N})} \\le C(N,\\underline{n}) \\Vert n_{1}-n_{2}\\Vert _{L^{2}(T^{d})},$ for any $n_{1},n_{2}$ such that $\\inf _{T^{d}} n_{1} \\ge n_{0} >0,~~\\inf _{T^{d}} n_{2} \\ge n_{0} >0,$ So, $M^{-1}$ is Lipschitz continuous in the sense of (2.8).", "Now the integral equation (2.9) can be rephrased as an ordinary differential equation on the finite-dimensional space $X_{N}$ $\\frac{d}{dt}(M[n(t)u_{N}(t)])= N[v,u_{N},n,B]$ where $n=F(u), B=G(u)$ and $\\begin{aligned}&<N[u,u_{N},n,B]>= \\int _{0}^{T} \\int _{T^{d}} (n_{N}u_{N}\\otimes u_{N}: \\nabla \\psi + (P(n_{N}+ P_{c}(n_{N})) {\\rm div} \\psi \\\\& - \\frac{\\hbar ^{2}}{2} (\\varphi ^{\\prime }(n_{N})\\Delta \\psi (n_{N}) \\nabla n_{N} \\psi + n_{N} \\varphi ^{\\prime } \\Delta \\varphi (n_{N}) {\\rm div} \\psi )dx+ \\lambda \\int _{0}^{T} \\int _{T^{d}}n_{N} \\nabla \\Delta ^{2s+1}(n_{N}u_{N})\\psi dx dt\\\\& \\lambda \\int _{0}^{T} \\int _{T^{d}} n_{N} \\Delta ^{s} {\\rm div} (n_{N}\\psi )\\cdot \\Delta ^{s+1} n_{N} dx - 2 \\int _{0}^{T} \\int _{T^{d}} \\mu (n_{N}) D(u_{N}) \\cdot \\nabla \\psi dx\\\\& -\\int _{0}^{T} \\int _{T^{d}} \\lambda (n_{N}) {\\rm div} n_{N} \\cdot {\\rm div} \\psi dx dt + \\nu _{b}\\int _{0}^{T} \\int _{T^{d}} (\\nabla \\times B_{N})\\times B_{N} \\cdot \\psi dx, ~\\psi \\in X_{N},\\end{aligned}$ The operator $N[u,u_{N},n,B]$ , defined for every $t\\in [0,T]$ as an operator from $X_{N}$ to $X_{N}^{\\star }$ is continuous in time.", "Then the existence of a unique solution to (2.9) can be obtained by using standard theory for systems of ordinary equations.", "In other words, for given $u$ , there exists a unique solution $u_{N}\\in C^{1}([0,T];X_{N})$ to (2.7).", "Integrating (2.9) over (0,t) yields the following nonlinear equation: $u_{N}= M^{-1}[F(u_{N})](t) (M[n_{0}] u_{0}+ \\int _{0}^{t} N(u_{N},u_{N}(s),n_{N},B_{N}))dt$ in $X_{N}$ .", "Because the operators $F, G, M^{-1}$ is Lipschitz continuous, this equation can be solved by evoking the fixed-pointed theorem of Banach on a short time interval $[0,T^{\\prime }]$ , where $T^{\\prime }\\le T$ , in the space $C^{0}([0,T^{\\prime }];X_{N})$ .", "In fact, we have even $u_{N}\\in C^{0}([0,T^{\\prime }];X_{N})$ .", "Thus, there exists a unique local-in-time solution $(n_{N},u_{N},B_{N})$ to (2.2),(2.7) and (2.4)." ], [ "Global existence of solutions", "In order to prove that the solution $(n_{N},u_{N},B_{N})$ constructed above exists on the whole time interval $[0,T]$ , it is sufficient to show that $u_{N}$ is bounded in $X_{N}$ on $[0,T^{\\prime }]$ by employing the energy estimate.", "Lemma 2.1 Let $T^{\\prime }\\le T$ , and let $n_{N}\\in C^{1}([0,T^{\\prime }];C^{3}(T^{d})), u_{N}\\in C^{1}([0,T^{\\prime }]; X_{N})$ and $B_{N}\\in L^{2}([0,T^{\\prime }]:H^{1}(T^{d}))\\cap L^{\\infty }([0,T^{\\prime }]; L^{2}(T^{d}))$ be a local-in-time solution to (2.2),(2.7), and (2.4) with $n=n_{N}, u=u_{N},B=B_{N}$ .", "Then $\\begin{aligned}& \\frac{d}{dt} E(n_{N}, u_{N}, B_{N}) + 2 \\int _{T^{d}} \\mu (n_{N})|\\nabla u_{N}|^{2}dx + \\int _{T^{d}} \\lambda (n_{N})|{\\rm div} u_{N}|^{2} dx + \\varepsilon \\int _{T^{d}} \\frac{1}{n}(P^{\\prime }(n)+P^{\\prime }_{c}(n))|\\nabla n|^{2} dx \\\\&+ \\nu _{b} \\int _{T^{d}} |\\nabla \\times B_{N}|^{2} dx + \\lambda \\int _{T^{d}} |\\Delta ^{s} \\nabla (n_{N}u_{N})|^{2} dx+ \\lambda \\varepsilon \\int _{T^{d}} |\\Delta ^{s+1} n_{N}|^{2}dx\\\\&+\\varepsilon \\int _{T^{d}}\\frac{\\hbar ^{2}}{2}\\varphi ^{\\prime } (n_{N}) \\Delta \\varphi (n_{N}) \\Delta n_{N} dx=0\\end{aligned}$ where $\\begin{aligned}& E(n_{N},u_{N},B_{N})= \\frac{1}{2}\\int _{T^{d}} n_{N}|u_{N}|^{2}dx + \\int _{T^{d}} [H(n_{N})+ H_{c}(n_{N})] dx\\\\&+ \\frac{\\hbar ^{2}}{2} \\int _{T^{d}} |\\nabla \\varphi (u_{N})|^{2}dx +\\frac{1}{2}\\int _{T^{d}} |B_{N}|^{2} dx + \\frac{1}{2}\\int _{T^{d}}\\frac{\\lambda }{2} |\\nabla ^{2s+1} n_{N}|^{2}dx,\\end{aligned}$ First we multipy (2.3) by $H^{\\prime }(n_{N})- \\frac{|u_{N}|^{2}}{2}- \\frac{\\hbar ^{2}}{2}\\varphi ^{\\prime } (n_{N}) \\Delta \\varphi (n_{N}) $ , integrate over $T^{d}$ , and integrate by parts: $\\begin{aligned}&0= \\int _{T^{d}} (\\partial _{t}H(n_{N})- \\frac{1}{2}|u_{N}|^{2}\\partial _{t} n_{N} + \\frac{\\hbar ^{2}}{2}\\partial _{t}|\\nabla \\varphi (n_{N})|^{2}- n_{N}(H^{\\prime \\prime }(n_{N})+H^{\\prime \\prime }_{c}(n_{N}))\\nabla n_{N}\\cdot u_{N}\\\\&+ n_{N}u_{N} \\cdot \\nabla u_{N} \\cdot u_{N} - \\frac{\\hbar ^{2}}{2}\\varphi ^{\\prime } (n_{N}) \\Delta \\varphi (n_{N}) {\\rm div} (n_{N}u_{N})+ \\varepsilon H^{\\prime \\prime }(n_{N})|\\nabla n_{N}|^{2} \\\\& - \\varepsilon \\nabla n_{N} \\cdot \\nabla u_{N}\\cdot u_{N}+ \\varepsilon \\frac{\\hbar ^{2}}{2}\\varphi ^{\\prime } (n_{N}) \\Delta \\varphi (n_{N}) \\Delta n_{N}) dx.\\end{aligned}$ Next, multipying the magnetic field equation (2.7) by $B_{N}$ we deduce that $\\int _{T^{d}} \\nabla \\times (u_{N}\\times B_{N})\\cdot B_{N} dx = \\frac{1}{2} \\int _{T^{d}}\\frac{d}{dt} |B_{N}|^{2} dx+ \\nu _{b} \\int _{T^{d}} |\\nabla \\times B_{N}|^{2}dx,$ Then using the test function $u=u_{N}, n=n_{N}, B=B_{N}=G(u_{N})$ in (2.9) and integrating by parts leads to $\\begin{aligned}&0= \\int _{T^{d}} (|u_{N}|^{2}\\partial _{t} n_{N} + \\frac{1}{2} n_{N}\\partial _{t}|u_{N}|^{2} - n_{N} u_{N} \\otimes u_{N}: \\nabla u_{N}+ (P^{\\prime }(n_{N})+P^{\\prime }_{c}(n_{N}) \\nabla n_{N}\\cdot u_{N}+ \\frac{\\lambda }{2}|\\nabla ^{2s+1} n|^{2} dx\\\\&-2\\int _{T^{d}} {\\rm div} (\\mu (n_{N})D(u_{N}))u_{N} dx -\\int _{T^{d}} \\nabla (\\lambda (n_{N}){\\rm div} u_{N})\\cdot u_{N} dx -\\frac{\\hbar ^{2}}{2} \\int _{T^{d}} n_{N} \\nabla (\\varphi ^{\\prime }\\Delta \\psi (n_{N})) u_{N} dx\\\\&-\\nu _{b} \\int _{T^{d}} (\\nabla \\times u_{N}) \\times B_{N} \\cdot B_{N} dx+\\lambda \\int _{T^{d}} |\\Delta ^{s} \\nabla (n_{N}u_{N})|^{2} dx+ \\lambda \\varepsilon \\int _{T^{d}} |\\Delta ^{s+1} n_{N}|^{2}dx\\end{aligned}$ Adding above three equations gives, since $n_{N}H^{\\prime \\prime }=p^{\\prime }(n_{N})$ .", "Thus the proof of Lemma 2.1 is finished.", "From Lemma 2.1 we have the following estimates: - the density estimates $\\begin{aligned}&\\Vert n_{N}\\Vert _{L^{\\infty }(0,T;L^{\\gamma ^{+}}(\\Omega ))} + \\Vert n_{N}\\Vert _{L^{\\infty }(0,T;L^{\\gamma ^{-}}(\\Omega ))}+\\Vert \\nabla \\varphi (n_{N})\\Vert _{L^{\\infty }(0,T;L^{2}(\\Omega ))} \\\\&+\\sqrt{\\varepsilon }\\Vert \\frac{1}{\\sqrt{n_{N}}}\\sqrt{\\frac{\\partial P_{c}}{\\partial n_{N}} }\\nabla n_{N}\\Vert _{L^{2}(0,T;L^{2}(\\Omega ))} \\le C ,\\end{aligned}$ - the velocity estimates $\\begin{aligned}&\\Vert \\sqrt{n_{N}}u_{N}\\Vert _{L^{\\infty }(0,T;L^{2}(\\Omega ))}+\\Vert \\sqrt{n_{N}}D( u_{N})\\Vert _{L^{2}(0,T;L^{2}(\\Omega ))}\\\\&+\\Vert \\sqrt{\\lambda }\\Delta ^{s}\\nabla (n_{N} u_{N})\\Vert _{L^{2}(0,T;L^{2}(\\Omega ))}\\le C ,\\end{aligned}$ By a interpolation inequality we can get the density $\\rho $ is bounded from below by a positive constant $\\begin{aligned}&\\Vert \\rho ^{-1}\\Vert _{L^{\\infty }((0,T)\\times \\Omega )}\\le \\Vert \\rho ^{-1}\\Vert _{L^{\\infty }((0,T);H^{2}})\\\\&\\le C(1+\\Vert \\nabla ^{3}\\rho \\Vert _{L^{\\infty }((0,T);L^{2}(\\Omega ))})^{3}(1+\\Vert \\rho ^{-1}\\Vert _{L^{\\infty }((0,T);L^{2}(\\Omega ))})^{4}\\le C(\\lambda ) ,\\end{aligned}$ above we require $\\gamma ^{-}>4$ and $2s+1\\ge 3$ .", "Combing with (2.15) we deduce the uniform bound for $\\mathbf {u}$ , thus we get a global approximating solution.", "The summarizing estimates (2.17)-(2.18) are uniform with the dimension N, thus we can extract the weakly convergent subsequences and pass the limit passage $N\\rightarrow \\infty $ in the Galerkin approximation." ], [ "This subsection is devoted to the limit passage $N\\rightarrow \\infty $ .", "Using estimates from the previous subsection we can extract weakly subsequences, whose limits satisfy the approximate system." ], [ "From (2.17)-(2.18) we deduce that $u_{N}\\rightarrow u ~~~weakly ~~in ~ L^{2}(0,T;W^{2s+1,2}(\\Omega ))$ and $n_{N}\\rightarrow n ~~~weakly ~~in ~ L^{2}(0,T;W^{2s+2,2}(\\Omega ))$ at least for a suitable subsequence.", "In addition the r.h.s.", "of the linear parabolic problem $\\begin{aligned}\\partial _{t} n+ div(n u)-\\varepsilon \\Delta n=0, \\\\\\rho (0,x)=\\rho _{\\lambda }^{0}(x),\\end{aligned}$ is uniformly bounded in $L^{2}(0,T;W^{2s,2}(\\Omega ))$ and the initial condition is sufficiently smooth, thus, applying he $L^{p}-L^{q}$ theory to this problem we conclude that $\\lbrace \\partial _{t}\\rho _{N}\\rbrace _{n=1}^{\\infty }$ is uniformly bounded in $L^{2}(0,T;W^{2s,2}(\\Omega ))$ .", "Hence, the standard compact embedding implies $\\rho _{N}\\rightarrow \\rho $ a.e.", "in $(0,T)\\times \\Omega $ and therefore passage to the limit in the approximate continuity equation is straightforward." ], [ "Having the strong convergence of the density, we start to identify the limit for $N\\rightarrow \\infty $ in the nonlinear terms of the momentum equation.", "The convective term.", "First, one observes that $\\rho _{N}\\mathbf {u}_{N}\\rightarrow \\rho \\mathbf {u} ~~~weakly^{*} ~~~in ~~~L^{\\infty }(0,T;L^{2}(\\Omega ))$ due to the uniform estimates (2.18) and the strong convergence of the density.", "Next, one can show that for any $\\phi \\in \\cap _{n=1}^{\\infty }X_{N}$ the family of functions $\\int _{\\Omega }\\rho _{N}\\mathbf {u}_{N}\\phi dx$ is bounded and equi-continuous in $C(0,T)$ , thus via the Arzela-Ascoli theorem and density of smooth functions in $L^{2}(\\Omega )$ we get that $\\rho _{N}\\mathbf {u}_{N}\\rightarrow \\rho \\mathbf {u} ~~~in ~~~C([0,T];L^{2}_{weak}(\\Omega ))$ Finally, by the compact embedding $L^{2}(\\Omega )\\subset W^{-1,2}(\\Omega )$ and the weak convergence of $\\mathbf {u}_{N}$ we verify that $\\rho _{N}\\mathbf {u}_{N}\\otimes \\mathbf {u}_{N}\\rightarrow \\rho \\mathbf {u} \\otimes \\mathbf {u} ~~~~~~weakly~~~in ~~~L^{2}((0,T)\\times \\Omega )$ The capillarity term.", "We write it in the form $\\int _{0}^{T}\\int _{\\Omega }\\rho _{N}\\nabla \\Delta ^{2s+1}\\rho _{N}\\cdot \\phi dx dt=\\int _{0}^{T}\\int _{\\Omega }\\Delta ^{s}div(\\rho _{N}\\phi )\\Delta ^{s+1}\\rho _{N} dx dt$ Due to (2.18) and the boundedness of the time derivative of $\\rho _{N}$ , we infer that $\\rho _{N}\\rightarrow \\rho ~~~~strongly~~ in~~ L^{2}(0,T;W^{2s+1,2}(\\Omega ))$ thus $\\int _{0}^{T}\\int _{\\Omega }\\Delta ^{s}div(\\rho _{N}\\phi )\\Delta ^{s+1}\\rho _{N}dx dt\\rightarrow \\int _{0}^{T}\\int _{\\Omega }\\Delta ^{s}div(\\rho \\phi )\\Delta ^{s+1}\\rho dx dt$ for any $\\phi \\in C^{\\infty }((0,T)\\times \\overline{\\Omega })$ .", "The momentum term.", "we write it in the form $-\\lambda \\int _{0}^{T}\\int _{\\Omega }\\rho _{N}\\Delta ^{2s+1}(\\rho _{N}\\mathbf {u}_{N})\\cdot \\phi dx dt=-\\lambda \\int _{0}^{T}\\int _{\\Omega }\\Delta ^{s}\\nabla (\\rho _{N}\\mathbf {u}_{N}):\\Delta ^{s}\\nabla (\\rho _{N}\\mathbf {\\phi }) dx dt$ so the convergence established in (3.1) and (3.6) are sufficient to pass to the limit here.", "Strong convergence of the density enables us to perform in the momentum equation (2.9) satisfied for any function $\\phi \\in C^{1}([0,T];(X_{N}))$ such that $\\phi (T)=0$ and by the density argument we can take all such test functions from $C^{1}([0,T];W^{2s+1}(\\Omega ))$ ." ], [ "Derivation of the B-D estimate", "At this level we are left with only two parameters of approximation: $\\varepsilon $ and $\\lambda $ .", "From the so-far obtained a-priori estimates only the ones following from (2.17) and (2.18) were independent of these parameters.", "Now we will have get more enough estimates for density and velocity from the B-D entropy energy inequality, we will prove the following lemma.", "Lemma 4.1 (Bresch-Desjardins type estimate) The following identity holds: $\\begin{aligned}& \\frac{d}{dt} \\int _{T^{d}} (\\frac{1}{2} n|u+ \\nabla \\phi (n)|^{2}+ H(n)+H_{c}(n)+ \\frac{\\hbar ^{2}}{2}|\\nabla \\varphi (n)|^{2}+ \\frac{1}{2}|B|^{2}) dx + \\int _{T^{d}} 2\\mu (n)|A(u)|^{2} dx \\\\&+ \\frac{\\hbar ^{2}}{2}\\int _{T^{d}} \\varphi ^{\\prime }(n)|\\Delta \\varphi (n)|^{2} dx+ \\nu _{b} \\int _{T^{d}} |\\nabla \\times B|^{2} dx + 2 \\int _{T^{d}} \\mu ^{\\prime }(n) (P^{\\prime }(n)+P^{\\prime }_{c}(n))\\frac{|\\nabla n|^{2}}{n} dx\\\\&+2\\lambda \\int _{\\Omega } \\Delta ^{s+1} n \\Delta ^{s} \\mu (n) dx =-2\\lambda \\int _{\\Omega } \\Delta ^{s}\\nabla (n u): \\Delta ^{s} \\nabla ^{2}\\mu (n) dx - \\varepsilon \\int _{\\Omega }{\\rm div}(nu)\\phi ^{^{\\prime }}(n)\\Delta n dx\\\\&+\\varepsilon \\int _{\\Omega }\\frac{|\\nabla \\phi (n)|^2}{2}\\Delta n dx-\\varepsilon \\int _{\\Omega }(\\nabla n \\cdot \\nabla ) u \\cdot \\nabla \\phi (n) dx+\\varepsilon \\int _{\\Omega } n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\Delta n) dx\\\\&+ \\int _{T^{d}} (\\nabla \\times B)\\times B \\cdot \\nabla \\phi (n) dx,\\end{aligned}$ in $\\mathcal {D}^{\\prime }(0,T)$ , where $\\nabla \\phi (n)=2\\frac{\\nabla \\mu (n)}{n} $ .", "The basic idea of the proof is to find the explicit form of the terms: $\\frac{d}{dt}\\int _{\\Omega }(\\frac{1}{2}n |u|^2+nu\\cdot \\nabla \\phi (n)+ \\frac{1}{2}n|\\nabla \\phi (n)|^2)dx.$ The first term can be evaluated by means of the main energy inequality, i.e.", "$\\begin{aligned}& \\frac{d}{dt}\\int _{T^{d}}(\\frac{1}{2} n|u|^{2}+ [H(n)+ H_{c}(n)] dx+ \\frac{\\hbar ^{2}}{2} |\\nabla \\varphi (u)|^{2} +\\frac{1}{2}|B|^{2} dx + \\frac{\\lambda }{2} |\\nabla ^{2s+1} n|^{2})dx\\\\& +2 \\int _{T^{d}} \\mu (n)|\\nabla u|^{2}dx + \\int _{T^{d}} \\lambda (n)|{\\rm div} u|^{2} dx + \\varepsilon \\int _{T^{d}} \\frac{1}{n}(P^{\\prime }(n)+P^{\\prime }_{c}(n))|\\nabla n|^{2} dx \\\\&+ \\nu _{b} \\int _{T^{d}} |\\nabla \\times B|^{2} dx + \\lambda \\int _{T^{d}} |\\Delta ^{s} \\nabla (nu)|^{2} dx+ \\lambda \\varepsilon \\int _{T^{d}} |\\Delta ^{s+1} n|^{2}dx\\\\&+\\varepsilon \\int _{T^{d}}\\frac{\\hbar ^{2}}{2}\\varphi ^{\\prime } (n) \\Delta \\varphi (n) \\Delta ndx=0,\\end{aligned}$ To get a relevant expression for third term in (4.2), we multiply the approximate continuity equation by $\\frac{|\\nabla \\phi (n)|^2}{2}$ and we obtain the following sequence of equalities $\\begin{aligned}& \\frac{d}{dt}\\int _{\\Omega }\\frac{1}{2}n|\\nabla \\phi (n)|^2dx \\\\& =\\int _{\\Omega }(n \\partial _{t}\\frac{|\\nabla \\phi (n)|^2}{2}-\\frac{|\\nabla \\phi (n)|^2}{2}div(n u)+\\varepsilon \\frac{|\\nabla \\phi (n)|^2}{2}\\Delta n) dx \\\\& = \\int _{\\Omega }(\\rho \\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\partial _{t}n)-\\frac{|\\nabla \\phi (n)|^2}{2}div(n u)+\\varepsilon \\frac{|\\nabla \\phi (n)|^2}{2}\\Delta n)dx,\\end{aligned}$ Using the approximate continuity equation, we get $\\begin{aligned}&\\int _{\\Omega }n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\partial _{t}n) dx \\\\& =\\int _{\\Omega }\\varepsilon n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\Delta n) dx-\\int _{\\Omega } \\rho \\nabla \\mathbf {u} : \\nabla \\phi (\\rho )\\otimes \\nabla \\phi (n) dx \\\\&-\\int _{\\Omega }n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)n div \\mathbf {u}) dx-\\int _{\\Omega } n\\mathbf {u} \\otimes \\nabla \\phi (n) :\\nabla ^{2}\\phi (n) dx,\\end{aligned}$ Integrating by parts the two last terms from the r.h.s.", "$\\begin{aligned}&\\int _{\\Omega }n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\partial _{t}n) dx \\\\& =\\int _{\\Omega }\\varepsilon n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\Delta n) dx-\\int _{\\Omega } n\\nabla u : \\nabla \\phi (n)\\otimes \\nabla \\phi (n) dx \\\\&+\\int _{\\Omega }n|\\nabla \\phi (n)|^{2} {\\rm div} u dx +\\int _{\\Omega } n^{2} \\phi ^{^{\\prime }}(n)\\Delta \\phi (n){\\rm div} u dx \\\\&+\\int _{\\Omega } |\\nabla \\phi (n)|^{2} {\\rm div}(n u) dx+\\int _{\\Omega } n u \\cdot \\nabla ( \\nabla \\phi (n)) \\cdot \\nabla (\\phi (n)) dx,\\end{aligned}$ Combining the three previous equalities we finally obtain $\\begin{aligned}& \\frac{d}{dt}\\int _{\\Omega }\\frac{1}{2}n|\\nabla \\phi (n)|^2dx \\\\& = \\int _{\\Omega }\\varepsilon n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\Delta n) dx-\\int _{\\Omega } n\\nabla u : \\nabla \\phi (n)\\otimes \\nabla \\phi (n) dx \\\\&+\\int _{\\Omega }n|\\nabla \\phi (n)|^{2} {\\rm div}u dx +\\int _{\\Omega } n^{2} \\phi ^{^{\\prime }}(n)\\Delta \\phi (n){\\rm div} u dx+\\int _{\\Omega }\\varepsilon \\frac{|\\nabla \\phi (n)|^2}{2}\\Delta n dx,\\end{aligned}$ In the above series of equalities, each one holds ponitwisely with respect to time due to the regularity of $n$ and $\\nabla \\phi $ .", "This is not the case of the middle integrant of (4.2), for which one should really think of weak in time formulation.", "Denote $V=W^{2s+1,2}(\\Omega ), ~~~~~~and ~~~~~\\mathbf {v}=n \\mathbf {u},~~~~h=\\nabla \\phi .$ We know that $\\mathbf {v}\\in L^{2}(o,T;V)$ and its weak derivative with respect to time variable $\\mathbf {v}^{^{\\prime }}\\in L^{2}(o,T;V^{*})$ where $V^{*}$ denotes the dual space to V. Moreover, $h\\in L^{2}(0,T;V)$ , $h^{^{\\prime }}\\in L^{2}(0,T;W^{2s-1,2}(\\Omega ))$ .", "Now, let $\\mathbf {v}_{m}$ , $h_{m}$ denote the standard mollifications in time of $\\mathbf {v}$ , $h$ respectively.", "By the properties of mollifiers we know that $\\mathbf {v}_{m},\\mathbf {v}_{m}^{^{\\prime }}\\in C^{\\infty }(0,T;V),~~~~~~~~~~~h_{m},h_{m}^{^{\\prime }}\\in C^{\\infty }(0,T;V)$ and $\\begin{aligned}\\mathbf {v}_{m}\\rightarrow v ~~~~~L^{2}(0,T;V),~~~~~h_{m}\\rightarrow h~~~~~~L^{2}(0,T;V), \\\\\\mathbf {v}_{m}^{^{\\prime }}\\rightarrow v^{^{\\prime }}~~~~~L^{2}(0,T;V^{*}),~~~~~h_{m}^{^{\\prime }}\\rightarrow h^{^{\\prime }}~~~~~~L^{2}(0,T;V^{*}),\\end{aligned}$ For these regularized sequences we may write $\\begin{aligned}\\frac{d}{dt}\\int _{\\Omega }\\mathbf {v}_{m}\\cdot h_{m} dx=\\frac{d}{dt}(\\mathbf {v}_{m},h_{m})_{V}=(\\mathbf {v}_{m}^{^{\\prime }},h_{m})_{V}+(\\mathbf {v}_{m},h_{m}^{^{\\prime }})_{V}, ~~~\\forall \\psi \\in \\mathcal {D}(0,T),\\end{aligned}$ Using the Riesz representation theorem we verify that $\\mathbf {v}_{m}^{^{\\prime }}\\in C^{\\infty }(0,T;V)$ uniquely determines the functional $\\Phi _{v_{m}^{^{\\prime }}}\\in V^{*}$ such that $(\\mathbf {v}_{m}^{^{\\prime }},\\psi )_{V}=(\\Phi _{v_{m}^{^{\\prime }}},\\psi )_{V^{*},V}=\\int _{\\Omega }\\mathbf {v}_{m}^{^{\\prime }}\\cdot \\psi dx$ , $\\forall \\psi \\in V$ ; for the second term from the r.h.s.", "of (4.11) we can simply replace $V=L^{2}(\\Omega )$ and thus we obtain $\\begin{aligned}-\\int _{0}^{T}(\\mathbf {v}_{m},h_{m})_{V}\\psi ^{^{\\prime }} dt= \\int _{0}^{T}(\\mathbf {v}_{m}^{^{\\prime }},h_{m})_{V^{*},V}\\psi dt +\\int _{0}^{T}(\\mathbf {v}_{m},h_{m}^{^{\\prime }})_{L^{2}(\\Omega )}\\psi dt~~~~\\forall \\psi \\in \\mathcal {D}(0,T),\\end{aligned}$ Observe that both integrands from the r.h.s.", "are uniformly bounded in $L^{1}(0,T)$ , thus, using (4.10), we let $m\\rightarrow \\infty $ to obtain $\\begin{aligned}\\frac{d}{dt}\\int _{\\Omega }\\mathbf {v}\\cdot h dx=(\\mathbf {v}^{^{\\prime }},h)_{V}+(\\mathbf {v},h^{^{\\prime }})_{V}, ~~~\\forall \\psi \\in \\mathcal {D}(0,T),\\end{aligned}$ Coming back to our original notation, this means that the operation $\\frac{d}{dt}\\int _{\\Omega }nu\\cdot \\nabla \\phi (n)dx =<\\partial _{t}(n u),\\nabla \\phi >_{V^{*},V}+\\int _{\\Omega }n u\\cdot \\partial _{t}\\nabla \\phi dx$ is well defined and is nothing but equality between two scalar distributions.", "By the fact that $\\partial _{t}\\nabla \\phi $ exists a.e.", "in $(0,T)\\times \\Omega $ we may use approximation to write $\\int _{\\Omega }nu\\cdot \\partial _{t}\\nabla \\phi (n)dx =\\int _{\\Omega }({\\rm div}(n u))^{2}\\phi ^{^{\\prime }}(n) dx- \\varepsilon \\int _{\\Omega }{\\rm div}(n u)\\phi ^{^{\\prime }}(n)\\Delta n dx$ whence the first term on the r.h.s.", "of (4.14) may be evaluated by testing the approximate momentum equation by $\\nabla n$ $\\begin{aligned}&<\\partial _{t}(n \\mathbf {u}),\\nabla \\phi (n)>_{V^{*},V}= -\\int _{\\Omega }(2\\mu (n)+\\lambda (n))\\Delta \\phi (n) {\\rm div} u dx +2\\int _{\\Omega }\\nabla u: \\nabla \\phi (n) \\otimes \\nabla \\mu (n) dx \\\\&-2 \\int _{\\Omega }\\nabla \\phi (n)\\cdot \\nabla \\mu (n) {\\rm div} u dx -\\int _{\\Omega }\\nabla \\phi (n)\\cdot \\nabla P dx -\\lambda \\int _{\\Omega } \\Delta ^{s+1}\\mu (n) \\Delta ^{s} {\\rm div} (n \\nabla \\phi (n))dx\\\\&-\\lambda \\int _{\\Omega } \\Delta ^{s}\\nabla (n u): \\Delta ^{s} \\nabla (n \\nabla \\phi (n))dx -\\int _{\\Omega }\\nabla \\phi (n)\\cdot {\\rm div}(n u\\otimes u) dx \\\\&-\\varepsilon \\int _{\\Omega }(\\nabla n \\cdot \\nabla ) u \\cdot \\nabla \\phi (n) dx-\\int _{\\Omega }\\varphi ^{\\prime }(n) |\\Delta \\varphi (n)|^{2} dx+ \\int _{T^{d}} (\\nabla \\times B)\\times B \\cdot \\phi (n) dx,\\end{aligned}$ Recalling the form of $\\phi (n)$ it can be deduced that the $\\begin{aligned}&\\frac{d}{dt}\\int _{\\Omega }(n u\\cdot \\nabla \\phi (n)+ \\frac{1}{2}n|\\nabla \\phi (n)|^2)dx+ \\int _{\\Omega }\\nabla \\phi (n)\\cdot \\nabla P dx+\\lambda \\int _{\\Omega } \\mu ^{\\prime }(n) \\Delta \\mu (n) \\Delta ^{s} \\mu (n) dx \\\\& = -\\int _{\\Omega }\\nabla \\phi (n)\\cdot {\\rm div}(n u\\otimes u) dx + \\int _{\\Omega }({\\rm div}(nu))^{2}\\phi ^{^{\\prime }}(n) dx-2\\lambda \\int _{\\Omega } \\Delta ^{s}\\nabla (n u): \\Delta ^{s} \\nabla ^{2}\\mu (n) dx \\\\&- \\varepsilon \\int _{\\Omega }{\\rm div}(nu)\\phi ^{^{\\prime }}(n)\\Delta n dx+\\varepsilon \\int _{\\Omega }\\frac{|\\nabla \\phi (n)|^2}{2}\\Delta n dx-\\varepsilon \\int _{\\Omega }(\\nabla n \\cdot \\nabla ) u \\cdot \\nabla \\phi (n) dx\\\\&+\\varepsilon \\int _{\\Omega } n\\nabla \\phi (n)\\cdot \\nabla (\\phi ^{^{\\prime }}(n)\\Delta n) dx-\\int _{\\Omega }\\varphi ^{\\prime }(n) |\\Delta \\varphi (n)|^{2} dx+ \\int _{T^{d}} (\\nabla \\times B)\\times B \\cdot \\nabla \\phi (n) dx,\\end{aligned}$ The first two terms from the r.h.s.", "of (4.17) can be transformed $\\begin{aligned}&\\int _{\\Omega }(({\\rm div}(n u))^{2}\\phi ^{^{\\prime }}(n) -\\nabla \\phi (n)\\cdot {\\rm div}(n u\\otimes u)) dx \\\\& = \\int _{\\Omega }(n^{2}\\phi ^{^{\\prime }}(n)({\\rm div} u)^{2}+n\\phi ^{^{\\prime }}\\cdot \\nabla n {\\rm div} u-n\\phi ^{^{\\prime }} \\nabla n( u\\cdot \\nabla u))dx \\\\& = 2\\int _{\\Omega }\\mu (n) \\partial _{i} u_{j}\\partial _{j} u_{i}dx = 2\\int _{\\Omega }\\mu (n) |D(u)|^{2} dx - 2\\int _{\\Omega }\\mu (n) (\\frac{\\partial _{i} u_{j}-\\partial _{j} u_{i}}{2})^{2}dx,\\end{aligned}$ thus, the assertion of Lemma 4.1 follows by adding (4.3) to (4.17).", "The main problem is to control the last term on the right hand side of (4.1), other terms can be easier to control.", "For this obstacle, we estimate as follow $\\begin{aligned}& 2|\\int _{T^{d}} (\\nabla \\times B)\\times B \\cdot \\frac{\\nabla \\mu (n)}{n}|\\le \\int _{T^{d}}\\frac{|\\nabla \\times B|^{2}}{\\varepsilon n^{2}}dx+ \\varepsilon \\int _{T^{d}}|\\nabla \\mu (n)\\times B|^{2} dx ,\\end{aligned}$ The first term of the right hand side will sent to the left hand side of equation and will we compensated with the term related to the resistivity thanks to the profiles condition introduced in (1.6).", "The dimension hypothesis appears at this point, in a 2-dimensional space, we can insure that $W^{1,1}\\subset L^{2}$ and this will be the main tool deal with the second term.", "we have $\\begin{aligned}\\Vert \\nabla \\mu (n)\\times B \\Vert _{L^{2}(T^{d})}^{2} &\\le C\\Vert \\nabla \\mu (n)\\times B\\Vert ^{2}_{W^{1,1}}\\\\& \\le C (\\Vert \\Delta \\mu (n)\\Vert _{L^{2}(T^{d})}^{2}\\Vert B\\Vert _{L^{T^{d}}}^{2}+ \\Vert \\nabla \\mu (n)\\Vert _{L^{2}(T^{d})}^{2}\\Vert \\nabla B\\Vert _{L^{2}(T^{d})}^{2}\\\\&+ \\Vert \\nabla \\mu (n)\\times B\\Vert _{L^{1}(T^{d})}^{2} ,\\end{aligned}$ But, from (2.12), we already know that $\\Vert B\\Vert _{L^{2}}$ and $\\Vert \\nabla \\mu (n)\\Vert _{L^{2}}$ are uniformly bounded by $\\Lambda _{0}$ , that is why we also get $\\Vert \\nabla \\mu (n)\\times B \\Vert _{L^{2}(T^{d})}^{2} \\le C(1+\\Vert \\Delta \\mu (n)\\Vert _{L^{2}}^{2}+\\Vert \\nabla \\times B\\Vert ^{2}_{L^{2}}),$ So we get, summing (4.20) and (4.21) and taking into account all these quantities, for $\\varepsilon $ small enough, we are considering here some coefficients $\\varepsilon <\\frac{1}{6}$ and such that $\\mu ^{\\prime }-C\\varepsilon $ still higher that a constant, say $\\delta $ .", "It also appears the necessary conditions on the constants $d_{0}$ and $d_{1}$ , to be high enough because we need to have $\\eta (n)-\\varepsilon ^{-1}n^{-2}-C\\varepsilon \\ge 0$ .", "To conclude, we apply a Gronwall's lemma, we will get B-D entropy energy estimates." ], [ "Estimates independent of $\\varepsilon ,\\lambda $ , passage to the limit {{formula:26f8cf68-0910-4793-85b3-4c16878d0ed2}}", "In this section we first present the new uniform bounds arising fron the estimate of B-D entropy, performed in this section, and then we let the last two approximation parameters to 0.", "Note that the limit passage $\\lambda \\rightarrow 0$ , $\\varepsilon \\rightarrow 0$ could be done in a single step.", "We complete the set uniform bounds by the following ones $\\sqrt{\\lambda } \\Vert \\Delta ^{s+1} n_{\\varepsilon ,\\lambda } \\Vert _{L^{2}((0,T)\\times \\Omega )}+\\Vert \\nabla \\phi (n_{\\varepsilon ,\\lambda }) \\Vert _{L^{2}((0,T)\\times \\Omega )}+\\Vert \\sqrt{\\frac{\\mu ^{\\prime }(n_{\\varepsilon ,\\lambda })(P^{\\prime }(n_{\\varepsilon ,\\lambda })+P^{\\prime }_{c}(n_{\\varepsilon ,\\lambda }))}{n_{\\varepsilon ,\\lambda }}}\\nabla n\\Vert _{L^{2}((0,T)\\times \\Omega )} \\le C$ moreover $\\Vert \\Delta \\mu (n_{\\varepsilon ,\\lambda })\\Vert _{L^{2}((0,T)\\times \\Omega )}\\le C$ The uniform estimates for the velocity vector field are the following ones $\\sqrt{\\lambda } \\Vert \\Delta ^{s} \\nabla (\\rho u_{\\varepsilon ,\\lambda }) \\Vert _{L^{2}((0,T)\\times \\Omega )}+\\Vert \\sqrt{\\mu (n_{\\varepsilon ,\\lambda })}\\nabla A(u_{\\varepsilon ,\\lambda })\\Vert _{L^{2}((0,T)\\times \\Omega )}\\le C$ and the constants from the r.h.s are independent of $\\varepsilon $ and $\\lambda $ .", "We now present several additional estimates of $n_{\\varepsilon ,\\lambda }$ and $u_{\\varepsilon ,\\lambda }$ based on imbedding of Sobolev spaces and simple interpolation inequalities." ], [ "Further estimates of $n$", "Lemma 5.1 $n_{\\varepsilon ,\\lambda }^{-1/2}~ is~ uniformly~ bounded~ in~ L^{\\infty }(0,T;L^{6}_{loc}(\\Omega ))\\cap L^{2}(0,T;H^{1}_{loc}(\\Omega )),$ $n_{\\varepsilon ,\\lambda }~ is~ uniformly~ bounded~ in~ L^{\\infty }(0,T;L^{p}_{loc}(\\Omega )),\\forall p<+\\infty .$ On the one hand, from (2.10) we know that $H_{c}(n_{\\varepsilon ,\\lambda })$ is uniformly bounded in $L^{\\infty }(0,T:L^{1}(\\Omega ))$ which implies that $n_{\\varepsilon ,\\lambda }^{-1/2}$ is bounded in $L^{\\infty }(0,T;L^{2\\gamma ^{-}})$ .", "On the other hand, there exist functions $\\zeta (n)=n$ for $n <1$ ,$\\zeta (n)=0$ for $n>1$ such that $\\nabla \\zeta (n)^{-\\frac{\\frac{1}{}}{2}}$ is bounded in $L^{2}(0,T;L^{2}(\\Omega ))$ .", "Then, thinking that $\\gamma ^{-}>1>\\alpha $ , we conclude that $\\nabla n_{\\varepsilon ,\\lambda }^{-1/2}$ is also bounded in $L^{2}(0,T;L^{2}(\\Omega ))$ .", "Since $\\nabla \\psi (n_{\\varepsilon ,\\lambda })$ is bounded in $L^{\\infty }(0,T;L^{2}(\\Omega ))$ , and $H(n_{\\varepsilon ,\\lambda })$ is uniformly bounded in $L^{\\infty }(0,T;L^{1}(\\Omega ))$ , thus we can use Sobolev embedding of $H^{1}(\\Omega )$ in $L^{p}(\\Omega )$ for all $p<+\\infty $ in the two dimension." ], [ "Estimate of the velocity vector field", "Lemma 5.2 $u_{\\varepsilon ,\\lambda }~ is~ uniformly~ bounded~ in~ L^{q_{1}}(0,T;W^{1,q_{2}}_{loc}(\\Omega )),~~q_{1}>\\frac{5}{3}~~ and~~ q_{2}>\\frac{15}{8},$ We use the Holder inequality to write $\\Vert \\nabla u_{\\varepsilon ,\\lambda }\\Vert _{L^{q_{1}}(0,T;L^{q_{3}}(\\Omega ))}\\le c(1+\\Vert \\zeta (n_{\\varepsilon ,\\lambda })^{-\\alpha /2}\\Vert _{L^{2j}(0,T;L^{6j}(\\Omega ))})\\Vert n_{\\varepsilon ,\\lambda }^{\\frac{\\alpha }{2}} \\nabla u_{\\varepsilon ,\\lambda } \\Vert _{L^{2}((0,T)\\times \\Omega )}.$ where $j=\\frac{\\gamma ^{-}+1-\\alpha }{\\alpha }, \\frac{1}{q_{1}}=\\frac{1}{2}+\\frac{1}{2j}, \\frac{1}{q_{3}}= \\frac{1}{2}+\\frac{1}{6j}$ .", "Therefore, the Korn inequality together with the Sobolev imbedding imply the lemma." ], [ "Magnetic field", "Thanks to estimates (2.10) and conditions on $\\eta $ that $B_{\\varepsilon ,\\lambda }~ is~ uniformly~ bounded~ in~ L^{\\infty }(0,T;L^{2}(\\Omega ))\\cap L^{2}(0,T;H^{1}(\\Omega )),$ By interpolation, we can also deduce that embedding the following result: Lemma 5.3 Let $\\beta $ be any parameter in $(0,1)$ and $p<+\\infty $ .", "$B_{\\varepsilon ,\\lambda }~ is~ uniformly~ bounded~ in~ L^{\\frac{2}{\\beta }}(0,T;L^{\\frac{2}{(\\frac{2}{p})\\alpha +1}}(\\Omega )),$" ], [ "With the B-D estimate at hand, especially with the bound on $\\Delta ^{s+1} n_{\\varepsilon ,\\lambda }$ in $L^{2}((0,T)\\times \\Omega )$ , which is now uniform with respect to $\\varepsilon $ , we may perform the limit passage similarly as in previous step.", "Indeed, the uniform estimates allow us to extract subsequences, such that $\\varepsilon \\Delta ^{s}\\nabla u_{\\varepsilon ,\\lambda }, \\varepsilon \\nabla n_{\\varepsilon ,\\lambda }, \\varepsilon \\Delta ^{s+1} n_{\\varepsilon ,\\lambda }\\rightarrow 0 ~~~ strongly ~~~~in~~~ L^{2}((0,T)\\times \\Omega )$ therefore $\\varepsilon \\nabla n_{\\varepsilon ,\\lambda } \\nabla u_{\\varepsilon ,\\lambda } \\rightarrow 0 ~~~ strongly ~~~~in~~~ L^{1}((0,T)\\times \\Omega )$" ], [ "For $n_{\\varepsilon ,\\lambda }$", "We know, thinks to (5.5), that $n_{\\varepsilon ,\\lambda }$ converges weakly to $n$ in $L^{\\infty }(0,T; L^{q}_{loc}(\\Omega ))$ , for all $q<+\\infty $ .", "To prove strong convergence on the density, we shall use the transport equation satisfying $\\mu (n)$ : $\\partial _{t}(\\mu (n))+ {\\rm div}(\\mu (n) u)+ \\frac{1}{2} \\lambda (n){\\rm div} u=0 ,$ Proving that $\\partial _{t}(\\phi \\mu (n))$ is bounded in $L^{2}(0,T; H^{-\\sigma _{0}}(\\Omega ))$ for any compactly supported $\\phi $ , we then conclude that $n_{\\varepsilon ,\\lambda }\\rightarrow n ~~in~C([0,T]; L^{q}_{loc}(\\Omega )),~~\\forall q<+\\infty ,$ From another point, to conclude to a compactness for $n_{\\varepsilon ,\\lambda }^{-1/2}$ in $C([0,T]; L^{q}_{loc}(\\Omega ))$ , for all $q<+\\infty $ , we must, in addition to (5.4), look at $\\partial _{t}(n^{-1/2})$ and try to show a boundedness in a space $L^{r}(0,T;H^{-\\sigma _{0}})$ with $r>1$ .", "From the transport equation we find $\\partial _{t}(n^{-1/2})-\\frac{3}{2}n^{-1/2}{\\rm div}u+{\\rm div}(n^{-1/2}u)=0,$ from which we can insure that $\\partial _{t}(n^{-1/2})$ is bounded in $L^{5/3}(0,T;W^{-1,\\frac{30}{11}}(\\Omega ))$ .", "Then, from (5.4), we can deduce that $n_{\\varepsilon ,\\lambda }^{-1/2}\\rightarrow n^{-1/2} ~~in~L^{p}(0,T; L^{q}_{loc}(\\Omega )),~~\\forall p<+\\infty ,\\forall q<6, ~~~in ~L^{2}(0,T; L^{q}_{loc}(\\Omega )),~~\\forall q<+\\infty ,$" ], [ "For $n_{\\varepsilon ,\\lambda }u_{\\varepsilon ,\\lambda }$", "We know that $n_{\\varepsilon ,\\lambda }u_{\\varepsilon ,\\lambda }$ converges weakly to $nu$ in $L^{\\infty }(0,T; L^{s<2}_{loc}(\\Omega ))$ as the product of $n_{\\varepsilon ,\\lambda }$ bounded in $L^{\\infty }(0,T; L^{r<\\infty }_{loc}(\\Omega ))$ and $\\sqrt{n_{\\varepsilon ,\\lambda }}u_{\\varepsilon ,\\lambda }$ bounded in $L^{\\infty }(0,T; L^{2}(\\Omega ))$ .", "To have compactness on $n_{\\varepsilon ,\\lambda }u_{\\varepsilon ,\\lambda }$ , we will of course use the momentum equation to assure that $\\partial _{t}(n_{\\varepsilon ,\\lambda }u_{\\varepsilon ,\\lambda })$ is bounded in $L^{p}_{loc}(0,T; H^{-\\sigma _{0}}(\\Omega ))$ for $p>1$ and $\\sigma _{0}$ large enough.", "To more precise on what is different in our system we shall forget the new term in the momentum equation related to the magnetic field, namely $\\nabla B\\times B$ .", "Using (5.8), we know that $\\nabla \\times B$ is bounded in $L^{2}(0,T;L^{2}(\\Omega ))$ , that is why we must have better than $L^{2}(0,T;L^{2}(\\Omega ))$ for $B$ and it is time to use Lemma 5.3.", "Indeed, for any $0<\\alpha <1$ we get the expected boundedness of $B$ in $L^{p}(0,T;L^{p}(\\Omega ))$ with $p>2$ so that $(\\nabla \\times B)\\times B$ is bounded in $B$ in $L^{q}(0,T;L^{q}(\\Omega ))$ with $q>1$ .", "Thus, we get $n_{\\varepsilon ,\\lambda }u_{\\varepsilon ,\\lambda }\\rightarrow nu ~~in~L^{p}(0,T; W^{-1,q}_{loc}(\\Omega )),~~\\forall p<+\\infty ,\\forall q<6,$ From (5.14) together with lemma 5.2, is the strong convergence of $\\int _{B}n_{\\varepsilon ,\\lambda } |u_{\\varepsilon ,\\lambda }|^{2}$ to $\\int _{B}n |u|^{2}$ , for all subset $B$ in $\\Omega $ .", "Moreover, since $\\sqrt{n_{\\varepsilon ,\\lambda }}u_{\\varepsilon ,\\lambda }$ converges weakly to $\\sqrt{n}u$ in $L^{\\infty }(0,T; L^{2}_{loc}(\\Omega ))$ , we insure that $\\sqrt{n_{\\varepsilon ,\\lambda }}u_{\\varepsilon ,\\lambda }\\rightarrow \\sqrt{n}u ~~in~L^{2}(0,T; L^{2}_{loc}(\\Omega )),$" ], [ "For the magnetic field $B_{\\varepsilon ,\\lambda }$", "We already know that the sequence $B_{\\varepsilon ,\\lambda }$ weakly converges the limit $B$ in $L^{\\infty }(0,T;L^{2}(\\Omega ))\\cap L^{2}(0,T;H^{1}(\\Omega ))$ .", "Let's now deal with $\\partial _{t}B$ in order to insure a strong convergence statement.", "Looking at equation (1.1c), we are lead to bound $u\\times B$ and $(\\xi _{b})\\nabla \\times B$ .", "For the first one, thinking to Lemma 5.2 and Lemma 5.3, we get $u\\times B$ bounded in $L^{p}_{loc}(0,T;L^{p})$ with $p>1$ what is enough comfortable.", "For the second, we write $\\xi _{b}\\nabla \\times B=\\sqrt{\\xi _{b}}\\sqrt{\\xi _{b}}\\nabla \\times B$ .", "We know that the term $\\sqrt{\\xi _{b}}\\nabla \\times B$ is bounded in $L^{2}(0,T;L^{2}(\\Omega ))$ and through conditions (1.6) and the bounds (37) or (38), we also $\\sqrt{\\xi _{b}}$ bounded in $L^{2}(0,T;L^{2}(\\Omega ))$ .", "This is just enough to conclude that $B_{t}$ is bounded in $L^{1}(0,T;W^{-1,1}(\\Omega ))$ .", "Then we get $B_{\\varepsilon ,\\lambda }\\rightarrow B ~~in~L^{p}(0,T; L^{2}(\\Omega )),~~\\forall p<+\\infty ,$" ], [ "Convergences", "    For the mass conservation, by the strong convergences of $n_{\\varepsilon ,\\lambda }$ to n in $C([0,T];L^{2}(\\Omega ))$ and the strong convergence of $\\sqrt{n_{\\varepsilon ,\\lambda }}u_{\\varepsilon ,\\lambda }$ in $L^{2}(0,T; L^{2}_{loc}(\\Omega ))$ .", "For the momentum equation, we have to justify how to pass the limit in the term $\\nabla \\times B_{\\varepsilon ,\\lambda } \\times B_{\\varepsilon ,\\lambda }$ .", "For that we should have a strong convergence of $B_{\\varepsilon ,\\lambda }$ to $B$ in $L^{2}(0,T; L^{2}_{loc}(\\Omega ))$ .", "Now is the time to deal with the magnetic field equation.", "It is clear for the term $\\partial _{t}B$ , now deal with $\\nabla \\times B \\times B$ and $\\nabla \\times B \\times B$ .", "With Lemma 5.2 and (5.16), we justify the convergence in the sense of distribution for the first one.", "The second one can be, one more time, written as the product of $\\sqrt{\\xi _{b}}\\nabla \\times B$ , weakly converging in $L^{2}(0,T; L^{2}_{loc}(\\Omega ))$ and $\\sqrt{\\eta }$ strongly convergence to $\\sqrt{\\eta }$ in $L^{2}(0,T; L^{2}_{loc}(\\Omega ))$ .", "tocsectionBibliography" ] ]
1606.05014
[ [ "Quasistationary solutions of scalar fields around accreting black holes" ], [ "Abstract Massive scalar fields can form long-lived configurations around black holes.", "These configurations, dubbed quasi-bound states, have been studied both in the linear and nonlinear regimes.", "In this paper we show that quasi-bound states can form in a dynamical scenario in which the mass of the black hole grows significantly due to the capture of infalling matter.", "We solve the Klein-Gordon equation numerically in spherical symmetry, mimicking the evolution of the spacetime through a sequence of analytic Schwarzschild black hole solutions of increasing mass.", "It is found that the frequency of oscillation of the quasi-bound states decreases as the mass of the black hole increases.", "In addition, accretion leads to a significative increase of the exponential decay of the scalar field energy due to the presence of terms of order higher than linear in the exponent.", "We compare the black hole mass growth rates used in our study with estimates from observational surveys and extrapolate our results to values of the scalar field masses consistent with models that propose scalar fields as dark matter in the universe.", "We show that even for unrealistically large mass accretion rates, quasi-bound states around accreting black holes can survive for cosmological timescales.", "Our results provide further support to the intriguing possibility of the existence of dark matter halos based on (ultra-light) scalar fields surrounding supermassive black holes in galactic centers." ], [ "Introduction", "There is compelling evidence that most nearby galaxies host supermassive black holes (SMBHs) in their centers, with masses in the range $10^6-10^9 M_{\\odot }$  [1], [2].", "Such SMBHs are expected to be the evolutionary result of the growth of seed BHs in high redshift galaxies through accretion episodes and mergers of massive BHs (MBH) binaries with masses somewhere in between those of stellar-origin BHs and SMBHs [3], [4].", "The discovery of supermassive luminous quasars at redshifts of $z > 6$ has shown that SMBHs with masses $\\sim 10^9 M_{\\odot }$ , must form extremely early on in the history of the universe [5], [6], [7].", "These SMBHs must grow rapidly in order to acquire its mass within a short period of $\\sim 1$ Gyr.", "Explaining the formation and evolution of SMBHs dwelling in the most powerful quasars when the Universe was less than 1 Gyr old (and of the regular and much smaller MBHs hidden in 13 Gyr old galaxies) is a pressing open issue [8].", "Proposed models involve the gravitational collapse of gas clouds or the collapse of supermassive stars in the early universe (a model hampered by the low masses of the initial seeds of first-generation Pop III stars [9]), the runaway growth by accretion onto Pop III BH, or mergers of smaller size BHs.", "In either scenario, the formation of SMBHs is a highly dynamical event amenable to gravitational wave investigations.", "Indeed, it is expected that eLISA will probe MBH binaries in the $10^3-10^7 M_{\\odot }$ range out to redshift $z>10$ through the detection of their gravitational waves in the mHz frequency band [10].", "While one can make a convincing case that SMBHs have grown largely through accretion, with the consequent energy emission observed in electromagnetic output, it has been argued that an exponential growth at the Eddington-limited e-folding time is too slow to grow stellar-mass BH seeds into the supermassive luminous quasars that are observed at $z \\sim 7$ [7].", "Some proposals to circumvent this issue invoke super-Eddington accretion for brief periods of time [11], the formation through accretion and merging of the first stellar remnants [12], [13] or via more massive seeds from the collapse of pre-galactic disks at high redshifts [14], [15], [16].", "In light of these proposals it is worth considering if rapid BH accretion may have any effect on the distribution of the associated dark matter halo when the latter is modeled as a scalar field.", "In the absence of accretion the existence of long-lasting scalar field configurations surrounding a (non-rotating) BH has been investigated in a number of recent papers, either in the test-field limit [17], [18], [19] or employing self-gravitating scalar fields [20], [21].", "These papers have shown that SMBHs at galactic centers do not represent a serious threat to dark matter models based on (ultra-light) scalar fields as a viable alternative to the usual description of dark matter in terms of weakly interacting massive particles.", "For both, scalar fields around SMBHs and axion-like scalar fields around primordial BHs, it has been found that scalar fields can survive for cosmological timescales [18].", "Despite these findings, and due to the rapid growth of accretion-powered SMBHs, it is worth investigating the chances of survival of the scalar field within such dynamical situation.", "This is the purpose of this paper.", "Here we study the properties of scalar field quasi-bound states in the background of an accreting spherically symmetric BH.", "Assuming that the BH mass grows adiabatically due to infalling matter we show that indeed, long-lasting, quasi-bound states can still form in such scenario.", "This paper is organized as follows: in Section  we lay out the mathematical and physical approach we use to carry out our investigation.", "In particular Section REF contains a brief synopsis of our numerical methodology.", "The results are presented and discussed in Section , while Section  contains the summary of our findings.", "Throughout the paper we employ geometrized units ($c=G=1$ ).", "Latin indices indicate spatial indices and hence run from 1 to 3 while Greek indices run from 0 to 3.", "Our setup considers a scalar field distribution $\\Phi $ of sufficiently small energy to neglect its self-gravity, i.e.", "the field can be regarded as a test-field.", "This configuration surrounds a BH which is assumed to be continuously accreting matter.", "The dynamics of the scalar field is governed by the Klein-Gordon equation, $\\Box \\Phi -\\mu ^2\\Phi =0 \\ ,$ where the D'Alambertian operator is defined by $\\Box :=(1/\\sqrt{-g})\\partial _{\\alpha }(\\sqrt{-g}g^{\\alpha \\beta }\\partial _{\\beta })$ .", "We follow the convention that $\\Phi $ is dimensionless and $\\mu $ , the mass of the scalar field, has dimensions of (length)$^{-1}$ .", "We write the spacetime metric $g_{\\alpha \\beta }$ as $ ds^2 & = & g_{\\alpha \\beta } dx^\\alpha dx^\\beta \\nonumber \\\\& = & - \\alpha ^2 dt^2 + \\gamma _{ij} (dx^i + \\beta ^i dt)(dx^j + \\beta ^j dt),$ where $\\alpha $ is the lapse function, $\\beta ^i$ the shift vector, and $\\gamma _{ij}$ the spatial metric.", "We adopt a conformal decomposition of the spatial metric $\\gamma _{ij}$ $ \\gamma _{ij} = e^{4 \\chi } \\hat{\\gamma }_{ij},$ where $\\psi := e^\\chi $ is the conformal factor, $\\hat{\\gamma }_{ij}$ the conformally related metric and $\\hat{\\gamma }$ its determinant.", "By assuming spherical symmetry the line element may be written as $dl^2 = e^{4\\chi } (a(t,r)dr^2+ r^2\\,b(t,r) d\\Omega ^2)\\,,$ with $d\\Omega ^2 = \\sin ^2\\theta d\\varphi ^2+d\\theta ^2$ being the solid angle element and $a(t,r)$ and $b(t,r)$ two independent metric functions.", "To solve the Klein-Gordon equation in spherical symmetry, we introduce two first-order fields, $\\Pi $ and $\\Psi $ , defined as: $\\Pi &:=& n^{\\alpha }\\partial _{\\alpha }\\Phi =\\frac{1}{\\alpha }(\\partial _{t}\\Phi -\\beta ^{r}\\partial _{r}\\Phi ) \\ ,\\\\\\Psi &:=&\\partial _{r}\\Phi \\ ,$ where $n^{\\alpha }$ is the unit vector normal to the surfaces of constant $t$ .", "Therefore, using Eq.", "(REF ) we obtain the following system of first-order equations: $\\partial _{t}\\Phi &=&\\beta ^{r}\\partial _{r}\\Phi +\\alpha \\Pi \\,,\\\\\\partial _{t}\\Psi &=&\\beta ^{r}\\partial _{r}\\Psi +\\Psi \\partial _{r}\\beta ^{r}+\\partial _{r}(\\alpha \\Pi ) \\,,\\\\\\partial _{t}\\Pi &=&\\beta ^{r}\\partial _{r}\\Pi +\\frac{\\alpha }{ae^{4\\chi }}[\\partial _{r}\\Psi \\nonumber \\\\&+&\\Psi \\biggl (\\frac{2}{r}-\\frac{\\partial _{r}a}{2a}+\\frac{\\partial {r}b}{b}+2\\partial _{r}\\chi \\biggl )\\biggl ]\\nonumber \\\\&+&\\frac{\\Psi }{ae^{4\\chi }}\\partial _{r}\\alpha +\\alpha K\\Pi - \\alpha \\mu ^{2}\\Phi \\,,$ where $K$ is the trace of the extrinsic curvature.", "The stress-energy tensor of the scalar field reads $T_{\\alpha \\beta }=\\partial _{\\alpha }\\Phi \\partial _{\\beta }\\Phi -\\frac{1}{2}g_{\\alpha \\beta }\\left(\\partial ^{\\sigma }\\Phi \\partial _{\\sigma }\\Phi +\\mu ^2\\Phi ^2 \\right)\\,.$ From this tensor we can compute the energy of the scalar field $E$ $E_{\\text{SF}}=\\int _{r_{\\rm {AH}}}^{\\infty }\\mathcal {E}_{\\text{SF}} dV \\,,$ where $r_{\\rm {AH}}$ is the radius of the apparent horizon and $\\mathcal {E}^{\\text{SF}}$ is given by $\\mathcal {E}_{\\text{SF}}&:=&n^{\\alpha }n^{\\beta }T_{\\alpha \\beta }=\\frac{1}{2}\\biggl (\\Pi ^{2}+\\frac{\\Psi ^{2}}{ae^{4\\chi }}\\biggl )+\\frac{1}{2}\\mu ^{2}\\Phi ^{2} \\,.$" ], [ "Analytic black hole solution", "The numerical evolution of a single BH using the so-called “moving puncture\" technique leads to a well-known time-independent, maximal slicing solution of Schwarzschild [22].", "It was shown in [23] that this solution can also be constructed analytically and be used as a test for numerical codes.", "We take advantage of this result to put forward the defining characteristic of the procedure we employ in the current investigation.", "Namely, we avoid evolving numerically the BH, computing instead a sequence of analytical solutions at each time step for different BH masses to mimic the BH growth for different accretion rates.", "This procedure allows us to simulate long episodes of accretion without resorting to a test fluid in order to achieve high accretion rates.", "The analytic solution is constructed by defining all quantities as a function of the gauge-invariant areal radius $R$ .", "We have to transform the solution into isotropic coordinates, as the latter are used in our numerical procedure, comparing the spatial metrics as a function of $R$ and the isotropic radius $r$ $\\alpha ^{-2}\\,dR^{2}+R^{2}d\\Omega ^{2}=\\psi ^{4}(dr^{2}+r^{2}d\\Omega ^{2})\\,.$ Thus, the isotropic radius $r$ is given as a function of $R$ by $r&=&\\biggl [\\frac{\\displaystyle 2R+M+(4R^{2}+4MR+3M^{2})^{1/2}}{4}\\biggl ]\\nonumber \\\\&\\times &\\biggl [\\frac{4+3\\sqrt{2})(2R-3M)}{8R+6M+3(8R^{2}+8MR+6M^{2})^{1/2}}\\biggl ]^{1/\\sqrt{2}}\\nonumber \\\\&=&R\\biggl [1-\\frac{M}{R}-\\frac{M^{2}}{2R^{2}}+...\\biggl ].$ In the limiting case $R\\rightarrow 3M/2$ , $r\\rightarrow 0$ .", "Correspondingly, the conformal factor is obtained from $\\psi &=&\\biggl [\\frac{4R}{\\displaystyle 2R+M+(4R^{2}+4MR+3M^{2})^{1/2}}\\biggl ]\\nonumber \\\\&\\times &\\biggl [\\frac{8R+6M+3(8R^{2}+8MR+6M^{2})^{1/2}}{4+3\\sqrt{2})(2R-3M)}\\biggl ]^{1/2\\sqrt{2}}\\,.$ Finally, the lapse function and the isotropic shift vector are respectively given by $\\alpha &=&\\biggl (1-\\frac{2M}{R}+\\frac{27M^{4}}{16R^{4}}\\biggl )^{1/2}\\,,\\\\\\beta ^{r}&=&\\frac{3\\sqrt{3}M^{2}}{4}\\frac{r}{R^{3}}\\,.$" ], [ "Adiabatic growth of the BH mass ", "We assume that the mass of the BH grows due to the capture of matter as e.g.", "falling in from an accretion disk.", "This infalling matter is assumed to interact with the scalar field only gravitationally, that is the quasi-bound states surrounding the BH can only be affected by the increase of the mass of the BH.", "We will employ a simple phenomenological law based on observational and theoretical grounds which allows us to reasonably incorporate the growth of the BH mass in our model.", "As mentioned before, the observational evidence of the existence of very luminous quasars, which implies BH masses of $\\sim 10^{8}-10^{9}M_{\\odot }$ at $z\\sim 6-7$  [24], [25], [7] demonstrates that SMBHs grow rapidly in a short span of time ($\\sim 10^{9}$ years).", "Moreover, cosmological simulations [26], [27], [28] suggest that SMBH seeds undergo an exponential growth phase at early times, $z\\gtrsim 4$ .", "Therefore, given a growth rate $\\dot{M}_{\\rm {BH}}$ , the mass of the BH will increase as $M = M_{0}\\,e^{\\dot{M}_{\\rm {BH}} t \\ },$ where $M_0$ is the initial BH mass and $t$ is the time in our simulations.", "The actual mechanism that produces the growth of the BH mass is not actually relevant for our study since we are interested in describing the evolution of the scalar field.", "Thus, assuming that it grows according to Eq.", "(REF ) seems quite convenient.", "Figure: Evolution of the BH mass for the different accretion rates indicated in the legend.", "Note that the cases M ˙ BH =0\\dot{M}_{\\rm {BH}}=0 and M ˙ BH =5×10 -7 \\dot{M}_{\\rm {BH}}=5\\times 10^{-7} practically overlap in the plot.", "Unless stated otherwise the same color criterionfor M ˙ BH \\dot{M}_{\\rm {BH}} is employed in the remaining figures of this paper.We first consider five different values for $\\dot{M}_{\\rm {BH}}$ , namely, $\\dot{M}_{\\rm {BH}}=\\lbrace 0, 5\\times 10^{-7},5\\times 10^{-6}, 5\\times 10^{-5}$ and $10^{-4\\rbrace }$ .", "The time evolution of the BH mass for the different growth (accretion) rates considered is plotted in Fig.", "REF .", "We note that our values of $\\dot{M}_{\\rm {BH}}$ , while small, are nevertheless orders of magnitude higher than realistic values inferred from observations, which are at most $\\dot{M}_{\\rm {BH}}\\sim 10^{-11}$ in our units [29].", "(We further expound on this issue in Section REF below.)", "Using such values would however render the numerical investigation prohibitively expensive.", "The first set of simulations reported in this paper evolve the scalar fields up to a final time $t=4\\times 10^4M$ , which is $\\sim 0.2$ s for $M=1\\,M_{\\odot }$ .", "Despite the evolutions are fairly long from the computational point of view, they are certainly not so on astrophysical grounds.", "Therefore, in order to study the effect of accretion on the scalar field evolution in affordable computational times, we have to resort to large enough growth rates.", "As we show below, our simulations indicate that even for the smaller $\\dot{M}_{\\rm {BH}}$ considered some slight differences appear by the end of the simulation.", "Nevertheless, longer, and computationally more expensive, simulations would be necessary to show the influence of the growth rate for the smaller values of $\\dot{M}_{\\rm {BH}}$ .", "In the second part of our study we change the computational setup of the problem by placing reflecting boundary conditions for the scalar field at some radius, i.e.", "placing a mirror beyond which the scalar field is required to vanish, as we did in [30].", "While evolving the scalar field in a cavity is certainly an unrealistic situation, it has some practical advantages as it allows us to investigate lower values of $\\dot{M}_{\\rm {BH}}$ and perform longer runs since the computational grid is significantly smaller than in the “natural\" setup (with outgoing boundary conditions at the outermost radial grid zone).", "Within this idealized setup the initial value of $\\dot{M}_{\\rm {BH}}$ can be as low as $\\dot{M}_{\\rm {BH}}=5\\times 10^{-9}$ .", "Despite this reduction only brings in practice about one order of magnitude gain in the final evolution time, it is nevertheless significant to better quantify the exponential decay in the energy of the scalar field, as we show below." ], [ "Initial data", "As initial data for the scalar field we set a Gaussian distribution of the form $\\Phi =A_0e^{-(r-r_0)^2/\\lambda ^2} \\ ,$ where $A_0$ is the initial amplitude, $r_0$ is the center of the Gaussian, and $\\lambda $ is its width.", "The auxiliary first order quantities are initialized as follows $\\Pi (t=0,r)&=&0 \\ , \\\\\\Psi (t=0,r) &=& -2\\frac{(r-r_0)}{\\lambda ^2}A_0e^{-(r-r_0)^2/\\lambda ^2} \\ .$ Likewise, we choose a conformally flat metric with $a=b=1$ together with a time symmetry condition, i.e.", "a vanishing extrinsic curvature, $K_{ij}=0$ .", "Figure: Left column: Time evolution of the scalar field with mass Mμ=0.1M\\mu =0.1 (top) and Mμ=0.2M\\mu =0.2 (bottom) and for different BH growth rates, M ˙ BH \\dot{M}_{\\rm {BH}}.", "Right column: Corresponding power spectra obtained from Fourier transforming the time series shown on the left.", "The units in the vertical axis are arbitrary.For the BH, we compute the Schwarzschild solution using the moving puncture technique from equations (REF )-() for the initial mass $M_{0}=1$ .", "Table: Initial parameters and most relevant quantities for the different distributions of scalar fields considered.From left to right the columns report: the BH mass growth rate, M ˙ BH \\dot{M}_{\\rm {BH}}, the scalar field mass, MμM\\mu ,the initial amplitude of the pulse, A 0 A_0, the real part of the angular frequency ω\\omega for the fundamental mode ofoscillation and the first overtone, the linear, quadratic, and cubic numerical fits of the decay rateof the modes (coefficients aa, bb and cc in the text), MsMs, and the final BH mass, M BH M_{\\rm {BH}}.The initial Gaussian pulse is located at r 0 =100Mr_0=100M withhalf-width λ=50\\lambda =50." ], [ "Numerical approach", "The solution of the Klein-Gordon evolution equation is computed with the same type of numerical techniques we have used in previous works.", "The reader is addressed in particular to Refs.", "[31], [20] for full details on those techniques.", "As a succinct summary we mention that the evolution equations are integrated using the second-order PIRK method introduced in [32], [33] which allows to handle singular terms that appear due to our choice of curvilinear coordinates.", "The spatial derivatives in the evolution equations are computed using a fourth-order centered finite difference approximation on a logarithmic grid except for the advection terms for which we adopt a fourth-order upwind scheme.", "We also use fourth-order Kreiss-Oliger dissipation to avoid high frequency noise appearing near the outer boundary.", "Again, we stress that in the approach adopted in this work, only the scalar field needs to be evolved numerically.", "The spacetime is updated following the algebraic equations from Section REF .", "Our simulations employ a logarithmic radial grid, as described in [21].", "We set the finest radial resolution close the origin, and a grid spacing of $\\Delta r=0.1M$ .", "In the first set of simulations, the outer boundary of the computational domain is placed at $r_{\\rm {max}}=4\\times 10^4M$ , far enough as to not affect the dynamics of the scalar field in the inner region during the entire simulation.", "The time step is chosen to $\\Delta t=0.5\\Delta r$ which guarantees long-term stable simulations.", "The final time of the numerical evolutions is $4\\times 10^4M$ .", "In the second set of simulations, corresponding to the mirrored states, the radial extension of the numerical domain can be significantly reduced.", "Specifically we place the outer boundary at $r_{\\rm {max}}=250M$ which allows us to use ten times less grid points than in the first setup.", "Finally, the time in the second set of simulations extends up to $8\\times 10^5M$ ." ], [ "Quasi-bound frequencies", "Quasi-bound states of scalar fields around BHs in the test-field approximation are configurations that have well defined (complex) frequencies.", "The real part gives the oscillation of the field and the imaginary part gives the rate of decay of the configuration.", "As mentioned in the Introduction, there is a growing body of work which has shown that for some choices of parameters quasi-bound states may survive for cosmological timescales around SMBHs and, thus, they are consistent with dark matter models based on (ultra-light) scalar fields [17], [18], [19], [20], [21].", "Such long-lasting, quasi-bound states have also been found to exist in the nonlinear regime [34].", "We turn next to discuss the effect of accretion on the quasi-bound states.", "We solve the Klein-Gordon system (REF ) using the initial data given by Eqs.", "(REF )-(REF ) in the rapidly changing gravitational field of an accreting BH.", "As done in [20], [21] we analyze the results of the simulations by extracting a time series for the scalar field amplitude at a set of observation points located at fixed radii $r_{\\rm {ext}}$ (typically at $r_{\\rm {ext}}=100M$ ).", "We identify the frequencies at which the scalar field oscillates by performing a Fast Fourier transform.", "It is worth mentioning that the values of the frequencies do not depend on the location of the observer.", "Our main results are summarized in Table REF .", "The first three columns report our different choices for the growth rate of the mass of the BH, $\\dot{M}_{\\rm {BH}}$ , the five different scalar field masses $M\\mu $ considered in this work, and the initial amplitude of the Gaussian pulse, $A_{0}$ .", "The reason we vary the initial amplitude of the pulse for the different values of $\\mu $ is to keep the initial scalar field energy (almost) constant for the different accretion rates considered, namely $E_{\\rm {SF}}^{0}=1.12\\times 10^{-4}$ .", "The oscillation frequencies of the scalar field are reported in columns 4 and 5 for the fundamental mode and the first overtone, respectively.", "The imaginary part of the frequencies, shown in columns 6 to 8, is the decay rate of the oscillations of the scalar field.", "From left to right these three columns report the linear, quadratic, and cubic parts of analytic fits to numerical data for the different models.", "The last column of Table REF also reports the mass of the BH at the end of the simulation, $M_{\\rm {BH}}$ , computed from Eq.", "(REF ), whose evolution is displayed in Fig.", "REF .", "Figure: Time evolution of the scalar field energy for different values of the BH mass growth rate and for thescalar field masses Mμ={0.05,0.08,0.10}M\\mu =\\lbrace 0.05,0.08,0.10\\rbrace , from top to bottom.", "The dashed black lines correspondto the exponential fit for the three models with higher BH mass growth rate.Figure: Time evolution of the scalar field energy for the masses Mμ={0.15,0.20}M\\mu =\\lbrace 0.15,0.20\\rbrace .Figure REF shows the time evolution of the scalar field amplitude and spectra extracted at $r_{\\rm {ext}}=100M$ .", "In the left panels we plot the evolution for two representative scalar field masses, namely $M\\mu =0.1$ (top) and $M\\mu =0.2$ (bottom), and for the different BH mass growth rates.", "The criterion for the color of the different curves and its relation to $\\dot{M}_{\\rm {BH}}$ follows from Fig.", "REF .", "For each panel, the decay of the amplitude of the oscillations of the scalar field is seen to depend strongly on $\\dot{M}_{\\rm {BH}}$ , regardless of $M\\mu $ .", "For the higher growth rates (orange and blue curves), the field amplitudes by the end of the simulations have almost practically vanished.", "Large field amplitudes and long-lasting oscillations are most noticeable for the three lower accretion rate models of our sample.", "The differences for $\\dot{M}_{\\rm {BH}}=5\\times 10^{-7}$ with respect to the non-accreting case become barely visible only for late times.", "The right panels of Fig.", "REF display the corresponding power spectra.", "They show well-defined oscillation frequencies, best seen in the insets, of the quasi-stationary states for the models with smaller BH mass growth rate (black, red and green curves), as expected from the bandwidth theorem, while the other two models attain wider peaks.", "As expected, in the non-accreting BH case, the oscillation frequencies shown in Fig.", "REF and reported in Table REF are in very good agreement with those found in [20] in the test-field regime and for the same scalar field masses.", "Despite in the current work we are not evolving the spacetime, while we did so in [20], the discarded back-reaction on the evolution of the scalar field is negligible because of the small energy of the scalar field in our models.", "Our results also indicate that the frequencies of the quasi-bound states decrease as the mass of the BH increases.", "One may infer such behavior from a classical mechanics analogy.", "Let us consider the solution of the one-dimensional wave equation in flat spacetime with moving boundaries.", "If the boundary moves with a constant velocity $v$ an explicit solution can be obtained.", "Let us consider the oscillation of a string along the $x$ direction.", "The string is constrained at two points, $x=0$ and $x=vt$ , and the amplitude of the wave is given by $h(t,x)$ .", "We are interested in the solution of the equation $\\partial _{tt} h(t,x) - \\partial _{xx}h(t,x) = 0, $ in the interval $0\\le x \\le vt$ , with the boundary conditions, $h(0,t) = h(vt,t)= 0$ and initial data, $h(x,0) = h_{0}(x)$ $\\partial _{t}h(0,x) = u_{0}(x)$ .", "The general solution of Eq.", "(REF ) is a periodic function in $\\log [(1-v)t]$ with period $T =\\log \\left(\\frac{1+v}{1-v}\\right)$ of the form $h(t,x) = \\sum _{n=-\\infty }^{+\\infty } C_{n} \\left\\lbrace \\exp \\left[i\\frac{2n\\pi }{T} \\tilde{\\alpha }\\right]- \\exp \\left[i\\frac{2n\\pi }{T} \\tilde{\\beta }\\right] \\right\\rbrace $ where $\\tilde{\\alpha }= \\log (t+x)$ and $\\tilde{\\beta }= \\log (t-x)$ and the coefficients $C_{n}$ are determined by the initial conditions.", "What is important to notice in this solution is that the frequency decreases as the velocity of the boundary increases.", "One may think that a similar process happens when the mass of the BH increases since that implies the displacement of the position of the horizon and, thus, the displacement of the position of the boundary that confines the scalar field." ], [ "Scalar field energy", "The time evolution of the energy of the scalar field, computed with Eq.", "(REF ), is shown in solid lines in Figs.", "REF and REF for all scalar field masses, $M\\mu =\\lbrace 0.05,0.08,0.10,0.15,0.20\\rbrace $ .", "The dashed lines in the three panels of Fig.", "REF correspond to analytic fits of the numerical data, that we discuss below.", "These two figures show that the initial energy of the scalar field is (almost) the same for all models considered.", "Both figures show that the decrease of the initial energy is more significant and faster the larger the scalar field mass and the larger the BH growth.", "Let us first discuss the lightest scalar field models.", "In figure REF , corresponding to $M\\mu =\\lbrace 0.05,0.08,0.10\\rbrace $ , we see that for the smallest, non-zero value of $\\dot{M}_{\\rm {BH}}$ , the decay of the scalar field energy is very close to the non-accreting case (compare red and black curves).", "For the higher values of the mass flux, the energy at early times begins decaying exponentially with a small slope but, at some point, the falling of scalar field energy onto the BH speeds up.", "The appearance of this effect depends on the BH mass growth rate, appearing earlier for higher $\\dot{M}_{\\rm {BH}}$ .", "After some time, the process stops and the scalar field energy settles to an almost constant value, given by the energy of the part of the scalar field that has escaped to infinity.", "There is, however, part of the scalar field still localized around the BH in the form of quasi-stationary bound states.", "On the other hand, Fig.", "REF shows that for $M\\mu =0.15$ and $0.2$ , the decay rate of the energy of the scalar field is much faster than for the lightest models and the energy decreases almost identically regardless of the BH mass growth rate.", "This becomes clear if we check the imaginary part of the frequency reported in column 6 of Table REF for these scalar field masses.", "The imaginary part of the frequency does not vary significantly when the growth of the BH mass is faster.", "However, as we mentioned before for the smaller values of the scalar field mass displayed in Fig.", "REF , the energy decay changes from an exponential decay with a linear exponent (the usual feature of a quasi-stationary state around a non-accreting Schwarzschild BH) to an exponential decay with both quadratic and cubic exponents, as shown by the analytic fits.", "These parts are entirely due to the growth of the BH and are orders of magnitude smaller than the linear part of the exponential decay.", "Therefore, they are only significant for sufficiently long times.", "Columns 6, 7, and 8 of Table REF report the values of the rate of decay according to the fit $\\exp [-c\\,t^3-b\\,t^{2}-a\\,t]$ , where $a$ , $b$ and $c$ are the linear, quadratic and cubic coefficients, respectively.", "The dashed black lines in Fig.", "REF fit part of the time evolution of the scalar field energy for the three models with higher BH growth rate.", "The values of the cubic, quadratic and linear coefficients increase with the scalar field mass, as can be inferred from Table REF .", "Table: Same as Table but for the simulation setup corresponding to a scalar field enclosed in acavity.", "The initial Gaussian pulse is, in this case, located at r 0 =50Mr_0=50M with half-width λ=25\\lambda =25.In order to further understand the influence of the growth of the BH mass in the evolution of the scalar field, we turn now to describe the results corresponding to the evolution of an additional set of 21 models.", "These models correspond to Schwarzschild-like BH spacetimes with 7 different BH mass growth rates, namely $\\dot{M}_{\\rm {BH}}=\\lbrace 0,\\,5\\times 10^{-9},\\,5\\times 10^{-8},\\,5\\times 10^{-7},\\,5\\times 10^{-6},\\,5\\times 10^{-5},\\,10^{-4}\\rbrace $ and we impose reflecting boundary conditions for the scalar field at some radius, as done in [30].", "At the location of the mirror, $r=r_{\\rm m}$ , and beyond, the scalar field $\\Phi $ is required to vanish.", "This leads to a discontinuity in the $\\Phi $ derivatives.", "In our simulations the mirror is located at $r_{\\rm {m}}=200M$ and the boundary conditions for the scalar field are $\\begin{split}\\Phi (r_{\\rm {m}})&=\\Psi (r_{\\rm {m}})=\\Pi (r_{\\rm {m}})=0,\\\\\\partial _{r}\\Phi (r_{\\rm {m}})&=\\partial _{r}\\Psi (r_{\\rm {m}})=\\partial _{r}\\Pi (r_{\\rm {m}})=0.\\end{split}$ We consider again a Gaussian distribution for the scalar field, with $r_{0}=50M$ and $\\lambda =25$ .", "As mentioned before, using a mirror is an idealized setup, yet it has pragmatic advantages since it allows us to perform longer runs and to study smaller growth rates.", "The spacetime is still given by the analytic solution of Section REF (hence, there are no reflections of the spacetime variables coming in from the outer boundary) and since the scalar field is now enclosed in a cavity no part of the scalar field propagates away from the BH.", "Therefore, all the oscillatory modes will be trapped in this case, not only those with $\\omega ^2<\\mu ^2$ , and the energy evolution will not be dominated by the asymptotic, scalar field energy minimum.", "In this idealized situation, the entire scalar field will fall into the BH with time.", "The results of this new set of simulations are similar to those discussed in the preceding section.", "They are summarized in Table REF for the lightest scalar field models analyzed, namely $M\\mu =\\lbrace 0.05,0.08,0.10\\rbrace $ .", "Again, the scalar field oscillates with a fundamental frequency and higher overtones (of which Table REF only reports the first one), but since the higher frequencies decay faster we end up with only the fundamental mode of oscillation.", "As for the cases without mirror, the real part of the frequency decreases with $\\dot{M}_{\\rm {BH}}$ .", "In the case of a non-accreting BH, the decay of the energy (and of the amplitude of the oscillations) is exponential with a single linear exponent as expected.", "As done before, in the case of accreting BHs and long evolution times, we fit the energy decay to the analytic formula $\\exp [-d\\,t^{4}-c\\,t^{3}-b\\,t^{2}-a\\,t]$ , with linear, quadratic, cubic and quartic terms.", "These parameters seem to depend on the BH mass growth rate and, probably, we could fit it to even higher order polynomials if we could reach even longer simulations.", "Indeed, some of the models reported in Table REF can already be fitted with higher order terms (even up to 5th or 6th, coefficients $e$ and $f$ respectively) for the current evolution times considered.", "The 6th order term is considered for only one particular model, even if it is very small, in order to keep the highest order term always positive and, therefore, drive the fit to tend to zero for $t\\rightarrow \\infty $ .", "The coefficients also depend on the scalar field mass, decreasing for smaller values of $\\mu $ .", "In Fig.", "REF we plot (solid lines) time evolutions of the scalar field energy for the mirrored states of models with masses $M\\mu =\\lbrace 0.05,0.08,0.10\\rbrace $ (from top to bottom) together with their corresponding fits (black dashed lines).", "The results discussed in the previous section become more clear in this idealized setup.", "Increasing the BH growth rate speeds up the decay of the energy.", "Moreover, the decay is longer than for the case of quasi-bound states without mirror and, contrary to what happens when there is an outgoing boundary, the energy does not relax to the value corresponding to that part of the scalar field that escapes to infinity." ], [ "Comparison with realistic BH mass growth rates", "We can estimate the rate of the BH mass growth due to accretion using the bolometric luminosity of an Active Galaxy Nuclei (AGN) as [35], [29] $L_{\\rm {bol}}=\\frac{\\eta \\dot{M}_{\\rm {BH}}}{(1-\\eta )}c^{2}\\,,$ where $L_{\\rm {bol}}$ is the bolometric luminosity, $\\dot{M}_{\\rm {BH}}$ is the BH mass growth rate, $c$ is the speed of light, and $\\eta $ is the radiative efficiency, for which we take the commonly adopted empirical value of 0.1 [36].", "Recently, bolometric luminosities larger than $10^{14}\\,L_{\\odot }$ have been discovered [29].", "According to Eq.", "(REF ), this corresponds to $\\dot{M}_{\\rm {BH}}\\sim 10^{-11}$ in our units.", "In figure REF we plot several BH growth rates in geometrized units of AGN bolometric luminosities as a function of the redshift $z$ .", "We use data from two different samples, namely that of [37], for redshifts between 0.01 and 2.224, and the luminosities of WISE-selected galaxies reported recently in [29], which include redshifts up to $z=4.593$ .", "The larger luminosities are obtained precisely for the higher redshifts, corresponding to the growth phase of SMBHs.", "Fig.", "REF shows that the maximum growth rate is only about two orders of magnitude smaller than the smallest $\\dot{M}_{\\rm {BH}}$ we can afford in our simulations.", "The influence of the accretion rate on the energy decay depends on the scalar field mass and decreases for smaller $M\\mu $ .", "Therefore, for the expected value of scalar field dark matter models, the actual value of $\\dot{M}_{\\rm {BH}}$ is expected to be fairly small.", "However, the growth of SMBH seeds to their actual masses, $\\sim 10^{9}M_{\\odot }$ , is believed to take place only during 1% of their lifetime, i.e.", "about $10^{9}$ years.", "Therefore, the role played by high order terms in the evolution of the energy could become significant during this phase and increase the amount of scalar field that may end up being absorbed by the BH during this period.", "Figure: AGN BH growth rates in geometrized units as a function of redshift.", "The dashed horizontal lines indicate the valuesof M ˙ BH ={5×10 -9 ( red ),5×10 -8 ( green ),5×10 -7 ( blue ),5×10 -6 ( magenta ),5×10 -5 ( cyan ),1×10 -4 ( orange )}\\dot{M}_{\\rm {BH}}=\\lbrace 5\\times 10^{-9}\\,(\\rm {red}),5\\times 10^{-8}\\,(\\rm {green}),5\\times 10^{-7}\\,(\\rm {blue}),5\\times 10^{-6}\\,(\\rm {magenta}),5\\times 10^{-5}\\,(\\rm {cyan}),1\\times 10^{-4}\\,(\\rm {orange})\\rbrace considered in our numerical study.", "The symbols correspond to observational datafrom , (red and blue points, respectively).", "Our values are about two orders of magnitude larger than the largest observationalvalue.Figure: Half-life of the quasiresonant frequencies of the quasi-bound states for different scalar field massesand different BH growth rates.", "The solid lines are least-square fits to the numerical data.Finally, we can extrapolate our results to the realistic case of ultra-light scalar field masses.", "In the case of a test-field the decay rate of the dynamical resonances is related to the imaginary part of the quasiresonant frequencies, and thus its half-life time $t_{1/2}$ is inversely proportional to $\\text{Im}(M\\omega )$  [18].", "For an accreting BH and the long evolution times reported in this work we have shown that terms higher than linear are important to capture the decay rate of the energy of the scalar field.", "We can then use the relation $E_{\\rm SF}=E^0_{\\rm SF}\\exp {[-(at+bt^2+ct^3)]}$ to solve for $t_{1/2}$ which corresponds to the time when the energy of the scalar field has decreased to half its initial value.", "The result of this exercise is depicted in Fig.", "REF , which shows $t_{1/2}$ as a function of $M\\mu $ for all five values of $\\dot{M}_{\\rm {BH}}$ used in the “unmirrored\" setup simulations.", "(We use in this figure the same color criterion for $\\dot{M}_{\\rm {BH}}$ as defined in Fig.", "REF .)", "The symbols in the figure indicate the values of $t_{1/2}$ for the lighter values of the scalar field mass ($M\\mu =0.05, 0.08$ and 0.1) used in our simulations.", "Following [18] we do a least squares fit to these data and extrapolate our results to lower values of $M\\mu $ .", "Fig.", "REF shows that even for high mass accretion rates, e.g.", "$\\dot{M}_{\\rm {BH}}=5\\times 10^{-7}$ (red curve) quasi-bound states around accreting BHs can survive for timescales significantly longer than the age of the Universe." ], [ "Conclusions", "We have studied the properties of scalar field quasi-bound states in the background of an accreting, spherically symmetric BH.", "We have assumed that the BH mass grows due to matter accretion, describing the effect of the increase of the BH mass on the properties of the quasi-bound states of the surrounding scalar field distribution.", "For our study we have solved the Klein-Gordon equation numerically, mimicking the evolution of the spacetime through a sequence of exact Schwarzschild BH solutions of increasing mass, using analytic expressions which depend only on the BH mass parameter.", "Our numerical approach has been limited to spherical symmetry and has been based on spherical coordinates and a PIRK numerical scheme for the time update of the Klein-Gordon equation.", "To study the effect of accretion on the scalar field evolution in affordable computational times, we have resorted to large growth rates, at best two orders of magnitude larger than the largest observational estimations.", "By performing a Fourier transform of the time series of our numerical data we have been able to characterize the scalar field states by their distinctive oscillation frequencies.", "It has been found that the frequencies decrease with increasing BH mass.", "Moreover, accretion results in a significative increase of the exponential decay of the scalar field energy due to the presence of terms of order higher than linear in the exponent.", "These terms are zero in the non-accreting Schwarzschild BH case, resulting in a linear exponential fit.", "Our computational setup has considered both outgoing and reflecting boundary conditions at the outer radial boundary, the latter describing a scalar field enclosed in a cavity.", "By imposing reflecting boundary conditions at a finite distance the scalar field does not escape to infinity and we can isolate the influence of accretion.", "Such configuration, albeit artificial, has helped us to study the higher order terms that appear in the exponential decay of the energy.", "Finally, we have compared our BH mass growth rates with estimates from observational surveys, and we have been able to extrapolate our results to realistic values of the scalar field mass $M\\mu $ .", "We have found that even for the high mass accretion rates considered in this work, e.g.", "$\\dot{M}_{\\rm {BH}}=5\\times 10^{-7}$ , quasi-bound states around accreting BHs can survive for cosmological timescales.", "The results obtained in this paper add further support to the intriguing possibility of the existence of dark matter halos based on (ultra-light) scalar fields surrounding SMBHs at galactic centers.", "This work was supported in part by the Spanish MINECO (AYA2013-40979-P), by the Generalitat Valenciana (PROMETEOII-2014-069), by the CONACyT-México, ICF-UNAM, and by the Max-Planck-Institut für Astrophysik.", "The computations have been performed at the Servei d'Informàtica de la Universitat de València." ] ]
1606.05146
[ [ "Probing proton intrinsic charm in photon or Z boson production\n accompanied by heavy jets at LHC" ], [ "Abstract We consider an observable very sensitive to the non-zero intrinsic charm (IC) contribution to the proton density.", "It is the ratio between the differential cross sections of the photon or $Z$-boson and $c$-jet production in the $pp$ collision, $\\gamma(Z) + c$, and the $\\gamma(Z)$ and the $b$-jet production.", "It is shown that this ratio can be approximately flat or increasing at large $\\gamma(Z)$ transverse momenta $p_T$ and their pseudo-rapidities $1.5 < \\eta < 2.4$ if the IC contribution is taken into account.", "On the contrary, in the absence of the IC this ratio decreases as $p_T$ grows.", "We also present the ratios of the cross sections integrated over $p_T$ as a function of the IC probability $w$.", "It is shown that these ratios are mostly independent on the theoretical uncertainties, and such predictions could therefore be much more promising for the search for the intrinsic charm signal at the LHC compared to the predictions for $p_T$-spectra, which significantly depend on these uncertainties." ], [ "Introduction", "The hypothesis of the intrinsic (or valence-like) heavy quark component, the quark Fock state $|uudQ{\\bar{Q}}\\rangle $[1], [2], [3], [4] in a proton suggested by Brodsky with coauthors[1], [2] (BHPS model) is intensively discussed in connection with an opportunity to verify it experimentally[5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [16].", "Up to now, there is a long-standing debate about the possible existence of the intrinsic charm (IC) and intrinsic strange (IS) quarks in a proton[7], [16], [17], [18].", "Thorough theoretical and experimental studies of these intrinsic heavy quark components would be very important for the experiments performed at the LHC.", "Recently it was shown that the possible existence of the intrinsic heavy quark components in the proton can be seen not only in the inclusive heavy flavor production at high energies[8], but also in the semi-inclusive production of prompt photons or vector bosons accompanied by heavy quark jets[9], [14].", "An experimental hint on possible existence of the IC contribution was observed in the Tevatron experiment on the prompt photon production in the association of the $c$ and $b$ jets in the $p{\\bar{p}}$ annihilation at $\\sqrt{s} = 1.98$  TeV[19], [20].", "It was shown that the description of the Tevatron data within the perturbative QCD (pQCD) could be significantly improved if the IC contributions were taken into account The photon transverse momentum ($p_T$ ) spectrum in the $\\gamma +c$ production and the ratio of the spectra in the $\\gamma +c$ and $\\gamma + b$ production measured at the Tevatron[22] are better described within the BHPS model[1], [2], which includes the IC contributions.", "According to the pQCD calculations[21], in the absence of the IC contribution this ratio decreases, when $p_T$ grows, while the Tevatron data show its flat behavior at large $p_T \\ge 100$  GeV[22].", "The possible IC signal can also be observed in the hard $pp$ production of the gauge bosons $Z$ or $W$ accompanied by heavy flavors.", "As it was shown[14], the ratio of the $Z + c$ and $W$ + heavy jet production cross sections maximizes the sensitivity to the IC component of the proton.", "Our early predictions about a possible intrinsic charm signal in the production of prompt photons or gauge bosons accompanied by heavy flavor jets concerned their transverse momenta distributions in the mid-rapidity region of $pp$ collisions at the LHC energies[9], [14].", "It was obtained with the IC probability about $w = 3.5$ %, which is the upper limit being due to constraints from the HERA data on the deep inelastic scattering.", "However, the upper limit of the IC probability in a proton is still very actively debated[7], [16], [17], [18].", "Therefore, in the present paper we focus mainly on the predictions for searching at any $w$ for the IC signal in the observables, which are very little sensitive to the theoretical uncertainties, namely, the ratios between the $\\gamma (Z) + c$ and $\\gamma (Z) + b$ cross sections in $pp$ collisions at the LHC energies.", "An important advantage of these observables is that many theoretical uncertainties, for example, heavy quark masses, the factorization and/or renormalization scales, are canceled, as will be demonstrated below.", "We show that the measure of these ratios is much more promising for the search for the IC signal.", "Below we perform the calculations in two ways.", "First, we use the parton-level Monte Carlo event generator mcfm[23], which implements the NLO pQCD calculations of associated $Z$ boson and heavy flavor jet production.", "The detailed description of the mcfm routine is available[23].", "To generate the prompt photon and heavy jet production cross sections, we apply the $k_T$ -factorization approach[24], [25], which becomes a commonly recognized tool in the high energy phenomenology.", "Our main motivation is that it gives a better description of the Tevatron data compared to the NLO pQCD calculations[21], as it was claimed[19], [20].", "We apply this approach to the associated $Z$ and heavy jet production to perform an independent cross-check of our resultsUnfortunately, the mcfm routine does not produce the prompt photon and heavy jet production cross sections..", "The outline of our paper is the following.", "In Sections 2 and 3 we recall basic ideas with a brief review of calculation steps.", "In Section 4 we present the numerical results of our calculations and a discussion.", "Finally, Section 5 contains our conclusions." ], [ "Intrinsic charm density in a proton as a function of IC probability $w$", "According to[6], [12], [26], the intrinsic charm distribution at the starting scale $\\mu _0^2$ as a function of $x$ can be presented in the following approximated form: $c_{int}(x,\\mu _0^2) = c_0 w x^2 \\left[(1-x)(1+10x+x^2)+6x(1+x)\\ln (x) \\right],$ where $w$ is the probability to find the Fock state $|uudc{\\bar{c}}\\rangle $ in the proton, $c_0$ is the normalization constant and the masses of the light quarks and the nucleon are neglectedable compared to the charm quark mass.", "The inclusion of the non-zero nucleon mass leads to a more complicated analytic form[27].", "According to the BHPS model[1], [2], the charm density in a proton is the sum of the extrinsic and intrinsic charm densities, $xc(x,\\mu _0^2) = xc_{\\it ext}(x,\\mu _0^2) + xc_{\\it int}(x,\\mu _0^2).$ The extrinsic, or ordinary quarks and gluons are generated on a short-time scale associated with the large-transverse-momentum processes.", "Their distribution functions satisfy the standard QCD evolution equations.", "Contrariwise, the intrinsic quarks and gluons can be associated with a bound-state hadron dynamics and one believes that they have a non-perturbative origin.", "The lifetime of this Fock-state should be much more than the interaction time of the hard probe[27].", "Some comments on this are presented below.", "It was argued[2] that existence, for example, of intrinsic heavy quark pairs $c{\\bar{c}}$ and $b{\\bar{b}}$ within the proton state can be due to the gluon-exchange and vacuum-polarization graphs.", "The charm density $xc(x,\\mu ^2)$ at an arbitrary scale $\\mu ^2$ is calculated using the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations[28].", "Let us stress here that both the intrinsic part $xc_{\\it int}$ and extrinsic one $xc_{\\it ext}$ depend on $\\mu ^2$ .", "In the general case, there is some mixing between two parts of (REF ) during the DGLAP evolution.", "However, such mixing is negligible[12], [13], especially at large $\\mu ^2$ and $x$ .", "It can be seen from comparison of our calculations of charmed quark densities presented in Fig.", "REF , where this mixing was included within the CTEQ[29] set, and Fig.", "2 of[12], when the mixing between two parts of the charm density was neglected.", "Our results on the total charm density $xc(x,\\mu ^2)$ are in good agreement with the calculations[12] in the whole kinematical region of $x$ because at $x<0.1$ the IC contribution is much smaller than the extrinsic one.", "Therefore, one can apply the DGLAP evolution separately to the first part $xc_{\\it ext}(x,\\mu _0^2)$ and the second part $xc_{int}(x,\\mu _0^2)$ of (REF ), as it was done in [12], [13].", "Such calculations were done by the CTEQ[29] and CT14[30] groups at some fixed values of the IC probability $w$ .", "Namely, the CTEQ group used $w = 1$ % and $w = 3.5$ %, and CT14 used $w = 1$ % and $w = 2$ %.", "Note that, according to the recent paper[27], the lifetime of the intrinsic charm should be more than the interaction time, at least, by a factor of about 5, when the quark Fock-state can be observed with the satisfactory accuracy.", "The ratio of these times is proportional to $Q^2$ or $p_T^2$[27].", "We will analyze the hard processes of $\\gamma (Z)$ production associated with heavy jets at LHC energies and $p_T^2\\ge $ 10$^4$ GeV$^2$ , when the lifetime of the intrinsic charm is much larger than the interaction time, where the intrinsic charm could be resolved.", "Taking into account that the IC probability $w$ enters into (REF ) as a constant in front of the function dependent on $x$ and $\\mu ^2$ , one can suggest a simple relation at any $w \\le w_{\\rm max}$ : $xc_{int}(x,\\mu ^2) = \\frac{w}{w_{\\rm max}}xc_{int}(x,\\mu ^2)|_{w = w_{\\rm max}}.$ Actually, that is the linear interpolation between two charm densities at the scale $\\mu ^2$ , obtained at $w = w_{\\rm max}$ and $w = 0$ .", "Later we adopt the charm distribution function from the CTEQ66M set[29].", "We assume $w_{\\rm max} = 3.5$ % everywhere, which corresponds to the CTEQ66c1 set[29].", "Additionally, we performed the three-point interpolation of the charmed quark distributions (over $w = 0$ , $w = 1$ % and $w = 3.5$ %, which correspond to the CTEQ66M, CTEQ66c0 and CTEQ66c1 sets, respectively).", "These results differ from the ones based on (REF ) by no more than $0.5$ %, thus giving us the confidence in our starting point.", "Below we apply the charmed quark density obtained by (REF ) and (REF ) to calculate the total and differential cross sections of associated prompt photon or $Z$ boson and heavy flavor jet production, $\\gamma (Z) + Q$ , at the LHC conditions.", "The suggested procedure to calculate $xc_{int}(x,\\mu ^2)$ at any $w\\le w_{\\rm max}$ allows us to reduce significantly the time for the calculation of these observables." ], [ "Theoretical approaches to the associated $\\gamma (Z) + Q$ production", "As was mentioned above, we perform the numerical calculations of the associated $\\gamma (Z) + Q$ production cross sections using the parton-level Monte Carlo event generator mcfm within the NLO pQCD as well as the $k_T$ -factorization QCD approach.", "The mcfm is able to calculate the processes, that involve the gauge bosons $Z$ or $W$ (see[23] for more information).", "In contrast to our early study of these processes[14] within the mcfm, we use this generator to calculate the differential and total cross sections of the $Z + c$ and $Z + b$ production in the $pp$ collision and their ratio as a function $w$ .", "The $k_T$ -factorization approach[24], [25] is based on the small-$x$ Balitsky-Fadin-Kuraev-Lipatov (BFKL)[31] gluon dynamics and provides solid theoretical grounds for the effects of the initial gluon radiation and the intrinsic parton transverse momentumA detailed description of the $k_T$ -factorization approach can be found, for example, in reviews[32].. Our main motivation to use here the $k_T$ -factorization formalism is that its predictions for the associated $\\gamma + Q$ production better agree with the Tevatron data compared to the NLO pQCD (see[19], [20]).", "The consideration is mainly based on the ${\\cal O}(\\alpha \\alpha _s)$ off-shell (depending on the transverse momenta of initial quarks and gluons) quark-gluon Compton-like scattering subprocess, see Fig.", "REF (a).", "Within this approach the transverse momentum dependent (TMD) parton densities include many high order corrections, while the partonic amplitudes are calculated within the leading order (LO) of QCD.", "The off-shell quark-gluon Compton scattering amplitude is calculated within the reggeized parton approach[33], [34], [35] based on the effective action formalism[36], which ensures the gauge invariance of the obtained amplitudes despite the off-shell initial quarks and gluonsHere we use the expressions derived earlier[37]..", "The TMD parton densities are calculated using the Kimber-Martin-Ryskin (KMR) approach, currently developed within the NLO[38].", "This approach is the formalism to construct the TMD quark and gluon densities from the known conventional parton distributions.", "The key assumption is that the $k_T$ dependence appears at the last evolution step, so that the DGLAP evolution can be used up to this step.", "Numerically, for the input we used parton densities derived in Section 2.", "Other details of these calculations are explained in[37].", "To improve the $k_T$ -factorization predictions at high transverse momenta, we take into account some ${\\cal O}(\\alpha \\alpha _s^2)$ contributions, namely $q \\bar{q} \\rightarrow V Q{\\bar{Q}}$ and $q Q \\rightarrow V q Q $ ones, where $V$ denotes the photon or the $Z$ boson, see Fig.", "REF (b) — (e).", "These contributions are significant at large $x$ and therefore can be calculated in the usual collinear QCD factorization scheme.", "Thus, we rely on the combination of two techniques that is most suitable." ], [ "Results and discussion", "Let us present the results of our calculations.", "First of all we describe our numerical input.", "Following to[39], we set the charmed and beauty quark masses $m_c = 1.4$  GeV, $m_b = 4.75$  GeV, the $Z$ -boson mass $m_Z = 91.1876$  GeV, and $\\sin ^2 \\theta _W = 0.23122$ .", "The chosen factorization and renormalization scales are $\\mu _R = \\mu _F = \\xi p_T$ or $\\mu _R = \\mu _F = \\xi m_T$ , where $p_T$ is the produced photon transverse momentum and $m_T$ is the $Z$ boson transverse mass.", "As usual, we vary the nonphysical parameter $\\xi $ between $1/2$ and 2 about the default value $\\xi = 1$ in order to estimate the scale uncertainties of our calculations.", "We employ the two-loop formula for the strong coupling constant with active quark flavors $n_f = 5$ at $\\Lambda _{\\rm QCD} = 226.2$  MeV and use the running QED coupling constant over a wide region of transverse momenta.", "The multidimensional integration in the $k_T$ -factorization calculations was performed by means of the Monte Carlo technique, using the vegas routine[40].", "In our calculations we also follow the conclusion obtained in our papers[9], [14] that the IC signal in the hard processes discussed here can be detected at ATLAS or CMS of the LHC in the forward rapidity region $1.5 < |\\eta |< 2.4$ and $p_T > 50$  GeV.", "Additionally, we require $|\\eta (Q)| < 2.4$ and $p_T(Q) > 25$  GeV, where $\\eta (Q)$ and $p_T(Q)$ are the pseudo-rapidity and transverse momentum of the heavy quark jet in a final state, as was done in[9], [14].", "The results of our calculations are shown in Figs.", "REF — REF .", "The transverse momentum distributions of photons and $Z$ bosons accompanied by the $c$ and $b$ quarks are presented in Figs.", "REF , REF and REF at the different IC probability $w$ (namely, $w = 0$ %, $w = 2 $ % and $w = 3.5$ %) at $\\sqrt{s} = 8$ and 13 TeV.", "One can see in Figs.", "REF and REF that the mcfm and $k_T$ -factorization predictions for $Z + Q$ production are very similar in the whole $p_T$ region, therefore below we will present the observables calculated within the $k_T$ -factorization approach only.", "The coincidence of these two calculations is due to effective allowance for the high-order corrections within the $k_T$ -factorization formalism (see, for example,[32] for more information).", "Both types of calculations predict a significant enhancement of $p_T$ distributions due to the IC terms at $p_T \\ge 100$  GeV, which is in agreement with the previous studies[9], [12], [14].", "The $p_T$ -spectrum ratios $\\sigma (\\gamma +c)/\\sigma (\\gamma +b)$ and $\\sigma (Z+c)/\\sigma (Z+b)$ versus $p_T$ at different $w$ are presented in Figs.", "REF and REF .", "One can see that in the absence of the IC contribution the ratio $\\sigma (\\gamma +c)/\\sigma (\\gamma +b)$ is about 3 at $p_T \\sim 100$  GeV and decreases down to 2 at $p_T \\sim 500$  GeV.", "This behavior is the same for both energies $\\sqrt{s} = 8$  TeV and $\\sqrt{s} = 13$  TeV.", "If one takes into account the IC contributions, this ratio becomes approximately flat at $w = 2$ % or even increasing up to about 4 at $w = 3.5$ %.", "It is very close to the Tevatron data[22]: the constant ratio $\\sigma (\\gamma + c) / \\sigma (\\gamma + b) \\sim 3.5 - 4.5$ measured in the $p\\bar{p}$ collisions at $110 < p_T < 300$  GeV and $\\sqrt{s} = 1.96$  TeV.", "However, this agreement cannot be treated as the IC indication due to huge experimental uncertainties (about 50%) and rather different kinematical conditions.", "If the IC contribution is included, the ratio $\\sigma (Z+c)/\\sigma (Z+b)$ also increases by a factor about 2 at $w = 3.5$ %, when the $Z$ boson transverse momentum grows from 100 GeV to 500 GeV (see Fig.", "REF ).", "In the absence of the IC terms this ratio slowly decreases.", "One can consider other observables which could be useful to detect the IC signal, the cross sections discussed above but integrated over $p_T > p_T^{\\rm min}$ , where $p_T^{\\rm min} \\ge 100$  GeV, and their ratios.", "Our predictions for such integrated cross sections versus the IC probability $w$ at $p_T^{\\rm min} = 100$ , 200 and 300 GeV for $\\sqrt{s} = 8$  TeV and $p_T^{\\rm min} = 200$ , 300 and 400 GeV for $\\sqrt{s} = 13$  TeV are shown in Figs.", "REF and REF .", "All the $p_T$ -spectra have a significant scale uncertainty as is shown in[14] (see also Figs.", "REF and REF ).", "According to[14], the ratio between the cross sections for the $Z+Q$ and $W+Q$ production in the $pp$ collision is less sensitive to the scale variation calculated within the mcfm.", "Nevertheless, the uncertainty in this ratio at large $p_T > 250$  GeV is about 40 — 50$\\%$ .", "In the present paper we check these results for the ratios $\\sigma (\\gamma +c)/\\sigma (\\gamma +b)$ and $\\sigma (Z+c)/\\sigma (Z+b)$ .", "In Figs.", "REF and REF (bottom) we present these ratios versus the IC probability $w$ calculated at different scales, when the cross sections of $\\gamma (Z) + Q$ production are integrated within the different intervals of transverse momentum.", "One can see a very small QCD scale uncertainty, especially at $\\sqrt{s} = 13$  TeV (bottom right), which is less than 1$\\%$ .", "In contrast, the scale uncertainty for the integrated $\\gamma (Z) + Q$ cross sections (see Figs.", "REF and REF , top) is significant and amounts to about 30 — 40%.", "The sizable difference between the scale uncertainties for the ratios $\\sigma (Z+Q)/\\sigma (W+Q)$ and $\\sigma (Z+c)/\\sigma (Z+b)$ is due to the different matrix elements for the $Z+Q$ and $W+Q $ production in $pp$ collisions, while the matrix elements for the $Z+c$ and $Z+b$ production are the same.", "It is important that the calculated ratios $\\sigma (\\gamma +c)/\\sigma (\\gamma +b)$ and $\\sigma (Z+c)/\\sigma (Z+b)$ can be used to determine the IC probability $w$ from the future LHC data.", "Moreover, these ratios are practically independent of the uncertainties of our calculations: actually, the curves corresponding to the usual scale variations as described above coincide with each other (see Figs.", "REF and REF , bottom).", "Therefore, we can recommend these observables as a test for the hypothesis of the IC component inside the proton." ], [ "Conclusion", "The transverse momentum spectra of the prompt photons and $Z$ bosons produced in association with the $c$ or $b$ jets in $pp$ collisions are calculated using the mcfm (NLO pQCD) and the $k_T$ -factorization approach at the LHC energies and pseudo-rapidites $1.5 < \\eta < 2.4$ using PDFs with and without the IC contribution.", "It is shown that these two approaches give similar results.", "We found that the contribution of the intrinsic charm can give a significant signal in the ratios $\\sigma (\\gamma + c) / \\sigma (\\gamma + b)$ and $\\sigma (Z + c) / \\sigma (Z + b)$ at forward pseudo-rapidities ($1.5 < \\eta <2.4$ ) corresponding to the ATLAS and CMS facilities.", "If the IC contributions are taken into account, the ratio $\\sigma (\\gamma + c) / \\sigma (\\gamma + b)$ as a function of the photon transverse momentum is approximately flat or increases at $p_T > 100$  GeV.", "The similar flat behavior of this ratio was observed in the $p{\\bar{p}}$ annihilation at the Tevatron.", "In the absence of the IC contributions this ratio decreases.", "Similarly, the ratio $\\sigma (Z + c) / \\sigma (Z + b)$ increases when the $Z$ boson transverse momentum grows if the IC contribution is included and slowly decreases in the absence of the IC terms.", "We argued that the ratio of the cross sections $\\gamma (Z) + c$ and $\\gamma (Z) + b$ integrated over $p_T > p_T^{\\rm min}$ with $p_T^{\\rm min} \\ge 100$  GeV can be used to determine the IC probability from the future LHC data.", "The advantage of the proposed ratios is that the theoretical uncertainties are very small, while the uncertainties for the $p_T$ -spectra of photons or $Z$ bosons produced in association with the $c$ or $b$ jets are large.", "Therefore, the search for the IC signal by analyzing the ratio $\\sigma (\\gamma /Z + c) / \\sigma (\\gamma /Z + b)$ can be more promising." ], [ "Acknowledgments", "We thank S.J.", "Brodsky, A.A. Glasov and D. Stump for extremely helpful discussions and recommendations in the study of this topic.", "The authors are grateful to H. Jung, P.M. Nadolsky for very useful discussions and comments.", "The authors are also grateful to L. Rotali for very constructive discussions.", "This work was supported in part by grant of the President of Russian Federation NS-7989.2016.2.", "A.V.L.", "is grateful to the DESY Directorate for the support within the framework of the Moscow — DESY project on Monte-Carlo implementation for HERA — LHC.", "Figure: The total charmed quark densitiy xc(x,μ 2 )xc(x,\\mu ^2) as a function of xx at different values of wwat μ 2 =10\\mu ^2 = 10 GeV 2 ^2 (top) and μ 2 =10 4 \\mu ^2 = 10^4 GeV 2 ^2 (bottom).The triple-dashed line is the IC contriubion at w=w= 1%, the dashed-double-dotted line corresponds tothe IC at w=w= 2%, the dashed-dotted curve is the IC at w=w= 3% and the double-dashed line corresponds tothe IC at w=w= 3.5 %." ] ]
1606.04882
[ [ "Two double poset polytopes" ], [ "Abstract To every poset P, Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P. This construction allows for deep insights into combinatorics by way of geometry and vice versa.", "Malvenuto and Reutenauer (2011) introduced 'double posets', that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled posets.", "Many combinatorial constructions can be naturally phrased in terms of double posets.", "We introduce the 'double order polytope' and the 'double chain polytope' and we amply demonstrate that they geometrically capture double posets, i.e., the interaction between the two partial orders.", "We describe the facial structures, Ehrhart polynomials, and volumes of these polytopes in terms of the combinatorics of double posets.", "We also describe a curious connection to Geissinger's valuation polytopes and we characterize 2-level polytopes among our double poset polytopes.", "Fulkerson's 'anti-blocking' polytopes from combinatorial optimization subsume stable set polytopes of graphs and chain polytopes of posets.", "We determine the geometry of Minkowski- and Cayley sums of anti-blocking polytopes.", "In particular, we describe a canonical subdivision of Minkowski sums of anti-blocking polytopes that facilitates the computation of Ehrhart (quasi-)polynomials and volumes.", "This also yields canonical triangulations of double poset polytopes.", "Finally, we investigate the affine semigroup rings associated to double poset polytopes.", "We show that they have quadratic Groebner bases, which gives an algebraic description of the unimodular flag triangulations described in the first part." ], [ "Introduction", "A (finite) partially ordered set (or poset, for short) is a finite set $P$ together with a reflexive, transitive, and anti-symmetric relation $\\preceq $ .", "The notion of partial order pervades all of mathematics and the enumerative and algebraic combinatorics of posets is underlying in computations in virtually all areas.", "In 1986, Stanley [34] defined two convex polytopes for every poset $P$ that, in quite different ways, geometrically capture combinatorial properties of $P$ .", "The order polytope $\\mathcal {O}(P)$ is set of all order preserving functions into the interval $[0,1]$ .", "That is, all functions $f : P\\rightarrow \\mathbb {R}$ such that $0 \\ \\le \\ f(a) \\ \\le \\ f(b) \\ \\le \\ 1$ for all $a,b \\in P$ with $a \\preceq b$ .", "Hence, $\\mathcal {O}(P)$ parametrizes functions on $P$ and many properties of $P$ are encoded in the boundary structure of $\\mathcal {O}(P)$ : faces of $\\mathcal {O}(P)$ are in correspondence with quotients of $P$ .", "In particular, the vertices of $\\mathcal {O}(P)$ are in bijection to filters of $P$ .", "But also metric and arithmetic properties of $\\mathcal {O}(P)$ can be determined from $P$ .", "The order polytope naturally has vertices in the lattice $\\mathbb {Z}^P$ and its Ehrhart polynomial $\\mathrm {Ehr}_{\\mathcal {O}(P)}(n) = |n\\mathcal {O}(P) \\cap \\mathbb {Z}^P|$ , up to a shift, coincides with the order polynomial $\\Omega _P(n)$ ; see Section REF for details.", "A full-dimensional simplex with vertices in a lattice $\\Lambda \\subset \\mathbb {R}^n$ is unimodular with respect to $\\Lambda $ if it has minimal volume.", "The normalized volume relative to $\\Lambda $ is the Euclidean volume scaled such that the volume of a unimodular simplex is 1.", "If the lattice is clear from the context, we denote the normalized volume by $\\operatorname{Vol}(\\mathcal {P})$ .", "By describing a canonical triangulation of $\\mathcal {O}(P)$ into unimodular simplices, Stanley showed that $\\operatorname{Vol}(\\mathcal {O}(P))$ is exactly the number of linear extensions of $P$ , that is, the number $e(P)$ of refinements of $\\preceq $ to a total order.", "We will review these results in more detail in Section REF .", "This bridge between geometry and combinatorics can, for example, be used to show that computing volume is hard (cf.", "[4]) and, conversely, geometric inequalities can be used on partially ordered sets; see [26], [34].", "The chain polytope $\\mathcal {C}({P})$ is the collection of functions $g : P\\rightarrow \\mathbb {R}_{\\ge 0}$ such that $g(a_1) + g(a_2) + \\cdots + g(a_k) \\ \\le \\ 1$ for all chains $a_1 \\prec a_2 \\prec \\cdots \\prec a_k$ in $P$ .", "In contrast to the order polytope, $\\mathcal {C}({P})$ does not determine $P$ .", "In fact, $\\mathcal {C}({P})$ is defined by the comparability graph of $P$ and bears strong relations to so-called stable set polytopes of perfect graphs; see Section REF .", "Surprisingly, it is shown in [34] that the chain polytope and the order polytope have the same Ehrhart polynomial and hence $\\operatorname{Vol}(\\mathcal {C}({P})) = \\operatorname{Vol}(\\mathcal {O}(P)) = e(P)$ , which shows that the number of linear extensions only depends on the comparability relation.", "Stanley's poset polytopes are very natural objects that appear in a variety of contexts in combinatorics and beyond; see [1], [25], [32], [11].", "Inspired by Stanley's labelled posets, Malvenuto and Reutenauer [28] introduced double poset in the context of combinatorial Hopf algebras.", "A double poset $\\mathbf {P}$ is a triple consisting of a finite ground set $P$ and two partial order relations $\\preceq _+$ and $\\preceq _-$ on $P$ .", "We will write $P_+ = (P,\\preceq _+)$ and $P_- = (P,\\preceq _-)$ to refer to the two underlying posets.", "If $\\preceq _-$ is a total order, then this corresponds to labelled poset in the sense of Stanley [33], which is the basis for the rich theory of $P$ -partitions.", "The combinatorial study of general double posets gained momentum in recent years with a focus on algebraic aspects; see, for example, [9], [10].", "The goal of this paper is to build a bridge to geometry by introducing two double poset polytopes that, like the chain- and the order polytope, geometrically reflect the combinatorial properties of double posets and, in particular, the interaction between the two partial orders." ], [ "Double order polytopes", "For a double poset $\\mathbf {P}= (P,\\preceq _\\pm )$ , we define the double order polytope as $\\mathcal {O}(\\mathbf {P}) \\ = \\ \\mathcal {O}(P,\\preceq _+,\\preceq _-) \\ := \\ \\operatorname{conv}\\bigl \\lbrace (2\\mathcal {O}(P_+)\\times \\lbrace 1\\rbrace ) \\cup (-2\\mathcal {O}(P_-) \\times \\lbrace -1\\rbrace ) \\bigr \\rbrace .$ This is a $(|P|+1)$ -dimensional polytope in $\\mathbb {R}^P\\times \\mathbb {R}$ .", "Its vertices are trivially in bijection to filters of $P_+$ and $P_-$ .", "This is a lattice polytope with respect to $\\mathbb {Z}^P\\times \\mathbb {Z}$ but we will mostly view $\\mathcal {O}(\\mathbf {P})$ as a lattice polytope with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ .", "That is, up to a translation by $(\\mathbf {0},1)$ , $\\mathcal {O}(\\mathbf {P})$ is the polytope $2 \\cdot \\operatorname{conv}\\bigl \\lbrace (\\mathcal {O}(P_+) \\times \\lbrace 1\\rbrace ) \\cup (-\\mathcal {O}(P_-) \\times \\lbrace 0\\rbrace ) \\bigr \\rbrace ,$ which is a lattice polytope with respect to $2\\mathbb {Z}^P\\times 2\\mathbb {Z}$ .", "In Section REF , we describe the facets of $\\mathcal {O}(\\mathbf {P})$ in terms of chains and cycles alternating between $P_+$ and $P_-$ and, for the important case of compatible double posets, we completely determine the facial structure in Section REF in terms of double Birkhoff lattices $\\mathcal {J}(\\mathbf {P}) := \\mathcal {J}(P_+) \\uplus \\mathcal {J}(P_-)$ .", "The double order polytope automatically has $2\\mathcal {O}(P_+)$ and $-2\\mathcal {O}(P_-)$ as facets.", "The non-trivial combinatorial structure is captured by the reduced double order polytope $\\overline{\\mathcal {O}}(\\mathbf {P}) \\ := \\ \\mathcal {O}(\\mathbf {P}) \\cap \\lbrace (f,t) : t = 0 \\rbrace \\ = \\ \\mathcal {O}(P_+) -\\mathcal {O}(P_-),$ which is a lattice polytope with respect to $\\mathbb {Z}^P$ by our choice of embedding.", "By placing $2\\mathcal {O}(P_+)$ and $-2\\mathcal {O}(P_-)$ at heights $+1$ and $-1$ , respectively, we made sure that $\\mathcal {O}(\\mathbf {P})$ always contains the origin.", "Every poset $(P,\\preceq )$ trivially induces a double poset $\\mathbf {P}_\\circ =(P,\\preceq ,\\preceq )$ and for an induced double poset, $\\mathcal {O}(\\mathbf {P}_\\circ )$ is centrally-symmetric and, up to a (lattice-preserving) shear, is the polytope $\\mathcal {O}(\\mathbf {P}_\\circ ) \\ \\cong \\ \\operatorname{conv}\\bigl \\lbrace (2\\mathcal {O}(P) \\times \\lbrace 1\\rbrace ) \\cup (2\\mathcal {O}(P^\\mathrm {op}) \\times \\lbrace -1\\rbrace ) \\bigr \\rbrace ,$ where $P^\\mathrm {op}$ is the poset with the opposite order.", "Geissinger [13] introduced a polytope associated to valuations on distributive lattices with values in $[0,1]$ .", "In Section REF , we show a surprising connection between Geissinger's valuation polytopes and polars of the (reduced) double order polytopes of $\\mathbf {P}_\\circ $ .", "We will review notions from the theory of double posets and emphasize their geometric counterparts." ], [ "Double chain-, Hansen-, and anti-blocking polytopes", "The double chain polytope associated to a double poset $\\mathbf {P}$ is the polytope $\\mathcal {C}({\\mathbf {P}}) \\ = \\ \\mathcal {C}({P,\\preceq _+,\\preceq _-}) \\ := \\ \\operatorname{conv}\\bigl \\lbrace (2\\mathcal {C}({P_+}) \\times \\lbrace 1\\rbrace ) \\cup (-2\\mathcal {C}({P_-}) \\times \\lbrace -1\\rbrace ) \\bigr \\rbrace .$ The reduced version $\\overline{\\mathcal {C}}(\\mathbf {P}) := \\mathcal {C}({P_+}) - \\mathcal {C}({P_-})$ is studied in Section  in the context of anti-blocking polytopes.", "According to Fulkerson [12], a full-dimensional polytope $\\mathcal {P}\\subseteq \\mathbb {R}_{\\ge 0}^n$ is anti-blocking if for any $q \\in \\mathcal {P}$ , it contains all points $p \\in \\mathbb {R}^n$ with $0 \\le p_i \\le q_i$ for $i=1,\\dots ,n$ .", "It is obvious from (REF ) that chain polytopes are anti-blocking.", "Anti-blocking polytopes are important in combinatorial optimization and, for example, contain stable set polytopes of graphs.", "For two polytopes $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ , we define the Cayley sum as the polytope ${\\mathcal {P}_1 \\boxplus \\mathcal {P}_2} \\ := \\ \\operatorname{conv}( \\mathcal {P}_1 \\times \\lbrace 1\\rbrace \\cup \\mathcal {P}_2 \\times \\lbrace -1\\rbrace )$ and we abbreviate ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2} := {\\mathcal {P}_1 \\boxplus -\\mathcal {P}_2}$ .", "Thus, $\\mathcal {O}(\\mathbf {P}) \\ = \\ {2\\mathcal {O}(P_+) \\boxminus 2\\mathcal {O}(P_-)} \\qquad \\text{ and }\\qquad \\mathcal {C}({\\mathbf {P}}) \\ = \\ {2\\mathcal {C}({P_+}) \\boxminus 2\\mathcal {C}({P_-})}.$ Section  is dedicated to a detailed study of the polytopes ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ as well as their sections $\\mathcal {P}_1 - \\mathcal {P}_2$ for anti-blocking polytopes $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}_{\\ge 0}^n$ .", "We completely determine the facets of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ in terms of $\\mathcal {P}_1,\\mathcal {P}_2$ in Section REF , which yields the combinatorics of $\\mathcal {C}({\\mathbf {P}})$ .", "In Section REF , we describe a canonical subdivision of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ and $\\mathcal {P}_1 - \\mathcal {P}_2$ for anti-blocking blocking polytopes $\\mathcal {P}_1,\\mathcal {P}_2$ .", "Moreover, if $\\mathcal {P}_1,\\mathcal {P}_2$ have regular, unimodular, or flag triangulations, then so has ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ (for an appropriately chosen affine lattice).", "The canonical subdivision enables us to give explicit formulas for the volume and the Ehrhart (quasi-)polynomials of these classes of polytopes.", "The chain polytope $\\mathcal {C}({P})$ only depends on the comparability graph $G(P)$ of $P$ and, more precisely, is the stable set polytope of $G(P)$ .", "Thus, $\\mathcal {C}({\\mathbf {P}})$ only depends on the double graph $(G(P_+),G(P_-))$ .", "For a graph $G$ , let $\\mathcal {P}_G$ be its stable set polytope; see Section REF for precise definitions.", "Lovász [27] characterized perfect graphs in terms of $\\mathcal {P}_G$ and Hansen [17] studied the polytopes $\\mathcal {H}(G) :={2\\mathcal {P}_G \\boxminus 2\\mathcal {P}_G}$ .", "If $G$ is perfect, then Hansen showed that the polar $\\mathcal {H}(G)^\\triangle $ is linearly isomorphic to $\\mathcal {H}(\\overline{G})$ where $\\overline{G}$ is the complement graph of $G$ .", "In Section REF , we generalize this result to all Cayley sums of anti-blocking polytopes." ], [ "2-level polytopes and volume", "A full-dimensional polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ is called $\\mathbf {2}$ -level if for any facet-defining hyperplane $H$ there is $t\\in \\mathbb {R}^n$ such that $H \\cup (t + H)$ contains all vertices of $P$ .", "The class of 2-level polytopes plays an important role in, for example, the study of centrally-symmetric polytopes [30], [17], polynomial optimization [14], [15], statistics [37], and combinatorial optimization [31].", "For example, Lovász [27] characterizes perfect graphs by the 2-levelness of their stable set polytopes and Hansen showed that $\\mathcal {H}(G)$ is 2-level if $G$ is perfect.", "In fact, we extend this to yet another characterization of perfect graphs in Corollary REF .", "This result implies that $\\mathcal {C}({\\mathbf {P}_\\circ })$ is 2-level for double posets induced by posets.", "However, it is in general not true that ${\\mathcal {P}_1 \\boxplus \\mathcal {P}_2}$ is 2-level if $\\mathcal {P}_1$ and $\\mathcal {P}_2$ are.", "A counterexample is the polytope ${\\Delta _{6,2} \\boxplus \\Delta _{6,4}}$ , where $\\Delta _{n,k}$ is the $(n,k)$ -hypersimplex.", "The starting point for this paper was the question for which double posets $\\mathbf {P}$ the polytopes $\\mathcal {O}(\\mathbf {P})$ and $\\mathcal {C}({\\mathbf {P}})$ are 2-level.", "Answers are given in Corollary REF , Proposition REF , and Corollary REF .", "A new class of 2-level polytopes comes from valuation polytopes; see Corollary REF .", "Sullivant [37] showed that 2-level lattice polytopes $\\mathcal {P}$ have the interesting property that any pulling triangulation that uses all lattice points in $\\mathcal {P}$ is unimodular.", "Hence, for 2-level lattice polytopes, the normalized volume is the number of simplices.", "In particular, $\\mathcal {O}(P)$ is 2-level and Stanley's canonical triangulation is a pulling triangulation.", "Stanley defined a piecewise linear homeomorphism between $\\mathcal {O}(P)$ and $\\mathcal {C}({P})$ whose domains of linearity are exactly the simplices of the canonical triangulation.", "Since this transfer map is lattice preserving, it follows that $\\mathrm {Ehr}_{\\mathcal {O}(P)}(n)= \\mathrm {Ehr}_{\\mathcal {C}({P})}(n)$ , which also implies the volume result.", "In Section , we generalize this transfer map to a lattice preserving PL homeomorphism $\\Psi _{\\mathbf {P}} : \\mathbb {R}^P\\times \\mathbb {R}\\rightarrow \\mathbb {R}^P\\times \\mathbb {R}$ for any compatible double poset $\\mathbf {P}$ .", "In particular, $\\mathcal {C}({\\mathbf {P}})$ is mapped to $\\mathcal {O}(\\mathbf {P})$ .", "This also transfers the canonical flag triangulation of $\\mathcal {C}({\\mathbf {P}})$ to a canonical flag triangulation of $\\mathcal {O}(\\mathbf {P})$ .", "Abstractly, the triangulation can be described in terms of a suitable subcomplex of the order complex of the double Birkhoff lattice $\\mathcal {J}(\\mathbf {P}) = \\mathcal {J}(P_+) \\uplus \\mathcal {J}(P_-)$ .", "In Section REF , we give explicit formulas for the Ehrhart polynomial and the volume of $\\mathcal {O}(\\mathbf {P})$ if $\\mathbf {P}$ is compatible and for $\\mathcal {C}({\\mathbf {P}})$ in general." ], [ "Double Hibi rings", "Hibi [19] studied rings associated to finite posets that give posets an algebraic incarnation and that are called Hibi rings.", "In modern language, the Hibi ring $\\mathcal {O}(P)]$ associated to a poset $P$ is the semigroup ring associated to $\\mathcal {O}(P)$ .", "Many properties of $P$ posses an algebraic counterpart and, in particular, Hibi exhibited a quadratic Gröbner basis for the associated toric ideal.", "In Section , we introduce the double Hibi rings $\\mathcal {O}(\\mathbf {P})]$ as suitable analogs for double posets, which are the semigroup rings associated to $\\mathcal {O}(\\mathbf {P})$ .", "We construct a quadratic Gröbner basis for the cases of compatible double posets.", "Using a result by Sturmfels [36], this shows the existence of a unimodular and flag triangulation of $\\mathcal {O}(\\mathbf {P})$ which coincides with the triangulation in Section .", "We also construct a quadratic Gröbner basis for the rings $\\mathcal {C}({\\mathbf {P}})]$ for arbitrary double posets and we remark on the algebraic implications for double posets.", "Acknowledgements.", "We would like to thank Christian Stump and Stefan Felsner for many helpful conversations regarding posets and we thank Vic Reiner for pointing out [28].", "We would also like to thank the referees for valuable suggestions.", "T. Chappell was supported by a Phase-I scholarship of the Berlin Mathematical School.", "T. Friedl and R. Sanyal were supported by the DFG-Collaborative Research Center, TRR 109 “Discretization in Geometry and Dynamics”.", "T. Friedl received additional support from the Dahlem Research School at Freie Universität Berlin." ], [ "Order polytopes", "Let $(P,\\preceq )$ be a poset.", "We write $\\widehat{P}$ for the poset obtained from $P$ by adjoining a minimum $\\widehat{0}$ and a maximum $\\widehat{1}$ .", "For an order relation $a \\prec b$ , we define a linear form $\\ell _{a,b} : \\mathbb {R}^P\\rightarrow \\mathbb {R}$ by $\\ell _{a,b}(f) \\ := \\ f(a) - f(b)$ for any $f \\in \\mathbb {R}^P$ .", "Moreover, for $a \\in P$ , we define $\\ell _{a,\\widehat{1}}(f):= f(a)$ and $\\ell _{\\widehat{0},a}(f) := -f(a)$ .", "With this notation, $f \\in \\mathbb {R}^P$ is contained in $\\mathcal {O}(P)$ if and only if $\\begin{aligned}\\ell _{a,b}(f) & \\ \\le \\ 0 \\quad \\text{ for all } a \\prec b,\\\\\\ell _{\\widehat{0},b}(f) & \\ \\le \\ 0 \\quad \\text{ for all } b \\in P, \\text{and}\\\\\\ell _{a,\\widehat{1}}(f) & \\ \\le \\ 1 \\quad \\text{ for all } a \\in P.\\end{aligned}$ Every nonempty face $F$ of $\\mathcal {O}(P)$ is of the form $F \\ = \\ \\mathcal {O}(P)^\\ell \\ := \\ \\lbrace f \\in \\mathcal {O}(P) : \\ell (f) \\ge \\ell (f^{\\prime }) \\text{ for all } f^{\\prime } \\in \\mathcal {O}(P) \\rbrace $ for some linear function $\\ell \\in (\\mathbb {R}^P)^*$ .", "Later, we want to identify $\\ell $ with its vector of coefficients and thus we write $\\ell (f) \\ = \\ \\sum _{a \\in P} \\ell _a f(a).$ Combinatorially, faces can be described using face partitions: To every face $F$ is an associated collection $B_1,\\dots ,B_m \\subseteq \\widehat{P}$ of nonempty and pairwise disjoint subsets that partition $\\widehat{P}$ .", "According to Stanley [34], a partition of $\\widehat{P}$ is a (closed) face partition if and only if each $(B_i, \\preceq )$ is a connected poset and $B_i\\preceq ^{\\prime } B_j :\\Leftrightarrow p_i \\preceq p_j$ for some $p_i \\in B_i, p_j \\in B_j$ is a partial order on $\\lbrace B_1,\\dots ,B_m\\rbrace $ .", "Of course, it is sufficient to remember the non-singleton parts and we define the reduced face partition of $F$ as $\\mathcal {B}(F) := \\lbrace B_i : |B_i| > 1 \\rbrace $ .", "The normal cone of a nonempty face $F \\subseteq \\mathcal {O}(P)$ is the polyhedral cone $N_P(F) \\ := \\ \\lbrace \\ell \\in (\\mathbb {R}^P)^* : F \\subseteq \\mathcal {O}(P)^\\ell \\rbrace .$ The following description of $N_P(F)$ follows directly from (REF ).", "Proposition 2.1 Let $P$ be a finite poset and $F \\subseteq \\mathcal {O}(P)$ a nonempty face with reduced face partition $\\mathcal {B}= \\lbrace B_1,\\dots ,B_k\\rbrace $ .", "Then $N_P(F) \\ = \\ \\operatorname{cone}\\lbrace \\ell _{a,b}: [a,b] \\subseteq B_i \\text{ forsome }i=1,\\dots ,k \\rbrace .$ We note the following simple but very useful consequence of this description.", "Corollary 2.2 Let $F \\subseteq \\mathcal {O}(P)$ be a nonempty face with reduced face partition $\\mathcal {B}= \\lbrace B_1,\\dots ,B_k \\rbrace $ .", "Then for every $\\ell \\in \\operatorname{relint}N_P(F)$ and $p \\in P$ the following hold: [(i)] if $p \\in \\min (B_i)$ for some $i$ , then $\\ell _p > 0$ ; if $p \\in \\max (B_i)$ for some $i$ , then $\\ell _p < 0$ ; if $p \\notin \\bigcup _i B_i$ , then $\\ell _p = 0$ .", "The vertices of $\\mathcal {O}(P)$ are exactly the indicator functions $\\mathbf {1}_\\mathsf {J}:P\\rightarrow \\lbrace 0,1\\rbrace $ where $\\mathsf {J}\\subseteq P$ is a filter.", "For a filter $\\mathsf {J}\\subseteq P$ , we write $\\widehat{\\mathsf {J}} := \\mathsf {J}\\cup \\lbrace \\widehat{1}\\rbrace $ for the filter induced in $\\widehat{P}$ .", "Proposition 2.3 Let $F \\subseteq \\mathcal {O}(P)$ be a face with (reduced) face partition $\\mathcal {B}= \\lbrace B_1,\\dots ,B_k\\rbrace $ and let $\\mathsf {J}\\subseteq P$ be a filter.", "Then $\\mathbf {1}_\\mathsf {J}\\in F$ if and only if $\\widehat{\\mathsf {J}} \\cap B_i \\ = \\ \\varnothing \\quad \\text{ or } \\quad \\widehat{\\mathsf {J}} \\cap B_i \\ = \\ B_i$ for all $i=1,\\dots ,k$ .", "That is, $\\mathbf {1}_\\mathsf {J}$ belongs to $F$ if and only if $\\widehat{\\mathsf {J}}$ does not separate any two comparable elements in $B_i$ , for all $i$ .", "Facets of double order polytopes Let $\\mathbf {P}= (P,\\preceq _\\pm )$ be a double poset.", "The double order polytope $\\mathcal {O}(\\mathbf {P})$ is a $(|P|+1)$ -dimensional polytope in $\\mathbb {R}^P\\times \\mathbb {R}$ with coordinates $(f,t)$ .", "It is obvious that the vertices of $\\mathcal {O}(\\mathbf {P})$ are exactly $(2\\mathbf {1}_{\\mathsf {J}_+},1), (-2\\mathbf {1}_{\\mathsf {J}_-},-1)$ for filters $\\mathsf {J}_+ \\subseteq P_+$ and $\\mathsf {J}_- \\subseteq P_-$ , respectively.", "To get the most out of our notational convention, for $\\sigma \\in \\lbrace -,+\\rbrace $ we define $-\\sigma \\ := \\ {\\left\\lbrace \\begin{array}{ll}- & \\text{ if } \\sigma = +\\\\+ & \\text{ if } \\sigma = -.\\end{array}\\right.", "}$ By construction, $2\\mathcal {O}(P_+) \\times \\lbrace 1\\rbrace $ and $-2\\mathcal {O}(P_-) \\times \\lbrace -1\\rbrace $ are facets that are obtained by maximizing the linear function $\\pm L_\\varnothing (f,t) := \\pm t$ over $\\mathcal {O}(\\mathbf {P})$ .", "We call the remaining facets vertical, as they are of the form ${F_+ \\boxminus F_-}$ , where $F_\\sigma \\subset \\mathcal {O}(P_\\sigma )$ are certain nonempty proper faces for $\\sigma = \\pm $ .", "The vertical facets are in bijection with the facets of the reduced double order polytope $\\overline{\\mathcal {O}}(\\mathbf {P}) = \\mathcal {O}(P_+) - \\mathcal {O}(P_-)$ .", "More precisely, if $F \\subset \\mathcal {O}(\\mathbf {P})$ is a facet, then there is a linear function $\\ell \\in (\\mathbb {R}^P)^*$ such that $F = {F_+ \\boxminus F_-}$ where $F_+ = \\mathcal {O}(P_+)^\\ell $ and $F_- = \\mathcal {O}(P_-)^{-\\ell }$ .", "This linear function is necessarily unique up to scaling and hence the faces $F_+,F_-$ are characterized by the property $\\operatorname{relint}N_{P_+}(F_+) \\ \\cap \\ \\operatorname{relint}-N_{P_-}(F_-) \\ = \\ \\mathbb {R}_{>0} \\cdot \\ell \\, .$ We will call a linear function $\\ell $ rigid if it satisfies (REF ) for a pair of faces $(F_+,F_-)$ .", "Our next goal is to give an explicit description of all rigid linear functions for $\\mathcal {O}(\\mathbf {P})$ which then yields a characterization of vertical facets.", "An alternating chain $C$ of a double poset $\\mathbf {P}= (P,\\preceq _\\pm )$ is a finite sequence of distinct elements $\\widehat{0}\\ = \\ p_0 \\ \\prec _{\\sigma } \\ p_1 \\ \\prec _{-\\sigma } \\ p_2 \\ \\prec _{\\sigma } \\ \\cdots \\ \\prec _{\\pm \\sigma } \\ p_k \\ = \\ \\widehat{1},$ where $\\sigma \\in \\lbrace \\pm \\rbrace $ .", "For an alternating chain $C$ , we define a linear function $\\ell _C$ by $\\ell _C(f) \\ := \\ \\sigma \\, \\sum _{i=1}^{k-1} (-1)^i f(p_i).$ Here, we severely abuse notation and interpret $\\sigma $ as $\\pm 1$ .", "Note that $\\ell _C \\equiv 0$ if $k=1$ and we call $C$ a proper alternating chain if $k > 1$ .", "An alternating cycle $C$ of $\\mathbf {P}$ is a sequence of length $2k$ of the form $p_0 \\ \\prec _{\\sigma } \\ p_1 \\ \\prec _{-\\sigma } \\ p_2 \\ \\prec _{\\sigma } \\ \\cdots \\ \\prec _{-\\sigma } \\ p_{2k} \\ = \\ p_0,$ where $\\sigma \\in \\lbrace \\pm \\rbrace $ and $p_i \\ne p_j$ for $0 \\le i < j < 2k$ .", "We similarly define a linear function associated to $C$ by $\\ell _C(f) \\ := \\ \\sigma \\, \\sum _{i=0}^{2k-1} (-1)^i f(p_i).$ Note that it is possible that a sequence of elements $p_1,p_2,\\dots ,p_k$ gives rise to two alternating chains, one starting with $\\prec _+$ and one starting with $\\prec _-$ .", "On the other hand, every alternating cycle of length $2k$ yields $k$ alternating cycles starting with $\\prec _+$ and $k$ alternating starting with $\\prec _-$ .", "Proposition 2.4 Let $\\mathbf {P}=(P,\\preceq _\\pm )$ be a double poset.", "If $\\ell $ is a rigid linear function for $\\mathcal {O}(\\mathbf {P})$ , then $\\ell = \\mu \\ell _C$ for some alternating chain or alternating cycle $C$ and $\\mu > 0$ .", "Let $F_+ = \\mathcal {O}(P_+)^{\\ell }$ and $F_- = \\mathcal {O}(P_-)^{-\\ell }$ be the two faces for which (REF ) holds and let $\\mathcal {B}_\\pm = \\lbrace B_{\\pm 1},B_{\\pm 2},\\dots \\rbrace $ be the corresponding reduced face partitions.", "We define a directed bipartite graph $G = (V_+\\cup V_-,E)$ with nodes $V_+ = \\lbrace p\\in P: \\ell _p > 0 \\rbrace $ and $V_-$ accordingly.", "If $\\widehat{1}$ is contained in some part of $\\mathcal {B}_+$ , then we add a corresponding node $\\widehat{1}_+$ to $V_-$ Consistently, we add a node $\\widehat{1}_-$ to $V_+$ if $\\widehat{1}$ it occurs in a part of $\\mathcal {B}_-$ .", "Note that $\\widehat{0}_-$ and $\\widehat{0}_+$ are distinct nodes.", "Similarly we add $\\widehat{0}_{+}$ to $V_+$ and $\\widehat{0}_{-}$ to $V_-$ if they appear in $\\mathcal {B}_+$ and $\\mathcal {B}_-$ , respectively.", "By Corollary REF , we have ensured that $\\max (B_{+i}) \\subseteq V_-$ and $\\max (B_{-i}) \\subseteq V_+$ for all $i$ .", "For $u \\in V_+$ and $v \\in V_-$ , we add the directed edge $uv \\in E$ if $u\\prec _+ v$ and $[u,v]_{P_+} \\subseteq B_{+i}$ for some $i$ .", "Similarly, we add the directed edge $vu \\in E$ if $v \\prec _- u$ and $[v,u]_{P_-}\\subseteq B_{-i}$ for some $i$ .", "We claim that every node $u$ except for maybe the special nodes $\\widehat{0}_{\\pm },\\widehat{1}_{\\pm }$ has an incoming and an outgoing edge.", "For example, if $u \\in V_+$ , then $\\ell _u > 0$ .", "By Corollary REF (iii), there is an $i$ such that $u \\in B_{+i}$ and by (ii), $u$ is not a maximal element in $B_{+i}$ .", "Thus, there is some $v \\in \\max (B_{+i})$ with $u \\prec _+ v$ and $uv$ is an edge.", "It follows that every longest path either yields an alternating cycle or a proper alternating chain.", "For an alternating cycle $C = (p_0 \\prec _+ \\cdots \\prec _- p_{2l})$ , we observe that $\\ell _C &\\ = \\ \\ell _{p_0,p_1} +\\ell _{p_2,p_3} +\\cdots +\\ell _{p_{2l-2},p_{2l-1}} \\text{ and } \\\\-\\ell _C &\\ = \\ \\ell _{p_1,p_2} +\\ell _{p_3,p_4} +\\cdots +\\ell _{p_{2l-1},p_{2l}}.$ Since for every $j$ , $[p_{2j},p_{2j+1}]_{P_+}$ is contained in some part of $\\mathcal {B}_+$ , we conclude that $\\ell _C \\in N_{P_+}(F_+)$ .", "Similarly, for all $j$ , $[p_{2j-1},p_{2j}]_{P_-}$ is contained in some part of $\\mathcal {B}_-$ , and hence $-\\ell _C \\in N_{P_-}(F_-)$ .", "Assuming that $\\ell $ is rigid then shows that $\\ell = \\mu \\ell _C$ for some $\\mu > 0$ .", "If $G$ does not contain a cycle, then let $C = (p_0,p_1,\\dots ,p_k)$ be a longest path in $G$ .", "In particular $p_0 = \\widehat{0}_{\\pm }$ and $p_k =\\widehat{1}_{\\pm }$ .", "The same reasoning applies and shows that $\\ell _C \\in N_{P_+}(F_+) \\cap -N_{P_-}(F_-)$ and hence $\\ell = \\mu \\ell _C$ for some $\\mu > 0$ .", "In general, not every alternating chain or cycle gives rise to a rigid linear function.", "Let $(P,\\preceq )$ be a poset that is not the antichain and define the double poset $\\mathbf {P}= (P,\\preceq ,\\preceq ^\\mathrm {op})$ , where $\\preceq ^\\mathrm {op}$ is the opposite order.", "In this case $\\mathcal {O}(\\mathbf {P})$ is, up to a shear, the ordinary prism over $\\mathcal {O}(P,\\preceq )$ .", "Hence, the vertical facets of $\\mathcal {O}(\\mathbf {P})$ are prisms over the facets of $\\mathcal {O}(P)$ .", "It follows from (REF ) that these facets correspond to cover relations in $P$ .", "Hence, every rigid $\\ell $ is of the form $\\ell = \\mu \\ell _{p,q}$ for cycles $p \\prec _+ q \\prec _- p$ where $p\\prec q$ is a cover relation in $P$ .", "We call a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ compatible if $P_+ = (P,\\preceq _+)$ and $P_- = (P,\\preceq _-)$ have a common linear extension.", "Note that a double poset is compatible if and only if it does not contain alternating cycles.", "Following [28], we call a double poset $\\mathbf {P}$ special if $\\preceq _-$ is a total order.", "At the other extreme, we call $\\mathbf {P}$ anti-special if $(P,\\preceq _-)$ is an anti-chain.", "A plane poset, as defined in [9] is a double poset $\\mathbf {P}=(P,\\preceq _+,\\preceq _-)$ such that distinct $a,b \\in P$ are $\\prec _+$ -comparable if and only if they are not $\\prec _-$ -comparable.", "For two posets $(P_1,\\preceq ^1)$ and $(P_2,\\preceq ^2)$ one classically defines the disjoint union $\\preceq _\\uplus $ and the ordinal sum $\\preceq _\\oplus $ as the posets on $P_1 \\uplus P_2$ as follows.", "For $a,b \\in P_1 \\uplus P_2$ set $a \\preceq _{\\uplus } b$ if $a,b \\in P_i$ and $a\\preceq ^i b$ for some $i \\in \\lbrace 1,2\\rbrace $ .", "For the ordinal sum, $\\preceq _\\oplus $ restricts to $\\preceq ^1$ and $\\preceq ^2$ on $P_1$ and $P_2$ respectively and $p_1 \\prec _\\oplus p_2$ for all $p_1 \\in P_1$ and $p_2 \\in P_2$ .", "The effect on order polytopes is $\\mathcal {O}(P_1 \\uplus P_2) = \\mathcal {O}(P_1) \\times \\mathcal {O}(P_2)$ and $\\mathcal {O}(P_1 \\oplus P_2)$ is a subdirect sum in the sense of McMullen [29].", "Malvenuto and Reutenauer [28] define the composition of two double posets $(P_1,\\prec ^1_\\pm )$ and $(P^{\\prime }_2,\\prec ^2_\\pm )$ as the double poset $(P,\\preceq _\\pm )$ such that $(P,\\preceq _+) = (P_1,\\prec ^1_+) \\uplus (P_2,\\prec ^2_+)$ and $(P,\\preceq _-) = (P_1,\\prec ^1_-) \\oplus (P_2,\\prec ^2_-)$ .", "The following is easily seen; for plane posets with the help of [9].", "Proposition 2.5 Anti-special and plane posets are compatible.", "Moreover, the composition of two compatible double posets is a compatible double poset.", "This defining property of compatible double posets assures us that in an alternating chain $p_i \\prec _\\sigma p_j$ implies $i < j$ for any $\\sigma \\in \\lbrace \\pm \\rbrace $ .", "In particular, a compatible double poset does not have alternating cycles.", "This also shows the following.", "Lemma 2.6 Let $\\mathbf {P}= (P,\\preceq _\\pm )$ be a compatible double poset.", "If $a_{i}\\prec _\\sigma a_{i+1} \\prec _{-\\sigma } \\cdots \\prec _{-\\tau } a_j \\prec _\\tau a_{j+1}$ is part of an alternating chain with $\\sigma ,\\tau \\in \\lbrace \\pm \\rbrace $ and $i < j$ , then there is no $b \\in P$ such that $a_i \\prec _\\sigma b\\prec _\\sigma a_{i+1}$ and $a_j \\prec _\\tau b \\prec _\\tau a_{j+1}$ .", "For compatible double posets, we can give complete characterization of facets.", "Theorem 2.7 Let $\\mathbf {P}$ a compatible double poset.", "A linear function $\\ell $ is rigid if and only if $\\ell \\in \\mathbb {R}_{>0} \\ell _C$ for some alternating chain $C$ .", "In particular, the facets of $\\mathcal {O}(\\mathbf {P})$ are in bijection with alternating chains.", "We already observed that $2\\mathcal {O}(P_+) \\times \\lbrace 1\\rbrace $ and $-2\\mathcal {O}(P_-)\\times \\lbrace -1\\rbrace $ correspond to the improper alternating chains $\\widehat{0}\\prec _\\sigma \\widehat{1}$ for $\\sigma = \\pm $ .", "By Proposition REF it remains to show that for any proper alternating chain $C$ the function $\\ell _C$ is rigid.", "We only consider the case that $C$ is an alternating chain of the form $\\widehat{0}\\ = \\ p_0 \\ \\prec _+ \\ p_1 \\ \\prec _- \\ p_2 \\ \\prec _+ \\ \\cdots \\ \\prec _+ \\ p_{2k-1} \\ \\prec _- \\ p_{2k} \\ \\prec _+ \\ p_{2k+1}\\ = \\ \\widehat{1}.$ The other cases can be treated analogously.", "Let $F_+ = \\mathcal {O}(P_+)^{\\ell _C}$ and and $F_- = \\mathcal {O}(P_-)^{-\\ell _C}$ be the corresponding faces with reduced face partitions $\\mathcal {B}_\\pm $ .", "Define $O = \\lbrace p_1, p_3, \\dots , p_{2k-1}\\rbrace $ and $E =\\lbrace p_2,p_4,\\dots ,p_{2k}\\rbrace $ .", "Then for any set $A \\subseteq P$ , we observe that $\\ell _C(\\mathbf {1}_A) = |E \\cap A| - |O \\cap A|$ .", "If $\\mathsf {J}$ is a filter of $P_+$ , then $p_{2i} \\in \\mathsf {J}$ implies $p_{2i+1} \\in \\mathsf {J}$ and hence $\\ell _C(\\mathbf {1}_\\mathsf {J}) \\le 1$ and thus $\\mathbf {1}_\\mathsf {J}\\in F_+$ if and only if $\\mathsf {J}$ does not separate $p_{2j}$ and $p_{2j+1}$ for $1 \\le j \\le k$ .", "Likewise, a filter $\\mathsf {J}\\subseteq P_-$ is contained in $F_-$ if and only if $\\mathsf {J}$ does not separate $p_{2j-1}$ and $p_{2j}$ for $1 \\le j\\le k$ .", "Lemma REF implies that $\\mathcal {B}_+ &\\ = \\ \\lbrace [p_0,p_1]_{P_+}, [p_2,p_3]_{P_+}, \\dots , [p_{2k},p_{2k+1}]_{P_+}\\rbrace \\text{ and}\\\\\\mathcal {B}_- &\\ = \\ \\lbrace [p_1,p_2]_{P_-}, [p_3,p_4]_{P_-},\\dots , [p_{2k-1},p_{2k}]_{P_-} \\rbrace .$ To show that $\\ell _C$ is rigid pick a linear function $\\ell (\\phi ) =\\sum _{p \\in P} \\ell _p \\phi (p)$ with $F_+ = \\mathcal {O}(P_+)^{\\ell }$ and $F_- =\\mathcal {O}(P_-)^{-\\ell }$ .", "Since the elements in $E$ and $O$ are exactly the minimal and maximal elements of the parts in $\\mathcal {B}_+$ , it follows from Corollary REF that $\\ell _p > 0$ if $p \\in E$ , $\\ell _p < 0$ for $p \\in O$ .", "By Lemma REF , it follows that if $q \\in (p_i,p_{i+1})_{P_+}$ , then $q$ is not contained in a part of the reduced face partition $\\mathcal {B}_-$ and vice versa.", "By Corollary REF (iii), it follows that $\\ell _p = 0$ for $p\\notin E \\cup O$ .", "Finally, $\\ell _{p_i} + \\ell _{p_{i+1}} = 0$ for all $1\\le i \\le 2k$ by Proposition REF and therefore $\\ell = \\mu \\ell _C$ for some $\\mu > 0 $ finishes the proof.", "Example 1 Let $\\mathbf {P}= (P,\\preceq _\\pm )$ be a compatible double poset with $|P| =n$ .", "Let $\\preceq _+ = \\preceq _- = \\preceq $ and $(P,\\preceq )$ be the $n$ -antichain.", "Then the only alternating chains are of the form $\\widehat{0}\\prec _\\sigma a \\prec _{-\\sigma } \\widehat{1}$ for $a \\in P$ .", "The double order polytope $\\mathcal {O}(\\mathbf {P})$ is the $(n+1)$ -dimensional cube with vertices $\\lbrace 0,2\\rbrace ^n \\times \\lbrace +1\\rbrace $ and $\\lbrace 0,-2\\rbrace ^n \\times \\lbrace -1\\rbrace $ .", "If $\\preceq _+ = \\preceq _- = \\preceq $ and $(P,\\preceq )$ is the $n$ -chain $[n]$ , then any alternating chain can be identified with an element in $\\lbrace -,+\\rbrace ^{n+1}$ .", "More precisely, $\\mathcal {O}(\\mathbf {P})$ is linearly isomorphic to the $(n+1)$ -dimensional crosspolytope $C_{n+1}^\\triangle = \\operatorname{conv}\\lbrace \\pm \\mathbf {e}_1,\\dots , \\pm \\mathbf {e}_{d+1}\\rbrace $ .", "Let $(P,\\preceq _+)$ be the $n$ -chain and $(P,\\preceq _-)$ be the $n$ -antichain.", "Then any alternating chain is of the form $\\widehat{0}\\prec _\\sigma a \\prec _{-\\sigma } \\widehat{1}$ for $\\sigma = \\pm $ and any relation $a \\prec _+ b$ be can be completed to a unique alternating chain.", "Thus, $\\mathcal {O}(\\mathbf {P})$ is a $(n+1)$ -dimensional polytope with $2^n + n + 1$ vertices and $\\binom{n}{2} + 2n + 2$ facets.", "The comb (see Figure REF ) is the poset $C_n$ on elements $\\lbrace a_1,\\dots ,a_n,b_1,\\dots ,b_n\\rbrace $ such that $a_i \\preceq a_j$ if $i \\le j$ and $b_i \\prec a_i$ for all $i,j \\in [n]$ .", "The $n$ -comb has $2^{n+1}-1$ filters and $3\\cdot 2^{n} - 2$ chains.", "Hence $\\mathcal {O}(C_n,\\preceq ,\\preceq )$ has $2^{n+2}-2$ vertices and $3\\cdot 2^{n+1} - 4$ facets.", "Generally, let $P_1, P_2$ be two posets and denote by $f_i$ and $c_i$ the number of filters and chains of $P_i$ for $i=1,2$ .", "Let $\\mathbf {P}_\\circ $ be the trivial double poset induced by $P_1 \\uplus P_2$ .", "Then $\\mathcal {O}(\\mathbf {P}_\\circ )$ has $2f_1f_2$ vertices and $2(c_1+c_2) - 2$ facets.", "Example 2 Consider the compatible 'XW'-double poset $\\mathbf {P}_{XW}$ on five elements, whose Hasse diagrams are given in Figure REF .", "The polytope $\\mathcal {O}(\\mathbf {P}_{XW})$ is six-dimensional with face vector $f(\\mathcal {O}(\\mathbf {P}_{XW})) \\ = \\ (21, 112, 247, 263, 135, 28).$ The facets correspond to the 28 alternating chains in $\\widehat{\\mathbf {P}}_{XW}$ , which are shown in Figure REF .", "Figure: The comb C n C_n.Figure: The 28 alternating chains in 𝐏 ^ XW \\widehat{\\mathbf {P}}_{XW}.For two particular types of posets, we wish to determine the combinatorics of $\\mathcal {O}(\\mathbf {P})$ in more detail.", "Example 3 (Dimension-2 posets) For $n \\ge 1$ , let $\\pi = (\\pi _1,\\pi _2,\\dots ,\\pi _n)$ be an ordered sequence of distinct numbers.", "We may define a partial order $\\preceq _\\pi $ on $[n]$ by $i \\prec _\\pi j$ if $i < j$ and $\\pi _i < \\pi _j$ .", "Following Dushnik and Miller [8], these are, up to isomorphism, exactly the posets of order dimension 2.", "A chain in $P_\\pi :=([n],\\preceq _\\pi )$ is a sequence $i_1 < i_2 < \\cdots < i_k$ with $\\pi _{i_1} < \\pi _{i_2} < \\cdots < \\pi _{i_k}$ .", "Thus, chains in $P_\\pi $ are in bijection to increasing subsequences of $\\pi $ .", "Conversely, one checks that filters (via their minimal elements) are in bijection to decreasing subsequences.", "It follows from Theorem REF that facets and vertices of $\\mathcal {O}([n],\\preceq _\\pi ,\\preceq _\\pi )$ are in 2-to-1 correspondence with increasing and decreasing sequences, respectively.", "Example 4 (Plane posets) Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "We may assume that $P= \\lbrace p_1,\\dots , p_n\\rbrace $ are labelled such that $p_i\\prec _\\sigma p_j$ for $\\sigma = +$ or $= -$ implies $i < j$ .", "By [10], $\\mathbf {P}$ is a plane poset, if and only if there is a sequence of distinct numbers $\\pi = (\\pi _1,\\pi _2,\\dots ,\\pi _n)$ such that for $p_i,p_j \\in P$ $\\begin{aligned}p_i \\prec _+ p_j & \\quad \\Longleftrightarrow \\quad i < j \\text{and } \\pi _i < \\pi _j \\text{ and }\\\\p_i \\prec _- p_j & \\quad \\Longleftrightarrow \\quad i < j \\text{and } \\pi _i > \\pi _j.\\end{aligned}$ This is to say, $P_+$ is canonically isomorphic to $([n], \\preceq _\\pi )$ and $P_-$ is canonically isomorphic to $([n],\\preceq _{-\\pi })$ .", "It follows that alternating chains in $\\mathbf {P}$ are in bijection to alternating sequences.", "That is, sequences $i_1 < i_2 < i_3 < \\cdots < i_k$ such that $\\pi _{i_1} < \\pi _{i_2} > \\pi _{i_3} < \\cdots $ .", "Hence, by Theorem REF , the facets of $\\mathcal {O}(\\mathbf {P})$ are in bijection to alternating sequences of $\\pi $ whereas the vertices are in bijection to increasing and decreasing sequences of $\\pi $ .", "As a consequence of the proof of Theorem REF we can determine a facet-defining inequality description of double order polytopes.", "For an alternating chain $C$ as in (REF ), let us write $\\operatorname{sgn}(C) = \\tau \\in \\lbrace -,+\\rbrace $ if the last relation in $C$ is $p_{k-1} \\prec _\\tau p_k=\\widehat{1}$ .", "Corollary 2.8 Let $\\mathbf {P}= (P,\\preceq _\\pm )$ be a compatible double poset.", "Then $\\mathcal {O}(\\mathbf {P})$ is the set of points $(f,t) \\in \\mathbb {R}^P\\times \\mathbb {R}$ such that $L_C(f,t) \\ := \\ \\ell _C(f) - \\operatorname{sgn}(C)\\, t \\ \\le \\ 1$ for all alternating chains $C$ of $\\mathbf {P}$ .", "Note that 0 is in the interior of $\\mathcal {O}(\\mathbf {P})$ .", "Hence by Theorem REF every facet-defining halfspace of $\\mathcal {O}(\\mathbf {P})$ is of the form $\\lbrace (\\phi ,t) : L(\\phi ,t) = \\mu \\ell _C +\\beta t \\le 1\\rbrace $ for some alternating chain $C$ and $\\mu , \\beta \\in \\mathbb {R}$ with $\\mu > 0$ .", "If $C$ is an alternating chain with $\\operatorname{sgn}(C) = +$ , then the maximal value of $\\ell _C$ over $2\\mathcal {O}(P_+)$ is 2 and 0 over $-2\\mathcal {O}(P_-)$ .", "The values are exchanged for $\\operatorname{sgn}(C) = -$ .", "It then follows that $\\mu = 1$ and $\\beta = -\\operatorname{sgn}(C)$ .", "With this, we can characterize the 2-level polytopes among compatible double order polytopes.", "Corollary 2.9 Let $\\mathbf {P}= (P,\\preceq _\\pm )$ be a compatible double poset.", "Then $\\mathcal {O}(\\mathbf {P})$ is 2-level if and only if $\\preceq _+ = \\preceq _-$ .", "In this case, the number of facets of $\\mathcal {O}(\\mathbf {P}_\\circ )$ is twice the number of chains in $(P,\\preceq )$ .", "If $\\preceq _+ = \\preceq _- = \\preceq $ , then every alternating chain is a chain in $P$ and conversely, every chain in $(P,\\preceq )$ gives rise to exactly two distinct alternating chains in $(P,\\preceq _\\pm )$ .", "In this case, it is straightforward to verify that the minimum of $\\ell _C$ over $2\\mathcal {O}(P)$ is $-2$ if $\\operatorname{sgn}(C) = +$ and 0 otherwise.", "The claim now follows from Corollary REF and together with Theorem REF also yields the number of facets.", "The converse follows from Proposition REF by noting that if both $(P,\\preceq _+,\\preceq _-)$ and $(P,\\preceq _-,\\preceq _+)$ are compatible and tertispecial then $\\preceq _+ = \\preceq _-$ .", "In [16] a double poset $(P,\\preceq _+,\\preceq _-)$ is called tertispecial if $a$ and $b$ are $\\prec _{-}$ -comparable whenever $a\\prec _+ b$ is a cover relation for $a,b \\in P$ .", "Proposition 2.10 Let $\\mathbf {P}= (P,\\preceq _\\pm )$ be a double poset.", "If $\\mathcal {O}(\\mathbf {P})$ is 2-level, then $\\mathbf {P}$ as well as $(P,\\preceq _-,\\preceq _+)$ are tertispecial.", "Let $\\sigma = \\pm $ and let $a \\prec _\\sigma b$ be a cover relation.", "The linear function $\\ell _{a,b}$ is facet defining for $\\mathcal {O}(P_\\sigma )$ and hence yields a facet for $\\mathcal {O}(\\mathbf {P})$ .", "If $a,b$ are not comparable in $P_{-\\sigma }$ , then the filters $\\varnothing , \\lbrace c \\in P: c \\succeq _-a\\rbrace $ and $\\lbrace c \\in P: c \\succeq _- b\\rbrace $ take three distinct values on $\\ell _{a,b}$ .", "Let us remark that the number of facets of a given double poset $\\mathbf {P}=(P,\\preceq _+,\\preceq _-)$ can be computed by the transfer-matrix method.", "Let us define the matrices $\\eta ^+, \\eta ^- \\in \\mathbb {R}^{\\widehat{P}\\times \\widehat{P}}$ by $\\eta ^\\sigma _{a,b} \\ := \\ {\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } a \\prec _\\sigma b\\\\0 & \\text{ otherwise}\\end{array}\\right.", "}$ for $a,b \\in \\widehat{P}$ and $\\sigma = \\pm $ .", "Then $(\\eta ^+\\eta ^-)^k_{\\widehat{0},\\widehat{1}}$ is the number of alternating chains of $\\mathbf {P}$ of length $k$ starting with $\\prec _+$ and ending with $\\prec _-$ .", "This shows the following.", "Corollary 2.11 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then the number of facets of $\\mathcal {O}(\\mathbf {P})$ is given by $\\left[ (\\mathrm {Id} - \\eta ^+\\eta ^-)^{-1}(\\mathrm {Id} + \\eta ^+) +(\\mathrm {Id} - \\eta ^-\\eta ^+)^{-1}(\\mathrm {Id} + \\eta ^-) \\right]_{\\widehat{0},\\widehat{1}}.$ Faces and embedded sublattices The Birkhoff lattice $\\mathcal {J}(P)$ of a finite poset $P$ is the distributive lattice given by the collection of filters of $P$ ordered by inclusion.", "A subposet $\\mathrm {L}\\subseteq \\mathcal {J}(P)$ is called an embedded sublattice if for any two filters $\\mathsf {J},\\mathsf {J}^{\\prime } \\in \\mathcal {J}(P)$ $\\mathsf {J}\\cup \\mathsf {J}^{\\prime }, \\mathsf {J}\\cap \\mathsf {J}^{\\prime } \\in \\mathrm {L}\\quad \\text{if andonly if} \\quad \\mathsf {J}, \\mathsf {J}^{\\prime } \\in \\mathrm {L}.$ For a subset $L \\subseteq \\mathcal {J}(P)$ of filters we write $F(L) :=\\operatorname{conv}(\\mathbf {1}_\\mathsf {J}: \\mathsf {J}\\in L)$ .", "Embedded sublattices give an alternative way to characterize faces of $\\mathcal {O}(P)$ .", "Theorem 2.12 ([38]) Let $P$ be a poset and $\\mathrm {L}\\subseteq \\mathcal {J}(P)$ a collection of filters.", "Then $F(\\mathrm {L})$ is a face of $\\mathcal {O}(P)$ if and only if $\\mathrm {L}$ is an embedded sublattice.", "We will generalize this description to the case of double order polytopes.", "Throughout this section, let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "We define $\\mathcal {J}(\\mathbf {P}) := \\mathcal {J}(P_+) \\uplus \\mathcal {J}(P_-)$ .", "For any subset $\\mathrm {L}\\subseteq \\mathcal {J}(\\mathbf {P})$ we will denote by $\\mathrm {L}_+$ the set $\\mathrm {L}\\cap \\mathcal {J}(P_+)$ and we define $\\mathrm {L}_-$ accordingly.", "Moreover, we shall write $\\overline{F}(\\mathrm {L}) \\ := \\ \\operatorname{conv}\\left( \\lbrace (2\\mathbf {1}_{\\mathsf {J}_+}, +1) : \\mathsf {J}_+ \\in \\mathrm {L}_+ \\rbrace \\cup \\lbrace (2\\mathbf {1}_{\\mathsf {J}_-}, -1) : \\mathsf {J}_- \\in \\mathrm {L}_- \\rbrace \\right)\\ \\subseteq \\ \\mathcal {O}(\\mathbf {P}).$ Thus, $\\overline{F}(\\mathrm {L}) = {2F(\\mathrm {L}_+) \\boxminus 2F(\\mathrm {L}_-)}$ .", "Theorem 2.13 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset and $\\mathrm {L}\\subseteq \\mathcal {J}(\\mathbf {P})$ .", "Then $\\overline{F}(\\mathrm {L})$ is a face of $\\mathcal {O}(\\mathbf {P})$ if and only if $\\mathrm {L}_+ \\subseteq \\mathcal {J}(P_+)$ and $\\mathrm {L}_- \\subseteq \\mathcal {J}(P_-)$ are embedded sublattices and for all filters $\\mathsf {J}_\\sigma \\subseteq \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathcal {J}(P_\\sigma )$ for $\\sigma = \\pm $ such that $\\mathsf {J}_+^{\\prime } \\setminus \\mathsf {J}_+ \\ = \\ \\mathsf {J}_-^{\\prime } \\setminus \\mathsf {J}_-$ it holds that $\\lbrace \\mathsf {J}_+,\\mathsf {J}_-\\rbrace \\subseteq \\mathrm {L}$ if and only if $\\lbrace \\mathsf {J}_+^{\\prime },\\mathsf {J}_-^{\\prime }\\rbrace \\subseteq \\mathrm {L}$ .", "We call a pair $\\mathrm {L}= \\mathrm {L}_+ \\uplus \\mathrm {L}_- \\subseteq \\mathcal {J}(\\mathbf {P})$ of embedded sublattice cooperating if they satisfy condition (ii) of Theorem REF above.", "We may also rephrase condition (ii) as follows.", "Lemma 2.14 Let $\\mathrm {L}_\\sigma \\subset \\mathcal {J}(P_\\sigma )$ be an embedded sublattice for $\\sigma = \\pm $ .", "Then $\\mathrm {L}_+, \\mathrm {L}_-$ are cooperating if and only if only if for any two filters $\\mathsf {J}_- \\in \\mathrm {L}_-, \\mathsf {J}_+ \\in \\mathrm {L}_+$ the following holds: For $A \\subseteq \\min (\\mathsf {J}_+) \\cap \\min (\\mathsf {J}_-)$ we have $\\mathsf {J}_- \\setminus A \\in \\mathrm {L}_- $ and $\\mathsf {J}_+ \\setminus A \\in \\mathrm {L}_+ $ , and for $B \\subseteq \\max (P_+ \\setminus \\mathsf {J}_+) \\cap \\max (P_-\\setminus \\mathsf {J}_-)$ we have $\\mathsf {J}_- \\cup B \\in \\mathrm {L}_- $ and $\\mathsf {J}_+ \\cup B \\in \\mathrm {L}_+ $ .", "It follows from the definition that for sets as stated, condition (ii) implies $\\mathsf {J}_\\sigma \\setminus A, \\mathsf {J}_\\sigma \\cup B \\in \\mathrm {L}_\\sigma $ for $\\sigma = \\pm $ .", "For the converse direction, let $\\mathsf {J}_\\sigma \\subseteq \\mathsf {J}_\\sigma ^{\\prime }$ such that $\\mathsf {J}_\\sigma ^{\\prime } \\in \\mathrm {L}_\\sigma $ for $\\sigma = \\pm $ .", "Assume that $D := \\mathsf {J}_+^{\\prime } \\setminus \\mathsf {J}_+ = \\mathsf {J}_-^{\\prime } \\setminus \\mathsf {J}_-$ .", "Then $A : = \\min (D) \\subseteq \\min (\\mathsf {J}_+^{\\prime }) \\cap \\min (\\mathsf {J}_-^{\\prime })$ and by (a), $\\mathsf {J}_\\sigma ^{\\prime } \\setminus A \\in \\mathrm {L}_\\sigma $ for $\\sigma = \\pm $ and induction on $|D|$ yields the claim.", "Theorem REF can be deduced from the description of facets in Theorem REF .", "We will give an alternative proof using Gröbner bases in Section .", "In conjunction with Theorem REF , we can read the dimension of $\\overline{F}(\\mathrm {L})$ from the cooperating pair $\\mathrm {L}$ .", "In the case of order polytopes, the canonical triangulation (see Section ) of $\\mathcal {O}(P)$ yields the following.", "Corollary 2.15 Let $F \\subseteq \\mathcal {O}(P)$ be a face with corresponding embedded sublattice $\\mathrm {L}\\subseteq \\mathcal {J}(P)$ .", "Then $\\dim F = l(\\mathrm {L}) - 1$ where $l(\\mathrm {L})$ is the length of a longest chain in $\\mathrm {L}$ .", "Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset and let $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ be a chain of filters in $(P,\\preceq _\\sigma )$ for $\\sigma = \\pm $ .", "The pair of chains $C = C_+ \\uplus C_-$ is non-interfering if $\\min (\\mathsf {J}_+) \\cap \\min (\\mathsf {J}_-) = \\varnothing $ for any $\\mathsf {J}_+ \\in C_+$ and $\\mathsf {J}_- \\in C_-$ .", "For $\\mathrm {L}\\subseteq \\mathcal {J}(\\mathbf {P})$ , we denote by $cl(\\mathrm {L})$ the maximum over $|C| = |C_+| + |C_-|$ where $C \\subseteq \\mathrm {L}$ is a pair of non-interfering chains.", "Figure: A maximal non-interfering set of filters in 𝐏 XW \\mathbf {P}_{XW}.", "A red orblue curve denotes the filter consisting of all elements above the curve.Corollary 2.16 Let $\\mathbf {P}$ be a compatible double poset and let $\\mathrm {L}\\subseteq \\mathcal {J}(\\mathbf {P})$ be a cooperating pair of embedded sublattices.", "Then $\\dim \\overline{F}(\\mathrm {L}) = cl(\\mathrm {L})-1$ .", "As a consequence of Theorem REF , $[\\mathbf {1}_{\\mathsf {J}},\\mathbf {1}_{\\mathsf {J}^{\\prime }}] \\subseteq \\mathcal {O}(P)$ is an edge if and only if $\\mathsf {J}\\subseteq \\mathsf {J}^{\\prime }$ are filters of $P$ such that $\\mathsf {J}^{\\prime } \\setminus \\mathsf {J}$ is a connected poset.", "Of course, this description captures all the horizontal edges of $\\mathcal {O}(\\mathbf {P})$ .", "The upcoming characterization of vertical edges follows from Theorem REF but we supply a direct proof.", "Corollary 2.17 Let $\\mathbf {P}$ be a compatible double poset and let $\\mathsf {J}_+ \\subseteq P_+$ and $\\mathsf {J}_- \\subseteq P_-$ be filters.", "Then $(2\\mathbf {1}_{\\mathsf {J}_+},+1)$ and $(-2\\mathbf {1}_{\\mathsf {J}_-},-1)$ are the endpoints of a vertical edge of $\\mathcal {O}(\\mathbf {P})$ if and only if $\\mathbf {1}_{\\mathsf {J}_+} - \\mathbf {1}_{\\mathsf {J}_-}$ is a vertex of $\\overline{\\mathcal {O}}(\\mathbf {P})$ if and only if $\\min (\\mathsf {J}_+) \\cap \\min (\\mathsf {J}_-) \\ = \\ \\varnothing \\quad \\text{and}\\quad \\max (P_+ \\setminus \\mathsf {J}_+) \\cap \\max (P_- \\setminus \\mathsf {J}_-) \\ = \\ \\varnothing .$ The first equivalence follows from the fact that $\\mathcal {O}(\\mathbf {P}) \\cap \\lbrace (\\phi ,t) : t = 0\\rbrace \\ = \\ (\\mathcal {O}(P_+)-\\mathcal {O}(P_-)) \\times \\lbrace 0\\rbrace $ and $\\mathbf {1}_{\\mathsf {J}_+} - \\mathbf {1}_{\\mathsf {J}_-}$ is the midpoint between $(2\\mathbf {1}_{\\mathsf {J}_+},+1)$ and $(-2\\mathbf {1}_{\\mathsf {J}_-},-1)$ .", "Before we come to the second claim, let us note that the face partition of a vertex $\\mathbf {1}_\\mathsf {J}$ for a poset $(P,\\preceq )$ is given by $\\lbrace \\mathsf {J},P\\setminus \\mathsf {J}\\rbrace $ .", "Thus, if $\\mathbf {1}_{\\mathsf {J}_+} - \\mathbf {1}_{\\mathsf {J}_-}$ is a vertex of $\\overline{\\mathcal {O}}(\\mathbf {P})$ , then there is a linear function $\\ell (f) = \\sum _{a\\in P} \\ell _a f(a)$ such that $\\mathcal {O}(P_+)^\\ell = \\lbrace \\mathbf {1}_{\\mathsf {J}_+}\\rbrace $ and $\\mathcal {O}(P_-)^{-\\ell } = \\lbrace \\mathbf {1}_{\\mathsf {J}_-}\\rbrace $ .", "Corollary REF then yields that $\\ell _a > 0$ for each $a \\in \\min (\\mathsf {J}_+)$ and $\\ell _a < 0$ for $a \\in \\min (\\mathsf {J}_-)$ .", "The same reasoning applies to $\\max (P_+\\setminus \\mathsf {J}_+)$ and $\\max (P_- \\setminus \\mathsf {J}_-)$ and shows necessity.", "Let $b \\in \\min (\\mathsf {J}_+)$ .", "If $b \\notin \\mathsf {J}_-$ , then the linear function $\\ell (f) := f(b)$ is maximized over $\\mathcal {O}(P_+)$ at every filter that contains $b$ and over $-\\mathcal {O}(P_-)$ at every filter that does not contain $b$ .", "If $b \\in \\mathsf {J}_-$ , then, by assumption, $b \\notin \\min (\\mathsf {J}_-)$ and there is some $p_2 \\in \\min (\\mathsf {J}_-)$ with $p_2\\prec _- b$ .", "Now, if $p_2 \\in \\mathsf {J}_+$ , then there is $p_3 \\in \\min (\\mathsf {J}_+)$ with $p_3 \\prec _+ p_2$ and so on.", "Compatibility now assures us that we get a descending alternating chain of the form $\\widehat{1}\\ \\succ _+ \\ b =: p_1 \\ \\succ _- \\ p_{2} \\ \\succ _+ \\ p_{3} \\ \\succ _- \\ \\cdots \\ \\succ _{\\sigma } \\ p_k \\ \\succ _{-\\sigma } \\ a \\ \\succ _\\sigma \\ \\widehat{0}$ where $p_2, p_4, p_6,\\ldots \\in \\min (\\mathsf {J}_-) \\cap \\mathsf {J}_+$ and $p_1,p_3, p_5,\\ldots \\in \\min (\\mathsf {J}_+) \\cap \\mathsf {J}_-$ and $a\\in \\min (\\mathsf {J}_{-\\sigma })\\setminus \\mathsf {J}_\\sigma $ .", "Consider the associated linear function $\\ell (f) \\ = \\ f(p_0) - f(p_1) + f(p_{2}) - \\cdots + (-1)^{k} f(p_k) +(-1)^{k+1} f(a)$ for $f\\in \\mathbb {R}^P$ .", "We claim that $\\ell (\\mathbf {1}_{\\mathsf {J}^{\\prime }_+}) \\le 1$ for each filter $\\mathsf {J}^{\\prime }_+ \\subseteq P_+$ and with equality if $b \\in \\mathsf {J}^{\\prime }_+$ .", "Indeed, if $p_{2i+1} \\in \\mathsf {J}^{\\prime }_+$ , then $p_{2i} \\in \\mathsf {J}^{\\prime }_+$ for all $i \\ge 1$ .", "Conversely, $\\ell (-\\mathbf {1}_{\\mathsf {J}^{\\prime }_-}) \\le 0 = \\ell (-\\mathbf {1}_{\\mathsf {J}_-})$ for each filter $\\mathsf {J}^{\\prime }_- \\subseteq P_-$ .", "This follows from the fact that $p_{2i}\\in \\mathsf {J}^{\\prime }_-$ implies $p_{2i-1} \\in \\mathsf {J}^{\\prime }_-$ for each $i \\ge 1$ .", "For $a \\in \\max (P_+\\setminus \\mathsf {J}_+)$ the situation is similar and we search for $b \\in \\max (P_- \\setminus \\mathsf {J}_-)$ with $a \\prec _- b$ in the case that $a \\notin \\mathsf {J}_-$ .", "This yields a linear function $\\ell \\in -N_{P_-}(\\mathbf {1}_{\\mathsf {J}_-})$ that is maximized over $\\mathcal {O}(P_+)$ at filters $\\mathbf {1}_{\\mathsf {J}^{\\prime }_+}$ with $a \\notin \\mathsf {J}^{\\prime }_+$ .", "Summing these linear functions for $b \\in \\min (\\mathsf {J}_+)$ and $a \\in \\max (P_+\\setminus \\mathsf {J}_+)$ yields a linear function $\\ell ^+$ with $\\mathcal {O}(P_+)^{\\ell ^+} =\\lbrace \\mathbf {1}_{\\mathsf {J}_+} \\rbrace $ and $\\mathbf {1}_{\\mathsf {J}_-} \\in \\mathcal {O}(P_-)^{-\\ell ^+}$ .", "Of course, the same reasoning applies to $\\mathsf {J}_-$ instead of $\\mathsf {J}_+$ and it follows that $\\ell ^+ - \\ell ^-$ is uniquely maximized at $\\mathbf {1}_{\\mathsf {J}_+} - \\mathbf {1}_{\\mathsf {J}_-}$ over $\\overline{\\mathcal {O}}(\\mathbf {P}) = \\mathcal {O}(P_+) -\\mathcal {O}(P_-)$ .", "Polars and valuation polytopes A real-valued valuation on a finite distributive lattice $(\\mathcal {J},\\vee ,\\wedge )$ is a function $h : \\mathcal {J}\\rightarrow \\mathbb {R}$ such that for any $a,b \\in \\mathcal {J}$ , $h(a \\vee b) \\ = \\ h(a) + h(b) - h(a \\wedge b)$ and $h(\\widehat{0}) = 0$ .", "Geissinger [13] studied the valuation polytope $\\mathrm {Val}(\\mathcal {J}) \\ := \\ \\lbrace h : \\mathcal {J}\\rightarrow [0,1] : h \\text{valuation} \\rbrace $ and conjectured that its vertices are exactly the valuations with values in $\\lbrace 0,1\\rbrace $ .", "This was shown by Dobbertin [7].", "Not much is known about the valuation polytope and Stanley's `5-'-Exercise [35] challenges the reader to find interesting combinatorial properties of $\\mathrm {Val}(\\mathcal {J})$ .", "In this section, we prove a curious relation between valuation polytopes and order polytopes.", "It follows from Birkhoff's fundamental theorem (cf.", "[35]) that any finite distributive lattice $\\mathcal {J}$ is of the form $\\mathcal {J}=\\mathcal {J}(P)$ , that is, it is the lattice of filters of some poset $P$ .", "In particular, for every valuation $h : \\mathcal {J}(P) \\rightarrow \\mathbb {R}$ there is a unique $h_0 : P\\rightarrow \\mathbb {R}$ such that $h(\\mathsf {J}) \\ = \\ \\sum _{a \\in \\mathsf {J}} h_0(a),$ for every filter $\\mathsf {J}\\subseteq P$ .", "Hence, $\\mathrm {Val}(\\mathcal {J})$ is linearly isomorphic to the $|P|$ -dimensional polytope $\\mathrm {Val}_0(P) \\ := \\ \\lbrace h_0 : P\\rightarrow \\mathbb {R}: 0 \\le h(\\mathsf {J}) \\le 1 \\text{ for all filters } \\mathsf {J}\\subseteq P\\rbrace .$ We denote by $S^\\triangle \\ = \\ \\lbrace \\ell \\in (\\mathbb {R}^n)^* : \\ell (s) \\le 1 \\text{ for all } s \\in S \\rbrace $ the polar of a set $S \\subset \\mathbb {R}^d$ .", "For a polytope $\\mathcal {P}\\subset \\mathbb {R}^d$ we write $\\mathrm {tprism}(\\mathcal {P}):={\\mathcal {P} \\boxminus \\mathcal {P}}\\subset \\mathbb {R}^{d+1}$ for the twisted prism of $\\mathcal {P}$ .", "Theorem 2.18 For any finite poset $P$ $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle \\ = \\ \\mathcal {O}(P,\\preceq ,\\preceq )^\\triangle \\ = \\ \\mathrm {tprism}(-\\mathrm {Val}_0(P)).$ For a chain $C = \\lbrace a_0 \\prec a_1 \\prec \\cdots \\prec a_k \\rbrace $ in $P$ , we define $\\ell ^{\\prime }_C(f) \\ := \\ \\sum _{i=0}^k (-1)^{k-i} f(a_i)$ and $L^{\\prime }_C(f,t) := \\ell ^{\\prime }_C(f) - t$ .", "It follows from Corollary REF and Corollary REF that $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle \\ = \\ \\operatorname{conv}( \\pm L^{\\prime }_C(f,t) : C \\subseteq P\\text{ chain} ).$ It is shown in Dobbertin [7] that $\\mathrm {Val}_0(P) \\ = \\ \\operatorname{conv}\\left( \\ell ^{\\prime }_C : C \\subseteq P\\text{ chain} \\right),$ from which the claim follows.", "As a direct consequence, we note the following.", "Corollary 2.19 Let $P$ be a finite poset.", "Then $\\mathrm {tprism}(\\mathrm {Val}(P))$ is 2-level.", "Since $\\mathcal {O}(\\mathbf {P}_\\circ )$ is centrally-symmetric and, by Corollary REF , 2-level, it follows that every vertex of $\\mathcal {O}(\\mathbf {P}_\\circ )$ takes the values $+1$ or $-1$ on every facet-defining linear function.", "The vertices correspond to facet normals under polarity, which shows that $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ is 2-level.", "Theorem REF now yields the claim.", "We can make the connection to valuations more transparent by considering valuations with values in $[-1,1]$ .", "Let $\\mathrm {Val}^\\pm (\\mathcal {J}(P))$ denote the corresponding polytope, then $\\mathrm {Val}_0^\\pm (P) \\ = \\ \\lbrace h_0 : P\\rightarrow \\mathbb {R}: -1 \\le h(\\mathsf {J}) \\le 1 \\text{ for all filters } \\mathsf {J}\\subseteq P\\rbrace \\ = \\ (\\mathcal {O}(P) \\cup -\\mathcal {O}(P))^\\triangle .$ Now, the convex hull of $\\mathcal {O}(P) \\cup -\\mathcal {O}(P)$ is exactly the image of $\\mathcal {O}(\\mathbf {P}_\\circ )$ under the projection $\\pi : \\mathbb {R}^P\\times \\mathbb {R}\\rightarrow \\mathbb {R}^P$ with $\\pi (f,t) = \\tfrac{1}{2}f$ .", "Hence, $\\mathrm {Val}_0^\\pm (P) \\ \\cong \\ \\pi (\\mathcal {O}(\\mathbf {P}_\\circ ))^\\triangle \\ \\cong \\ \\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle \\cap \\mathrm {im}(\\pi ^*) \\ \\cong \\ \\mathrm {tprism}(-2\\mathrm {Val}_0(P))\\cap (\\mathbb {R}^P\\times \\lbrace 0\\rbrace ),$ by Theorem REF .", "If we now view $\\mathrm {tprism}(-\\mathrm {Val}_0(P))$ as a Cayley sum, we obtain Corollary 2.20 For any poset $P$ $\\mathrm {Val}_0^\\pm (P) \\ = \\ \\mathrm {Val}_0(P) -\\mathrm {Val}_0(P).$ A polytope $\\mathcal {P}$ with vertices in a lattice $\\Lambda \\subset \\mathbb {R}^n$ is reflexive if $\\mathcal {P}^\\triangle $ is a lattice polytope with respect to the dual lattice $\\Lambda ^\\vee := \\lbrace \\ell \\in (\\mathbb {R}^n)^* : \\ell (x) \\in \\mathbb {Z}\\text{ for all} x \\in \\Lambda \\rbrace $ .", "For two polytopes $\\mathcal {P},\\mathcal {Q}\\subset \\mathbb {R}^n$ , write $\\Gamma (\\mathcal {P},\\mathcal {Q}):=\\operatorname{conv}(\\mathcal {P}\\cup -\\mathcal {Q})$ .", "Thus, $\\Gamma (\\mathcal {P},\\mathcal {Q})$ is the projection of ${\\mathcal {P} \\boxminus \\mathcal {Q}}$ onto the first $n$ coordinates.", "The polytopes $\\Gamma (\\mathcal {O}(P),\\mathcal {O}(P))$ where studied by Hibi, Matsuda, and Tsuchiya [22], [21] in the context of Gorenstein polytopes, i.e.", "lattice polytopes $\\mathcal {P}$ such that $r\\mathcal {P}$ is reflexive for some $r \\in \\mathbb {Z}_{>0}$ .", "By taking polars, we obtain the following from (REF ) and Corollary REF .", "Corollary 2.21 For any poset $P$ , $\\Gamma (\\mathcal {O}(P),\\mathcal {O}(P)) \\ = \\ (\\mathrm {Val}_0(P) -\\mathrm {Val}_0(P))^\\triangle .$ In particular, $\\Gamma (\\mathcal {O}(P),\\mathcal {O}(P))$ is reflexive.", "An explicit description of the face lattices of $\\mathrm {Val}(P)$ , $\\mathrm {Val}^\\pm (P)$ as well as $\\Gamma (\\mathcal {O}(P),\\mathcal {O}(P))$ can be obtained from Theorem REF .", "This theorem also yields information about the polars of $\\mathcal {O}(P,\\preceq _+,\\preceq _-)$ for compatible double posets.", "For a poset that is not compatible, the next result shows that the origin is not contained in the interior of $\\mathcal {O}(\\mathbf {P})$ and hence the polar is not bounded.", "Proposition 2.22 Let $\\mathbf {P}$ be a double poset.", "Then $\\mathcal {O}(\\mathbf {P})$ contains the origin in its interior if and only if $\\mathbf {P}$ is compatible.", "If $\\mathbf {P}$ is compatible, then Corollary REF shows that $\\mathbf {0}$ strictly satisfies all facet-defining inequalities.", "If $\\mathbf {P}$ is not compatible, then it contains an alternating cycle $C$ .", "It follows easily that $\\ell _{C} \\le 0$ on $\\mathcal {O}(P_+)$ and $-\\mathcal {O}(P_-)$ and hence $\\mathcal {O}(\\mathbf {P})$ is contained in the negative halfspace of $H = \\lbrace (f,t) : \\ell _{C}(f) \\le 0\\rbrace $ .", "Moreover, $\\mathbf {0} \\in H\\cap \\mathcal {O}(\\mathbf {P})$ , which shows that $\\mathbf {0} \\notin \\operatorname{relint}\\mathcal {O}(\\mathbf {P})$ .", "Anti-blocking polytopes Anti-blocking polytopes and Minkowski sums A polytope $\\mathcal {P}\\subset \\mathbb {R}_{\\ge 0}^n$ is called anti-blocking if $q \\in \\mathcal {P}\\ \\text{ and } \\ 0 \\le p \\le q \\quad \\Longrightarrow \\quad p \\in \\mathcal {P},$ where $p \\le q$ refers to componentwise order in $\\mathbb {R}^n$ .", "The notion of anti-blocking polyhedra was introduced by Fulkerson [12] in connection with min-max-relations in combinatorial optimization; our main reference for anti-blocking polytopes is Schrijver [31].", "In this section, we consider the Cayley sums ${\\mathcal {P} \\boxminus \\mathcal {Q}} \\ = \\ \\operatorname{conv}( \\mathcal {P}\\times \\lbrace 1\\rbrace \\cup (-\\mathcal {Q}) \\times \\lbrace -1\\rbrace ),$ where $\\mathcal {P}$ and $\\mathcal {Q}$ are anti-blocking polytopes.", "As before, we write $\\mathrm {tprism}(\\mathcal {P})$ for ${\\mathcal {P} \\boxminus \\mathcal {P}}$ .", "Our main source of examples will be the class of stable set polytopes: For a graph $G = (V,E)$ , a stable set is a subset $S \\subseteq V$ such that $\\binom{S}{2} \\cap E = \\varnothing $ .", "For simplicity, we will assume that $V = [n]$ and we write $\\mathbf {1}_S \\in \\lbrace 0,1\\rbrace ^n$ for the characteristic vector of a stable set $S$ .", "The stable set polytope of $G$ is the anti-blocking polytope $\\mathcal {P}_G \\ := \\ \\operatorname{conv}( \\mathbf {1}_S : S \\subseteq V \\text{ stable set} ) \\ \\subseteq \\ \\mathbb {R}^n.$ The class of perfect graphs is particularly interesting in this respect.", "Lovász [27] characterized perfect graphs in terms of their stable set polytopes and we use his characterization as a definition of perfect graphs.", "A clique of a graph $G = (V,E)$ is a subset $C\\subseteq V$ such that $\\binom{C}{2} \\subseteq E$ .", "For a vector $x \\in \\mathbb {R}^n$ and a subset $J \\subseteq [n]$ , we write $x(J) = \\sum _{j \\in J} x_j$ .", "Theorem 3.1 ([27]) A graph $G = ([n],E)$ is perfect if and only if $\\mathcal {P}_G \\ = \\ \\lbrace x \\in \\mathbb {R}^n : x \\ge 0, x(C) \\le 1 \\text{ for all cliques} C \\subseteq [n] \\rbrace .$ In this language, we can express the chain polytope of a poset $P$ as a stable set polytope: The comparability graph $G(P)$ of a poset $(P,\\preceq )$ is the undirected graph with vertex set $P$ and edge set $\\lbrace xy : x \\prec y \\text{ or } y \\prec x \\rbrace $ .", "Note that cliques in $G(P)$ are exactly the chains of $P$ .", "For a poset $P= ([n],\\preceq )$ the comparability graph $G(P)$ is perfect and hence $\\mathcal {C}({P}) \\ = \\ \\lbrace x \\in \\mathbb {R}^n : x \\ge 0, x(C) \\le 1 \\text{ for allchains } C \\subseteq [n] \\rbrace \\ = \\ \\mathcal {P}_{G(P)}.$ If $\\mathcal {P}\\subset \\mathbb {R}^n$ is an anti-blocking polytope, then there are $\\mathbf {c}_1,\\dots ,\\mathbf {c}_r \\in \\mathbb {R}_{\\ge 0}^n$ such that $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {c}_1,\\dots ,\\mathbf {c}_r\\rbrace ^{\\downarrow } \\ := \\ \\mathbb {R}^n_{\\ge 0} \\cap (\\operatorname{conv}(\\mathbf {c}_1,\\dots ,\\mathbf {c}_r) - \\mathbb {R}^n_{\\ge 0}).$ The unique minimal such set, denoted by $V^\\downarrow (P)$ , is given by the minimal elements of the vertex set of $\\mathcal {P}$ with respect to the partial order $\\le $ .", "It also follows from (REF ) and the Minkowski–Weyl theorem that there is a minimal collection $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {R}^n_{\\ge 0}$ such that $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for all }i=1,\\dots ,s\\rbrace $ For a polytope $\\mathcal {Q}\\subseteq \\mathbb {R}_{\\ge 0}^n$ , its associated anti-blocking polytope is the set $A({\\mathcal {Q}}) \\ := \\ \\lbrace \\mathbf {d}\\in \\mathbb {R}_{\\ge 0}^n : \\langle {\\mathbf {d},\\mathbf {x}}\\rangle \\le 1 \\text{ for all } \\mathbf {x}\\in \\mathcal {Q}\\rbrace .$ The following is the structure theorem for anti-blocking polytopes akin to the bipolar theorem for convex bodies.", "Theorem 3.2 ([31]) Let $\\mathcal {P}\\subset \\mathbb {R}^n$ be a full-dimensional anti-blocking polytope with $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {c}_1,\\dots ,\\mathbf {c}_r\\rbrace ^{\\downarrow } &\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0,\\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for all } i=1,\\dots ,s\\rbrace \\\\\\multicolumn{2}{l}{\\text{for some $\\mathbf {c}_1,\\dots ,\\mathbf {c}_r,\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {R}_{\\ge 0}^n$.", "Then}}\\\\A({\\mathcal {P}}) \\ = \\ \\lbrace \\mathbf {d}_1,\\dots ,\\mathbf {d}_s\\rbrace ^{\\downarrow } &\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {c}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for all } i=1,\\dots ,r\\rbrace .$ In particular, $A({A({\\mathcal {P}})}) = \\mathcal {P}$ .", "Before we come to our first result regarding Cayley- and Minkowski-sums of anti-blocking polytopes, we note the following fact.", "We write $V(\\mathcal {P})$ for the vertex set of a polytope $\\mathcal {P}$ .", "Proposition 3.3 Let $\\mathcal {P}_1, \\mathcal {P}_2$ be two full-dimensional anti-blocking polytopes.", "Then the vertices of $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ are exactly $(V(\\mathcal {P}_1) \\cup V(-\\mathcal {P}_2)) \\setminus \\lbrace \\mathbf {0}\\rbrace $ .", "For a polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ and a vector $\\mathbf {c}\\in \\mathbb {R}^n$ , we denote by $\\mathcal {P}^\\mathbf {c}$ the face of $\\mathcal {P}$ that maximizes the linear function $\\mathbf {x}\\mapsto \\langle {\\mathbf {c},\\mathbf {x}}\\rangle $ .", "It suffices to show that every $\\mathbf {v}\\in V(\\mathcal {P}_1) \\setminus \\lbrace \\mathbf {0}\\rbrace $ is a vertex of $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ .", "Let $\\mathbf {c}\\in \\mathbb {R}^n$ such that $\\mathcal {P}_1^\\mathbf {c}= \\lbrace \\mathbf {v}\\rbrace $ .", "Since $\\mathbf {v}\\ne 0$ , there is some $\\mathbf {d}\\in \\mathbb {R}^n_{\\ge 0}$ such that $\\langle {\\mathbf {d}, \\mathbf {u}_1}\\rangle \\le 1$ for all $\\mathbf {u}_1 \\in \\mathcal {P}_1$ and $\\langle {\\mathbf {d}, \\mathbf {v}}\\rangle = 1$ .", "Hence, for any $\\mu \\ge 0$ , $\\mathcal {P}_1^{\\mathbf {c}+ \\mu \\mathbf {d}} = \\lbrace v\\rbrace $ .", "Now, $\\langle {\\mathbf {d},-\\mathbf {u}_2}\\rangle \\le 0$ for all $\\mathbf {u}_2 \\in P_2$ .", "In particular, for $\\mu >0$ sufficiently large, $\\langle {\\mathbf {c}+ \\mu \\mathbf {d},\\mathbf {u}_2}\\rangle \\ \\le \\ \\langle {\\mathbf {c},\\mathbf {u}_2}\\rangle \\ < \\ \\mu +\\langle {\\mathbf {c},\\mathbf {v}}\\rangle \\ = \\ \\langle {\\mathbf {c}+ \\mu \\mathbf {d},\\mathbf {v}}\\rangle ,$ which shows that $\\mathbf {v}$ uniquely maximizes $\\langle {\\mathbf {c}+ \\mu \\mathbf {d},\\mathbf {u}}\\rangle $ over $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ .", "For $\\mathbf {d}\\in \\mathbb {R}^n_{\\ge 0}$ and $I \\subseteq [n]$ , we write ${\\mathbf {d}}^{[I]}$ for the vector with $({\\mathbf {d}}^{[I]})_j \\ = \\ {\\left\\lbrace \\begin{array}{ll}d_j & \\text{ for } j \\in I \\\\0 & \\text{ otherwise}.\\end{array}\\right.", "}$ Theorem 3.4 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $(\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle \\ = \\ \\operatorname{conv}( A({\\mathcal {P}_1}) \\cup -A({\\mathcal {P}_2}) ).$ Moreover, $({2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2})^\\triangle \\ = \\ {-A({\\mathcal {P}_2}) \\boxminus -A({\\mathcal {P}_1})}.$ Let us denote the right-hand side of the first equation by $\\mathcal {Q}$ .", "Note that $\\langle {\\mathbf {u}_1,-\\mathbf {v}_2}\\rangle \\le 0$ for $\\mathbf {u}_1 \\in A({P_1})$ and $\\mathbf {v}_2 \\in P_2$ .", "This shows that $\\langle {\\mathbf {u}_1,\\mathbf {v}}\\rangle \\le 1$ for all $\\mathbf {v}\\in P_1 - P_2$ .", "By symmetry, this yields $\\mathcal {Q}\\subseteq (\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle $ .", "For the converse, observe that every vertex of $\\mathcal {Q}$ is of the form ${\\mathbf {d}}^{[I]}$ with $\\mathbf {d}\\in V^\\downarrow (A({\\mathcal {P}_1}))\\cup -V^\\downarrow (A({\\mathcal {P}_2}))$ .", "It follows that $\\mathbf {z}\\in \\mathcal {Q}^\\triangle $ if and only if $\\langle {{\\mathbf {d}}^{[I]}, \\mathbf {z}}\\rangle \\le 1$ for all $ \\mathbf {d}\\in V^\\downarrow (A({\\mathcal {P}_1}))\\cup -V^\\downarrow (A({\\mathcal {P}_2}))$ and all $I \\subseteq [n]$ .", "For $\\mathbf {z}\\in \\mathcal {Q}^\\triangle $ write $\\mathbf {z}= \\mathbf {z}^1 - \\mathbf {z}^2$ with $\\mathbf {z}^1,\\mathbf {z}^2 \\ge 0$ and $\\operatorname{supp}(\\mathbf {z}^1) \\cap \\operatorname{supp}(\\mathbf {z}^2) = \\varnothing $ , where for any $\\mathbf {z}=(z_1,\\dots ,z_n)\\in \\mathbb {R}^n$ we set $\\operatorname{supp}(\\mathbf {z}):=\\lbrace i:z_i \\ne 0\\rbrace $ .", "We claim that $\\mathbf {z}^i \\in \\mathcal {P}_i$ for $i=1,2$ .", "Indeed, let $I = \\operatorname{supp}(\\mathbf {z}^1)$ .", "Then for any $\\mathbf {d}\\in V^\\downarrow (\\mathcal {P}_1)$ we have $\\langle {\\mathbf {d},\\mathbf {z}^1}\\rangle \\ = \\ \\langle {{\\mathbf {d}}^{[I]},\\mathbf {z}}\\rangle \\ \\le \\ 1$ and hence $\\mathbf {z}^1 \\in \\mathcal {P}_1$ .", "Applying the same argument to $\\mathbf {z}^2$ shows that $\\mathbf {z}\\in \\mathcal {P}_1 - \\mathcal {P}_2$ and hence $(\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle \\subseteq \\mathcal {Q}$ .", "For the second claim, note that any linear function on $\\mathbb {R}^n \\times \\mathbb {R}$ that maximizes on a vertical facet of ${2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2}$ is of the form $\\alpha _\\mathbf {d}\\langle {\\mathbf {d},\\mathbf {x}}\\rangle + \\delta _\\mathbf {d}t$ for $\\mathbf {d}$ a vertex of $(\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle $ and some $\\alpha _\\mathbf {d}, \\delta _\\mathbf {d}\\in \\mathbb {R}$ with $\\alpha _\\mathbf {d}> 0$ .", "By the first claim and Proposition REF , it follows that $\\mathbf {d}\\in (V(A({\\mathcal {P}_1})) \\cup V(-A({\\mathcal {P}_2}))) \\setminus \\lbrace \\mathbf {0}\\rbrace $ .", "If $\\mathbf {d}\\in V(A({\\mathcal {P}_1})) \\setminus \\lbrace \\mathbf {0}\\rbrace $ , then $\\langle {\\mathbf {d},\\mathbf {u}_1}\\rangle \\le 1$ is tight for $\\mathbf {u}_1 \\in \\mathcal {P}_1$ whereas $\\langle {\\mathbf {d},-\\mathbf {u}_2}\\rangle \\le 0$ is tight for $-\\mathbf {u}_2 \\in -\\mathcal {P}_2$ .", "Hence, $\\langle {\\mathbf {d},\\mathbf {x}}\\rangle - t \\ \\le \\ 1$ is the corresponding facet-defining halfspace.", "Similarly, if $-\\mathbf {d}\\in -V(A({\\mathcal {P}_1})) \\setminus \\lbrace \\mathbf {0}\\rbrace $ , then $\\langle {-\\mathbf {d},x}\\rangle + t \\ \\le \\ 1$ is facet-defining.", "Together with the two horizontal facets $\\langle {\\mathbf {0},\\mathbf {x}}\\rangle \\pm t \\le 1$ this yields an inequality description of $({-A({\\mathcal {P}_2}) \\boxminus -A({\\mathcal {P}_1})})^\\triangle $ , which proves the second claim.", "Theorem REF together with Theorem REF has a nice implication that was used in [30] in connection with Hansen polytopes.", "Corollary 3.5 For any full-dimensional anti-blocking polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ , the polytope ${\\mathcal {P} \\boxminus A({\\mathcal {P}})}$ is linearly isomorphic to its polar $({\\mathcal {P} \\boxminus A({\\mathcal {P}})})^\\triangle $ .", "In particular, ${\\mathcal {P} \\boxminus A({\\mathcal {P}})}$ is self-dual.", "Stable set polytopes of double graphs and double chain polytopes A double graph is a triple $\\mathbf {G}= (V,E_+,E_-)$ consisting of a node set $V$ with two sets of edges $E_+,E_- \\subseteq \\binom{V}{2}$ .", "Again, we write $G_+ = (V, E_+)$ and $G_- = (V, E_-)$ to denote the two underlying ordinary graphs.", "The results of the preceding sections prompt the definition of stable set polytope of a double graph $\\mathcal {P}_{\\mathbf {G}} \\ := \\ {2\\mathcal {P}_{G_+} \\boxminus 2\\mathcal {P}_{G_-}}.$ For a double graph $\\mathbf {G}$ , define the complement graph as $\\overline{\\mathbf {G}} = (V,E_-^c,E_+^c)$ .", "Then Theorem REF implies the following relation.", "Corollary 3.6 Let $\\mathbf {G}$ be a perfect double graph.", "Then $\\mathcal {P}_{\\mathbf {G}}^\\triangle $ is linearly isomorphic to $\\mathcal {P}_{\\overline{\\mathbf {G}}}$ .", "We have $\\mathcal {P}_{\\mathbf {G}}^\\triangle =({2\\mathcal {P}_{G_+} \\boxminus 2\\mathcal {P}_{G_-}})^\\triangle \\ = \\ {-A({\\mathcal {P}_{G_-}}) \\boxminus -A({\\mathcal {P}_{G_+}})} \\ = \\ {-\\mathcal {P}_{\\overline{G}_-} \\boxminus -\\mathcal {P}_{\\overline{G}_+}} \\ \\cong \\ \\mathcal {P}_{\\overline{\\mathbf {G}}}.$ In particular, a double poset $\\mathbf {P}= (P,\\preceq _\\pm )$ gives rise to a double graph $\\mathbf {G}(\\mathbf {P}) = (G(P_+),G(P_-))$ and the double chain polytope of $\\mathbf {P}$ is simply $\\mathcal {C}({\\mathbf {P}}) = \\mathcal {P}_{\\mathbf {G}(\\mathbf {P})}$ , the double chain polytope of $\\mathbf {P}$ .", "Theorem REF directly gives a facet description of the double chain polytope.", "Note that compatibility is not required.", "Theorem 3.7 Let $\\mathbf {P}$ be a double poset and $\\mathcal {C}({\\mathbf {P}})$ its double chain polytope.", "Then $(g,t) \\in \\mathbb {R}^P\\times \\mathbb {R}$ is contained in $\\mathcal {C}({\\mathbf {P}})$ if and only if $\\sum _{a \\in C_+} g(a) - t \\ \\le \\ 1 \\quad \\text{ and } \\quad \\sum _{a \\in C_-} -g(a) + t \\ \\le \\ 1,$ where $C_+ \\subseteq P_+$ and $C_- \\subseteq P_-$ ranges of all chains.", "For the usual order- and chain polytope, Hibi and Li [20] showed that $\\mathcal {O}(P)$ has at most as many facets as $\\mathcal {C}({P})$ and equality holds if and only if $P$ does not contain the 5-element poset with Hasse diagram 'X'.", "This is different in the case of double poset polytopes.", "Corollary 3.8 Let $(P,\\preceq )$ be a poset.", "Then $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ have the same number of facets.", "Alternating chains in $\\mathbf {P}_\\circ $ are in bijection to twice the number of chains in $P$ .", "However, it is not true that $\\mathcal {O}(\\mathbf {P}_\\circ )$ is always combinatorially isomorphic to $\\mathcal {C}({\\mathbf {P}_\\circ })$ .", "Example 5 Let $P$ be the 5-element poset with Hasse diagram 'X'.", "Then the face vectors of $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ are $f(\\mathcal {O}(\\mathbf {P}_\\circ )) \\ &= \\ (16, 88, 204, 240, 144, 36) \\\\f(\\mathcal {C}({\\mathbf {P}_\\circ })) \\ &= \\ (16, 88, 222, 276, 162, 36).$ Hibi and Li [20] conjectured that $f(\\mathcal {O}(P))\\le f(\\mathcal {C}({P}))$ componentwise.", "Computations suggest that the same relation should hold for the double poset polytopes of induced double posets.", "Conjecture 1 Let $\\mathbf {P}= (P,\\preceq ,\\preceq )$ be a double poset induced by a poset $(P,\\preceq )$ .", "Then $f_i(\\mathcal {O}(\\mathbf {P})) \\ \\le \\ f_i(\\mathcal {C}({\\mathbf {P}}))$ for $0 \\le i \\le |P|$ .", "An extension of the conjecture to general compatible double posets fails, as the following example shows.", "Example 6 Let $\\mathbf {A}_n$ be an alternating chain of length $n$ , that is, $\\mathbf {P}$ is a double poset on elements $a_1,a_2,\\dots ,a_{n+1}$ with cover relations $a_1 \\ \\prec _+ \\ a_2 \\ \\prec _- \\ a_3 \\ \\prec _+ \\cdots $ It follows from Theorem REF that the number of facets of $\\mathcal {C}({\\mathbf {A}_n})$ is $3n+4$ .", "Since $\\mathbf {A}_n$ is compatible, then by Theorem REF the number of facets of $\\mathcal {O}(\\mathbf {A}_n)$ equals the number of alternating chains which is easily computed to be $\\binom{n+3}{2}+1$ .", "Thus, for $n \\ge 3$ , the alternating chains $\\mathbf {A}_n$ fail Conjecture REF for the number of facets.", "For $n=3$ , we explicitly compute $f(\\mathcal {O}(\\mathbf {A}_3)) \\ &= \\ ( 21, 70, 95, 60, 16) \\quad \\text{and} \\\\f(\\mathcal {C}({\\mathbf {A}_3})) \\ &= \\ (21, 67, 86, 51, 13).$ Every graph $G = (V,E)$ trivially gives rise to a double graph $\\mathbf {G}_\\circ =(V,E,E)$ .", "Thus, the Hansen polytope of a graph $G$ is the polytope $\\mathcal {H}(G) = \\mathcal {P}_{\\mathbf {G}_\\circ }$ .", "Theorem REF then yields a strengthening of the main result of Hansen [17].", "Note that for the complement graph $\\overline{G} = (V,E^c)$ , it follows that a subset $S\\subseteq V$ is a stable set of $G$ if and only if $S$ is a clique of $\\overline{G}$ and vice versa.", "Corollary 3.9 ([17]) Let $G$ be a perfect graph.", "Then $\\mathcal {H}(G)$ is 2-level and $\\mathcal {H}(G)^\\triangle $ is affinely isomorphic to $\\mathcal {H}(\\overline{G})$ .", "By Theorem REF and Theorem  REF $\\mathcal {H}(G)^\\triangle \\ = \\ {-A({\\mathcal {P}_G}) \\boxminus -A({\\mathcal {P}_G})} \\ = \\ {-\\mathcal {P}_{\\overline{G}} \\boxminus -\\mathcal {P}_{\\overline{G}}} \\ \\cong \\ \\mathcal {H}(\\overline{G}),$ which proves the second claim.", "A vertex of $\\mathcal {H}(G)^\\triangle $ is of the form $\\mathbf {d}= \\pm (-\\mathbf {1}_C,1)$ for some clique $C$ of $G$ .", "Thus, for any vertex $\\mathbf {v}= \\pm (2\\mathbf {1}_S,1) \\in \\mathcal {H}(G)$ , where $S$ is a stable set of $G$ , we compute $\\langle {\\mathbf {d},\\mathbf {v}}\\rangle = \\pm ( 1 - 2|S \\cap C|) = \\pm 1$ .", "Example 7 (Double chain polytopes of dimension-two posets) Following Example REF , let $\\pi _+,\\pi _-\\in \\mathbb {Z}^n$ be two integer sequences with associated posets $P_{\\pi _+}$ and $P_{\\pi _-}$ of order dimension two.", "Consider the double posets $\\mathbf {P}=(P_{\\pi _+},P_{\\pi _-})$ and $-\\mathbf {P}=(P_{-\\pi _-},P_{-\\pi _+})$ .", "We have $\\overline{\\mathbf {G}(\\mathbf {P})} \\ = \\ (\\overline{G(P_{\\pi _-})},\\overline{G(P_{\\pi _+})}) \\ = \\ (G(P_{-\\pi _-}),G(P_{-\\pi _+})) \\ = \\ \\mathbf {G}(-\\mathbf {P})$ and hence $\\mathcal {C}({\\mathbf {P}})^\\triangle \\ \\cong \\ \\mathcal {C}({-\\mathbf {P}})$ by Corollary REF .", "However, it is not necessarily true that $\\mathcal {O}(\\mathbf {P})^\\triangle \\cong \\mathcal {O}(-\\mathbf {P})$ , as can be checked for the double poset induced by the $X$ -poset; cf.", "Example REF .", "Example 8 (Double chain polytopes of plane posets) Let $\\mathbf {P}$ be a plane double poset.", "By the last example, the double chain polytope $\\mathcal {C}({\\mathbf {P}})$ is linearly equivalent to its polar $\\mathcal {C}({\\mathbf {P}})^\\triangle $ .", "Among the 2-level polytopes, independence polytopes of perfect graphs play a distinguished role.", "The following observation, due to Samuel Fiorini (personal communication), characterizes 2-level anti-blocking polytopes.", "Proposition 3.10 Let $\\mathcal {P}$ be a full-dimensional anti-blocking polytope.", "Then $\\mathcal {P}$ is 2-level if and only if $\\mathcal {P}$ is linearly isomorphic to $\\mathcal {P}_G$ for some perfect graph $G$ .", "The origin is a vertex of $\\mathcal {P}$ and, since $\\mathcal {P}$ is full-dimensional and anti-blocking, its neighbors are $ \\alpha _1 \\mathbf {e}_1, \\dots , \\alpha _1 \\mathbf {e}_n$ are for some $\\alpha _i > 0$ .", "After a linear transformation, we can assume that $\\alpha _1 = \\cdots = \\alpha _n = 1$ .", "Since $\\mathcal {P}$ is 2-level, $\\mathcal {P}=\\lbrace \\mathbf {x}\\in \\mathbb {R}_{\\ge 0}^n : \\langle {\\mathbf {d}_i, \\mathbf {x}}\\rangle \\le 1 \\text{ for } i=1,\\dots ,s\\rbrace $ where $\\mathbf {d}_i \\in \\lbrace 0,1\\rbrace ^n$ for all $i=1,\\dots ,s$ .", "Let $G = ([n],E)$ be the minimal graph with cliques $\\operatorname{supp}(\\mathbf {d}_i)$ for all $i=1,\\dots ,s$ .", "That is, $E = \\bigcup _i \\binom{\\operatorname{supp}(\\mathbf {d}_i)}{2}$ .", "We have $\\mathcal {P}_G \\subseteq \\mathcal {P}$ .", "Conversely, any vertex of $\\mathcal {P}$ is of the form $\\mathbf {1}_S$ for some $S\\subseteq [n]$ and $\\langle {\\mathbf {d}_i,\\mathbf {1}_S}\\rangle = |\\operatorname{supp}(\\mathbf {d}_i) \\cap S| \\le 1$ shows that $\\mathcal {P}\\subseteq \\mathcal {P}_G$ .", "This implies a characterization of the 2-level polytopes among Cayley sums of anti-blocking polytopes.", "Corollary 3.11 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $\\mathcal {P}= {\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ is 2-level if and only if $\\mathcal {P}$ is affinely isomorphic to $\\mathcal {H}(G)$ for some perfect graph $G$ .", "Sufficiency is Hansen's result (Corollary REF ).", "For necessity, observe that $\\mathcal {P}_1$ and $\\mathcal {P}_2$ are faces and hence have to be 2-level.", "By the proof of Proposition REF , we may assume that $\\mathcal {P}_1 = \\mathcal {P}_{G_1}$ for some perfect graph $G_1$ and $\\mathcal {P}_2 = A\\mathcal {P}_{G_2}$ for some perfect $G_2$ and a diagonal matrix $A\\in \\mathbb {R}^{n\\times n}$ with diagonal entries $a_i>0$ for $i\\in [n]$ .", "We will proceed in two steps: We first prove that $A$ must be the identity matrix and then show that $G_1=G_2$ .", "For every $i\\in [n]$ the inequality $x_i\\ge 0$ is facet-defining for $\\mathcal {P}_1$ .", "Hence it induces a facet-defining inequality for $\\mathcal {P}$ , which must be of the form $\\ell _i\\ := \\ -b_ix_i+t \\ \\le \\ 1$ for some $b_i>0$ , where $t$ denotes the last coordinate in $\\mathbb {R}^{n+1}$ .", "Observe that $\\ell _i$ takes the values 1 and $1-b_i$ on the vertices $\\lbrace \\mathbf {0},\\mathbf {e}_i\\rbrace \\times \\lbrace 1\\rbrace $ of the face $\\mathcal {P}_1\\times \\lbrace 1\\rbrace $ .", "On the other hand, on $\\lbrace \\mathbf {0},-a_i\\mathbf {e}_i\\rbrace \\times \\lbrace -1\\rbrace \\subset -\\mathcal {P}_2\\times \\lbrace -1\\rbrace $ , the values are $-1$ and $-1+a_ib_i$ .", "Now 2-levelness implies $a_i=1$ and $b_i=2$ .", "It now follows from Theorem REF that the facet-defining inequalities for $\\mathcal {P}$ are $\\begin{split}2\\mathbf {1}_{C_1}(\\mathbf {x})-t \\ &\\le \\ 1\\text{ and}\\\\-2\\mathbf {1}_{C_2}(\\mathbf {x})+t\\ &\\le \\ 1,\\end{split}$ where $C_1$ and $C_2$ are cliques in $G_1$ and $G_2$ , respectively.", "By 2-levelness each of these linear functions takes the values $-1$ and 1 on the vertices of $P$ .", "This easily implies that every clique in $G_1$ must be a clique in $G_2$ and conversely.", "Hence $G_1=G_2$ .", "Canonical Subdivisions We now turn to the canonical subdivisions of $\\mathcal {P}_1 - \\mathcal {P}_2$ and ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ for anti-blocking polytopes $\\mathcal {P}_1,\\mathcal {P}_2$ .", "A subdivision of $\\mathcal {P}= \\mathcal {P}_1 - \\mathcal {P}_2$ is a collection of polytopes $\\mathcal {Q}^1,\\dots ,\\mathcal {Q}^m \\subseteq \\mathcal {P}$ each of dimension $\\dim \\mathcal {P}$ such that $\\mathcal {P}= \\mathcal {Q}^1\\cup \\cdots \\cup \\mathcal {Q}^m$ and $\\mathcal {Q}^i \\cap \\mathcal {Q}^j$ is a face of both for all $i \\ne j$ .", "We call the subdivision mixed if each $\\mathcal {Q}^i$ is of the form $\\mathcal {Q}^i_1 - \\mathcal {Q}^i_2$ where $\\mathcal {Q}^i_j$ is a vertex-induced subpolytope of $\\mathcal {P}_j$ for $j=1,2$ .", "Finally, a mixed subdivision is exact if $\\dim \\mathcal {Q}^i = \\dim \\mathcal {Q}^i_1 + \\dim \\mathcal {Q}^i_2$ .", "That is, $\\mathcal {Q}^i$ is linearly isomorphic to the Cartesian product $\\mathcal {Q}^i_1 \\times \\mathcal {Q}^i_2$ .", "For a full-dimensional anti-blocking polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ , every index set $J \\subseteq [n]$ defines a distinct face $\\mathcal {P}|_J := \\lbrace x \\in \\mathcal {P}: x_j = 0 \\text{ for } j\\notin J \\rbrace $ .", "This is an anti-blocking polytope of dimension $|J|$ .", "For disjoint $I,J \\subseteq [n]$ , the polytopes $\\mathcal {P}_1|_I, \\mathcal {P}_2|_J$ lie in orthogonal subspaces and $\\mathcal {P}_1|_I - \\mathcal {P}_2|_J$ is in fact a Cartesian product.", "In this case, the Cayley sum ${\\mathcal {P}_1|_I \\boxminus \\mathcal {P}_2|_J}$ is called a join and denoted by $ \\mathcal {P}_1|_I * \\mathcal {P}_2|_J$ .", "As with the Cartesian product, the combinatorics of $ \\mathcal {P}_1|_I * \\mathcal {P}_2|_J$ is completely determined by the combinatorics of $ \\mathcal {P}_1|_I$ and $\\mathcal {P}_2|_J$ .", "Lemma 3.12 Let $\\mathcal {P}_1, \\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $\\mathcal {P}_1 - \\mathcal {P}_2$ has a regular exact mixed subdivision with cells $\\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ for all $J \\subseteq [n]$ .", "In particular, ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ has a regular subdivision into joins $\\mathcal {P}_1|_{J} * \\mathcal {P}_2|_{J^c}$ for all $J \\subseteq [n]$ .", "We call the subdivisions of Lemma REF the canonical subdivisions of $\\mathcal {P}_1 - \\mathcal {P}_2$ and ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ , respectively.", "By the Cayley trick [6], it is suffices to prove only the first claim.", "The subdivision of $\\mathcal {P}_1 - \\mathcal {P}_2$ is very easy to describe: Let us first note that the polytopes $\\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ for $J \\subseteq [n]$ only meet in faces.", "Hence, we only need to verify that they cover $\\mathcal {P}_1 - \\mathcal {P}_2$ .", "It suffices to show that for any point $\\mathbf {x}\\in \\mathcal {P}_1 - \\mathcal {P}_2$ with $x_i \\ne 0$ for all $i$ , there is a $J \\subseteq [n]$ with $\\mathbf {x}\\in \\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ .", "Let $\\mathbf {x}_1,\\mathbf {x}_2 \\in \\mathbb {R}_{\\ge 0}^n$ with $\\mathbf {x}= \\mathbf {x}_1 - \\mathbf {x}_2$ and $\\operatorname{supp}(\\mathbf {x}_1) \\cap \\operatorname{supp}(\\mathbf {x}_2) =\\varnothing $ .", "We claim that $\\mathbf {x}_i \\in \\mathcal {P}_i$ for $i=1,2$ .", "Indeed, if $\\mathbf {x}=\\mathbf {y}_1 - \\mathbf {y}_2$ for some $\\mathbf {y}_i \\in \\mathcal {P}_i$ , then $0 \\le \\mathbf {x}_i \\le \\mathbf {y}_i$ and $\\mathbf {x}_i \\in \\mathcal {P}_i$ by (REF ).", "In particular, $\\mathbf {x}_1 \\in \\mathcal {P}_1 |_J$ and $\\mathbf {x}_2 \\in \\mathcal {P}_2 |_{J^c}$ and therefore $\\mathbf {x}\\in \\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ .", "To show regularity, let $\\omega : \\mathbb {R}^n \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ be the linear function such that $\\omega (\\mathbf {e}_i,0) = -\\omega (0,\\mathbf {e}_j) = 1$ for all $i,j = 1,\\dots ,n$ .", "Then $\\omega $ induces a mixed subdivision by picking for every point $\\mathbf {x}\\in \\mathcal {P}_1 - \\mathcal {P}_2$ , the unique cell $F_1 - F_2$ such that $\\mathbf {x}= \\mathbf {x}_1 - \\mathbf {x}_2$ with $\\mathbf {x}_i \\in \\operatorname{relint}F_i$ and $(\\mathbf {x}_1,\\mathbf {x}_2)$ minimizes $\\omega $ over the set $\\lbrace (\\mathbf {y}_1,\\mathbf {y}_2) \\in \\mathcal {P}_1 \\times \\mathcal {P}_2 : \\mathbf {x}= \\mathbf {y}_1 - \\mathbf {y}_2 \\rbrace ;$ see Section 9.2.2 of de Loera et al.", "[6] for more details.", "If $\\omega $ is not generic, one has to be careful as the minimizer is not necessarily unique but in our case, we observe that for any $\\mathbf {y}_i \\in \\mathcal {P}_i$ with $\\mathbf {x}= \\mathbf {y}_1 - \\mathbf {y}_2$ we have $\\omega (\\mathbf {y}_1,\\mathbf {y}_2) >\\omega (\\mathbf {x}_1,\\mathbf {x}_2)$ for all $(\\mathbf {y}_i,\\mathbf {y}_2)\\ne (\\mathbf {x}_1,\\mathbf {x}_2)$ with $(\\mathbf {x}_1,\\mathbf {x}_2)$ defined above.", "Figure: The canonical subdivision of 𝒫 1 -𝒫 2 \\mathcal {P}_1 - \\mathcal {P}_2 for two(random) anti-blocking polytopes 𝒫 1 ,𝒫 2 ⊂ℝ ≥0 3 \\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^3_{\\ge 0}.We define a triangulation of a polytope to be a subdivision into simplices without new vertices.", "For a polytope with vertices in an affine lattice $\\mathbb {A}$ , a triangulation is unimodular if each simplex is unimodular or, equivalently, has normalized volume $=1$ .", "A triangulation is flag if any minimal non-face is of dimension 1.", "This property implies that the underlying simplicial complex is completely determined by its graph.", "Theorem 3.13 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes with subdivisions $\\mathcal {S}_1$ and $\\mathcal {S}_2$ , respectively.", "For $J \\subseteq [n]$ , let $\\mathcal {S}_i|_J := \\lbrace S \\cap \\mathcal {P}_i|_J : S \\in \\mathcal {S}_i \\rbrace $ be the restriction of $\\mathcal {S}_i$ to $\\mathcal {P}_i|_J$ for $i=1,2$ .", "Then $\\mathcal {S}\\ := \\ \\bigcup _{J \\subseteq [n]} \\mathcal {S}_1|_J * \\mathcal {S}_2|_{J^c}$ is a subdivision of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ .", "In particular, [(i)] If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are regular, then $\\mathcal {S}$ is regular.", "If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are unimodular triangulations with respect to $\\Lambda $ , then $\\mathcal {S}$ is a unimodular triangulation with respect to the affine lattice $\\Lambda \\times (2\\mathbb {Z}+ 1)$ .", "If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are flag, then $\\mathcal {S}$ is flag.", "Note that (iii) also holds if the triangulations use more lattice points than just the vertices.", "For the first claim, observe that $\\mathcal {S}_i|_J$ is a subdivision of the face $\\mathcal {P}_i|_J$ .", "By [6], $\\mathcal {S}_1|_J *\\mathcal {S}_2|_{J^c}$ is a subdivision of $\\mathcal {P}_1|_J * \\mathcal {P}_2|_{J^c}$ .", "Hence, $\\mathcal {S}$ is a refinement of the canonical subdivision of Lemma REF .", "If $\\mathcal {S}_i$ is a regular subdivision of $\\mathcal {P}_i$ , then there are weights $\\omega _i : V(\\mathcal {P}_i) \\rightarrow \\mathbb {R}$ for $i=1,2$ .", "By adding a constant weight to every vertex if necessary, we can assume that $\\omega _1(\\mathbf {v}_1) >0$ and $\\omega _2(\\mathbf {v}_2) < 0$ for all $\\mathbf {v}_1 \\in V(\\mathcal {P}_1)$ and $\\mathbf {v}_2 \\in V(\\mathcal {P}_2)$ .", "Again using the Cayley trick, it is easily seen that $\\omega :V({\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}) \\rightarrow \\mathbb {R}$ given by $\\omega (\\mathbf {v}_1,+1) :=\\omega _1(\\mathbf {v}_1)$ and $\\omega (\\mathbf {v}_2,-1) := \\omega _2(\\mathbf {v}_2)$ induces $\\mathcal {S}$ .", "Claim (ii) simply follows from the fact that the join of two unimodular simplices is unimodular.", "For claim (iii), let $\\sigma = \\sigma _1 \\uplus \\sigma _2 \\subseteq V({\\mathcal {P}_1 \\boxminus \\mathcal {P}_2})$ be a minimal non-face.", "Since $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are flag, it follows that $\\sigma _1 \\in \\mathcal {S}_1$ and $\\sigma _2 \\in \\mathcal {S}_2$ .", "Thus, there vertices $v_i \\in \\sigma _i$ for $i=1,2$ such that $\\operatorname{supp}(v_1) \\cap \\operatorname{supp}(v_2) \\ne \\varnothing $ but $\\sigma \\setminus \\lbrace v_i\\rbrace $ is a face for $i=1$ and $i=2$ .", "But then $\\lbrace v_1,v_2\\rbrace $ is already a non-face and the claim follows.", "The theorem has some immediate consequences.", "Corollary 3.14 Let $\\mathcal {P}_1, \\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional anti-blocking polytopes with vertices in a given lattice.", "If $\\mathcal {P}_1, \\mathcal {P}_2$ have unimodular triangulations, then $\\mathcal {P}_1 - \\mathcal {P}_2$ and $\\Gamma (\\mathcal {P}_1,\\mathcal {P}_2) =\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ also have unimodular triangulations.", "By Theorem REF and the Cayley trick, $\\mathcal {P}_1 - \\mathcal {P}_2$ has a mixed subdivision into Cartesian products of unimodular simplices.", "Products of unimodular simplices are 2-level and, for example by [37], have unimodular triangulations.", "The polytope $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ inherits a triangulation from the upper or lower hull of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ , which has a unimodular triangulation by Theorem REF .", "Corollary 3.15 Let $\\mathbf {G}$ be a perfect double graph.", "Then $\\mathcal {P}_{\\mathbf {G}}$ , $\\mathcal {P}_{G_+} -\\mathcal {P}_{G_-}$ , and $\\Gamma (\\mathcal {P}_{G_+},\\mathcal {P}_{G_-})$ have regular unimodular triangulations.", "By Theorem REF , both polytopes $\\mathcal {P}_{G_+}$ and $\\mathcal {P}_{G_-}$ are 2-level and by [37] have unimodular triangulations.", "The result now follows from the previous corollary.", "Lattice points and volume Lemma REF directly implies a formula for the (normalized) volume of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ in terms of the volumes of the anti-blocking polytopes $\\mathcal {P}_1,\\mathcal {P}_2$ .", "Corollary 3.16 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $\\operatorname{vol}(\\mathcal {P}_1 - \\mathcal {P}_2) \\ = \\ \\sum _{J \\subseteq [n]} \\operatorname{vol}(\\mathcal {P}_1|_J)\\operatorname{vol}(\\mathcal {P}_2|_{J^c}).$ If $\\mathcal {P}_1$ and $\\mathcal {P}_2$ have unimodular triangulations with respect to a lattice $\\Lambda $ , then the normalized volume of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ with respect to the affine lattice $\\Lambda \\times (2\\mathbb {Z}+1)$ is $\\operatorname{Vol}({\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}) \\ = \\ \\sum _{J \\subseteq [n]}\\operatorname{Vol}(\\mathcal {P}_1|_J)\\operatorname{Vol}(\\mathcal {P}_2|_{J^c}).$ Both claims follow from Lemma REF .", "For the second statement, note that Theorem REF yields that ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ has a unimodular triangulation and hence its normalized volume is the number of simplices of maximal dimension, which is the number in the right-hand side.", "If $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ are rational anti-blocking polytopes, then so are ${2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2}$ and $\\mathcal {P}_1 - \\mathcal {P}_2$ .", "Our next goal is to determine their Ehrhart quasi-polynomials for a particular interesting case.", "We briefly recall the basics of Ehrhart theory; for more see, for example, [2], [3].", "If $\\mathcal {P}\\subset \\mathbb {R}^n$ is a $d$ -dimensional polytope with rational vertex coordinates, then the function $\\mathrm {Ehr}_\\mathcal {P}(k) := |k\\mathcal {P}\\cap \\mathbb {Z}^n|$ agrees with a quasi-polynomial of degree $d$ .", "We will identify $\\mathrm {Ehr}_\\mathcal {P}(k)$ with this quasi-polynomial, called the Ehrhart quasi-polynomial.", "If $\\mathcal {P}$ has its vertices in $\\mathbb {Z}^n$ , then $\\mathrm {Ehr}_\\mathcal {P}(k)$ is a polynomial of degree $d$ .", "If $\\mathcal {P}$ is full-dimensional, then the leading coefficient of $\\mathrm {Ehr}_\\mathcal {P}(k)$ is $\\operatorname{vol}(\\mathcal {P})$ .", "We will need the following fundamental result of Ehrhart theory.", "Theorem 3.17 (Ehrhart–Macdonald theorem) Let $\\mathcal {P}\\subset \\mathbb {R}^n$ be a rational polytope of dimension $d$ , then $(-1)^d \\mathrm {Ehr}_{\\mathcal {P}}(-k) \\ = \\ | \\operatorname{relint}(k\\mathcal {P}) \\cap \\mathbb {Z}^n|.$ We call an anti-blocking polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ dual integral if $A(\\mathcal {P})$ has all vertices in $\\mathbb {Z}^n$ .", "By Theorem REF , this means that there are $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {Z}_{\\ge 0}^n$ such that $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for }i=1,\\dots ,s \\rbrace .$ Corollary 3.18 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional rational anti-blocking polytopes.", "If $\\mathcal {P}_1$ is dual integral, then $\\mathrm {Ehr}_{\\mathcal {P}_1 - \\mathcal {P}_2}(k) \\ = \\ \\sum _{J \\subseteq [n]}(-1)^{|J|}\\mathrm {Ehr}_{\\mathcal {P}_1|_J}(-k-1) \\mathrm {Ehr}_{\\mathcal {P}_2|_{J^c}}(k).$ The Corollary is simply deduced from Theorem REF and the following stronger assertion.", "For a set $S \\subset \\mathbb {R}^n$ , let us write $E(S) := |S \\cap \\mathbb {Z}^n|$ .", "Theorem 3.19 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional rational anti-blocking polytopes and assume that $\\mathcal {P}_1$ is dual integral.", "For any $a,b \\in \\mathbb {Z}_{>0}$ $E(a\\mathcal {P}_1 - b\\mathcal {P}_2) \\ = \\ | (a\\mathcal {P}_1 - b\\mathcal {P}_2) \\cap \\mathbb {Z}^n | \\ = \\ \\sum _{J \\subseteq [n]} E(\\operatorname{relint}((a+1)\\mathcal {P}_1))\\cdot E(b\\mathcal {P}_2).$ It follows from Lemma REF that for any $a,b \\in \\mathbb {Z}_{>0}$ , $a\\mathcal {P}_1 - b\\mathcal {P}_2 \\ = \\ \\bigcup _{J \\subseteq [n]} (a\\mathcal {P}_1|_J -b\\mathcal {P}_2|_{J^c}).$ For $J \\subseteq [n]$ , the cell $ a\\mathcal {P}_1|_J - b\\mathcal {P}_2|_{J^c}$ is contained in the orthant $\\mathbb {R}_{\\ge 0}^J \\times \\mathbb {R}_{\\le 0}^{J^c}$ .", "It is easy to see that $\\mathbb {Z}^n \\ = \\ \\biguplus _{J \\subseteq [n]} \\mathbb {Z}^J_{> 0} \\times \\mathbb {Z}^{J^c}_{\\le 0}$ is a partition and for each $J \\subseteq [n]$ $(a\\mathcal {P}_1 - b\\mathcal {P}_2) \\cap (\\mathbb {Z}^J_{> 0} \\times \\mathbb {Z}^{J^c}_{\\le 0})\\ = \\ (a\\mathcal {P}_1|_J - b\\mathcal {P}_2|_{J^c}) \\cap (\\mathbb {Z}^J_{> 0} \\times \\mathbb {Z}^{J^c}_{\\le 0})\\ = \\ (a\\mathcal {P}_1|_J \\cap \\mathbb {Z}_{>0}^J) - (b\\mathcal {P}_2|_{J^c} \\cap \\mathbb {Z}^{J^c}).$ If $\\mathcal {P}_1$ is dual integral, then $\\mathcal {P}_1|_J$ is dual integral.", "Thus, for a fixed $J$ , there are $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {Z}_{\\ge 0}^{J}$ such that $(a\\mathcal {P}_1|_J \\cap \\mathbb {Z}^J_{>0})&\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {Z}^J : \\mathbf {x}> 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le a \\rbrace \\\\&\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {Z}^J : \\mathbf {x}> 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle < a+1 \\rbrace \\ = \\ \\operatorname{relint}((a+1) \\mathcal {P}_1|_J) \\cap \\mathbb {Z}^J.$ This proves the result.", "Clearly, it would be desirable to apply Corollary REF to the case that $\\mathcal {P}_1$ is a lattice polytope as well as dual integral.", "Proposition 3.20 Let $\\mathcal {P}\\subset \\mathbb {R}^n$ be a full-dimensional dual-integral anti-blocking polytope with vertices in $\\mathbb {Z}^n$ .", "Then $\\mathcal {P}= \\mathcal {P}_G$ for some perfect graph $G$ .", "Let $\\mathcal {P}$ be given by $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for} i=1,\\dots ,s \\rbrace $ for some $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {Z}_{\\ge 0}^n$ .", "Since $\\mathcal {P}$ is full-dimensional and a lattice polytope, it follows that $\\mathbf {e}_1,\\dots ,\\mathbf {e}_n \\in \\mathcal {P}$ and for any $1 \\le j \\le s$ we compute $0 \\ \\le \\ \\langle {\\mathbf {d}_j, \\mathbf {e}_i}\\rangle \\ \\le \\ 1$ for all $i$ and since the $\\mathbf {d}_j$ are integer vectors, it follows that $\\mathbf {d}_j = \\mathbf {1}_{C_j}$ for some $C_j \\subset [n]$ .", "Consequently, the vertices of $\\mathcal {P}$ are in $\\lbrace 0,1\\rbrace ^n$ and $\\mathcal {P}$ is 2-level.", "By Proposition REF , $\\mathcal {P}= \\mathcal {P}_G$ for some perfect graph $G$ .", "This severely limits the applicability of Corollary REF to lattice anti-blocking polytopes.", "On the other hand, we do not know of many results regarding the Ehrhart polynomials or even volumes of stable set polytopes of perfect graphs; see also the next section.", "Theorem 3.21 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional rational anti-blocking polytopes such that $\\mathcal {P}_1$ is dual integral.", "Then for $\\mathcal {P}:= {2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2}$ $\\mathrm {Ehr}_\\mathcal {P}(k) \\ = \\ | k \\mathcal {P}\\cap \\mathbb {Z}^{n+1}| \\ = \\ \\sum _{J \\subseteq [n]} (-1)^{|J|}\\sum _{s=-k}^k\\mathrm {Ehr}_{\\mathcal {P}_1|_J}( s-k-1 ) \\cdot \\mathrm {Ehr}_{\\mathcal {P}_2|_{J^c}}( k+s).$ For $k > 0$ , $k\\mathcal {P}\\ = \\ \\operatorname{conv}( 2k\\mathcal {P}_1 \\times \\lbrace k\\rbrace \\cup -2k\\mathcal {P}_2 \\times \\lbrace -k\\rbrace ).$ In particular, if $(\\mathbf {p},t)$ is a lattice point in $k\\mathcal {P}$ , then $-k \\le t\\le k$ .", "For fixed $t$ , $\\lbrace \\mathbf {p}\\in \\mathbb {Z}^n : (\\mathbf {p},t) \\in k\\mathcal {P}\\rbrace \\ = \\ \\left((k-t)\\mathcal {P}_1 - (k+t)\\mathcal {P}_2\\right) \\cap \\mathbb {Z}^n.$ Theorems REF and REF then complete the proof.", "Triangulations and transfers If $\\mathbf {P}= \\mathbf {P}_\\circ = (P,\\preceq ,\\preceq )$ is induced by a single poset, then Corollaries REF and REF assure us that $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ are 2-level and [37] implies that both polytopes have unimodular triangulations with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ .", "In this section we give explicit triangulations of the double chain polytope $\\mathcal {C}({\\mathbf {P}})$ and, in the compatible case, of the double order polytope $\\mathcal {O}(\\mathbf {P})$ .", "These triangulations will be regular, unimodular, and flag.", "To that end, we will generalize Stanley's approach [34] from poset polytopes to double poset polytopes.", "We put the triangulation to good use and explicitly compute the Ehrhart polynomial and the volume of $\\mathcal {C}({\\mathbf {P}})$ and, in case that $\\mathbf {P}$ is compatible, of $\\mathcal {O}(\\mathbf {P})$ .", "Triangulations of double poset polytopes Recall from the introduction that for a poset $(P,\\preceq )$ , the order polytope $\\mathcal {O}(P)$ parametrizes all order preserving maps $f : P\\rightarrow [0,1]$ .", "Any $f \\in \\mathcal {O}(P)$ induces a partial order $P_f =(P,\\preceq _f)$ by $a \\prec _f b$ if $a \\prec b$ or, when $a,b$ are incomparable, if $f(a) < f(b)$ .", "Clearly, $\\preceq _f$ refines $\\preceq $ and hence $\\mathcal {O}(P_f) \\subseteq \\mathcal {O}(P)$ .", "Since filters in $P_f$ are filters in $P$ , $\\mathcal {O}(P_f)$ is a vertex-induced subpolytope of $\\mathcal {O}(P)$ .", "If $f$ is generic, that is, $f(a) \\ne f(b)$ for all $a \\ne b$ , then $\\preceq _f$ is a total order and $\\mathcal {O}(P_f)$ is a unimodular simplex of dimension $|P|$ .", "Stanley showed that the collection of all these simplices constitute a unimodular triangulation of $\\mathcal {O}(P)$ .", "More precisely, this canonical triangulation of $\\mathcal {O}(P)$ is given by the order complex $\\Delta (\\mathcal {J}(P))$ of $\\mathcal {J}(P)$ , i.e., the collection of chains in the Birkhoff lattice of $P$ ordered by inclusion.", "Since a collection of filters $\\mathsf {J}_0,\\dots ,\\mathsf {J}_k$ is not a chain if and only if $\\mathsf {J}_i\\lnot \\subseteq \\mathsf {J}_j$ and $\\mathsf {J}_j \\lnot \\subseteq \\mathsf {J}_i$ for some $0\\le i,j \\le k$ , the canonical triangulation is flag.", "Stanley [34] elegantly transferred the canonical triangulation of $\\mathcal {O}(P)$ to $\\mathcal {C}({P})$ in the following sense.", "Define the transfer map $\\phi _P: \\mathcal {O}(P) \\rightarrow \\mathcal {C}({P})$ by $(\\phi _Pf)(b) \\ := \\ \\min \\lbrace f(b) - f(a) : a \\prec b \\rbrace ,$ for $f \\in \\mathcal {O}(P)$ and $b \\in P$ .", "This is a piecewise linear map and the domains of linearity are exactly the full-dimensional simplices $\\mathcal {O}(P_f)$ for generic $f$ .", "In particular, $\\phi _P(\\mathbf {1}_\\mathsf {J}) =\\mathbf {1}_{\\min (\\mathsf {J})}$ for any filter $\\mathsf {J}\\subseteq P$ , which shows that $\\phi _P$ maps $\\mathcal {O}(P)$ into $\\mathcal {C}({P})$ .", "To show that $\\phi _P$ is a PL homeomorphism of the two polytopes, Stanley gives an explicit inverse $\\psi _P: \\mathcal {C}({P}) \\rightarrow \\mathcal {O}(P)$ by $(\\psi _Pg)(b) \\ := \\ \\max \\lbrace g(a_0) + \\cdots + g(a_{k-1}) +g(a_k) : a_0 \\prec \\cdots \\prec a_{k-1} \\prec a_k \\preceq b \\rbrace ,$ for any $g \\in \\mathcal {C}({P})$ .", "Note that our definition of $\\psi _P$ differs from that in [34] in that we do not require that the chain has to end in $b$ .", "This will be important later.", "It can be easily checked that $\\psi _P$ is an inverse to $\\phi _P$ .", "Hence, the simplices $\\operatorname{conv}( \\mathbf {1}_{\\min (\\mathsf {J}_0)}, \\dots , \\mathbf {1}_{\\min (\\mathsf {J}_k)} ) \\quad \\text{for } \\quad \\lbrace \\mathsf {J}_0 \\subseteq \\cdots \\subseteq \\mathsf {J}_k \\rbrace \\in \\Delta (\\mathcal {J}(P))$ constitute a flag triangulation of $\\mathcal {C}({P})$ .", "We will follow the same approach as Stanley but, curiously, it will be simpler to start with a triangulation of $\\mathcal {C}({\\mathbf {P}})$ .", "Recall from Section REF that a pair of chains $C = C_+ \\uplus C_-$ with $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ is non-interfering if $\\min (\\mathsf {J}_+)\\cap \\min (\\mathsf {J}_-) = \\varnothing $ for any $\\mathsf {J}_\\sigma \\in C_\\sigma $ for $\\sigma = \\pm $ .", "Corollary 4.1 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then a regular triangulation of $\\mathcal {C}({\\mathbf {P}})$ is given as follows: The $(k-1)$ -dimensional simplices are in bijection to non-interfering pairs of chains $C = C_+ \\uplus C_+ \\subseteq \\mathcal {J}(\\mathbf {P})$ with $|C| = |C_+| +|C_-| = k$ .", "Moreover, the triangulation is regular, unimodular (with respect to $\\mathbb {A}$ ), and flag.", "The canonical triangulation of $\\mathcal {C}({P_\\sigma })$ is regular, unimodular, and flag for $\\sigma =\\pm $ .", "As described above, its $(l_\\sigma -1)$ -simplices are in bijection to chains $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ of length $|C_\\sigma | = l_\\sigma $ .", "More precisely, the simplex corresponding to $C_\\sigma $ is given by $F(C_\\sigma ) = \\operatorname{conv}( \\mathbf {1}_{\\min (\\mathsf {J}_\\sigma )} : \\mathsf {J}_\\sigma \\in C_\\sigma ).$ By Theorem REF applied to $\\mathcal {C}({\\mathbf {P}}) ={2\\mathcal {C}({P_+}) \\boxminus 2\\mathcal {C}({P_-})}$ , it follows that a unimodular and flag triangulation is given by the joins $2F(C_+) * 2F(C_-)$ for all chains $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ such that $F(C_+)$ and $F(C_-)$ lie in complementary coordinate subspaces.", "This, however, is exactly the case when $\\min (\\mathsf {J}_+) \\cap \\min (\\mathsf {J}_-) = \\varnothing $ for all $\\mathsf {J}_\\sigma \\in C_\\sigma $ for $\\sigma =\\pm $ .", "Corollary REF gives a canonical triangulation that combinatorially can be described as a subcomplex of $\\Delta (\\mathcal {J}(\\mathbf {P})) =\\Delta (\\mathcal {J}(P_+)) * \\Delta (\\mathcal {J}(P_-))$ , called the non-interfering complex $\\Delta ^\\mathrm {ni}(\\mathbf {P}) \\ := \\ \\lbrace C : C = C_+ \\uplus C_- \\in \\Delta (\\mathcal {J}(\\mathbf {P})) ,\\, C \\text{ non-interfering} \\rbrace .$ Associating $\\Delta (\\mathcal {J}(P))$ to a poset $P$ is very natural and can be motivated, for example, from an algebraic-combinatorial approach to the order polynomial (cf. [3]).", "It would be very interesting to know if the association $\\mathbf {P}$ to $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ is equally natural from a purely combinatorial perspective.", "Given a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ , we define a piecewise linear map $\\Psi _{\\mathbf {P}} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ by $\\Psi _{\\mathbf {P}}(g) \\ := \\ \\psi _{P_+}(g) \\ - \\ \\psi _{P_-}(-g),$ for any $g \\in \\mathbb {R}^P$ .", "Here, we use that $\\psi _{}$ , as given in (REF ), is defined on all of $\\mathbb {R}^P$ with the following important property: For $g \\in \\mathbb {R}^P$ , let us write $g = g^+ - g^-$ , where $g^+,g^- \\in \\mathbb {R}_{\\ge 0}^P$ with disjoint supports.", "Then $\\psi _{P_\\sigma }(g) = \\psi _{P_\\sigma }(g^+)$ for $\\sigma = \\pm $ .", "Thus, $\\Psi _{\\mathbf {P}}(g) \\ = \\ \\psi _{P_+}(g^+) - \\psi _{P_-}(g^-),$ for any $g \\in \\mathbb {R}^P$ .", "In particular, $\\Psi _{\\mathbf {P}}$ takes $\\lambda \\mathcal {C}({P_+}) - \\mu \\mathcal {C}({P_-})$ into $\\lambda \\mathcal {O}(P_+) - \\mu \\mathcal {O}(P_-)$ for any $\\lambda ,\\mu \\ge 0$ .", "Indeed, for any pair of antichains $A_\\sigma \\subseteq P_\\sigma $ , first observe that $\\mathbf {1}_{A_+} - \\mathbf {1}_{A_-} =\\mathbf {1}_{A_+\\setminus A_-} - \\mathbf {1}_{A_-\\setminus A_+}$ .", "Hence, it suffices to assume that $A_+ \\cap A_- = \\varnothing $ .", "We compute $\\Psi _{\\mathbf {P}}(\\mathbf {1}_{A_+} - \\mathbf {1}_{A_-} ) \\ = \\ \\mathbf {1}_{\\mathsf {J}_+} -\\mathbf {1}_{\\mathsf {J}_-},$ where for $\\sigma = \\pm $ , $\\mathsf {J}_\\sigma \\subseteq P_\\sigma $ is the filter generated by $A_\\sigma $ .", "If $\\mathbf {P}$ is a compatible double poset, then Corollary REF implies that $\\Psi _{\\mathbf {P}}$ is a surjection on vertex sets.", "Lemma 4.2 Let $\\mathbf {P}=(\\mathcal {P},\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then $\\Psi _{\\mathbf {P}} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ is a lattice-preserving piecewise linear isomorphism.", "It follows directly from (REF ) that $\\Psi _{\\mathbf {P}}$ is piecewise linear.", "To show that $\\Psi _{\\mathbf {P}}$ is an isomorphism, we explicitly construct for $f\\in \\mathbb {R}^P$ a $g \\in \\mathbb {R}^P$ such that $\\Psi _{\\mathbf {P}}(g)=f$ .", "Since $\\mathbf {P}$ is compatible, we can assume that $P= \\lbrace a_1,\\dots ,a_n\\rbrace $ such that $a_i \\prec _+ a_j$ or $a_i \\prec _- a_j$ implies $i < j$ .", "It follows from (REF ) that $\\Psi _{\\mathbf {P}}(g^{\\prime })(a_1) = g^{\\prime }(a_1)$ for any $g^{\\prime } \\in \\mathbb {R}^P$ and hence, we can set $g(a_1) := f(a_1)$ .", "Now assume that $g$ is already defined on $D_k := \\lbrace a_1,\\dots , a_k \\rbrace $ for some $k$ .", "For $g^{\\prime }\\in \\mathbb {R}^P$ observe that $\\psi _{P_+}(g^{\\prime })(a_{k+1}) \\ = \\ \\max (g^{\\prime }(a_{k+1}),0) + r$ where $r=0$ or $r = \\psi _{P_+}(g^{\\prime })(a_i)$ for some $i \\le k$ .", "Analogously, $\\psi _{P_-}(-g^{\\prime })(a_{k+1}) \\ = \\ \\max (-g^{\\prime }(a_{k+1}),0) + s$ where $s=0$ or $s = \\psi _{P_-}(-g^{\\prime })(a_j)$ for some $j \\le k$ .", "Thus, we set $g(a_{k+1})\\ := \\ f(a_{k+1}) - r + s$ This uniquely determines $g$ by induction on $k$ .", "To prove that $\\Psi _{\\mathbf {P}}$ is lattice-preserving, observe that by (REF ) we have $\\Psi _{\\mathbf {P}}(\\mathbb {Z}^P)\\subseteq \\mathbb {Z}^P$ .", "Moreover, if $f=\\Psi _{\\mathbf {P}}(g)$ with $f\\in \\mathbb {Z}^P$ and the above construction shows that $g\\in \\mathbb {Z}^P$ .", "Hence, $\\Psi _{\\mathbf {P}}(\\mathbb {Z}^P)\\subseteq \\mathbb {Z}^P$ , which finishes the proof.", "Using the notation from (REF ) in Section REF , the lemma shows that $\\lbrace \\overline{F}(C) : C \\in \\ \\Delta ^\\mathrm {ni}(\\mathbf {P}) \\rbrace $ is a realization of the flag simplicial complex $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ by unimodular simplices inside $\\mathcal {O}(\\mathbf {P})$ .", "Using Gröbner bases in Section , we will show the following result.", "Theorem 4.3 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then the map $(g,t) \\mapsto (\\Psi _{\\mathbf {P}}(g),t)$ is a piecewise linear homeomorphism from $\\mathbb {R}^{P}\\times \\mathbb {R}$ to itself that preserves the lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ .", "In particular, it maps $\\mathcal {C}({\\mathbf {P}})$ to $\\mathcal {O}(\\mathbf {P})$ and hence $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ is a regular, unimodular, and flag triangulation of $\\mathcal {O}(\\mathbf {P})$ .", "By the previous lemma, (REF ) is a realization of $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ in $\\mathcal {O}(\\mathbf {P})$ without new vertices.", "Moreover, every maximal simplex contains the edge $e = \\operatorname{conv}\\lbrace (\\mathbf {0},1), (\\mathbf {0},-1)\\rbrace $ .", "Hence, it suffices to show that for every maximal simplex in $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ , the supporting hyperplane of every facet not containing $e$ is supporting for $\\mathcal {O}(\\mathbf {P})$ .", "Let $C = \\lbrace \\mathsf {J}_{+0} \\subset \\cdots \\subset \\mathsf {J}_{+k}, \\mathsf {J}_{-0}\\subset \\cdots \\subset \\mathsf {J}_{-l}\\rbrace $ be two maximal non-interfering chains.", "Set $A_{+i} := \\min (\\mathsf {J}_{+i})$ for $1 \\le i \\le k$ and $A_{-0},\\dots ,A_{-l}$ likewise.", "It follows that $P_1 = \\bigcup A_{+i}$ and $P_2 = \\bigcup A_{-j}$ give a partition of $P$ .", "In particular, since $C$ was maximal, we have that $\\lbrace a^+_{k-i-1}\\rbrace = A_{+i} \\setminus A_{+(i-1)}$ and $P_1 = \\lbrace a^+_1,\\dots ,a^+_k\\rbrace $ .", "In particular, if $a^+_s\\prec _+ a^+_t$ , then $s < t$ .", "The same argument yields $P_2 =\\lbrace a^-_1,\\dots ,a^-_l\\rbrace $ and the labelling is a linear extension of $(P_2,\\preceq _-)$ .", "We focus on $P_1$ ; the argument for $P_2$ is analogous.", "Pick the maximal chain $D$ in $(P_1,\\preceq _+)$ starting in $a^+_k$ .", "Then $A_{+i} \\cap D\\ne \\varnothing $ for all $i > 0$ and hence $\\lbrace (g,t) \\in \\mathbb {R}^{P_1}:\\langle {\\mathbf {1}_D,g}\\rangle = 1\\rbrace $ is the hyperplane for the maximal simplex in the triangulation of $\\mathcal {C}({P_1,\\preceq _+})$ corresponding to $A_{+0},\\dots ,A_{+k}$ and not containing the origin.", "Thus, one of the two hyperplanes supporting a facet of the simplex in $\\mathcal {C}({\\mathbf {P}})$ corresponding to $C$ is given by $H := \\lbrace (g,t) \\in \\mathbb {R}^P\\times \\mathbb {R}:\\ell (g,t) = 1 \\rbrace $ where $\\ell (g,t) = \\langle {\\mathbf {1}_D,g}\\rangle - t$ .", "Now, $\\Psi _{\\mathbf {P}}$ is linear on the simplex $C$ in $\\mathcal {C}({\\mathbf {P}})$ and can be easily inverted.", "Since $\\mathbf {P}$ is compatible, we can find a linear extension $\\sigma : P\\rightarrow \\lbrace 1,\\dots ,|P|\\rbrace $ that respects the constructed linear extensions on $P_1$ and $P_2$ .", "On the image of $C$ under $\\Psi _{\\mathbf {P}}$ , the inverse is given by the linear transformation $T :\\mathbb {R}^P\\times \\mathbb {R}\\rightarrow \\mathbb {R}^P\\times \\mathbb {R}$ with $T(f,t) = (f^{\\prime },t)$ and $f^{\\prime } : P\\rightarrow \\mathbb {R}$ is defined as follows.", "If $b \\in P_1$ , then by $f^{\\prime }(b) =f(b) - f(\\bar{b})$ , $\\bar{b} \\prec _+ b$ is a cover relation and $\\sigma (\\bar{b})$ is maximal.", "If $b \\in P_2$ , we choose $\\bar{b}$ covered by $b$ in with respect to $\\preceq _-$ .", "It can now be checked that $\\ell \\circ T = L_C$ for some alternating chain $C$ .", "Thus $H$ is supporting for $\\mathcal {O}(\\mathbf {P})$ and the map $\\Psi _{\\mathbf {P}}$ maps $\\mathcal {C}({\\mathbf {P}})$ onto $\\mathcal {O}(\\mathbf {P})$ .", "Theorem REF does not extend to the non-compatible case as the following example shows.", "Example 9 Consider the double poset $\\mathbf {P}= ([2],\\le ,\\ge )$ , that is, $P_+$ is the 2-chain $\\lbrace 1,2\\rbrace $ and $P_-$ is the opposite poset.", "Then $\\mathcal {C}({P_+})= \\mathcal {C}({P_-}) = T := \\lbrace x \\in \\mathbb {R}^2 : x \\ge 0, x_1 + x_2 \\le 1\\rbrace $ and $\\mathcal {C}({\\mathbf {P}})$ is a three-dimensional octahedron with volume $\\frac{16}{3}$ .", "Any triangulation of the octahedron has at least four simplices.", "In contrast, $\\mathcal {O}(P_-) = \\mathbf {1} - \\mathcal {O}(P_+)$ and hence $\\mathcal {O}(\\mathbf {P})$ is linearly isomorphic to a prism over a triangle with volume 4.", "Any triangulation of the prism has exactly 3 tetrahedra.", "Volumes and Ehrhart polynomials The canonical subdivision of $\\mathcal {O}(P)$ makes it easy to compute its volume.", "For a generic $f \\in \\mathcal {O}(P)$ , there is a unique linear extension $\\sigma : P\\rightarrow \\lbrace 1,2,\\dots ,d\\rbrace $ where $d := |P|$ such that $\\mathcal {O}(P_f) \\ = \\ \\lbrace h \\in \\mathbb {R}^P : 0 \\le h(\\sigma ^{-1}(1)) \\le \\cdots \\le h(\\sigma ^{-1}(d)) \\le 1 \\rbrace .$ In particular, the full-dimensional simplex $\\mathcal {O}(P_f)$ is unimodular relative to $\\mathbb {Z}^P\\subseteq \\mathbb {R}^P$ and has volume $\\operatorname{vol}(\\mathcal {O}(P_f)) =\\frac{1}{|P|!", "}$ .", "If we denote by $e(P)$ the number of linear extensions of $P$ , then Stanley [34] showed the following.", "Corollary 4.4 $\\operatorname{Vol}(\\mathcal {O}(P)) = |P|!", "\\cdot \\operatorname{vol}(\\mathcal {O}(P)) = e(P)$ .", "For the Ehrhart polynomial $\\mathrm {Ehr}_{\\mathcal {O}(P)}(n)$ of $\\mathcal {O}(P)$ it suffices to interpret the lattice points in $n \\mathcal {O}(P)$ for $n > 0$ .", "Every point in $n\\mathcal {O}(P) \\cap \\mathbb {Z}^P$ corresponds to an order preserving map $\\phi : P\\rightarrow [n+1]$ .", "Counting order preserving maps is classical [35]: the order polynomial $\\Omega _P(n)$ of $P$ counts the number of order preserving maps into $n$ -chains.", "The strict order polynomial $\\Omega ^\\circ _P(n)$ counts the number of strictly order preserving maps $f : P\\rightarrow [n]$ , that is, $f(a) < f(b)$ for $a \\prec b$ .", "The transfer map $\\phi _P$ as well as its inverse $\\psi _P$ (given in (REF ) and (REF ), respectively) both take lattice points to lattice points and hence, together with Theorem REF , yield the following result.", "Corollary 4.5 Let $P$ be a finite poset.", "Then for every $n > 0$ $\\Omega _P(n+1)& \\ = \\ \\mathrm {Ehr}_{\\mathcal {O}(P)}(n) \\ = \\ \\mathrm {Ehr}_{\\mathcal {C}({P})}(n)\\\\\\multicolumn{2}{l}{\\text{ and }}\\\\(-1)^{|P|}\\Omega ^\\circ _P(n-1) &\\ = \\ \\mathrm {Ehr}_{\\mathcal {O}(P)}(-n) \\ = \\ \\mathrm {Ehr}_{\\mathcal {C}({P})}(-n).$ In particular, $\\operatorname{vol}(\\mathcal {O}(P)) = \\operatorname{vol}(\\mathcal {C}({P}))$ .", "This is an interesting result as it implies that the number of linear extensions of a poset $P$ only depends on the comparability graph $G(P)$ .", "Theorem 4.6 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then $\\overline{\\mathcal {C}}(\\mathbf {P})$ is a lattice polytope with respect to $\\mathbb {Z}^P$ and $\\mathrm {Ehr}_{\\overline{\\mathcal {C}}(\\mathbf {P})}(n-1) \\ &= \\ \\sum _{P= P_1 \\uplus P_2}\\Omega ^\\circ _{(P_1,\\preceq _+)}(n-1) \\cdot \\Omega _{(P_2,\\preceq _-)}(n) \\text{ and }\\\\\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ &= \\ \\sum _{P= P_1 \\uplus P_2} \\binom{|P|}{|P_1|}e(P_1,\\preceq _+)\\cdot e(P_2,\\preceq _-).$ Since $\\mathcal {C}({P}) = \\mathcal {P}_{G(P)}$ is a dual integral anti-blocking polytope, the first identity follows from Corollary REF and Corollary REF .", "The second identity follows from Corollary REF and Corollary REF .", "Notice from Theorem REF we can also deduce a closed formula for the Ehrhart polynomial of $\\mathcal {C}({\\mathbf {P}})$ with respect to the lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ and, by substituting $\\frac{1}{2}k$ for $k$ , also with respect to the affine lattice $\\mathbb {A}$ .", "These formulas are not very enlightening and instead we record the normalized volume.", "Note that the minimal Euclidean volume of a full-dimensional simplex with vertices in $\\mathbb {A}= \\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ is $\\frac{2^{|P|+1}}{(|P|+1)!", "}$ .", "Corollary 4.7 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then the normalized volume with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ is $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{P= P_1 \\uplus P_2}e(P_1,\\preceq _+) \\cdot e(P_2,\\preceq _-).$ We leave it to the reader to give direct combinatorial interpretations of the volume and the Ehrhart polynomials for double posets.", "It follows directly from (REF ) that $\\Psi _{P_{\\pm }} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ maps lattice points to lattice points.", "If $\\mathbf {P}$ is compatible, then the proof of Lemma REF asserts that $\\Psi _{P_{\\pm }}$ is in fact lattice preserving.", "Hence, we record an analog to Corollary REF .", "Corollary 4.8 If $\\mathbf {P}$ is a compatible double poset, then $\\mathcal {O}(\\mathbf {P})$ and $\\mathcal {C}({\\mathbf {P}})$ have the same Ehrhart polynomials and hence the same volumes.", "The formulas of Theorem REF are particularly simple when $\\mathbf {P}$ is special or anti-special.", "We illustrate these cases at some simple examples.", "Example 10 For the 'XW'-double poset we have $\\operatorname{Vol}(\\mathcal {O}(\\mathbf {P}_{XW})) \\ = \\ \\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}_{XW}}))\\ = \\ \\tfrac{6!", "}{2^6} \\operatorname{vol}(\\mathcal {C}({\\mathbf {P}_{XW}}))\\ = \\ 128$ and $\\operatorname{Vol}(\\overline{\\mathcal {O}}(\\mathbf {P}_{XW}))= \\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P}_{XW}))= 6!", "\\operatorname{vol}(\\overline{\\mathcal {C}}(\\mathbf {P}_{XW}))= 880$ .", "Example 11 As the following examples are all compatible, the given values also give the normalized volumes of the respective (reduced) double order polytopes.", "Let $\\mathbf {P}= ([d],\\le ,\\le )$ be the double chain on $d$ elements.", "Then $\\mathcal {C}({\\mathbf {P}})$ is a crosspolytope and $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) = 2^d$ and it follows from Vandermonde's identity that $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ d!", "\\operatorname{vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i}^2 \\ = \\ \\binom{2d}{d}.$ If $\\mathbf {P}$ is the double anti-chain on $d$ elements, then $\\mathcal {C}({\\mathbf {P}})$ is isomorphic to $[0,2]^{d} \\times [-1,1]$ and its normalized volume is $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\tfrac{(d+1)!", "}{2^{d+1}}\\operatorname{vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i} i!(d-i)!", "\\ = \\ (d+1)!.$ Likewise, $\\overline{\\mathcal {C}}(\\mathbf {P})$ is isomorphic to $[-1,1]^d$ , which can be decomposed into $2^d$ unit cubes.", "Consequently, its normalized volume is $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i}^2 i!(d-i)!", "\\ = \\ 2^dd!.$ Let $\\mathbf {P}$ be the double poset such that $P_+$ is the $d$ -chain and $P_-$ is the $d$ -antichain.", "Then $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{i=0}^d \\frac{d!}{i!", "}$ is the number of choices of ordered subsets of a $d$ -set.", "Moreover $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d\\binom{d}{i}^2 i!$ is the number of partial permutation matrices, i.e.", "$0/1$ -matrices of size $d$ with at most one nonzero entry per row and column.", "Indeed, such a matrix is uniquely identified by an $i$ -by-$i$ permutation matrix and a choice of $i$ rows and $i$ columns in which it is embedded.", "For the comb $C_n$ , the number of linear extensions is $e(C_n) =(2n-1)!", "!$ .", "Let $\\mathbf {P}$ be the double poset induced by the comb $C_n$ .", "Then an induction argument shows that $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ 4^n\\, n!.$ It would be nice to have a bijective proof of this equality.", "Let $\\mathbf {P}_\\circ = (P,\\preceq ,\\preceq )$ be a compatible double poset induced by a poset $(P,\\preceq )$ .", "By Corollary REF , the polytopes $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ have the same normalized volume.", "Since both polytopes are 2-level, this means that the number of maximal simplices in any pulling triangulation of $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ coincides.", "From Theorem REF , we know that $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ is the twisted prism over the valuation polytope associated to $P$ .", "On the other hand, we know from Corollary REF that $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ is linearly isomorphic to the Hansen polytope $\\mathcal {H}(\\overline{G(P)})$ .", "Moreover, $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ and $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ are both 2-level and it is enticing to conjecture that their normalized volumes also agree.", "Unfortunately, this is not the case.", "For the poset $P$ on 5 elements whose Hasse diagram is the letter 'X', any pulling triangulation of $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ has 324 simplices whereas for $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ pulling triangulations have 320 simplices.", "Gröbner bases and triangulations Double Hibi rings Hibi [19] associated to any finite poset $(P,\\preceq )$ a ring $\\mathcal {O}(P)]$ , nowadays called Hibi ring, that algebraically reflects many of the order-theoretic properties of $P$ .", "The ring $\\mathcal {O}(P)]$ is defined as the graded subring of the polynomial ring $S = t, s_a : a \\in P]$ generated by the elements $t \\cdot s^\\mathsf {J}$ , where $s^\\mathsf {J}\\ := \\ \\prod _{a \\in \\mathsf {J}} s_a,$ ranges over all filters $\\mathsf {J}\\subseteq P$ .", "For example, Hibi showed that $\\mathcal {O}(P)]$ is a normal Cohen–Macaulay domain of dimension $|P|+1$ and that $\\mathcal {O}(P)]$ is Gorenstein if and only if $P$ is a graded poset.", "By definition, Hibi rings are toric and hence they have the following quotient description.", "Let $R =x_\\mathsf {J}: \\mathsf {J}\\in \\mathcal {J}(P)]$ be the polynomial ring with variables indexed by filters and define the homogeneous ring map $\\phi : R \\rightarrow S$ by $\\phi (x_\\mathsf {J}) = t\\, s_\\mathsf {J}$ .", "Then $\\mathcal {O}(P)] \\cong R /\\mathrm {I}_{\\mathcal {O}(P)}$ where $\\mathrm {I}_{\\mathcal {O}(P)} = \\ker \\phi $ is a toric ideal.", "Hibi elegantly described a reduced Gröbner basis of $\\mathrm {I}_{\\mathcal {O}(P)}$ in terms of $\\mathcal {J}(P)$ .", "Fix a total order $\\le $ on the variables of $R$ such that $x_\\mathsf {J}\\le x_{\\mathsf {J}^{\\prime }}$ whenever $\\mathsf {J}\\subseteq \\mathsf {J}^{\\prime }$ and let $\\le _{rev}$ denote the induced reverse lexicographic order on $R$ .", "For $f \\in R$ , we write $\\textrm {in}_{\\le _{rev}}(f)$ for its leading term with respect to $\\le _{rev}$ and we will underline leading terms in what follows.", "Theorem 5.1 ([18]) Let $(P,\\preceq )$ be a finite poset.", "Then the collection $\\underline{x_{\\mathsf {J}} \\, x_{\\mathsf {J}^{\\prime }}} \\ - \\ x_{\\mathsf {J}\\cap \\mathsf {J}^{\\prime }} \\, x_{\\mathsf {J}\\cup \\mathsf {J}^{\\prime }} \\quad \\text{with }\\mathsf {J},\\mathsf {J}^{\\prime }\\in \\mathcal {J}(P)\\text{ incomparable}$ is a reduced Gröbner basis of $\\mathrm {I}_{\\mathcal {O}(P)}$ .", "The binomials (REF ) are called Hibi relations.", "In light of the previous sections, the natural question that we will address now is regarding an algebraic counterpart of the Hibi rings for double posets.", "For a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ , we define the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ as the subalgebra of the Laurent ring $\\hat{S} := t_-,t_+,s_a,s_a^{-1} : a \\in P]$ spanned by the elements $t_+\\cdot s^{\\mathsf {J}}$ for filters $\\mathsf {J}\\in \\mathcal {J}(P_+)$ and $t_- \\cdot (s^\\mathsf {J})^{-1}$ for filters $\\mathsf {J}\\in \\mathcal {J}(P_-)$ .", "This is the affine semigroup ring associated to $\\mathcal {O}(\\mathbf {P})$ with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ .", "Up to a translation by $(\\mathbf {0},1)$ and the lattice isomorphism $2\\mathbb {Z}^P\\times 2\\mathbb {Z}\\cong \\mathbb {Z}^P\\times \\mathbb {Z}$ , the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ is the affine semigroup ring of $\\operatorname{conv}\\bigl \\lbrace (\\mathcal {O}(P_+) \\times \\lbrace 1\\rbrace ) \\cup (-\\mathcal {O}(P_-) \\times \\lbrace 0\\rbrace )\\bigr \\rbrace ,$ with respect to the usual lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ .", "In particular, the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ is graded of Krull dimension $|P|+1$ .", "Moreover, since the double order polytope $\\mathcal {O}(\\mathbf {P})$ is reflexive by Corollary REF , it follows that $\\mathcal {O}(\\mathbf {P})]$ is a Gorenstein domain for any compatible double poset $\\mathbf {P}$ .", "The rings $\\mathcal {O}(\\mathbf {P})]$ as well as affine semigroup rings associated to the double chain polytopes $\\mathcal {C}({\\mathbf {P}})$ as treated at the end of Section REF were also considered by Hibi and Tsuchiya [24].", "Set $\\hat{R} := x_{\\mathsf {J}_+},x_{\\mathsf {J}_-} : \\mathsf {J}_+ \\in \\mathcal {J}(P_+), \\mathsf {J}_+ \\in \\mathcal {J}(P_+)]$ and define the monomial map $\\hat{\\phi }: \\hat{R} \\rightarrow \\hat{S}$ by $\\hat{\\phi }(x_{\\mathsf {J}_+}) \\ = \\ t_+\\, s^{\\mathsf {J}_+}\\quad \\text{ and } \\quad \\hat{\\phi }(x_{\\mathsf {J}_-}) \\ = \\ t_-\\, (s^{\\mathsf {J}_-})^{-1}.$ The corresponding toric ideal $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})} = \\ker \\hat{\\phi }$ is then generated by the binomials $\\underline{x_{\\mathsf {J}_{+1}} x_{\\mathsf {J}_{+2}} \\dots x_{\\mathsf {J}_{+k_+}} \\cdot x_{\\mathsf {J}_{-1}} x_{\\mathsf {J}_{-2}} \\dots x_{\\mathsf {J}_{-k_-}}}-x_{\\mathsf {J}^{\\prime }_{+1}} x_{\\mathsf {J}^{\\prime }_{+2}} \\dots x_{\\mathsf {J}^{\\prime }_{+k_+}} \\cdot x_{\\mathsf {J}^{\\prime }_{-1}} x_{\\mathsf {J}^{\\prime }_{-2}} \\dots x_{\\mathsf {J}^{\\prime }_{-k_-}},$ for filters $\\mathsf {J}_{+1}, \\dots , \\mathsf {J}_{+k_+}, \\mathsf {J}^{\\prime }_{+1}, \\dots ,\\mathsf {J}^{\\prime }_{+k_+} \\in \\mathcal {J}(P_+)$ and $\\mathsf {J}_{-1}, \\dots , \\mathsf {J}_{-k_-},\\mathsf {J}^{\\prime }_{-1}, \\dots , \\mathsf {J}^{\\prime }_{-k_-} \\in \\mathcal {J}(P_-)$ .", "Again, fix a total order $\\le $ on the variables of $\\hat{R}$ such that for $\\sigma = \\pm $ $x_{\\mathsf {J}_\\sigma } < x_{\\mathsf {J}^{\\prime }_\\sigma }$ for any filters $\\mathsf {J}_\\sigma ,\\mathsf {J}^{\\prime }_\\sigma \\in \\mathcal {J}(P_\\sigma )$ with $\\mathsf {J}_\\sigma \\subset \\mathsf {J}^{\\prime }_\\sigma $ , and $x_{\\mathsf {J}_+} < x_{\\mathsf {J}_-}$ for any filters $\\mathsf {J}_+ \\in \\mathcal {J}(P_+)$ and $\\mathsf {J}_- \\in \\mathcal {J}(P_-)$ , and denote by $\\le _{rev}$ the reverse lexicographic term order on $\\hat{R}$ induced by this order on the variables.", "Theorem 5.2 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then a Gröbner basis for $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ is given by the binomials $\\underline{x_{\\mathsf {J}_\\sigma }\\, x_{\\mathsf {J}_\\sigma ^{\\prime }}} \\ &- \\ x_{\\mathsf {J}_\\sigma \\cup \\mathsf {J}_\\sigma ^{\\prime }} \\, x_{\\mathsf {J}_\\sigma \\cap \\mathsf {J}_\\sigma ^{\\prime }}\\\\\\multicolumn{2}{l}{\\text{for incomparable filters $\\mathsf {J}_\\sigma ,\\mathsf {J}_\\sigma ^{\\prime }\\in \\mathcal {J}(P_\\sigma )$ and $\\sigma = \\pm $, and }}\\\\\\underline{x_{\\mathsf {J}_+}\\, x_{\\mathsf {J}_-}} \\ &- \\ x_{\\mathsf {J}_+\\setminus A} \\, x_{\\mathsf {J}_-\\setminus A}$ for filters $\\mathsf {J}_+ \\in \\mathcal {J}(P_+), \\mathsf {J}_- \\in \\mathcal {J}(P_-)$ such that $A := \\min (\\mathsf {J}_+)\\cap \\min (\\mathsf {J}_-) \\ne \\varnothing $ .", "It is clear that binomials of the form (REF ) and () are contained in $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ and hence it suffices to show that their leading terms generate $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ .", "For this, let us take a closer look at the combinatorics of $\\hat{\\phi }$ .", "Let $\\mathcal {G}$ be the collection of binomial given in (REF ) and () and let $f = \\underline{m_1} - m_2$ be an element of the form (REF ).", "By reducing $f$ by the binomial of (REF ), we can view $f$ as a quadruple $\\begin{aligned}\\mathsf {J}_{+1} \\subset \\mathsf {J}_{+2} \\subset \\cdots \\subset \\mathsf {J}_{+k_+} &\\quad \\qquad \\mathsf {J}_{-1} \\subset \\mathsf {J}_{+2} \\subset \\cdots \\subset \\mathsf {J}_{-k_-}\\\\\\mathsf {J}_{+1}^{\\prime } \\subset \\mathsf {J}_{+2}^{\\prime } \\subset \\cdots \\subset \\mathsf {J}_{+k_+}^{\\prime } &\\quad \\qquad \\mathsf {J}_{-1}^{\\prime } \\subset \\mathsf {J}_{+2}^{\\prime } \\subset \\cdots \\subset \\mathsf {J}_{-k_-}^{\\prime }.\\end{aligned}$ From the definition of $\\hat{\\phi }$ it follows that such a quadruple defines a binomial in $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ if and only if for any $q \\in P$ $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r} \\rbrace - \\max \\lbrace s : q \\notin \\mathsf {J}_{-s} \\rbrace \\ = \\ \\max \\lbrace r : q \\notin \\mathsf {J}_{+r}^{\\prime } \\rbrace - \\max \\lbrace s : q \\notin \\mathsf {J}_{-s}^{\\prime }\\rbrace .$ and we note the following implication.", "Lemma 5.3 Let the collection of filters in (REF ) correspond to a binomial $f \\in \\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ and let $q \\in P$ .", "Then there is some $1\\le i \\le k_+$ such that $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ if and only if there is some $1 \\le j \\le k_-$ such that $q \\in \\mathsf {J}_{-j} \\setminus \\mathsf {J}_{-j}^{\\prime }$ .", "If $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ , then $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r} \\rbrace < i$ and $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r}^{\\prime } \\rbrace \\ge i$ and (REF ) implies that $q \\in \\mathsf {J}_{-j} \\setminus \\mathsf {J}_{-j}^{\\prime }$ for some $j$ .", "The other direction is identical.", "We call $q \\in P$ moving if it satisfies one of the two equivalent conditions of Lemma REF .", "Let $f = \\underline{m_1} - m_2 \\in \\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ be a binomial represented by a collection of filters given by (REF ).", "If $k_- = 0$ or $k_+= 0$ , then the Hibi relations (REF ) for $P_-$ or $P_+$ together with Theorem REF yields the result.", "Thus, we assume that $k_-,k_+ > 0$ and we need to show that there are filters $\\mathsf {J}_{+i}$ and $\\mathsf {J}_{-j}$ such that $\\min (\\mathsf {J}_{+i}) \\cap \\min (\\mathsf {J}_{-j}) \\ne \\varnothing $ .", "Observe that there is at least one moving element.", "Indeed, $\\mathsf {J}_{+1}\\nsubseteq \\mathsf {J}_{+1}^{\\prime }$ and hence $\\mathsf {J}_{+1} \\setminus \\mathsf {J}_{+1}^{\\prime } \\ne \\varnothing $ .", "Otherwise, $x_{\\mathsf {J}_{+1}} <x_{\\mathsf {J}_{+1}^{\\prime }}$ and the reverse lexicographic term order $\\le _{rev}$ would not select $m_1$ as the lead term of $f$ .", "Among all moving elements, choose $q$ to be minimal with respect to $\\preceq _+$ and $\\preceq _-$ .", "Since $\\mathbf {P}$ is a compatible double poset, such a $q$ exists.", "But then, if $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ , then $q \\in \\min (\\mathsf {J}_{+i})$ .", "The same holds true for $\\mathsf {J}_{-j}$ and shows that $\\underline{m_1}$ is divisible by the leading term of a binomial of type ().", "Gröbner bases, faces, and triangulations In light of the regular and unimodular triangulation of $\\mathcal {O}(P)$ given in [34] (and recalled in Section REF ), the Hibi ring $\\mathcal {O}(P)]$ is exactly the affine semigroup ring associated to $\\mathcal {O}(P)$ .", "That is, $\\mathcal {O}(P)]$ is the standard graded $-algebra associated to thenormal affine semigroup$$\\lbrace (f,k) \\in \\mathbb {Z}^P\\times \\mathbb {Z}: k \\ge 0, f \\in k \\mathcal {O}(P)\\rbrace .$$$ For a lattice polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ , Sturmfels [36] described a beautiful relationship between regular triangulations of $\\mathcal {P}$ and radicals of initial ideals of the toric ideal $\\mathrm {I}_\\mathcal {P}$ .", "It follows from Theorem REF that $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is a squarefree ideal generated by quadratic monomials.", "Appealing to [36], this yields the following refinement of Theorem REF .", "Corollary 5.4 Let $\\mathbf {P}$ be a compatible double poset.", "Then $\\mathcal {O}(\\mathbf {P})$ has a regular triangulation whose underlying simplicial complex is exactly $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ .", "The initial ideal $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is already radical and Theorem 8.3 of [36] yields that $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is the Stanley-Reisner ideal of a regular triangulation of $\\mathcal {O}(\\mathbf {P})$ .", "Hence, a collection $C = C_1 \\uplus C_2 \\subseteq \\mathcal {J}(\\mathbf {P})$ forms a simplex in the triangulation of $\\mathcal {O}(\\mathbf {P})$ if and only if $\\prod _{\\mathsf {J}_+ \\in C_+} x_{\\mathsf {J}_+}\\prod _{\\mathsf {J}_- \\in C_-} x_{\\mathsf {J}_-} \\ \\notin \\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}).$ Translating the conditions given in Theorem REF , this is the case if and only if $C_\\sigma = C \\cap \\mathcal {J}(P_\\sigma )$ is a chain of filters for $\\sigma = \\pm $ and $C_+, C_-$ are non-interfering chains.", "This is exactly the definition of the flag complex $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ .", "Using the orbit-cone correspondence for affine toric varieties (see, for example, [5]), we can give an algebraic perspective on Theorem REF .", "We are in a particularly nice situation as the polytopes we consider have unimodular triangulations and hence the affine semigroup rings are generated in degree 1 by the vertices of the underlying polytope.", "Lemma 5.5 Let $V \\subset \\Lambda $ be a finite set of lattice points and $\\mathcal {P}=\\operatorname{conv}(V)$ the corresponding lattice polytope.", "If $\\mathrm {I}\\subseteq x_v : v\\in V]$ is the toric ideal of the homogenization $\\lbrace (v,1) : v\\in V \\rbrace \\subseteq \\Lambda \\times \\mathbb {Z}$ , then for any subset $U \\subseteq V$ , we have that $\\operatorname{conv}(U)$ is a face of $\\mathcal {P}$ with $\\operatorname{conv}(U)\\cap V = U$ if and only if $f(\\mathbf {1}_U) \\ = \\ 0 \\qquad \\text{ for all } f \\in \\mathrm {I}.$ Let $\\mathrm {L}\\subseteq \\mathcal {J}(\\mathbf {P})$ .", "Then for $\\sigma = \\pm $ and $\\mathsf {J}_\\sigma , \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathcal {J}(P_\\sigma )$ Lemma REF and (REF ) of Theorem REF states that $\\mathsf {J}_\\sigma , \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathrm {L}_\\sigma \\quad \\Longleftrightarrow \\quad \\mathsf {J}_\\sigma \\cup \\mathsf {J}_\\sigma ^{\\prime },\\mathsf {J}_\\sigma \\cap \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathrm {L}_\\sigma .$ That is, if and only if $\\mathrm {L}_\\sigma $ is an embedded Thus, $\\mathrm {L}_\\sigma $ is an embedded sublattice of $\\mathcal {J}(P_\\sigma )$ .", "The same reasoning shows that the conditions imposed by () are equivalent to those of Lemma REF .", "We can also use Sturmfels' result in the other direction to find Gröbner bases.", "For a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ we may define the subring $\\mathcal {C}({\\mathbf {P}})] \\subseteq \\hat{R}$ generated by the monomials $t_+ s^{\\min (\\mathsf {J}_+)}$ and $t_+ (s^{\\min (\\mathsf {J}_-)})^{-1}$ for filters $\\mathsf {J}_+ \\subseteq P_+$ and $\\mathsf {J}_- \\subseteq P_-$ .", "The corresponding toric ideal $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ is contained in the ring $T = x_{A_+},x_{A_-} ]$ , where $A_\\sigma $ ranges over all anti-chains in $P_\\sigma $ for $\\sigma = \\pm $ .", "Since $\\mathcal {O}(\\mathbf {P})$ is the stable set polytope of the perfect double graph $G(\\mathbf {P})$ , it follows from Corollary REF that $\\mathcal {C}({\\mathbf {P}})]$ is the normal affine semigroup ring associated to the lattice polytope $\\mathcal {C}({\\mathbf {P}})$ .", "To describe a Gröbner basis for, we introduce the following notation.", "For $\\sigma = \\pm $ and two antichains $A,A^{\\prime } \\subseteq P_\\sigma $ define $A \\sqcup A^{\\prime }\\ := \\ \\min (A \\cup A^{\\prime })$ and $A \\sqcap A^{\\prime } \\ := \\ (A \\cap A^{\\prime }) \\cup (\\max (A \\cup A^{\\prime }) \\setminus \\min (A \\cup A^{\\prime })).$ For a subset $S \\subseteq P$ and $\\sigma = \\pm $ , we write $\\langle S \\rangle _\\sigma := \\lbrace a \\in P: a\\succeq _\\sigma s \\text{ for some } s \\in S \\rbrace $ for the filter in $P_\\sigma $ generated by $S$ .", "Theorem 5.6 Let $\\mathbf {P}$ be a double poset.", "Then a Gröbner basis for $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ is given by the binomials $\\underline{x_A \\, x_{A^{\\prime }}} \\ &- \\ x_{A \\sqcup A^{\\prime }}\\, x_{A \\sqcap A^{\\prime }} \\qquad \\langle A \\rangle _\\sigma ,\\langle A \\rangle _\\sigma \\in \\mathcal {J}(P_\\sigma ) \\text{ incomparable}\\\\\\multicolumn{2}{l}{\\text{ for antichains $A,A^{\\prime } \\subset P_\\sigma $ for $\\sigma = \\pm $and }}\\\\\\underline{x_{A_+} \\, x_{A_-}} \\ &- \\ x_{A_+ \\setminus A_-}x_{A_-\\setminus A_+}\\qquad \\text{ for antichains } A_\\sigma \\subseteq P_\\sigma .$ It is easy to verify that the given binomials are contained in $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ .", "Moreover, the leading monomials are exactly the minimal non-faces of the unimodular triangulation of Corollary REF .", "The result now follows from Theorem 8.3 in [36].", "Remark 1 Reformulated in the language of double posets, Hibi, Matsuda, and Tsuchiya [22], [21], [23] computed related Gröbner bases of the toric ideals associated with the polytopes $\\Gamma (\\mathcal {O}(P_+),\\mathcal {O}(P_-))$ (in the compatible case), $\\Gamma (\\mathcal {C}({P_+}),\\mathcal {C}({P_-}))$ , and $\\Gamma (\\mathcal {O}(P_+),\\mathcal {C}({P_-}))$ for a double poset $\\mathbf {P}$ .", "See the paragraph before Corollary REF for notation." ], [ "Anti-blocking polytopes and Minkowski sums", "A polytope $\\mathcal {P}\\subset \\mathbb {R}_{\\ge 0}^n$ is called anti-blocking if $q \\in \\mathcal {P}\\ \\text{ and } \\ 0 \\le p \\le q \\quad \\Longrightarrow \\quad p \\in \\mathcal {P},$ where $p \\le q$ refers to componentwise order in $\\mathbb {R}^n$ .", "The notion of anti-blocking polyhedra was introduced by Fulkerson [12] in connection with min-max-relations in combinatorial optimization; our main reference for anti-blocking polytopes is Schrijver [31].", "In this section, we consider the Cayley sums ${\\mathcal {P} \\boxminus \\mathcal {Q}} \\ = \\ \\operatorname{conv}( \\mathcal {P}\\times \\lbrace 1\\rbrace \\cup (-\\mathcal {Q}) \\times \\lbrace -1\\rbrace ),$ where $\\mathcal {P}$ and $\\mathcal {Q}$ are anti-blocking polytopes.", "As before, we write $\\mathrm {tprism}(\\mathcal {P})$ for ${\\mathcal {P} \\boxminus \\mathcal {P}}$ .", "Our main source of examples will be the class of stable set polytopes: For a graph $G = (V,E)$ , a stable set is a subset $S \\subseteq V$ such that $\\binom{S}{2} \\cap E = \\varnothing $ .", "For simplicity, we will assume that $V = [n]$ and we write $\\mathbf {1}_S \\in \\lbrace 0,1\\rbrace ^n$ for the characteristic vector of a stable set $S$ .", "The stable set polytope of $G$ is the anti-blocking polytope $\\mathcal {P}_G \\ := \\ \\operatorname{conv}( \\mathbf {1}_S : S \\subseteq V \\text{ stable set} ) \\ \\subseteq \\ \\mathbb {R}^n.$ The class of perfect graphs is particularly interesting in this respect.", "Lovász [27] characterized perfect graphs in terms of their stable set polytopes and we use his characterization as a definition of perfect graphs.", "A clique of a graph $G = (V,E)$ is a subset $C\\subseteq V$ such that $\\binom{C}{2} \\subseteq E$ .", "For a vector $x \\in \\mathbb {R}^n$ and a subset $J \\subseteq [n]$ , we write $x(J) = \\sum _{j \\in J} x_j$ .", "Theorem 3.1 ([27]) A graph $G = ([n],E)$ is perfect if and only if $\\mathcal {P}_G \\ = \\ \\lbrace x \\in \\mathbb {R}^n : x \\ge 0, x(C) \\le 1 \\text{ for all cliques} C \\subseteq [n] \\rbrace .$ In this language, we can express the chain polytope of a poset $P$ as a stable set polytope: The comparability graph $G(P)$ of a poset $(P,\\preceq )$ is the undirected graph with vertex set $P$ and edge set $\\lbrace xy : x \\prec y \\text{ or } y \\prec x \\rbrace $ .", "Note that cliques in $G(P)$ are exactly the chains of $P$ .", "For a poset $P= ([n],\\preceq )$ the comparability graph $G(P)$ is perfect and hence $\\mathcal {C}({P}) \\ = \\ \\lbrace x \\in \\mathbb {R}^n : x \\ge 0, x(C) \\le 1 \\text{ for allchains } C \\subseteq [n] \\rbrace \\ = \\ \\mathcal {P}_{G(P)}.$ If $\\mathcal {P}\\subset \\mathbb {R}^n$ is an anti-blocking polytope, then there are $\\mathbf {c}_1,\\dots ,\\mathbf {c}_r \\in \\mathbb {R}_{\\ge 0}^n$ such that $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {c}_1,\\dots ,\\mathbf {c}_r\\rbrace ^{\\downarrow } \\ := \\ \\mathbb {R}^n_{\\ge 0} \\cap (\\operatorname{conv}(\\mathbf {c}_1,\\dots ,\\mathbf {c}_r) - \\mathbb {R}^n_{\\ge 0}).$ The unique minimal such set, denoted by $V^\\downarrow (P)$ , is given by the minimal elements of the vertex set of $\\mathcal {P}$ with respect to the partial order $\\le $ .", "It also follows from (REF ) and the Minkowski–Weyl theorem that there is a minimal collection $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {R}^n_{\\ge 0}$ such that $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for all }i=1,\\dots ,s\\rbrace $ For a polytope $\\mathcal {Q}\\subseteq \\mathbb {R}_{\\ge 0}^n$ , its associated anti-blocking polytope is the set $A({\\mathcal {Q}}) \\ := \\ \\lbrace \\mathbf {d}\\in \\mathbb {R}_{\\ge 0}^n : \\langle {\\mathbf {d},\\mathbf {x}}\\rangle \\le 1 \\text{ for all } \\mathbf {x}\\in \\mathcal {Q}\\rbrace .$ The following is the structure theorem for anti-blocking polytopes akin to the bipolar theorem for convex bodies.", "Theorem 3.2 ([31]) Let $\\mathcal {P}\\subset \\mathbb {R}^n$ be a full-dimensional anti-blocking polytope with $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {c}_1,\\dots ,\\mathbf {c}_r\\rbrace ^{\\downarrow } &\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0,\\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for all } i=1,\\dots ,s\\rbrace \\\\\\multicolumn{2}{l}{\\text{for some $\\mathbf {c}_1,\\dots ,\\mathbf {c}_r,\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {R}_{\\ge 0}^n$.", "Then}}\\\\A({\\mathcal {P}}) \\ = \\ \\lbrace \\mathbf {d}_1,\\dots ,\\mathbf {d}_s\\rbrace ^{\\downarrow } &\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {c}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for all } i=1,\\dots ,r\\rbrace .$ In particular, $A({A({\\mathcal {P}})}) = \\mathcal {P}$ .", "Before we come to our first result regarding Cayley- and Minkowski-sums of anti-blocking polytopes, we note the following fact.", "We write $V(\\mathcal {P})$ for the vertex set of a polytope $\\mathcal {P}$ .", "Proposition 3.3 Let $\\mathcal {P}_1, \\mathcal {P}_2$ be two full-dimensional anti-blocking polytopes.", "Then the vertices of $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ are exactly $(V(\\mathcal {P}_1) \\cup V(-\\mathcal {P}_2)) \\setminus \\lbrace \\mathbf {0}\\rbrace $ .", "For a polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ and a vector $\\mathbf {c}\\in \\mathbb {R}^n$ , we denote by $\\mathcal {P}^\\mathbf {c}$ the face of $\\mathcal {P}$ that maximizes the linear function $\\mathbf {x}\\mapsto \\langle {\\mathbf {c},\\mathbf {x}}\\rangle $ .", "It suffices to show that every $\\mathbf {v}\\in V(\\mathcal {P}_1) \\setminus \\lbrace \\mathbf {0}\\rbrace $ is a vertex of $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ .", "Let $\\mathbf {c}\\in \\mathbb {R}^n$ such that $\\mathcal {P}_1^\\mathbf {c}= \\lbrace \\mathbf {v}\\rbrace $ .", "Since $\\mathbf {v}\\ne 0$ , there is some $\\mathbf {d}\\in \\mathbb {R}^n_{\\ge 0}$ such that $\\langle {\\mathbf {d}, \\mathbf {u}_1}\\rangle \\le 1$ for all $\\mathbf {u}_1 \\in \\mathcal {P}_1$ and $\\langle {\\mathbf {d}, \\mathbf {v}}\\rangle = 1$ .", "Hence, for any $\\mu \\ge 0$ , $\\mathcal {P}_1^{\\mathbf {c}+ \\mu \\mathbf {d}} = \\lbrace v\\rbrace $ .", "Now, $\\langle {\\mathbf {d},-\\mathbf {u}_2}\\rangle \\le 0$ for all $\\mathbf {u}_2 \\in P_2$ .", "In particular, for $\\mu >0$ sufficiently large, $\\langle {\\mathbf {c}+ \\mu \\mathbf {d},\\mathbf {u}_2}\\rangle \\ \\le \\ \\langle {\\mathbf {c},\\mathbf {u}_2}\\rangle \\ < \\ \\mu +\\langle {\\mathbf {c},\\mathbf {v}}\\rangle \\ = \\ \\langle {\\mathbf {c}+ \\mu \\mathbf {d},\\mathbf {v}}\\rangle ,$ which shows that $\\mathbf {v}$ uniquely maximizes $\\langle {\\mathbf {c}+ \\mu \\mathbf {d},\\mathbf {u}}\\rangle $ over $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ .", "For $\\mathbf {d}\\in \\mathbb {R}^n_{\\ge 0}$ and $I \\subseteq [n]$ , we write ${\\mathbf {d}}^{[I]}$ for the vector with $({\\mathbf {d}}^{[I]})_j \\ = \\ {\\left\\lbrace \\begin{array}{ll}d_j & \\text{ for } j \\in I \\\\0 & \\text{ otherwise}.\\end{array}\\right.", "}$ Theorem 3.4 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $(\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle \\ = \\ \\operatorname{conv}( A({\\mathcal {P}_1}) \\cup -A({\\mathcal {P}_2}) ).$ Moreover, $({2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2})^\\triangle \\ = \\ {-A({\\mathcal {P}_2}) \\boxminus -A({\\mathcal {P}_1})}.$ Let us denote the right-hand side of the first equation by $\\mathcal {Q}$ .", "Note that $\\langle {\\mathbf {u}_1,-\\mathbf {v}_2}\\rangle \\le 0$ for $\\mathbf {u}_1 \\in A({P_1})$ and $\\mathbf {v}_2 \\in P_2$ .", "This shows that $\\langle {\\mathbf {u}_1,\\mathbf {v}}\\rangle \\le 1$ for all $\\mathbf {v}\\in P_1 - P_2$ .", "By symmetry, this yields $\\mathcal {Q}\\subseteq (\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle $ .", "For the converse, observe that every vertex of $\\mathcal {Q}$ is of the form ${\\mathbf {d}}^{[I]}$ with $\\mathbf {d}\\in V^\\downarrow (A({\\mathcal {P}_1}))\\cup -V^\\downarrow (A({\\mathcal {P}_2}))$ .", "It follows that $\\mathbf {z}\\in \\mathcal {Q}^\\triangle $ if and only if $\\langle {{\\mathbf {d}}^{[I]}, \\mathbf {z}}\\rangle \\le 1$ for all $ \\mathbf {d}\\in V^\\downarrow (A({\\mathcal {P}_1}))\\cup -V^\\downarrow (A({\\mathcal {P}_2}))$ and all $I \\subseteq [n]$ .", "For $\\mathbf {z}\\in \\mathcal {Q}^\\triangle $ write $\\mathbf {z}= \\mathbf {z}^1 - \\mathbf {z}^2$ with $\\mathbf {z}^1,\\mathbf {z}^2 \\ge 0$ and $\\operatorname{supp}(\\mathbf {z}^1) \\cap \\operatorname{supp}(\\mathbf {z}^2) = \\varnothing $ , where for any $\\mathbf {z}=(z_1,\\dots ,z_n)\\in \\mathbb {R}^n$ we set $\\operatorname{supp}(\\mathbf {z}):=\\lbrace i:z_i \\ne 0\\rbrace $ .", "We claim that $\\mathbf {z}^i \\in \\mathcal {P}_i$ for $i=1,2$ .", "Indeed, let $I = \\operatorname{supp}(\\mathbf {z}^1)$ .", "Then for any $\\mathbf {d}\\in V^\\downarrow (\\mathcal {P}_1)$ we have $\\langle {\\mathbf {d},\\mathbf {z}^1}\\rangle \\ = \\ \\langle {{\\mathbf {d}}^{[I]},\\mathbf {z}}\\rangle \\ \\le \\ 1$ and hence $\\mathbf {z}^1 \\in \\mathcal {P}_1$ .", "Applying the same argument to $\\mathbf {z}^2$ shows that $\\mathbf {z}\\in \\mathcal {P}_1 - \\mathcal {P}_2$ and hence $(\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle \\subseteq \\mathcal {Q}$ .", "For the second claim, note that any linear function on $\\mathbb {R}^n \\times \\mathbb {R}$ that maximizes on a vertical facet of ${2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2}$ is of the form $\\alpha _\\mathbf {d}\\langle {\\mathbf {d},\\mathbf {x}}\\rangle + \\delta _\\mathbf {d}t$ for $\\mathbf {d}$ a vertex of $(\\mathcal {P}_1 - \\mathcal {P}_2)^\\triangle $ and some $\\alpha _\\mathbf {d}, \\delta _\\mathbf {d}\\in \\mathbb {R}$ with $\\alpha _\\mathbf {d}> 0$ .", "By the first claim and Proposition REF , it follows that $\\mathbf {d}\\in (V(A({\\mathcal {P}_1})) \\cup V(-A({\\mathcal {P}_2}))) \\setminus \\lbrace \\mathbf {0}\\rbrace $ .", "If $\\mathbf {d}\\in V(A({\\mathcal {P}_1})) \\setminus \\lbrace \\mathbf {0}\\rbrace $ , then $\\langle {\\mathbf {d},\\mathbf {u}_1}\\rangle \\le 1$ is tight for $\\mathbf {u}_1 \\in \\mathcal {P}_1$ whereas $\\langle {\\mathbf {d},-\\mathbf {u}_2}\\rangle \\le 0$ is tight for $-\\mathbf {u}_2 \\in -\\mathcal {P}_2$ .", "Hence, $\\langle {\\mathbf {d},\\mathbf {x}}\\rangle - t \\ \\le \\ 1$ is the corresponding facet-defining halfspace.", "Similarly, if $-\\mathbf {d}\\in -V(A({\\mathcal {P}_1})) \\setminus \\lbrace \\mathbf {0}\\rbrace $ , then $\\langle {-\\mathbf {d},x}\\rangle + t \\ \\le \\ 1$ is facet-defining.", "Together with the two horizontal facets $\\langle {\\mathbf {0},\\mathbf {x}}\\rangle \\pm t \\le 1$ this yields an inequality description of $({-A({\\mathcal {P}_2}) \\boxminus -A({\\mathcal {P}_1})})^\\triangle $ , which proves the second claim.", "Theorem REF together with Theorem REF has a nice implication that was used in [30] in connection with Hansen polytopes.", "Corollary 3.5 For any full-dimensional anti-blocking polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ , the polytope ${\\mathcal {P} \\boxminus A({\\mathcal {P}})}$ is linearly isomorphic to its polar $({\\mathcal {P} \\boxminus A({\\mathcal {P}})})^\\triangle $ .", "In particular, ${\\mathcal {P} \\boxminus A({\\mathcal {P}})}$ is self-dual." ], [ "Stable set polytopes of double graphs and double chain polytopes", "A double graph is a triple $\\mathbf {G}= (V,E_+,E_-)$ consisting of a node set $V$ with two sets of edges $E_+,E_- \\subseteq \\binom{V}{2}$ .", "Again, we write $G_+ = (V, E_+)$ and $G_- = (V, E_-)$ to denote the two underlying ordinary graphs.", "The results of the preceding sections prompt the definition of stable set polytope of a double graph $\\mathcal {P}_{\\mathbf {G}} \\ := \\ {2\\mathcal {P}_{G_+} \\boxminus 2\\mathcal {P}_{G_-}}.$ For a double graph $\\mathbf {G}$ , define the complement graph as $\\overline{\\mathbf {G}} = (V,E_-^c,E_+^c)$ .", "Then Theorem REF implies the following relation.", "Corollary 3.6 Let $\\mathbf {G}$ be a perfect double graph.", "Then $\\mathcal {P}_{\\mathbf {G}}^\\triangle $ is linearly isomorphic to $\\mathcal {P}_{\\overline{\\mathbf {G}}}$ .", "We have $\\mathcal {P}_{\\mathbf {G}}^\\triangle =({2\\mathcal {P}_{G_+} \\boxminus 2\\mathcal {P}_{G_-}})^\\triangle \\ = \\ {-A({\\mathcal {P}_{G_-}}) \\boxminus -A({\\mathcal {P}_{G_+}})} \\ = \\ {-\\mathcal {P}_{\\overline{G}_-} \\boxminus -\\mathcal {P}_{\\overline{G}_+}} \\ \\cong \\ \\mathcal {P}_{\\overline{\\mathbf {G}}}.$ In particular, a double poset $\\mathbf {P}= (P,\\preceq _\\pm )$ gives rise to a double graph $\\mathbf {G}(\\mathbf {P}) = (G(P_+),G(P_-))$ and the double chain polytope of $\\mathbf {P}$ is simply $\\mathcal {C}({\\mathbf {P}}) = \\mathcal {P}_{\\mathbf {G}(\\mathbf {P})}$ , the double chain polytope of $\\mathbf {P}$ .", "Theorem REF directly gives a facet description of the double chain polytope.", "Note that compatibility is not required.", "Theorem 3.7 Let $\\mathbf {P}$ be a double poset and $\\mathcal {C}({\\mathbf {P}})$ its double chain polytope.", "Then $(g,t) \\in \\mathbb {R}^P\\times \\mathbb {R}$ is contained in $\\mathcal {C}({\\mathbf {P}})$ if and only if $\\sum _{a \\in C_+} g(a) - t \\ \\le \\ 1 \\quad \\text{ and } \\quad \\sum _{a \\in C_-} -g(a) + t \\ \\le \\ 1,$ where $C_+ \\subseteq P_+$ and $C_- \\subseteq P_-$ ranges of all chains.", "For the usual order- and chain polytope, Hibi and Li [20] showed that $\\mathcal {O}(P)$ has at most as many facets as $\\mathcal {C}({P})$ and equality holds if and only if $P$ does not contain the 5-element poset with Hasse diagram 'X'.", "This is different in the case of double poset polytopes.", "Corollary 3.8 Let $(P,\\preceq )$ be a poset.", "Then $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ have the same number of facets.", "Alternating chains in $\\mathbf {P}_\\circ $ are in bijection to twice the number of chains in $P$ .", "However, it is not true that $\\mathcal {O}(\\mathbf {P}_\\circ )$ is always combinatorially isomorphic to $\\mathcal {C}({\\mathbf {P}_\\circ })$ .", "Example 5 Let $P$ be the 5-element poset with Hasse diagram 'X'.", "Then the face vectors of $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ are $f(\\mathcal {O}(\\mathbf {P}_\\circ )) \\ &= \\ (16, 88, 204, 240, 144, 36) \\\\f(\\mathcal {C}({\\mathbf {P}_\\circ })) \\ &= \\ (16, 88, 222, 276, 162, 36).$ Hibi and Li [20] conjectured that $f(\\mathcal {O}(P))\\le f(\\mathcal {C}({P}))$ componentwise.", "Computations suggest that the same relation should hold for the double poset polytopes of induced double posets.", "Conjecture 1 Let $\\mathbf {P}= (P,\\preceq ,\\preceq )$ be a double poset induced by a poset $(P,\\preceq )$ .", "Then $f_i(\\mathcal {O}(\\mathbf {P})) \\ \\le \\ f_i(\\mathcal {C}({\\mathbf {P}}))$ for $0 \\le i \\le |P|$ .", "An extension of the conjecture to general compatible double posets fails, as the following example shows.", "Example 6 Let $\\mathbf {A}_n$ be an alternating chain of length $n$ , that is, $\\mathbf {P}$ is a double poset on elements $a_1,a_2,\\dots ,a_{n+1}$ with cover relations $a_1 \\ \\prec _+ \\ a_2 \\ \\prec _- \\ a_3 \\ \\prec _+ \\cdots $ It follows from Theorem REF that the number of facets of $\\mathcal {C}({\\mathbf {A}_n})$ is $3n+4$ .", "Since $\\mathbf {A}_n$ is compatible, then by Theorem REF the number of facets of $\\mathcal {O}(\\mathbf {A}_n)$ equals the number of alternating chains which is easily computed to be $\\binom{n+3}{2}+1$ .", "Thus, for $n \\ge 3$ , the alternating chains $\\mathbf {A}_n$ fail Conjecture REF for the number of facets.", "For $n=3$ , we explicitly compute $f(\\mathcal {O}(\\mathbf {A}_3)) \\ &= \\ ( 21, 70, 95, 60, 16) \\quad \\text{and} \\\\f(\\mathcal {C}({\\mathbf {A}_3})) \\ &= \\ (21, 67, 86, 51, 13).$ Every graph $G = (V,E)$ trivially gives rise to a double graph $\\mathbf {G}_\\circ =(V,E,E)$ .", "Thus, the Hansen polytope of a graph $G$ is the polytope $\\mathcal {H}(G) = \\mathcal {P}_{\\mathbf {G}_\\circ }$ .", "Theorem REF then yields a strengthening of the main result of Hansen [17].", "Note that for the complement graph $\\overline{G} = (V,E^c)$ , it follows that a subset $S\\subseteq V$ is a stable set of $G$ if and only if $S$ is a clique of $\\overline{G}$ and vice versa.", "Corollary 3.9 ([17]) Let $G$ be a perfect graph.", "Then $\\mathcal {H}(G)$ is 2-level and $\\mathcal {H}(G)^\\triangle $ is affinely isomorphic to $\\mathcal {H}(\\overline{G})$ .", "By Theorem REF and Theorem  REF $\\mathcal {H}(G)^\\triangle \\ = \\ {-A({\\mathcal {P}_G}) \\boxminus -A({\\mathcal {P}_G})} \\ = \\ {-\\mathcal {P}_{\\overline{G}} \\boxminus -\\mathcal {P}_{\\overline{G}}} \\ \\cong \\ \\mathcal {H}(\\overline{G}),$ which proves the second claim.", "A vertex of $\\mathcal {H}(G)^\\triangle $ is of the form $\\mathbf {d}= \\pm (-\\mathbf {1}_C,1)$ for some clique $C$ of $G$ .", "Thus, for any vertex $\\mathbf {v}= \\pm (2\\mathbf {1}_S,1) \\in \\mathcal {H}(G)$ , where $S$ is a stable set of $G$ , we compute $\\langle {\\mathbf {d},\\mathbf {v}}\\rangle = \\pm ( 1 - 2|S \\cap C|) = \\pm 1$ .", "Example 7 (Double chain polytopes of dimension-two posets) Following Example REF , let $\\pi _+,\\pi _-\\in \\mathbb {Z}^n$ be two integer sequences with associated posets $P_{\\pi _+}$ and $P_{\\pi _-}$ of order dimension two.", "Consider the double posets $\\mathbf {P}=(P_{\\pi _+},P_{\\pi _-})$ and $-\\mathbf {P}=(P_{-\\pi _-},P_{-\\pi _+})$ .", "We have $\\overline{\\mathbf {G}(\\mathbf {P})} \\ = \\ (\\overline{G(P_{\\pi _-})},\\overline{G(P_{\\pi _+})}) \\ = \\ (G(P_{-\\pi _-}),G(P_{-\\pi _+})) \\ = \\ \\mathbf {G}(-\\mathbf {P})$ and hence $\\mathcal {C}({\\mathbf {P}})^\\triangle \\ \\cong \\ \\mathcal {C}({-\\mathbf {P}})$ by Corollary REF .", "However, it is not necessarily true that $\\mathcal {O}(\\mathbf {P})^\\triangle \\cong \\mathcal {O}(-\\mathbf {P})$ , as can be checked for the double poset induced by the $X$ -poset; cf.", "Example REF .", "Example 8 (Double chain polytopes of plane posets) Let $\\mathbf {P}$ be a plane double poset.", "By the last example, the double chain polytope $\\mathcal {C}({\\mathbf {P}})$ is linearly equivalent to its polar $\\mathcal {C}({\\mathbf {P}})^\\triangle $ .", "Among the 2-level polytopes, independence polytopes of perfect graphs play a distinguished role.", "The following observation, due to Samuel Fiorini (personal communication), characterizes 2-level anti-blocking polytopes.", "Proposition 3.10 Let $\\mathcal {P}$ be a full-dimensional anti-blocking polytope.", "Then $\\mathcal {P}$ is 2-level if and only if $\\mathcal {P}$ is linearly isomorphic to $\\mathcal {P}_G$ for some perfect graph $G$ .", "The origin is a vertex of $\\mathcal {P}$ and, since $\\mathcal {P}$ is full-dimensional and anti-blocking, its neighbors are $ \\alpha _1 \\mathbf {e}_1, \\dots , \\alpha _1 \\mathbf {e}_n$ are for some $\\alpha _i > 0$ .", "After a linear transformation, we can assume that $\\alpha _1 = \\cdots = \\alpha _n = 1$ .", "Since $\\mathcal {P}$ is 2-level, $\\mathcal {P}=\\lbrace \\mathbf {x}\\in \\mathbb {R}_{\\ge 0}^n : \\langle {\\mathbf {d}_i, \\mathbf {x}}\\rangle \\le 1 \\text{ for } i=1,\\dots ,s\\rbrace $ where $\\mathbf {d}_i \\in \\lbrace 0,1\\rbrace ^n$ for all $i=1,\\dots ,s$ .", "Let $G = ([n],E)$ be the minimal graph with cliques $\\operatorname{supp}(\\mathbf {d}_i)$ for all $i=1,\\dots ,s$ .", "That is, $E = \\bigcup _i \\binom{\\operatorname{supp}(\\mathbf {d}_i)}{2}$ .", "We have $\\mathcal {P}_G \\subseteq \\mathcal {P}$ .", "Conversely, any vertex of $\\mathcal {P}$ is of the form $\\mathbf {1}_S$ for some $S\\subseteq [n]$ and $\\langle {\\mathbf {d}_i,\\mathbf {1}_S}\\rangle = |\\operatorname{supp}(\\mathbf {d}_i) \\cap S| \\le 1$ shows that $\\mathcal {P}\\subseteq \\mathcal {P}_G$ .", "This implies a characterization of the 2-level polytopes among Cayley sums of anti-blocking polytopes.", "Corollary 3.11 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $\\mathcal {P}= {\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ is 2-level if and only if $\\mathcal {P}$ is affinely isomorphic to $\\mathcal {H}(G)$ for some perfect graph $G$ .", "Sufficiency is Hansen's result (Corollary REF ).", "For necessity, observe that $\\mathcal {P}_1$ and $\\mathcal {P}_2$ are faces and hence have to be 2-level.", "By the proof of Proposition REF , we may assume that $\\mathcal {P}_1 = \\mathcal {P}_{G_1}$ for some perfect graph $G_1$ and $\\mathcal {P}_2 = A\\mathcal {P}_{G_2}$ for some perfect $G_2$ and a diagonal matrix $A\\in \\mathbb {R}^{n\\times n}$ with diagonal entries $a_i>0$ for $i\\in [n]$ .", "We will proceed in two steps: We first prove that $A$ must be the identity matrix and then show that $G_1=G_2$ .", "For every $i\\in [n]$ the inequality $x_i\\ge 0$ is facet-defining for $\\mathcal {P}_1$ .", "Hence it induces a facet-defining inequality for $\\mathcal {P}$ , which must be of the form $\\ell _i\\ := \\ -b_ix_i+t \\ \\le \\ 1$ for some $b_i>0$ , where $t$ denotes the last coordinate in $\\mathbb {R}^{n+1}$ .", "Observe that $\\ell _i$ takes the values 1 and $1-b_i$ on the vertices $\\lbrace \\mathbf {0},\\mathbf {e}_i\\rbrace \\times \\lbrace 1\\rbrace $ of the face $\\mathcal {P}_1\\times \\lbrace 1\\rbrace $ .", "On the other hand, on $\\lbrace \\mathbf {0},-a_i\\mathbf {e}_i\\rbrace \\times \\lbrace -1\\rbrace \\subset -\\mathcal {P}_2\\times \\lbrace -1\\rbrace $ , the values are $-1$ and $-1+a_ib_i$ .", "Now 2-levelness implies $a_i=1$ and $b_i=2$ .", "It now follows from Theorem REF that the facet-defining inequalities for $\\mathcal {P}$ are $\\begin{split}2\\mathbf {1}_{C_1}(\\mathbf {x})-t \\ &\\le \\ 1\\text{ and}\\\\-2\\mathbf {1}_{C_2}(\\mathbf {x})+t\\ &\\le \\ 1,\\end{split}$ where $C_1$ and $C_2$ are cliques in $G_1$ and $G_2$ , respectively.", "By 2-levelness each of these linear functions takes the values $-1$ and 1 on the vertices of $P$ .", "This easily implies that every clique in $G_1$ must be a clique in $G_2$ and conversely.", "Hence $G_1=G_2$ ." ], [ "Canonical Subdivisions", "We now turn to the canonical subdivisions of $\\mathcal {P}_1 - \\mathcal {P}_2$ and ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ for anti-blocking polytopes $\\mathcal {P}_1,\\mathcal {P}_2$ .", "A subdivision of $\\mathcal {P}= \\mathcal {P}_1 - \\mathcal {P}_2$ is a collection of polytopes $\\mathcal {Q}^1,\\dots ,\\mathcal {Q}^m \\subseteq \\mathcal {P}$ each of dimension $\\dim \\mathcal {P}$ such that $\\mathcal {P}= \\mathcal {Q}^1\\cup \\cdots \\cup \\mathcal {Q}^m$ and $\\mathcal {Q}^i \\cap \\mathcal {Q}^j$ is a face of both for all $i \\ne j$ .", "We call the subdivision mixed if each $\\mathcal {Q}^i$ is of the form $\\mathcal {Q}^i_1 - \\mathcal {Q}^i_2$ where $\\mathcal {Q}^i_j$ is a vertex-induced subpolytope of $\\mathcal {P}_j$ for $j=1,2$ .", "Finally, a mixed subdivision is exact if $\\dim \\mathcal {Q}^i = \\dim \\mathcal {Q}^i_1 + \\dim \\mathcal {Q}^i_2$ .", "That is, $\\mathcal {Q}^i$ is linearly isomorphic to the Cartesian product $\\mathcal {Q}^i_1 \\times \\mathcal {Q}^i_2$ .", "For a full-dimensional anti-blocking polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ , every index set $J \\subseteq [n]$ defines a distinct face $\\mathcal {P}|_J := \\lbrace x \\in \\mathcal {P}: x_j = 0 \\text{ for } j\\notin J \\rbrace $ .", "This is an anti-blocking polytope of dimension $|J|$ .", "For disjoint $I,J \\subseteq [n]$ , the polytopes $\\mathcal {P}_1|_I, \\mathcal {P}_2|_J$ lie in orthogonal subspaces and $\\mathcal {P}_1|_I - \\mathcal {P}_2|_J$ is in fact a Cartesian product.", "In this case, the Cayley sum ${\\mathcal {P}_1|_I \\boxminus \\mathcal {P}_2|_J}$ is called a join and denoted by $ \\mathcal {P}_1|_I * \\mathcal {P}_2|_J$ .", "As with the Cartesian product, the combinatorics of $ \\mathcal {P}_1|_I * \\mathcal {P}_2|_J$ is completely determined by the combinatorics of $ \\mathcal {P}_1|_I$ and $\\mathcal {P}_2|_J$ .", "Lemma 3.12 Let $\\mathcal {P}_1, \\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $\\mathcal {P}_1 - \\mathcal {P}_2$ has a regular exact mixed subdivision with cells $\\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ for all $J \\subseteq [n]$ .", "In particular, ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ has a regular subdivision into joins $\\mathcal {P}_1|_{J} * \\mathcal {P}_2|_{J^c}$ for all $J \\subseteq [n]$ .", "We call the subdivisions of Lemma REF the canonical subdivisions of $\\mathcal {P}_1 - \\mathcal {P}_2$ and ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ , respectively.", "By the Cayley trick [6], it is suffices to prove only the first claim.", "The subdivision of $\\mathcal {P}_1 - \\mathcal {P}_2$ is very easy to describe: Let us first note that the polytopes $\\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ for $J \\subseteq [n]$ only meet in faces.", "Hence, we only need to verify that they cover $\\mathcal {P}_1 - \\mathcal {P}_2$ .", "It suffices to show that for any point $\\mathbf {x}\\in \\mathcal {P}_1 - \\mathcal {P}_2$ with $x_i \\ne 0$ for all $i$ , there is a $J \\subseteq [n]$ with $\\mathbf {x}\\in \\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ .", "Let $\\mathbf {x}_1,\\mathbf {x}_2 \\in \\mathbb {R}_{\\ge 0}^n$ with $\\mathbf {x}= \\mathbf {x}_1 - \\mathbf {x}_2$ and $\\operatorname{supp}(\\mathbf {x}_1) \\cap \\operatorname{supp}(\\mathbf {x}_2) =\\varnothing $ .", "We claim that $\\mathbf {x}_i \\in \\mathcal {P}_i$ for $i=1,2$ .", "Indeed, if $\\mathbf {x}=\\mathbf {y}_1 - \\mathbf {y}_2$ for some $\\mathbf {y}_i \\in \\mathcal {P}_i$ , then $0 \\le \\mathbf {x}_i \\le \\mathbf {y}_i$ and $\\mathbf {x}_i \\in \\mathcal {P}_i$ by (REF ).", "In particular, $\\mathbf {x}_1 \\in \\mathcal {P}_1 |_J$ and $\\mathbf {x}_2 \\in \\mathcal {P}_2 |_{J^c}$ and therefore $\\mathbf {x}\\in \\mathcal {P}_1|_{J} - \\mathcal {P}_2|_{J^c}$ .", "To show regularity, let $\\omega : \\mathbb {R}^n \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ be the linear function such that $\\omega (\\mathbf {e}_i,0) = -\\omega (0,\\mathbf {e}_j) = 1$ for all $i,j = 1,\\dots ,n$ .", "Then $\\omega $ induces a mixed subdivision by picking for every point $\\mathbf {x}\\in \\mathcal {P}_1 - \\mathcal {P}_2$ , the unique cell $F_1 - F_2$ such that $\\mathbf {x}= \\mathbf {x}_1 - \\mathbf {x}_2$ with $\\mathbf {x}_i \\in \\operatorname{relint}F_i$ and $(\\mathbf {x}_1,\\mathbf {x}_2)$ minimizes $\\omega $ over the set $\\lbrace (\\mathbf {y}_1,\\mathbf {y}_2) \\in \\mathcal {P}_1 \\times \\mathcal {P}_2 : \\mathbf {x}= \\mathbf {y}_1 - \\mathbf {y}_2 \\rbrace ;$ see Section 9.2.2 of de Loera et al.", "[6] for more details.", "If $\\omega $ is not generic, one has to be careful as the minimizer is not necessarily unique but in our case, we observe that for any $\\mathbf {y}_i \\in \\mathcal {P}_i$ with $\\mathbf {x}= \\mathbf {y}_1 - \\mathbf {y}_2$ we have $\\omega (\\mathbf {y}_1,\\mathbf {y}_2) >\\omega (\\mathbf {x}_1,\\mathbf {x}_2)$ for all $(\\mathbf {y}_i,\\mathbf {y}_2)\\ne (\\mathbf {x}_1,\\mathbf {x}_2)$ with $(\\mathbf {x}_1,\\mathbf {x}_2)$ defined above.", "Figure: The canonical subdivision of 𝒫 1 -𝒫 2 \\mathcal {P}_1 - \\mathcal {P}_2 for two(random) anti-blocking polytopes 𝒫 1 ,𝒫 2 ⊂ℝ ≥0 3 \\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^3_{\\ge 0}.We define a triangulation of a polytope to be a subdivision into simplices without new vertices.", "For a polytope with vertices in an affine lattice $\\mathbb {A}$ , a triangulation is unimodular if each simplex is unimodular or, equivalently, has normalized volume $=1$ .", "A triangulation is flag if any minimal non-face is of dimension 1.", "This property implies that the underlying simplicial complex is completely determined by its graph.", "Theorem 3.13 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes with subdivisions $\\mathcal {S}_1$ and $\\mathcal {S}_2$ , respectively.", "For $J \\subseteq [n]$ , let $\\mathcal {S}_i|_J := \\lbrace S \\cap \\mathcal {P}_i|_J : S \\in \\mathcal {S}_i \\rbrace $ be the restriction of $\\mathcal {S}_i$ to $\\mathcal {P}_i|_J$ for $i=1,2$ .", "Then $\\mathcal {S}\\ := \\ \\bigcup _{J \\subseteq [n]} \\mathcal {S}_1|_J * \\mathcal {S}_2|_{J^c}$ is a subdivision of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ .", "In particular, [(i)] If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are regular, then $\\mathcal {S}$ is regular.", "If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are unimodular triangulations with respect to $\\Lambda $ , then $\\mathcal {S}$ is a unimodular triangulation with respect to the affine lattice $\\Lambda \\times (2\\mathbb {Z}+ 1)$ .", "If $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are flag, then $\\mathcal {S}$ is flag.", "Note that (iii) also holds if the triangulations use more lattice points than just the vertices.", "For the first claim, observe that $\\mathcal {S}_i|_J$ is a subdivision of the face $\\mathcal {P}_i|_J$ .", "By [6], $\\mathcal {S}_1|_J *\\mathcal {S}_2|_{J^c}$ is a subdivision of $\\mathcal {P}_1|_J * \\mathcal {P}_2|_{J^c}$ .", "Hence, $\\mathcal {S}$ is a refinement of the canonical subdivision of Lemma REF .", "If $\\mathcal {S}_i$ is a regular subdivision of $\\mathcal {P}_i$ , then there are weights $\\omega _i : V(\\mathcal {P}_i) \\rightarrow \\mathbb {R}$ for $i=1,2$ .", "By adding a constant weight to every vertex if necessary, we can assume that $\\omega _1(\\mathbf {v}_1) >0$ and $\\omega _2(\\mathbf {v}_2) < 0$ for all $\\mathbf {v}_1 \\in V(\\mathcal {P}_1)$ and $\\mathbf {v}_2 \\in V(\\mathcal {P}_2)$ .", "Again using the Cayley trick, it is easily seen that $\\omega :V({\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}) \\rightarrow \\mathbb {R}$ given by $\\omega (\\mathbf {v}_1,+1) :=\\omega _1(\\mathbf {v}_1)$ and $\\omega (\\mathbf {v}_2,-1) := \\omega _2(\\mathbf {v}_2)$ induces $\\mathcal {S}$ .", "Claim (ii) simply follows from the fact that the join of two unimodular simplices is unimodular.", "For claim (iii), let $\\sigma = \\sigma _1 \\uplus \\sigma _2 \\subseteq V({\\mathcal {P}_1 \\boxminus \\mathcal {P}_2})$ be a minimal non-face.", "Since $\\mathcal {S}_1$ and $\\mathcal {S}_2$ are flag, it follows that $\\sigma _1 \\in \\mathcal {S}_1$ and $\\sigma _2 \\in \\mathcal {S}_2$ .", "Thus, there vertices $v_i \\in \\sigma _i$ for $i=1,2$ such that $\\operatorname{supp}(v_1) \\cap \\operatorname{supp}(v_2) \\ne \\varnothing $ but $\\sigma \\setminus \\lbrace v_i\\rbrace $ is a face for $i=1$ and $i=2$ .", "But then $\\lbrace v_1,v_2\\rbrace $ is already a non-face and the claim follows.", "The theorem has some immediate consequences.", "Corollary 3.14 Let $\\mathcal {P}_1, \\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional anti-blocking polytopes with vertices in a given lattice.", "If $\\mathcal {P}_1, \\mathcal {P}_2$ have unimodular triangulations, then $\\mathcal {P}_1 - \\mathcal {P}_2$ and $\\Gamma (\\mathcal {P}_1,\\mathcal {P}_2) =\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ also have unimodular triangulations.", "By Theorem REF and the Cayley trick, $\\mathcal {P}_1 - \\mathcal {P}_2$ has a mixed subdivision into Cartesian products of unimodular simplices.", "Products of unimodular simplices are 2-level and, for example by [37], have unimodular triangulations.", "The polytope $\\operatorname{conv}(\\mathcal {P}_1 \\cup -\\mathcal {P}_2)$ inherits a triangulation from the upper or lower hull of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ , which has a unimodular triangulation by Theorem REF .", "Corollary 3.15 Let $\\mathbf {G}$ be a perfect double graph.", "Then $\\mathcal {P}_{\\mathbf {G}}$ , $\\mathcal {P}_{G_+} -\\mathcal {P}_{G_-}$ , and $\\Gamma (\\mathcal {P}_{G_+},\\mathcal {P}_{G_-})$ have regular unimodular triangulations.", "By Theorem REF , both polytopes $\\mathcal {P}_{G_+}$ and $\\mathcal {P}_{G_-}$ are 2-level and by [37] have unimodular triangulations.", "The result now follows from the previous corollary.", "Lattice points and volume Lemma REF directly implies a formula for the (normalized) volume of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ in terms of the volumes of the anti-blocking polytopes $\\mathcal {P}_1,\\mathcal {P}_2$ .", "Corollary 3.16 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be full-dimensional anti-blocking polytopes.", "Then $\\operatorname{vol}(\\mathcal {P}_1 - \\mathcal {P}_2) \\ = \\ \\sum _{J \\subseteq [n]} \\operatorname{vol}(\\mathcal {P}_1|_J)\\operatorname{vol}(\\mathcal {P}_2|_{J^c}).$ If $\\mathcal {P}_1$ and $\\mathcal {P}_2$ have unimodular triangulations with respect to a lattice $\\Lambda $ , then the normalized volume of ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ with respect to the affine lattice $\\Lambda \\times (2\\mathbb {Z}+1)$ is $\\operatorname{Vol}({\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}) \\ = \\ \\sum _{J \\subseteq [n]}\\operatorname{Vol}(\\mathcal {P}_1|_J)\\operatorname{Vol}(\\mathcal {P}_2|_{J^c}).$ Both claims follow from Lemma REF .", "For the second statement, note that Theorem REF yields that ${\\mathcal {P}_1 \\boxminus \\mathcal {P}_2}$ has a unimodular triangulation and hence its normalized volume is the number of simplices of maximal dimension, which is the number in the right-hand side.", "If $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ are rational anti-blocking polytopes, then so are ${2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2}$ and $\\mathcal {P}_1 - \\mathcal {P}_2$ .", "Our next goal is to determine their Ehrhart quasi-polynomials for a particular interesting case.", "We briefly recall the basics of Ehrhart theory; for more see, for example, [2], [3].", "If $\\mathcal {P}\\subset \\mathbb {R}^n$ is a $d$ -dimensional polytope with rational vertex coordinates, then the function $\\mathrm {Ehr}_\\mathcal {P}(k) := |k\\mathcal {P}\\cap \\mathbb {Z}^n|$ agrees with a quasi-polynomial of degree $d$ .", "We will identify $\\mathrm {Ehr}_\\mathcal {P}(k)$ with this quasi-polynomial, called the Ehrhart quasi-polynomial.", "If $\\mathcal {P}$ has its vertices in $\\mathbb {Z}^n$ , then $\\mathrm {Ehr}_\\mathcal {P}(k)$ is a polynomial of degree $d$ .", "If $\\mathcal {P}$ is full-dimensional, then the leading coefficient of $\\mathrm {Ehr}_\\mathcal {P}(k)$ is $\\operatorname{vol}(\\mathcal {P})$ .", "We will need the following fundamental result of Ehrhart theory.", "Theorem 3.17 (Ehrhart–Macdonald theorem) Let $\\mathcal {P}\\subset \\mathbb {R}^n$ be a rational polytope of dimension $d$ , then $(-1)^d \\mathrm {Ehr}_{\\mathcal {P}}(-k) \\ = \\ | \\operatorname{relint}(k\\mathcal {P}) \\cap \\mathbb {Z}^n|.$ We call an anti-blocking polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ dual integral if $A(\\mathcal {P})$ has all vertices in $\\mathbb {Z}^n$ .", "By Theorem REF , this means that there are $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {Z}_{\\ge 0}^n$ such that $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for }i=1,\\dots ,s \\rbrace .$ Corollary 3.18 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional rational anti-blocking polytopes.", "If $\\mathcal {P}_1$ is dual integral, then $\\mathrm {Ehr}_{\\mathcal {P}_1 - \\mathcal {P}_2}(k) \\ = \\ \\sum _{J \\subseteq [n]}(-1)^{|J|}\\mathrm {Ehr}_{\\mathcal {P}_1|_J}(-k-1) \\mathrm {Ehr}_{\\mathcal {P}_2|_{J^c}}(k).$ The Corollary is simply deduced from Theorem REF and the following stronger assertion.", "For a set $S \\subset \\mathbb {R}^n$ , let us write $E(S) := |S \\cap \\mathbb {Z}^n|$ .", "Theorem 3.19 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional rational anti-blocking polytopes and assume that $\\mathcal {P}_1$ is dual integral.", "For any $a,b \\in \\mathbb {Z}_{>0}$ $E(a\\mathcal {P}_1 - b\\mathcal {P}_2) \\ = \\ | (a\\mathcal {P}_1 - b\\mathcal {P}_2) \\cap \\mathbb {Z}^n | \\ = \\ \\sum _{J \\subseteq [n]} E(\\operatorname{relint}((a+1)\\mathcal {P}_1))\\cdot E(b\\mathcal {P}_2).$ It follows from Lemma REF that for any $a,b \\in \\mathbb {Z}_{>0}$ , $a\\mathcal {P}_1 - b\\mathcal {P}_2 \\ = \\ \\bigcup _{J \\subseteq [n]} (a\\mathcal {P}_1|_J -b\\mathcal {P}_2|_{J^c}).$ For $J \\subseteq [n]$ , the cell $ a\\mathcal {P}_1|_J - b\\mathcal {P}_2|_{J^c}$ is contained in the orthant $\\mathbb {R}_{\\ge 0}^J \\times \\mathbb {R}_{\\le 0}^{J^c}$ .", "It is easy to see that $\\mathbb {Z}^n \\ = \\ \\biguplus _{J \\subseteq [n]} \\mathbb {Z}^J_{> 0} \\times \\mathbb {Z}^{J^c}_{\\le 0}$ is a partition and for each $J \\subseteq [n]$ $(a\\mathcal {P}_1 - b\\mathcal {P}_2) \\cap (\\mathbb {Z}^J_{> 0} \\times \\mathbb {Z}^{J^c}_{\\le 0})\\ = \\ (a\\mathcal {P}_1|_J - b\\mathcal {P}_2|_{J^c}) \\cap (\\mathbb {Z}^J_{> 0} \\times \\mathbb {Z}^{J^c}_{\\le 0})\\ = \\ (a\\mathcal {P}_1|_J \\cap \\mathbb {Z}_{>0}^J) - (b\\mathcal {P}_2|_{J^c} \\cap \\mathbb {Z}^{J^c}).$ If $\\mathcal {P}_1$ is dual integral, then $\\mathcal {P}_1|_J$ is dual integral.", "Thus, for a fixed $J$ , there are $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {Z}_{\\ge 0}^{J}$ such that $(a\\mathcal {P}_1|_J \\cap \\mathbb {Z}^J_{>0})&\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {Z}^J : \\mathbf {x}> 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le a \\rbrace \\\\&\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {Z}^J : \\mathbf {x}> 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle < a+1 \\rbrace \\ = \\ \\operatorname{relint}((a+1) \\mathcal {P}_1|_J) \\cap \\mathbb {Z}^J.$ This proves the result.", "Clearly, it would be desirable to apply Corollary REF to the case that $\\mathcal {P}_1$ is a lattice polytope as well as dual integral.", "Proposition 3.20 Let $\\mathcal {P}\\subset \\mathbb {R}^n$ be a full-dimensional dual-integral anti-blocking polytope with vertices in $\\mathbb {Z}^n$ .", "Then $\\mathcal {P}= \\mathcal {P}_G$ for some perfect graph $G$ .", "Let $\\mathcal {P}$ be given by $\\mathcal {P}\\ = \\ \\lbrace \\mathbf {x}\\in \\mathbb {R}^n : \\mathbf {x}\\ge 0, \\langle {\\mathbf {d}_i,\\mathbf {x}}\\rangle \\le 1 \\text{ for} i=1,\\dots ,s \\rbrace $ for some $\\mathbf {d}_1,\\dots ,\\mathbf {d}_s \\in \\mathbb {Z}_{\\ge 0}^n$ .", "Since $\\mathcal {P}$ is full-dimensional and a lattice polytope, it follows that $\\mathbf {e}_1,\\dots ,\\mathbf {e}_n \\in \\mathcal {P}$ and for any $1 \\le j \\le s$ we compute $0 \\ \\le \\ \\langle {\\mathbf {d}_j, \\mathbf {e}_i}\\rangle \\ \\le \\ 1$ for all $i$ and since the $\\mathbf {d}_j$ are integer vectors, it follows that $\\mathbf {d}_j = \\mathbf {1}_{C_j}$ for some $C_j \\subset [n]$ .", "Consequently, the vertices of $\\mathcal {P}$ are in $\\lbrace 0,1\\rbrace ^n$ and $\\mathcal {P}$ is 2-level.", "By Proposition REF , $\\mathcal {P}= \\mathcal {P}_G$ for some perfect graph $G$ .", "This severely limits the applicability of Corollary REF to lattice anti-blocking polytopes.", "On the other hand, we do not know of many results regarding the Ehrhart polynomials or even volumes of stable set polytopes of perfect graphs; see also the next section.", "Theorem 3.21 Let $\\mathcal {P}_1,\\mathcal {P}_2 \\subset \\mathbb {R}^n$ be two full-dimensional rational anti-blocking polytopes such that $\\mathcal {P}_1$ is dual integral.", "Then for $\\mathcal {P}:= {2\\mathcal {P}_1 \\boxminus 2\\mathcal {P}_2}$ $\\mathrm {Ehr}_\\mathcal {P}(k) \\ = \\ | k \\mathcal {P}\\cap \\mathbb {Z}^{n+1}| \\ = \\ \\sum _{J \\subseteq [n]} (-1)^{|J|}\\sum _{s=-k}^k\\mathrm {Ehr}_{\\mathcal {P}_1|_J}( s-k-1 ) \\cdot \\mathrm {Ehr}_{\\mathcal {P}_2|_{J^c}}( k+s).$ For $k > 0$ , $k\\mathcal {P}\\ = \\ \\operatorname{conv}( 2k\\mathcal {P}_1 \\times \\lbrace k\\rbrace \\cup -2k\\mathcal {P}_2 \\times \\lbrace -k\\rbrace ).$ In particular, if $(\\mathbf {p},t)$ is a lattice point in $k\\mathcal {P}$ , then $-k \\le t\\le k$ .", "For fixed $t$ , $\\lbrace \\mathbf {p}\\in \\mathbb {Z}^n : (\\mathbf {p},t) \\in k\\mathcal {P}\\rbrace \\ = \\ \\left((k-t)\\mathcal {P}_1 - (k+t)\\mathcal {P}_2\\right) \\cap \\mathbb {Z}^n.$ Theorems REF and REF then complete the proof.", "Triangulations and transfers If $\\mathbf {P}= \\mathbf {P}_\\circ = (P,\\preceq ,\\preceq )$ is induced by a single poset, then Corollaries REF and REF assure us that $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ are 2-level and [37] implies that both polytopes have unimodular triangulations with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ .", "In this section we give explicit triangulations of the double chain polytope $\\mathcal {C}({\\mathbf {P}})$ and, in the compatible case, of the double order polytope $\\mathcal {O}(\\mathbf {P})$ .", "These triangulations will be regular, unimodular, and flag.", "To that end, we will generalize Stanley's approach [34] from poset polytopes to double poset polytopes.", "We put the triangulation to good use and explicitly compute the Ehrhart polynomial and the volume of $\\mathcal {C}({\\mathbf {P}})$ and, in case that $\\mathbf {P}$ is compatible, of $\\mathcal {O}(\\mathbf {P})$ .", "Triangulations of double poset polytopes Recall from the introduction that for a poset $(P,\\preceq )$ , the order polytope $\\mathcal {O}(P)$ parametrizes all order preserving maps $f : P\\rightarrow [0,1]$ .", "Any $f \\in \\mathcal {O}(P)$ induces a partial order $P_f =(P,\\preceq _f)$ by $a \\prec _f b$ if $a \\prec b$ or, when $a,b$ are incomparable, if $f(a) < f(b)$ .", "Clearly, $\\preceq _f$ refines $\\preceq $ and hence $\\mathcal {O}(P_f) \\subseteq \\mathcal {O}(P)$ .", "Since filters in $P_f$ are filters in $P$ , $\\mathcal {O}(P_f)$ is a vertex-induced subpolytope of $\\mathcal {O}(P)$ .", "If $f$ is generic, that is, $f(a) \\ne f(b)$ for all $a \\ne b$ , then $\\preceq _f$ is a total order and $\\mathcal {O}(P_f)$ is a unimodular simplex of dimension $|P|$ .", "Stanley showed that the collection of all these simplices constitute a unimodular triangulation of $\\mathcal {O}(P)$ .", "More precisely, this canonical triangulation of $\\mathcal {O}(P)$ is given by the order complex $\\Delta (\\mathcal {J}(P))$ of $\\mathcal {J}(P)$ , i.e., the collection of chains in the Birkhoff lattice of $P$ ordered by inclusion.", "Since a collection of filters $\\mathsf {J}_0,\\dots ,\\mathsf {J}_k$ is not a chain if and only if $\\mathsf {J}_i\\lnot \\subseteq \\mathsf {J}_j$ and $\\mathsf {J}_j \\lnot \\subseteq \\mathsf {J}_i$ for some $0\\le i,j \\le k$ , the canonical triangulation is flag.", "Stanley [34] elegantly transferred the canonical triangulation of $\\mathcal {O}(P)$ to $\\mathcal {C}({P})$ in the following sense.", "Define the transfer map $\\phi _P: \\mathcal {O}(P) \\rightarrow \\mathcal {C}({P})$ by $(\\phi _Pf)(b) \\ := \\ \\min \\lbrace f(b) - f(a) : a \\prec b \\rbrace ,$ for $f \\in \\mathcal {O}(P)$ and $b \\in P$ .", "This is a piecewise linear map and the domains of linearity are exactly the full-dimensional simplices $\\mathcal {O}(P_f)$ for generic $f$ .", "In particular, $\\phi _P(\\mathbf {1}_\\mathsf {J}) =\\mathbf {1}_{\\min (\\mathsf {J})}$ for any filter $\\mathsf {J}\\subseteq P$ , which shows that $\\phi _P$ maps $\\mathcal {O}(P)$ into $\\mathcal {C}({P})$ .", "To show that $\\phi _P$ is a PL homeomorphism of the two polytopes, Stanley gives an explicit inverse $\\psi _P: \\mathcal {C}({P}) \\rightarrow \\mathcal {O}(P)$ by $(\\psi _Pg)(b) \\ := \\ \\max \\lbrace g(a_0) + \\cdots + g(a_{k-1}) +g(a_k) : a_0 \\prec \\cdots \\prec a_{k-1} \\prec a_k \\preceq b \\rbrace ,$ for any $g \\in \\mathcal {C}({P})$ .", "Note that our definition of $\\psi _P$ differs from that in [34] in that we do not require that the chain has to end in $b$ .", "This will be important later.", "It can be easily checked that $\\psi _P$ is an inverse to $\\phi _P$ .", "Hence, the simplices $\\operatorname{conv}( \\mathbf {1}_{\\min (\\mathsf {J}_0)}, \\dots , \\mathbf {1}_{\\min (\\mathsf {J}_k)} ) \\quad \\text{for } \\quad \\lbrace \\mathsf {J}_0 \\subseteq \\cdots \\subseteq \\mathsf {J}_k \\rbrace \\in \\Delta (\\mathcal {J}(P))$ constitute a flag triangulation of $\\mathcal {C}({P})$ .", "We will follow the same approach as Stanley but, curiously, it will be simpler to start with a triangulation of $\\mathcal {C}({\\mathbf {P}})$ .", "Recall from Section REF that a pair of chains $C = C_+ \\uplus C_-$ with $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ is non-interfering if $\\min (\\mathsf {J}_+)\\cap \\min (\\mathsf {J}_-) = \\varnothing $ for any $\\mathsf {J}_\\sigma \\in C_\\sigma $ for $\\sigma = \\pm $ .", "Corollary 4.1 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then a regular triangulation of $\\mathcal {C}({\\mathbf {P}})$ is given as follows: The $(k-1)$ -dimensional simplices are in bijection to non-interfering pairs of chains $C = C_+ \\uplus C_+ \\subseteq \\mathcal {J}(\\mathbf {P})$ with $|C| = |C_+| +|C_-| = k$ .", "Moreover, the triangulation is regular, unimodular (with respect to $\\mathbb {A}$ ), and flag.", "The canonical triangulation of $\\mathcal {C}({P_\\sigma })$ is regular, unimodular, and flag for $\\sigma =\\pm $ .", "As described above, its $(l_\\sigma -1)$ -simplices are in bijection to chains $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ of length $|C_\\sigma | = l_\\sigma $ .", "More precisely, the simplex corresponding to $C_\\sigma $ is given by $F(C_\\sigma ) = \\operatorname{conv}( \\mathbf {1}_{\\min (\\mathsf {J}_\\sigma )} : \\mathsf {J}_\\sigma \\in C_\\sigma ).$ By Theorem REF applied to $\\mathcal {C}({\\mathbf {P}}) ={2\\mathcal {C}({P_+}) \\boxminus 2\\mathcal {C}({P_-})}$ , it follows that a unimodular and flag triangulation is given by the joins $2F(C_+) * 2F(C_-)$ for all chains $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ such that $F(C_+)$ and $F(C_-)$ lie in complementary coordinate subspaces.", "This, however, is exactly the case when $\\min (\\mathsf {J}_+) \\cap \\min (\\mathsf {J}_-) = \\varnothing $ for all $\\mathsf {J}_\\sigma \\in C_\\sigma $ for $\\sigma =\\pm $ .", "Corollary REF gives a canonical triangulation that combinatorially can be described as a subcomplex of $\\Delta (\\mathcal {J}(\\mathbf {P})) =\\Delta (\\mathcal {J}(P_+)) * \\Delta (\\mathcal {J}(P_-))$ , called the non-interfering complex $\\Delta ^\\mathrm {ni}(\\mathbf {P}) \\ := \\ \\lbrace C : C = C_+ \\uplus C_- \\in \\Delta (\\mathcal {J}(\\mathbf {P})) ,\\, C \\text{ non-interfering} \\rbrace .$ Associating $\\Delta (\\mathcal {J}(P))$ to a poset $P$ is very natural and can be motivated, for example, from an algebraic-combinatorial approach to the order polynomial (cf. [3]).", "It would be very interesting to know if the association $\\mathbf {P}$ to $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ is equally natural from a purely combinatorial perspective.", "Given a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ , we define a piecewise linear map $\\Psi _{\\mathbf {P}} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ by $\\Psi _{\\mathbf {P}}(g) \\ := \\ \\psi _{P_+}(g) \\ - \\ \\psi _{P_-}(-g),$ for any $g \\in \\mathbb {R}^P$ .", "Here, we use that $\\psi _{}$ , as given in (REF ), is defined on all of $\\mathbb {R}^P$ with the following important property: For $g \\in \\mathbb {R}^P$ , let us write $g = g^+ - g^-$ , where $g^+,g^- \\in \\mathbb {R}_{\\ge 0}^P$ with disjoint supports.", "Then $\\psi _{P_\\sigma }(g) = \\psi _{P_\\sigma }(g^+)$ for $\\sigma = \\pm $ .", "Thus, $\\Psi _{\\mathbf {P}}(g) \\ = \\ \\psi _{P_+}(g^+) - \\psi _{P_-}(g^-),$ for any $g \\in \\mathbb {R}^P$ .", "In particular, $\\Psi _{\\mathbf {P}}$ takes $\\lambda \\mathcal {C}({P_+}) - \\mu \\mathcal {C}({P_-})$ into $\\lambda \\mathcal {O}(P_+) - \\mu \\mathcal {O}(P_-)$ for any $\\lambda ,\\mu \\ge 0$ .", "Indeed, for any pair of antichains $A_\\sigma \\subseteq P_\\sigma $ , first observe that $\\mathbf {1}_{A_+} - \\mathbf {1}_{A_-} =\\mathbf {1}_{A_+\\setminus A_-} - \\mathbf {1}_{A_-\\setminus A_+}$ .", "Hence, it suffices to assume that $A_+ \\cap A_- = \\varnothing $ .", "We compute $\\Psi _{\\mathbf {P}}(\\mathbf {1}_{A_+} - \\mathbf {1}_{A_-} ) \\ = \\ \\mathbf {1}_{\\mathsf {J}_+} -\\mathbf {1}_{\\mathsf {J}_-},$ where for $\\sigma = \\pm $ , $\\mathsf {J}_\\sigma \\subseteq P_\\sigma $ is the filter generated by $A_\\sigma $ .", "If $\\mathbf {P}$ is a compatible double poset, then Corollary REF implies that $\\Psi _{\\mathbf {P}}$ is a surjection on vertex sets.", "Lemma 4.2 Let $\\mathbf {P}=(\\mathcal {P},\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then $\\Psi _{\\mathbf {P}} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ is a lattice-preserving piecewise linear isomorphism.", "It follows directly from (REF ) that $\\Psi _{\\mathbf {P}}$ is piecewise linear.", "To show that $\\Psi _{\\mathbf {P}}$ is an isomorphism, we explicitly construct for $f\\in \\mathbb {R}^P$ a $g \\in \\mathbb {R}^P$ such that $\\Psi _{\\mathbf {P}}(g)=f$ .", "Since $\\mathbf {P}$ is compatible, we can assume that $P= \\lbrace a_1,\\dots ,a_n\\rbrace $ such that $a_i \\prec _+ a_j$ or $a_i \\prec _- a_j$ implies $i < j$ .", "It follows from (REF ) that $\\Psi _{\\mathbf {P}}(g^{\\prime })(a_1) = g^{\\prime }(a_1)$ for any $g^{\\prime } \\in \\mathbb {R}^P$ and hence, we can set $g(a_1) := f(a_1)$ .", "Now assume that $g$ is already defined on $D_k := \\lbrace a_1,\\dots , a_k \\rbrace $ for some $k$ .", "For $g^{\\prime }\\in \\mathbb {R}^P$ observe that $\\psi _{P_+}(g^{\\prime })(a_{k+1}) \\ = \\ \\max (g^{\\prime }(a_{k+1}),0) + r$ where $r=0$ or $r = \\psi _{P_+}(g^{\\prime })(a_i)$ for some $i \\le k$ .", "Analogously, $\\psi _{P_-}(-g^{\\prime })(a_{k+1}) \\ = \\ \\max (-g^{\\prime }(a_{k+1}),0) + s$ where $s=0$ or $s = \\psi _{P_-}(-g^{\\prime })(a_j)$ for some $j \\le k$ .", "Thus, we set $g(a_{k+1})\\ := \\ f(a_{k+1}) - r + s$ This uniquely determines $g$ by induction on $k$ .", "To prove that $\\Psi _{\\mathbf {P}}$ is lattice-preserving, observe that by (REF ) we have $\\Psi _{\\mathbf {P}}(\\mathbb {Z}^P)\\subseteq \\mathbb {Z}^P$ .", "Moreover, if $f=\\Psi _{\\mathbf {P}}(g)$ with $f\\in \\mathbb {Z}^P$ and the above construction shows that $g\\in \\mathbb {Z}^P$ .", "Hence, $\\Psi _{\\mathbf {P}}(\\mathbb {Z}^P)\\subseteq \\mathbb {Z}^P$ , which finishes the proof.", "Using the notation from (REF ) in Section REF , the lemma shows that $\\lbrace \\overline{F}(C) : C \\in \\ \\Delta ^\\mathrm {ni}(\\mathbf {P}) \\rbrace $ is a realization of the flag simplicial complex $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ by unimodular simplices inside $\\mathcal {O}(\\mathbf {P})$ .", "Using Gröbner bases in Section , we will show the following result.", "Theorem 4.3 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then the map $(g,t) \\mapsto (\\Psi _{\\mathbf {P}}(g),t)$ is a piecewise linear homeomorphism from $\\mathbb {R}^{P}\\times \\mathbb {R}$ to itself that preserves the lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ .", "In particular, it maps $\\mathcal {C}({\\mathbf {P}})$ to $\\mathcal {O}(\\mathbf {P})$ and hence $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ is a regular, unimodular, and flag triangulation of $\\mathcal {O}(\\mathbf {P})$ .", "By the previous lemma, (REF ) is a realization of $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ in $\\mathcal {O}(\\mathbf {P})$ without new vertices.", "Moreover, every maximal simplex contains the edge $e = \\operatorname{conv}\\lbrace (\\mathbf {0},1), (\\mathbf {0},-1)\\rbrace $ .", "Hence, it suffices to show that for every maximal simplex in $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ , the supporting hyperplane of every facet not containing $e$ is supporting for $\\mathcal {O}(\\mathbf {P})$ .", "Let $C = \\lbrace \\mathsf {J}_{+0} \\subset \\cdots \\subset \\mathsf {J}_{+k}, \\mathsf {J}_{-0}\\subset \\cdots \\subset \\mathsf {J}_{-l}\\rbrace $ be two maximal non-interfering chains.", "Set $A_{+i} := \\min (\\mathsf {J}_{+i})$ for $1 \\le i \\le k$ and $A_{-0},\\dots ,A_{-l}$ likewise.", "It follows that $P_1 = \\bigcup A_{+i}$ and $P_2 = \\bigcup A_{-j}$ give a partition of $P$ .", "In particular, since $C$ was maximal, we have that $\\lbrace a^+_{k-i-1}\\rbrace = A_{+i} \\setminus A_{+(i-1)}$ and $P_1 = \\lbrace a^+_1,\\dots ,a^+_k\\rbrace $ .", "In particular, if $a^+_s\\prec _+ a^+_t$ , then $s < t$ .", "The same argument yields $P_2 =\\lbrace a^-_1,\\dots ,a^-_l\\rbrace $ and the labelling is a linear extension of $(P_2,\\preceq _-)$ .", "We focus on $P_1$ ; the argument for $P_2$ is analogous.", "Pick the maximal chain $D$ in $(P_1,\\preceq _+)$ starting in $a^+_k$ .", "Then $A_{+i} \\cap D\\ne \\varnothing $ for all $i > 0$ and hence $\\lbrace (g,t) \\in \\mathbb {R}^{P_1}:\\langle {\\mathbf {1}_D,g}\\rangle = 1\\rbrace $ is the hyperplane for the maximal simplex in the triangulation of $\\mathcal {C}({P_1,\\preceq _+})$ corresponding to $A_{+0},\\dots ,A_{+k}$ and not containing the origin.", "Thus, one of the two hyperplanes supporting a facet of the simplex in $\\mathcal {C}({\\mathbf {P}})$ corresponding to $C$ is given by $H := \\lbrace (g,t) \\in \\mathbb {R}^P\\times \\mathbb {R}:\\ell (g,t) = 1 \\rbrace $ where $\\ell (g,t) = \\langle {\\mathbf {1}_D,g}\\rangle - t$ .", "Now, $\\Psi _{\\mathbf {P}}$ is linear on the simplex $C$ in $\\mathcal {C}({\\mathbf {P}})$ and can be easily inverted.", "Since $\\mathbf {P}$ is compatible, we can find a linear extension $\\sigma : P\\rightarrow \\lbrace 1,\\dots ,|P|\\rbrace $ that respects the constructed linear extensions on $P_1$ and $P_2$ .", "On the image of $C$ under $\\Psi _{\\mathbf {P}}$ , the inverse is given by the linear transformation $T :\\mathbb {R}^P\\times \\mathbb {R}\\rightarrow \\mathbb {R}^P\\times \\mathbb {R}$ with $T(f,t) = (f^{\\prime },t)$ and $f^{\\prime } : P\\rightarrow \\mathbb {R}$ is defined as follows.", "If $b \\in P_1$ , then by $f^{\\prime }(b) =f(b) - f(\\bar{b})$ , $\\bar{b} \\prec _+ b$ is a cover relation and $\\sigma (\\bar{b})$ is maximal.", "If $b \\in P_2$ , we choose $\\bar{b}$ covered by $b$ in with respect to $\\preceq _-$ .", "It can now be checked that $\\ell \\circ T = L_C$ for some alternating chain $C$ .", "Thus $H$ is supporting for $\\mathcal {O}(\\mathbf {P})$ and the map $\\Psi _{\\mathbf {P}}$ maps $\\mathcal {C}({\\mathbf {P}})$ onto $\\mathcal {O}(\\mathbf {P})$ .", "Theorem REF does not extend to the non-compatible case as the following example shows.", "Example 9 Consider the double poset $\\mathbf {P}= ([2],\\le ,\\ge )$ , that is, $P_+$ is the 2-chain $\\lbrace 1,2\\rbrace $ and $P_-$ is the opposite poset.", "Then $\\mathcal {C}({P_+})= \\mathcal {C}({P_-}) = T := \\lbrace x \\in \\mathbb {R}^2 : x \\ge 0, x_1 + x_2 \\le 1\\rbrace $ and $\\mathcal {C}({\\mathbf {P}})$ is a three-dimensional octahedron with volume $\\frac{16}{3}$ .", "Any triangulation of the octahedron has at least four simplices.", "In contrast, $\\mathcal {O}(P_-) = \\mathbf {1} - \\mathcal {O}(P_+)$ and hence $\\mathcal {O}(\\mathbf {P})$ is linearly isomorphic to a prism over a triangle with volume 4.", "Any triangulation of the prism has exactly 3 tetrahedra.", "Volumes and Ehrhart polynomials The canonical subdivision of $\\mathcal {O}(P)$ makes it easy to compute its volume.", "For a generic $f \\in \\mathcal {O}(P)$ , there is a unique linear extension $\\sigma : P\\rightarrow \\lbrace 1,2,\\dots ,d\\rbrace $ where $d := |P|$ such that $\\mathcal {O}(P_f) \\ = \\ \\lbrace h \\in \\mathbb {R}^P : 0 \\le h(\\sigma ^{-1}(1)) \\le \\cdots \\le h(\\sigma ^{-1}(d)) \\le 1 \\rbrace .$ In particular, the full-dimensional simplex $\\mathcal {O}(P_f)$ is unimodular relative to $\\mathbb {Z}^P\\subseteq \\mathbb {R}^P$ and has volume $\\operatorname{vol}(\\mathcal {O}(P_f)) =\\frac{1}{|P|!", "}$ .", "If we denote by $e(P)$ the number of linear extensions of $P$ , then Stanley [34] showed the following.", "Corollary 4.4 $\\operatorname{Vol}(\\mathcal {O}(P)) = |P|!", "\\cdot \\operatorname{vol}(\\mathcal {O}(P)) = e(P)$ .", "For the Ehrhart polynomial $\\mathrm {Ehr}_{\\mathcal {O}(P)}(n)$ of $\\mathcal {O}(P)$ it suffices to interpret the lattice points in $n \\mathcal {O}(P)$ for $n > 0$ .", "Every point in $n\\mathcal {O}(P) \\cap \\mathbb {Z}^P$ corresponds to an order preserving map $\\phi : P\\rightarrow [n+1]$ .", "Counting order preserving maps is classical [35]: the order polynomial $\\Omega _P(n)$ of $P$ counts the number of order preserving maps into $n$ -chains.", "The strict order polynomial $\\Omega ^\\circ _P(n)$ counts the number of strictly order preserving maps $f : P\\rightarrow [n]$ , that is, $f(a) < f(b)$ for $a \\prec b$ .", "The transfer map $\\phi _P$ as well as its inverse $\\psi _P$ (given in (REF ) and (REF ), respectively) both take lattice points to lattice points and hence, together with Theorem REF , yield the following result.", "Corollary 4.5 Let $P$ be a finite poset.", "Then for every $n > 0$ $\\Omega _P(n+1)& \\ = \\ \\mathrm {Ehr}_{\\mathcal {O}(P)}(n) \\ = \\ \\mathrm {Ehr}_{\\mathcal {C}({P})}(n)\\\\\\multicolumn{2}{l}{\\text{ and }}\\\\(-1)^{|P|}\\Omega ^\\circ _P(n-1) &\\ = \\ \\mathrm {Ehr}_{\\mathcal {O}(P)}(-n) \\ = \\ \\mathrm {Ehr}_{\\mathcal {C}({P})}(-n).$ In particular, $\\operatorname{vol}(\\mathcal {O}(P)) = \\operatorname{vol}(\\mathcal {C}({P}))$ .", "This is an interesting result as it implies that the number of linear extensions of a poset $P$ only depends on the comparability graph $G(P)$ .", "Theorem 4.6 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then $\\overline{\\mathcal {C}}(\\mathbf {P})$ is a lattice polytope with respect to $\\mathbb {Z}^P$ and $\\mathrm {Ehr}_{\\overline{\\mathcal {C}}(\\mathbf {P})}(n-1) \\ &= \\ \\sum _{P= P_1 \\uplus P_2}\\Omega ^\\circ _{(P_1,\\preceq _+)}(n-1) \\cdot \\Omega _{(P_2,\\preceq _-)}(n) \\text{ and }\\\\\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ &= \\ \\sum _{P= P_1 \\uplus P_2} \\binom{|P|}{|P_1|}e(P_1,\\preceq _+)\\cdot e(P_2,\\preceq _-).$ Since $\\mathcal {C}({P}) = \\mathcal {P}_{G(P)}$ is a dual integral anti-blocking polytope, the first identity follows from Corollary REF and Corollary REF .", "The second identity follows from Corollary REF and Corollary REF .", "Notice from Theorem REF we can also deduce a closed formula for the Ehrhart polynomial of $\\mathcal {C}({\\mathbf {P}})$ with respect to the lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ and, by substituting $\\frac{1}{2}k$ for $k$ , also with respect to the affine lattice $\\mathbb {A}$ .", "These formulas are not very enlightening and instead we record the normalized volume.", "Note that the minimal Euclidean volume of a full-dimensional simplex with vertices in $\\mathbb {A}= \\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ is $\\frac{2^{|P|+1}}{(|P|+1)!", "}$ .", "Corollary 4.7 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then the normalized volume with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ is $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{P= P_1 \\uplus P_2}e(P_1,\\preceq _+) \\cdot e(P_2,\\preceq _-).$ We leave it to the reader to give direct combinatorial interpretations of the volume and the Ehrhart polynomials for double posets.", "It follows directly from (REF ) that $\\Psi _{P_{\\pm }} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ maps lattice points to lattice points.", "If $\\mathbf {P}$ is compatible, then the proof of Lemma REF asserts that $\\Psi _{P_{\\pm }}$ is in fact lattice preserving.", "Hence, we record an analog to Corollary REF .", "Corollary 4.8 If $\\mathbf {P}$ is a compatible double poset, then $\\mathcal {O}(\\mathbf {P})$ and $\\mathcal {C}({\\mathbf {P}})$ have the same Ehrhart polynomials and hence the same volumes.", "The formulas of Theorem REF are particularly simple when $\\mathbf {P}$ is special or anti-special.", "We illustrate these cases at some simple examples.", "Example 10 For the 'XW'-double poset we have $\\operatorname{Vol}(\\mathcal {O}(\\mathbf {P}_{XW})) \\ = \\ \\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}_{XW}}))\\ = \\ \\tfrac{6!", "}{2^6} \\operatorname{vol}(\\mathcal {C}({\\mathbf {P}_{XW}}))\\ = \\ 128$ and $\\operatorname{Vol}(\\overline{\\mathcal {O}}(\\mathbf {P}_{XW}))= \\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P}_{XW}))= 6!", "\\operatorname{vol}(\\overline{\\mathcal {C}}(\\mathbf {P}_{XW}))= 880$ .", "Example 11 As the following examples are all compatible, the given values also give the normalized volumes of the respective (reduced) double order polytopes.", "Let $\\mathbf {P}= ([d],\\le ,\\le )$ be the double chain on $d$ elements.", "Then $\\mathcal {C}({\\mathbf {P}})$ is a crosspolytope and $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) = 2^d$ and it follows from Vandermonde's identity that $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ d!", "\\operatorname{vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i}^2 \\ = \\ \\binom{2d}{d}.$ If $\\mathbf {P}$ is the double anti-chain on $d$ elements, then $\\mathcal {C}({\\mathbf {P}})$ is isomorphic to $[0,2]^{d} \\times [-1,1]$ and its normalized volume is $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\tfrac{(d+1)!", "}{2^{d+1}}\\operatorname{vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i} i!(d-i)!", "\\ = \\ (d+1)!.$ Likewise, $\\overline{\\mathcal {C}}(\\mathbf {P})$ is isomorphic to $[-1,1]^d$ , which can be decomposed into $2^d$ unit cubes.", "Consequently, its normalized volume is $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i}^2 i!(d-i)!", "\\ = \\ 2^dd!.$ Let $\\mathbf {P}$ be the double poset such that $P_+$ is the $d$ -chain and $P_-$ is the $d$ -antichain.", "Then $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{i=0}^d \\frac{d!}{i!", "}$ is the number of choices of ordered subsets of a $d$ -set.", "Moreover $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d\\binom{d}{i}^2 i!$ is the number of partial permutation matrices, i.e.", "$0/1$ -matrices of size $d$ with at most one nonzero entry per row and column.", "Indeed, such a matrix is uniquely identified by an $i$ -by-$i$ permutation matrix and a choice of $i$ rows and $i$ columns in which it is embedded.", "For the comb $C_n$ , the number of linear extensions is $e(C_n) =(2n-1)!", "!$ .", "Let $\\mathbf {P}$ be the double poset induced by the comb $C_n$ .", "Then an induction argument shows that $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ 4^n\\, n!.$ It would be nice to have a bijective proof of this equality.", "Let $\\mathbf {P}_\\circ = (P,\\preceq ,\\preceq )$ be a compatible double poset induced by a poset $(P,\\preceq )$ .", "By Corollary REF , the polytopes $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ have the same normalized volume.", "Since both polytopes are 2-level, this means that the number of maximal simplices in any pulling triangulation of $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ coincides.", "From Theorem REF , we know that $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ is the twisted prism over the valuation polytope associated to $P$ .", "On the other hand, we know from Corollary REF that $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ is linearly isomorphic to the Hansen polytope $\\mathcal {H}(\\overline{G(P)})$ .", "Moreover, $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ and $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ are both 2-level and it is enticing to conjecture that their normalized volumes also agree.", "Unfortunately, this is not the case.", "For the poset $P$ on 5 elements whose Hasse diagram is the letter 'X', any pulling triangulation of $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ has 324 simplices whereas for $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ pulling triangulations have 320 simplices.", "Gröbner bases and triangulations Double Hibi rings Hibi [19] associated to any finite poset $(P,\\preceq )$ a ring $\\mathcal {O}(P)]$ , nowadays called Hibi ring, that algebraically reflects many of the order-theoretic properties of $P$ .", "The ring $\\mathcal {O}(P)]$ is defined as the graded subring of the polynomial ring $S = t, s_a : a \\in P]$ generated by the elements $t \\cdot s^\\mathsf {J}$ , where $s^\\mathsf {J}\\ := \\ \\prod _{a \\in \\mathsf {J}} s_a,$ ranges over all filters $\\mathsf {J}\\subseteq P$ .", "For example, Hibi showed that $\\mathcal {O}(P)]$ is a normal Cohen–Macaulay domain of dimension $|P|+1$ and that $\\mathcal {O}(P)]$ is Gorenstein if and only if $P$ is a graded poset.", "By definition, Hibi rings are toric and hence they have the following quotient description.", "Let $R =x_\\mathsf {J}: \\mathsf {J}\\in \\mathcal {J}(P)]$ be the polynomial ring with variables indexed by filters and define the homogeneous ring map $\\phi : R \\rightarrow S$ by $\\phi (x_\\mathsf {J}) = t\\, s_\\mathsf {J}$ .", "Then $\\mathcal {O}(P)] \\cong R /\\mathrm {I}_{\\mathcal {O}(P)}$ where $\\mathrm {I}_{\\mathcal {O}(P)} = \\ker \\phi $ is a toric ideal.", "Hibi elegantly described a reduced Gröbner basis of $\\mathrm {I}_{\\mathcal {O}(P)}$ in terms of $\\mathcal {J}(P)$ .", "Fix a total order $\\le $ on the variables of $R$ such that $x_\\mathsf {J}\\le x_{\\mathsf {J}^{\\prime }}$ whenever $\\mathsf {J}\\subseteq \\mathsf {J}^{\\prime }$ and let $\\le _{rev}$ denote the induced reverse lexicographic order on $R$ .", "For $f \\in R$ , we write $\\textrm {in}_{\\le _{rev}}(f)$ for its leading term with respect to $\\le _{rev}$ and we will underline leading terms in what follows.", "Theorem 5.1 ([18]) Let $(P,\\preceq )$ be a finite poset.", "Then the collection $\\underline{x_{\\mathsf {J}} \\, x_{\\mathsf {J}^{\\prime }}} \\ - \\ x_{\\mathsf {J}\\cap \\mathsf {J}^{\\prime }} \\, x_{\\mathsf {J}\\cup \\mathsf {J}^{\\prime }} \\quad \\text{with }\\mathsf {J},\\mathsf {J}^{\\prime }\\in \\mathcal {J}(P)\\text{ incomparable}$ is a reduced Gröbner basis of $\\mathrm {I}_{\\mathcal {O}(P)}$ .", "The binomials (REF ) are called Hibi relations.", "In light of the previous sections, the natural question that we will address now is regarding an algebraic counterpart of the Hibi rings for double posets.", "For a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ , we define the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ as the subalgebra of the Laurent ring $\\hat{S} := t_-,t_+,s_a,s_a^{-1} : a \\in P]$ spanned by the elements $t_+\\cdot s^{\\mathsf {J}}$ for filters $\\mathsf {J}\\in \\mathcal {J}(P_+)$ and $t_- \\cdot (s^\\mathsf {J})^{-1}$ for filters $\\mathsf {J}\\in \\mathcal {J}(P_-)$ .", "This is the affine semigroup ring associated to $\\mathcal {O}(\\mathbf {P})$ with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ .", "Up to a translation by $(\\mathbf {0},1)$ and the lattice isomorphism $2\\mathbb {Z}^P\\times 2\\mathbb {Z}\\cong \\mathbb {Z}^P\\times \\mathbb {Z}$ , the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ is the affine semigroup ring of $\\operatorname{conv}\\bigl \\lbrace (\\mathcal {O}(P_+) \\times \\lbrace 1\\rbrace ) \\cup (-\\mathcal {O}(P_-) \\times \\lbrace 0\\rbrace )\\bigr \\rbrace ,$ with respect to the usual lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ .", "In particular, the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ is graded of Krull dimension $|P|+1$ .", "Moreover, since the double order polytope $\\mathcal {O}(\\mathbf {P})$ is reflexive by Corollary REF , it follows that $\\mathcal {O}(\\mathbf {P})]$ is a Gorenstein domain for any compatible double poset $\\mathbf {P}$ .", "The rings $\\mathcal {O}(\\mathbf {P})]$ as well as affine semigroup rings associated to the double chain polytopes $\\mathcal {C}({\\mathbf {P}})$ as treated at the end of Section REF were also considered by Hibi and Tsuchiya [24].", "Set $\\hat{R} := x_{\\mathsf {J}_+},x_{\\mathsf {J}_-} : \\mathsf {J}_+ \\in \\mathcal {J}(P_+), \\mathsf {J}_+ \\in \\mathcal {J}(P_+)]$ and define the monomial map $\\hat{\\phi }: \\hat{R} \\rightarrow \\hat{S}$ by $\\hat{\\phi }(x_{\\mathsf {J}_+}) \\ = \\ t_+\\, s^{\\mathsf {J}_+}\\quad \\text{ and } \\quad \\hat{\\phi }(x_{\\mathsf {J}_-}) \\ = \\ t_-\\, (s^{\\mathsf {J}_-})^{-1}.$ The corresponding toric ideal $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})} = \\ker \\hat{\\phi }$ is then generated by the binomials $\\underline{x_{\\mathsf {J}_{+1}} x_{\\mathsf {J}_{+2}} \\dots x_{\\mathsf {J}_{+k_+}} \\cdot x_{\\mathsf {J}_{-1}} x_{\\mathsf {J}_{-2}} \\dots x_{\\mathsf {J}_{-k_-}}}-x_{\\mathsf {J}^{\\prime }_{+1}} x_{\\mathsf {J}^{\\prime }_{+2}} \\dots x_{\\mathsf {J}^{\\prime }_{+k_+}} \\cdot x_{\\mathsf {J}^{\\prime }_{-1}} x_{\\mathsf {J}^{\\prime }_{-2}} \\dots x_{\\mathsf {J}^{\\prime }_{-k_-}},$ for filters $\\mathsf {J}_{+1}, \\dots , \\mathsf {J}_{+k_+}, \\mathsf {J}^{\\prime }_{+1}, \\dots ,\\mathsf {J}^{\\prime }_{+k_+} \\in \\mathcal {J}(P_+)$ and $\\mathsf {J}_{-1}, \\dots , \\mathsf {J}_{-k_-},\\mathsf {J}^{\\prime }_{-1}, \\dots , \\mathsf {J}^{\\prime }_{-k_-} \\in \\mathcal {J}(P_-)$ .", "Again, fix a total order $\\le $ on the variables of $\\hat{R}$ such that for $\\sigma = \\pm $ $x_{\\mathsf {J}_\\sigma } < x_{\\mathsf {J}^{\\prime }_\\sigma }$ for any filters $\\mathsf {J}_\\sigma ,\\mathsf {J}^{\\prime }_\\sigma \\in \\mathcal {J}(P_\\sigma )$ with $\\mathsf {J}_\\sigma \\subset \\mathsf {J}^{\\prime }_\\sigma $ , and $x_{\\mathsf {J}_+} < x_{\\mathsf {J}_-}$ for any filters $\\mathsf {J}_+ \\in \\mathcal {J}(P_+)$ and $\\mathsf {J}_- \\in \\mathcal {J}(P_-)$ , and denote by $\\le _{rev}$ the reverse lexicographic term order on $\\hat{R}$ induced by this order on the variables.", "Theorem 5.2 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then a Gröbner basis for $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ is given by the binomials $\\underline{x_{\\mathsf {J}_\\sigma }\\, x_{\\mathsf {J}_\\sigma ^{\\prime }}} \\ &- \\ x_{\\mathsf {J}_\\sigma \\cup \\mathsf {J}_\\sigma ^{\\prime }} \\, x_{\\mathsf {J}_\\sigma \\cap \\mathsf {J}_\\sigma ^{\\prime }}\\\\\\multicolumn{2}{l}{\\text{for incomparable filters $\\mathsf {J}_\\sigma ,\\mathsf {J}_\\sigma ^{\\prime }\\in \\mathcal {J}(P_\\sigma )$ and $\\sigma = \\pm $, and }}\\\\\\underline{x_{\\mathsf {J}_+}\\, x_{\\mathsf {J}_-}} \\ &- \\ x_{\\mathsf {J}_+\\setminus A} \\, x_{\\mathsf {J}_-\\setminus A}$ for filters $\\mathsf {J}_+ \\in \\mathcal {J}(P_+), \\mathsf {J}_- \\in \\mathcal {J}(P_-)$ such that $A := \\min (\\mathsf {J}_+)\\cap \\min (\\mathsf {J}_-) \\ne \\varnothing $ .", "It is clear that binomials of the form (REF ) and () are contained in $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ and hence it suffices to show that their leading terms generate $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ .", "For this, let us take a closer look at the combinatorics of $\\hat{\\phi }$ .", "Let $\\mathcal {G}$ be the collection of binomial given in (REF ) and () and let $f = \\underline{m_1} - m_2$ be an element of the form (REF ).", "By reducing $f$ by the binomial of (REF ), we can view $f$ as a quadruple $\\begin{aligned}\\mathsf {J}_{+1} \\subset \\mathsf {J}_{+2} \\subset \\cdots \\subset \\mathsf {J}_{+k_+} &\\quad \\qquad \\mathsf {J}_{-1} \\subset \\mathsf {J}_{+2} \\subset \\cdots \\subset \\mathsf {J}_{-k_-}\\\\\\mathsf {J}_{+1}^{\\prime } \\subset \\mathsf {J}_{+2}^{\\prime } \\subset \\cdots \\subset \\mathsf {J}_{+k_+}^{\\prime } &\\quad \\qquad \\mathsf {J}_{-1}^{\\prime } \\subset \\mathsf {J}_{+2}^{\\prime } \\subset \\cdots \\subset \\mathsf {J}_{-k_-}^{\\prime }.\\end{aligned}$ From the definition of $\\hat{\\phi }$ it follows that such a quadruple defines a binomial in $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ if and only if for any $q \\in P$ $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r} \\rbrace - \\max \\lbrace s : q \\notin \\mathsf {J}_{-s} \\rbrace \\ = \\ \\max \\lbrace r : q \\notin \\mathsf {J}_{+r}^{\\prime } \\rbrace - \\max \\lbrace s : q \\notin \\mathsf {J}_{-s}^{\\prime }\\rbrace .$ and we note the following implication.", "Lemma 5.3 Let the collection of filters in (REF ) correspond to a binomial $f \\in \\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ and let $q \\in P$ .", "Then there is some $1\\le i \\le k_+$ such that $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ if and only if there is some $1 \\le j \\le k_-$ such that $q \\in \\mathsf {J}_{-j} \\setminus \\mathsf {J}_{-j}^{\\prime }$ .", "If $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ , then $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r} \\rbrace < i$ and $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r}^{\\prime } \\rbrace \\ge i$ and (REF ) implies that $q \\in \\mathsf {J}_{-j} \\setminus \\mathsf {J}_{-j}^{\\prime }$ for some $j$ .", "The other direction is identical.", "We call $q \\in P$ moving if it satisfies one of the two equivalent conditions of Lemma REF .", "Let $f = \\underline{m_1} - m_2 \\in \\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ be a binomial represented by a collection of filters given by (REF ).", "If $k_- = 0$ or $k_+= 0$ , then the Hibi relations (REF ) for $P_-$ or $P_+$ together with Theorem REF yields the result.", "Thus, we assume that $k_-,k_+ > 0$ and we need to show that there are filters $\\mathsf {J}_{+i}$ and $\\mathsf {J}_{-j}$ such that $\\min (\\mathsf {J}_{+i}) \\cap \\min (\\mathsf {J}_{-j}) \\ne \\varnothing $ .", "Observe that there is at least one moving element.", "Indeed, $\\mathsf {J}_{+1}\\nsubseteq \\mathsf {J}_{+1}^{\\prime }$ and hence $\\mathsf {J}_{+1} \\setminus \\mathsf {J}_{+1}^{\\prime } \\ne \\varnothing $ .", "Otherwise, $x_{\\mathsf {J}_{+1}} <x_{\\mathsf {J}_{+1}^{\\prime }}$ and the reverse lexicographic term order $\\le _{rev}$ would not select $m_1$ as the lead term of $f$ .", "Among all moving elements, choose $q$ to be minimal with respect to $\\preceq _+$ and $\\preceq _-$ .", "Since $\\mathbf {P}$ is a compatible double poset, such a $q$ exists.", "But then, if $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ , then $q \\in \\min (\\mathsf {J}_{+i})$ .", "The same holds true for $\\mathsf {J}_{-j}$ and shows that $\\underline{m_1}$ is divisible by the leading term of a binomial of type ().", "Gröbner bases, faces, and triangulations In light of the regular and unimodular triangulation of $\\mathcal {O}(P)$ given in [34] (and recalled in Section REF ), the Hibi ring $\\mathcal {O}(P)]$ is exactly the affine semigroup ring associated to $\\mathcal {O}(P)$ .", "That is, $\\mathcal {O}(P)]$ is the standard graded $-algebra associated to thenormal affine semigroup$$\\lbrace (f,k) \\in \\mathbb {Z}^P\\times \\mathbb {Z}: k \\ge 0, f \\in k \\mathcal {O}(P)\\rbrace .$$$ For a lattice polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ , Sturmfels [36] described a beautiful relationship between regular triangulations of $\\mathcal {P}$ and radicals of initial ideals of the toric ideal $\\mathrm {I}_\\mathcal {P}$ .", "It follows from Theorem REF that $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is a squarefree ideal generated by quadratic monomials.", "Appealing to [36], this yields the following refinement of Theorem REF .", "Corollary 5.4 Let $\\mathbf {P}$ be a compatible double poset.", "Then $\\mathcal {O}(\\mathbf {P})$ has a regular triangulation whose underlying simplicial complex is exactly $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ .", "The initial ideal $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is already radical and Theorem 8.3 of [36] yields that $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is the Stanley-Reisner ideal of a regular triangulation of $\\mathcal {O}(\\mathbf {P})$ .", "Hence, a collection $C = C_1 \\uplus C_2 \\subseteq \\mathcal {J}(\\mathbf {P})$ forms a simplex in the triangulation of $\\mathcal {O}(\\mathbf {P})$ if and only if $\\prod _{\\mathsf {J}_+ \\in C_+} x_{\\mathsf {J}_+}\\prod _{\\mathsf {J}_- \\in C_-} x_{\\mathsf {J}_-} \\ \\notin \\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}).$ Translating the conditions given in Theorem REF , this is the case if and only if $C_\\sigma = C \\cap \\mathcal {J}(P_\\sigma )$ is a chain of filters for $\\sigma = \\pm $ and $C_+, C_-$ are non-interfering chains.", "This is exactly the definition of the flag complex $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ .", "Using the orbit-cone correspondence for affine toric varieties (see, for example, [5]), we can give an algebraic perspective on Theorem REF .", "We are in a particularly nice situation as the polytopes we consider have unimodular triangulations and hence the affine semigroup rings are generated in degree 1 by the vertices of the underlying polytope.", "Lemma 5.5 Let $V \\subset \\Lambda $ be a finite set of lattice points and $\\mathcal {P}=\\operatorname{conv}(V)$ the corresponding lattice polytope.", "If $\\mathrm {I}\\subseteq x_v : v\\in V]$ is the toric ideal of the homogenization $\\lbrace (v,1) : v\\in V \\rbrace \\subseteq \\Lambda \\times \\mathbb {Z}$ , then for any subset $U \\subseteq V$ , we have that $\\operatorname{conv}(U)$ is a face of $\\mathcal {P}$ with $\\operatorname{conv}(U)\\cap V = U$ if and only if $f(\\mathbf {1}_U) \\ = \\ 0 \\qquad \\text{ for all } f \\in \\mathrm {I}.$ Let $\\mathrm {L}\\subseteq \\mathcal {J}(\\mathbf {P})$ .", "Then for $\\sigma = \\pm $ and $\\mathsf {J}_\\sigma , \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathcal {J}(P_\\sigma )$ Lemma REF and (REF ) of Theorem REF states that $\\mathsf {J}_\\sigma , \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathrm {L}_\\sigma \\quad \\Longleftrightarrow \\quad \\mathsf {J}_\\sigma \\cup \\mathsf {J}_\\sigma ^{\\prime },\\mathsf {J}_\\sigma \\cap \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathrm {L}_\\sigma .$ That is, if and only if $\\mathrm {L}_\\sigma $ is an embedded Thus, $\\mathrm {L}_\\sigma $ is an embedded sublattice of $\\mathcal {J}(P_\\sigma )$ .", "The same reasoning shows that the conditions imposed by () are equivalent to those of Lemma REF .", "We can also use Sturmfels' result in the other direction to find Gröbner bases.", "For a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ we may define the subring $\\mathcal {C}({\\mathbf {P}})] \\subseteq \\hat{R}$ generated by the monomials $t_+ s^{\\min (\\mathsf {J}_+)}$ and $t_+ (s^{\\min (\\mathsf {J}_-)})^{-1}$ for filters $\\mathsf {J}_+ \\subseteq P_+$ and $\\mathsf {J}_- \\subseteq P_-$ .", "The corresponding toric ideal $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ is contained in the ring $T = x_{A_+},x_{A_-} ]$ , where $A_\\sigma $ ranges over all anti-chains in $P_\\sigma $ for $\\sigma = \\pm $ .", "Since $\\mathcal {O}(\\mathbf {P})$ is the stable set polytope of the perfect double graph $G(\\mathbf {P})$ , it follows from Corollary REF that $\\mathcal {C}({\\mathbf {P}})]$ is the normal affine semigroup ring associated to the lattice polytope $\\mathcal {C}({\\mathbf {P}})$ .", "To describe a Gröbner basis for, we introduce the following notation.", "For $\\sigma = \\pm $ and two antichains $A,A^{\\prime } \\subseteq P_\\sigma $ define $A \\sqcup A^{\\prime }\\ := \\ \\min (A \\cup A^{\\prime })$ and $A \\sqcap A^{\\prime } \\ := \\ (A \\cap A^{\\prime }) \\cup (\\max (A \\cup A^{\\prime }) \\setminus \\min (A \\cup A^{\\prime })).$ For a subset $S \\subseteq P$ and $\\sigma = \\pm $ , we write $\\langle S \\rangle _\\sigma := \\lbrace a \\in P: a\\succeq _\\sigma s \\text{ for some } s \\in S \\rbrace $ for the filter in $P_\\sigma $ generated by $S$ .", "Theorem 5.6 Let $\\mathbf {P}$ be a double poset.", "Then a Gröbner basis for $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ is given by the binomials $\\underline{x_A \\, x_{A^{\\prime }}} \\ &- \\ x_{A \\sqcup A^{\\prime }}\\, x_{A \\sqcap A^{\\prime }} \\qquad \\langle A \\rangle _\\sigma ,\\langle A \\rangle _\\sigma \\in \\mathcal {J}(P_\\sigma ) \\text{ incomparable}\\\\\\multicolumn{2}{l}{\\text{ for antichains $A,A^{\\prime } \\subset P_\\sigma $ for $\\sigma = \\pm $and }}\\\\\\underline{x_{A_+} \\, x_{A_-}} \\ &- \\ x_{A_+ \\setminus A_-}x_{A_-\\setminus A_+}\\qquad \\text{ for antichains } A_\\sigma \\subseteq P_\\sigma .$ It is easy to verify that the given binomials are contained in $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ .", "Moreover, the leading monomials are exactly the minimal non-faces of the unimodular triangulation of Corollary REF .", "The result now follows from Theorem 8.3 in [36].", "Remark 1 Reformulated in the language of double posets, Hibi, Matsuda, and Tsuchiya [22], [21], [23] computed related Gröbner bases of the toric ideals associated with the polytopes $\\Gamma (\\mathcal {O}(P_+),\\mathcal {O}(P_-))$ (in the compatible case), $\\Gamma (\\mathcal {C}({P_+}),\\mathcal {C}({P_-}))$ , and $\\Gamma (\\mathcal {O}(P_+),\\mathcal {C}({P_-}))$ for a double poset $\\mathbf {P}$ .", "See the paragraph before Corollary REF for notation." ], [ "Triangulations and transfers", "If $\\mathbf {P}= \\mathbf {P}_\\circ = (P,\\preceq ,\\preceq )$ is induced by a single poset, then Corollaries REF and REF assure us that $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ are 2-level and [37] implies that both polytopes have unimodular triangulations with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ .", "In this section we give explicit triangulations of the double chain polytope $\\mathcal {C}({\\mathbf {P}})$ and, in the compatible case, of the double order polytope $\\mathcal {O}(\\mathbf {P})$ .", "These triangulations will be regular, unimodular, and flag.", "To that end, we will generalize Stanley's approach [34] from poset polytopes to double poset polytopes.", "We put the triangulation to good use and explicitly compute the Ehrhart polynomial and the volume of $\\mathcal {C}({\\mathbf {P}})$ and, in case that $\\mathbf {P}$ is compatible, of $\\mathcal {O}(\\mathbf {P})$ ." ], [ "Triangulations of double poset polytopes", "Recall from the introduction that for a poset $(P,\\preceq )$ , the order polytope $\\mathcal {O}(P)$ parametrizes all order preserving maps $f : P\\rightarrow [0,1]$ .", "Any $f \\in \\mathcal {O}(P)$ induces a partial order $P_f =(P,\\preceq _f)$ by $a \\prec _f b$ if $a \\prec b$ or, when $a,b$ are incomparable, if $f(a) < f(b)$ .", "Clearly, $\\preceq _f$ refines $\\preceq $ and hence $\\mathcal {O}(P_f) \\subseteq \\mathcal {O}(P)$ .", "Since filters in $P_f$ are filters in $P$ , $\\mathcal {O}(P_f)$ is a vertex-induced subpolytope of $\\mathcal {O}(P)$ .", "If $f$ is generic, that is, $f(a) \\ne f(b)$ for all $a \\ne b$ , then $\\preceq _f$ is a total order and $\\mathcal {O}(P_f)$ is a unimodular simplex of dimension $|P|$ .", "Stanley showed that the collection of all these simplices constitute a unimodular triangulation of $\\mathcal {O}(P)$ .", "More precisely, this canonical triangulation of $\\mathcal {O}(P)$ is given by the order complex $\\Delta (\\mathcal {J}(P))$ of $\\mathcal {J}(P)$ , i.e., the collection of chains in the Birkhoff lattice of $P$ ordered by inclusion.", "Since a collection of filters $\\mathsf {J}_0,\\dots ,\\mathsf {J}_k$ is not a chain if and only if $\\mathsf {J}_i\\lnot \\subseteq \\mathsf {J}_j$ and $\\mathsf {J}_j \\lnot \\subseteq \\mathsf {J}_i$ for some $0\\le i,j \\le k$ , the canonical triangulation is flag.", "Stanley [34] elegantly transferred the canonical triangulation of $\\mathcal {O}(P)$ to $\\mathcal {C}({P})$ in the following sense.", "Define the transfer map $\\phi _P: \\mathcal {O}(P) \\rightarrow \\mathcal {C}({P})$ by $(\\phi _Pf)(b) \\ := \\ \\min \\lbrace f(b) - f(a) : a \\prec b \\rbrace ,$ for $f \\in \\mathcal {O}(P)$ and $b \\in P$ .", "This is a piecewise linear map and the domains of linearity are exactly the full-dimensional simplices $\\mathcal {O}(P_f)$ for generic $f$ .", "In particular, $\\phi _P(\\mathbf {1}_\\mathsf {J}) =\\mathbf {1}_{\\min (\\mathsf {J})}$ for any filter $\\mathsf {J}\\subseteq P$ , which shows that $\\phi _P$ maps $\\mathcal {O}(P)$ into $\\mathcal {C}({P})$ .", "To show that $\\phi _P$ is a PL homeomorphism of the two polytopes, Stanley gives an explicit inverse $\\psi _P: \\mathcal {C}({P}) \\rightarrow \\mathcal {O}(P)$ by $(\\psi _Pg)(b) \\ := \\ \\max \\lbrace g(a_0) + \\cdots + g(a_{k-1}) +g(a_k) : a_0 \\prec \\cdots \\prec a_{k-1} \\prec a_k \\preceq b \\rbrace ,$ for any $g \\in \\mathcal {C}({P})$ .", "Note that our definition of $\\psi _P$ differs from that in [34] in that we do not require that the chain has to end in $b$ .", "This will be important later.", "It can be easily checked that $\\psi _P$ is an inverse to $\\phi _P$ .", "Hence, the simplices $\\operatorname{conv}( \\mathbf {1}_{\\min (\\mathsf {J}_0)}, \\dots , \\mathbf {1}_{\\min (\\mathsf {J}_k)} ) \\quad \\text{for } \\quad \\lbrace \\mathsf {J}_0 \\subseteq \\cdots \\subseteq \\mathsf {J}_k \\rbrace \\in \\Delta (\\mathcal {J}(P))$ constitute a flag triangulation of $\\mathcal {C}({P})$ .", "We will follow the same approach as Stanley but, curiously, it will be simpler to start with a triangulation of $\\mathcal {C}({\\mathbf {P}})$ .", "Recall from Section REF that a pair of chains $C = C_+ \\uplus C_-$ with $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ is non-interfering if $\\min (\\mathsf {J}_+)\\cap \\min (\\mathsf {J}_-) = \\varnothing $ for any $\\mathsf {J}_\\sigma \\in C_\\sigma $ for $\\sigma = \\pm $ .", "Corollary 4.1 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then a regular triangulation of $\\mathcal {C}({\\mathbf {P}})$ is given as follows: The $(k-1)$ -dimensional simplices are in bijection to non-interfering pairs of chains $C = C_+ \\uplus C_+ \\subseteq \\mathcal {J}(\\mathbf {P})$ with $|C| = |C_+| +|C_-| = k$ .", "Moreover, the triangulation is regular, unimodular (with respect to $\\mathbb {A}$ ), and flag.", "The canonical triangulation of $\\mathcal {C}({P_\\sigma })$ is regular, unimodular, and flag for $\\sigma =\\pm $ .", "As described above, its $(l_\\sigma -1)$ -simplices are in bijection to chains $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ of length $|C_\\sigma | = l_\\sigma $ .", "More precisely, the simplex corresponding to $C_\\sigma $ is given by $F(C_\\sigma ) = \\operatorname{conv}( \\mathbf {1}_{\\min (\\mathsf {J}_\\sigma )} : \\mathsf {J}_\\sigma \\in C_\\sigma ).$ By Theorem REF applied to $\\mathcal {C}({\\mathbf {P}}) ={2\\mathcal {C}({P_+}) \\boxminus 2\\mathcal {C}({P_-})}$ , it follows that a unimodular and flag triangulation is given by the joins $2F(C_+) * 2F(C_-)$ for all chains $C_\\sigma \\subseteq \\mathcal {J}(P_\\sigma )$ such that $F(C_+)$ and $F(C_-)$ lie in complementary coordinate subspaces.", "This, however, is exactly the case when $\\min (\\mathsf {J}_+) \\cap \\min (\\mathsf {J}_-) = \\varnothing $ for all $\\mathsf {J}_\\sigma \\in C_\\sigma $ for $\\sigma =\\pm $ .", "Corollary REF gives a canonical triangulation that combinatorially can be described as a subcomplex of $\\Delta (\\mathcal {J}(\\mathbf {P})) =\\Delta (\\mathcal {J}(P_+)) * \\Delta (\\mathcal {J}(P_-))$ , called the non-interfering complex $\\Delta ^\\mathrm {ni}(\\mathbf {P}) \\ := \\ \\lbrace C : C = C_+ \\uplus C_- \\in \\Delta (\\mathcal {J}(\\mathbf {P})) ,\\, C \\text{ non-interfering} \\rbrace .$ Associating $\\Delta (\\mathcal {J}(P))$ to a poset $P$ is very natural and can be motivated, for example, from an algebraic-combinatorial approach to the order polynomial (cf. [3]).", "It would be very interesting to know if the association $\\mathbf {P}$ to $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ is equally natural from a purely combinatorial perspective.", "Given a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ , we define a piecewise linear map $\\Psi _{\\mathbf {P}} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ by $\\Psi _{\\mathbf {P}}(g) \\ := \\ \\psi _{P_+}(g) \\ - \\ \\psi _{P_-}(-g),$ for any $g \\in \\mathbb {R}^P$ .", "Here, we use that $\\psi _{}$ , as given in (REF ), is defined on all of $\\mathbb {R}^P$ with the following important property: For $g \\in \\mathbb {R}^P$ , let us write $g = g^+ - g^-$ , where $g^+,g^- \\in \\mathbb {R}_{\\ge 0}^P$ with disjoint supports.", "Then $\\psi _{P_\\sigma }(g) = \\psi _{P_\\sigma }(g^+)$ for $\\sigma = \\pm $ .", "Thus, $\\Psi _{\\mathbf {P}}(g) \\ = \\ \\psi _{P_+}(g^+) - \\psi _{P_-}(g^-),$ for any $g \\in \\mathbb {R}^P$ .", "In particular, $\\Psi _{\\mathbf {P}}$ takes $\\lambda \\mathcal {C}({P_+}) - \\mu \\mathcal {C}({P_-})$ into $\\lambda \\mathcal {O}(P_+) - \\mu \\mathcal {O}(P_-)$ for any $\\lambda ,\\mu \\ge 0$ .", "Indeed, for any pair of antichains $A_\\sigma \\subseteq P_\\sigma $ , first observe that $\\mathbf {1}_{A_+} - \\mathbf {1}_{A_-} =\\mathbf {1}_{A_+\\setminus A_-} - \\mathbf {1}_{A_-\\setminus A_+}$ .", "Hence, it suffices to assume that $A_+ \\cap A_- = \\varnothing $ .", "We compute $\\Psi _{\\mathbf {P}}(\\mathbf {1}_{A_+} - \\mathbf {1}_{A_-} ) \\ = \\ \\mathbf {1}_{\\mathsf {J}_+} -\\mathbf {1}_{\\mathsf {J}_-},$ where for $\\sigma = \\pm $ , $\\mathsf {J}_\\sigma \\subseteq P_\\sigma $ is the filter generated by $A_\\sigma $ .", "If $\\mathbf {P}$ is a compatible double poset, then Corollary REF implies that $\\Psi _{\\mathbf {P}}$ is a surjection on vertex sets.", "Lemma 4.2 Let $\\mathbf {P}=(\\mathcal {P},\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then $\\Psi _{\\mathbf {P}} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ is a lattice-preserving piecewise linear isomorphism.", "It follows directly from (REF ) that $\\Psi _{\\mathbf {P}}$ is piecewise linear.", "To show that $\\Psi _{\\mathbf {P}}$ is an isomorphism, we explicitly construct for $f\\in \\mathbb {R}^P$ a $g \\in \\mathbb {R}^P$ such that $\\Psi _{\\mathbf {P}}(g)=f$ .", "Since $\\mathbf {P}$ is compatible, we can assume that $P= \\lbrace a_1,\\dots ,a_n\\rbrace $ such that $a_i \\prec _+ a_j$ or $a_i \\prec _- a_j$ implies $i < j$ .", "It follows from (REF ) that $\\Psi _{\\mathbf {P}}(g^{\\prime })(a_1) = g^{\\prime }(a_1)$ for any $g^{\\prime } \\in \\mathbb {R}^P$ and hence, we can set $g(a_1) := f(a_1)$ .", "Now assume that $g$ is already defined on $D_k := \\lbrace a_1,\\dots , a_k \\rbrace $ for some $k$ .", "For $g^{\\prime }\\in \\mathbb {R}^P$ observe that $\\psi _{P_+}(g^{\\prime })(a_{k+1}) \\ = \\ \\max (g^{\\prime }(a_{k+1}),0) + r$ where $r=0$ or $r = \\psi _{P_+}(g^{\\prime })(a_i)$ for some $i \\le k$ .", "Analogously, $\\psi _{P_-}(-g^{\\prime })(a_{k+1}) \\ = \\ \\max (-g^{\\prime }(a_{k+1}),0) + s$ where $s=0$ or $s = \\psi _{P_-}(-g^{\\prime })(a_j)$ for some $j \\le k$ .", "Thus, we set $g(a_{k+1})\\ := \\ f(a_{k+1}) - r + s$ This uniquely determines $g$ by induction on $k$ .", "To prove that $\\Psi _{\\mathbf {P}}$ is lattice-preserving, observe that by (REF ) we have $\\Psi _{\\mathbf {P}}(\\mathbb {Z}^P)\\subseteq \\mathbb {Z}^P$ .", "Moreover, if $f=\\Psi _{\\mathbf {P}}(g)$ with $f\\in \\mathbb {Z}^P$ and the above construction shows that $g\\in \\mathbb {Z}^P$ .", "Hence, $\\Psi _{\\mathbf {P}}(\\mathbb {Z}^P)\\subseteq \\mathbb {Z}^P$ , which finishes the proof.", "Using the notation from (REF ) in Section REF , the lemma shows that $\\lbrace \\overline{F}(C) : C \\in \\ \\Delta ^\\mathrm {ni}(\\mathbf {P}) \\rbrace $ is a realization of the flag simplicial complex $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ by unimodular simplices inside $\\mathcal {O}(\\mathbf {P})$ .", "Using Gröbner bases in Section , we will show the following result.", "Theorem 4.3 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then the map $(g,t) \\mapsto (\\Psi _{\\mathbf {P}}(g),t)$ is a piecewise linear homeomorphism from $\\mathbb {R}^{P}\\times \\mathbb {R}$ to itself that preserves the lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ .", "In particular, it maps $\\mathcal {C}({\\mathbf {P}})$ to $\\mathcal {O}(\\mathbf {P})$ and hence $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ is a regular, unimodular, and flag triangulation of $\\mathcal {O}(\\mathbf {P})$ .", "By the previous lemma, (REF ) is a realization of $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ in $\\mathcal {O}(\\mathbf {P})$ without new vertices.", "Moreover, every maximal simplex contains the edge $e = \\operatorname{conv}\\lbrace (\\mathbf {0},1), (\\mathbf {0},-1)\\rbrace $ .", "Hence, it suffices to show that for every maximal simplex in $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ , the supporting hyperplane of every facet not containing $e$ is supporting for $\\mathcal {O}(\\mathbf {P})$ .", "Let $C = \\lbrace \\mathsf {J}_{+0} \\subset \\cdots \\subset \\mathsf {J}_{+k}, \\mathsf {J}_{-0}\\subset \\cdots \\subset \\mathsf {J}_{-l}\\rbrace $ be two maximal non-interfering chains.", "Set $A_{+i} := \\min (\\mathsf {J}_{+i})$ for $1 \\le i \\le k$ and $A_{-0},\\dots ,A_{-l}$ likewise.", "It follows that $P_1 = \\bigcup A_{+i}$ and $P_2 = \\bigcup A_{-j}$ give a partition of $P$ .", "In particular, since $C$ was maximal, we have that $\\lbrace a^+_{k-i-1}\\rbrace = A_{+i} \\setminus A_{+(i-1)}$ and $P_1 = \\lbrace a^+_1,\\dots ,a^+_k\\rbrace $ .", "In particular, if $a^+_s\\prec _+ a^+_t$ , then $s < t$ .", "The same argument yields $P_2 =\\lbrace a^-_1,\\dots ,a^-_l\\rbrace $ and the labelling is a linear extension of $(P_2,\\preceq _-)$ .", "We focus on $P_1$ ; the argument for $P_2$ is analogous.", "Pick the maximal chain $D$ in $(P_1,\\preceq _+)$ starting in $a^+_k$ .", "Then $A_{+i} \\cap D\\ne \\varnothing $ for all $i > 0$ and hence $\\lbrace (g,t) \\in \\mathbb {R}^{P_1}:\\langle {\\mathbf {1}_D,g}\\rangle = 1\\rbrace $ is the hyperplane for the maximal simplex in the triangulation of $\\mathcal {C}({P_1,\\preceq _+})$ corresponding to $A_{+0},\\dots ,A_{+k}$ and not containing the origin.", "Thus, one of the two hyperplanes supporting a facet of the simplex in $\\mathcal {C}({\\mathbf {P}})$ corresponding to $C$ is given by $H := \\lbrace (g,t) \\in \\mathbb {R}^P\\times \\mathbb {R}:\\ell (g,t) = 1 \\rbrace $ where $\\ell (g,t) = \\langle {\\mathbf {1}_D,g}\\rangle - t$ .", "Now, $\\Psi _{\\mathbf {P}}$ is linear on the simplex $C$ in $\\mathcal {C}({\\mathbf {P}})$ and can be easily inverted.", "Since $\\mathbf {P}$ is compatible, we can find a linear extension $\\sigma : P\\rightarrow \\lbrace 1,\\dots ,|P|\\rbrace $ that respects the constructed linear extensions on $P_1$ and $P_2$ .", "On the image of $C$ under $\\Psi _{\\mathbf {P}}$ , the inverse is given by the linear transformation $T :\\mathbb {R}^P\\times \\mathbb {R}\\rightarrow \\mathbb {R}^P\\times \\mathbb {R}$ with $T(f,t) = (f^{\\prime },t)$ and $f^{\\prime } : P\\rightarrow \\mathbb {R}$ is defined as follows.", "If $b \\in P_1$ , then by $f^{\\prime }(b) =f(b) - f(\\bar{b})$ , $\\bar{b} \\prec _+ b$ is a cover relation and $\\sigma (\\bar{b})$ is maximal.", "If $b \\in P_2$ , we choose $\\bar{b}$ covered by $b$ in with respect to $\\preceq _-$ .", "It can now be checked that $\\ell \\circ T = L_C$ for some alternating chain $C$ .", "Thus $H$ is supporting for $\\mathcal {O}(\\mathbf {P})$ and the map $\\Psi _{\\mathbf {P}}$ maps $\\mathcal {C}({\\mathbf {P}})$ onto $\\mathcal {O}(\\mathbf {P})$ .", "Theorem REF does not extend to the non-compatible case as the following example shows.", "Example 9 Consider the double poset $\\mathbf {P}= ([2],\\le ,\\ge )$ , that is, $P_+$ is the 2-chain $\\lbrace 1,2\\rbrace $ and $P_-$ is the opposite poset.", "Then $\\mathcal {C}({P_+})= \\mathcal {C}({P_-}) = T := \\lbrace x \\in \\mathbb {R}^2 : x \\ge 0, x_1 + x_2 \\le 1\\rbrace $ and $\\mathcal {C}({\\mathbf {P}})$ is a three-dimensional octahedron with volume $\\frac{16}{3}$ .", "Any triangulation of the octahedron has at least four simplices.", "In contrast, $\\mathcal {O}(P_-) = \\mathbf {1} - \\mathcal {O}(P_+)$ and hence $\\mathcal {O}(\\mathbf {P})$ is linearly isomorphic to a prism over a triangle with volume 4.", "Any triangulation of the prism has exactly 3 tetrahedra." ], [ "Volumes and Ehrhart polynomials", "The canonical subdivision of $\\mathcal {O}(P)$ makes it easy to compute its volume.", "For a generic $f \\in \\mathcal {O}(P)$ , there is a unique linear extension $\\sigma : P\\rightarrow \\lbrace 1,2,\\dots ,d\\rbrace $ where $d := |P|$ such that $\\mathcal {O}(P_f) \\ = \\ \\lbrace h \\in \\mathbb {R}^P : 0 \\le h(\\sigma ^{-1}(1)) \\le \\cdots \\le h(\\sigma ^{-1}(d)) \\le 1 \\rbrace .$ In particular, the full-dimensional simplex $\\mathcal {O}(P_f)$ is unimodular relative to $\\mathbb {Z}^P\\subseteq \\mathbb {R}^P$ and has volume $\\operatorname{vol}(\\mathcal {O}(P_f)) =\\frac{1}{|P|!", "}$ .", "If we denote by $e(P)$ the number of linear extensions of $P$ , then Stanley [34] showed the following.", "Corollary 4.4 $\\operatorname{Vol}(\\mathcal {O}(P)) = |P|!", "\\cdot \\operatorname{vol}(\\mathcal {O}(P)) = e(P)$ .", "For the Ehrhart polynomial $\\mathrm {Ehr}_{\\mathcal {O}(P)}(n)$ of $\\mathcal {O}(P)$ it suffices to interpret the lattice points in $n \\mathcal {O}(P)$ for $n > 0$ .", "Every point in $n\\mathcal {O}(P) \\cap \\mathbb {Z}^P$ corresponds to an order preserving map $\\phi : P\\rightarrow [n+1]$ .", "Counting order preserving maps is classical [35]: the order polynomial $\\Omega _P(n)$ of $P$ counts the number of order preserving maps into $n$ -chains.", "The strict order polynomial $\\Omega ^\\circ _P(n)$ counts the number of strictly order preserving maps $f : P\\rightarrow [n]$ , that is, $f(a) < f(b)$ for $a \\prec b$ .", "The transfer map $\\phi _P$ as well as its inverse $\\psi _P$ (given in (REF ) and (REF ), respectively) both take lattice points to lattice points and hence, together with Theorem REF , yield the following result.", "Corollary 4.5 Let $P$ be a finite poset.", "Then for every $n > 0$ $\\Omega _P(n+1)& \\ = \\ \\mathrm {Ehr}_{\\mathcal {O}(P)}(n) \\ = \\ \\mathrm {Ehr}_{\\mathcal {C}({P})}(n)\\\\\\multicolumn{2}{l}{\\text{ and }}\\\\(-1)^{|P|}\\Omega ^\\circ _P(n-1) &\\ = \\ \\mathrm {Ehr}_{\\mathcal {O}(P)}(-n) \\ = \\ \\mathrm {Ehr}_{\\mathcal {C}({P})}(-n).$ In particular, $\\operatorname{vol}(\\mathcal {O}(P)) = \\operatorname{vol}(\\mathcal {C}({P}))$ .", "This is an interesting result as it implies that the number of linear extensions of a poset $P$ only depends on the comparability graph $G(P)$ .", "Theorem 4.6 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then $\\overline{\\mathcal {C}}(\\mathbf {P})$ is a lattice polytope with respect to $\\mathbb {Z}^P$ and $\\mathrm {Ehr}_{\\overline{\\mathcal {C}}(\\mathbf {P})}(n-1) \\ &= \\ \\sum _{P= P_1 \\uplus P_2}\\Omega ^\\circ _{(P_1,\\preceq _+)}(n-1) \\cdot \\Omega _{(P_2,\\preceq _-)}(n) \\text{ and }\\\\\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ &= \\ \\sum _{P= P_1 \\uplus P_2} \\binom{|P|}{|P_1|}e(P_1,\\preceq _+)\\cdot e(P_2,\\preceq _-).$ Since $\\mathcal {C}({P}) = \\mathcal {P}_{G(P)}$ is a dual integral anti-blocking polytope, the first identity follows from Corollary REF and Corollary REF .", "The second identity follows from Corollary REF and Corollary REF .", "Notice from Theorem REF we can also deduce a closed formula for the Ehrhart polynomial of $\\mathcal {C}({\\mathbf {P}})$ with respect to the lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ and, by substituting $\\frac{1}{2}k$ for $k$ , also with respect to the affine lattice $\\mathbb {A}$ .", "These formulas are not very enlightening and instead we record the normalized volume.", "Note that the minimal Euclidean volume of a full-dimensional simplex with vertices in $\\mathbb {A}= \\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ is $\\frac{2^{|P|+1}}{(|P|+1)!", "}$ .", "Corollary 4.7 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a double poset.", "Then the normalized volume with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ is $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{P= P_1 \\uplus P_2}e(P_1,\\preceq _+) \\cdot e(P_2,\\preceq _-).$ We leave it to the reader to give direct combinatorial interpretations of the volume and the Ehrhart polynomials for double posets.", "It follows directly from (REF ) that $\\Psi _{P_{\\pm }} : \\mathbb {R}^P\\rightarrow \\mathbb {R}^P$ maps lattice points to lattice points.", "If $\\mathbf {P}$ is compatible, then the proof of Lemma REF asserts that $\\Psi _{P_{\\pm }}$ is in fact lattice preserving.", "Hence, we record an analog to Corollary REF .", "Corollary 4.8 If $\\mathbf {P}$ is a compatible double poset, then $\\mathcal {O}(\\mathbf {P})$ and $\\mathcal {C}({\\mathbf {P}})$ have the same Ehrhart polynomials and hence the same volumes.", "The formulas of Theorem REF are particularly simple when $\\mathbf {P}$ is special or anti-special.", "We illustrate these cases at some simple examples.", "Example 10 For the 'XW'-double poset we have $\\operatorname{Vol}(\\mathcal {O}(\\mathbf {P}_{XW})) \\ = \\ \\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}_{XW}}))\\ = \\ \\tfrac{6!", "}{2^6} \\operatorname{vol}(\\mathcal {C}({\\mathbf {P}_{XW}}))\\ = \\ 128$ and $\\operatorname{Vol}(\\overline{\\mathcal {O}}(\\mathbf {P}_{XW}))= \\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P}_{XW}))= 6!", "\\operatorname{vol}(\\overline{\\mathcal {C}}(\\mathbf {P}_{XW}))= 880$ .", "Example 11 As the following examples are all compatible, the given values also give the normalized volumes of the respective (reduced) double order polytopes.", "Let $\\mathbf {P}= ([d],\\le ,\\le )$ be the double chain on $d$ elements.", "Then $\\mathcal {C}({\\mathbf {P}})$ is a crosspolytope and $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) = 2^d$ and it follows from Vandermonde's identity that $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ d!", "\\operatorname{vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i}^2 \\ = \\ \\binom{2d}{d}.$ If $\\mathbf {P}$ is the double anti-chain on $d$ elements, then $\\mathcal {C}({\\mathbf {P}})$ is isomorphic to $[0,2]^{d} \\times [-1,1]$ and its normalized volume is $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\tfrac{(d+1)!", "}{2^{d+1}}\\operatorname{vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i} i!(d-i)!", "\\ = \\ (d+1)!.$ Likewise, $\\overline{\\mathcal {C}}(\\mathbf {P})$ is isomorphic to $[-1,1]^d$ , which can be decomposed into $2^d$ unit cubes.", "Consequently, its normalized volume is $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d \\binom{d}{i}^2 i!(d-i)!", "\\ = \\ 2^dd!.$ Let $\\mathbf {P}$ be the double poset such that $P_+$ is the $d$ -chain and $P_-$ is the $d$ -antichain.", "Then $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ \\sum _{i=0}^d \\frac{d!}{i!", "}$ is the number of choices of ordered subsets of a $d$ -set.", "Moreover $\\operatorname{Vol}(\\overline{\\mathcal {C}}(\\mathbf {P})) \\ = \\ \\sum _{i=0}^d\\binom{d}{i}^2 i!$ is the number of partial permutation matrices, i.e.", "$0/1$ -matrices of size $d$ with at most one nonzero entry per row and column.", "Indeed, such a matrix is uniquely identified by an $i$ -by-$i$ permutation matrix and a choice of $i$ rows and $i$ columns in which it is embedded.", "For the comb $C_n$ , the number of linear extensions is $e(C_n) =(2n-1)!", "!$ .", "Let $\\mathbf {P}$ be the double poset induced by the comb $C_n$ .", "Then an induction argument shows that $\\operatorname{Vol}(\\mathcal {C}({\\mathbf {P}})) \\ = \\ 4^n\\, n!.$ It would be nice to have a bijective proof of this equality.", "Let $\\mathbf {P}_\\circ = (P,\\preceq ,\\preceq )$ be a compatible double poset induced by a poset $(P,\\preceq )$ .", "By Corollary REF , the polytopes $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ have the same normalized volume.", "Since both polytopes are 2-level, this means that the number of maximal simplices in any pulling triangulation of $\\mathcal {O}(\\mathbf {P}_\\circ )$ and $\\mathcal {C}({\\mathbf {P}_\\circ })$ coincides.", "From Theorem REF , we know that $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ is the twisted prism over the valuation polytope associated to $P$ .", "On the other hand, we know from Corollary REF that $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ is linearly isomorphic to the Hansen polytope $\\mathcal {H}(\\overline{G(P)})$ .", "Moreover, $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ and $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ are both 2-level and it is enticing to conjecture that their normalized volumes also agree.", "Unfortunately, this is not the case.", "For the poset $P$ on 5 elements whose Hasse diagram is the letter 'X', any pulling triangulation of $\\mathcal {C}({\\mathbf {P}_\\circ })^\\triangle $ has 324 simplices whereas for $\\mathcal {O}(\\mathbf {P}_\\circ )^\\triangle $ pulling triangulations have 320 simplices." ], [ "Double Hibi rings", "Hibi [19] associated to any finite poset $(P,\\preceq )$ a ring $\\mathcal {O}(P)]$ , nowadays called Hibi ring, that algebraically reflects many of the order-theoretic properties of $P$ .", "The ring $\\mathcal {O}(P)]$ is defined as the graded subring of the polynomial ring $S = t, s_a : a \\in P]$ generated by the elements $t \\cdot s^\\mathsf {J}$ , where $s^\\mathsf {J}\\ := \\ \\prod _{a \\in \\mathsf {J}} s_a,$ ranges over all filters $\\mathsf {J}\\subseteq P$ .", "For example, Hibi showed that $\\mathcal {O}(P)]$ is a normal Cohen–Macaulay domain of dimension $|P|+1$ and that $\\mathcal {O}(P)]$ is Gorenstein if and only if $P$ is a graded poset.", "By definition, Hibi rings are toric and hence they have the following quotient description.", "Let $R =x_\\mathsf {J}: \\mathsf {J}\\in \\mathcal {J}(P)]$ be the polynomial ring with variables indexed by filters and define the homogeneous ring map $\\phi : R \\rightarrow S$ by $\\phi (x_\\mathsf {J}) = t\\, s_\\mathsf {J}$ .", "Then $\\mathcal {O}(P)] \\cong R /\\mathrm {I}_{\\mathcal {O}(P)}$ where $\\mathrm {I}_{\\mathcal {O}(P)} = \\ker \\phi $ is a toric ideal.", "Hibi elegantly described a reduced Gröbner basis of $\\mathrm {I}_{\\mathcal {O}(P)}$ in terms of $\\mathcal {J}(P)$ .", "Fix a total order $\\le $ on the variables of $R$ such that $x_\\mathsf {J}\\le x_{\\mathsf {J}^{\\prime }}$ whenever $\\mathsf {J}\\subseteq \\mathsf {J}^{\\prime }$ and let $\\le _{rev}$ denote the induced reverse lexicographic order on $R$ .", "For $f \\in R$ , we write $\\textrm {in}_{\\le _{rev}}(f)$ for its leading term with respect to $\\le _{rev}$ and we will underline leading terms in what follows.", "Theorem 5.1 ([18]) Let $(P,\\preceq )$ be a finite poset.", "Then the collection $\\underline{x_{\\mathsf {J}} \\, x_{\\mathsf {J}^{\\prime }}} \\ - \\ x_{\\mathsf {J}\\cap \\mathsf {J}^{\\prime }} \\, x_{\\mathsf {J}\\cup \\mathsf {J}^{\\prime }} \\quad \\text{with }\\mathsf {J},\\mathsf {J}^{\\prime }\\in \\mathcal {J}(P)\\text{ incomparable}$ is a reduced Gröbner basis of $\\mathrm {I}_{\\mathcal {O}(P)}$ .", "The binomials (REF ) are called Hibi relations.", "In light of the previous sections, the natural question that we will address now is regarding an algebraic counterpart of the Hibi rings for double posets.", "For a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ , we define the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ as the subalgebra of the Laurent ring $\\hat{S} := t_-,t_+,s_a,s_a^{-1} : a \\in P]$ spanned by the elements $t_+\\cdot s^{\\mathsf {J}}$ for filters $\\mathsf {J}\\in \\mathcal {J}(P_+)$ and $t_- \\cdot (s^\\mathsf {J})^{-1}$ for filters $\\mathsf {J}\\in \\mathcal {J}(P_-)$ .", "This is the affine semigroup ring associated to $\\mathcal {O}(\\mathbf {P})$ with respect to the affine lattice $\\mathbb {A}= 2\\mathbb {Z}^P\\times (2\\mathbb {Z}+ 1)$ .", "Up to a translation by $(\\mathbf {0},1)$ and the lattice isomorphism $2\\mathbb {Z}^P\\times 2\\mathbb {Z}\\cong \\mathbb {Z}^P\\times \\mathbb {Z}$ , the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ is the affine semigroup ring of $\\operatorname{conv}\\bigl \\lbrace (\\mathcal {O}(P_+) \\times \\lbrace 1\\rbrace ) \\cup (-\\mathcal {O}(P_-) \\times \\lbrace 0\\rbrace )\\bigr \\rbrace ,$ with respect to the usual lattice $\\mathbb {Z}^P\\times \\mathbb {Z}$ .", "In particular, the double Hibi ring $\\mathcal {O}(\\mathbf {P})]$ is graded of Krull dimension $|P|+1$ .", "Moreover, since the double order polytope $\\mathcal {O}(\\mathbf {P})$ is reflexive by Corollary REF , it follows that $\\mathcal {O}(\\mathbf {P})]$ is a Gorenstein domain for any compatible double poset $\\mathbf {P}$ .", "The rings $\\mathcal {O}(\\mathbf {P})]$ as well as affine semigroup rings associated to the double chain polytopes $\\mathcal {C}({\\mathbf {P}})$ as treated at the end of Section REF were also considered by Hibi and Tsuchiya [24].", "Set $\\hat{R} := x_{\\mathsf {J}_+},x_{\\mathsf {J}_-} : \\mathsf {J}_+ \\in \\mathcal {J}(P_+), \\mathsf {J}_+ \\in \\mathcal {J}(P_+)]$ and define the monomial map $\\hat{\\phi }: \\hat{R} \\rightarrow \\hat{S}$ by $\\hat{\\phi }(x_{\\mathsf {J}_+}) \\ = \\ t_+\\, s^{\\mathsf {J}_+}\\quad \\text{ and } \\quad \\hat{\\phi }(x_{\\mathsf {J}_-}) \\ = \\ t_-\\, (s^{\\mathsf {J}_-})^{-1}.$ The corresponding toric ideal $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})} = \\ker \\hat{\\phi }$ is then generated by the binomials $\\underline{x_{\\mathsf {J}_{+1}} x_{\\mathsf {J}_{+2}} \\dots x_{\\mathsf {J}_{+k_+}} \\cdot x_{\\mathsf {J}_{-1}} x_{\\mathsf {J}_{-2}} \\dots x_{\\mathsf {J}_{-k_-}}}-x_{\\mathsf {J}^{\\prime }_{+1}} x_{\\mathsf {J}^{\\prime }_{+2}} \\dots x_{\\mathsf {J}^{\\prime }_{+k_+}} \\cdot x_{\\mathsf {J}^{\\prime }_{-1}} x_{\\mathsf {J}^{\\prime }_{-2}} \\dots x_{\\mathsf {J}^{\\prime }_{-k_-}},$ for filters $\\mathsf {J}_{+1}, \\dots , \\mathsf {J}_{+k_+}, \\mathsf {J}^{\\prime }_{+1}, \\dots ,\\mathsf {J}^{\\prime }_{+k_+} \\in \\mathcal {J}(P_+)$ and $\\mathsf {J}_{-1}, \\dots , \\mathsf {J}_{-k_-},\\mathsf {J}^{\\prime }_{-1}, \\dots , \\mathsf {J}^{\\prime }_{-k_-} \\in \\mathcal {J}(P_-)$ .", "Again, fix a total order $\\le $ on the variables of $\\hat{R}$ such that for $\\sigma = \\pm $ $x_{\\mathsf {J}_\\sigma } < x_{\\mathsf {J}^{\\prime }_\\sigma }$ for any filters $\\mathsf {J}_\\sigma ,\\mathsf {J}^{\\prime }_\\sigma \\in \\mathcal {J}(P_\\sigma )$ with $\\mathsf {J}_\\sigma \\subset \\mathsf {J}^{\\prime }_\\sigma $ , and $x_{\\mathsf {J}_+} < x_{\\mathsf {J}_-}$ for any filters $\\mathsf {J}_+ \\in \\mathcal {J}(P_+)$ and $\\mathsf {J}_- \\in \\mathcal {J}(P_-)$ , and denote by $\\le _{rev}$ the reverse lexicographic term order on $\\hat{R}$ induced by this order on the variables.", "Theorem 5.2 Let $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ be a compatible double poset.", "Then a Gröbner basis for $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ is given by the binomials $\\underline{x_{\\mathsf {J}_\\sigma }\\, x_{\\mathsf {J}_\\sigma ^{\\prime }}} \\ &- \\ x_{\\mathsf {J}_\\sigma \\cup \\mathsf {J}_\\sigma ^{\\prime }} \\, x_{\\mathsf {J}_\\sigma \\cap \\mathsf {J}_\\sigma ^{\\prime }}\\\\\\multicolumn{2}{l}{\\text{for incomparable filters $\\mathsf {J}_\\sigma ,\\mathsf {J}_\\sigma ^{\\prime }\\in \\mathcal {J}(P_\\sigma )$ and $\\sigma = \\pm $, and }}\\\\\\underline{x_{\\mathsf {J}_+}\\, x_{\\mathsf {J}_-}} \\ &- \\ x_{\\mathsf {J}_+\\setminus A} \\, x_{\\mathsf {J}_-\\setminus A}$ for filters $\\mathsf {J}_+ \\in \\mathcal {J}(P_+), \\mathsf {J}_- \\in \\mathcal {J}(P_-)$ such that $A := \\min (\\mathsf {J}_+)\\cap \\min (\\mathsf {J}_-) \\ne \\varnothing $ .", "It is clear that binomials of the form (REF ) and () are contained in $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ and hence it suffices to show that their leading terms generate $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ .", "For this, let us take a closer look at the combinatorics of $\\hat{\\phi }$ .", "Let $\\mathcal {G}$ be the collection of binomial given in (REF ) and () and let $f = \\underline{m_1} - m_2$ be an element of the form (REF ).", "By reducing $f$ by the binomial of (REF ), we can view $f$ as a quadruple $\\begin{aligned}\\mathsf {J}_{+1} \\subset \\mathsf {J}_{+2} \\subset \\cdots \\subset \\mathsf {J}_{+k_+} &\\quad \\qquad \\mathsf {J}_{-1} \\subset \\mathsf {J}_{+2} \\subset \\cdots \\subset \\mathsf {J}_{-k_-}\\\\\\mathsf {J}_{+1}^{\\prime } \\subset \\mathsf {J}_{+2}^{\\prime } \\subset \\cdots \\subset \\mathsf {J}_{+k_+}^{\\prime } &\\quad \\qquad \\mathsf {J}_{-1}^{\\prime } \\subset \\mathsf {J}_{+2}^{\\prime } \\subset \\cdots \\subset \\mathsf {J}_{-k_-}^{\\prime }.\\end{aligned}$ From the definition of $\\hat{\\phi }$ it follows that such a quadruple defines a binomial in $\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ if and only if for any $q \\in P$ $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r} \\rbrace - \\max \\lbrace s : q \\notin \\mathsf {J}_{-s} \\rbrace \\ = \\ \\max \\lbrace r : q \\notin \\mathsf {J}_{+r}^{\\prime } \\rbrace - \\max \\lbrace s : q \\notin \\mathsf {J}_{-s}^{\\prime }\\rbrace .$ and we note the following implication.", "Lemma 5.3 Let the collection of filters in (REF ) correspond to a binomial $f \\in \\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ and let $q \\in P$ .", "Then there is some $1\\le i \\le k_+$ such that $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ if and only if there is some $1 \\le j \\le k_-$ such that $q \\in \\mathsf {J}_{-j} \\setminus \\mathsf {J}_{-j}^{\\prime }$ .", "If $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ , then $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r} \\rbrace < i$ and $\\max \\lbrace r:q \\notin \\mathsf {J}_{+r}^{\\prime } \\rbrace \\ge i$ and (REF ) implies that $q \\in \\mathsf {J}_{-j} \\setminus \\mathsf {J}_{-j}^{\\prime }$ for some $j$ .", "The other direction is identical.", "We call $q \\in P$ moving if it satisfies one of the two equivalent conditions of Lemma REF .", "Let $f = \\underline{m_1} - m_2 \\in \\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}$ be a binomial represented by a collection of filters given by (REF ).", "If $k_- = 0$ or $k_+= 0$ , then the Hibi relations (REF ) for $P_-$ or $P_+$ together with Theorem REF yields the result.", "Thus, we assume that $k_-,k_+ > 0$ and we need to show that there are filters $\\mathsf {J}_{+i}$ and $\\mathsf {J}_{-j}$ such that $\\min (\\mathsf {J}_{+i}) \\cap \\min (\\mathsf {J}_{-j}) \\ne \\varnothing $ .", "Observe that there is at least one moving element.", "Indeed, $\\mathsf {J}_{+1}\\nsubseteq \\mathsf {J}_{+1}^{\\prime }$ and hence $\\mathsf {J}_{+1} \\setminus \\mathsf {J}_{+1}^{\\prime } \\ne \\varnothing $ .", "Otherwise, $x_{\\mathsf {J}_{+1}} <x_{\\mathsf {J}_{+1}^{\\prime }}$ and the reverse lexicographic term order $\\le _{rev}$ would not select $m_1$ as the lead term of $f$ .", "Among all moving elements, choose $q$ to be minimal with respect to $\\preceq _+$ and $\\preceq _-$ .", "Since $\\mathbf {P}$ is a compatible double poset, such a $q$ exists.", "But then, if $q \\in \\mathsf {J}_{+i} \\setminus \\mathsf {J}_{+i}^{\\prime }$ , then $q \\in \\min (\\mathsf {J}_{+i})$ .", "The same holds true for $\\mathsf {J}_{-j}$ and shows that $\\underline{m_1}$ is divisible by the leading term of a binomial of type ().", "Gröbner bases, faces, and triangulations In light of the regular and unimodular triangulation of $\\mathcal {O}(P)$ given in [34] (and recalled in Section REF ), the Hibi ring $\\mathcal {O}(P)]$ is exactly the affine semigroup ring associated to $\\mathcal {O}(P)$ .", "That is, $\\mathcal {O}(P)]$ is the standard graded $-algebra associated to thenormal affine semigroup$$\\lbrace (f,k) \\in \\mathbb {Z}^P\\times \\mathbb {Z}: k \\ge 0, f \\in k \\mathcal {O}(P)\\rbrace .$$$ For a lattice polytope $\\mathcal {P}\\subset \\mathbb {R}^n$ , Sturmfels [36] described a beautiful relationship between regular triangulations of $\\mathcal {P}$ and radicals of initial ideals of the toric ideal $\\mathrm {I}_\\mathcal {P}$ .", "It follows from Theorem REF that $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is a squarefree ideal generated by quadratic monomials.", "Appealing to [36], this yields the following refinement of Theorem REF .", "Corollary 5.4 Let $\\mathbf {P}$ be a compatible double poset.", "Then $\\mathcal {O}(\\mathbf {P})$ has a regular triangulation whose underlying simplicial complex is exactly $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ .", "The initial ideal $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is already radical and Theorem 8.3 of [36] yields that $\\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})})$ is the Stanley-Reisner ideal of a regular triangulation of $\\mathcal {O}(\\mathbf {P})$ .", "Hence, a collection $C = C_1 \\uplus C_2 \\subseteq \\mathcal {J}(\\mathbf {P})$ forms a simplex in the triangulation of $\\mathcal {O}(\\mathbf {P})$ if and only if $\\prod _{\\mathsf {J}_+ \\in C_+} x_{\\mathsf {J}_+}\\prod _{\\mathsf {J}_- \\in C_-} x_{\\mathsf {J}_-} \\ \\notin \\textrm {in}_{\\le _{rev}}(\\mathrm {I}_{\\mathcal {O}(\\mathbf {P})}).$ Translating the conditions given in Theorem REF , this is the case if and only if $C_\\sigma = C \\cap \\mathcal {J}(P_\\sigma )$ is a chain of filters for $\\sigma = \\pm $ and $C_+, C_-$ are non-interfering chains.", "This is exactly the definition of the flag complex $\\Delta ^\\mathrm {ni}(\\mathbf {P})$ .", "Using the orbit-cone correspondence for affine toric varieties (see, for example, [5]), we can give an algebraic perspective on Theorem REF .", "We are in a particularly nice situation as the polytopes we consider have unimodular triangulations and hence the affine semigroup rings are generated in degree 1 by the vertices of the underlying polytope.", "Lemma 5.5 Let $V \\subset \\Lambda $ be a finite set of lattice points and $\\mathcal {P}=\\operatorname{conv}(V)$ the corresponding lattice polytope.", "If $\\mathrm {I}\\subseteq x_v : v\\in V]$ is the toric ideal of the homogenization $\\lbrace (v,1) : v\\in V \\rbrace \\subseteq \\Lambda \\times \\mathbb {Z}$ , then for any subset $U \\subseteq V$ , we have that $\\operatorname{conv}(U)$ is a face of $\\mathcal {P}$ with $\\operatorname{conv}(U)\\cap V = U$ if and only if $f(\\mathbf {1}_U) \\ = \\ 0 \\qquad \\text{ for all } f \\in \\mathrm {I}.$ Let $\\mathrm {L}\\subseteq \\mathcal {J}(\\mathbf {P})$ .", "Then for $\\sigma = \\pm $ and $\\mathsf {J}_\\sigma , \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathcal {J}(P_\\sigma )$ Lemma REF and (REF ) of Theorem REF states that $\\mathsf {J}_\\sigma , \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathrm {L}_\\sigma \\quad \\Longleftrightarrow \\quad \\mathsf {J}_\\sigma \\cup \\mathsf {J}_\\sigma ^{\\prime },\\mathsf {J}_\\sigma \\cap \\mathsf {J}_\\sigma ^{\\prime } \\in \\mathrm {L}_\\sigma .$ That is, if and only if $\\mathrm {L}_\\sigma $ is an embedded Thus, $\\mathrm {L}_\\sigma $ is an embedded sublattice of $\\mathcal {J}(P_\\sigma )$ .", "The same reasoning shows that the conditions imposed by () are equivalent to those of Lemma REF .", "We can also use Sturmfels' result in the other direction to find Gröbner bases.", "For a double poset $\\mathbf {P}= (P,\\preceq _+,\\preceq _-)$ we may define the subring $\\mathcal {C}({\\mathbf {P}})] \\subseteq \\hat{R}$ generated by the monomials $t_+ s^{\\min (\\mathsf {J}_+)}$ and $t_+ (s^{\\min (\\mathsf {J}_-)})^{-1}$ for filters $\\mathsf {J}_+ \\subseteq P_+$ and $\\mathsf {J}_- \\subseteq P_-$ .", "The corresponding toric ideal $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ is contained in the ring $T = x_{A_+},x_{A_-} ]$ , where $A_\\sigma $ ranges over all anti-chains in $P_\\sigma $ for $\\sigma = \\pm $ .", "Since $\\mathcal {O}(\\mathbf {P})$ is the stable set polytope of the perfect double graph $G(\\mathbf {P})$ , it follows from Corollary REF that $\\mathcal {C}({\\mathbf {P}})]$ is the normal affine semigroup ring associated to the lattice polytope $\\mathcal {C}({\\mathbf {P}})$ .", "To describe a Gröbner basis for, we introduce the following notation.", "For $\\sigma = \\pm $ and two antichains $A,A^{\\prime } \\subseteq P_\\sigma $ define $A \\sqcup A^{\\prime }\\ := \\ \\min (A \\cup A^{\\prime })$ and $A \\sqcap A^{\\prime } \\ := \\ (A \\cap A^{\\prime }) \\cup (\\max (A \\cup A^{\\prime }) \\setminus \\min (A \\cup A^{\\prime })).$ For a subset $S \\subseteq P$ and $\\sigma = \\pm $ , we write $\\langle S \\rangle _\\sigma := \\lbrace a \\in P: a\\succeq _\\sigma s \\text{ for some } s \\in S \\rbrace $ for the filter in $P_\\sigma $ generated by $S$ .", "Theorem 5.6 Let $\\mathbf {P}$ be a double poset.", "Then a Gröbner basis for $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ is given by the binomials $\\underline{x_A \\, x_{A^{\\prime }}} \\ &- \\ x_{A \\sqcup A^{\\prime }}\\, x_{A \\sqcap A^{\\prime }} \\qquad \\langle A \\rangle _\\sigma ,\\langle A \\rangle _\\sigma \\in \\mathcal {J}(P_\\sigma ) \\text{ incomparable}\\\\\\multicolumn{2}{l}{\\text{ for antichains $A,A^{\\prime } \\subset P_\\sigma $ for $\\sigma = \\pm $and }}\\\\\\underline{x_{A_+} \\, x_{A_-}} \\ &- \\ x_{A_+ \\setminus A_-}x_{A_-\\setminus A_+}\\qquad \\text{ for antichains } A_\\sigma \\subseteq P_\\sigma .$ It is easy to verify that the given binomials are contained in $\\mathrm {I}_{\\mathcal {C}({\\mathbf {P}})}$ .", "Moreover, the leading monomials are exactly the minimal non-faces of the unimodular triangulation of Corollary REF .", "The result now follows from Theorem 8.3 in [36].", "Remark 1 Reformulated in the language of double posets, Hibi, Matsuda, and Tsuchiya [22], [21], [23] computed related Gröbner bases of the toric ideals associated with the polytopes $\\Gamma (\\mathcal {O}(P_+),\\mathcal {O}(P_-))$ (in the compatible case), $\\Gamma (\\mathcal {C}({P_+}),\\mathcal {C}({P_-}))$ , and $\\Gamma (\\mathcal {O}(P_+),\\mathcal {C}({P_-}))$ for a double poset $\\mathbf {P}$ .", "See the paragraph before Corollary REF for notation." ] ]
1606.04938
[ [ "Performance Analysis of Target Parameters Estimation Using Multiple\n Widely Separated Antenna Arrays" ], [ "Abstract Target parameter estimation performance is investigated for a radar employing a set of widely separated transmitting and receiving antenna arrays.", "Cases with multiple extended targets are considered under two signal model assumptions: stochastic and deterministic.", "The general expressions for the corresponding Cramer-Rao lower bound (CRLB) and the asymptotic properties of the maximum-likelihood (ML) estimator are derived for a radar with $M_t$ arrays of $L_t$ transmitting elements and $M_r$ arrays of $L_r$ receiving elements for both types of signal models.", "It is shown that for an infinitely large product $M_tM_r$, and a finite $L_r$, the ML estimator is consistent and efficient under the stochastic model, while the deterministic model requires $M_tM_r$ to be finite and $L_r$ to be infinitely large in order to guarantee consistency and efficiency.", "Monte Carlo simulations further investigate the estimation performance of the proposed radar configuration in practical scenarios with finite $M_tM_r$ and $L_r$, and a fixed total number of available receiving antenna elements, $M_r L_r$.", "The numerical results demonstrate that grouping receiving elements into properly sized arrays reduces the mean squared error (MSE) and decreases the threshold SNR.", "In the numerical examples considered, the preferred configurations employ $M_t M_r > 1$.", "In fact, when $M_t M_r$ becomes too small, due to the loss of the geometric gain, the estimation performance becomes strongly dependent on the particular scenario and can degrade significantly, while the CRLB may become a poor prediction of the MSE even for high SNR.", "This suggests it may be advantageous to employ approaches where neither $M_tM_r$ nor $L_r$ are too small." ], [ "Performance Analysis of Target Parameters Estimation Using Multiple Widely Separated Antenna Arrays Peter ˜Khomchuk, ˜Student ˜Member, ˜IEEE, Rick ˜S.", "˜Blum, ˜Fellow, ˜IEEE, Igal ˜Bilik, ˜Member, ˜IEEE This work was supported by the National Science Foundation under Grant No.", "ECCS-1405579." ] ]
1606.05043
[ [ "Einstein's Approach to Statistical Mechanics: The 1902-04 Papers" ], [ "Abstract We summarize the papers published by Einstein in the Annalen der Physik in the years 1902-04 on the derivation of the properties of thermal equilibrium on the basis of the mechanical equations of motion and of the calculus of probabilities.", "We point out the line of thought that led Einstein to an especially economical foundation of the discipline, and to focus on fluctuations of the energy as a possible tool for establishing the validity of this foundation.", "We also sketch a comparison of Einstein's approach with that of Gibbs, suggesting that although they obtained similar results, they had different motivations and interpreted them in very different ways." ], [ "Introduction", "By the end of June 1902, just after being accepted as Technical Assistant level III at the Federal Patent Office in Bern, Albert Einstein, 23, sent to the renowned journal Annalen der Physik a manuscript with the bold title “Kinetic Theory of Thermal Equilibrium and of the Second Law of Thermodynamics” [1].", "In the introduction, he explains that he wishes to fill a gap in the foundations of the general theory of heat, “for one has not yet succeeded in deriving the laws of thermal equilibrium and the second law of thermodynamics using only the equations of mechanics and the probability calculus”.", "He also announces “an extension of the second law that is of importance for the application of thermodynamics”.", "Finally, he will provide “the mathematical expression of the entropy from the standpoint of mechanics”.", "Einstein's papers and their translations are available on the Princeton University Press site [2].", "In the following two years Einstein followed this line of research publishing a paper each year [3], [4].", "The third one, entitled “On the general molecular theory of heat”, submitted on March 27, 1904, opened a new path, by tacitly extending the results obtained for a general mechanical system (with a large, but finite, number of degrees of freedom) to the case of black-body radiation.", "In pursuing this line of research Einstein found an unexpected result, that pointed at an inconsistency between the current understanding of the processes of light emission and absorption and the statistical approach.", "To resolve this inconsistency, in the first paper [5] of his “Annus Mirabilis” 1905, Einstein renounced the detailed picture of light emission and adsorption provided by Maxwell's equations, maintaining his statistical approach, in particular the statistical interpretation of entropy.", "He introduced therefore the concept of light quanta, presented as a “heuristic point of view”.", "The importance of the 1902–04 papers on the molecular theory of heat in Einstein's intellectual development and in the advance of physics has been stressed by Kuhn [6], when he states that What brought Einstein to the blackbody problem in 1904 and to Planck in 1906 was the coherent development of a research program begun in 1902, a program so nearly independent of Planck's that it would almost certainly have led to the blackbody law even if Planck had never lived.", "In spite of their importance, the 1902–04 papers have received comparatively little attention.", "One of the reasons was the publication in 1902 of Gibbs' Elementary Principles in Statistical Mechanics.", "[7] This book is considered, especially since the publication of the influential book by R. C. Tolman [8], as the founding text of the discipline.", "Einstein himself contributed to the neglect of the 1902-1904 papers.", "In his answer to Paul Hertz' criticism of his derivation of the second principle [9], he says I only wish to add that the road taken by Gibbs in his book, which consists in one’s starting directly from the canonical ensemble, is in our opinion preferable to the road I took.", "If I had known Gibbs' book at that time, I would have not published these papers at all, but I would have limited myself to the treatment of a few points.", "In his scientific autobiography [10] Einstein returned to this point, saying Not acquainted with the earlier investigations by Boltzmann and Gibbs, which had appeared earlier and actually exhausted the subject, I developed the statistical mechanics and molecular-kinetic theory of thermodynamics which was based on the former.", "My major aim in this was to find facts which would guarantee as much as possible the existence of atoms of definite size.", "The last sentence of this quotation highlights the different attitude of Einstein with respect to Gibbs.", "Einstein aims at using the statistical approach to establish the reality of atoms, while Gibbs aims at a rational foundation of thermodynamics, and consequently focuses on the regularities which emerge in systems with many degrees of freedom.", "Einstein's papers contain a more direct and fundamental approach to the statistical mechanics of equilibrium, and could actually suggest a didactically effective path to the introduction of the fundamental ideas of the field.", "We shall therefore attempt to to ease their reading by summarizing them, pointing out in particular the differences between Einstein's and Gibbs' points of view.", "We shall not try to discuss all the detailed analyses of the papers which have appeared in the literature (beyond Kuhn's work [6], one can also read [11], [12], [13], [14], [15], [16]), but shall only refer to the more interesting observations." ], [ "Kinetic theory of thermal equilibrium and of the second principle of thermodynamics", "The first two papers [1], [3] have a very similar structure.", "The second paper aims to widen the scope of the first, by attempting to consider “general” dynamical systems and irreversible processes.", "We shall follow the first paper, and we shall then briefly review the points in which the second paper differs.", "We adapt Einsteins discussion to modern notation.", "Einstein begins by considering a general physical system as represented by a mechanical system with many coordinates $q=(q_{1},\\ldots ,q_{n})$ and the corresponding momenta $p=(p_{1},\\ldots ,p_{n})$ , obeying the canonical equations of motion with a time-independent Hamiltonian that is the sum of a potential energy (function of the $q$ 's alone) and of a kinetic energy that is a quadratic function of the $p$ 's, whose coefficients are arbitrary functions of the $q$ 's (and is implicitly supposed to be positive definite).", "Following Gibbs, we shall call the $p$ 's and $q$ 's collectively as the phase variables, and the space they span the phase space.", "Einstein then considers a very large number $N$ of such systems, with the same Hamiltonian, whose energies $E$ lie between two very close values $\\overline{E}$ and $\\overline{E}+\\delta E$ .", "He then looks for the stationary distribution of these systems in phase space.", "Here Einstein introduces a strong mechanical hypothesis by assuming that, apart from the energy, there is no other function defined on the phase space that is constant in time.This is the fundamental hypothesis linking the mechanical and the statistical aspects of the problem.", "It is probably inspired by the consideration of monocyclic systems, introduced by Helmholtz [17] and discussed by Boltzmann in [18].", "Cf.", "[19] and [20].", "He argues that this condition is equivalent to the requirement that the stationary distribution of the systems in phase space depends only on the value of the energy.", "He proves indeed that if there are other functions $\\phi (q,p)$ that are constants of the motion, the stationary distribution is not uniquely identified by the value of the energy, but does not attempt to prove the converse.", "He then shows that Liouville's theorem implies that the local density of systems in phase space is constant in time and therefore, by the mentioned hypothesis, must be a function of the energy alone.", "Since the energies of all $N$ systems are infinitely close to one another, this density must be uniform on the region of phase space defined by the corresponding value of the Hamiltonian.", "In this way Einstein has defined what is now called the microcanonical ensemble, i.e., the distribution in phase space which is uniform when the energy of the system lies between two closely lying values, and vanishes otherwise.", "Einstein now turns to the consideration of thermal equilibrium between one system $\\mathsf {S}$ and one $\\Sigma $ considerably larger.Einstein actually considers two systems with the same number of degrees of freedom, but where the energy contained in $\\Sigma $ is considerably larger.", "Apparently the equipartition theorem, which he derives in § 6 of the paper, led him to realize the awkwardness of this restriction, and he drops it in the second paper.", "The second system acts as a thermal reservoir, and the first one as a thermometer.", "He assumes that the total energy $\\mathcal {E}$ of the global system $\\mathsf {S}\\cup \\Sigma $ can be written as $\\mathcal {E}=E+H,$ up to negligible terms, where $E$ pertains to $\\mathsf {S}$ and $H$ to $\\Sigma $ .", "Let the phase variables of $\\mathsf {S}$ be denoted by $(p,q)$ and those of of $\\Sigma $ by $(\\pi ,\\chi )$ .", "The question is now to find the distribution of the phase variables of $\\mathsf {S}$ when the energy of the global system lies between $\\mathcal {E}_{0}$ and $\\mathcal {E}_{0}+\\delta \\mathcal {E}$ , while the phase variables of $\\Sigma $ can take on any values.", "As pointed out by Uffink [15], this problem was considered several times by Boltzmann, who almost always solved it by taking an ideal gas for $\\Sigma $ and explicitly evaluating the resulting phase-space integral.", "Einstein instead introduces an elegant trick which leads directly to the desired result.", "Let us consider an infinitesimally small domain $g$ in the phase space of the global system $\\mathsf {S}\\cup \\Sigma $ , with energy $\\mathcal {E}$ between $\\mathcal {E}_{0}$ and $\\mathcal {E}_{0}+\\delta \\mathcal {E}$ .", "Then the number $d N$ of systems of the ensemble which are found in $g$ is $d N = A \\int _{g}d p\\,d q\\; d\\pi \\, d\\chi ,$ where $A$ is a constant.", "Actually one can choose instead of $A$ any function of the total energy $\\mathcal {E}$ which takes the value $A$ for $\\mathcal {E}=\\mathcal {E}_{0}$ .", "Let us thus setEinstein actually uses the notation $2h$ instead of $\\beta $ , which is now the traditional choice.", "$A = A^{\\prime } \\,e^{-\\beta \\, \\mathcal {E}_{0}}=A^{\\prime }\\,e^{-\\beta \\,E}e^{-\\beta \\,H },$ where $\\beta $ is a constant.", "Thus the number $d N^{\\prime }$ of systems such that the phase variables of $\\mathsf {S}$ lie in a region of volume $d p\\;d q$ around the point $(p,q)$ , while the variables of $\\Sigma $ can have any value, as long as $\\mathcal {E}$ lies between $\\mathcal {E}_{0}$ and $\\mathcal {E}_{0}+\\delta \\mathcal {E}$ , is given by $d N^{\\prime } = A^{\\prime } e^{-\\beta E}\\,d p\\, d q\\int e^{-\\beta H}\\,d \\pi \\, d \\chi ,$ where the integral runs over all values of the phase variables of $\\Sigma $ such that the values of its Hamiltonian $H$ lie between $H_{0}$ and $H_{0}+\\delta \\mathcal {E}$ , and $H_{0}=\\mathcal {E}_{0}-E.$ The value of the constant $\\beta $ can be fixed by requiring that the integral appearing on the right-hand side of equation (REF ) be independent of $E$ .", "Indeed, once $\\delta \\mathcal {E}$ is fixed, the integral can be considered as a function $\\Phi (H)$ of $H$ alone.", "Thus, since $E\\ll \\mathcal {E}_{0}$ , we have $\\Phi (H_{0})=\\Phi (\\mathcal {E}_{0}-E)\\simeq \\Phi (\\mathcal {E}_{0})-E\\,\\Phi ^{\\prime }(\\mathcal {E}_{0}),$ where $\\Phi ^{\\prime }$ is the derivative of $\\Phi $ with respect to its argument.", "Thus $\\Phi ^{\\prime }(\\mathcal {E}_{0})=0$ .", "We can write however $\\Phi (H)=e^{-\\beta H}\\cdot \\omega (H),$ where $\\omega (H)=\\int d\\pi _{1}\\cdots d\\chi _{n}$ , with the integral extended to the region in phase space such that the energy of $\\Sigma $ lies between $H$ and $H+\\delta \\mathcal {E}$ .", "The condition now reads $e^{-\\beta \\mathcal {E}_{0}}\\omega (\\mathcal {E}_{0})\\left[-\\beta +\\frac{\\omega ^{\\prime }(\\mathcal {E}_{0})}{\\omega (\\mathcal {E}_{0})}\\right]=0,$ where $\\omega ^{\\prime }$ is the derivative of $\\omega $ with respect to is argument.", "We therefore obtain the required condition for $\\beta $ in the form $\\beta = \\frac{\\omega ^{\\prime }(\\mathcal {E}_{0})}{\\omega (\\mathcal {E}_{0})}.$ Einstein now turns to show that the quantity $\\beta $ is always positive.", "He first derives a lemma, by considering a general (positive definite) quadratic function $\\varphi (x_{1},\\ldots ,x_{n})$ of $n$ variables (where $n$ is large enough), and defining the function $z(y)$ by the integral $z(y)=\\int d x_{1}\\cdots d x_{n},$ where the integral is extended to all points for which $\\varphi $ lies between $y$ and $y+\\Delta $ , where $\\Delta $ is fixed.", "He then easily shows that, for $n\\ge 3$ , $z(y)$ is an increasing function of $y$ .", "Let us now denote by $\\Gamma (H)$ the phase space available to the larger system $\\Sigma $ when the values of its Hamiltonian lie between $H$ and $H+\\delta \\mathcal {E}$ .", "The Hamiltonian of $\\Sigma $ is given by the sum of the potential energy, that depends only on the coordinates, and of the kinetic energy, which is a quadratic form in the momenta, whose coefficients depend only on the coordinates.", "Let $H_{0}$ and $H_{1}$ be two values of $H$ , with $H_{1}>H_{0}$ , and let $\\Gamma (H_{0})$ and $\\Gamma (H_{1})$ be the corresponding available space regions.", "Let $Q(H_{0})$ be the region of coordinate space such that the potential energy of the system is smaller than $H_{0}$ .", "Thus if the point $(\\pi ,\\chi )$ belongs to $\\Gamma (H_{0})$ , the point $(\\chi )$ belongs to $Q(H_{0})$ .", "Within $\\Gamma (H_{1})$ let us identify the region $\\Gamma ^{\\prime }(H_{1})$ where the coordinates $\\chi $ belong to $Q(H_{0})$ .", "Thus, for each such values of the coordinates, since the total energy is larger than $H_{0}$ , the kinetic energy must be larger.", "Therefore, by the lemma on the monotonic increase of $z(y)$ with $y$ , for each such point in coordinate space, the volume available to the momenta is larger for $\\Gamma ^{\\prime }(H_{1})$ than for $\\Gamma (H_{0})$ .", "Integrating over the coordinates we obtain that the volume of $\\Gamma ^{\\prime }(H_{1})$ must be larger than that of $\\Gamma (H_{0})$ .", "Since the volume of the region of $\\Gamma (H_{1})$ that does not belong to $\\Gamma ^{\\prime }(H_{1})$ cannot be negative, the volume of $\\Gamma (H_{1})$ must be larger than that of $\\Gamma (H_{0})$ , i.e., the function $\\omega (H)$ increases with $H$ , and $\\beta $ given by the above expression must be positive.", "Now, Einstein derives what is now known as the zero-th law of thermodynamics.", "Since $\\beta $ depends only on the state of $\\Sigma $ , but determines the distribution of $\\mathsf {S}$ in state space, independently on how $\\Sigma $ and $\\mathsf {S}$ interact, it follows that if a given system $\\Sigma $ interacts with two small system $\\mathsf {S}$ and $\\mathsf {S}^{\\prime }$ and is in equilibrium with them, $\\mathsf {S}$ and $\\mathsf {S}^{\\prime }$ must have the same value of $\\beta $ .", "In particular, if $\\mathsf {S}$ and $\\mathsf {S}^{\\prime }$ are mechanically identical, the average value of any arbitrary observable function $A(p,q)$ must be equal in $\\mathsf {S}$ and $\\mathsf {S}^{\\prime }$ .", "Einstein then calls $\\mathsf {S}$ and $\\mathsf {S}^{\\prime }$ thermometers, $\\beta $ the temperature function and the average of $A$ the temperature measure.", "Then Einstein goes on to prove the converse result, namely that if two systems that have the same values of $\\beta $ are put in contact, they will be in thermal equilibrium.", "He considers two systems, $\\Sigma _{1}$ and $\\Sigma _{2}$ , weakly interacting.", "Let each of them be in contact with an (infinitesimally) small thermometer $\\mathsf {S}_{1}$ and $\\mathsf {S}_{2}$ .", "The temperature measures $A_{1}$ and $A_{2}$ in each thermometer will be the same, since we are in fact dealing with a single interacting system in thermal equilibrium, and therefore also the corresponding temperature functions $\\beta _{1}$ and $\\beta _{2}$ will be equal.", "Let the interaction terms between $\\Sigma _{1}$ and $\\Sigma _{2}$ be slowly brought to zero.", "Then the readings of the thermometers will remain equal, but now the reading of $\\mathsf {S}_{1}$ deals only with $\\Sigma _{1}$ and that of $\\mathsf {S}_{2}$ only with $\\Sigma _{2}$ .", "The process is reversible, since we are dealing with a sequence of thermal equilibrium states.", "Thus, by reversing it, we obtain the required result.", "As an immediate consequence, we obtain that if $\\Sigma _{1}$ and $\\Sigma _{2}$ are in thermal equilibrium, and so are $\\Sigma _{2}$ and $\\Sigma _{3}$ , then $\\Sigma _{1}$ and $\\Sigma _{3}$ are in thermal equilibrium, since they share the same value of $\\beta $ .", "Einstein concludes this section with the intriguing remark: I would like to note here that until now we have made use of the assumption that our systems are mechanical only inasmuch as we have applied Liouville's theorem and the energy principle.", "Probably the basic laws of the theory of heat can be developed for systems that are defined in a much more general way.", "We will not attempt to do this here, but will rely on the equations of mechanics.", "We will not deal here with the important question as to how far the train of thought can be separated from the model employed and generalized.", "Uffink [15] has remarked that “this quote indicates (with hindsight) a remarkable underestimation of the logical dependence of [Einstein's] approach on the ergodic hypothesis.” But the passage shows, as also stressed by Uffink, that already in 1902 Einstein was considering the need to extend the statistical approach beyond its application to mechanical systems, no matter how general they can be conceived.", "A simple calculation allows Einstein to derive the equipartition theorem in the following form.", "Let the kinetic energy of a system be represented by a quadratic expression of the form $K=\\frac{1}{2}\\left(\\alpha _{1}p_{1}^{2}+\\cdots +\\alpha _{n}p_{n}^{2}\\right),$ where $\\alpha _{i}$ , $i=1,\\ldots ,n$ , are positive constants or functions of the coordinates $q$ .", "This form can always be reached from a general quadratic expression by a suitable canonical transformation.", "The $p$ variables had been denoted as “momentoids” by Boltzmann.", "Then the average of $K$ at equilibrium is given by $\\left<{K}\\right>=\\frac{n}{2\\beta }.$ In particular, this result implies that the kinetic energy of a single molecule in an ideal gas is equal to $3/(2\\beta )$ on average.", "Kinetic theory teaches us that this quantity is proportional to the product of the pressure and the volume per particle in an ideal gas.", "Since this is proportional to the absolute temperature $T$ , we obtain $\\frac{1}{\\beta }=k_{\\mathrm {B}} T=\\frac{\\omega (H)}{\\omega ^{\\prime }(H)},$ where $k_{\\mathrm {B}}$ is Boltzmann's constant and $\\omega (H)$ is the volume of phase space contained by the equal-energy surfaces of $\\Sigma $ corresponding to the values $H$ and $H+\\delta \\mathcal {E}$ .", "Having found the relation between $\\beta $ and the temperature, Einstein proceeds to the derivation of the second law of thermodynamics, which he here limits to the statement of the integrability of heat divided by the absolute temperature.", "He switches back to a Lagrangian setting, describing the system's state by the coordinates $q$ and their time derivatives $\\dot{q}$ , and introduces externally applied forces.", "These forces are split into ones derived from a potential depending on the $q$ 's, and others that allow for heat transfer.", "The first ones are assumed to vary slowly with time, while the second ones change very rapidly.", "The infinitesimal heat $\\delta Q$ is defined as the work of the second type of forces.", "Then a reversible transformation is one in which the system is led from an equilibrium state with given values of $\\beta $ and of the volume $V$ to one with the values $\\beta +\\delta \\beta $ and $V+\\delta V$ .", "Here Einstein tacitly assumes that the time average of the relevant quantities in a slow transformation can be obtained by averaging the same quantity over the distribution of the $N$ systems in phase space.", "He thus finds that $\\frac{\\delta Q}{T}=d\\left(\\frac{\\left<{E}\\right>-F}{T}\\right),$ where $\\left<{E}\\right>$ is the average total energy of the system, and $F$ is a constant introduced so that the distribution $P(p, q)= e^{\\beta (E(p,q)-F)}$ is normalized.", "Einstein remarks that this expression contains the total energy, and is independent of its splitting into kinetic and potential terms.This will be the starting point of his 1903 paper.", "One can readily integrate this expression, obtaining an explicit form of the entropy $S$ : $S =\\frac{\\left<{E}\\right>-F}{T}=\\frac{\\left<{E}\\right>}{T}+k_{\\mathrm {B}}\\log \\int e^{-\\beta E(p,q)}\\,d p\\,d q+\\text{const.", "}$ Now Einstein states the announced generalization of the second principle.", "It is worth quoting this short paragraph in its entirety.", "Einstein denotes by $V_{a}$ the potential of the conservative forces performing the reversible transformation.", "He then states No assumptions had to be made about the nature of the forces that correspond to the potential $V_{a}$ [the conservative ones], not even that such forces occur in nature.", "Thus, the mechanical theory of heat requires that we arrive at correct results if we apply Carnot's principle to ideal processes, which can be produced from the observed processes by introducing arbitrarily chosen $V_{a}$ 's.", "Of course, the results obtained from the theoretical consideration of those processes can have real meaning only when the ideal auxiliary forces $V_{a}$ no longer appear in them.", "Thus the strategy which led to the establishment of the Einstein relation in Brownian motion, in the 1905 paper, is already sketched in this one." ], [ "A theory of the foundations of thermodynamics", "In his 1903 memoir, entitled “A theory of the foundations of thermodynamics” [3], Einstein asks whether kinetic theory is essential for the derivation of the postulates of thermal equilibrium and of the entropy concept, or whether “assumptions of a more general nature” could be sufficient.", "He goes on therefore to consider a general dynamical system whose state is identified by a collection $p$ of variables $p=(p_{1},\\ldots ,p_{n})$ , which correspond to both coordinates and momenta for a mechanical system, and evolve by a general system of equations of motion of the kind $\\frac{d p_{i}}{d t}=\\varphi _{i}(p_{1},\\ldots ,p_{n});\\qquad i=1,\\ldots ,n.$ Assuming that the system allows for a unique integral of motion, the energy $E(p)$ , he then introduces the equilibrium postulate, according to which a “physical system” eventually reaches a time-independent macroscopic state, in which any “perceptible quantity” assumes a time-independent value.", "Einstein then looks for the stationary distribution of a collection of $N$ systems, with $N$ large.", "Each system evolves according to equations (REF ) and has an energy between $E$ and $E+\\delta E$ .", "He claims that the equilibrium postulate, along with the absence of any integral of motion beyond the energy, implies the existence of a well-defined probability distribution in $p$ -space.", "Einstein's argument reads Starting at an arbitrary point of time and throughout time $\\mathcal {T}$ , we consider a physical system which is represented by the equations (REF ) and has the energy $E$ .", "If we imagine having chosen some arbitrary region $\\Gamma $ of the state variables $p_{1}\\ldots p_{n}$ , then at a given instant of time $\\mathcal {T}$ the values of the variables $p_{1}\\ldots p_{n}$ will lie within the chosen region $\\Gamma $ or outside it; hence, during a fraction of the time $\\mathcal {T}$ , which we will call $\\tau $ , they will lie in the chosen region $\\Gamma $ .", "Our condition then reads as follows: If the $p_{1}\\ldots p_{n}$ are state variables of a physical system, i.e., of a system that assumes a stationary state, then for each region $\\Gamma $ the quantity $\\tau /\\mathcal {T}$ has a definite limiting value for $\\mathcal {T}=\\infty $ .", "For each infinitesimally small region this value is infinitesimally small.", "Thus the stationary distribution is identified by a function $\\epsilon (p_{1},\\ldots , p_{n})$ such that the number $d N$ of systems which at any given instant in time are found in the infinitesimal region $g$ located around $(p_{1},\\ldots ,p_{n})$ is given by $d N = \\epsilon (p_{1},\\ldots ,p_{n})\\,d p_{1}\\cdots d p_{n}.$ If this is true at a given instant $t$ , then at a close instant $t+d t$ one has $d N_{t+d t}=d N_{t}-\\left(\\sum _{\\nu =1}^{n}\\frac{\\partial (\\epsilon \\varphi _{\\nu })}{\\partial p_{\\nu }}\\right)d p_{1}\\cdots d p_{n}.$ Since $d N_{t+d t}=d N_{t}$ , by the stationarity of the distribution, one must have $\\sum _{\\nu =1}^{n}\\frac{\\partial (\\epsilon \\varphi _{\\nu })}{\\partial p_{\\nu }}=0.$ Then $-\\sum _{\\nu =1}^{n}\\frac{\\partial \\varphi _{\\nu }}{\\partial p_{\\nu }}=\\sum _{\\nu =1}^{n}\\frac{\\partial \\log \\epsilon }{\\partial p_{\\nu }}\\varphi _{\\nu }=\\frac{d\\log \\epsilon }{d t}.$ The solution of equation (REF ) is $\\epsilon =\\exp \\left[-\\int d t\\;\\sum _{\\nu =1}^{n}\\frac{\\partial \\varphi _{\\nu }}{\\partial p_{\\nu }}+\\psi (E)\\right],$ where $\\psi (E)$ is a time-independent integration constant that, by the previous hypotheses, can only depend on the $p$ 's via the energy $E$ .", "One thus obtains $\\epsilon =\\text{const.", "}\\times \\exp \\left[-\\int dt\\;\\sum _{\\nu =1}^{n}\\frac{\\partial \\varphi _{\\nu }}{\\partial p_{\\nu }}\\right]=\\text{const.", "}\\; e^{-m},$ where $m$ is given by $m = \\int dt\\;\\sum _{\\nu =1}^{n}\\frac{\\partial \\varphi _{\\nu }}{\\partial p_{\\nu }}.$ Einstein now assumes that it is possible to introduce new state variables, denoted by $\\pi _{1},\\ldots ,\\pi _{n}$ , such that the factor $e^{-m}$ is cancelled by the Jacobian of the transformation.", "With this transformation, one obtains a uniform stationary distribution in phase space.", "However it is clear that this transformation cannot be performed unless $m$ is time-independent, which implies $d(\\log \\epsilon )/d t=0$ throughout, i.e., a form of Liouville's theorem.", "The oversight was realized by Einstein in March 1903, as witnessed by a letter to Michele Besso, [21] quoted by Uffink [15]: If you look at my paper more closely, you will find that the assumption of the energy principle & of the fundamental atomistic idea alone does not suffice for an explanation of the second law; instead, coordinates $p$ must exist for the representation of things, such that for every conceivable total system $\\sum \\partial \\phi _{\\nu }/\\partial p_{\\nu }=0$ .", "[...] If that is true, then the entire generalization attained in my last paper consists in the elimination of the concept of force as well as in the fact that $E$ can possess an arbitrary form (not completely)?", "The sections that immediately follow, on the distribution of a system in contact with a reservoir, on the absolute temperature and thermal equilibrium, and on the definition of “infinitely slow” (quasistationary) processes, are not fundamentally different from the corresponding sections of the 1902 memoir.", "The derivation of the mechanical expression of the entropy is however slightly different, in particular because the possibility of resorting to the Lagrangian formulation is no longer available.", "Einstein considers a situation in which the functions $\\varphi _{\\nu }$ which appear on the right-hand side of the equations (REF ) depend not only on the coordinates $p_{\\nu }$ , but also on some parameters $\\lambda $ .", "He then considers an infinitely-slow infinitesimal transformation, subdividing it into an isopycnic process, in which the $\\lambda $ 's are kept constant, but the system is put in thermal contact with a system at a different temperature, and an adiabatic process, in which the system is isolated, but the $\\lambda $ 's are allowed to vary.", "The energy change $dE$ is given in general by $dE=\\sum \\frac{\\partial E}{\\partial \\lambda }d\\lambda +\\sum _{\\nu }\\frac{\\partial E}{\\partial p_{\\nu }}dp_{\\nu }.$ In an isopycnic process the first term on the right-hand side of this equation vanishes, but the second term can be different from zero, since the equations of motion (REF ), which conserve $E$ , do not hold when the system is not isolated.", "In an adiabatic process, on the other hand, the second term vanishes, since the equations of motion (REF ) satisfy energy conservation, but at the same time one has $dQ=0$ .", "One can therefore write in general $dQ=\\sum _{\\nu }\\frac{\\partial E}{\\partial p_{\\nu }}\\,dp_{\\nu }.$ Therefore, in the expression for the change of energy in an infinitely slow process given in equation (REF ), one can identify the second term in the right-hand side with the infinitesimal heat exchange $dQ$ , and the first one, accordingly, with the infinitesimal work.", "Einstein has thus obtained a mechanical expression of the first principle of thermodynamics.", "Let us now denote by $W(p_{1},\\ldots ,p_{n})$ the probability distribution in phase space of the system when it is in equilibrium with an external body with a temperature function given by $\\beta $ .", "As derived by Einstein in § 3 of the paper, along the lines of its 1902 paper, it is given by $dW=e^{c-\\beta E}\\,dp_{1}\\cdots dp_{n},$ where the constant $c$ is defined by the normalization condition $\\int dW=\\int e^{c-\\beta E}\\,dp_{1}\\cdots dp_{n}=1.$ Let us assume that after the transformation, the system is in equilibrium with a body with temperature function $\\beta +d\\beta $ , while the parameters $\\lambda $ assume the values $\\lambda +d\\lambda $ .", "Then the normalization condition assumes the form $\\int \\exp \\left[c+dc -(\\beta +d\\beta )\\left(E+\\sum \\frac{\\partial E}{\\partial \\lambda }d\\lambda \\right)\\right]\\,dp_{1}\\cdots dp_{n}=1.$ One thus obtains, to first order, $\\begin{split}&\\int \\left(dc - E\\,d\\beta -\\beta \\sum \\frac{\\partial E}{\\partial \\lambda }d\\lambda \\right)\\, e^{c-\\beta E}\\,dp_{1}\\cdots dp_{n}=0.\\end{split}$ Einstein now argues that the expression in parentheses can be considered as a constant, “because the system's energy $E$ never differs markedly from a fixed average before and after the process”, and thus obtains $dc - E\\,d\\beta -\\beta \\sum \\frac{\\partial E}{\\partial \\lambda }d\\lambda =0.$ Since $E\\,d\\beta +\\beta \\sum \\frac{\\partial E}{\\partial \\lambda }d\\lambda =d\\left(\\beta E\\right)-\\beta \\sum _{\\nu }\\frac{\\partial E}{\\partial p_{\\nu }}dp_{\\nu }=d\\left(\\beta E\\right)-\\beta \\,dQ,$ where equation (REF ) has been substituted, Einstein obtains the relation $\\beta \\,dQ=d(\\beta E-c),$ and thus, since $1/\\beta =k_\\mathrm {B}T$ , $\\frac{dQ}{T}=d\\left(\\frac{E}{T}-k_{\\mathrm {B}}c\\right)=dS,$ from which he obtains the expression of the entropy $S=\\frac{E}{T}-k_{\\mathrm {B}}c=\\frac{E}{T}+k_{\\mathrm {B}}\\log \\int e^{-E/k_\\mathrm {B}T}\\,dp_{1}\\cdots dp_{n}.$ It is interesting to remark that in the 1902 paper Einstein had derived a similar expression of the heat exchanged $dQ$ involving the average values of the kinetic and potential energies, while here Einstein states that the values of the energy $E$ which matter are not very different from their mean value.", "This assumption is unnecessary because the relation (REF ) holds if $E$ is understood as the mean value of the energy, which is enough to reach Einstein's goals.", "Moreover, Einstein has not yet derived this property of the energy distribution.", "We shall see that this assumption also leads Einstein to a quite dubious result in the next discussion, where he attempts to establish the property of entropy increase.", "In our opinion, Einstein later reconsidered this argument and was led therefore to investigate the fluctuations of energy, which he discusses in his next paper.", "Einstein now attempts to prove that the entropy does not decrease in transformations involving an adiabatically isolated system.", "He goes on from the probability distribution of a single system in its phase space, when the value of its energy is fixed, to the distribution of a collection of a very large number $N$ of such systems with the same value of the energy.", "Dividing the phase space in $\\ell $ regions $g_{i}$ , $i=1,\\ldots ,\\ell $ of equal volume, Einstein looks for the probability of that $n_{1}$ systems fall in $g_{1}$ , ..., $n_{\\ell }$ systems fall in $g_{\\ell }$ .", "The result is obviously $W=\\left(\\frac{1}{\\ell }\\right)^{N}\\frac{N!", "}{n_{1}!\\cdots n_{\\ell }!", "}.$ One then has, by Stirling's formula, $\\begin{split}\\log W &=\\text{const.", "}-\\sum _{i} n_{i} \\log n_{i} \\simeq \\text{const.", "}-\\int \\rho \\log \\rho \\;dp_{1}\\cdots dp_{n},\\end{split}$ where $\\rho $ is the density of systems in the $p$ -space, when $\\ell \\rightarrow \\infty $ .", "It would have been a simple step to connect explicitly this expression to the entropy by means of Boltzmann's formula, but Einstein does not do it.", "He instead uses it first to show that this expression reaches a maximum when $\\rho $ is constant on the whole region of phase space in which the energy has the assigned value.", "He then argues that if the density $\\rho $ differs noticeably from a constant (for states of a given value of the energy), it will be possible to find distributions with a larger value of $W$ .", "In this case, if we follow in time the ensemble, the distribution will change with time, and since “we will have to assume that always more probable distributions will follow upon improbable ones, i.e., that $W$ increases until the distribution of states has become constant and $W$ a maximum”.", "Thus, if the distribution changes from $\\rho $ to $\\rho ^{\\prime }$ as time goes by, and the probability correspondingly increases from $W$ to $W^{\\prime }$ , the integral on the right-hand side of equation (REF ) decreases.", "He then argues that if the values of $\\log \\rho $ (when $\\rho $ does not essentially vanish) are close to uniform, and the probability increases, one obtains the relation $-\\log \\rho ^{\\prime }\\ge -\\log \\rho .$ This equation cannot be true without qualification, due to the normalization condition, and is however unnecessary for Einstein's argument in the immediately following section.", "See, e.g., the discussion in [15].", "This is probably one of the points which led Einstein, in retrospect, to reconsider the assumption that the values of the energy which have non-vanishing probability are close to constant, and to evaluate the energy fluctuations.", "Einstein then takes advantage of this result to obtain the law of entropy increase in the following way.", "He considers a finite number of systems $\\sigma _{1},\\ldots ,\\sigma _{\\nu },\\ldots $ , that together form an isolated system with state variables $p^{(1)}_{1},\\ldots ,p^{(1)}_{n_{1}},\\ldots ,$$p_{1}^{(\\nu )},\\ldots ,p_{n_{\\nu }}^{(\\nu )},\\ldots $ , such that $n=\\sum _{\\nu }n_{\\nu }$ .", "System $\\sigma _{\\nu }$ is initially in equilibrium at a temperature $T_{\\nu }=1/k_{\\mathrm {B}}\\beta _{\\nu }$ , and is therefore described by the distribution $dw_{\\nu }=e^{c_{\\nu }-\\beta _{\\nu }E_{\\nu }}\\,dp_{1}^{(\\nu )}\\cdots dp_{n_{\\nu }}^{(\\nu )}.$ Then the distribution of the global system is given by $dw=\\prod _{\\nu }dw_{\\nu }=e^{\\sum (c_{\\nu }-\\beta _{\\nu }E_{\\nu })}\\,dp_{1}\\cdots dp_{n}.$ Let us assume that the systems are now allowed to interact among themselves, and that at the end of the process a new equilibrium is reached, characterized by the temperature parameters $\\beta ^{\\prime }_{\\nu }$ , etc.", "We then have, at the end of the process, $dw^{\\prime }=\\prod _{\\nu }dw^{\\prime }_{\\nu }=e^{\\sum (c^{\\prime }_{\\nu }-\\beta ^{\\prime }_{\\nu }E^{\\prime }_{\\nu })}\\,dp_{1}\\cdots dp_{n}.$ Einstein now introduces an ensemble of a very large number $N$ of global systems $\\Sigma $ to argue that, since $W$ always increases, the distributions $\\rho &= N \\,e^{\\sum (c_{\\nu }-\\beta E_{\\nu })};\\\\\\rho ^{\\prime } &= N \\,e^{\\sum (c^{\\prime }_{\\nu }-\\beta ^{\\prime } E^{\\prime }_{\\nu })};$ satisfy equation (REF ), i.e., $\\sum \\left(c^{\\prime }_{\\nu }-\\beta ^{\\prime }_{\\nu } E^{\\prime }_{\\nu }\\right)\\ge \\sum (c_{\\nu }-\\beta _{\\nu }E_{\\nu }).$ But this implies, by equation (REF ), $\\sum S^{\\prime }_{\\nu }\\ge \\sum S_{\\nu }.$ Again, the detour by equation (REF ) is disputable and unnecessary.", "Indeed, it is sufficient to use equation (REF ) to obtain equation (REF ) where $E$ is now taken as the mean value of the energy, and the result would follow.", "The observations made after equation (REF ) also apply here.", "However, the main weakness of the argument lies in the petitio principii that the probability $W$ of the ensemble distribution should always increase.", "This objection was raised by Paul Hertz in 1910 [22], and Einstein soon acknowledged [9] that the objection was “fully founded”.", "In the closing section of this paper Einstein applies these results to a simple description of a thermal engine connected in turn to several heat reservoirs to derive the second principle in the form of Clausius." ], [ "On the general molecular theory of heat", "A change of pace is easily noticed already in the first lines of the 1904 paper, entitled “On the general molecular theory of heat.”[4] Here he refers to his previous papers, in which he had spoken of the “kinetic theory of heat” as laying the foundations of thermodynamics, by the less specific expression of “molecular theory of heat”.", "The paper contains several results worth mentioning, as announced at the end of the introduction First, I derive an expression for the entropy of a system, which is completely analogous to the expression found by Boltzmann for ideal gases and assumed by Planck in his theory of radiation.", "Then I give a simple derivation of the second law.", "After that I examine the meaning of a universal constant, which plays an important role in the general molecular theory of heat.", "I conclude with an application of the theory to black-body radiation, which yields a most interesting relationship between the above-mentioned universal constant, which is determined by the magnitudes of the elementary quanta of matter and electricity, and the order of magnitude of the radiation wave-lengths, without recourse to special hypotheses.", "These results are obtained as independent developments of the theory reported in the previous two papers.", "In the previous papers he had derived the canonical expression of entropy, namely $S=\\frac{E}{T}+k_{\\mathrm {B}}\\int e^{-E/k_\\mathrm {B}T}\\,dp_{1}\\cdots dp_{n},$ where $(p_{1},\\ldots ,p_{n})$ are the general state variables of the system, and $E$ is the value of the internal energy.", "In § 1 of this paper Einstein derives the expression we now call microcanonical, which is related to the density of states of energy $E$ , $\\omega (E)$ , by the relation $S=k_{\\mathrm {B}}\\log [\\omega (E)].$ He obtains this result by integrating the relation between the temperature and $\\omega (E)$ previously derived: $\\frac{1}{k_{\\mathrm {B}}T}=\\frac{\\omega ^{\\prime }(E)}{\\omega (E)},$ where one assumes that the system's energy lies between $E$ and $E+\\delta E$ .", "Note, however, that in the previous papers $\\omega (E)$ was the energy density of the thermal reservoir, while this relation is tacitly applied here to the energy density of the system.", "Interestingly, in this paper Einstein defines for the first time the density of states $\\omega (E)$ in the now customary way, by $\\omega (E)\\,\\delta E=\\int _{E}^{E+\\delta E}d p_{1}\\cdots d p_{n},$ while in the previous papers he kept including the $\\delta E$ factor in its definition.", "The “derivation” of the second law in § 2 suffers again, as in the 1903 paper, from the petitio principii of the assumption that more improbable states never follow more probable ones.", "The calculation is now simpler, but the result is also more restricted.", "First Einstein formulates the zero-th laws law of thermodynamics by assuming that if a system is in contact with an environment at temperature $T_{0}$ it acquires the temperature $T_{0}$ and keeps it from then on.", "However, according to the molecular theory of heat, this is not absolutely true, but true only with some approximation.", "In particular the probability $W\\,\\delta E$ that the energy of such a system has a value lying between $E$ and $E+\\delta E$ at an arbitrary point in time is given by $W\\,\\delta E=C \\, e^{-E/k_\\mathrm {B}T_{0}}\\,\\omega (E) \\,\\delta E,$ where $C$ is a constant.", "Einstein argues that this distribution is very sharply peaked and that, because of the previous result, it can also be written in the form $W\\,\\delta E=C\\, \\exp \\left[\\frac{1}{k_{\\mathrm {B}}}\\left(S-\\frac{E}{T_{0}}\\right)\\right]\\,\\delta E,$ where $S=S(E)$ is the value of the entropy pertaining to the value $E$ of the internal energy.", "Note that here again the property of the distribution of being sharply peaked is not needed, and anyway has not yet been derived.", "More interestingly, as far as we know, this is the first statement of Einstein's principle of fluctuations, which relates the probability of an energy fluctuation in a thermodynamic system to the difference in the expression $\\mathcal {F}(E,T)=E-TS(E)$ , which is now known as the availability.", "Now Einstein considers a system made of several such subsystems, all in contact with a large similar system at the temperature $T_{0}$ .", "The probability $\\mathfrak {W}$ of a given distribution $(E_{1},\\ldots ,E_{\\ell })$ of the energy among these subsystems is given by $\\mathfrak {W}\\propto \\exp \\left[\\frac{1}{k_{\\mathrm {B}}}\\left(\\sum _{i=1}^{\\ell }S_{i}-\\frac{1}{T_{0}}\\sum _{i=1}^{\\ell }E_{i}\\right)\\right].$ Let the reservoirs exchange energy, maybe by the assistance of cyclic machines, reaching an energy distribution $(E^{\\prime }_{1},\\ldots ,E^{\\prime }_{\\ell })$ .", "The corresponding probability is given by $\\mathfrak {W}^{\\prime }\\propto \\exp \\left[\\frac{1}{k_{\\mathrm {B}}}\\left(\\sum _{i=1}^{\\ell }S^{\\prime }_{i}-\\frac{1}{T_{0}}\\sum _{i=1}^{\\ell }E^{\\prime }_{i}\\right)\\right].$ Assuming again that less probable states are followed by more probable ones, one must have $\\mathfrak {W}^{\\prime }\\ge \\mathfrak {W}.$ Since $\\sum _{i}E_{i}$ is conserved, this equation implies $\\sum _{i=1}^{\\ell }S^{\\prime }_{i}\\ge \\sum _{i=1}^{\\ell }S_{i}.$ It is hard for us to make sense of this derivation.", "The results seems restricted to systems in contact with a reservoir with a given temperature $T_{0}$ , and therefore it is by no means general.", "In particular the inequality among the $\\mathfrak {W}$ 's cannot be absolutely satisfied without violating the normalization of probabilities, just as in the case of equation (REF ).", "The most interesting part is the way in which Einstein treats the distribution of energies among the system as a collective state of a system made of several subsystems and, at the same time, as one possible macroscopic state of a system governed by a canonical distribution at the temperature $T_{0}$ .", "This device will be put to use in the 1910 work on critical fluctuations.", "[23] The physical interpretation of the constant $\\kappa =k_{\\mathrm {B}}/2$ is obtained by Einstein in § 3 by evaluating, via his equipartition theorem, the kinetic energy of a mechanical system of $n$ particles, and by relating the resulting expression to the one obtained by the kinetic theory for the ideal gas.", "He thus obtains an explicit estimate of $\\kappa =6.5\\cdot 10^{-17}{erg\\, K^{-1}}$ , corresponding to $k_{\\mathrm {B}}=1.3\\cdot 10^{-23}{J\\,K^{-1}}$ .", "The discrepancy with modern values is due to the use of the value $N_{\\mathrm {A}}=6.4\\cdot 10^{23}{mol^{-1}}$ for Avogadro's number, that Einstein found in O. E. Meyer's book.", "[24] In § 4, under the title “General meaning of the constant $\\kappa $ ” Einstein discusses the fluctuations of the energy in the canonical ensemble, deriving the relation between the specific heat and the amplitude of energy fluctuations as $\\left<{E^{2}}\\right>-\\left<{E}\\right>^{2}=k_{\\mathrm {B}}T^{2}\\frac{d\\left<{E}\\right>}{d T},$ where $\\left<{\\ldots }\\right>$ denotes the canonical average.", "Gibbs had obtained the same expression in [7], but pointed out almost immediately that these fluctuations were not observable.", "With $\\epsilon $ , $\\epsilon _{p}$ and $\\epsilon _{q}$ the total, kinetic and potential energies respectively, and denoting averages by a bar, he states [7] It follows that to human experience and observation with respect to such an ensemble as we are considering, or with respect to systems which may be regarded as taken at random from such an ensemble, when the number of degrees of freedom is of such order of magnitude as the number of molecules in the bodies subject to our observation and experiment, $\\epsilon -\\bar{\\epsilon }$ , $\\epsilon _{p}-\\bar{\\epsilon }_{p}$ , $\\epsilon _{q}-\\bar{\\epsilon }_{q}$ would be in general vanishing quantities, since such experience would not be wide enough to embrace the more considerable divergencies from the mean values, and such observation not nice enough to distinguish the ordinary divergencies.", "In other words, such ensembles would appear to human observation as ensembles of uniform energy, and in which the potential and kinetic energies (supposing that there were means of easing these quantities separately) had each separately uniform values.", "Characteristically, Einstein instead goes over immediately to look for a system in which these fluctuations could be observed and he finds that the blackbody radiation could provide such a system.", "It is worth quoting his reasoning [4] If the linear dimensions of a space filled with temperature radiation are very large in comparison with the wavelength corresponding to the maximum energy of the radiation at the temperature in question, then the mean energy fluctuation will obviously be very small in comparison with the mean radiation energy of that space.", "In contrast, if the radiation space is of the same order of magnitude as that wavelength, then the energy fluctuation will be of the same order of magnitude as the energy of the radiation of the radiation space.", "Einstein pauses only one moment before proceeding to the application of his molecular theory of heat to black-body radiation [4] Of course, one can object that we are not permitted to assert that a radiation space should be viewed as a system of the kind we have assumed, not even if the applicability of the general molecular theory is conceded.", "Perhaps one would have to assume, for example, that the boundaries of the space vary with its electromagnetic state.", "However, these circumstances need not be considered, as we are dealing with orders of magnitude only.", "Einstein can thus evaluate the size $\\left<{\\epsilon ^{2}}\\right>$ of the energy fluctuations $\\epsilon =E-\\left<{E}\\right>$ from equation (REF ) and from the Stefan-Boltzmann law $\\left<{E}\\right>=a\\, v\\, T^{4},$ where $a=7.06\\cdot 10^{-15}\\,{erg\\,cm^{-3}\\,K^{-4}}$ is the radiation constant, $T$ is the absolute temperature, and $v$ is the cavity volume.", "Then, the linear dimensions of a cavity for which $\\left<{\\epsilon ^{2}}\\right>\\simeq \\left<{E}\\right>$ are given by $\\@root 3 \\of {v}=\\frac{1}{T}\\@root 3 \\of {\\frac{4 k_{\\mathrm {B}}}{a}}=\\frac{0.42}{T},$ which compares well (in order of magnitude) with the expression $\\lambda _{\\max }=0.293/T$ obtained from Planck's law (both lengths are expressed in cm, and $T$ is expressed in Kelvin).", "However, in the following months, trying to explicitly apply his theory to that system, he will encounter a paradox, which he will brilliantly overcome by renouncing the classical picture of the emission and adsorption of light, based on Maxwell's equations, and by introducing the concept of the light quanta.", "[5] But that is another story, which has already been told many times." ], [ "Einstein and Gibbs", "One usually takes for granted that the research projects pursued by Einstein in these three papers, and by Gibbs in his 1902 book [7] were equivalent, and that the more mathematically refined argumentation contained in the latter made Einstein's approach redundant.", "A closer scrutiny shows however fundamental differences in their approaches, and makes Einstein's approach more attractive to present-day physicists.", "Gibbs program focuses in understanding the properties of ensembles of mechanical systems, i.e., of systems whose dynamical equations are given, but whose initial conditions are only given in a probability distribution.", "He gives this discipline the name of “statistical mechanics”.", "He stresses that its relevance goes beyond establishing a foundation of thermodynamics [7] But although, as a matter of history, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light in departments quite outside of thermodynamics.", "Indeed, statistical mechanics laws are more general than those of thermodynamics [7] The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results.", "The laws of statistical mechanics apply to conservative systems of any number of degrees of freedom, and are exact.", "On the other hand, according to Gibbs, our ignorance of the basic constitution of material bodies make unreliable our inferences based on supposed models of matter, even when derived by the methods of statistical mechanics [7] In the present state of science, it seems hardly possible to frame a dynamic theory of molecular action which shall embrace the phenomena of thermodynamics, of radiation, and of the electrical manifestations which accompany the union of atoms.", "[...] Even if we confine our attention to the phenomena distinctively thermodynamic, we do not escape difficulties in as simple a matter as the number of degrees of freedom of a diatomic gas.", "It is well known that while theory would assign to the gas six degrees of freedom per molecule, in our experiments on specific heat we cannot account for more than five.", "Certainly, one is building on an insecure foundation, who rests his work on hypotheses concerning the constitution of matter.", "Gibbs therefore attempts to reduce his goal to a purely mathematical treatment [7] Difficulties of this kind have deterred the author from attempting to explain the mysteries of nature, and have forced him to be contented with the more modest aim of deducing some of the more obvious propositions relating to the statistical branch of mechanics.", "Here, there can be no mistake in regard to the agreement of the hypotheses with the facts of nature, for nothing is assumed in that respect.", "The only error into which one can fall, is the want of agreement between the premises and the conclusions, and this, with care, one may hope, in the main, to avoid.", "One can therefore only hope to establish analogies between quantities which are defined within statistical mechanics, and those which are empirically encountered in thermodynamics [7] We meet with other quantities, in the development of the subject, which, when the number of degrees of freedom is very great, coincide sensibly with the modulus, and with the average index of probability, taken negatively, in a canonical ensemble, and which, therefore, may also be regarded as corresponding to temperature and entropy.", "The relations of the laws of statistical mechanics with thermodynamics is further discussed in [7] A very little study of the statistical properties of conservative systems of a finite number of degrees of freedom is sufficient to make it appear, more or less distinctly, that the general laws of thermodynamics are the limit toward which the exact laws of such systems approximate, when their number of degrees of freedom is indefinitely increased.", "And the problem of finding the exact relations, as distinguished from the approximate, for systems of a great number of degrees of freedom, is practically the same as that of finding the relations which hold for any number of degrees of freedom, as distinguished from those which have been established on an empirical basis for systems of a great number of degrees of freedom.", "The enunciation and proof of these exact laws, for systems of any finite number of degrees of freedom, has been a principal object of the preceding discussion.", "But it should be distinctly stated that, if the results obtained when the numbers of degrees of freedom are enormous coincide sensibly with the general laws of thermodynamics, however interesting and significant this coincidence may be, we are still far from having explained the phenomena of nature with respect to these laws.", "For, as compared with the case of nature, the systems which we have considered are of an ideal simplicity.", "[...] The phenomena of radiant heat, which certainly should not be neglected in any complete system of thermodynamics, and the electrical phenomena associated with the combination of atoms, seem to show that the hypothesis of systems of a finite number of degrees of freedom is inadequate for the explanation of the properties of bodies.", "In Gibbs' approach, the probability distribution is a datum of the problem, while in Einstein's one it is one of the unknowns.", "The greatest difference is that Gibbs starts from the equal a priori probability postulate, while for Einstein what is important is to evaluate time averages and these are replaced by phase space averages through an ergodic hypothesis.", "Thus Gibbs is allowed to introduce the canonical distribution a priori, as a particularly simple one, endowed with interesting properties, in particular because it factorizes when one considers the collection of two or more mechanically independent systems [7] The distribution [...] seems to represent the most simple case conceivable, since it has the property that when the system consists of parts with separate energies, the laws of the distribution in phase of the separate parts are of the same nature, a property which enormously simplifies the discussion, and is the foundation of extremely important relations to thermodynamics.", "On the contrary, for Einstein, the canonical distribution is the distribution which describes the mechanical state of a system in contact with a thermal reservoir at a given temperature, while the “simplest” distribution is rather the microcanonical, which represents the state of an isolated system at equilibrium.", "And the former is derived from the latter.", "Einstein's 1910 lecture notes on the Kinetic Theory of Heat at the University of Zurich show, in Navarro's words [14], how his approach allowed him to proceed to the systematic application of statistical mechanics, once the canonical distribution is attained, to a large variety of fields.", "This is a sample list of the applications presented in the lecture notes: paramagnetism, Brownian motion, magnetic properties of solids, electron theory of metals, thermoelectricity, particle suspensions and viscosity.", "Gibbs invented, instead, a method whereby he could find no direct physical application other than the detection of the already mentioned thermodynamic analogies.", "Had Gibbs lived longer (he died the year after the publication of Elementary Principles) this might have changed.", "But, given his rigorous and extremely cautious attitude, any assumption on the issue is enormously risky.", "Even more strikingly, in Einstein's hands, deviations from the expected behavior become a tool for the investigation of the microscopic dynamics.", "This difference in attitude was already highlighted above, in the discussion of energy fluctuations, but the clearest example is the 1905 paper on light emission and adsorption, [5] where one notably reads This relation,It is the relation now known as Jean's radiation law.", "found as a condition for the dynamical equilibrium, not only fails to agree with the experiments, but also intimates that in our model a well-defined distribution of the energy between ether and matter is out of the question.", "[...] In the following, we shall treat the “black-body radiation” in connection with the experiments, without establishing it on any model of the production or propagation of the radiation.", "Thus Einstein brackets the contemporary models of light adsorption and propagation, but maintains the statistical interpretation of entropy.", "He then evaluates the radiation entropy from the empirical distribution law and interprets it in terms of the statistical approach as describing the coexistence of point-like particles in a given volume (cf. [26]).", "This paper was soon followed by the equally bold application of Planck's radiation theory to the specific heats of solids [27]." ], [ "Concluding remarks", "We presented Einstein's approach to statistical mechanics in contrast to the one taken by Gibbs.", "The results are equivalent since both are based on Boltzmann's contributions.", "Gibbs' starting point is the equal a priori probability hypothesis in phase space that leads to the microcanonical probability density for an ensemble (of representative systems, according to Tolman [8]).", "Einstein, on the other hand, starts by stating that what is important is the evaluation of time averages of appropriate quantities.", "These can be replaced by averages of the same quantities over an unknown density function over the phase space, with the help of an ergodic hypothesis.", "Einstein introduces the assumption that the energy is the only conserved quantity to play the role of the ergodic hypothesis.", "Using this assumption and Lioville's theorem, Einstein shows that the unknown density function mentioned before must be constant on the energy shell, that is it must be the microcanonical distribution.", "From there, the interpretation of the canonical distribution is different: for Gibbs, it is the simplest distribution, which leads to describe as statistically independent systems which are physically independent, while for Einstein it is the distribution which describes the state of a system in contact with a reservoir.", "Thus the index of the canonical distribution (as defined by Gibbs) is analogous to the temperature for Gibbs, but can be identified with the temperature for Einstein.", "It is also interesting to remark that in several points Einstein states (without proof) that the distribution of energy values in the canonical ensemble is sharply peaked, and deduces from this some dubious inequalities for the probability density itself.", "Only in the 1904 paper he explicitly evaluates the size of fluctuations, obtaining a result already derived by Gibbs.", "Then, while Gibbs had stressed the non-observability of energy fluctuations in macroscopic systems (thus contributing to the “rational foundation of thermodynamics”), Einstein points at the use of fluctuations as a tool for investigating microscopic dynamics (as he did, in particular, in [28], where he hinted at the dual wave-particle nature of radiation by interpreting the two terms appearing the expression of energy fluctuations).", "What interest can a present-day reader find in these papers?", "We think that they sketch a very neat road map for the introduction of the basic concepts of statistical mechanics, focusing on their heuristic value.", "One first focuses on isolated systems and identifies the microcanonical ensemble as the equilibrium distribution by means of the thermal equilibrium principle.", "For this step, Einstein's reasoning given above, based on the postulate of the absence of integrals of motion beyond the energy, is excellent.", "Then, one looks at a small part of such an isolated system, and one shows that the corresponding distribution is the canonical one.", "Finally, one identifies the mechanical expressions of temperature, infinitesimal heat and, by integration, of entropy.", "All these steps can be tersely traced by following, more or less closely, Einstein's path.", "At this point, the focus can be shifted to the evaluation of fluctuations, which allow on the one hand to recover the equivalence of ensembles for large enough systems and, by the same token, to identify situations in which the underlying molecular reality shows up in the behavior of macroscopic systems (like, e.g., in Brownian motion).", "This road map has been more or less followed by several modern textbooks on statistical mechanics, but we think that it would be fair to stress that it had first been sketched in the papers we described.", "In any case, we will be satisfied if the present note encourages some colleagues to have a look at these papers, in which the first steps in the making of a giant are recorded." ], [ "Acknowledgments", "LP was introduced to critical phenomena by Leo's lectures in the 1971 Varenna School, and RR fondly remembers Leo's course in the Escuela Mexicana de Física Estadística, which had a great influence on the Statistical Physics group at UNAM.", "Both authors dedicate this work to Leo's memory.", "LP is grateful to Jeferson Arenzon for encouraging him to present his ideas on Einstein's 1902–04 works." ] ]
1606.04890
[ [ "Advanced-Retarded Differential Equations in Quantum Photonic Systems" ], [ "Abstract We propose the realization of photonic circuits whose dynamics is governed by advanced-retarded differential equations.", "Beyond their mathematical interest, these photonic configurations enable the implementation of quantum feedback and feedforward without requiring any intermediate measurement.", "We show how this protocol can be applied to implement interesting delay effects in the quantum regime, as well as in the classical limit.", "Our results elucidate the potential of the protocol as a promising route towards integrated quantum control systems on a chip." ], [ "Introduction", "In advanced-retarded (A-R) differential equations, or mixed functional differential equations, the derivative of the associated function explicitly depends on itself evaluated at different advanced-retarded values of the variable [1], [2], [3], [4], [5].", "In order to solve such A-R equations either analytically or numerically, we require the knowledge of the solution history out of the domain of the equation.", "In many scientific disciplines A-R differential equations are used to describe phenomena containing feedback and feedforward interactions in their evolution [6], [7], [8].", "In physics, for instance, A-R equations can be used to model dynamical systems exhibiting certain symmetry in the evolution.", "As a prominent examples one may mention the application of A-R equations [9] in Wheeler-Feynman absorber theory [10], [11] and in the propagation of waves in discrete spatial systems [12].", "In the context of Quantum Mechanics, the implementation of feedback is more intricate than in the classical case due to the sensitivity of quantum systems to measurements.", "In this regard, a set of techniques has been developed for the realization of feedback-dependent systems, each of them employing different resources such as dynamical delays [13], [14], [15], machine learning optimization [16], weak measurements [17], [18], including quantum memristors [19], [20], and projective measurements for digital feedback [21], among others.", "Certainly, the inclusion of feedback or memory effects in quantum dynamical systems has extended the scope of quantum protocols, and it has allowed for the study and reproduction of more complex phenomena.", "Therefore, devising schemes for engineering Hamiltonians that display advanced-retarded dynamics is of great relevance.", "Along these lines, the field of non-Markovian quantum dynamics focuses on the study of effective equations that govern the evolution of systems interacting with environments depending on previous times [22], [23], [24], [25].", "As a result, estimation of non-Markovianity sheds light on the memory content of the systems under study.", "In this article we show that photonic lattices can be used to effectively tailor the dynamics of classical and quantum light fields in an advanced-retarded fashion.", "Our strategy is to exploit the duality between light propagation in space and time evolution [26].", "Our A-R photonic approach exploits the isomorphism existing between the steady state of judiciously-designed photonic waveguide circuits and solutions of A-R differential equations.", "We foresee that the inherent versatility of the proposed system will make the implementation of feedback and feedforward noticeably simple, in both quantum and classical frameworks, and thus may pave the way to interesting applications in integrated quantum technologies.", "In order to introduce our protocol, we start by considering the following first order linear and non-autonomous A-R differential equation $i\\frac{dx(t)}{dt}=\\beta (t)x(t)+\\kappa ^{-}(t)x(t-\\tau )+\\kappa ^{+}(t)x(t+\\tau ),$ with $\\kappa ^+ (t)=\\kappa ^- (t+\\tau )$ , and boundary conditions $\\kappa ^- (0< t < \\tau ) = \\kappa ^+ ((N-1)\\tau < t < N\\tau )=0$ .", "Associating the functions $x(t)$ , $x(t\\pm \\tau )$ with the mode amplitude of monochromatic waves traversing the $j$ -th and $j\\pm 1$ -th waveguides, $a_{j}(z)$ and $a_{j\\pm 1}(z)$ , of an array of $N$ evanescently coupled waveguides, each supporting a single mode and having a “time\" dependent propagation constant $\\beta (z)$ , we obtain a system governed by a set of $N$ differential equations $i\\frac{da_j}{dz}=\\beta (z) a_j (z) + \\kappa _{j, j+1} (z) a_{j+1} (z) + \\kappa _{j,j-1} (z) a_{j-1} (z),$ with $j\\in [1,N]$ .", "In the quantum optics regime, the dynamics of single photons traversing this type of devices is governed by a set of Heisenberg equations that are isomorphic to Eq.", "(REF ).", "The only difference is that in the quantum case, the mode amplitude $a_{j}$ is replaced by the creation operators $a_{j}^{\\dagger }$ [27].", "In order to make Eq.", "(REF ) isomorphic to Eq.", "(REF ), we must impose a continuity condition $a_j (0)=a_{j-1}(\\tau )$ within the interval $j \\in [2, N-1]$ .", "Physically, this condition implies that the mode field $a_{j-1}$ , at the propagation distance of $(j-1)\\tau $ ($\\tau $ is the length of the waveguides), has to be fed back into the input of the $j$ -th waveguide.", "Furthermore, the finiteness of the waveguide array imposes the boundary conditions $\\kappa _{1,0}=\\kappa _{N,N+1}=0$ .", "Additionally, we establish the aforementioned mapping of the independent variable $t$ into the spatial coordinate $z$ .", "Therefore, our protocol requires the implementation of waveguide lattices endowed with input-output connections as illustrated in Fig.", "REF .", "We point out that the phase introduced via the fiber propagation can be traced and disregarded given that it can be made equal for all fiber connections, being therefore a global phase.", "Figure: Scheme of the proposed implementation.", "Illustration of the chip for implementing the photonic simulator in Eq.", "(), where the arrows represent the input and output ports and the lines inside and outside the chip represent the waveguides and fiber connections respectively.The analogy we present here can be exploited with classical electromagnetic fields or single photons.", "The motivation for doing a quantum simulation is the possibility to embed the advanced-retarded dynamics in more general quantum protocols allowing the implementation of quantum feedback and feedforward.", "In the classical case, the initial condition $a_1 (0)$ is implemented by continuously injecting light into the system.", "This condition is crucial to establish the isomorphism between the light dynamics in the waveguide array and the solution of Eq.", "(REF ).", "As a result, the solution of Eq.", "(REF ) is obtained in the stationary regime of our photonic system.", "Once the intensity is measured, the modulus square of the solution is obtained by merging the intensity evolution of each waveguide in a single variable.", "See Fig.", "REF for a demonstration of the potentiality of our protocol, in which we analyze the setup depicted in Fig.", "REF .", "The light dynamics occurring in such an array is governed by Eq.", "(REF ).", "Figure: Numerical simulations of the dynamics.", "(a) Intensity evolution for an array having N=6N=6 waveguides and constant lattice parameters β=1\\beta ~=~1, κ=β+N\\kappa ~=~\\sqrt{\\beta +N} and τ=1\\tau =1.", "(b) Intensity of all the waveguides concatenated in a single curve, which represents the absolute square of the solution of Eq.", "().An interesting point to highlight here is the existence of a $z$ reversal symmetry in the simulation with respect to the central point of the evolution, $z_c=N\\tau /2$ , for constant lattice parameters.", "This relation holds for the modulus square of the solution, $a a^{*}(z_c +z)=a a^{*} (z_c -z)$ .", "Consequently, after the system reaches the steady state, it simultaneously fulfills the periodic boundary condition, $a(0)=a(N\\tau )$ , where we have neglected a global phase factor.", "This property combined with the space-time analogy opens a possible framework for the study and implementation of Closed and Open Timelike Curve gates [28], [29], [30], [31], [32], [33], [34]." ], [ "Non-Autonomous Equations", "A more general scenario arises when considering space-dependent parameters in Eq.", "(REF ), which in the context of photonic lattices, means that the system becomes dynamic.", "The dynamic character is achieved with modulations of the refractive index of individual waveguides.", "We consider an implementable system conformed by a periodic variation $\\beta (z)=\\beta _{0}+\\epsilon \\cos (\\omega z)$ , where $\\beta _{0}$ is a constant, $\\epsilon $ is the modulation amplitude, and $\\omega $ stands for the modulation frequency along $z$ .", "For illustration purposes, we provide a numerical calculation of this particular photonics simulator in the Supplementary Fig.", "1.", "Notice that in the context of advanced-retarded equations the time dependence allows us to encode non-autonomous equations, which are hard to compute in general." ], [ "Systems of Equations", "We now turn our attention to demonstrate how our protocol can be extended to provide solutions of systems of A-R equations.", "To this end, we encode every unknown function in a waveguide array, and place all the involved arrays close to each other in such a way that light fields traversing the system can tunnel from array to array.", "In this manner, all functions are self-coupled and coupled to others, enriching the dynamics of the systems.", "In order to explain the operating principle of the protocol, we focus on the simplest case of two variables.", "Note that in this case, we can relate each array to a component of a qubit to be implemented in the dual-rail encoding with a single photon.", "As a first possibility, we can put together two lattices in which the feedback takes place as depicted in Fig.", "REF a.", "In this scenario, the light can hop to neighboring sites as well as to the sites of the adjacent array.", "This arrangement enables the time evolution simulation of a single qubit Hamiltonian combined with two terms corresponding to advanced and retarded couplings $i|\\dot{\\psi }(t)\\rangle =H(t) |\\psi (t)\\rangle + H^+ (t)|\\psi (t+\\tau )\\rangle + H^- (t) |\\psi (t-\\tau )\\rangle $ $H(t)=\\left( \\begin{array}{cc} \\beta _x (t)&q(t)\\\\q(t)&\\beta _y (t) \\end{array} \\right), \\qquad H^+(t)=\\left( \\begin{array}{cc} \\kappa ^+_x (t)&d^+_{xy}(t)\\\\d^+_{yx}(t)&\\kappa ^+_y (t) \\end{array} \\right), \\qquad H^-(t)=\\left( \\begin{array}{cc} \\kappa ^-_x (t)&d^-_{xy}(t)\\\\d^-_{yx}(t)&\\kappa ^-_y (t) \\end{array} \\right).$ A second configuration arises by mixing the connectors as shown in Fig.", "REF b.", "In this case we flip the qubit in the advanced and retarded times.", "Even though the equation exhibits the same structure as Eq.", "(REF ), the forward and backward Hamiltonians, $H^+$ and $H^-$ are the same matrices given in Eq.", "(REF ) multiplied to the left by $\\sigma _x$ .", "Here $H(t)$ , $H^+ (t)$ and $H^- (t)$ depend on the propagation constants, on the coupling between waveguides belonging to different arrays and the coupling between the guides of the same array.", "The vertical coupling is refereed to as $q$ while the transverse coupling constants are represented by $\\kappa $ and the labels $(x,y)$ correspond to the plane where the arrays are located.", "Moreover, $d$ accounts for the diagonal coupling.", "In addition to the conditions for each array in (REF ), the following consistency and boundary conditions are fulfilled, $\\nonumber H^+ ((n-1)\\tau < t < n\\tau )= H^- (0< t < \\tau )=0, \\qquad d^+_{yx}(t)=d^-_{xy}(t+\\delta ), \\qquad d^+_{xy}(t)=d^-_{yx}(t+\\delta ).$ In Supplementary Fig.", "2 a numerical calculation of Eq.", "(REF ) is provided.", "Even though our formalism is based mainly on classical optics, one can emulate a quantum system (qubit) via the encoding of multiple A-R equations, as shown in this section.", "Moreover, one may also include quantum effects via the loading of the chip with genuinely quantum photonic states, e.g., two-photon or few-photon Fock states, which will introduce quantum effects as photon bunching, in combination with advanced-retarded physics.", "A thorough analysis of this last possibility is, however, outside of the scope of this article.", "Figure: Systems of equations.", "Scheme of the chip in a perpendicular plane showing the input output connections and the parameters of the simulation.", "Here β\\beta is the propagation constant, qq is the vertical coupling constant, κ\\kappa is the horizontal coupling constant and dd the diagonal coupling constant.", "a) The scheme in which each plane is associated with a component of the qubit simulates Eq. ().", "b) The crossed links allow for a stronger temporal mixing of the qubit components in the derivative.", "This situation corresponds to the second example of Eq.", "()." ], [ "Multiple Delays", "We next consider a variant of Eq.", "(REF ) with multiple delays.", "This configuration arises when we allow each waveguide to couple to multiple neighbors and reordering the feedback connectors.", "The first non-trivial example is a two-time A-R equation, Eq.", "(REF ).", "$i\\dot{x}(t)= \\beta (t)x(t)+\\kappa ^+ (t) x(t+\\tau )+\\kappa ^{++} (t) x(t+2\\tau )+ \\kappa ^{-} (t) x(t-\\tau ) + \\kappa ^{--} (t) x(t-2 \\tau ).$ Experimentally, the arrangement can be engineered by fabricating the waveguides in a zig-zag configuration.", "The resulting equation shares the structure of an A-R differential equation with additional feedback and feedforward terms.", "For this particular system the coupling coefficients are related as $k^{++}(t)=k^{--}(t+2\\tau ), \\qquad k^{+}(t)=k^{-}(t+\\tau ),$ with the appropriate boundary conditions.", "See Supplementary Fig.", "3 for a numerical simulation of Eq.", "(REF )." ], [ "Higher Order Equations", "One last generalization of the A-R simulator consists in introducing complex dynamical parameters.", "This can be achieved by combining our feedback technique with Bloch oscillator lattices [35], [36], [37].", "These types of arrays can be implemented by including a transverse ramping on the potential of the waveguides or by curving the waveguide arrays.", "Provided the evolution equations for the Bloch oscillator array, $i \\dot{a_n}+ n\\beta {z}a_{n} + \\kappa ^+ a_{n+1} + \\kappa ^- a_{n-1}=0$ , and making the formal transformation $a_n=\\tilde{a}_{n}(z)\\exp (in\\phi (z))$ , with $\\phi (z)=\\int _{0}^z\\beta (z^{\\prime })dz^{\\prime }$ , one can show that it is formally equivalent to a system endowed with complex coefficients, $i\\dot{\\tilde{a}}_n + \\exp (in\\phi (z))\\kappa ^+ \\tilde{a}_{n+1} + \\exp (-in\\phi (z))\\kappa ^-\\tilde{a}_{n-1}$ .", "The inclusion of arbitrary complex parameters could be used to enhance the versatility of the protocol.", "Furthermore, a complete control of the coupling constants would allow to simulate higher order equations via systems of first order equations.", "Even though the toolbox introduced here is valid for simulating diverse physical configurations, it can be generalized by an appropriate mathematical treatment.", "Let $M(t)$ be the matrix containing the information about the propagation constant and couplings among the waveguides defined by $i \\dot{a}(t)=M(t) a(t)$ , $F$ the matrix encoding the input-output connections, and $\\alpha $ the initial state independent of the feedback and feedforward mechanism, such that $a(0)=F a(\\tau ) + \\alpha $ .", "Consider now that the dynamical equation is solvable in terms of the evolution operator $U(t)$ , $a(t)=U(t) a(0)$ .", "Our goal is to determine the complete initial condition $a(0)$ in terms of the independent initial condition $\\alpha $ , evolution operator $U(t)$ and input-output matrix $F$ .", "The consistency relations at $a(\\tau )$ impose a set of equations that $a(0)$ has to fulfill, $a(\\tau )=U(\\tau ) a(0)$ , $a(0)=F a(\\tau ) + \\alpha \\Rightarrow a(0)=\\left( \\mathbb {1}-FU(\\tau ) \\right)^{-1} \\alpha , \\qquad a(t)=U(t)( \\mathbb {1}-FU(\\tau ))^{-1}\\alpha .$ Notice that this relation holds for any $\\alpha $ , allowing the input of quantum states superposed in more than one waveguide, and is also valid for different configurations of couplings $U$ and connections $F$ , limited by the existence of the inverse of $(\\mathbb {1}-F U(\\tau ))$ .", "Moreover, we can think of different experimental conditions, in which the connections happen at distinct evolution times $\\tau _i$ , $a(0)=\\sum F_i a(\\tau _i)+\\alpha $ , resulting in $a(t)=U(t)( \\mathbb {1}-\\sum F_i U(\\tau _i))^{-1}\\alpha $ ." ], [ "Implementation Analysis", "We analyze now the main limitations of the protocol regarding its interpretation and experimental realization.", "Regarding the quantum equation, although the probability is not normalized during the dynamics, it is normalized in the input and output points.", "Therefore, initialization and measurements retrieve the correct interpretation, even if the particle undergoes forward and backward jumps on time, an effect that could be useful for the simulation of absorbing potentials [38].", "A natural source of errors is given by the waveguide losses damaging the quantum state in the time elapsed until the photon escapes from the chip.", "This time is related with the population in the solution, which is unknown before the experiment is realized.", "Identifying the relation between the resonances and the dynamical parameters could be helpful for estimating the population in the stationary state, and therefore the total experimental time for achieving the stationary state in the classical case.", "See Fig.", "REF for a scheme of the error depending on the distance from the stationary state.", "Figure: Error analysis.", "We depict the decimal logarithm of the error as a function of time for three runs of the simulation with different distance with respect to the stationary state.", "The fact that the effective interaction between photons is null makes possible the analogy between the stationary state solution and the accumulation of solutions for an initial excitation combined until the initial population has escaped from the output port.", "Therefore, the distance is calculated as the norm of the population that remains in the chip.", "The dynamical constants of the system are equivalents to the ones in Fig.", ".We have to take into account that the losses introduced by the fiber connections will break the continuity condition allowing us to simulate Eq.", "(REF ) in terms of Eq.", "(REF ).", "The length and propagation constant of this fibers have to be tuned so that no phase is introduced in the evolution.", "Additionally, the space dependence of propagation and coupling constants is limited by the experimentally realizable functions.", "The degrees of freedom to be considered are the writing precision for modifying the propagation constants and the spatial path of each waveguide for modifying the coupling constants.", "In conclusion, we have developed a flexible but realistic toolbox for implementing advanced-retarded differential equations in integrated quantum photonics circuits.", "We have shown that our analogy enables the simulation of time dependent systems of advanced-retarded equations, which in the context of quantum information can be employed to realize feedback and feedforward driven dynamics.", "Therefore, we consider that our work enhances the versatility of quantum simulators in the abstract mathematical direction and in terms of applications for retrospective and prospective quantum memory.", "See the included Supplemental Information for supporting content.", "We acknowledge support from Spanish MINECO/FEDER FIS2015-69983-P; Ramón y Cajal Grant RYC-2012-11391; UPV/EHU UFI 11/55 and EHUA14/04; Basque Government BFI-2012-322 and IT986-16; a UPV/EHU postdoctoral fellowship; Deutsche Forschungsgemeinschaft (grants SZ 276/7-1, SZ 276/9-1, SZ 276/12-1, BL 574/13-1, GRK 2101/1); and the German Ministry for Science and Education (grant 03Z1HN31).", "U. A.-R. and A. P.-L. made the numerical simulations while U. A.-R., A. P.-L., I. L. E., M. G., M. S., L. L., A. S., and E. S. developed the protocol and wrote the manuscript.", "The authors declare no competing financial interests.", "This document provides supplementary information to Advanced-Retarded Differential Equations in Quantum Photonic Systems.", "In particular, we provide examples of the genuine physical situation developed in the manuscript each of them associated with a given A.-R. equation, which show the versatility of our analog photonics simulator.", "Figure: Supplementary Figure 1.", "Numerical Simulation of Eq.", "(1)(1) with N=6N=6, β=1\\beta =1, ϵ=1\\epsilon =1, κ=7\\kappa =\\sqrt{7}, ω=2\\omega =2.", "(a) Intensity in the stationary state in the waveguides array.", "(b) Modulus square of the solution as a function of time.", "(c) Decimal Logarithm of the error of the simulation with respect to the solution of the A-R equation.Consider the oscillatory time dependence of the propagation constant, $\\beta $ , for the single variable A-R equation, Eq.", "$(1)$ in the manuscript.", "We numerically simulate this system for a lattice of $N=6$ waveguides, $\\beta =1$ , $\\kappa =\\sqrt{7}$ , $\\epsilon =1$ and $\\omega =2$ , see Fig.", "REF .", "We have selected this non-autonomous advanced-retarded equation to show the existence resonant solutions, in which a high amount of light gets trapped in the chip.", "Notice that although a high population is achieved in the stationary state the theoretical error is still small.", "Figure: Supplementary Figure 2.", "Numerical Simulation of Eq.", "(3)(3) with N=5N=5, β x =1\\beta _x=1, β y =2\\beta _y=2, κ x =3\\kappa _x=3, κ y =1\\kappa _y=1, q=1q=1, d=1d=1, τ=1\\tau =1 for the initial state |ψ(0)〉=|0〉.|\\psi (0)\\rangle =|0\\rangle .", "(a) Waveguides intensity in the xx plane corresponding to the first component of the qubit.", "(b) Waveguides intensity in the yy plane corresponding to the second component of the qubit.", "(c) Modulus square of the quantum state as a function of time.In Fig.", "REF we show a numerical simulation of the array proposed to simulate systems of A-R equations given by Eq.", "$(3)$ .", "The dynamical parameters are the following ones, $N=5$ , $\\beta _x=1$ , $\\beta _y=2$ , $\\kappa _x=3$ , $\\kappa _y=1$ , $q=1$ , $d=1$ and $\\tau =1$ for an initial state $|\\psi (0)\\rangle = |0\\rangle $ .", "The theoretical error is smaller than $10^{-2}$ for the complete time evolution.", "We have selected this parameters to show that highly asymmetric solutions are also possible for time independent equations even with the limitation induced by the physical constrains in the coupling constants $\\kappa $ .", "Figure: Supplementary Figure 3.", "Numerical simulation of Eq.", "(6)(6) with N=5N=5, β=1\\beta =1, κ=5\\kappa =5 and τ=1\\tau =1.", "(a) Intensity in the stationary state in the waveguides array.", "(b) Modulus square of the solution as a function of time.", "(c) Decimal Logarithm of the error of the simulation with respect to the solution of the A-R equation.Eq.", "$(6)$ in the manuscript describes the evolution of a system driven by two feedback and two forward terms.", "See Fig.", "REF for a numerical simulation of this equation with $N=5$ , $\\beta =1$ , $\\kappa =5$ and $\\tau =1$ ." ] ]
1606.05143
[ [ "On Properties of Differential Inclusions with Prox-regular Sets" ], [ "Abstract In this paper, some regularity properties of solutions of the following differential inclusion \\begin{equation}\\nonumber \\left\\{ \\begin{array}{l} \\dot{x}(t) \\in f\\big(x(t)\\big) -N_{C}\\big(x(t)\\big)\\; {\\rm a.e.}", "\\; t \\in [0,+\\infty), x(0) = x_0\\in C, \\end{array}\\right.", "\\end{equation} are analyzed where $f: H\\to H$ is Lipschitz continuous and $C$ is closed, uniformly prox-regular subset of a Hilbet space $H$.", "Here $N_{C}(\\cdot)$ denotes the proximal normal cone of $C$.", "This work can be considered as an improvement of [Hantoute-Mazade 2013] since these properties are established without the additional tangential condition at each point in $C$." ], [ "Introduction", "In the seventies, sweeping processes are introduced and deeply studied by J. J. Moreau through the series of papers [12], [13], [14], [15], [16] which plays an important role in elasto-plasticity, quasi-statics, dynamics, especially in mechanics [17], [18], [3].", "Roughly speaking, a point is swept by a moving closed convex set $C(t)$ in a Hilbert space $H$ and can be formulated in the form of differential inclusion as follows $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) \\in -N_{C(t)}(x(t))\\; {\\rm a.e.}", "\\; t \\in [0,T],\\\\ \\\\\\end{array}x(0) = x_0\\in C(0),\\right.$ where $N_{C(t)}(\\cdot )$ denotes the normal cone of $C(t)$ in the sense of convex analysis.", "When the systems are perturbed, it is natural to study the following variant $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) \\in -N_{C(t)}(x(t))+F(t,x(t))\\; {\\rm a.e.}", "\\; t \\in [0,T],\\\\ \\\\\\end{array}x(0) = x_0\\in C(0),\\right.$ where $F: \\mathbb {R}^+\\times H \\rightarrow 2^H$ is a set-valued mapping with nonempty weakly compact convex values in $H$ .", "For example, to study the planning procedures in mathematical economy, C. Henry [10] introduced and proved the existence of solutions in finite dimension of the system $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) \\in P_{T_{C}\\big (x(t)\\big )}\\big (F(x(t))\\big )\\; {\\rm a.e.}", "\\; t \\in [0,T],\\\\ \\\\x(0) = x_0\\in C,\\end{array}\\right.$ where $F:\\mathbb {R}^n\\rightarrow 2^{\\mathbb {R}^n}$ is upper semi-continuous with nonempty, convex, compact values and $C$ is a closed, convex set in $\\mathbb {R}^n$ .", "Here $T, P$ denote the tangent cone and projection operators, respectively.", "Later B. Cornet [6] extended the system $(\\ref {henry})$ for the case $C\\subset \\mathbb {R}^n$ is Clarke tangentially regular and reduced to $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) \\in F\\big (x(t)\\big ) -N_{C}\\big (x(t)\\big )\\; {\\rm a.e.}", "\\; t \\in [0,T],\\\\ \\\\x(0) = x_0\\in C.\\end{array}\\right.$ There are numerous results for various variants of sweeping processes in literature but most of them are about the existence of solutions (see, e.g., [4], [7], [8], [22]).", "In this paper, we are interested in properties of solutions of the differential inclusion $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) \\in f\\big (x(t)\\big ) -N_{C}\\big (x(t)\\big )\\; {\\rm a.e.}", "\\; t \\in [0,+\\infty ),\\\\ \\\\x(0) = x_0\\in C,\\end{array}\\right.$ where $f: H\\rightarrow H$ is Lipschitz continuous and $C$ is closed, uniformly prox-regular subset of a Hilbet space $H$ .", "It is known that $(\\ref {main})$ has a unique locally absolutely continuous solution $x(\\cdot )$ on $ [0,+\\infty )$ (see [7] for example).", "However, it is also important to know more regularity properties of solutions, even the asymptotic behaviour, to understand better the systems.", "In [9], the authors considered this direction for the same problem.", "The main properties are the right differentiable of the solution and $\\dot{x}^+(\\cdot )$ is right continuous at each $t\\ge 0$ , which later play an important role in studying Lyapunov functions as well as asymptotic behaviour of solutions.", "However, these properties are obtained in [9] under the tangential condition: $f(x)\\in T(C,x)$ for all $x\\in C.$ The condition is unnecessary since if $C$ is closed, convex then $N_C(\\cdot )$ is maximal monotone operator and thus we do not need such kind of condition [2].", "It motivates us to establish the same properties but without the additional tangential condition.", "The paper is organized as follow.", "In section , we recall some basic notations, definitions and results which are used throughout the paper.", "Some regularities properties of solutions are established without tangential condition in section $\\ref {section3}$ .", "Some conclusions and perspectives end the paper in section ." ], [ "Notations and Mathematical Background", "Let us begin with some notations used in the paper.", "Let $H$ be a Hilbert space.", "Denote by $\\langle \\cdot ,\\cdot \\rangle $ , $\\Vert \\cdot \\Vert $ the scalar product and the corresponding norm in $H$ .", "Denote by $I$ the identity operator, by $\\mathbb {B}$ the unit ball in $H$ and $\\mathbb {B}_r=r\\mathbb {B},\\; \\mathbb {B}_r(x)=x+r\\mathbb {B}$ .", "The distance from a point $s$ to a closed set $C$ is denoted by ${ d}(s,C)$ or ${ d}_C(s)$ and ${ d}(s,C)=\\inf _{x\\in C} \\Vert s-x\\Vert .$ Denote by $C^0$ the set of minimal norm elements of $C$ , $i.", "e.$ $C^0=\\lbrace c\\in C: \\Vert c\\Vert =\\inf _{c^{\\prime }\\in C} \\Vert c^{\\prime }\\Vert \\rbrace .$ It is know that if $C$ is closed and convex then $C^0$ contains exactly one element.", "The set of all points in $C$ that are nearest to $s$ is denoted by ${\\rm Proj}(C,s)=\\lbrace x\\in C: \\Vert s-x\\Vert ={ d}(s,C)\\rbrace .$ When ${\\rm Proj}(C,s)=\\lbrace x\\rbrace $ , we can write $x={\\rm proj}(C,s)$ to emphasize the single-valued property.", "Let $x\\in {\\rm Proj}(C,s)$ and $t\\ge 0$ , then the vector $t(s-x)$ is called proximal normal to $C$ at $x$ .", "The set of all such vectors is a cone, called proximal normal cone of $C$ at $x$ and denoted by $N^P(C,x)$ .", "It is a known result [5], [20] that $\\xi \\in N^P(C,x)$ if and only if there exist some $\\sigma >0, \\delta >0$ such that $\\langle \\xi , y-x \\rangle \\le \\delta \\Vert y-x\\Vert ^2\\;\\;{\\rm for\\;all\\;}y\\in C\\cap \\mathbb {B}_\\sigma (x).$ The Fréchet normal cone $N^F(\\cdot )$ , the limiting normal cone $N^L(\\cdot )$ and the Clarke normal cone $N^C(\\cdot )$ are defined respectively as follows: $N^F(C,x)=\\lbrace \\xi \\in H: \\forall \\delta >0, \\exists \\sigma >0 \\;s.\\; t.\\; \\langle \\xi , y-x \\rangle \\le \\delta \\Vert y-x\\Vert \\;{\\rm for\\;all\\;}y\\in C\\cap \\mathbb {B}_\\sigma (x)\\rbrace .$ $N^L(C,x)&=&\\lbrace \\xi \\in H:\\exists \\; \\xi _n\\rightarrow \\xi \\;{\\rm weakly\\;and}\\; \\xi _n\\in N^P(C,x_n), x_n\\rightarrow x\\;{\\rm in}\\; C\\rbrace \\\\&=&\\lbrace \\xi \\in H:\\exists \\; \\xi _n\\rightarrow \\xi \\;{\\rm weakly\\;and}\\; \\xi _n\\in N^F(C,x_n), x_n\\rightarrow x\\;{\\rm in}\\; C\\rbrace .$ $N^C(C,x)=\\overline{{\\rm co}}N^L(C,x).$ If $x\\notin C$ , one has $N^P(C,x)=N^F(C,x)=N^L(C,x)=N^C(C,x)=\\emptyset $ and for all $x\\in C$ : $N^P(C,x)\\subset N^F(C,x)\\subset N^L(C,x)\\subset N^C(C,x).$ If $C$ is convex then these normal cones are coincide.", "It is in fact still true for prox-regular sets, which are defined as follows.", "Then we can write only $N(C,x)$ for simplicity.", "Definition 2.1 The closed set $C$ is called $r-prox-regular$ iff each point $s$ in the $r$ -enlargement of $C$ $U_{r}(C)=\\lbrace w\\in H: { d}(w,C)<r\\rbrace ,$ has a unique nearest point ${\\rm proj}(C,s)$ and the mapping ${\\rm proj}(C,\\cdot )$ is continuous in $U_{r}(C)$ .", "Proposition 2.1 [19], [22] Let $C$ be a closed set in $H$ .", "The followings are equivalent: 1) $C$ is $r-prox-regular$ .", "2) For all $x\\in C$ and $\\xi \\in N^L(C,x)$ such that $\\Vert \\xi \\Vert \\le r$ , we have $x={\\rm proj}(C,x+\\xi ).$ 3) For all $x\\in C$ and $\\xi \\in N^L(C,x)$ , we have $\\langle \\xi , y-x\\rangle \\le \\frac{\\Vert \\xi \\Vert }{2r}\\Vert y-x\\Vert ^2\\;\\;\\forall \\; y\\in C.$ 4) (Hypo-monotonicity) For all $x,x^{\\prime }\\in C$ , $\\xi \\in N^L(C,x)$ , $\\xi ^{\\prime }\\in N^L(C,x^{\\prime })$ and $\\xi ,\\xi ^{\\prime }\\in \\mathbb {B}_{r}$ we have $\\langle \\xi -\\xi ^{\\prime },x-x^{\\prime } \\rangle \\ge -\\Vert x-x^{\\prime }\\Vert ^2.$ If $r=+\\infty $ , then $C$ is convex.", "Some examples of prox-regular sets [4]: The finite union of disjoint intervals is non-convex but uniformly $r$ -prox-regular and $r$ depends on the distances between the intervals.", "More generally, any finite union of disjoint convex subsets in $H$ is non-convex but uniformly $r$ -prox-regular and $r$ depends on the distances between the sets.", "We finish the section with a version of Gronwall's inequality (see, e.g., Lemma 4.1 in [21]).", "Lemma 2.1 Let $T>0$ be given and $a(\\cdot ),b(\\cdot )\\in L^1([t_0,t_0+T];\\mathbb {R})$ with $b(t)\\ge 0$ for almost all $t\\in [t_0,t_0+T].$ Let the absolutely continuous function $w: [t_0,t_0+T]\\rightarrow \\mathbb {R}_+$ satisfy: $(1-\\alpha )w^{\\prime }(t)\\le a(t)w(t)+b(t)w^\\alpha (t),\\;\\; a. e. \\;t\\in [t_0,t_0+T],$ where $0\\le \\alpha <1$ .", "Then for all $t\\in [t_0,t_0+T]$ : $w^{1-\\alpha }(t)\\le w^{1-\\alpha }(t_0){\\rm exp}\\Big (\\int _{t_0}^t a(\\tau )d\\tau \\Big )+\\int _{t_0}^t{\\rm exp}\\Big (\\int _{s}^t a(\\tau )d\\tau \\Big )b(s)ds.$" ], [ "Main Results", "Let us first recall the existence and uniqueness result of $(\\ref {main})$ (see, e.g., [7]).", "Theorem 3.1 Let $H$ be a Hilbert space and $C$ be a closed, r-prox-regular set.", "Let $f:H\\rightarrow H$ be a k-Lipschitz continuous function.", "Then for each $x_0\\in C$ , the following differential inclusion $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) \\in f\\big (x(t)\\big ) -N_{C}\\big (x(t)\\big )\\; {\\rm a.e.}", "\\; t \\in [0,+\\infty ),\\\\ \\\\x(0) = x_0\\in C,\\end{array}\\right.$ has a unique locally absolutely continuous solution $x(\\cdot )$ .", "In addition, we have $\\Vert \\dot{x}(t) -f\\big (x(t)\\big )\\Vert \\le \\Vert f\\big (x(t)\\big )\\Vert \\; {\\rm for}\\; a. e. \\; t\\ge 0.$ Let $x(\\cdot )$ be the unique solution of $(\\ref {main})$ satisfying $x(0)=x_0$ .", "Define $v: \\mathbb {R}_+\\rightarrow H$ by $v(t):=\\Big (f\\big (x(t)\\big )-N\\big (C,x(t)\\big )\\Big )^0$ and $v_0:=v(0)=\\big (f(x_0)-N(C,x_0)\\big )^0.$ By using similar arguments as in Lemma 1.8 [11], we have the following lemma.", "Lemma 3.1 We have $\\Vert v_0\\Vert \\le \\liminf _{t\\rightarrow 0^+}\\Vert v(t)\\Vert .$ Proof.", "If $\\liminf _{t\\rightarrow 0^+}\\Vert v(t)\\Vert =+\\infty $ then the conclusion holds.", "If $\\liminf _{t\\rightarrow 0^+}\\Vert v(t)\\Vert =\\gamma < +\\infty ,$ then there exists a sequence $(t_n)_{n\\ge 1}$ such that $t_n \\rightarrow 0^+$ and $\\lim _{n\\rightarrow +\\infty }\\Vert v(t_n)\\Vert =\\gamma .$ In particular, the sequence $\\big (v(t_n)\\big )_{n\\ge 1}$ is bounded hence there exist a subsequence $\\big (v(t_{n_k})\\big )_{k\\ge 1}$ and $\\xi \\in H$ such that $\\big (v(t_{n_k})\\big )_{k\\ge 1}$ converges weakly to $\\xi $ .", "Recall that $v(t_{n_k})=\\Big (f\\big (x(t_{n_k})\\big )-N\\big (C;x(t_{n_k})\\big )\\Big )^0\\in f\\big (x(t_{n_k})\\big )-N\\big (C;x(t_{n_k})\\big ).$ Hence $f\\big (x(t_{n_k})\\big )-v(t_{n_k})\\in N\\big (C;x(t_{n_k})\\big ).$ We can find some $\\beta >0$ such that $\\Vert f\\big (x(t_{n_k})\\big )-v(t_{n_k})\\Vert \\le \\beta $ for all $k\\ge 1.$ Using the prox-regularity of $C$ , one has $\\langle f(x\\big (t_{n_k})\\big )-v(t_{n_k}), c-x(t_{n_k}) \\rangle \\le \\frac{\\beta }{2r}\\Vert c-x(t_{n_k})\\Vert ^2 \\;{\\rm for\\; all}\\; c\\in C, \\;k\\ge 1.$ Let $k\\rightarrow +\\infty $ , we get $\\langle f(x_0)-\\xi ,c-x_0\\rangle \\le \\frac{\\beta }{2r}\\Vert c-x_0\\Vert ^2 \\;{\\rm for\\; all}\\; c\\in C.$ Thus $f(x_0)-\\xi \\in N(C;x_0)$ or equivalently $\\xi \\in f(x_0)-N(C;x_0)$ .", "Then $\\Vert \\xi \\Vert \\le \\liminf _{k\\rightarrow +\\infty }\\Vert v(t_{n_k})\\Vert = \\liminf _{n\\rightarrow +\\infty }\\Vert v(t_n)\\Vert =\\gamma ,$ due to the weak lower semicontinuity of the norm and the conclusion follows.", "Lemma 3.2 Let $x(\\cdot )$ be the unique solution of $(\\ref {main})$ satisfying $x(0)=x_0$ .", "Then one has $\\limsup _{t\\rightarrow 0^+}\\Vert \\frac{x(t)-x_0}{t}\\Vert \\le \\Vert v_0\\Vert ,$ where $v_0=\\big (f(x_0)-N(C,x_0)\\big )^0=f(x_0)-{\\rm proj}\\big (f(x_0), N_C(x_0)\\big ).$ Proof.", "We have $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) - f\\big (x(t)\\big )\\in - N_{C}\\big (x(t)\\big )\\; {\\rm a.e.}", "\\; t \\in [0,+\\infty ),\\\\ \\\\v_0-f(x_0)\\in - N_{C}(x_0),\\end{array}\\right.$ and $\\Vert \\dot{x}(t) -f\\big (x(t)\\big )\\Vert \\le \\Vert f\\big (x(t)\\big )\\Vert \\; {\\rm for}\\; a. e. \\; t\\ge 0.$ Using the prox-regularity of $C$ and Proposition REF , one has $\\langle \\dot{x}(t)-f\\big (x(t)\\big )-v_0+f(x_0), x(t)-x_0 \\rangle \\le \\frac{1}{r}\\big (\\Vert f\\big (x(t)\\big )\\Vert +\\Vert v_0-f(x_0)\\Vert \\big )\\Vert x(t)-x_0\\Vert ^2.$ Combining with the k-Lipschitz continuity of $f(\\cdot )$ , one deduces that $\\frac{1}{2}\\frac{d}{dt}\\Vert x(t)-x_0\\Vert ^2\\le \\Vert v_0\\Vert \\Vert x(t)-x_0\\Vert +a(t)\\Vert x(t)-x_0\\Vert ^2,$ where $a(t)=k+\\frac{1}{r}\\big (\\Vert f\\big (x(t)\\big )\\Vert +\\Vert v_0-f(x_0)\\Vert \\big )$ .", "Using Gronwall's inequality (Lemma REF ), one obtains for all $t\\ge 0$ that $\\Vert x(t)-x_0\\Vert \\le \\Vert v_0\\Vert \\int _0^t {\\rm exp}\\Big (\\int _s^t a(\\tau )d\\tau \\Big )ds.$ Hence $\\limsup _{t\\rightarrow 0^+}\\Vert \\frac{x(t)-x_0}{t}\\Vert \\le \\Vert v_0\\Vert \\limsup _{t\\rightarrow 0^+}\\frac{1}{t}\\int _0^t {\\rm exp}\\Big (\\int _s^t a(\\tau )d\\tau \\Big )ds=\\Vert v_0\\Vert .$ Lemma 3.3 Let $x(\\cdot ),y(\\cdot )$ be the unique solution of $(\\ref {main})$ satisfying initial conditions $x(0)=x_0,y(0)=y_0$ respectively.", "Then for all $t\\ge 0:$ $\\Vert x(t)-y(t)\\Vert \\le \\Vert x(0)-y(0)\\Vert {\\rm exp}\\Big (\\int _0^tb(s)ds\\Big )\\;\\;t\\ge 0,$ where $b(t)=k+\\frac{1}{r}(\\Vert f(x(t))\\Vert +\\Vert f(y(t))\\Vert ).$ In particular, for $a.", "e.\\;t\\ge 0$ , one has $\\Vert \\dot{x}(t)\\Vert \\le \\Vert v_0\\Vert {\\rm exp}\\Big (\\int _0^t\\big (k+\\frac{2\\Vert f(x(s))\\Vert }{r}\\big )ds\\Big ),$ where $v_0$ is defined in Lemma REF .", "Proof.", "Using the prox-regularity of $C$ and Lipschitz continuity of $f(\\cdot )$ similarly as above, we have $\\frac{1}{2}\\frac{d}{dt}\\Vert x(t)-y(t)\\Vert ^2\\le b(t)\\Vert x(t)-y(t)\\Vert ^2\\;\\;a. e. \\; t\\ge 0,$ where $b(t)=k+\\frac{1}{r}\\big (\\Vert f\\big (x(t)\\big )\\Vert +\\Vert f\\big (y(t)\\big )\\Vert \\big ).$ Then the Gronwall's inequality (Lemma REF ) implies $(\\ref {2sol})$ .", "Given some $h>0$ , and we take $y(0)=x(h)$ then $y(t)=x(t+h)$ for all $t\\ge 0.$ From $(\\ref {2sol})$ , we deduce that $\\Vert \\frac{x(t+h)-x(t)}{h}\\Vert \\le \\Vert \\frac{x(h)-x(0)}{h}\\Vert {\\rm exp}\\Big (\\int _0^t\\big (k+\\frac{\\Vert f(x(s))\\Vert +\\Vert f(x(s+h))\\Vert }{r}\\big )ds\\Big ) \\; {\\rm for\\;all\\;}t\\ge 0.$ Fixed some $t_0\\ge 0$ such that $\\dot{x}(t_0)$ exists.", "Taking the limsup of both sides of $(\\ref {estidiff})$ as $h\\rightarrow 0^+$ and using Lemma REF , one gets $\\Vert \\dot{x}(t_0)\\Vert \\le \\Vert v_0\\Vert {\\rm exp}\\Big (\\int _0^{t_0}\\big (k+\\frac{2\\Vert f(x(s))\\Vert }{r}\\big )ds\\Big ).$ Thus $(\\ref {boundedde})$ follows.", "Now, we are ready for the main result which states that the solution is right differentiable and $\\dot{x}^+(\\cdot )$ is right continuous at each $t\\ge 0$ .", "We also recall an important property (Theorem REF -i) acquired in Proposition 2.6 [9] by using a different approach.", "Theorem 3.2 Let $x(\\cdot )$ be the unique solution of the system satisfying $x(0)=x_0$ .", "Then we have: $(i)$ $\\dot{x}(t)=v(t)=\\Big (f\\big (x(t)\\big )-N\\big (C,x(t)\\big )\\Big )^0$ for almost every $t\\in [0,+\\infty ).$ $(ii)$ For all $t^*\\in [0,+\\infty )$ , the right derivative $\\dot{x}^+(t^*)$ exists and $\\dot{x}^+(t^*)=\\Big (f\\big (x(t^*)\\big )-N_C\\big (x(t^*)\\big )\\Big )^0.$ Furthermore $\\dot{x}^+(\\cdot )$ is continuous on the right.", "Proof.", "Let $E=\\lbrace t\\in [0,+\\infty ): \\dot{x}(t) \\;{\\rm exists}\\rbrace $ .", "It is clear that the Lebesgue measure of $[0,+\\infty )\\setminus E$ is zero.", "$(i)$ Fixed $t_0\\in E$ .", "Let $y(\\cdot )$ be the unique solution of the system with initial condition $y(0)=x(t_0).$ Then $y(t)=x(t+t_0)$ for all $t\\ge 0.$ Applying Lemma $\\ref {limsup}$ , we get $\\limsup _{t\\rightarrow 0^+}\\Vert \\frac{y(t)-y(0)}{t}\\Vert \\le \\Vert \\Big (f\\big (y(0)\\big )-N\\big (C,y(0)\\big )\\Big )^0\\Vert ,$ or equivalently $\\limsup _{t\\rightarrow 0^+}\\Vert \\frac{x(t+t_0)-x(t_0)}{t}\\Vert \\le \\Vert \\Big (f\\big (x(t_0)\\big )-N\\big (C,x(t_0)\\big )\\Big )^0\\Vert .$ Hence $\\Vert \\dot{x}(t_0)\\Vert \\le \\Vert \\Big (f\\big (x(t_0))-N(C,x(t_0)\\big )\\Big )^0\\Vert .$ On the other hand $\\dot{x}(t_0)\\in f\\big (x(t_0)\\big )-N\\big (C,x(t_0)\\big )$ , thus $\\dot{x}(t_0)=\\Big (f\\big (x(t_0)\\big )-N\\big (C,x(t_0)\\big )\\Big )^0.$ $(ii)$ Due to the property of semi-group, it is sufficient to prove for $t^*=0.$ Using $(i)$ and $(\\ref {boundedde})$ of Lemma REF , for all $t\\in E$ , we have $\\Vert v(t)\\Vert \\le \\Vert v_0\\Vert {\\rm exp}\\Big (\\int _0^t\\big (k+\\frac{2\\Vert f(x(s))\\Vert }{r}\\big )ds\\Big ),$ where $v(t)= \\Big (f\\big (x(t)\\big )-N\\big (C,x(t)\\big )\\Big )^0$ .", "It implies that $\\limsup _{t\\rightarrow 0^+, t\\in E}\\Vert v(t)\\Vert \\le \\Vert v_0\\Vert .$ On the other hand, Lemma REF deduces that $\\Vert v_0\\Vert \\le \\liminf _{t\\rightarrow 0^+}\\Vert v(t)\\Vert \\le \\liminf _{t\\rightarrow 0^+,t\\in E}\\Vert v(t)\\Vert .$ From $(\\ref {a1})$ and $(\\ref {a2})$ , we obtain $\\lim _{t\\rightarrow 0^+,t\\in E}\\Vert v(t)\\Vert = \\Vert v_0\\Vert .$ Thus for any sequence $(t_n)_{n\\ge 1}\\subset E$ and $t_n\\rightarrow 0$ , we have $\\Vert v(t_n)\\Vert \\rightarrow \\Vert v_0\\Vert \\;{\\rm as}\\; n\\rightarrow +\\infty .$ Then $\\big (v(t_n)\\big )_{n\\ge 1}$ is bounded and therefore there exists some $v^*\\in H$ such that a subsequence $(v(t_{n_k}))_{k\\ge 1}$ converges weakly to $v^*$ when $k\\rightarrow +\\infty .$ Similarly as in Lemma REF , we can prove that $v^*\\in f(x_0)-N(C;x_0)$ .", "On the other hand $\\Vert v^*\\Vert \\le \\liminf _{k\\rightarrow +\\infty }\\Vert v(t_{n_k})\\Vert = \\lim _{k\\rightarrow +\\infty }\\Vert v(t_n)\\Vert =\\Vert v_0\\Vert ,$ due to $(\\ref {v_n}).$ Thus, we must have $v^*=v_0$ and the set of weak cluster point of $\\big (v(t_n)\\big )_{n\\ge 1}$ contains only $v_0$ .", "It implies that $v(t_n)$ converges weakly to $v_0$ .", "Combining with $(\\ref {v_n})$ , one deduces that $v(t_n)$ converges strongly to $v_0$ .", "In conclusion $\\lim _{t\\rightarrow 0^+,t\\in E}v(t)=v_0.$ Due to the absolute continuity of $x(\\cdot )$ and $(i)$ , for all $h>0$ , we have $x(h)-x_0=\\int _0^h\\dot{x}(s)ds=\\int _0^hv(s)ds,$ where $v(\\cdot )$ is locally integrable and satisfying $(\\ref {rcon})$ .", "Now we prove that $\\lim _{h\\rightarrow 0^+}\\frac{1}{h}\\int _0^hv(s)ds=v_0.$ Indeed, given $\\epsilon >0$ .", "From $(\\ref {rcon})$ , there exists $\\delta >0$ such that for all $s\\in E, s\\le \\delta $ then $\\Vert v(s)-v_0\\Vert \\le \\epsilon $ .", "Hence for all $h\\le \\delta $ : $\\Vert \\frac{1}{h}\\int _0^hv(s)ds-v_0\\Vert \\le \\frac{1}{h}\\int _0^h\\Vert v(s)-v_0\\Vert ds=\\frac{1}{h}\\int _{ [0,h]\\cap E}\\Vert v(s)-v_0\\Vert ds\\le \\frac{\\epsilon }{h}\\int _{ [0,h]\\cap E} ds=\\epsilon .$ So we have $(\\ref {aver})$ and thus from $(\\ref {differ})$ , the right derivative $\\dot{x}^+(0)$ exists and $\\dot{x}^+(0)=v_0=\\big (f(x_0)-N(C,x_0)\\big )^0.$ It implies for all $t\\ge 0$ that $\\dot{x}^+(t)=v(t)=\\Big (f\\big (x(t)\\big )-N\\big (C,x(t)\\big )\\Big )^0.$ Then taking the limit both sides of $(\\ref {estidiff})$ , we deduce for all $t\\ge 0$ that $\\Vert \\dot{x}^+(t)\\Vert \\le \\Vert \\dot{x}^+(0)\\Vert {\\rm exp}\\Big (\\int _0^t\\big (k+\\frac{2\\Vert f(x(s))\\Vert }{r}\\big )ds\\Big ),$ or equivalently $\\Vert v(t)\\Vert \\le \\Vert v_0\\Vert {\\rm exp}\\Big (\\int _0^t\\big (k+\\frac{2\\Vert f(x(s))\\Vert }{r}\\big )ds\\Big ).$ Therefore $\\limsup _{t\\rightarrow 0^+}\\Vert v(t)\\Vert \\le \\Vert v_0\\Vert .$ Combining with $(\\ref {a2})$ , we obtain $\\lim _{t\\rightarrow 0^+}\\Vert v(t)\\Vert = \\Vert v_0\\Vert .$ Similar as $(\\ref {rcon})$ , we can prove that $\\lim _{t\\rightarrow 0^+}v(t)= v_0.$ It means that $\\dot{x}^+(\\cdot )$ is right continuous at 0 and due to the property of semi-group, it is right continuous at any $t\\ge 0.$ Now we consider the case $f(\\cdot )=-\\nabla V(\\cdot )$ where $V$ is $C^{1,+}$ function (i.e., $V$ is differentiable and $\\nabla V$ is Lipschitz continuous) and study some asymptotic properties of the solutions.", "The system then can be considered as an extension of “gradient equation\" [1].", "Proposition 3.1 Let $V: H \\rightarrow \\mathbb {R}$ be a $C^{1,+}$ function.", "Let $x(\\cdot )$ be the solution of the system $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) \\in -\\nabla V\\big (x(t)\\big ) -N\\big (C,x(t)\\big )\\; {\\rm a.e.}", "\\; t \\in [0,+\\infty ),\\\\ \\\\x(0) = x_0\\in C.\\end{array}\\right.$ Then we have $\\frac{d}{dt}V\\big (x(t)\\big )+\\Vert \\dot{x}(t)\\Vert ^2=0, \\;{\\rm for}\\; a. e. \\;t\\ge 0.$ In particular, $V$ is a Lyapunov function of the system.", "Furthermore $(i)$ if $V$ is coercive, i.e., $V(x)\\rightarrow +\\infty \\;\\;{\\rm as}\\;\\; \\Vert x\\Vert \\rightarrow +\\infty ,$ then $x(\\cdot )$ is bounded on $\\mathbb {R}_+$ .", "$(ii)$ if $V$ is bounded from below on $C$ then $\\lim _{t\\rightarrow +\\infty }V\\big (x(t)\\big )=V_\\infty $ exists and $\\dot{x}\\in L^{2}([0,+\\infty );H)$ with $\\int _0^{+\\infty }\\Vert \\dot{x}(s)\\Vert ^2ds=V(x_0)-V_\\infty .$ $(iii)$ If $V$ is convex and bounded from below on $C$ then $V_\\infty =\\displaystyle \\inf _{y\\in C} V(y).$ Proof.", "Fixed some $t\\ge 0$ such that $(i)$ of Theorem REF holds, i.e., $\\dot{x}(t)=\\Big (-\\nabla V\\big (x(t)\\big )-N_C\\big (x(t)\\big )\\Big )^0=-\\nabla V\\big (x(t)\\big )-{\\rm proj}\\Big (N\\big (C,x(t)\\big ); -\\nabla V\\big (x(t)\\big )\\Big )$ .", "Then $&&\\big \\langle \\dot{x}(t)+ \\nabla V\\big (x(t)\\big ), \\dot{x}(t) \\big \\rangle \\\\&=&\\big \\langle -{\\rm proj}\\Big (N\\big (C,x(t)\\big ); -\\nabla V\\big (x(t)\\big )\\Big ), \\nabla V(x(t))-{\\rm proj}\\Big (N\\big (C,x(t)\\big ); -\\nabla V\\big (x(t)\\big )\\Big )\\big \\rangle \\\\&=&0.$ Note that $\\frac{d}{dt}V\\big (x(t)\\big )=\\big \\langle \\nabla V\\big (x(t)\\big ), \\dot{x}(t) \\big \\rangle $ and $(\\ref {lya})$ follows.", "In particular, we have $\\frac{d}{dt}V\\big (x(t)\\big )\\le 0$ for a.e.", "$t\\ge 0.$ It means that $V$ is a Lyapunov function of the system.", "Then $(i)$ and $(ii)$ follow classically.", "$(iii)$ Fix some $y\\in C$ and consider the function $\\varphi (t)=\\frac{1}{2}\\Vert x(t)-y\\Vert ^2.$ Due to the $r$ -prox-regularity of $C$ and the fact that $\\dot{x}(t) +\\nabla V\\big (x(t)\\big )\\in -N\\big (C,x(t)\\big )\\; {\\rm a.e.}", "\\; t \\in [0,+\\infty )$ , one has $\\langle \\dot{x}(t) +\\nabla V\\big (x(t)\\big ), x(t)-y \\rangle \\le \\frac{\\Vert \\nabla V\\big (x(t)\\big )\\Vert }{r}\\Vert x(t)-y\\Vert ^2.$ Thus $\\dot{\\varphi }(t)=\\langle \\dot{x}(t), x(t)-y \\rangle &\\le & \\frac{2\\Vert \\nabla V\\big (x(t)\\big )\\Vert }{r} \\varphi (t)+\\langle \\nabla V\\big (x(t)\\big ), y-x(t) \\rangle \\\\&\\le & \\frac{2\\Vert \\nabla V\\big (x(t)\\big )\\Vert }{r} \\varphi (t)+V(y)-V\\big (x(t)\\big ),$ due to the convexity of $V$ .", "Using Gronwall's inequality (Lemma REF ), for all $t\\ge 0$ one obtains $0\\le \\varphi (t)&\\le & \\varphi (0){\\rm exp}\\Big (\\int _0^t \\frac{2\\Vert \\nabla V(x(\\tau ))\\Vert }{r}d\\tau \\Big )+\\int _0^t {\\rm exp}\\Big (\\int _s^t \\frac{2\\Vert \\nabla V\\big (x(\\tau )\\big )\\Vert }{r}d\\tau \\Big )[V(y)-V\\big (x(s)\\big )]ds\\\\&\\le &{\\rm exp}\\Big (\\int _0^t \\frac{2\\Vert \\nabla V(x(\\tau ))\\Vert }{r}d\\tau \\Big ) \\big [\\varphi (0)+t\\Big (V(y)-V\\big (x(t)\\big )\\Big )\\big ],$ since $ V\\big (x(s)\\big )\\ge V\\big (x(t))\\big )$ for all $s\\in [0,t]$ .", "It implies that $V\\big (x(t)\\big )\\le V(y)+\\frac{\\varphi (0)}{t}.$ Let $t\\rightarrow +\\infty $ , one gets $V_\\infty \\le V(y).$ Since $y$ is arbitrary in $C$ , it deduces that $V_\\infty \\le \\displaystyle \\inf _{y\\in C}V(y).$ On the other hand $V\\big (x(t)\\big )\\ge \\displaystyle \\inf _{y\\in C}V(y)$ since $x(t)\\in C$ for all $t\\ge 0.$ Hence $V_\\infty \\ge \\displaystyle \\inf _{y\\in C}V(y)$ .", "Therefore $V_\\infty = \\displaystyle \\inf _{y\\in C}V(y)$ , it means the trajectory is minimizing for $V$ on $C$ ." ], [ "Conclusion", "In this paper, we have established some important regularity properties for a class of differential inclusions involving normal cone operator of prox-regular sets without tangential assumption.", "Some asymptotic behaviours of the solutions are also studied.", "It is interesting to consider properties of solutions of sweeping process with prox-regular sets, where $C$ can depend on time and even the state.", "It is out of scope of the current work and will be considered in the future.", "$\\mathbf {Acknowledgments}$ The author would like to acknowledge the referees for their careful reading and insightful suggestions.", "The research is supported by Fondecyt Project 3150332." ] ]
1606.05197
[ [ "Atomic configuration and properties of austenitic steels at finite\n temperature: The effect of longitudinal spin fluctuations" ], [ "Abstract High temperature atomic configurations of fcc Fe-Cr-Ni alloys with alloy composition close to austenitic steel are studied in statistical thermodynamic simulations with effective interactions obtained in ab initio calculations.", "The latter are done taking longitudinal spin fluctuations (LSF) into consideration within a quasiclassical phenomenological model.", "It is demonstrated that magnetic state affects greatly the alloy properties and in particular, it is shown that the LSF substantially modify the bonding and interatomic interactions of fcc Fe-Cr-Ni alloys even at ambient conditions.", "The calculated atomic short-range order (SRO) is in reasonable agreement with existing experimental data for Fe0.56}Cr0.21Ni0.23, which has strong preference for the (001) type ordering between Ni and Cr atoms.", "A similar ordering tendency is found for the Fe0.75Cr0.17Ni0.08 alloy composition, which approximately corresponds to the widely used 304 and 316 austenitic steel grades." ], [ "Introduction", "Austenitic steels based on Fe-Cr-Ni system are in extensive use in different domestic and industrial applications due to their excellent corrosion resistant and mechanical properties.", "They usually consist of more than three elements added on a purpose or accidently, however, Cr and Ni are the usual alloying elements: Cr provides the corrosion resistance, and its content is usually in between 15–25 at.", "%, while Ni stabilizes the fcc structure, and its content is as a rule within 5-15 at.% (can be up to 35 at.% in some special grades).", "Although these alloys have been for decades under development and investigation, accurate description of their finite temperature properties at the atomic and electronic structure levels is still a challenging task.", "One of the main obstacles in getting accurate ab initio picture of fcc Fe-Cr-Ni alloys is their non-trivial magnetism.", "The magnetic phase diagram of Fe$_{80-x}$ Ni$_x$ Cr$_{20}$ ($10 \\le x \\le 30$ ) alloys have been determined in Ref.", "magphd.", "At low temperatures, one can find antiferromagnetic, spin-glass, ferromagnetic and a mixture of ferromagnetic and spin glass states for a certain range of compositions.", "In particular, the antiferromagnetic state is the low temperature magnetic state with the Neel temperature of about 20-40 K if concentration of Ni is within 5-15 at.", "%.", "This means that in practice austenitic steels are in paramagnetic state at ambient conditions.", "The problem is that this paramagnetic state is a highly non-trivial phenomenon, which is extremely difficult to model accurately on the first-principles level since a static disordered local moment model breaks down due to disappearance of the local magnetic moments on Cr and Ni atoms in the density functional theory (DFT) calculations.", "At the same time, their magnitude deviates substantially from zero due to longitudinal spin fluctuations on short time scales, producing a significant effect upon all the properties of steels.", "In principle, such magnetic excitations can be accounted for using advanced ab initio methods like, for instance, dynamical mean-field theory.", "[2] However, their applications to steels, which are multicomponent random alloys, is too cumbersome.", "In this paper, we therefore use a simplified formalism for the LSF developed in Ref.", "ruban07lsf,ruban13.", "It is a classical high temperature limit of the spin-fluctuation theory[5], [6] where the contribution due to thermally induced spin fluctuations are considered within a phenomenological model based on a classical magnetic Hamiltonian.", "Although it breaks down at low temperatures where quantum effects are important, it provides a reasonable account of the LSF at elevated temperatures.", "[4] Using this model of the LSF, we consider properties of Fe-Cr-Ni alloys and, in particular, atomic ordering at elevated temperatures.", "The experimental information on the atomic short range order (SRO) in austenitic steels is scarce.", "A quite detailed investigation of the atomic SRO in fcc Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ was done by Cenedese et al.", "[7] by thermal neutron diffuse scattering from single crystals.", "The atomic SRO has been also measured in alloys with similar compositions, Fe$_{82-x}$ Cr$_{18}$ Ni$_{x}$ ($x=$ 15, 20, and 25 wt.%) by Braude et al.", "using x-ray diffuse scattering technique.", "[8] However, some results of this investigation are contradictory as is discussed below.", "Recently, the phase stability of ternary fcc and bcc Fe-Cr-Ni alloys has been investigated in Ref.", "wrobel15 using a combination of DFT, cluster expansion (CE), and magnetic cluster expansion (MCE) techniques.", "These authors have found good agreement of different calculated properties with experimental data, including results for the atomic SRO in Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ alloy as stated by the authors.", "At the same time, the results presented in Table VII of Ref.", "wrobel15 show some obvious problems for the calculated atomic SRO in this alloy.", "Namely, the calculated atomic SRO is too strong compared with the experimental data,[7] especially taking into consideration the fact that calculations are for 1300 K, while in the experiment, the samples were annealed first for 1 hour at 1273 K and then for 10 hours at 773 K. Of course, it might well be that 10 hours is not enough to fully equilibrate this alloy at 773 K, nevertheless a substantial rearrangement can be expected on a local scale of at least several interatomic distances.", "In fact, the authors of Ref.", "wrobel15 find ordering transition at 1550 K for Fe$_{50}$ Cr$_{25}$ Ni$_{25}$ .", "At this temperature the diffusion is very fast, and had such a high-temperature transition really existed, it or the corresponding ordered phase would be definitely seen or detected in different kind of experiments.", "This means that there is a problem with the existing theoretical description of the atomic SRO in austenitic steels.", "To solve it is one of the aims of the present investigation.", "Another aim is to demonstrate the role of the LSF in austentic steels at finite temperatures.", "For that purpose, we consider here two alloy: Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ alloy, whose composition is close to the widely used 304 and 316 steel grades, and Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ , whose atomic SRO has been obtained experimentally[7] and just recently calculated using ab initio theory.", "[9] We also outline a technique for calculating chemical and magnetic exchange interactions within the exact muffin-tin orbital (EMTO) method.", "[10], [11]" ], [ "First-principles methodology", "Electronic structure calculations of random fcc Fe-Cr-Ni alloys have been done using the coherent potential approximation (CPA)[12] and locally self-consistent Green's function (LSGF) technique,[13] which accurately accounts for the local environment effects in random alloys.", "Both these techniques have been used within the EMTO method[10], [11] referenced here as EMTO-CPA[14] and ELSGF[15], respectively.", "The EMTO-CPA calculations have been done by the Lyngby version of the Green's function EMTO code,[16] where the screened Coulomb interactions in the single-site DFT-CPA approximation[17] and screened generalized perturbation method (SGPM)[18], [19], [20] are implemented (See Appendix ).", "In particular, the contributions of the screened Coulomb interactions in the DFT-CPA to the one-electron potential of alloy components, $V^i_{\\rm scr}$ , and to the total energy, $E_{\\rm scr}$ , are:[17] $V^i_{\\rm scr} &=& -e^2 \\alpha ^0_i\\frac{\\bar{q}_i}{S}\\nonumber \\\\E_{\\rm scr} = \\sum _i c_i E^i_{\\rm scr}; \\quad E^i_{\\rm scr} &=& -e^2 \\frac{1}{2}\\alpha ^0_i\\beta _{\\rm scr}\\frac{\\bar{q}^2_i}{S}.$ where $\\bar{q}_i$ and $\\alpha _i^0$ are the net charge of the atomic sphere of the $i$ th alloy component in the single-site CPA calculations and its on-site screening constant, which are different for different alloy components in multicomponent alloys, $S$ the Wigner-Seitz radius, $\\beta _{\\rm scr}$ the average on-site screening constants, which accounts for the electrostatic multipole moment energy contribution due to inhomogeneous local environment of different sites in random alloy.", "The self-consistent electronic structure calculations have been done within the local density approximation using Perdew and Wang functional,[21] while the total energy have been calculated using the full charged density technique[11] in the PBE generalized gradient approximation.", "[22] In the Brillouin zone integration, a 32$\\times $ 32$\\times $ 32 Monkhorst-Pack grid have been used.", "[23] All the calculations have been done with $l_{max}=3$ for partial waves and the electronic core states were recalculated at every iteration during the self-consistent calculations for valence electrons.", "The on-site and inter-site screening constants needed in the EMTO-CPA and SGPM calculations were obtained for Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ and Fe$_{50}$ Cr$_{25}$ Ni$_{25}$ random alloys in a 864-atom supercell (6$\\times $ 6$\\times $ 6 translations of the fcc 4-atom cubic unit cell) ELSGF calculations.", "After the configurational optimization, the first six Warren-Cowley SRO parameters of the supercells were less than 0.002 (in absolute value) and next 2, for the 7th and 8th coordination shells, less than 0.015 for all three pairs of the alloy components.", "The calculations were done in the disordered local moment state[24], [25] as is implemented in the ELSGF code[15] with the LSF at 800 K (see the next section).", "The local interaction zone (LIZ) has contained the first two coordination shells of the central site (of the LIZ).", "The on-site screening constants have been determined as $ \\alpha ^0_i = \\frac{S \\langle V^{\\rm Mad}_i \\rangle }{e^2 \\langle q_i \\rangle } ,$ where $\\langle q_i \\rangle $ and $\\langle V^{\\rm Mad}_i \\rangle $ are the conditional average of the net charges, $q_i$ , and the Madelung potentials, $V^{\\rm Mad}_i$ of the $i$ -th component in the supercell.", "The calculated on-site screening constants, $\\alpha ^0_i$ , vary very little with alloy composition, lattice constant, and temperature due to thermal electronic and magnetic excitations due to the LSF.", "They are approximately equal to 0.725, 0.777, and 0.823 for Fe, Cr, and Ni, respectively, while $\\beta _{\\rm scr}$ is about 1.14.", "These values of the screening constants have been used in all the DFT-CPA calculations.", "The intersite screening constants, $\\alpha ^{ij}_{p}$ , needed in the calculations of the electrostatic contribution, $V^{ij-\\rm scr}_{p}$ to the SGPM potential at the $p$ th coordination shell for the $i$ -$j$ pair of alloy components: $ V^{ij-\\rm scr}_{p} = e^2\\alpha ^{ij}_{p}\\frac{\\bar{q}^2_{ij}}{S} ,$ where $\\bar{q}_{ij}=\\bar{q}_i - \\bar{q}_j$ .", "Screening constants $\\alpha ^{ij}_{p}$ have been obtained in the supercell ELSGF calculations random alloy from the screening charge by exchanging the corresponding alloy components ($i$ and $j$ in some specific sites having random local environment on average) as is described in Ref.", "screening." ], [ "Longitudinal spin fluctuations in\nFe-Cr-Ni alloys", "The description of the paramagnetic state in fcc alloys in DFT calculations at ambient conditions requires an additional modeling, which takes into consideration thermally induced longitudinal spin fluctuations.", "[26], [27], [28], [29], [3], [4] Here, we follow the formalism developed in Refs.", "ruban07lsf and ruban13.", "The main idea is to consider the LSF as an entropic effect within a static one-electron consideration.", "In this case, the magnitude of the local magnetic moment induced by the entropy can be obtained, to a first approximation, from the longitudinal spin fluctuation energy in the corresponding statistical thermodynamic simulations.", "The longitudinal spin fluctuation energy is the energy of embedding an impurity having a given magnetic moment $m_i$ into the DLM effective medium with a certain choice of the average local magnetic moments of alloy components.", "[3] Since, the average local magnetic moment at a given temperature is not known in advance, the statistical simulations have to be done in a self-consistent way.", "Obviously, this is an extremely time consuming scheme in the case of multicomponent alloys.", "However, it can be greatly simplified by using an approximate expression for the entropy of longitudinal spin fluctuations:[4] $ S_i^{\\rm lsf} = 3 \\ln m_i ,$ which is valid in the classical high-temperature limit for the quadratic form of the LSF energy.", "Although the later is approximately true only for Ni and Cr, while for Fe there is a non zero local magnetic moment for equilibrium lattice constants of austenitic steels, we have used this expression for Fe too in order to keep continuous description of the magnetic energy for small lattice constants.", "Figure: (Color online)Longitudinal spin fluctuation energies of alloy componentsin Fe 75 _{75}Cr 17 _{17}Ni 08 _{08} in the DLM state obtainedfor the Fe, Cr, and Ni local magnetic moments 1.69, 0.82, and0.38 μ B \\mu _{\\rm B}, respectively.In Fig.", "REF , the longitudinal spin fluctuation energy for Fe, Cr, and Ni in Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ in the DLM effective medium due to the LSF is shown.", "In these calculations, the local magnetic moment of one of the components have been changed while all the other were kept fixed to the following magnitudes: 1.69 $\\mu _{\\rm B}$ for Fe, 0.82 $\\mu _{\\rm B}$ for Cr, and 0.38 $\\mu _{\\rm B}$ for Ni, which approximately corresponds to their local magnetic moments at 300 K obtained using Eq.", "(REF ) in the self-consistent calculations.", "One can see, that the LSF energy curves for Ni and Cr resemble parabola.", "In the case of Fe, the LSF energy has minimum at $\\approx $ 1.4 $\\mu _B$ .", "Thus, at least a fourth order polynomial is needed to get a qualitative behavior of the LSF energy in the latter case.", "However, this approximate scheme works reasonably well even for the LSF induced magnetic moment of Fe.", "In Fig.", "REF , the local magnetic moments of Ni, Cr, and Fe are shown as function of temperature, which have been obtained in the single-site mean-field approximation from the LSF energies, $E_i^{\\rm LSF}$ , presented in Fig.", "REF as: $ m_i = 1/Z_i \\int E_i^{\\rm LSF}(m) m^3 dm ,$ where $Z_i$ is the partial statistical sum for alloy component $i$ : $ Z_i = \\int E_i^{\\rm LSF}(m) m^2 dm ,$ Figure: (Color online)Local magnetic moments of alloy components inin Fe 0.75 _{0.75}Cr 0.17 _{0.17}Ni 0.08 _{0.08} due to LSF obtained fromthe LSF energies presented in Fig.", "as a function of temperature.It can be substantially simplified without losing much accuracy (which is not on the demand in semiquantitative modeling) by just calculating the energy of the DLM state with varying magnitude of the local magnetic moment of one of the alloy components while keeping m of the others fixed to the one which corresponds to the chosen temperature.", "For instance, using (REF ) one finds that the local magnetic moments in Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ at 300 K are 1.69, 0.82, and 0.38 $\\mu _B$ for Fe, Cr, and Ni, respectively, while they are 1.52, 0.77, and 0.35 $\\mu _B$ if (REF ) is used with the corresponding spin-fluctuation energies.", "As one can see, even for Fe, Eq.", "(REF ) produces quite reasonable results only slightly overestimating local magnetic moment.", "Let us note that without LSF the local magnetic moment of Fe is about 1.4 $\\mu _B$ while Cr and Ni becomes non-spin polarised." ], [ "Lattice parameter and elastic constants of\nFe$_{75}$ Cr{{formula:46880020-d746-4e5c-a585-26a272670399}} Ni{{formula:c31135b3-65a3-4479-97b6-4c667121145f}} \nat ambient conditions", "The importance of the LSF in austenitic steels even at room temperature can be seen in the first-principles calculations of the lattice constant and bulk modulus of Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ alloy.", "In Fig.", "REF , we show the total energies (up to an arbitrary constant) of random Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ alloy obtained in the EMTO-CPA calculations in the DLM paramagnetic state with and without LSF.", "In the DLM calculations, the magnetic entropy of Fe has been accounted for using the entropy of the ideal paramagnetic gas ($S_{mag}^{\\rm IP} = \\ln (m_{Fe} + 1)$ ).", "The one-electron thermal excitations have been included using Fermi-Dirac distribution function.", "[30] Figure: (Color online)The electronic and magnetic free energies ofFe 75 _{75}Cr 17 _{17}Ni 08 _{08} at 300 K obtainedin the DLM calculations with and without LSF.", "The free energiesare shifted to some arbitrary constant.Figure: (Color online)Local magnetic moments of Fe, Cr, and Ni inFe 75 _{75}Cr 17 _{17}Ni 08 _{08} in the DLM calculationswith and without LSF.If LSF are not included, the free energy curve exhibits quite irregular behaviour around Wigner-Seitz (WS) radius of 2.6 a.u.", "(or lattice constant about 3.52 Å), which can be traced down to the abrupt change of the magnitude of magnetic moment of Fe, which is clearly seen in Fig.", "REF .", "The equilibrium WS sphere radii (without phonon contribution) is 2.618 a.u.", "and bulk modulus is about 210 gPa, which is too large compared to the room temperature bulk modulus of austenitic steels with similar composition, which is usually in the range of 140-170 GPa.", "[31] The inclusion of the LSF, apart from smoothing the total energy curve, leads to the increase of the equilibrium WS radius to 2.623 a.u.", "or lattice constant of 3.55 Å and to a substantial decrease of the bulk modulus: 161 GPa.", "In the Debye-Grüneisen model, the room temperature lattice constant then comes out to be $~$ 3.57 Å and bulk modulus 155 GPa, which are in reasonable agreement with experimental data.", "[31] The calculated room temperature shear elastic constants, $c^{\\prime }$ and $c_{44}$ : 35 and 138 GPa, respectively, are also in good agreement with the existing experimental data: 38 and 121 GPa.", "[31]" ], [ "Effective interactions", "Finite temperature atomic configuration of Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ and Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ has been obtained in Monte Carlo simulations using the following configurational Hamiltonian: $ H &=& \\frac{1}{2} \\sum _p \\sum _{\\alpha , \\beta \\ne \\delta }V^{(2)-\\alpha \\beta [\\delta ]}_p\\sum _{ij \\in p} \\delta c^{\\alpha }_i \\delta c^{\\beta }_j + \\\\ \\nonumber &&\\frac{1}{3} \\sum _t \\sum _{\\alpha , \\beta , \\gamma \\ne \\delta }V^{(3)-\\alpha \\beta \\gamma [\\delta ]}_t\\sum _{i,j,k} \\delta c^{\\alpha }_i \\delta c^{\\beta }_j \\delta c^{\\gamma }_k + h.o.t.", ".$ Here, the summation is performed over alloy different type of clusters ($p$ and $t$ stands for indexes of the pairs and triangles), alloy components (designated by Greek letters) and lattice sites ($i$ , $j$ , and $k$ ); $V^{(2)-\\alpha \\beta [\\delta ]}_p$ and $V^{(3)-\\alpha \\beta \\gamma [\\delta ]}_t$ are the pair- and three-site effective interactions, which have been determined using the SGPM implemented in the Lyngby version of the EMTO-CPA code (see Appendix for details), and $\\delta c^{\\alpha }_i = c^{\\alpha }_i - c^{\\alpha }$ is the concentration fluctuation of the $\\alpha $ component from its average concentration in alloy, $c^{\\alpha }$ at site $i$ .", "The contribution from pair interactions in (REF ) can be reduced to a quasibinary form: $ H^{(2)} = -\\frac{1}{2} \\sum _p \\sum _{\\alpha \\ne \\beta }\\widetilde{V}^{(2)-\\alpha \\beta }_p\\sum _{ij \\in p} \\delta c^{\\alpha }_i \\delta c^{\\beta }_j ,$ where $\\widetilde{V}^{(2)-\\alpha \\beta }_p$ are the usual binary effective interactions describing the mutual ordering of $\\alpha $ and $\\beta $ atoms and connected with the multipcomponent effective pair interactions, $V^{(2)-\\alpha \\beta [\\delta ]}_p$ as[32], [33] $V^{(2)-\\alpha \\beta [\\delta ]}_p = \\frac{1}{2} \\left[\\widetilde{V}^{(2)-\\alpha \\delta }_p +\\widetilde{V}^{(2)-\\beta \\delta }_p -\\widetilde{V}^{(2)-\\alpha \\beta }_p \\right] .$ The advantage of such a quasibinary representation is its direct connection to the Hamiltonians and interactions of the binary systems of the components composing the multicomponent one.", "Unfortunately, the contribution from multisite interactions cannot be reduced to a similar quasibinary form due to the presence of additional indexes of alloy components.", "In first-principles calculations, the interactions entering Eqs.", "(REF ) and (REF ) have been obtained using a DLM six component model of the three-component alloys: two components with the opposite orientation of magnetic moment for each alloy component.", "The chemical interactions have been obtained in the magnetic-moment averaged form for each alloy component in the DLM-CPA calculations.", "The corresponding expressions for the interactions in the EMTO-CPA method are given in Appendix .", "The effective interactions of the above Hamiltonian are not only concentration dependent, but they also depend on the temperature due to the temperature dependence of the equilibrium volume of the alloy, its magnetic state, and the local magnetic moments of its components in the DLM-LSF state.", "As is demonstrated below such a temperature dependence should be taking into consideration in atomistic modeling at high temperatures.", "In Fig.", "REF , the quasibinary effective pair interactions, $\\widetilde{V}^{(2)-\\alpha \\beta }_p$ , in Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ and Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ of Hamiltonian (REF ) are shown.", "They have been obtained for the high temperature ( 800–1000K) lattice constant of 3.615 Å.", "Two sets of interactions have been obtained in the DLM-LSF state at 800 K, while one set of interactions, for Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ alloy was calculated in the DLM calculations without LSF.", "In this case, there is no magnetic moment on Ni and Cr atoms.", "As one can see, the LSF produce quite substantial effect in the case of Fe-Ni and Fe-Cr effective interactions, although it is not that pronounced in the case of Ni-Cr interactions.", "At the same time, the concentration dependence seems to be quite moderate, within the range of the accuracy of the SGPM calculations.", "The dependence of the effective interactions on the lattice constant is in fact quite strong (not shown in the figure).", "For instance, the nearest-neighbor effective interactions obtained in the DLM state for the theoretical 0 K lattice constant, 3.55 Å, are 5.89, -0.62, and 10.05 mRy for Fe-Cr, Fe-Ni and Fe-Cr pairs, respectively, which are quite different from those for the high temperature lattice constant of 3.615 Å: 3.96, 0.30, and 8.24 mRy.", "Obviously, the external and internal parameters cannot be disregarded in finding theoretical atomic configuration of austenitic steels at finite temperatures.", "Figure: (Color online)Quasibinary effective pair interactions inFe 75 _{75}Cr 17 _{17}Ni 08 _{08} (open circles) andFe 56 _{56}Cr 21 _{21}Ni 23 _{23} (filled diamonds) obtained at800 K in the DLM-LSF calculations.Small filled circles show the results for Fe 56 _{56}Cr 21 _{21}Ni 23 _{23}without LSF (which means DLM calculations for Fe andnon-spin-polorized for Cr and Ni).Effective interactions presented in Fig.", "REF look qualitatively similar for all the pairs: the strongest interaction of the ordering type at the first coordination shell following the next strongest interaction with the opposite sign at the second coordination shell.", "However, as we will see below, there cannot be mutual ordering of all the pairs due to the fact that the pairs with the strongest ordering energies, specifically NiCr at the first coordination shell, will be ordering first.", "Let us note that the strong ordering interaction of Ni and Cr atoms at the first coordination shell is almost entirely due to the screened Coulomb interaction, which is of an order of 8–9 mRy, while the one-electron contribution is only about 0.2–0.5 mRy.", "In the case of Fe-Ni and Fe-Cr pairs, the nearest neighbor screened Coulomb interaction is relatively small, about 1 and 2 mRy, respectively.", "This is an expected result, since the screened Coulomb interactions are proportional to the charge transfer between the corresponding alloy components, which is in its turn determined mostly by their mutual size difference.", "The latter is obviously the largest for Ni and Cr, while it is relatively small for Fe-Cr and Fe-Ni pairs.", "The size difference of alloy components leads to the appearance of local lattice relaxations, which can be accounted for in the configurational Hamiltonian by the so-called strain-induced interactions.", "Unfortunately, it is impossible to calculate accurately these interactions in ternary Fe-Cr-Ni alloys, especially at high temperatures in the paramagnetic state with LSF.", "Therefore in this work, we use a simple qualitative model.", "First of all, we disregard the strain induced interactions for Fe-Ni and Fe-Cr pairs, which should be small anyway.", "As for the Ni-Cr strain-induced interactions, we take them from Ref.", "rahaman14 where they have been obtained for Ni-Cr binary alloys.", "Although this is a quite rough approximation, it is used here to test a qualitative effect of such interactions onto the ordering in austenitic steels.", "Multisite interactions are relatively small in these alloys.", "The strongest 3-site interaction is of the 114-type (for the nomenclature of the multisite interactions see Ref.", "rahaman14), i.e.", "for the cluster of the three nearest-neighbors on the line in the closed packed [110] direction (there are 8 such interactions in general for a given 3-site cluster, which correspond to different combinations of alloy indexes).", "It is especially strong for the CrCrCr[Ni] configuration: 3.11 and 2.86 mRy in Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ and Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ , respectively.", "This specific interaction corresponds actually to the same type of interaction in Ni-Cr binary alloy where it is in fact also the strongest 3-site interaction,[34] although the value of the interaction is quite reduced in ternary Fe-Cr-Ni alloys compared to that in binary Ni-Cr alloys.", "The strongest 4-site interactions are for the tetrahedron of the nearest neighbors and for the four nearest neighbor sites along the closed packed [110] direction, which are also the strongest interactions in the Ni-Cr system.", "[34] In ternary alloys, however, the quisibinary Fe-Ni 4-site interactions (FeFeFeFe[Ni]) are comparable with the corresponding quasibinary Ni-Cr interactions (CrCrCrCr[Ni]), while the interactions with mixed alloy component indexes are almost an order of magnitude smaller.", "[35] In order to check how the SGPM works, for this particular system, the ordering energy of the Fe$_2$ NiCr-L1$_2$ m ordered structure in the DLM state (without LSF) have been calculated for the lattice constant of 3.615 Å from the EMTO total energies and from SGPM interactions for Fe$_{50}$ Ni$_{25}$ Cr$_{25}$ random alloy.", "In the direct calculations the ordering energy is $-$ 1 mRy/atom, while it is about $-$ 2 mRy/atom from the pair, 3-site and 4-site SGPM interactions.", "The agreement is reasonable, considering the smallness of the ordering energy, and the fact that magnetic moment in the ordered state, $~1.0$ $\\mu _{\\rm B}$ , is different from that in the random alloy, $~1.6$ $\\mu _{\\rm B}$ .", "Such a small ordering energy ($~$ 160 K) also means that this structure can hardly be formed at 650 K as predicted in Ref.", "wrobel15.", "One of the reasons why the ordering strength is greatly exaggerated in Ref.", "wrobel15 is the fact that its authors are using concentration independent cluster expansion for 0 K enthalpies of formation obtained in the ferromagnetic state (with ferromagnetic alignment of Fe and Ni magnetic moments and antiferromagnetic one of Cr and Fe(Ni)) in the high temperature Monte Carlo simulations.", "To show the effect of the magnetic state upon the effective interactions, we compare in Fig.", "REF the effective pair interactions for random Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ alloy obtained in the DLM-LSF state at 800 K and in the ferromagnetic (FM) state (again, the magnetic moment of Cr is antiferromagnetically aligned with those of Fe and Ni).", "Figure: (Color online)Effective pair interactions in Fe 56 _{56}Cr 21 _{21}Ni 23 _{23} obtainedin the DLM-LSF state at 800 K and in the ferromagnetic(FM) state.Although the Ni-Cr effective pair interactions are approximately the same in both states, it is obviously not the case of the Fe-Cr and Fe-Ni effective pair interactions: the strongest nearest neighbor interaction for both pairs is more than twice as large in the FM state as in the DLM-LSF state.", "Such a strong dependence of the effective interactions on the magnetic state means that the corresponding modeling of austenitic steels at finite temperatures should be done in the relevant to this temperature magnetic state.", "Let us note that the effect of the magnetic state upon chemical interactions in fcc Fe-Ni alloys have been already studied in Ref.", "ruban07inv,ekholm10.", "In particular, as has been demonstrated in Ref.", "ruban07inv, the DLM state leads to a significant drop of the strongest nearest neighbor interaction in Invar Fe$_{65}$ Ni$_{35}$ alloy, which makes, in the end, the ordering of this alloy impossible at temperatures relevant to its preparation.", "It is interesting that the interactions obtained in Ref.", "ruban07inv for the Invar binary alloy are very close to those for Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ shown in Fig.", "REF .", "It has been also demonstrated[37] that there is a substantial reduction of the effective chemical interactions even in the finite temperature FM state due to a reduce magnetization, which is very similar to the case of bcc Fe-Cr alloys.", "[38] It is obvious that such a coupling between magnetic state and effective interactions should be properly taken into consideration in the corresponding thermodynamic modeling at high temperatures." ], [ "Atomic ordering in austenitic steels", "Monte Carlo calculations of atomic alloy configuration in Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ and Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ alloys have been done using a simulation box containing 12$\\times $ 12$\\times $ 12($\\times $ 4) sites.", "In both cases, the first 21 effective pair interactions have been used with the contribution from the Ni-Cr strain-induced interactions taken from Ref.", "rahaman14.", "We have also used the four strongest 3-site interactions of the 111, 112, 113 and 114-type (see Ref.", "rahaman14 for the nomenclature), and two 4-site interactions, for the tetrahedron of the nearest neighbors and the nearest neighbor sites along the closed packed [110] direction.", "The calculated atomic SRO parameters at 800 K are presented in Table REF together with the experimental data from Ref.", "cenedese84.", "Although the agreement is only qualitative, one should take into consideration the fact that the strain-induced interactions are considered only quite approximately in this work.", "Besides, the experimental values seem to be quite sensitive to the model used in the analysis of the diffuse scattering intensities.", "Nevertheless, the picture of atomic ordering is quite clear and consistent: the strongest ordering is between Ni and Cr nearest and next nearest neighbors.", "The calculated type of ordering is (100), which is the same as in the experiment, but it is different from the (1$\\frac{1}{2}$ 0) type in the ordered Ni$_2$ Cr phase.", "However, as has been shown in Ref.", "rahaman14, the effective cluster interactions and ordering are sensitive to the alloy composition in fcc Ni-Cr alloys, and the (100)-type ordering is consistent with the results for equiatomic Ni-Cr alloys.", "The next in the strength of ordering are the pairs of Fe and Cr atoms at the first coordination shell.", "Although the type of ordering in theoretical calculations, (100) is not consistent with the experimental one, (1$\\frac{1}{2}$ 0), one can again notice that the type of ordering in the experiment depends on the fitting model.", "[7] We have also neglected the Fe-Cr strain-induced interactions, which may produce a certain effect.", "And finally, Fe and Ni atoms repel each other at the first coordination shell exhibiting there a kind of \"phase separation\" behavior, although the corresponding effective pair interaction is of an ordering type.", "As has been already mentioned, the Fe-Ni effective pair interactions in binary Invar alloy, Fe$_{65}$ Ni$_{35}$ ,[36] are very close to those obtained in this work for austenite and shown in Fig.", "REF , where the slight ordering tendency is observed.", "The difference between the atomic ordering of Fe and Ni atoms in Invar alloy and ternary Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ alloy is thus entirely due to compositional restrictions in the latter case, which do not allow atoms of all the types to establish their best local environment.", "Table: Warren-Cowley short-range-order parameters,α lmn αβ \\alpha _{lmn}^{\\alpha \\beta }, determined in Monte Carlo simulationsfor Fe 56 _{56}Cr 21 _{21}Ni 23 _{23} Ni at 800 K. The experimentalparameters are taken from Ref.", "cenedese84.It should be noted that the experimental data for the SRO parameters in Fe$_{82-x}$ Cr$_{18}$ Ni$_{x}$ ($x=$ 15, 20, and 25 wt.%) alloys obtained by Braude et al.", "[8] for temperatures $~$ 1400 K, seem to be quite strange.", "For instance, the Warren-Cowley SRO parameter for Ni-Cr pairs is about $-$ 0.3 for the first (110) and second (200) coordination shells, while it is about  0.15 for the fourth (220) coordination shell.", "It is hard to imagine that kind of ordering behavior and it also contradicts experimental data by Cenedese et al..[7] The only common point is the fact that the ordering tendency between pairs of alloy components decreases in the order: Ni-Cr, Fe-Cr, and Fe-Ni.", "Figure: (Color online)Warren-Cowley short-range order parameters inFe 75 _{75}Cr 17 _{17}Ni 08 _{08} (open circles) andFe 56 _{56}Cr 21 _{21}Ni 23 _{23} (filled diamonds) at 800 Kobtained in Monte Carlo simulations.The closeness of effective interactions and compositions for Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ and Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ alloys means that the atomic SRO should be also very similar.", "In Fig.", "REF , we show the atomic SRO in these alloys at 800 K. One can see that this is indeed the case: there is quite strong ordering of Ni-Cr atoms.", "In both cases the values of the SRO parameters are quite large (in absolute value), which means that these alloys should not be that far from an order-disorder phase transition.", "Indeed, upon decreasing temperature the phase transition into the (Fe,Ni)$_3$ Cr-L1$_2$ -like structure is observed at about 520 K for the Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ alloy composition and at 480 K for the Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ alloy composition, although most probably these transition temperatures are overestimated since the theoretical atomic SRO is stronger than experimental one.", "The ordered structure is shown in Fig.", "REF for the case of Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ alloy (a similar structure is observed in the Monte Carlo simulations for Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ ).", "Figure: (Color online)Atomic structure of Fe 56 _{56}Cr 21 _{21}Ni 23 _{23} at 400 Kobtained in Monte Carlo simulations.", "Type of atoms arecolorcoded in the following way: Cr is yellow, Fe is red,and Ni is grayblue.As one can see, Cr atoms occupy a simple cubic positions of the fcc lattice, while Ni and Fe atoms occupy the remaining (face-centred) positions forming a peculiar mutual ordering.", "If the temperature in Monte Carlo simulations is lowed further, there will another phase transition at about 170 K leading to a mixture of two ordered phases: Fe$_3$ Ni$_3$ Cr$_2$ and some not so clear ordering paten of the remaining Fe and Cr atoms.", "However, this last transition is hardly relevant to the reality, where low temperature magnetic state can change substantially the interactions, and the atomic diffusion is absent on the usual practical human time scale." ], [ "Summary", "An accurate ab initio-based description of the finite-temperature properties of austenitic stainless steels require taking the thermally induced LSF into consideration.", "In this work, the simplest model has been used, which is based on a classical consideration of the spin-fluctuation energy, which is a rough statical single-site mean-field approximation.", "It is demonstrated, however, that it allows one to bring such properties as bulk modulus in close agreement with experimental data.", "The simulations of atomic configuration of austenitic steels done for two compositions, Fe$_{56}$ Cr$_{21}$ Ni$_{23}$ and Fe$_{75}$ Cr$_{17}$ Ni$_{08}$ , show that there is substantial degree of the atomic SRO at high temperatures.", "These results are in reasonable agreement with the existing experimental data.", "[7] The theoretical calculations by Wrobel et al.", "[9] predict much stronger ordering tendency due to the use of the low-temperature magnetic ground state in the cluster expansion.", "In this paper, we show that Fe-Cr and Fe-Ni effective pair interactions at the first coordination shell are approximately twice as large as those in the DLM(-LSF) state.", "In order to establish a truly quantitative picture of the atomic ordering and the properties of austenitic steels at finite temperature, further investigation is needed into the finite temperature magnetism of these alloys and its use in the atomistic scale modeling of their structural and vibrational properties." ], [ "SGPM interactions in the EMTO-CPA method", "The one-electron contribution to the GPM interactions within the EMTO-CPA Green's function formalism can be obtained from the one-electron energy, which in this case is (for details of the EMTO-CPA parameters see [11]) $ E_{one-el} = \\frac{1}{2 \\pi i} \\oint z \\langle G(z)\\rangle dz ,$ where $\\langle G(z)\\rangle $ is the CPA average Green's function of alloy (for Brave lattice to simplify notations): $ \\langle G(z)\\rangle = \\int _{\\rm BZ} d {\\mathbf {k}}\\sum _{LL^{\\prime }}\\tilde{g}_{L^{\\prime };L}({\\mathbf {k}},z)\\dot{S}_{L; L^{\\prime }}({\\mathbf {k}},z) + s.s.c.$ Here, only multisite term is explicitly shown, while single-site contribution (s.s.c) is omitted since it does not contribute to the intersite chemical and magnetic interactions.", "Summations are running over sublattices ($ij$ ), indexes of angular momentum ($L,L^{\\prime }$ ) and integration is done over k-points of the Brillouin zone.", "$\\dot{S}_{LL^{\\prime }}({\\mathbf {k}},z)$ is the energy derivative of the slope matrix, $S_{LL^{\\prime }}({\\mathbf {k}},z$ ), which depend on energy, $z$ , and k-point, and $\\tilde{g}_{LL^{\\prime }}({\\mathbf {k}},z)$ is the k-point resolved CPA path operator: $\\tilde{g}_{LL^{\\prime }}({\\mathbf {k}},z) =\\frac{1}{S_{LL^{\\prime }}({\\mathbf {k}},z) - \\tilde{D}_L(z)} ,$ where $\\tilde{D}_L(z)$ is the coherent potential function of the EMTO method obtained self-consistently from the following CPA set of equations: $ &&\\widetilde{g}_{LL^{\\prime }}(z) =\\int d {\\mathbf {k}} \\tilde{g}_{LL^{\\prime }}({\\mathbf {k}},z) \\equiv \\widetilde{g} \\\\&&g^{\\alpha } = \\widetilde{g} + \\widetilde{g} \\left[D^{\\alpha } - \\tilde{D}\\right]g^{\\alpha } \\\\&&\\widetilde{g} = \\sum _{\\alpha } c^{\\alpha } g^{\\alpha }$ In the last two equations, we have omitted angular momentum index and energy dependence of $\\widetilde{g}$ and $D$ .", "The chemical and magnetic exchange interactions then can be found using the force theorem [39] either for chemical fluctuations at some particular site (relative to the CPA effective medium)[18], [40], [41], [42], [43], [20] or by introducing small displacement of the direction of the spin relative to unperturbed spin orientation at this site[44], [45], [46] and then finding the change of the one-electron energy (REF ) by expanding the multisite part of the Green's function in (REF ).", "The resulting expressions are similar to those in the KKR or KKR-ASA methods.", "[20] For instance, the GPM quasibinary effective pair interactions in a multicomponent alloy are $ \\widetilde{V}^{(2)-\\alpha \\beta -1}_p = -\\frac{1}{\\pi } \\Im \\int ^{E_F}{ \\rm Tr} \\left[ \\Delta t^{\\alpha \\beta } \\widetilde{g}_{ij}\\Delta t^{\\alpha \\beta } \\widetilde{g}_{ji} \\right] dE ,$ where $\\Delta t^{\\alpha \\beta } = t^{\\alpha } - t^{\\beta }$ , and $t^{\\alpha }$ has the meaning of the single-site scattering t-matrixes, which in the EMTO method are $t^{\\alpha } = \\left[ 1 + \\widetilde{g}(\\tilde{D} - D^{\\alpha }) \\right]^{-1}(\\tilde{D} - D^{\\alpha }) ,$ which actually satisfy the CPA equation: $\\sum _{\\alpha } c^{\\alpha } t^{\\alpha } = 0.$ Another quantity entering (REF ) is the CPA scattering path operator between $i$ and $j$ sites, which belong to the coordination shell $p$ : $ \\widetilde{g}_{ji}(z) = \\int _{\\rm BZ}\\tilde{g}({\\mathbf {k}},z) e^{i {\\mathbf {k}}({\\mathbf {R}}_i - {\\mathbf {R}}_j)}d {\\mathbf {k}} ,$ where angular momentum indexes are omitted.", "In general, the interaction of order $n$ of Hamiltonian (REF ) is defined as $V^{(n)- \\alpha \\beta \\ldots \\gamma [\\delta ] }_{f} =-\\frac{1}{\\pi } \\Im \\int ^{E_F} { \\rm Tr}\\left[ t^{\\alpha } \\widetilde{g}_{ij} t^{\\beta } \\widetilde{g}_{jk}\\ldots \\widetilde{g}_{lk} t^{\\gamma } \\right] dE ,$ where $i$ , $j$ , $\\ldots $ , and $k$ are the sites of the cluster $f$ .", "As a matter of fact, this is only one specific contribution to this interaction, and in order to get the total interaction for $n>3$ , one should sum over all possible paths connecting sites of the cluster.", "This formalism can be easily generalized to the case of paramagnetic alloys described by the DLM, or alloys with partial non-zero magnetization within partial DLM (PDLM) approach.", "In this case, the expressions for the effective interactions remain the same, but $t^{\\alpha }$ entering the corresponding formulas are modified.", "In particular, if alloy component A is in the PDLM state with magnetization $m < 1$ , it is presented as an alloy with spin-up and spin-down orientation, A$^{\\uparrow }_{x}$ A$^{\\downarrow }_{1-x}$ , where $m = 2x -1$ (assuming that $x \\ge 0.5$ , and $x = 0.5$ corresponds to the DLM paramagnetic state).", "It can be shown that corresponding magnetic averaging (like it is done, for instance, in Ref.", "ruban04,rahaman11) results in this case in $t^{A} = x t^{A^{\\uparrow }} + (1-x) t^{A^{\\downarrow }}$ for every spin component.", "Finally, in order to get the screened pair effective interaction at the coordination shell $p$ , one should add the corresponding screening contribution, $V^{scr-\\alpha \\beta }_p$ , as it is defined in (REF ), so that $\\widetilde{V}^{(2)-\\alpha \\beta }_p =\\widetilde{V}^{(2)-\\alpha \\beta -1}_p + V^{scr-\\alpha \\beta }_p .$" ], [ "Magnetic exchange interactions in the EMTO-CPA method", "Magnetic exchange interactions, $J_p$ of Heisenberg Hamiltonian $ H^H = -\\sum _p \\sum _{ij \\in p}J_p {\\mathbf {e}}_i {\\mathbf {e}}_j ,$ where ${\\mathbf {e}}_i$ is the spin variable at site $i$ , can be derived using the so-called magnetic force theorem[45], [46] and has the following form for elementary solid closely resembling Eq.", "(REF ): $ J_p = -\\frac{1}{\\pi } \\Im \\int ^{E_F}{ \\rm Tr} \\left[ \\Delta g_{ij}^{\\uparrow }\\Delta g_{ji}^{\\downarrow } \\right] dE ,$ where $\\Delta = D^{\\uparrow } - D^{\\downarrow }$ , and $g_{ij}^{\\uparrow (\\downarrow )}$ are the intersite path operator for spin-up (and spin-down) states as they determined in (REF ).", "In the case of a random alloy, magnetic exchange interactions between $\\alpha $ and $\\beta $ alloy components, $J_p^{\\alpha \\beta }$ , is determined as[46] $ J_p^{\\alpha \\beta } = -\\frac{1}{\\pi } \\Im \\int ^{E_F}{ \\rm Tr} \\left[ \\Delta ^{\\alpha } \\bar{g}_{ij}^{\\alpha \\beta \\uparrow }\\Delta ^{\\beta } \\bar{g}_{ji}^{\\beta \\alpha \\downarrow } \\right] dE ,$ where $\\Delta ^{\\alpha } = D^{\\alpha \\uparrow } - D^{\\alpha \\downarrow }$ and $\\bar{g}_{ij}^{\\alpha \\beta } =\\left[ 1 + \\widetilde{g}(\\tilde{D} - D^{\\alpha }) \\right]^{-1}\\widetilde{g}_{ij} \\\\ \\nonumber \\left[ 1 + (\\tilde{D} - D^{\\beta })\\widetilde{g} \\right]^{-1} .$ Here, it is assumed that alloy is homogeneous and thus there is no site-dependence of the potential parameters (although the generalization to inhomogeneous alloys is straightforward).", "The DLM state is a special one since magnetic exchange interactions in the DLM state are equal to the corresponding GPM interactions $8 J_p = - \\widetilde{V}^{(2)}_p$ ,[46], [48] which can be proved analytically.", "On the other hand, it follows from the comparison of the Ising and Heisenberg Hamiltonians and the fact that the last one is reduced to the Ising one for collinear magnetic configurations.", "This also means that GPM provides an easy way to calculated higher order magnetic interactions.", "[20], [48] AVR acknowledges the support of the Swedish Research Council (VR project 2015-05538), the European Research Council grant, the VINNEX center Hero-m, financed by the Swedish Governmental Agency for Innovation Systems (VINNOVA), Swedish industry, and the Royal Institute of Technology (KTH).", "Calculations have been done using NSC (Linköping) and PDC (Stockholm) resources provided by the Swedish National Infrastructure for Computing (SNIC).", "The support by the Austrian Federal Government (in particular from Bundesministerium fŸr Verkehr, Innovation und Technologie and Bundesministerium fŸr Wirtschaft, Familie und Jugend) represented by …sterreichische Forschungsfšrderungsgesellschaft mbH and the Styrian and the Tyrolean Provincial Government, represented by Steirische Wirtschaftsfšrderungsgesellschaft mbH and Standortagentur Tirol, within the framework of the COMET Funding Programme is also gratefully acknowledged." ] ]
1606.05096
[ [ "Optimal Liquidation under Partial Information with Price Impact" ], [ "Abstract We study the optimal liquidation problem in a market model where the bid price follows a geometric pure jump process whose local characteristics are driven by an unobservable finite-state Markov chain and by the liquidation rate.", "This model is consistent with stylized facts of high frequency data such as the discrete nature of tick data and the clustering in the order flow.", "We include both temporary and permanent effects into our analysis.", "We use stochastic filtering to reduce the optimal liquidation problem to an equivalent optimization problem under complete information.", "This leads to a stochastic control problem for piecewise deterministic Markov processes (PDMPs).", "We carry out a detailed mathematical analysis of this problem.", "In particular, we derive the optimality equation for the value function, we characterize the value function as continuous viscosity solution of the associated dynamic programming equation, and we prove a novel comparison result.", "The paper concludes with numerical results illustrating the impact of partial information and price impact on the value function and on the optimal liquidation rate." ], [ "Introduction", "In financial markets, traders frequently face the task of selling a large amount of a given asset over a short time period.", "This has led to a large literature on optimal portfolio execution.", "The existing work can be divided into two classes: market impact models and order book models.", "In a market impact model one directly specifies the impact of a given trading strategy on the bid price of the asset.", "The fundamental price (the price if the trader is inactive) is usually modelled as a diffusion process such as Brownian motion.", "In an order book model, instead, one specifies the dynamics of the limit order book.", "This is more complex but gives an explanation of the price impact in terms of fundamental quantities.", "Portfolio liquidation strategies are executed at a high trading frequency.", "Hence a sound market impact model should be consistent with key stylized facts of high frequency data as discussed for instance by [19] or [24].", "First, on very fine time scales the bid price of an asset is best described by a pure jump process, since in reality prices move on a discrete grid defined by the tick size.", "Second, the order flow is clustered in time: there are random periods with a lot of buy orders or with a lot of sell orders, interspersed by quieter times with less trading activity.", "[24] attributes this to the fact that many observed orders are components of a larger parent order that is executed in small blocks.", "A further reason for the clustering in the inter-event times are random fluctuations in the arrival rate of new information, see, e.g. [5].", "Third, the distribution of returns over short time intervals is strongly non-Gaussian but has heavy tails and a large mass around zero; to a certain extent this is a consequence of the first two stylized facts.", "Finally, there is permanent price impact, that is the implementation of a liquidation strategy pushes prices downwards.", "To capture these stylized facts we model the bid price as marked point process with Markov switching whose local characteristics (intensity and jump size distribution) depend on the trader's current liquidation rate $\\nu _t$ and on the value $Y_t$ of a finite state Markov chain $Y$ .", "The fact that the local characteristics depend on $\\nu _t$ is used to model permanent price impact.", "Markov switching allows us to reproduce the observed clustering in the order flow.", "Our framework encompasses models with a high intensity of downward jumps in one state of $Y$ and a high intensity of upward jumps in another state of $Y$ and models where inter-event times are given by a mixture of exponential distributions.", "We view the process $Y$ as an abstract modelling device that generates clustering and assume therefore that $Y$ is unobservable by the trader.", "This is consistent with the fact that economic sources for clustering such as the trading activity of other investors are not directly observable.", "Markov modulated marked point processes with partial information (without price impact) were considered previously in the statistical modelling of high frequency data, see for instance [45], [26], or [17]; however, we are the first to study optimal liquidation in such a setting.", "The first step in the analysis of a control problem with partial information is to derive an equivalent problem under full information via stochastic filtering.", "Hence we have to determine the dynamics of the conditional distribution of $Y_t$ given the bid price history up to time $t$ .", "Note that this provides a further rationale for modelling the bid price as a marked point process: the strong non-normality of short-period returns implies that it is very problematic to use high frequency data as input for the numerical solution of the filtering equations in the classical setup where observations are modelled as a Brownian motion with drift, as the resulting filters become extremely unstable.", "Instead one should take the structure of the observation process seriously and work in a point process model.", "We use the reference probability approach to derive the filtering equations for our model.", "In this way we circumvent the issue that the information available to the investor depends on her liquidation strategy.", "We end up with a control problem whose state process $X$ consists of the stock price, the inventory level, and the filter process.", "We provide a detailed mathematical analysis of this problem.", "The form of the asset return dynamics implies that $X$ is a piecewise deterministic Markov process (PDMP) so that we rely on control theory for PDMPs; a general introduction to this theory is given in [29] or in [12].", "We establish the dynamic programming equation for the value function and we derive conditions on the data of the problem that guarantee the continuity of the value function.", "This requires a careful analysis of the behaviour of the value function close to the boundary of the state space.", "As a further step we characterize the value function as the unique continuous viscosity solution of the Hamilton-Jacobi-Bellman (HJB) partial integro-differential equation associated with the problem and we give an example showing that in general the HJB equation does not admit a classical solution.", "Moreover, we prove a novel comparison theorem for the HJB equation which is valid in more general PDMP setups.", "A comparison principle is necessary to ensure the convergence of numerical schemes to the value function, see [9].", "The paper closes with a section on applications.", "We discuss properties of the optimal liquidation rate and of the expected liquidation profit and we use a finite difference approximation of the HJB equation to analyze the influence of the temporary and permanent price impact parameters on the form of the optimal liquidation rate.", "Among others, we find that for certain parameter constellations the optimal strategy displays a surprising gambling behaviour of the trader that cannot be guessed upfront and we give an economic interpretation that is based on the form of the HJB equation.", "Moreover, we study the additional liquidation profit from the use of a filtering model, and we report results from a small calibration study that provides further support for our model.", "We continue with a brief discussion of the existing literature.", "Starting with market impact models, the first contribution is [15] who analyze the optimal portfolio execution problem for a risk-neutral agent in a model with linear and purely permanent price impact where the fundamental price follows an arithmetic random walk.", "This model has been generalized by [2] who consider also risk aversion and temporary price impact.", "Since then, market impact models have been extensively studied.", "Important contributions include [38], [44], [43], [6], [37].", "Recently, [20] studied an extension of the Almgren Chriss model with a self-exciting temporary price impact.", "All these models work in a (discretized) diffusion framework.", "In the order book literature on the other hand, a few contributions based on point process models exist.", "[13] analyze the optimal portfolio execution problem in a model with discrete order flow represented by a Poisson process with observable intensity.", "The price impact is purely temporary and is represented in terms of a cost function.", "[11] consider the same setting with a standard Poisson process and solve the cost minimization problem by using tools from the control theory of PDMPs.", "A further order book model based on point process methodology is [14].", "There it is assumed that the trader uses limit orders and that she can control the intensity of the order flow by choosing the spread at which she is willing to trade.", "Additional contributions based on diffusion models are [1], [41], [18].", "For a detailed overview we refer to the surveys [36], [35] or [19].", "From a methodological point of view our analysis is also related to the literature on expected utility maximization or hedging for pure jump process such as [10] or [40].", "Important contributions to the control theory of PDMPs include [29], [31], [4], [33], [11], [25].", "Viscosity solutions for PDMP control problems were previously considered in [30].", "The outline of the paper is the following.", "In Section , we introduce our model, the main assumptions and the optimization problem.", "In Section , we derive the filtering equations for our model.", "Section contains the mathematical analysis of the optimization problem via PDMP techniques.", "In Section we provide a viscosity solution characterization of the value function.", "Finally, in Section , we present the results of our numerical experiments.", "The appendix contains additional proofs." ], [ "The Model", "Throughout we work on the filtered probability space $(\\Omega ,\\mathcal {F},\\mathbb {F},\\mathbf {P})$ , where the filtration $\\mathbb {F}=\\lbrace \\mathcal {F}_t\\rbrace _{t\\ge 0}$ satisfies the usual conditions.", "Here $\\mathbb {F}$ is the global filtration, i.e.", "all considered processes are $\\mathbb {F}$ -adapted, and $\\mathbf {P}$ is the historical probability measure.", "We consider a trader who wants to liquidate $w_0>0$ units of a given security (referred to as the stock in the sequel) over the period $[0, T]$ for a given time horizon $T$ .", "We denote the bid price process by $S = (S_t)_{0 \\le t \\le T}$ and $\\mathbb {F}^S$ is the filtration generated by $S$ .", "In what follows, we assume that $\\mathbb {F}^S$ satisfies the usual conditions.", "We assume that the trader sells the shares at a nonnegative $\\mathbb {F}^S$ -adapted rate $ \\nu = (\\nu _t)_{0 \\le t \\le T}$ such that for every $t \\in [0,T]$ , $\\nu _t\\in [0, \\nu ^{\\text{max}}]$ for a given positive constant $\\nu ^{\\text{max}}$ .", "Hence her inventory, i.e.", "the amount of shares she holds at time $t \\in [0,T]$ , is given by $W_t = w_0 -\\int _0^t \\nu _{u}\\mathrm {d}u, \\quad t \\in [0,T].$ Modelling the inventory as an absolutely continuous process corresponds to the situation where the trader is frequently submitting small sell orders.", "By taking $\\nu $ to be nonnegative, we confine the trader to pure selling strategies; the motivation for imposing the upper bound $\\nu ^{\\text{max}}$ on the liquidation rate is discussed in Section  below.", "The goal of the trader is to maximize the expected revenue from her trading strategy.", "We assume that the implementation of the liquidation strategy generates temporary and permanent price impact, where permanent price impact is the impact of trading on the dynamics of $S$ and temporary price impact is the impact of trading on the execution price of the current trade.", "subsection20mm-0.25cm Dynamics of the bid price.", "In order to reproduce stylized facts of high frequency data such as the path structure of asset prices and the clustering of the order flow, we model the bid price as a Markov-modulated geometric finite activity pure jump process.", "Let $Y = (Y_{t})_{0 \\le t \\le T}$ be a continuous-time finite-state Markov chain on $(\\Omega ,\\mathcal {F},\\mathbb {F},\\mathbf {P})$ with state space $\\mathcal {E} = \\lbrace e_1,e_2,...,e_K\\rbrace $ ($e_k$ is $k$ -th unit vector in $\\mathbb {R}^K$ ), generator matrix $ Q=(q^{ij})_{i,j=1,\\dots ,K}$ and initial distribution $\\pi _0=(\\pi _0^1,\\cdots ,\\pi _0^K)$ .", "We assume that the bid price has the dynamics $\\mathrm {d}S_t = S_{t^-} \\mathrm {d}R_t, \\quad S_0=s\\in (0, \\infty ),$ where the return process $R=(R_t)_{0 \\le t \\le T}$ is a finite activity pure jump process.", "We assume that $\\Delta R_t:=R_t- R_{t^-} > - 1$ so that $S$ is strictly positive.", "Denote by $\\mu ^R$ the random measure associated with $R$ , defined by $ \\mu ^R(\\mathrm {d}t, \\mathrm {d}z):= \\!\\!\\sum _{u \\ge 0, \\Delta R_u\\ne 0}\\delta _{\\lbrace u, \\Delta R_u\\rbrace }(\\mathrm {d}t, \\mathrm {d}z),\\,$ and by $\\eta ^{\\mathbf {P}}$ the $(\\mathbb {F},\\mathbf {P})$ -dual predictable projection (or compensating random measure) of $\\mu ^R$ .", "We assume that $\\eta ^{\\mathbf {P}}$ is absolutely continuous and of the form $\\eta ^\\mathbf {P}(t,Y_{t^-},\\nu _{t^-};\\mathrm {d}z) \\mathrm {d}t$ , for a finite measure $\\eta ^\\mathbf {P}(t,e,\\nu ;\\mathrm {d}z)$ on $\\mathbb {R}$ .", "Moreover, we assume that the processes $R$ and $Y$ have no common jumps, so that $R$ and $Y$ are orthogonal, $[R,Y]_t \\equiv 0$ for all $t \\in [0, T]$ , $\\mathbf {P}$ -a.s.", "The measure $\\eta ^\\mathbf {P}(t,e,\\nu ;\\mathrm {d}z)$ is a crucial quantity as it determines the law of the bid price with respect to filtration $\\mathbb {F}$ under $\\mathbf {P}$ .", "The fact that $\\eta ^{\\mathbf {P}} $ depends on the current liquidation rate serves to model permanent price impact; the dependence of $\\eta ^{\\mathbf {P}} $ on $Y_{t-}$ can be used to reproduce the clustering in inter-event durations observed in high frequency data and to model the feedback effect from the trading activity of the rest of the market.", "Finally, time-dependence of $\\eta ^{\\mathbf {P}} $ can be used to model the strong intra-day seasonality patterns observed for high frequency data.", "These aspects are explained in more detail in Example REF below.", "Now we turn to the semimartingale decomposition of the bid price with respect to the full information filtration $\\mathbb {F}$ .", "Denote for all $(t,e, \\nu )\\in [0,T]\\times \\mathcal {E} \\times [0, \\nu ^{\\text{max}}]$ , the mean of ${\\eta }^\\mathbf {P}$ by $\\overline{\\eta }^\\mathbf {P}(t,e, \\nu ): = \\int _{\\mathbb {R}} z \\, \\eta ^\\mathbf {P}(t,e, \\nu ; \\mathrm {d}z)\\,;$ $\\overline{\\eta }^\\mathbf {P}(t,e, \\nu )$ exists under Assumption REF below.", "Fix some liquidation strategy $\\nu $ .", "Then the martingale part $M^R$ of the return process is given by $M^R_t = R_t - \\int _0^t \\overline{\\eta }^\\mathbf {P}(s,Y_{s-}, \\nu _{s-})\\mathrm {d}s $ , for all $t\\in [0,T]$ , and the $\\mathbb {F}$ -semimartingale decomposition of $S$ equals $S_t = S_0 + \\int _0^t S_{s-} \\mathrm {d}M^R_s + \\int _0^t S_{s-} \\overline{\\eta }^\\mathbf {P}(s,Y_{s-}, \\nu _{s-})\\mathrm {d}s\\,, \\quad t\\in [0,T]\\,.$ It is well-known that the semimartingale decomposition of $S$ with respect to the trader's filtration $\\mathbb {F}^S$ is obtained by projecting the process $\\overline{\\eta }^\\mathbf {P}(t,Y_{t^-}, \\nu _{t^-})$ onto $\\mathbb {F}^S$ .", "In the sequel we assume that for all $(t,e) \\in [0,T] \\times \\mathcal {E}$ , the mapping $\\nu \\mapsto \\overline{\\eta }^\\mathbf {P}(t,e, \\nu )$ is decreasing on $[0, \\infty )$ , that is selling pushes the price down on average.", "Furthermore, we make the following regularity assumption.", "Assumption 2.1 (Properties of $\\eta ^\\mathbf {P}$ ) There is a deterministic finite measure $\\eta ^{\\mathbf {Q}}$ on $\\mathbb {R}$ whose support, denoted by $\\text{supp}(\\eta )$ , is a compact subset of $(-1, \\infty )$ , such that for all $(t,e,\\nu ) \\in [0,T] \\times \\mathcal {E}\\times [0, \\infty )$ the measure $\\eta ^{\\mathbf {P}} (t,e, \\nu ; \\mathrm {d}z)$ is equivalent to $\\eta ^{\\mathbf {Q}}(\\mathrm {d}z) $ .", "Furthermore, for every $\\nu ^{\\text{max}}< \\infty $ there is some constant $M > 0$ such that $ M^{-1} < \\frac{\\mathrm {d}\\eta ^\\mathbf {P}(t,e, \\nu ) }{\\mathrm {d}\\eta ^\\mathbf {Q}}(z) < M \\text{ for all } (t, e, \\nu ) \\in [0,T] \\times \\mathcal {E} \\times [0, \\nu ^{\\text{max}}]\\,.$ The assumption implies that for every $\\nu ^{\\text{max}}$ there is a $ \\lambda ^{\\text{max}} < \\infty $ such that $\\sup \\lbrace \\eta ^\\mathbf {P}(t,e,\\nu ;\\mathbb {R}) \\colon (t,e,\\nu ) \\in [0,T]\\times \\mathcal {E}\\times [0 , \\nu ^{\\text{max}}]\\rbrace \\le \\lambda ^{\\text{max}} \\,;$ in particular the counting process associated to the jumps of $S$ is $\\mathbf {P}$ -nonexplosive.", "Moreover, it provides a sufficient condition for the existence of a reference probability measure, i.e.", "a probability measure $\\mathbf {Q}$ equivalent to $\\mathbf {P}$ on $(\\Omega , \\mathcal {F}_T)$ , such that under $\\mathbf {Q}$ , $\\mu ^R$ is a Poisson random measure with intensity measure $\\eta ^{\\mathbf {Q}}(\\mathrm {d}z)$ , independent of $Y$ and $\\nu $ .", "This is needed in the analysis of the filtering problem of the trader in Section .", "Note that the equivalence of $\\eta ^\\mathbf {P}$ and $\\eta ^\\mathbf {Q}$ implies that for all $(t,e,\\nu ) \\in [0,T] \\times \\mathcal {E}\\times [0, \\infty )$ the support of $\\eta ^{\\mathbf {P}}$ is equal to $\\text{supp}(\\eta )$ .", "The assumption that $\\text{supp}(\\eta )$ is compact is not restrictive, since in reality the bid price moves only by a few ticks at a time.", "The following examples serve to illustrate our framework; they will be taken up in our numerical experiments in Section .", "Example 2.2 Consider the case where the return process $R$ follows a bivariate point process, i.e.", "there are two possible jump sizes, $\\Delta R\\in \\lbrace - \\theta , \\theta \\rbrace $ for some $\\theta >0$ .", "In this example we assume that the dynamics of $S$ is independent of $Y$ and $t$ .", "Moreover, the intensity $\\lambda ^+$ of an upward jump is constant and equal to $c^{\\text{up}}>0$ , and the intensity $\\lambda ^-$ of a downward jump depends on the rate of trading and is given by $\\lambda ^-(\\nu ) = c^{\\text{down}}(1+ a \\nu )$ for constants $c^{\\text{down}}, a > 0$ .", "Note that, with this choice of $\\lambda ^-$ , the intensity of a downward jump in $S$ is linearly increasing in the liquidation rate $\\nu $ .", "The function $\\overline{\\eta }^\\mathbf {P}$ from (REF ) is independent of $t$ and $e$ and linearly decreasing in $\\nu $ ; it is given by $\\overline{\\eta }^\\mathbf {P}(\\nu ) = \\theta (c^{\\text{up}}- c^{\\text{down}}(1+a \\nu ))$ .", "Linear models for the permanent price impact are frequently considered in the literature as they have theoretical and empirical advantages; see for instance [3] or [35].", "Example 2.3 Now we generalize Example REF and allow $\\eta ^\\mathbf {P}$ to depend on the state process $Y$ .", "We consider a two-state Markov chain $Y$ with the state space $\\mathcal {E}=\\lbrace e_1,e_2\\rbrace $ and we assume that $e_1$ is a `good' state and $e_2$ a `bad' state in the following sense: in state $e_1$ the intensity of an upward move of the stock is larger than in state $e_2$ ; the intensity of a downward move on the other hand is larger in state $e_2$ than in $e_1$ .", "We therefore choose constants $c^{\\text{up}}_1 > c^{\\text{up}}_2 >0$ , $c^{\\text{down}}_2 > c^{\\text{down}}_1>0$ and a price impact parameter $a> 0$ and we set for $i=1,2$ , $\\lambda ^+ (e_i, \\nu ) = (c^{\\text{up}}_1, c^{\\text{up}}_2) e_i \\; \\text{ and } \\lambda ^- (e_i, \\nu ) = (1 + a \\nu ) (c^{\\text{down}}_1, c^{\\text{down}}_2) e_i.$ Then, $\\eta ^\\mathbf {P}( e_i, \\nu ,\\mathrm {d}z) = \\lambda ^+ (e_i, \\nu ) \\delta _{\\lbrace \\theta \\rbrace }(\\mathrm {d}z) +\\lambda ^- (e_i, \\nu ) \\delta _{ \\lbrace -\\theta \\rbrace }(\\mathrm {d}z)$ , for $i=1,2$ .", "Since $c^{\\text{up}}_1 > c^{\\text{up}}_2$ , in state $e_1$ one has on average more buy orders; this might represent a scenario where another trader is executing a large buy program.", "Similarly, since $c^{\\text{down}}_2 > c^{\\text{down}}_1$ , there are on average more sell orders in state $e_2$ , for instance because another trader is executing a large sell program.", "The form of $\\eta ^\\mathbf {P}$ implies that the permanent price impact is linear and proportional to the intensity of a downward move and hence larger in the `bad' state $e_2$ than in the good state $e_1$ .", "Note that within our general setup this example could be enhanced in a number of ways.", "For instance, the transition intensities $ c^{\\text{up}}_i$ and $c^{\\text{down}}_i$ and the liquidity parameter $a$ could be made time dependent to reflect the fact that on most markets trading activity during the day is $U$ -shaped with more trades occurring at the beginning and the end of a day than in the middle.", "Moreover, one could introduce an additional state where the market is moving sideways, or one could consider the case where the liquidity parameter $a$ depends on $Y$ .", "Remark 2.4 (Calibration.)", "We briefly discuss a potential approach for parameter estimation in our setup.", "For $\\nu \\equiv 0$ the model is a hidden Markov model with point process observation.", "It is therefore natural to use the expectation maximization (EM) methodology for Markov modulated point processes as described in [32] or in [27] to estimate the generator matrix of $Y$ and parameters of the compensator$\\eta ^\\mathbf {P}$ .", "A numerical case study with simulated and real data in the context of Example REF is given in Section REF .", "Using proprietary data on the performance of equity sales, [3] find empirical support for a linear permanent price impact function; the parameter $a$ can be estimated by regressing price changes on trading volume.", "subsection20mm-0.25cm The optimization problem.", "In this section we specify the ingredients of the traders optimization problem in detail." ], [ "Liquidation strategies.", "We assume that the state process $Y$ is not directly observable by the trader.", "Instead, she observes the price process $S$ and knows the model parameters, so that information available to her is carried by filtration $\\mathbb {F}^S$ or, equivalently, by the filtration generated by the return process $R$ .", "Hence we assume that the trader uses only liquidation strategies that are $\\mathbb {F}^S$ -adapted.", "Moreover we impose a bound on the maximal speed of trading: we fix some constant $\\nu ^{\\text{max}}> w_0/T$ and we call a liquidation strategy $\\nu $ admissible if $\\nu $ is $\\mathbb {F}^S$ -adapted and if $\\nu _t \\in [0,\\nu ^{\\text{max}}]$ for all $t \\in [0,T]$ $\\mathbf {P}$ -a.s.", "Note that the condition $\\nu ^{\\text{max}}> w_0/T$ ensures that it is feasible for the trader to liquidate the whole inventory over the period $[0,T]$ .", "The assumption of a bounded liquidation rate merits a discussion.", "From a mathematical point of view a bound on the liquidation rate facilitates the application of results for the control of piecewise deterministic Markov processes, since in this theory it is typically assumed that the strategies take values in a compact control space.", "Moreover, without this assumption the viscosity solution characterization of the value function (see Theorem REF below) does not hold.", "A counterexample is given in Section , where we show that for unbounded liquidation rate the value function is a strict supersolution of the corresponding dynamic programming equation, cf.", "Remark REF .", "Finally, the upper bound on $\\nu _t$ ensures that under Assumption REF for every admissible strategy $\\nu $ a return process $R$ with compensating measure $\\eta ^\\mathbf {P}(t, Y_{t-}, \\nu _{t-}; \\mathrm {d}z) \\mathrm {d}t$ (and hence the bid price process (REF )) exists.", "From a financial point of view an upper bound on $\\nu _t$ is reasonable, as trading at infinite speed would correspond to large block transactions; allowing such transactions at some time point $t < T$ would require an explicit model for market resiliency.", "It is however not clear how to determine $\\nu ^{\\text{max}}$ empirically.", "In Proposition REF below we therefore show that $J^{*,m}$ , the optimal liquidation value if the trader uses $\\mathbb {F}^S$ -adapted strategies with $\\nu _t \\le m$ for all $t$ , is bounded independently of $m$ .", "The sequence $\\lbrace J^{*,m}\\rbrace _{m \\in \\mathbb {N}}$ is obviously increasing, since a higher $m$ means that the trader can optimize over a larger set of strategies.", "Hence, $\\lbrace J^{*,m}\\rbrace _{m \\in \\mathbb {N}}$ is Cauchy.", "This implies that optimal proceeds from liquidation are nearly independent of the precise numerical value chosen for $\\nu ^{\\text{max}}$ .", "In order to further support this argument we present results of numerical experiments in the framework of Example REF .", "Table REF displays the value function $J^{*,m}$ for varying $\\nu ^{\\text{max}}$ (expressed as multiple of the initial inventory $w_0$ ) and for fixed $w_0 = 6000$ .", "The value grows in $\\nu ^{\\text{max}}$ , but for $\\nu ^{\\text{max}}>2w_0$ , the additional gain is small.", "For details on the numerical analysis we refer to Section .", "Table: The expected proceeds from liquidationfor varying ν max \\nu ^{\\text{max}} and fixed w 0 =6000w_0=6000.", "Details on the numerical methodology are given in Section ." ], [ "Objective of the trader.", "To account for the case where not all shares have been sold prior to time $T$ we specify the liquidation value of the remaining share position $W_T$ .", "This liquidation value is of the form $h(W_T)S_T$ for some increasing, continuous and concave function $h$ with $h(w) \\le w$ and $h(0)=0$ .", "For instance, the choice $h(w) = \\frac{w}{1 + \\vartheta w}$ for some $\\vartheta >0$ models the situation where the liquidation value that is strictly smaller than the book value, reflecting the limited liquidity of the market for the stock.Note that when considering a block transaction at the horizon date $T$ we do not need to model market resiliency or permanent price impact as the model `ends' at $T$ .", "For $\\theta \\rightarrow \\infty $ we obtain the limit $h(w) \\equiv 0$ ; this models the situation where a block transaction at the terminal date is prohibitively expensive.", "We model the temporary price impact by a nonnegative, continuous increasing function $f$ , so that the proceeds from liquidation are given for every $t \\in [0,T]$ by $\\int _0^t \\nu _s S_s (1-f(\\nu _s)) \\mathrm {d}s$ .", "For instance, [3] propose a power function of the form $f(\\nu ) = c_f\\nu ^\\varsigma $ and they estimate the coefficient $\\varsigma \\approx 0.6$ .", "Now we define the time $\\tau $ as the minimum of the first time the inventory is completely liquidated and the horizon $T$ : $\\tau :=\\inf \\lbrace t\\ge 0 : \\ W_t\\le 0\\rbrace \\wedge T\\,.$ Denote by $\\rho $ the (subjective) discount rate of the trader.", "Consider an admissible strategy $\\nu $ and denote the corresponding bid price by $ S^{\\nu }$ .", "The expected discounted value of the proceeds generated by the liquidation strategy $\\nu $ is equal to $J(\\nu ) = \\mathbb {E}\\left(\\int _0^\\tau e^{-\\rho u} \\nu _u S_u^{\\nu } (1 -f(\\nu _{u})) \\mathrm {d}u + e^{-\\rho \\tau } S_\\tau ^{\\nu } h( W_\\tau ) \\right)\\,.$ The trader wants to maximize (REF ) over all admissible strategies; the corresponding optimal value is denoted by $J^*$ , or, if we want to emphasize the dependence on the upper bound on the liquidation rate, by $ J^{*,\\nu ^{\\text{max}}}$ .", "Note that the form of the objective function in (REF ) implies that the trader is risk neutral.", "Risk neutrality is frequently assumed in the literature on optimal order execution, see for instance [15].", "Moreover, in our setup the risk the trader may take is limited as we consider pure selling strategies and as the time period $[0,T]$ is fairly short.", "The next proposition shows that , if the trader uses $\\mathbb {F}^S$ -adapted strategies with $\\nu _t \\le m$ for all $t\\in [0,T]$ , the optimal liquidation value $J^{*,m}$ is bounded independently of $m$ .", "Proposition 2.5 Suppose that Assumption REF holds and that the function $(t,e,\\nu )\\rightarrow \\overline{\\eta }^\\mathbf {P}(t,e, \\nu )$ from (REF ) is decreasing in $\\nu $ , and set $\\overline{\\eta } = 0 \\vee \\sup \\lbrace \\overline{\\eta }^\\mathbf {P}(t,e, 0) - \\rho \\colon t \\in [0,T], e \\in \\mathcal {E}\\rbrace .$ Then $\\sup _{m >0} J^{*,m} \\le w_0 S_0 e^{ \\overline{\\eta }T}$ .", "Note that the upper bound on $J^*$ corresponds to the liquidation value of the inventory in a frictionless model where the expected value of the bid price grows at the maximum rate $ \\overline{\\eta } + \\rho $ .", "Fix some $\\mathbb {F}^S$ -adapted strategy $\\nu $ with values in $[0,m]$ and let $\\widetilde{S}^{\\nu }_t = e^{-\\rho t} S_t^{\\nu } $ .", "Since $W_t = w_0 - \\int _0^t \\nu _s \\mathrm {d}s$ we get by partial integration that $\\int _0^\\tau \\nu _s \\widetilde{S}_s^{\\nu } \\mathrm {d}s = - \\int _0^\\tau \\widetilde{S}_s^{\\nu } \\mathrm {d}W_s =S_0 w_0 - \\widetilde{S}_\\tau ^{\\nu } W_\\tau + \\int _0^\\tau W_s \\mathrm {d}\\widetilde{S}^{\\nu }_s\\,.$ Since $h(w) \\le w$ and $f(\\nu ) \\ge 0$ we thus get that $\\int _0^\\tau \\!", "\\nu _u \\widetilde{S}_u^{\\nu } (1 -f(\\nu _{u})) \\mathrm {d}u + \\widetilde{S}_\\tau ^{\\nu } h( W_\\tau )\\le \\int _0^\\tau \\nu _u \\widetilde{S}_u^{\\nu } \\mathrm {d}u + \\widetilde{S}_\\tau ^{\\nu } W_\\tau =S_0 w_0 + \\int _0^\\tau W_u \\mathrm {d}\\widetilde{S}^{\\nu }_u .$ Now $ \\int _0^\\tau W_u \\mathrm {d}\\widetilde{S}^{\\nu }_u = \\int _0^\\tau W_u \\widetilde{S}^{\\nu }_u \\mathrm {d}M_u^R +\\int _0^\\tau W_u \\widetilde{S}^{\\nu }_u ( \\overline{\\eta }^\\mathbf {P}(u,Y_{u-}, \\nu _{u-}) -\\rho ) \\mathrm {d}u$ .", "Moreover, $\\int _0^{\\cdot \\wedge \\tau } W_u \\widetilde{S}^{\\nu }_u \\mathrm {d}M_u^R$ is a true martingale: as $0 \\le W_u \\le w_0$ , a similar argument as in the proof of Lemma REF shows that this process is of integrable quadratic variation.", "Since $\\overline{\\eta }^\\mathbf {P}(u,Y_{u-}, \\nu _{u-}) -\\rho \\le \\overline{\\eta }$ , $\\tau \\le T$ and $W_u \\le w_0$ , we get $J(\\nu ) &\\le S_0 w_0 + \\mathbb {E} \\Big ( \\int _0^\\tau W_u \\widetilde{S}^{\\nu }_u (\\overline{\\eta }^\\mathbf {P}(u,Y_{u-}, \\nu _{u-}) - \\rho )\\mathrm {d}u \\Big )\\\\ & \\le S_0 w_0 + \\mathbb {E} \\Big ( \\int _0^T w_0 \\widetilde{S}^{\\nu }_u \\overline{\\eta }\\, \\mathrm {d}u \\Big )\\,.$ Next we show that $\\mathbb {E} \\big ( \\widetilde{S}_t^{\\nu } \\big ) \\le S_0 e^{\\overline{\\eta } t} $ .", "To this end, note that by Lemma REF , $\\int _0^\\cdot S_{s-}^{\\nu } \\mathrm {d}M_s^R$ is a true martingale so that $ \\mathbb {E} \\big ( \\widetilde{S}_t^{\\nu } \\big ) = S_0 +\\mathbb {E} \\Big ( \\int _0^t \\widetilde{S}^{\\nu }_u (\\overline{\\eta }^\\mathbf {P}(u,Y_{u-}, \\nu _{u-}) - \\rho ) \\mathrm {d}u \\Big )\\le S_0 + \\overline{\\eta } \\int _0^t \\mathbb {E} \\big ( \\widetilde{S}_u^{\\nu } \\big ) \\mathrm {d}u,$ and the claim follows from the Gronwall inequality.", "Using (REF ) we finally get that $J(\\nu ) \\le S_0 w_0 (1 + \\int _0^T \\overline{\\eta } e^{\\overline{\\eta } u} \\mathrm {d}u ) = S_0 w_0 e^{\\overline{\\eta } T}$ , and hence the result." ], [ "Partial Information and Filtering", "Considering $\\mathbb {F}^S$ -adapted investment strategies results in an optimal control problem under partial information.", "The standard approach to dealing with such problems is to introduce the filter for the Markov chain as additional state variable of the control problem.", "In this section we therefore derive the filtering equations for our model.", "Filtering for point process observations is for instance considered in [23], [32], [34], [21], [22].", "This literature is mostly based on the innovations approach.", "In this paper, instead, we address the filtering problem via the reference probability approach.", "This methodology relies on the existence of an equivalent probability measure such that the observation process is driven by a random measure with dual predictable projection independent of the Markov chain, see for instance [16].", "The reference probability approach permits us to overcome the difficulties caused by the fact that the observation process $S$ is affected by the liquidation strategy chosen by the trader.", "subsection20mm-0.25cm Reference probability.", "We start from a filtered probability space $(\\Omega ,\\mathcal {F},\\mathbb {F},\\mathbf {Q})$ that supports a Markov chain $Y$ with state space $\\mathcal {E}$ and generator matrix $Q$ , and an independent Poisson random measure $\\mu ^R$ with compensator $\\eta ^\\mathbf {Q}(\\mathrm {d}z) \\mathrm {d}t$ as in Assumption REF .2; $\\mathbf {Q}$ is known as the reference probability measure.", "Note that under $\\mathbf {Q}$ the dynamics of $S$ and $R$ are independent of the liquidation strategy $\\nu $ so that the filtration $\\mathbb {F}^S$ is exogenously given.", "Moreover, the independence of $Y$ and $\\mu ^R$ implies that $R$ and $Y$ have no common jumps.", "For $(t,e,\\nu ,z) \\in [0,T]\\times \\mathcal {E}\\times [0,\\nu ^{\\text{max}}] \\times \\text{supp}(\\eta ) $ , we define the function $\\beta $ by $ \\beta (t,e, \\nu , z): =\\frac{\\mathrm {d}\\eta ^\\mathbf {P}(t,e, \\nu ;\\mathrm {d}z) }{\\mathrm {d}\\eta ^\\mathbf {Q}(\\mathrm {d}z)}(z) - 1\\,,$ i.e.", "$\\beta (t,e, \\nu , z)+1$ is the Radon-Nikodym derivative of the measure $\\eta ^\\mathbf {P}(t,e, \\nu ; \\mathrm {d}z)$ with respect to $\\eta ^\\mathbf {Q}(\\mathrm {d}z)$ .", "Fix now an admissible liquidation strategy $\\nu $ and define for $t \\in [0,T]$ the stochastic exponential $\\widetilde{Z}$ by $ \\widetilde{Z}_{t}=1+ \\int _0^t \\int _{\\mathbb {R}} \\widetilde{Z}_{s^-} \\beta ( s,Y_{s^-},\\nu _{s^-}, z) \\left( \\mu ^R(\\mathrm {d}s,\\mathrm {d}z) -\\eta ^\\mathbf {Q}(\\mathrm {d}z) \\mathrm {d}s \\right).$ Then we have the following result.", "Lemma 3.1 Let Assumption REF prevail.", "Then the process $\\widetilde{Z}$ is a strictly positive martingale with $\\mathbb {E}^\\mathbf {Q}\\big ( \\widetilde{Z}_T\\big ) =1.$ Define a measure $\\mathbf {P}$ on $\\mathcal {F}_T$ by setting $\\frac{\\mathrm {d}\\mathbf {P}}{\\mathrm {d}\\mathbf {Q}}\\big |_{\\mathcal {F}_T} = \\widetilde{Z}_T$ .", "Then $\\mathbf {P}$ and $\\mathbf {Q}$ are equivalent and, under $\\mathbf {P}$ , the random measure $\\mu ^{R}$ has the compensator $\\eta ^\\mathbf {P}$ .", "The proof of the lemma is postponed to Appendix .", "subsection20mm-0.25cm Filtering equations.", "For a function $f\\colon \\mathcal {E} \\rightarrow \\mathbb {R}$ , we introduce the filter $\\pi (f)$ as the optional projection of the process $f(Y)$ on the filtration $\\mathbb {F}^S$ , i.e.", "$\\pi (f)$ is a càdlàg process such that for all $t \\in [0,T]$ , it holds that $\\pi _t(f)=\\mathbb {E}\\left(f(Y_t)\\mid \\mathcal {F}^S_t\\right)$ .", "Note that $f(Y_t) = \\langle \\mathbf {f},Y_t \\rangle $ for all $t \\in [0,T]$ , where $\\langle \\ , \\ \\rangle $ denotes the scalar product on $\\mathbb {R}^K$ and $\\mathbf {f}_i = f(e_i)$ , $i \\in \\lbrace 1, \\dots , K\\rbrace $ , so that functions of the Markov chain can be identified with $K$ -vectors.", "Let for all $t \\in [0,T]$ and $i \\in \\lbrace 1, \\dots , K\\rbrace $ , $\\pi ^i_t:=\\mathbb {E}\\left({\\mathbf {1}}_{\\lbrace Y_t=e_i\\rbrace }\\mid \\mathcal {F}^S_t\\right)$ .", "Then, we can represent the filter as $\\pi _{t}(f)=\\sum _{i=1}^K \\mathbf {f}_i\\pi ^i_t=\\langle \\mathbf {f},\\pi _t \\rangle , \\quad t \\in [0,T].$ The objective of this section is to derive the dynamics of the process $\\pi = (\\pi ^1, \\dots , \\pi ^K)$ .", "To this end, we first observe that by the Kallianpur-Striebel formula we have $\\displaystyle \\pi _t(f):=\\frac{p_t(f)}{p_t(1)}$ for all $t \\in [0,T]$ , where $p(f)$ denotes the unnormalized version of the filter, which is defined by $p_t(f):=\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}_t \\langle \\mathbf {f},Y_t \\rangle \\mid \\mathcal {F}^S_t\\right), \\ t \\in [0,T]\\,.$ The dynamics of $p(f)$ is given in the next theorem.", "Theorem 3.2 (The Zakai equation) Suppose Assumption REF holds and let $f\\colon \\mathcal {E}\\rightarrow \\mathbb {R}$ .", "Then, for all $t \\in [0,T]$ , the unnormalized filter (REF ) solves the equation: $p_t(f)=\\pi _0(f)+\\int _0^tp_s(Qf)\\mathrm {d}s+\\int _0^t\\int _\\mathbb {R}p_{s^-}\\left( \\beta (z) f\\right)(\\mu ^R(\\mathrm {d}s, \\mathrm {d}z)-\\eta ^\\mathbf {Q}_s(\\mathrm {d}z)\\mathrm {d}s),$ where $p_{t^-}\\left( \\beta (z) f\\right)=\\mathbb {E}^{\\mathbf {Q}}\\left(f(Y_{t^-}) \\widetilde{Z}_{t^-} \\beta (t,Y_{t^-},\\nu _{t^-},z)\\mid \\mathcal {F}^S_t\\right)$ and $p_t(Qf)=\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}_t \\langle Q \\mathbf {f},Y_t \\rangle \\mid \\mathcal {F}^S_t\\right)$ .", "We now provide the general idea of the proof, details are given in Appendix .", "Consider the process $\\widetilde{Z}$ defined in (REF ) and some function $f\\colon \\mathcal {E}\\rightarrow \\mathbb {R}$ .", "Then by Itô's formula the product $\\widetilde{Z}_t f(Y_t)$ has the following $(\\mathbf {Q},\\mathbb {F})$ -semimartingale decomposition $\\widetilde{Z}_t f(Y_t)= &f(Y_0)+ \\int _0^t \\widetilde{Z}_s \\langle Q\\mathbf {f},Y_t \\rangle \\mathrm {d}s+\\int _0^t\\widetilde{Z}_s \\mathrm {d}M^f_s\\\\& + \\int _0^t\\widetilde{Z}_s f(Y_s)\\int _\\mathbb {R}\\beta (s, Y_{s^-}, \\nu _{s^-},z)\\left(\\mu ^R(\\mathrm {d}s, \\mathrm {d}z)-\\eta ^\\mathbf {Q}_{s}(\\mathrm {d}z)\\mathrm {d}s\\right),$ where $M^f=(M^f)_{t \\in [0,T]}$ is the true $(\\mathbb {F}, \\mathbf {Q})$ -martingale appearing in the semimartingale decomposition of $f(Y)$ .", "Taking the conditional expectation with respect to $\\mathcal {F}^S_t$ yields the result, since it can be shown that $\\mathbb {E}^\\mathbf {Q}\\big ( \\int _0^t\\widetilde{Z}_s \\mathrm {d}M^f_s \\mid \\mathcal {F}_t^S \\big ) =0$ .", "We introduce the notation $\\pi _{t^-}(\\eta ^\\mathbf {P}(\\mathrm {d}z)):=\\sum _{i=1}^K\\pi ^i_{t^-}\\eta ^\\mathbf {P}(t,e_i,\\nu _t,\\mathrm {d}z), \\quad t\\in [0,T].$ By applying [16] it is easy to see that $\\pi _{t^-}(\\eta ^\\mathbf {P}(\\mathrm {d}z))\\mathrm {d}t$ provides the $(\\mathbb {F}^S, \\mathbf {P})$ -dual predictable projection of the measure $\\mu ^R$ .", "The next proposition provides the dynamics of the conditional state probabilities.", "Proposition 3.3 The process $\\pi $ solves the $\\pi _t^i=\\pi _0^i+\\int _0^t\\sum _{j=1}^Kq^{ji}\\pi _{s}^j\\mathrm {d}s+\\int _0^t\\int _{\\mathbb {R}} \\pi _{s^-}^i u^i(s,\\nu _{s^-},\\pi _s, z)(\\mu ^R(\\mathrm {d}s, \\mathrm {d}z)-\\pi _{s^-}(\\eta ^{\\mathbf {P}}(\\mathrm {d}z))\\mathrm {d}s),$ for every $t \\in [0,T]$ and $ 1 \\le i \\le K$ , where $\\displaystyle u^i(t,\\nu ,\\pi ,z):=\\frac{(\\mathrm {d}\\eta ^{\\mathbf {P}}(t,e_i,\\nu )/\\mathrm {d}\\eta ^{\\mathbf {Q}})(z)}{\\sum _{j=1}^K \\pi ^j ({\\mathrm {d}\\eta ^{\\mathbf {P}}(t,e_j,\\nu )}/{\\mathrm {d}\\eta ^\\mathbf {Q}} )( z)}-1$ .", "By the Kallianpur-Striebel formula we have that $\\displaystyle \\pi _t(f):=\\frac{p_t(f)}{p_t(1)}$ , for every $t \\in [0,T]$ .", "Then, by (REF ) and Itô formula we get the dynamics of the normalized filter $\\pi (f)$ .", "The claimed result is obtained by setting $f(Y_t)={\\mathbf {1}}_{\\lbrace Y_t=e_i \\rbrace }$ , for every $i \\in \\lbrace 1, \\dots , K\\rbrace $ .", "Note that the filtering equation (REF ) does not depend on the particular choice of $\\eta ^\\mathbf {Q}$ ." ], [ "Filter equations for Example ", "In the following we give the dynamics of $\\pi $ for Example REF .", "For a two-state Markov chain it is sufficient to specify the dynamics of $\\pi = \\pi ^1$ , since $\\pi ^2=1-\\pi ^1$ .", "Define two point processes by $ N_t^{\\text{up}} = \\sum _{T_n \\le t} 1_{\\lbrace \\Delta R_{T_n} = \\theta \\rbrace }$ and $ N_t^{\\text{down}} = \\sum _{T_n \\le t} 1_{\\lbrace \\Delta R_{T_n} = - \\theta \\rbrace } $ , for all $t \\in [0,T]$ , that count the upward and the downward jumps of the return process.", "It is easily seen that for every $(\\nu , \\pi , z)\\in [0, \\nu ^{\\text{max}}]\\times [0,1]\\times \\lbrace -\\theta , \\theta \\rbrace $ , the function $u^1$ is given by $u^1 (\\nu , \\pi , z ) = \\frac{ \\lambda ^+(e_1, \\nu )}{\\pi \\lambda ^+ (e_1, \\nu ) + (1-\\pi ) \\lambda ^+ (e_2, \\nu )} 1_{\\lbrace z = \\theta \\rbrace }+ \\frac{\\lambda ^-(e_1, \\nu )}{\\pi \\lambda ^- (e_1, \\nu ) + (1-\\pi ) \\lambda ^- (e_2, \\nu )} 1_{\\lbrace z = - \\theta \\rbrace }.$ By Corollary REF we then get the following equation for $\\pi _t = \\pi _t^1$ : $\\mathrm {d}\\pi _t &= \\left(q^{11}\\pi _t + q^{21}(1-\\pi _t)\\right)\\mathrm {d}t\\\\&+\\pi _t(1-\\pi _t)\\left((\\lambda ^+(e_1, \\nu _t)+\\lambda ^-(e_1, \\nu _t))- (\\lambda ^+(e_2, \\nu _t)+\\lambda ^-(e_2, \\nu _t))\\right) \\mathrm {d}t\\\\&+ \\pi _{t-} \\Big (\\frac{\\lambda ^+(e_1, \\nu _t)}{\\pi _{t-}\\lambda ^+(e_1, \\nu _t)+(1-\\pi _{t-})\\lambda ^+(e_2, \\nu _t)}-1\\Big )\\mathrm {d}N_t^{\\text{up}}\\\\&+ \\pi _{t-} \\Big (\\frac{\\lambda ^-( e_1, \\nu _t)}{\\pi _{t-}\\lambda ^-(e_1, \\nu _t)+(1-\\pi _{t-})\\lambda ^-(e_2, \\nu _t)}-1\\Big ) \\mathrm {d}N_t^{\\text{down}}.$" ], [ "Control Problem I: Analysis via PDMPs", "We begin with a brief overview of our analysis of the control problem (REF ).", "In Proposition REF below we show that the Kushner-Stratonovich equation (REF ) has a unique solution.", "Then standard arguments ensure that the original control problem under incomplete information is equivalent to a control problem under complete information with state process equal to the $(K+2)$ -dimensional process $X:=(W,S,\\pi )$ .", "This process is a PDMP in the sense of [29], that is a trajectory of $X$ consists of a deterministic part which solves an ordinary differential equation (ODE), interspersed by random jumps.", "Therefore, to solve the optimal liquidation problem we apply control theory for PDMPs.", "This theory is based on the observation that a control problem for a PDMP is discrete in time: loosely speaking, at every jump-time of the process one chooses a control policy to be followed up to the next jump time or until maturity.", "Therefore, one can identify the control problem for the PDMP with a control problem for a discrete-time, infinite-horizon Markov decision model (MDM).", "Using this connection we show that the value function of the optimal liquidation problem is continuous and that is the unique solution of the dynamic programming or optimality equation for the MDM.", "These results are the basis for the viscosity-solution characterization of the value function in Section .", "subsection20mm-0.25cm Optimal liquidation as a control problem for a PDMP.", "From the viewpoint of the trader endowed with the filtration $\\mathbb {F}^S$ , the state of the economic system at time $t\\in [0,T]$ is given by $X_t=(W_t,S_t,\\pi _t)$ .", "Since it is more convenient to work with autonomous Markov processes we include time into the state and define $\\widetilde{X}_t:=(t,X_t)$ .", "The state space of $\\widetilde{X}$ is $\\widetilde{\\mathcal {X}}=[0,T]\\times \\mathcal {X}$ where $ \\mathcal {X}=[0, w_0]\\times \\mathbb {R}^+\\times \\mathcal {S}^K $ with $\\mathcal {S}^K$ being the $K$ -dimensional simplex.", "Let $\\nu $ be the liquidation strategy followed by the trader.", "It follows from (REF ), (REF ), and from the fact that the bid price is a pure jump process that between jump times the state process follows the ODE $\\mathrm {d}\\widetilde{X}_t= g (\\widetilde{X}_t,\\nu _t)\\mathrm {d}t $ , where the vector field $ g (\\widetilde{x},\\nu )\\in \\mathbb {R}^{K+3}$ is given by $ g^1 (\\widetilde{x},\\nu )=1$ , $ g^2 (\\widetilde{x},\\nu )=-\\nu $ , $g^3 (\\widetilde{x},\\nu )=0$ , and for $k=1,\\dots ,K$ , $ g^{k+3} (\\widetilde{x},\\nu )=\\sum _{j=1}^K q^{jk}\\pi ^j-\\pi ^k\\sum _{j=1}^K \\pi ^j \\int _{\\mathbb {R}} u^k(t,\\nu ,\\pi ,z)\\eta ^\\mathbf {P}(t,e_j,\\nu ,\\mathrm {d}z).$ For our analysis we need the following regularity property of $g$ .", "Lemma 4.1 Under Assumption REF , the function $g$ is Lipschitz continuous in $\\widetilde{x}$ uniformly in $(t,\\nu ) \\in [0,T] \\times [0,\\nu ^{\\text{max}}] $ ; the Lipschitz constant is denoted by $K_g$ .", "The proof is postponed to Appendix .", "The jump rate of the state process $\\widetilde{X}$ is given by $\\lambda (\\widetilde{X}_{t^-}, \\nu _{t^-})$ , $t \\in (0,T]$ , where for every $(\\widetilde{x}, \\nu )\\in \\widetilde{\\mathcal {X}}\\times [0, \\nu ^{\\text{max}}]$ , $\\lambda (\\widetilde{x},\\nu )=\\lambda (t,w,s,\\pi ,\\nu ):=\\sum _{j=1}^K \\pi ^j \\eta ^\\mathbf {P}(t,e_j,\\nu ,\\mathbb {R}).$ Next, we identify the transition kernel $Q_{\\widetilde{X}}$ that governs the jumps of $\\widetilde{X}$ .", "Denote by $\\lbrace T_n\\rbrace _{n \\in \\mathbb {N}}$ the sequence of jump times of $\\widetilde{X}$ .", "It follows from (REF ) that for any measurable function $f \\colon \\widetilde{\\mathcal {X}} \\rightarrow \\mathbb {R}^+ $ , $ Q_{\\widetilde{X}} f (\\widetilde{x}, \\nu ) &:= E\\big ( f(\\widetilde{X}_{T_n}) \\mid T_n = t, X_{T_n -} =x, \\nu _{T_n -} = \\nu \\big ) = \\frac{1}{\\lambda (\\widetilde{x},\\nu )} \\overline{Q}_{\\widetilde{X}} f(\\widetilde{x}, \\nu ) \\,,$ where the unnormalized kernel $\\overline{ Q}_{\\widetilde{X}}$ is given by $\\overline{ Q}_{\\widetilde{X}} f (\\widetilde{x}, \\nu ) =\\sum _{j=1}^K \\pi ^j \\int _{\\mathbb {R}} f \\big ( t,w,s(1+z),\\pi ^{1}(1+u^{1}),\\dots ,\\pi ^{K}(1+u^{K}) \\big ) \\eta ^\\mathbf {P}(t,e_j,\\nu ,\\mathrm {d}z).$ Here $u^i$ is short for $u^i (t,\\nu ,\\pi ,z)$ .", "Summarizing, $\\widetilde{X}$ is a PDMP with characteristics given by the vector field $g$ , the jump rate $\\lambda $ and the transition kernel $Q_{\\widetilde{X}}$ .", "It is standard in control theory for PDMPs to work with so-called open-loop controls.", "In the current context this means that the trader chooses at each jump time $T_n<\\tau $ a liquidation policy $\\nu ^n$ to be followed up to $T_{n+1}\\wedge \\tau $ .", "This policy may depend on the state $\\widetilde{X}_{T_n}=(T_n,X_{T_n})$ .", "Definition 4.2 Denote by $\\mathcal {A}$ the set of measurable mappings $\\alpha \\colon [0,T]\\rightarrow [0,\\nu ^{\\text{max}}]$ .", "An admissible open loop liquidation strategy is a sequence of mappings $\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}}$ with $\\nu ^n:\\widetilde{\\mathcal {X}}\\rightarrow \\mathcal {A}$ ; the liquidation rate at time $t$ is given by $\\nu _t=\\sum _{n=0}^\\infty {\\mathbf {1}}_{(T_n\\wedge \\tau , T_{n+1}\\wedge \\tau ]} (t) \\nu ^n (t- T_n,\\widetilde{X}_{T_n}).$ It follows from [16] that an admissible strategy as defined in Section  is of the form given in Definition REF , but for $\\mathcal {F}^S_{T_n}$ measurable mappings $\\nu ^n \\colon \\Omega \\rightarrow \\mathcal {A}$ for every $n \\in \\mathbb {N}$ , that $\\nu ^n $ may depend on the entire history of the system.", "General results for Markov decision models (see [12]) show that the expected profit of the trader stays the same if instead we consider the smaller class of admissible open loop strategies, so that we may restrict ourselves to this class.", "Proposition 4.3 Let Assumption REF hold.", "For every admissible liquidation strategy $\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}}$ and every initial value $\\widetilde{x}$ , a unique PDMP with characteristics $g$ , $\\lambda $ , and $Q_{\\widetilde{X}}$ as above exists.", "In particular the Kushner-Stratonovic equation (REF ) has a unique solution.", "Lemma REF implies that for $\\alpha \\in \\mathcal {A}$ the ODE $\\mathrm {d}\\widetilde{X}_t= g (\\widetilde{X}_t,\\alpha _t)\\mathrm {d}t $ has a unique solution so that between jumps the state process is well-defined.", "At any jump time $T_n$ , $\\widetilde{X}_{T_n}$ is uniquely defined in terms of observable data $(T_n, \\Delta R_{T_n})$ .", "Moreover, since the jump intensity is bounded by $\\lambda ^\\text{max}$ , jump times cannot accumulate.", "Denote by $\\mathbf {P}^{\\lbrace \\nu ^n\\rbrace }_{(t,x)}$ (equiv.", "$\\mathbf {P}^{\\lbrace \\nu ^n\\rbrace }_{\\widetilde{x}}$ ) the law of the state process provided that $X_t=x\\in \\mathcal {X}$ and that the trader uses the open-loop strategy $\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}}$ .", "The reward function associated to an admissible liquidation strategy $\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}} $ is defined by $V\\left(t,x,\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}} \\right)=\\mathbb {E}_{(t,x)}^{\\lbrace \\nu ^n\\rbrace }\\Big ( \\int _t^{\\tau } e^{-\\rho (u-t)} \\nu _u S_u(1-f(\\nu _u)) \\mathrm {d}u + e^{\\rho (\\tau -t)}h\\left(W_{\\tau }\\right)S_{\\tau } \\Big ),$ and the value function of the liquidation problem under partial information is $&V(t,x) =\\sup \\left\\lbrace V\\left(t,x,\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}} \\right)\\colon \\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}} \\ \\text{admissible liquidation strategy} \\right\\rbrace .", "\\quad {}$ Remark 4.4 Note that the compensator $\\eta ^\\mathbf {P}$ and the dynamics of the filter $\\pi $ are independent of the current bid price $s$ , and that the payoff of a liquidation strategy $\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}}$ is positively homogeneous in $s$ .", "This implies that the reward and the value function of the liquidation problem are positively homogeneous in $s$ and, in particular, $ V(t,w,s,\\pi ) = s V(t,w,1,\\pi ).$ subsection20mm-0.25cm Associated Markov decision model.", "The optimization problem in (REF ) is discrete in time since the control policy is chosen at the discrete time points $T_n$ , $n \\in \\mathbb {N}$ , and the value of the state process at these time points forms a discrete-time Markov chain (for $T_n < \\tau $ ).", "Hence (REF ) can be rewritten as a control problem in an infinite horizon Markov decision model.", "The state process of the MDM is given by the sequence $\\lbrace L_n\\rbrace _{n \\in \\mathbb {N}} $ of random variables with $L_n=\\widetilde{X}_{T_n}\\ \\text{ for } \\ T_n<\\tau \\ \\text{ and } \\ L_n=\\bar{\\Delta }\\ \\text{ for } \\ T_n\\ge \\tau , \\quad n\\in \\mathbb {N}\\,,$ where $\\bar{\\Delta }$ is the cemetery state.", "In order to derive the transition kernel of the sequence $\\lbrace L_n\\rbrace _{n \\in \\mathbb {N}} $ and the reward function of the MDM, we introduce some notation.", "For a function $\\alpha \\in \\mathcal {A}$ we denote by $\\widetilde{\\varphi }^{\\alpha }_t(\\widetilde{x})$ or by $\\widetilde{\\varphi }_t (\\alpha , \\widetilde{x})$ the flow of the initial value problem $\\frac{\\mathrm {d}}{\\mathrm {d}t} \\widetilde{X}(t)=g \\big (\\widetilde{X}(t), \\alpha _t\\big )$ with initial condition $ \\widetilde{X}(0) =\\widetilde{x}.$ Whenever we want to make the dependence on time explicit we write $\\widetilde{\\varphi }^{\\alpha }$ in the form $(t,\\varphi ^{\\alpha })$ .", "Moreover, we define the function $\\lambda ^{\\alpha }_u$ by $\\lambda ^{\\alpha }_u(\\widetilde{x}) =\\lambda (\\widetilde{\\varphi }^{\\alpha }_u(\\widetilde{x}),\\alpha _u)=\\lambda ((t+u,\\varphi ^{\\alpha }_u),\\alpha _u)\\, \\quad u \\in [0,T-t],$ and we let $\\Lambda ^{\\alpha }_u(\\widetilde{x}) = \\int _0^u \\lambda ^{\\alpha }_v(\\widetilde{x})\\mathrm {d}v$ .", "Next we take a closer look at the boundary of $\\widetilde{X}$ .", "First note that the process $\\pi $ takes values in the hyperplane $ \\mathcal {H}^K = \\lbrace x \\in \\mathbb {R}^K \\colon \\sum _{i=1}^K x_i = 1 \\rbrace $ , so that $\\widetilde{\\mathcal {X}}$ is contained in the set $\\mathcal {H}= \\mathbb {R}^3 \\times \\mathcal {H}^K$ , which is a hyperplane of $\\mathbb {R}^{K+3}$ .", "When considering the boundary or the interior of the state space we always refer to the relative boundary or the relative interior with respect to $\\mathcal {H}$ .", "Of particular interest to us is the active boundary $\\Gamma $ of the state space, that is the part of the boundary of $\\widetilde{\\mathcal {X}}$ which can be reached by the flow $\\widetilde{\\varphi }^{\\alpha }_\\cdot (\\widetilde{x})$ starting in an interior point $\\tilde{x}\\in \\operatorname{int}(\\widetilde{\\mathcal {X}})$ .", "The boundary of $\\widetilde{\\mathcal {X}}$ can only be reached if $w=0$ , if $t=T$ , or if the filter process reaches the boundary of the $K$ -dimensional simplex.", "The latter is not possible: indeed, if $\\pi _0^i>0$ , then $\\pi _t^i>0$ for all $t\\in [0,T]$ , since there is a positive probability that the Markov chain has not changed its state and since the conditional distribution of $Y_t$ given $\\mathcal {F}_t^S$ is equivalent to the unconditional distribution of $Y_t$ by the Kallianpur-Striebel formula.", "Hence the active boundary equals $\\Gamma = \\Gamma _1 \\cup \\Gamma _2$ , where $ \\Gamma _1 = [0,T] \\times \\lbrace 0\\rbrace \\times (0, \\infty ) \\times \\mathcal {S}^K_0 \\ \\text{ and } \\ \\Gamma _2 = \\lbrace T\\rbrace \\times [0, w_0] \\times (0, \\infty ) \\times \\mathcal {S}^K_0,$ and where $\\mathcal {S}^K_0$ is the interior of $\\mathcal {S}^K$ , i.e.", "$\\mathcal {S}^K_0:=\\lbrace x\\in \\mathcal {S}^K \\colon x_i > 0 \\text{ for all } i\\rbrace $ .", "In (REF ) $\\Gamma _1$ is the lateral part of the active boundary corresponding to an inventory level equal to zero, and $\\Gamma _2$ is the terminal boundary corresponding to the exit from the state space at maturity $T$ .", "In the sequel we denote the first exit time of the flow $\\widetilde{\\varphi }^{\\alpha }_\\cdot (\\widetilde{x})$ from $\\widetilde{\\mathcal {X}}$ by $\\tau ^{\\varphi }=\\tau ^{\\varphi }(\\widetilde{x}, \\alpha )=\\inf \\lbrace u\\ge 0: \\widetilde{\\varphi }^{\\alpha }_u(\\widetilde{x}) \\in \\Gamma \\rbrace \\,.$ Notice that the stopping time $\\tau $ defined in (REF ) corresponds to the first time the state process $\\widetilde{X}$ reaches the active boundary $\\Gamma $ .", "Using similar arguments as in [12] or in [29], it is easily seen that the transition kernel $Q_L$ of the sequence $\\lbrace L_n\\rbrace _{n \\in \\mathbb {N}} $ is given by $Q_L f \\big ( (t,x), \\alpha \\big ) = \\int _0^{ \\tau ^\\varphi (\\widetilde{x})} \\!", "\\!", "e^{- \\Lambda ^\\alpha _u (\\widetilde{x})} \\overline{Q}_{\\widetilde{X}} f(u+t, \\varphi _u(\\widetilde{x}), \\alpha _u \\big ) \\mathrm {d}u + e^{- \\Lambda _{\\tau ^\\varphi }^\\alpha (\\tilde{x})} f(\\bar{\\Delta }) \\,;$ we omit the details.", "Moreover, since the cemetery state is absorbing, $Q_L {\\mathbf {1}}_{\\lbrace \\bar{\\Delta }\\rbrace } (\\bar{\\Delta }, \\alpha )=1$ .", "Finally we define the one-period reward function $r \\colon \\widetilde{\\mathcal {X}}\\times \\mathcal {A}\\rightarrow \\mathbb {R}^+$ by $r(\\widetilde{x}, \\alpha )=\\int _0^{\\tau ^{\\varphi }}e^{-\\rho u}e^{-\\Lambda ^{\\alpha }_u(\\widetilde{x})}\\alpha _u s (1 - f(\\alpha _u))\\mathrm {d}u + e^{-\\rho \\tau ^\\varphi } e^{-\\Lambda ^{\\alpha }_{\\tau ^{\\alpha }}(\\widetilde{x})}h(w_{\\tau ^{\\varphi }})s,$ and $w_{\\tau ^{\\varphi }}$ the inventory-component of $\\widetilde{\\varphi }^{\\alpha }$ , and we set $r(\\bar{\\Delta }) =0$ .", "For an admissible strategy $\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}} $ we set $ J_{\\infty }^{\\lbrace \\nu ^n\\rbrace }(\\widetilde{x})=\\mathbb {E}_{\\widetilde{x}}^{\\lbrace \\nu ^n\\rbrace }\\Big ( \\sum _{n=0}^{\\infty }r\\left( L_n,\\nu ^n(L^n)\\right) \\Big )$ , and $J_{\\infty }(\\widetilde{x}):=\\sup \\left\\lbrace J_{\\infty }^{\\lbrace \\nu ^n\\rbrace }(\\widetilde{x}):\\lbrace \\nu ^n\\rbrace _{n \\in \\mathbb {N}} \\ \\text{admissible liquidation strategy} \\right\\rbrace .$ The next lemma shows that the MDM with transition kernel $Q_L$ and one-period reward $r(L,\\alpha )$ is equivalent to the optimization problem (REF ).", "Lemma 4.5 For every admissible strategy $\\lbrace \\nu ^n\\rbrace $ it holds that $V^{\\lbrace \\nu ^n\\rbrace }=J_{\\infty }^{\\lbrace \\nu ^n\\rbrace }$ .", "Hence $V=J_{\\infty }$ , and the control problems (REF ) and (REF ) are equivalent.", "The proof is similar to the proof of [29] and is therefore omitted.", "subsection20mm-0.25cm The Bellman equation.", "In this section we study the Bellman equation for the value function $V$ .", "Define for $\\alpha \\in \\mathcal {A}$ and a measurable function $v\\colon \\widetilde{\\mathcal {X}}\\rightarrow \\mathbb {R}^+$ the function $\\mathcal {L}v(\\cdot ,\\alpha )$ by $ \\mathcal {L}v( \\widetilde{x},\\alpha )=r(\\widetilde{x},\\alpha )+ Q_L v (\\widetilde{x}, \\alpha ), \\; \\widetilde{x} \\in \\widetilde{\\mathcal {X}}.$ The maximal reward operator $\\mathcal {T}$ is then given by $\\mathcal {T}v(\\widetilde{x})=\\sup _{\\alpha \\in \\mathcal {A}}\\mathcal {L}v(\\widetilde{x},\\alpha )$ .", "Since the one-period reward function is nonnegative we have a so-called positive MDM and it follows from [12] that the value function satisfies the so-called Bellman or optimality equation $ V(\\widetilde{x}) = \\mathcal {T} V(\\widetilde{x}), \\quad \\widetilde{x} \\in \\widetilde{\\mathcal {X}},$ that is $V$ is a fixed point of the operator $\\mathcal {T}$ .", "In order to characterize $V$ as viscosity solution of the HJB equation associated with the PDMP $\\widetilde{X}$ (see Section ) we need a stronger result.", "We want to show: i) that the value function $V$ is the unique fixed point of $\\mathcal {T}$ in a suitable function class $\\mathcal {M}$ ; ii) that for a starting point $v^0 \\in \\mathcal {M} $ iterations of the form $v^{n+1} = \\mathcal {T} v^n$ , $n \\in \\mathbb {N}1$ , converge to $V$ ; and iii) that $V$ is continuous on $\\widetilde{\\mathcal {X}}$ .", "Points i) and ii) follow from the next lemma.", "Lemma 4.6 Define for $\\gamma >0$ , the function $b \\colon \\widetilde{\\mathcal {X}} \\cup \\lbrace \\bar{\\Delta }\\rbrace \\rightarrow \\mathbb {R}^+$ by $b(\\widetilde{x})=b(t,x):=s w e^{\\gamma (T-t)}$ , $\\widetilde{x} \\in \\widetilde{\\mathcal {X}}$ , and $b(\\bar{\\Delta }) = 0$ .", "Then under Assumption REF , $ b$ is a bounding function for the MDM with transition kernel $Q_L$ and reward function $r$ , that is there are constants $c_r, c_Q $ such that for all $(\\widetilde{x},\\alpha )\\in \\widetilde{\\mathcal {X}}\\times \\mathcal {A}$ .", "$ |r(\\widetilde{x},\\alpha )| \\le c_r b(\\widetilde{x}) \\ \\text{ and } \\ Q_L b (\\widetilde{x}, \\alpha ) \\le c_Q b(\\widetilde{x})\\,.$ Moreover, for $\\gamma $ sufficiently large it holds that $c_Q <1$ , that is the MDM is contracting.", "The proof is postponed to Appendix .", "In the sequel we denote by $\\mathcal {B}_b$ the set of functions $\\mathcal {B}_b:=\\big \\lbrace v:\\widetilde{\\mathcal {X}}\\rightarrow \\mathbb {R} \\text{ such that } \\sup \\nolimits _{\\widetilde{x} \\in \\widetilde{\\mathcal {X}}} \\big |{ v(\\widetilde{x}) }{ b(\\widetilde{x})}\\big | <\\infty \\big \\rbrace \\,, $ and we define for $v\\in \\mathcal {B}_b$ the norm $\\Vert v\\Vert _b=\\sup _{\\widetilde{x}\\in \\widetilde{\\mathcal {X}}} |{v(\\widetilde{x})}{ b(\\widetilde{x})}|$ .", "Then the following holds, see [12]: a) $(\\mathcal {B}_b,\\Vert \\cdot \\Vert )_b$ is a Banach space; b) $\\mathcal {T} (\\mathcal {B}_b) \\subset \\mathcal {B}_b$ ; c) $\\Vert \\mathcal {T}v-\\mathcal {T}u \\Vert _b \\le c_Q \\Vert v-u \\Vert _b$ .", "If the MDM is contracting, the maximal reward operator is a contraction on $(\\mathcal {B}_b,\\Vert \\cdot \\Vert )_b$ , and the value function is an element of $\\mathcal {B}_b$ .", "Banach's fixed point theorem thus gives properties i.)", "and ii.)", "above with $\\mathcal {M} = \\mathcal {B}_b$ .", "In order to establish property iii.)", "(continuity of $V$ ) we observe that the set $\\mathcal {C}_b := \\lbrace v \\in \\mathcal {B}_b \\colon \\text{ $v$ is continuous}\\rbrace $ is a closed subset of $(\\mathcal {B}_b,\\Vert \\cdot \\Vert )_b$ , see [12].", "Moreover, we show in Proposition REF that under certain continuity conditions (see Assumptions REF  and REF ), $\\mathcal {T} $ maps $\\mathcal {C}_b$ into itself.", "Hence it follows from Banach's fixed point theorem that $V \\in \\mathcal {C}_b$ .", "Assumption 4.7 1.", "The measure $\\eta ^j(t,\\nu ;\\mathrm {d}z)$ for $j \\in \\lbrace 1, \\dots , K\\rbrace $ is continuous in the weak topology, i.e.", "for all bounded and continuous $\\phi $ , the mapping $ (t, \\nu ) \\mapsto \\int _\\mathbb {R}\\phi (z) \\eta ^j(\\mathrm {d}z) $ is continuous on $[0,T] \\times [0, \\nu ^{\\text{max}}]$ .", "2.", "For the functions $u^j$ introduced in Corollary REF the following holds: for any sequence $\\lbrace (t^n, \\nu ^n, \\pi ^n)\\rbrace _{n \\in \\mathbb {N}}$ with $ (t^n, \\nu ^n, \\pi ^n)\\in [0,T) \\times [0, \\nu ^{\\text{max}}] \\times \\mathcal {S}^K$ for every $n \\in \\mathbb {N}$ , such that $\\displaystyle (t^n, \\nu ^n, \\pi ^n) \\xrightarrow[n \\rightarrow \\infty ]{} (t, \\nu , \\pi )$ , one has $\\lim _{n \\rightarrow \\infty } \\ \\sup \\nolimits _{z \\in {\\rm supp}(\\eta )} |u^j(t^n, \\nu ^n, \\pi ^n,z) - u^j(t, \\nu , \\pi , z) | =0\\,.$ Proposition 4.8 Suppose that Assumptions REF  and REF hold and let $v \\in \\mathcal {C}_b$ .", "Then $\\mathcal {T} v \\in C_b$ .", "Consider some sequence $\\widetilde{x}_n \\rightarrow \\widetilde{x}$ for $n \\rightarrow \\infty $ .", "Since $ | \\mathcal {T}v (\\widetilde{x}_n) - \\mathcal {T}v (\\widetilde{x}) | \\le \\sup _{\\alpha \\in \\mathcal {A}} | \\mathcal {L}v( \\widetilde{x}_n,\\alpha ) -\\mathcal {L}v( \\widetilde{x},\\alpha )|,$ it suffices to estimate the difference $ \\sup _{\\alpha \\in \\mathcal {A}} | \\mathcal {L}v( \\widetilde{x}_n,\\alpha ) -\\mathcal {L}v( \\widetilde{x},\\alpha )|$ .", "First, note that by the Lipschitz continuity of $g$ , established in Lemma REF , we have $\\big | \\widetilde{\\varphi }_t^{\\alpha } (\\widetilde{x}_n) - \\widetilde{\\varphi }_t^{\\alpha } (\\widetilde{x}) \\big | \\le |\\widetilde{x}_n - \\widetilde{x}| + K_g \\int _0^t \\big | \\widetilde{\\varphi }_u^{\\alpha } (\\widetilde{x}_n) - \\widetilde{\\varphi }_u^{\\alpha } (\\widetilde{x}) \\big | \\,\\mathrm {d}u\\,.$ Gronwall inequality hence yields that $\\sup \\nolimits _{t \\in [0,T], \\alpha \\in \\mathcal {A}} \\big | \\widetilde{\\varphi }_t^{\\alpha } (\\widetilde{x}_n) - \\widetilde{\\varphi }_t^{\\alpha } (\\widetilde{x}) \\big | \\le |\\widetilde{x}_n - \\widetilde{x}| e^{K_g T}\\,,$ and thus uniform convergence for $n \\rightarrow \\infty $ of the flow $\\widetilde{\\varphi }^{\\alpha } (\\widetilde{x}_n)$ to $\\widetilde{\\varphi }^{\\alpha } (\\widetilde{x})$ .", "This does however not imply that $\\tau ^{\\varphi _n}$ , the entrance time of $\\widetilde{\\varphi }^{\\alpha } (\\widetilde{x}_n)$ into the active boundary of the state space, converges to $\\tau ^\\varphi $ for $n \\rightarrow \\infty $ .", "To deal with this issue we distinguish two cases: Case 1.", "The flow $\\widetilde{\\varphi }_\\cdot ^{\\alpha } (\\widetilde{x})$ exits the state space $\\widetilde{\\mathcal {X}}$ at the terminal boundary $\\Gamma _2$ (see (REF )).", "This implies that $\\tau ^\\varphi = T-t$ and that the inventory level $w_u$ is strictly positive for $u < T-t$ .", "We therefore conclude from (REF ) that $\\tau ^{\\varphi _n}$ converges to $T-t$ .", "Under Assumptions REF and REF the uniform convergence $\\lim _{n \\rightarrow \\infty } \\sup _{\\alpha \\in \\mathcal {A}} | \\mathcal {L}v( \\widetilde{x}_n,\\alpha ) -\\mathcal {L}v( \\widetilde{x},\\alpha )| =0$ thus follows immediately using the definition of $r$ and the continuity of the mapping $(\\widetilde{x}, \\nu ) \\mapsto \\bar{Q} v(\\widetilde{x}, \\nu )$ established in Lemma REF , see Appendix .", "Case 2.", "The flow $\\widetilde{\\varphi }_\\cdot ^{\\alpha } (\\widetilde{x})$ exits $\\widetilde{\\mathcal {X}}$ at the lateral boundary $\\Gamma _1$ so that $w_{\\tau ^\\varphi } =0$ .", "In that case (REF ) implies that $\\liminf _{n \\rightarrow \\infty } \\tau ^{\\varphi _n} \\ge \\tau ^\\varphi $ ; it is however possible that this inequality is strict.", "We first show continuity of the reward function for that case.", "We decompose $r(\\widetilde{x}_n,\\alpha )$ as follows, setting $\\rho =0$ for simplicity: $ r(\\widetilde{x}_n,\\alpha ) & = s \\int _{0}^{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n} } \\hspace{-5.69046pt} e^{-\\Lambda ^{\\alpha }_u (\\widetilde{x}_n)} \\alpha _{u} (1 - f(\\alpha _u)) \\mathrm {d}u \\\\& + s \\int ^{\\tau ^{\\varphi _n}}_{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n} } \\hspace{-5.69046pt} e^{-\\Lambda ^{\\alpha }_u (\\widetilde{x_n})} \\alpha _{u} (1 - f(\\alpha _u)) \\mathrm {d}u +s e^{-\\Lambda ^{\\alpha }_{\\tau ^{\\varphi _n}}(\\widetilde{x}_n)}h(w_{\\tau ^{\\varphi _n}}).", "$ Now it follows from (REF ) that the integral in (REF ) converges for $n \\rightarrow \\infty $ to $r(\\widetilde{x},\\alpha )$ uniformly in $\\alpha \\in \\mathcal {A}$ .", "The terms in () are bounded from above by $ s w_{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n}}$ ; this can be shown via a similar partial integration argument as in the proof of Lemma REF .", "Moreover, $w_{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n}}$ converges uniformly in $\\alpha \\in \\mathcal {A}$ to $w_{\\tau ^\\varphi }=0 $ , so that () converges to zero.", "Next we turn to the transition kernel.", "We decompose $Q_L v$ : $Q_L v\\big (\\widetilde{x}_n,\\alpha \\big ) &= \\int _0^{ \\tau ^\\varphi \\wedge \\tau ^{\\varphi _n}} \\hspace{-8.5359pt} e^{- \\Lambda _u^{\\alpha } (\\tilde{x}_n)} \\overline{Q} v(\\widetilde{\\varphi }_u(\\widetilde{x}_n), \\alpha _u) \\mathrm {d}u+ \\int ^{\\tau ^{\\varphi _n}}_{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n} } \\hspace{-8.5359pt} e^{- \\Lambda _u^\\alpha (\\tilde{x}_n)} \\overline{Q} v(\\widetilde{\\varphi }_u^{\\alpha }(\\widetilde{x}_n), \\alpha _u \\big ) \\mathrm {d}u\\,.$ For $n \\rightarrow \\infty $ , the first integral converges to $Q_L v\\big (\\widetilde{x},\\alpha \\big )$ using (REF ) and the continuity of the mapping $(\\widetilde{x}, \\nu ) \\mapsto \\overline{Q} v(\\widetilde{x}, \\nu )$ (Lemma REF ).", "To estimate the second term note that $\\overline{Q} v(\\widetilde{x}, \\nu ) \\le \\Vert v\\Vert _b s w \\lambda ({\\widetilde{x},\\nu })$ (as $\\frac{1}{\\lambda }\\bar{Q}$ is a probability transition kernel), so that the integral is bounded by $ \\Vert v\\Vert _b s w_{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n}} \\int ^{\\tau ^{\\varphi _n}}_{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n} } \\hspace{-2.84544pt} \\lambda ^{\\alpha }_u e^{- \\Lambda _u^\\alpha (\\widetilde{x}_n)} \\mathrm {d}u \\le \\Vert v\\Vert _b s w_{\\tau ^{\\varphi }\\wedge \\tau ^{\\varphi _n}}\\,, $ and the last term converges to zero for $n \\rightarrow \\infty $ , uniformly in $\\alpha \\in \\mathcal {A}$ .", "Remark 4.9 Note that existing continuity results for $\\mathcal {L}v( \\cdot ,\\alpha )$ such as [29] make the assumption that the flow $\\varphi ^{\\alpha }$ reaches the active boundary at a uniform speed, independent of the chosen control.", "In order to ensure this hypothesis in our framework we would have to impose a strictly positive lower bound on the admissible liquidation rate.", "This is an economically implausible restriction of the strategy space which is why we prefer to rely on a direct argument.", "We summarize the results of this section in the following theorem.", "Theorem 4.10 Suppose that Assumptions REF  and REF hold.", "Then the value function $V$ is continuous on $\\widetilde{\\mathcal {X}}$ and satisfies the boundary conditions $V(\\widetilde{x}) = 0 $ for $\\widetilde{x}$ in the lateral boundary $\\Gamma _1$ and $V(T, x) = s h(w)$ .", "Moreover, $V$ is the unique solution of the Bellman or optimality equation $ V= \\widetilde{\\mathcal {T}} V$ in $\\mathcal {B}_b$ ." ], [ "Control Problem II: Viscosity Solutions ", "In this section we show that the value function is a viscosity solution of the standard HJB equation associated with the controlled Markov process $(W, \\pi )$ and we derive a comparison principle for that equation.", "These results are crucial to ensure the convergence of suitable numerical schemes for the HJB equation and thus for the numerical solution of the optimal liquidation problem.", "In Section  we provide an example which shows that in general the HJB equation does not admit a classical solution.", "subsection20mm-0.25cm Viscosity solution characterization.", "As a first step we write down the Bellman equation and we use the positive homogeneity of $V$ in the bid price (see Remark REF ) to eliminate $s$ from the set of state variables.", "Define $ \\widetilde{\\mathcal {Y}} = [0,T] \\times [0, w_0] \\times \\mathcal {S}^K $ and denote by $\\operatorname{int} \\widetilde{\\mathcal {Y}}$ and $\\partial \\widetilde{\\mathcal {Y}}$ the relative interior and the relative boundary of $\\widetilde{\\mathcal {Y}}$ with respect to the hyperplane $\\mathbb {R}^2 \\times \\mathcal {H}^K$ .", "For $\\widetilde{y} \\in \\widetilde{\\mathcal {Y}} $ we set $ V^{\\prime } (\\widetilde{y}) = V^{\\prime }(t,w,\\pi ) := V(t, w,1, \\pi )\\,,$ so that the value function satisfies the relation $V(\\widetilde{x}) = s V^{\\prime }(\\widetilde{y})$ .", "For $\\nu \\in [0, \\nu ^{\\text{max}}]$ , $\\widetilde{y} \\in \\widetilde{\\mathcal {Y}}$ , and any measurable function $\\Psi \\colon \\widetilde{\\mathcal {Y}} \\rightarrow \\mathbb {R}^+$ , define $\\overline{ Q}^{\\prime } \\Psi ( \\widetilde{y} , \\nu ) := \\sum _{j=1}^K \\pi ^j \\int _{\\mathbb {R}} (1+z) \\Psi \\left(t,w,(\\pi ^i(1+u^i(t,\\pi ,\\nu ,z)))_{i=1,\\dots ,K} \\right) \\eta ^{\\mathbf {P}}(t,e_j,\\nu ,\\mathrm {d}z)$ and note that $\\overline{Q} V ( \\widetilde{x}, \\nu ) = s \\overline{ Q}^{\\prime } V^{\\prime } (\\widetilde{y},\\nu )$ .", "From now on we denote by $\\widetilde{\\varphi }_u^\\alpha (\\widetilde{y}) $ the flow of the vector field $g$ with price component $g^3$ omitted, and we write $\\tau ^{\\varphi }$ for the first time this flow reaches the active boundary of $\\widetilde{\\mathcal {Y}}$ given by $\\Gamma := [0,T] \\times \\lbrace 0\\rbrace \\times \\mathcal {S}^K_0 \\cup \\lbrace T\\rbrace \\times [0, w_0] \\times \\mathcal {S}^K_0\\,$ of $\\widetilde{\\mathcal {Y}}$ .", "By positive homogeneity, the Bellman equation for $V$ reduces to the following optimality equation for $V^\\prime $ : $\\begin{split}V^\\prime (\\widetilde{y}) &= \\sup _{\\alpha \\in A} \\Big \\lbrace \\int _0^{\\tau ^{\\varphi } } \\hspace{-2.84544pt}e^{-(\\rho u +\\Lambda ^\\alpha _u(\\widetilde{y}))} \\big ( \\alpha _u (1 - f(\\alpha _u)) + \\overline{ Q}^\\prime V^\\prime ( \\widetilde{\\varphi }_u^\\alpha (\\widetilde{y}) , \\alpha _u ) \\big ) \\mathrm {d}u \\\\& \\quad + e^{- (\\rho \\tau ^{\\varphi }+ \\Lambda _{\\tau ^{\\varphi }}^\\alpha (\\widetilde{y}) )} h (w_{\\tau ^{\\varphi }} ) \\Big \\rbrace .\\end{split}$ For $\\Psi \\colon \\widetilde{\\mathcal {Y}} \\rightarrow \\mathbb {R}^+$ bounded, define the function $\\ell ^\\Psi \\colon \\widetilde{\\mathcal {Y}} \\times [0, \\nu ^{\\text{max}}] \\rightarrow \\mathbb {R}^+ $ and the operator $\\mathcal {T}^{\\prime }$ by $ \\ell ^\\Psi (\\widetilde{y}, \\nu ) &= \\nu (1- f(\\nu )) + \\overline{Q}^{\\prime } \\Psi ( \\widetilde{y} , \\nu )\\,, \\\\\\mathcal {T}^\\prime \\Psi (\\widetilde{y}) &= \\sup _{\\alpha \\in A} \\Big \\lbrace \\int _0^{\\tau ^{\\varphi } } \\hspace{-5.69046pt}e^{-(\\rho u + \\Lambda ^\\alpha _u(\\widetilde{y}))} \\ell ^{\\Psi }\\big (\\widetilde{\\varphi }_u^\\alpha (\\widetilde{y}), \\alpha _u) \\mathrm {d}u+ e^{- (\\rho \\tau ^{\\varphi }+ \\Lambda _{\\tau ^{\\varphi }}^\\alpha (\\widetilde{y}) )}h(w_{\\tau ^{\\varphi }}) \\Big \\rbrace .$ Note that for fixed $\\Psi $ , $v^\\Psi := \\mathcal {T}^\\prime \\Psi $ is the value function of a deterministic exit-time optimal control problem with instantaneous reward $\\ell ^\\Psi $ and boundary value $h$ .", "Viscosity solutions for this problem are studied extensively in [8].", "Moreover, the optimality equation (REF ) for $V^{\\prime }$ can be written as the fixed point equation $V^\\prime = \\mathcal {T}^\\prime V^\\prime $ .", "[30] observed that this can be used to obtain a viscosity solution characterization of the value function in a PDMP control problem, and we now explain how this idea applies in our framework.", "Define for $\\Psi \\colon \\widetilde{\\mathcal {Y}} \\rightarrow \\mathbb {R}^+$ the function $ F_\\Psi \\colon \\widetilde{\\mathcal {Y}} \\times \\mathbb {R}^+ \\times \\mathbb {R}^{K+2} \\rightarrow \\mathbb {R}$ by $ F_\\Psi ( \\widetilde{y}, v, p) = - \\sup \\big \\lbrace -(\\rho +\\lambda ( \\widetilde{y}, \\nu )) v + g(\\widetilde{y}, \\nu )^{\\prime } p + \\ell ^{\\Psi } (\\widetilde{y}, \\nu ) \\colon \\nu \\in [0, \\nu ^{\\text{max}}] \\big \\rbrace \\,.$ The dynamic programming equation associated with the control problem () is $ F_{\\Psi } \\big ( \\widetilde{y}, v^\\Psi (\\widetilde{y}), \\nabla v^\\Psi (\\widetilde{y}) \\big ) = 0 \\text{ for } \\widetilde{y} \\in \\text{int} \\widetilde{\\mathcal {Y}}, \\quad v^\\Psi (\\widetilde{y}) = h(\\widetilde{y}) \\text{ for } \\widetilde{y} \\in \\partial \\widetilde{\\mathcal {Y}} \\,.$ Moreover, since $V^\\prime = \\mathcal {T}^\\prime V^\\prime $ , we expect that $V^{\\prime }$ solves in a suitable sense the equation $ F_{V^{\\prime }} \\big ( \\widetilde{y}, V^{\\prime }(\\widetilde{y}), \\nabla V^{\\prime } (\\widetilde{y}) \\big ) = 0, \\text{ for } \\widetilde{y} \\in \\text{int} \\widetilde{\\mathcal {Y}}, \\quad V^{\\prime }(\\widetilde{y}) = h(\\widetilde{y}) \\text{ for } \\widetilde{y} \\in \\partial \\widetilde{\\mathcal {Y}}.$ Remark 5.1 Notice that, equations (REF ) and (REF ) differs in the sense that in (REF ) the function $F_\\Psi $ enters with $\\Psi $ fixed, whereas in (REF ) one works with the function $F_{V^{\\prime }}$ .", "This reflects the fact that control problem () associated to equation (REF ) has an exogenously given running cost, while in the optimization problem (REF ), leading to equation (REF ), function $V^{\\prime }$ is the solution of a fixed point equation, and therefore the running cost is endogenous.", "There are two issues with equations (REF ) and (REF ): $v^\\Psi $ and $V^{\\prime }$ are typically not $\\mathcal {C}^1$ functions, and the value of these functions on the non-active part $\\partial \\widetilde{Y} \\setminus \\Gamma $ of the boundary is determined endogenously.", "Following [8] we therefore work with the following notion of viscosity solutions.", "Definition 5.2 1.", "A bounded upper semi-continuous (u.s.c.)", "function $v$ on $\\widetilde{\\mathcal {Y}}$ is a viscosity subsolution of (REF ), if for all $\\phi \\in \\mathcal {C}^1(\\widetilde{\\mathcal {Y}}) $ and all local maxima ${\\widetilde{y}}_0 \\in \\widetilde{\\mathcal {Y}} $ of $v - \\phi $ one has $ \\begin{split}F_{\\Psi } \\big ( \\widetilde{y}_0, v(\\widetilde{y}_0), \\nabla \\phi (\\widetilde{y}_0) \\big ) &\\le 0 \\text{ for }\\widetilde{y}_0 \\in \\text{int} \\widetilde{\\mathcal {Y}} , \\\\\\min \\big \\lbrace F_{\\Psi } \\big ( \\widetilde{y}_0, v(\\widetilde{y}_0), \\nabla \\phi (\\widetilde{y}_0) \\big ), v(\\widetilde{y}_0) - h(\\widetilde{y}_0 )\\big \\rbrace & \\le 0 \\text{ for } \\widetilde{y}_0 \\in \\partial \\widetilde{\\mathcal {Y}} .\\end{split}$ A bounded lower semi-continuous (l.s.c.)", "function $u$ on $\\widetilde{\\mathcal {Y}}$ is a viscosity supersolution of (REF ), if for all $\\phi \\in \\mathcal {C}^1(\\widetilde{\\mathcal {Y}}) $ and all local minima ${\\widetilde{y}}_0 \\in \\widetilde{\\mathcal {Y}} $ of $u - \\phi $ one has $ \\begin{split}F_{\\Psi } \\big ( \\widetilde{y}_0, u(\\widetilde{y}_0), \\nabla \\phi (\\widetilde{y}_0) \\big ) &\\ge 0 \\text{ for }\\widetilde{y}_0 \\in \\text{int} \\widetilde{\\mathcal {Y}} , \\\\\\max \\big \\lbrace F_{\\Psi } \\big ( \\widetilde{y}_0, u(\\widetilde{y}_0), \\nabla \\phi (\\widetilde{y}_0) \\big ), u(\\widetilde{y}_0) - h(\\widetilde{y}_0 )\\big \\rbrace & \\ge 0 \\text{ for } \\widetilde{y}_0 \\in \\partial \\widetilde{\\mathcal {Y}}.\\end{split}$ A viscosity solution $v^\\Psi $ of (REF ) is either a continuous function on $\\widetilde{\\mathcal {Y}}$ that is both a sub and a supersolution of (REF ), or a bounded function with u.s.c.", "and l.s.c.", "envelopes that are a sub and a supersolution of (REF ).", "2.", "A bounded u.s.c.", "function $v$ on $\\widetilde{\\mathcal {Y}}$ is a viscosity subsolution of (REF ), if the relation (REF ) holds for $F= F_v$ .", "Similarly, a bounded l.s.c.", "function $u$ on $\\widetilde{\\mathcal {Y}}$ is a viscosity supersolution of (REF ), if (REF ) holds for $F= F_u$ .", "Finally, $V^{\\prime }$ is a viscosity solution of (REF ), if it is both a sub and a supersolution of that equation.", "Note that Definition REF allows for the case that $v^\\Psi (\\widetilde{y}_0 ) \\ne h(\\widetilde{y}_0)$ for certain boundary points $\\widetilde{y}_0\\in \\partial \\widetilde{\\mathcal {Y}}$ .", "In particular, if $F_{\\Psi } \\big ( \\widetilde{y}_0, v^\\Psi (\\widetilde{y}_0), \\nabla v^\\Psi (\\widetilde{y}_0) \\big ) =0 $ in the viscosity sense, (REF ) and (REF ) hold irrespectively of the value of $h(\\widetilde{y}_0)$ .", "Theorem 5.3 Suppose that Assumptions REF and REF hold.", "Then the value function $V^{\\prime }$ is a continuous viscosity solution of (REF ) in $\\widetilde{\\mathcal {Y}}$ .", "Moreover, a comparison principle holds for (REF ): if $v\\ge 0 $ is a subsolution and $u \\ge 0 $ a supersolution of (REF ) such that $v(\\widetilde{y})/w$ and $u(\\widetilde{y})/w$ are bounded on $\\widetilde{\\mathcal {Y}}$ and such that $v = u = h$ on the active boundary $\\Gamma $ of $\\widetilde{\\mathcal {Y}}$ , then $v \\le u $ on $\\operatorname{int}\\widetilde{\\mathcal {Y}}$ .", "It follows that $V^{\\prime } $ is the only continuous viscosity solution of (REF ).", "First, by Theorem REF , $V^{\\prime }$ is continuous.", "Moreover, [8] implies that $V^{\\prime }$ is a viscosity solution of (REF ) with $\\Psi = V^\\prime $ and hence of equation (REF ).", "Next we prove the comparison principle.", "In order to establish the inequality $v \\le u$ we use an inductive argument based on the monotonicity of $\\mathcal {T}^\\prime $ and on a comparison result for (REF ).", "Let $u_0 := u$ and define $u_1=\\mathcal {T}u_0$ .", "It follows from [8] that $u_1$ is a viscosity solution of (REF ) with $\\Psi = u_0$ .", "Moreover, $u_1(\\widetilde{y})/w$ is bounded on $\\widetilde{\\mathcal {Y}}$ so that $u_1 = h$ on $\\Gamma $ .", "Since $u_0$ is a supersolution of (REF ) it is also a supersolution of (REF ) with $\\Psi = u_0$ .", "[8] gives the inequality $u_1 \\le u_0$ on $\\operatorname{int}\\widetilde{ \\mathcal {Y}}$ , since the functions $u^+$ and $u^-$ defined in that theorem coincide in our case.", "Define now inductively $u_{n}=\\mathcal {T}^{\\prime } u_{n-1}$ , and suppose that $u_{n}\\le u_{n-1}$ .", "Then, using the monotonicity of $\\mathcal {T}^{\\prime }$ , we have $u_{n+1}=\\mathcal {T}^{\\prime } u_n \\le \\mathcal {T}^{\\prime } u_{n-1}=u_n.$ This proves that $u_{n +1}\\le u_{n} $ for every $n\\in \\mathbb {N}$ .", "Moreover, as explained in Section , the sequence $\\lbrace u_n\\rbrace _{n \\in \\mathbb {N}}$ converges to $V^\\prime $ , so that $u_n \\ge V^\\prime $ for all $n$ .", "In the same way we can construct a sequence of functions $\\lbrace v^n\\rbrace $ with $v_0 =v$ such that $v^n \\uparrow V^{\\prime }$ , and we conclude that $v \\le V^\\prime \\le u$ .", "The remaining statements are clear.", "Remark 5.4 Note that the results in [30] do not apply directly in our case since their assumptions on the behaviour of the vector field $g$ on the lateral boundary are not satisfied in our model, see also the discussion in Remark REF .", "Moreover, [30] do not give a comparison principle for (REF ).", "Finally, we write the dynamic programming equation (REF ) explicitly.", "To this end, we use the fact that $\\lambda ( \\widetilde{y}, \\nu ) = \\sum _{k=1}^K \\pi ^k \\eta ^\\mathbf {P}(t,e_k,\\nu ,\\mathbb {R})$ , the definition of $g$ , and the definition of $l^{V^{\\prime }}$ in (REF ) to obtain $0 &= \\frac{\\partial V^{\\prime }}{\\partial t}(t,w,\\pi ) + \\sup \\big \\lbrace H (\\nu , t,w,\\pi , V^{\\prime }, \\nabla V^{\\prime }) \\colon \\nu \\in [0, \\nu ^{\\text{max}}]\\big \\rbrace \\,, \\text{ with }$ $ \\begin{split}H (\\nu , t,w,\\pi , V^{\\prime }, \\nabla V^{\\prime }) &= -\\rho V^{\\prime } + \\nu (1-f(\\nu )) - \\nu \\frac{\\partial V^{\\prime }}{\\partial w}(t,w,\\pi ) \\\\&+\\sum _{k,j=1}^K\\frac{\\partial V^{\\prime }}{\\partial \\pi ^k}(t,w,\\pi )\\pi ^j \\Big ( q^{jk}- \\pi ^k\\int _{\\mathbb {R}} u^k(t,\\nu ,\\pi , z) \\eta ^{\\mathbf {P}}(t,e_j,\\nu ,\\mathrm {d}z)\\Big )\\\\&+\\sum _{j=1}^K \\pi ^j\\int _\\mathbb {R}\\Delta V^{\\prime }(t,w,\\pi ,z) \\eta ^{\\mathbf {P}}(t,e_j,\\nu ,\\mathrm {d}z) ,\\end{split}$ and $\\Delta V^{\\prime }(t,w,\\pi ,z):=(1+z) V^{\\prime }\\left(t,w, (\\pi ^{i}(1+u^{i}(t,\\nu ,\\pi ,z)))_{i=1,...,K}\\right)- V^{\\prime }(t,w,\\pi ).$ This equation coincides with the standard HJB equation associated with the controlled Markov process $(W,\\pi )$ .", "The advantage of using viscosity solution theory is that we are able to give a mathematical meaning to this equation even if $V^{\\prime }$ is merely continuous.", "This is relevant in our context.", "Indeed, in the next section we present a simple example where $V^{\\prime }$ is not $\\mathcal {C}^1$ .", "subsection20mm-0.25cm A counterexample.", "We now give an example within a setup where the value function is a viscosity solution of the dynamic programming equation but not a classical solution.", "Precisely, we work in the context of Example REF with linear permanent price impact and deterministic compensator $\\eta ^\\mathbf {P}$ .", "For simplicity we let $\\rho =0$ , $s=1$ , $h(w)\\equiv 0$ , $f(\\nu ) \\equiv 0$ (zero terminal liquidation value and no temporary price impact).", "Moreover, we assume that $c^{\\text{up}}<c^{\\text{down}}$ .", "The function $\\overline{\\eta }^\\mathbf {P}$ from (REF ) is thus given by $\\overline{\\eta }^\\mathbf {P}(\\nu ):=\\theta (c^{\\text{up}}-c^{\\text{down}}(1+a\\nu ))$ and $\\overline{\\eta }^\\mathbf {P}(\\nu ) < 0$ for $\\nu >0$ .", "It follows that $S^{\\nu }$ is a supermartingale for any admissible $\\nu $ , and we conjecture that it is optimal to sell as fast as possible to reduce the loss due to the falling bid price.", "Denote by $\\tau (w):= w/\\nu ^{\\text{max}}$ the minimal time necessary to liquidate the inventory $w$ .", "The optimal strategy is thus given by $\\nu ^*_t=\\nu ^{{\\max }}{\\mathbf {1}}_{\\left[0, \\tau (w_0)\\wedge T\\right]}(t)$ .", "Moreover, for $t< \\tau (w_0) \\wedge T$ one has $\\overline{\\eta }^\\mathbf {P}(\\nu _t)=\\overline{\\eta }^\\mathbf {P}(\\nu ^{\\max })$ and $\\mathbb {E}(S_t^*) = \\exp \\big (t \\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}})\\big )$ .", "Hence we get that $J(\\nu ^*) = \\int _0^{\\tau (w_0) \\wedge T} \\hspace{-2.84544pt}\\nu ^{\\max } \\exp \\big ( u \\, \\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}})\\big )\\mathrm {d}u\\,.$ Solving this integral we get the following candidate for the value function $V^{\\prime }(t, w) : =\\frac{\\nu ^{\\text{max}}}{\\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}})} \\Big \\lbrace \\exp \\big ( \\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}}) (\\tau (w) \\wedge (T-t))\\big ) - 1 \\Big \\rbrace , \\quad (t,w)\\in [0,T]\\times [0, \\nu ^{\\text{max}}].$ In order to verify that $V^{\\prime }$ is in fact the value function we show that $V^{\\prime }$ is a viscosity solution of the HJB equation REF .", "In the current setting this equation becomes $-\\frac{\\partial V^{\\prime }}{\\partial t}-\\sup \\Big \\lbrace \\nu -\\nu \\frac{\\partial V^{\\prime }}{\\partial w}-\\overline{\\eta }^\\mathbf {P}(\\nu )V^{\\prime } \\colon {\\nu \\in [0, \\nu ^{\\max }]} \\Big \\rbrace =0.$ First note that $V^{\\prime }$ satisfies the correct terminal and boundary conditions.", "Define the set $ G:= \\lbrace (t,w) \\in [0,T] \\times [0,w_0] \\colon \\tau (w) = (T-t) \\rbrace \\,.$ The function $V^{\\prime }$ is $\\mathcal {C}^1$ on $ [0,T] \\times [0,w_0] \\setminus G$ , and it is a classical solution of (REF ) on this set.", "However $V^{\\prime }$ is not differentiable on $G$ and hence not a classical solution everywhere.", "Fix some point $(\\overline{t}, \\overline{w}) \\in G$ .", "In order to show that $V^{\\prime }$ is a viscosity solution of the (REF ) we need to verify the subsolution property in this point.", "(For the supersolution property there is nothing to show as there is no $\\mathcal {C}^1$ -function $\\phi $ such that $V^{\\prime }-\\phi $ has a local minimum in $(\\overline{t},\\overline{w})$ .)", "Consider $\\phi \\in \\mathcal {C}^1$ such that $V^{\\prime }-\\phi $ has a local maximum in $(\\overline{t},\\overline{w})$ .", "By considering the left and right derivatives of the functions $t \\mapsto (V^{\\prime }-\\phi ) (t,\\overline{w})$ respectively $w \\mapsto (V^{\\prime }-\\phi ) (\\overline{t}, w)$ we get the following inequalities for the partial derivatives of $\\phi $ $ -\\nu ^{\\max }e^{\\overline{\\eta }^\\mathbf {P}(\\nu ^{\\max })(T-\\overline{t})}\\le \\frac{\\partial \\phi }{\\partial t} (\\overline{t}, \\overline{w}) \\le 0 \\; \\text{ and }\\;0 \\le \\frac{\\partial \\phi }{\\partial w} (\\overline{t}, \\overline{w}) \\le \\exp \\big ( \\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}}) \\tau (\\overline{w}) \\big ) \\,.$ Moreover, it holds on $G$ that $V^{\\prime }(t,w )=\\frac{\\nu ^{\\max }}{\\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}})} \\big \\lbrace \\exp \\big ( \\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}})(T-t)\\big ) -1\\big \\rbrace $ .", "As $w = \\nu ^{\\text{max}}(T-t)$ on $G$ , differentiating with respect to $t$ gives that $ \\Big ( \\frac{\\partial \\phi }{\\partial t}-\\nu ^{\\max }\\frac{\\partial \\phi }{\\partial w} \\Big )(\\overline{t}, \\overline{w}) =-\\nu ^{\\text{max}}\\exp \\big (\\overline{\\eta }^\\mathbf {P}(\\nu ^{\\text{max}})(T-\\overline{t})\\big ) .$ Applying the inequalities for $\\frac{\\partial \\phi }{\\partial w}$ we get that $\\sup \\Big \\lbrace \\nu -\\nu \\frac{\\partial \\phi }{\\partial w}+\\overline{\\eta }^\\mathbf {P}(\\nu )V^{\\prime } \\colon {\\nu \\in [0,\\nu ^{\\max }]}\\Big \\rbrace &=\\nu ^{\\max }\\Big (-\\frac{\\partial \\phi }{\\partial w}+ e^{\\overline{\\eta }^\\mathbf {P}(\\nu ^{\\max })(T-t)}\\Big ).$ Using (REF ) this gives $-\\frac{\\partial \\phi }{\\partial t}-\\sup \\big \\lbrace \\nu -\\nu \\frac{\\partial \\phi }{\\partial w}+\\overline{\\eta }^\\mathbf {P}(\\nu )V^{\\prime } \\colon \\nu \\in [0,\\nu ^{\\max }]\\big \\rbrace =0$ and hence the subsolution property.", "Remark 5.5 It can be shown that for $\\nu ^{\\max }\\rightarrow \\infty $ the value function $V^{\\prime }$ from (REF ) converges to $V^{\\prime , \\infty } (t,w) :=- \\frac{1}{\\theta c^{\\text{down}}a} \\big ( \\exp (-w \\theta a c^{\\text{down}}-1 \\big )$ and that $V^{\\prime , \\infty } $ is a strict (classical) supersolution of equation (REF ).", "Hence $V^{\\prime , \\infty } $ is the value function of the optimal liquidation problem for $\\nu ^{\\text{max}}= \\infty $ , and we conclude that without an upper bound on the liquidation rate the viscosity-solution characterization from Theorem REF does not hold in general." ], [ "Examples and numerical results", "In this section we study the optimal liquidation rate and the expected liquidation profit in our model.", "For concreteness we work in the framework of Example REF , that is the example where $\\eta ^\\mathbf {P}$ depends on the liquidation strategy as well as on a two-state Markov chain.", "We focus on two different research questions: (i) the influence of model parameters on the form of the optimal liquidation rate; (ii) the additional liquidation profit from the use of stochastic filtering and a comparison to classical approaches.", "Moreover, we report the results of a small calibration study." ], [ "Numerical method.", "Since the HJB equation cannot be solved analytically, we resort to numerical methods.", "We apply an explicit finite difference scheme to solve the HJB equation and to compute the corresponding liquidation strategy.", "First, we turn the HJB equation into an initial value problem via time reversion.", "Given a time discretization $0=t_0<\\dots <t_k<\\dots <t_m=T$ we set $V^{\\prime }_{t_0}=h$ , and given $V^{\\prime }_{t_k}$ , we approximate the liquidation strategy as follows: $\\nu _{t_k}^\\ast (w,\\pi ):= \\operatorname{arg max}_{\\nu \\in [0, \\nu ^{\\text{max}}]} H (\\nu , t_k,w,\\pi , V^{\\prime }_{t_k}, \\nabla ^{\\text{disc}} V^{\\prime }_{t_k}) \\,,$ where $\\nabla ^{\\text{disc}}$ is the gradient operator with derivatives replaced by suitable finite differences.", "In a slight abuse of language we refer to $\\nu _{t_k}^\\ast $ from (REF ) as the optimal liquidation rate.", "With this we obtain the next time iterate of the value function, $V^{\\prime }_{t_{k+1}}=V^{\\prime }_{t_k}+(t_{k+1}-t_k)\\,H (\\nu _{t_k}^\\ast , t_k,w,\\pi , V^{\\prime }_{t_k}, \\nabla ^{\\text{disc}} V^{\\prime }_{t_k})\\,.$ Since the comparison principle holds, as shown in Theorem REF , and the value function is the unique viscosity solution of our HJB equation, we get convergence of the proposed procedure to the value function by similar arguments as in [9], [28]; details are omitted.", "subsection20mm-0.25cm Optimal liquidation rate.", "We start by computing the optimal liquidation rate $\\nu _{t_k}^\\ast $ for Example REF , assuming that the temporary price impact is of the form $f(\\nu ) = c_f\\nu ^\\varsigma $ for $\\varsigma >0$ .", "Since $\\pi _t^1+\\pi ^2_t=1$ for all $t\\in [0,T]$ , we can eliminate the process $\\pi ^2$ from the set of state variables.", "In the sequel we denote by $\\pi _t$ the conditional probability of being in the good state $e_1$ at time $t$ and by $V^{\\prime }(t,w,\\pi )$ the value function evaluated at the point $(t,w,(\\pi ,1-\\pi ))$ .", "To compute $\\nu _{t_k}^\\ast $ we substitute the functions $u_i$ given in (REF ) and the dynamics of the process $(\\pi _t)_{t \\in [0,T]} $ from (REF ) into the general HJB equation (REF ).", "Denote by $\\pi ^{\\text{post}}_t = \\frac{\\pi _t c^{\\text{down}}_1}{\\pi _t c^\\text{down}_1+(1-\\pi _t) c^{\\text{down}}_2}, \\quad t \\in [0,T],$ the updated (posterior) probability of state $e_1$ given that a downward jump occurs at $t$ .", "Moreover, denote the discretized partial derivatives of $V^{\\prime }$ appearing in (REF ) by $ \\frac{\\delta V^{\\prime }}{\\delta w}$ and $ \\frac{\\delta V^{\\prime }}{\\delta \\pi }$ .", "Substitution into (REF ) leads to $\\nu ^\\ast _{t_k} &= \\operatorname{arg max}_{\\nu \\in [0, \\nu ^{\\text{max}}]} \\big \\lbrace \\nu (1-c_f\\nu ^\\varsigma ) - \\nu C(t_k,w,\\pi ) \\big \\rbrace , \\;\\text{ with } \\\\C (t_k,w,\\pi ) &= \\frac{\\delta V^{\\prime }}{\\delta w} (t_k,w,\\pi ) + \\frac{\\delta V^{\\prime }}{\\delta \\pi }(t_k,w,\\pi ) \\pi (1-\\pi ) a(c^\\text{down}_1 - c^\\text{down}_2) \\\\& - \\big \\lbrace (1-\\theta ) V^{\\prime }(t_k,w,\\pi ^{\\text{post}})-V^{\\prime }(t_k,w,\\pi )\\big \\rbrace (\\pi c^\\text{down}_1+(1-\\pi )c^\\text{down}_2)a\\,.$ Maximizing (REF ) with respect to $\\nu $ , we get that $\\nu ^\\ast _{t_k} = 0$ if $C (t_k,w,\\pi ) > 1$ ; for $C (t_k,w,\\pi ) \\le 1$ one has $\\nu ^\\ast _{t_k} = \\widetilde{\\nu }^\\ast \\wedge \\nu ^{\\text{max}}$ , where $\\widetilde{\\nu }^\\ast $ solves the equation $ 1- c_f(\\varsigma +1) \\nu ^\\varsigma = C(t_k,w,\\pi )\\,.$ This characterization of $\\nu ^\\ast _{t_k}$ is very intuitive: $1-c_f(\\varsigma +1) \\nu ^\\varsigma $ gives the marginal liquidation benefit due to an increase in $\\nu $ and $C(t_k,w,\\pi )$ can be viewed as marginal cost of an increase in $\\nu $ (see below).", "For $C (t_k,w,\\pi ) \\le 1$ , $\\widetilde{\\nu }^\\ast $ is found by equating marginal benefit and marginal cost; for $C (t_k,w,\\pi ) > 1$ the marginal benefit is smaller than the marginal cost for all $\\nu \\ge 0$ and $\\nu ^*_{t_k} = 0$ .", "The optimal liquidation rate $\\nu ^*_{t_k}$ is thus determined by the marginal cost $C(t_k,w, \\pi )$ , and we now give an economic interpretation of the terms in ().", "First, $\\frac{\\delta V^{\\prime }}{\\delta w} $ is a marginal opportunity cost, since selling inventory reduces the amount that can be liquidated in the future.", "Moreover, it holds that $\\nonumber - \\big ( (1-\\theta ) V^{\\prime }(t_k,w,\\pi ^{\\text{post}})-V^{\\prime }(t_k,w,\\pi )\\big ) &= \\theta V^{\\prime }(t_k,w,\\pi ^{\\text{post}}) - \\big ( V^{\\prime }(t_k,w, \\pi ^{\\text{post}}) - V^{\\prime }(t_k,w, \\pi )\\big )\\,.$ The term $\\theta V^{\\prime }(t_k,w,\\pi ^{\\text{post}})$ gives the reduction in the expected liquidation value due to a downward jump in the return process, and $(\\pi c^\\text{down}_1+(1-\\pi )c^\\text{down}_2)a$ is the marginal increase in the intensity of a downward jump, so that the term $ \\theta V(t_k,w,\\pi ^{\\text{post}}) (\\pi c^\\text{down}_1+(1-\\pi )c^\\text{down}_2)a$ measures the marginal cost due to permanent price impact; in the sequel we refer to (REF ) as illiquidity cost.", "Finally, note that $\\pi ^{\\text{post}}- \\pi = \\frac{ \\pi (1-\\pi ) (c^\\text{down}_1 - c^\\text{down}_2)}{\\pi c^\\text{down}_1+(1-\\pi )c^\\text{down}_2} \\,.$ Hence the remaining terms in () are equal to $ - \\Big ( V^{\\prime }(t_k,w, \\pi ^{\\text{post}}) - V^{\\prime }(t_k,w, \\pi ) - \\frac{\\delta V^{\\prime }}{\\delta \\pi }(t_k,w,\\pi )(\\pi ^{\\text{post}}- \\pi ) \\Big ) a(\\pi c^{\\text{down}}_1+(1-\\pi )c^{\\text{down}}_2)\\,.$ Simulations indicate that $V^{\\prime }$ is convex in $\\pi $ ; this is quite natural as it implies that uncertainty about the true state reduces the optimal liquidation value.", "It follows that (REF ) is negative which leads to an increase in the optimal liquidation rate (REF ).", "Since $\\pi ^{\\text{post}}- \\pi $ is largest for $\\pi \\approx 0.5$ , this effect is most pronounced if the investor is uncertain about the true state.", "Hence (REF ) can be viewed as an uncertainty correction that makes the trader sell faster if he is uncertain about the true state." ], [ "Numerical analysis and varying price impact parameters.", "To gain further insight into the structure of the optimal liquidation rate we resort to numerical experiments.", "We work with the parameter set given in Table REF .", "Moreover, we set the liquidation value $h(w)\\equiv 0$ , that is we assume that block transactions at the horizon date are prohibitively expensive.", "Without loss of generality we set $s=1$ , so that the expected liquidation profit is equal to $V^{\\prime }$ .", "Table: Parameter values used in numerical experiments.First, we discuss the form of the optimal liquidation rate for varying size of the temporary price impact, that is for varying $c_f$ , keeping the permanent price impact parameter $a$ constant at the moderate value $a= 7\\times 10^{-6}$ .", "Figure REF shows the liquidation rate at $t=0$ for the cases of no, intermediate, and large temporary price impact as a function of $w$ and $\\pi $ .", "The figure is a contour plot: white areas correspond to $\\nu _0 =0$ ; black areas correspond to selling at maximum speed ($\\nu _0 = 18000$ ); grey areas correspond to selling at a moderate speed, see also the color bars below the graphs.", "Comparing the graphs we see that for higher temporary price impact (high $c_f$ ) the trader tends to trade more evenly over the state space to keep the cost due to the temporary price impact small.", "Note that for $c_f\\rightarrow 0$ the liquidation strategy converges to a bang-bang type strategy.", "The optimal policy is then characterized by two regions: a sell region, where the trader sells at the maximum speed, and a wait region, where she does not sell at all.", "This reaction of $\\nu ^*_{t_k}$ to variations in $c_f$ can also be derived theoretically by inspection of (REF ).", "Figure: Contour plot of the liquidation policy as a function of ww (abscissa) and π\\pi (ordinate) for c f =5×10 -11 c_f=5\\times 10^{-11} (left), c f =5×10 -5 c_f=5\\times 10^{-5} (middle), and c f =10 -5 c_f=10^{-5} (right) and t=0t=0 for Example .Now we study the impact of the permanent price impact $a$ on the form of the optimal liquidation rate.", "Figure REF shows that for moderate $a$ the liquidation rate is decreasing in $\\pi $ and increasing in the inventory level.", "The situation changes when the permanent price impact becomes large.", "The left plot in Figure REF depicts the sell and wait regions under partial information in dependence of the inventory level $w$ and the filter probability $\\pi $ for $a=7 \\times 10^{-5}$ , and essentially without temporary price impact.", "For this value of $a$ the sell region forms a band from low values of $w$ and $\\pi $ to high values of $w$ and $\\pi $ .", "In particular, for large $w$ and small $\\pi $ there is a gambling region where the trader does not sell, even if a small value of $\\pi $ means that the bid price is trending downward (recall that $\\pi $ gives the probability that $Y$ is in the good state).", "In the presence of a temporary price impact (right plot of Figure REF ) the qualitative behaviour of the liquidation rate is similar to the case without temporary price impact, but the transition from waiting to selling at the maximum rate is smooth.", "Figure: Contour plot of the liquidation policy as a function of ww (abscissa) and π\\pi (ordinate) for c f =5×10 -11 c_f=5\\times 10^{-11} (left) and c f =10 -5 c_f= 10^{-5} (right) for a=7×10 -5 a= 7 \\times 10^{-5} and t=0t=0 for Example .The observed form of $\\nu ^*_{t_k}$ has the following explanation.", "Our numerical experiments show that for the chosen parameter values $V^{\\prime }$ is almost linear in $\\pi $ , so that the uncertainty correction (REF ) is negligible.", "Hence the liquidation rate $\\nu ^\\ast _{t_k}$ is determined by the interplay of the opportunity cost $\\frac{\\delta V^{\\prime }}{\\delta w} (t_k,w,\\pi )$ and of the illiquidity cost (REF ).", "We found that the opportunity cost is increasing in $\\pi $ .", "This is very intuitive: in the good state the investor expects an increase in the expected bid price which makes additional inventory more valuable.", "Moreover, we found that $\\frac{\\delta V^{\\prime }}{\\delta w} (t_k,w,\\pi )$ is decreasing in $w$ , that is the optimal liquidation problem has decreasing returns to scale.", "The illiquidity cost has the opposite monotonicity behaviour: it is increasing in $w$ (as it is proportional to $V^{\\prime }(t_k,w,\\pi ^{\\text{post}}) $ ) and, for the given parameters, decreasing in $\\pi $ .", "Now for small values of $a$ the opportunity cost dominates the illiquidity cost for all $(w,\\pi )$ and $C(t_k,w,\\pi )$ is increasing in $\\pi $ and decreasing in $w$ .", "By (REF ), the liquidation rate is thus decreasing in $\\pi $ and increasing in $w$ , which is in line with the monotonicity behaviour observed in Figure REF .", "If $a$ is large the situation is more involved.", "The opportunity cost dominates for small $w$ , leading to a liquidation rate that is decreasing in $\\pi $ .", "For large $w$ the illiquidity cost dominates, $C$ is decreasing in $\\pi $ , and the optimal liquidation rate is increasing in $\\pi $ .", "For $w$ large enough this effect is strong enough to generate the unexpected gambling region observed in Figure REF ." ], [ "Impact of other model components.", "In reality the support of $\\eta ^\\mathbf {P}$ is larger than $\\lbrace - \\theta , \\theta \\rbrace $ as the price may jump by more than one tick.", "Hence it is important to test the sensitivity of $\\nu _{t_k}^\\ast $ with respect to the precise form of the support.", "To this end, we computed the optimal strategy for a different parameter set $\\tilde{\\theta }, {\\tilde{c}}^{\\text{up}}_i , {\\tilde{c}}^{\\text{down}}_i$ , $i=1,2$ with $\\tilde{\\theta }= 2 \\theta $ and ${\\tilde{c}}^{\\text{up}}_i = 0.5 c^{\\text{up}}_i$ , ${\\tilde{c}}^{\\text{down}}_i = 0.5 c^{\\text{down}}_i$ , $i=1,2$ .", "Note that for the new parameters the support of $ \\eta ^\\mathbf {P}$ is different but the expected return of the bid price in each of the two states is the same.", "We found that the liquidation value and the optimal strategy were nearly identical to the original case.", "This shows that our approach is quite robust with respect to the exact form of the support of $\\eta ^\\mathbf {P}$ and justifies the use of a simple model with only two possible values for the jump size of $R$ .", "subsection20mm-0.25cm Gain from filtering and comparison to classical approaches.", "In this section we compare the expected proceeds of the optimal liquidation rate to the expected proceeds of a trader who mistakenly uses a model with deterministic $\\eta ^\\mathbf {P}$ as in Example REF .", "We use the following parameters for the deterministic model: $c^{\\text{up}} = 0.5 c_1^{\\text{up}} + 0.5 c_2^{\\text{up}}$ , $c^{\\text{down}} = 0.5 c_1^{\\text{down}} + 0.5 c_2^{\\text{down}}$ , that is the trader ignores regime switching but works with the stationary distribution of the Markov chain throughout.", "To compute the resulting liquidation rate $\\nu ^{\\ast ,\\text{det}}_{t_k}$ , we consider the HJB equation for the value function $V^{\\prime ,\\text{det}}$ for Example REF , which is given by $\\frac{\\partial V^{\\prime ,\\text{det}}}{\\partial t} ( - \\rho V^{\\prime ,\\text{det}}+ \\sup _{\\nu \\in [0,\\nu ^{\\text{max}}]}\\Big \\lbrace \\nu (1-c_f\\nu ^\\varsigma ) - \\nu \\frac{\\partial V^{\\prime ,\\text{det}}}{\\partial w} - \\bar{\\eta }^\\mathbf {P}(\\nu ) V^{\\prime ,\\text{det}}(t,w) \\Big \\rbrace =0,$ where, for the given model specification, $\\bar{\\eta }^\\mathbf {P}(\\nu ) = \\theta c^{\\text{down}}a $ .", "Then $\\nu ^{\\ast ,\\text{det}}_{t_k}$ is the maximizer in (REF ) (with partial derivatives replaced by finite differences) and depends only on time and inventory level.", "Numerical results of the performance comparisons how that, for a time horizon of two days, the expected gain from the use of filtering is $113.38$ .", "For a longer liquidation horizon of four days the effect becomes even stronger with $167.35$ .", "This shows that the additional complexity of using a filtering model may be worthwhile.", "Remark 6.1 (Comparison to [2]) It is interesting that the optimal liquidation rate $\\nu ^{\\ast ,\\text{det}_{t_k}}$ is identical to the optimal rate in a geometric version of the well-known model of [2], referred to as geometric AC-model in the sequel.", "In particular, the performance comparison applies also to the case where the investor uses this classical model.", "In the geometric AC-model it is assumed that that the bid price has dynamics $ dS_t^{\\nu } = \\bar{\\eta }^\\mathbf {P}(\\nu _t) S_t^{\\nu } dt + \\sigma S_t^{\\nu } d B_t\\,,$ for a Brownian motion $B$ .", "By standard arguments the HJB equation for the value function $V^\\text{AC} $ of the optimal liquidation problem in the geometric AC-model is $ \\frac{\\partial V^\\text{AC}}{\\partial t} - \\rho V^\\text{AC} + \\sup _{\\nu \\in [0,\\nu ^{\\text{max}}]}\\Big \\lbrace s\\nu (1-c_f\\nu ^\\varsigma ) - \\nu \\frac{\\partial V^\\text{AC}}{\\partial w} - \\bar{\\eta }^\\mathbf {P}(\\nu ) s \\frac{\\partial V^\\text{AC}}{\\partial s} + \\frac{1}{2} \\sigma ^2 s^2 \\frac{\\partial ^2 V^\\text{AC}}{\\partial s^2} \\Big \\rbrace =0\\,.$ Moreover, since $V^\\text{AC}$ is homogeneous in $s$ , $V^\\text{AC}(t,s,w) = s V^{\\prime , \\text{AC}}(t,w)$ .", "It follows that $\\frac{\\partial ^2 V^\\text{AC}}{\\partial s^2} =0$ , and the HJB equation for $V^{\\prime , \\text{AC}}$ reduces to (REF ).", "Hence the optimal liquidation rate in the geometric AC model and in the jump-model with deterministic compensator coincide.", "The equivalence between the jump model and the geometric AC-model (REF ) holds only for the case where the compensator is deterministic: a model of the form (REF ) with drift driven by an unobservable Markov chain would lead to a diffusion equation for the filter and hence to a control problem for diffusion processes.", "subsection20mm-0.25cm Model calibration.", "Finally we report the results of a small calibration study.", "We used a robust version of the EM algorithm to estimate the parameters of the bid price dynamics for the model specification from Example REF ; see [27] for details on the methodology.", "First, in order to test the performance of the algorithm we ran a study with simulated data for two different parameter sets.", "In set 1 we use the parameters from Table REF ; in set 2 we work with $c^{\\text{up}}_1 =c^{\\text{up}}_2=c^{\\text{down}}_1=c^{\\text{down}}_2 =1000$ , that is we consider a situation without Markov switching in the true data-generating process.", "However, the EM algorithm allows for different parameters in the two states.", "Hence this is a test, if the EM methodology points out spurious regime changes and trading opportunities which are not really in the data.", "The outcome of this exercise is presented in Figure REF , where we plot the hidden trajectory of $Y$ together with the filter estimate $\\widehat{Y}$ generated from the simulated data using the estimated model parameters.", "We see that in the left plot the filter nicely picks up the regime change, in the right plot the estimate $\\widehat{Y}_t$ is close to $1.5$ throughout, that is the estimated model correctly indicates that there is no Markov switching in the data.", "Finally we applied the algorithm to bid price data from the share price of Google, sampled at a frequency of one second.", "The EM estimates are $\\widehat{c}^\\text{up}_1 = 2128$ , $\\widehat{c}^\\text{up}_2 = 1751$ , $\\widehat{c}^\\text{down}_1 = 1769$ , $\\widehat{c}^\\text{down}_2 = 1888$ , which shows the same qualitative behaviour as the values used in our simulation study.", "A trajectory of the ensuing filter is given in Fig REF .", "One would need an extensive empirical study to confirm and refine these results, but this is beyond the scope of the present paper.", "Figure: A trajectory of the Markov chain YY (dashed) and of the corresponding filter Y ^\\widehat{Y} (straight line) computed using the parameter estimates from the EM algorithm as input.", "Left plot: results for parameter set 1 (with Markov switching); right plot: results for parameter set 2 (no Markov switching) In the graphs state e 1 e_1 (e 2 e_2) is represented by the value 1 (the value 2), and Y ^ t =π t 1+(1-π t )2\\widehat{Y}_t = \\pi _t 1 + (1-\\pi _t)2.The estimated parameters for parameter set  1 are as follows: c ^ 1 up =993\\widehat{c}^\\text{up}_1 = 993; c ^ 2 up =875\\widehat{c}^\\text{up}_2 = 875; c ^ 1 down =842\\widehat{c}^\\text{down}_1 = 842; c ^ 2 down =960\\widehat{c}^\\text{down}_2 = 960.", "For parameter set 2 we obtained c ^ 1 up =940\\widehat{c}^\\text{up}_1 = 940; c ^ 2 up =941\\widehat{c}^\\text{up}_2 = 941; c ^ 1 down =9445\\widehat{c}^\\text{down}_1 = 9445; c ^ 2 down =957\\widehat{c}^\\text{down}_2 = 957.Figure: Trajectory of Y ^\\widehat{Y} computed from the Google share price on 2012-06-21, sampled at a frequency of one second.", "(Data are from the LOBSTER database, see https://lobsterdata.com)" ], [ "Acknowledgements", "A part of this article was written while K. Colaneri was affiliated with the Department of Economics, University of Perugia.", "Support by the Vienna Science and Technology Fund (WWTF) through project MA14-031 is gratefully acknowledged.", "M. Szölgyenyi is supported by the AXA Research Fund grant “Numerical Methods for Stochastic Differential Equations with Irregular Coefficients with Applications in Risk Theory and Mathematical Finance\".", "A part of this article was written while M. Szölgyenyi was affiliated with the Institute of Statistics and Mathematics, Vienna University of Economics and Business.", "The authors are grateful for the excellent research assistance by Camilla Damian." ], [ "Setup and filtering: proofs and additional results", "Lemma A.1 Suppose that Assumption REF holds.", "Fix $m > w_0/T$ and consider some $\\mathbb {F}^S$ -adapted strategy $\\nu $ with values in $[0,m]$ .", "Define $C:= 0 \\vee \\sup \\big \\lbrace \\int _\\mathbb {R}(z^2 + 2z) \\eta ^\\mathbf {P}(t,e,\\nu ,dz)\\colon (t, e, \\nu ) \\in [0,T] \\times \\mathcal {E} \\times [0 , m] \\big \\rbrace \\,.", "$ Then $C < \\infty $ , $\\mathbb {E}((S_t^{\\nu })^2) \\le S_0^2 e^{C t}$ , and $(\\int _0^t S_{s-}^{\\nu } \\mathrm {d}M_s^R)_{0\\le t\\le T}$ is a true martingale.", "To ease the notation we write $S_t$ for $S_t^{\\nu }$ .", "We begin with the bound on $S_t^2$ .", "First note that $C$ is finite by Assumption REF .", "At a jump time $T_n$ of $R$ it holds that $S_{T_n} = S_{T_{n}-}(1 + \\Delta R_{T_n})$ and therefore $S_{T_n}^2 - S_{T_n -}^2 = S_{T_n -}^2 \\Delta R_{T_n} ^2 + 2 S_{T_n -}^2 \\Delta R_{T_n}\\,.$ Hence $ S_t^2 = S_0^2 + \\int _0^t \\int _\\mathbb {R}S_{s-}^2 (z^2 + 2z) \\mu ^R(\\mathrm {d}z,\\mathrm {d}s)$ and we get $\\mathbb {E}(S_t^2) & = S_0^2 + \\mathbb {E} \\Big ( \\int _0^t \\int _\\mathbb {R}S_{s}^2 (z^2 +2z) \\eta ^\\mathbf {P}(s, Y_{s-} \\nu _{s-}, \\mathrm {d}z) \\, \\mathrm {d}s \\Big )\\\\& \\le S_0^2 + C \\int _0^t \\mathbb {E} \\big ( S_s^2\\big ) \\mathrm {d}s\\,,$ so that $\\mathbb {E}((S_t^{\\nu })^2) \\le S_0^2 e^{C t}$ by the Gronwall inequality.", "To show that $\\int _0^{\\cdot } S_{s-} \\mathrm {d}M_s^R$ is a true martingale we show that this process has integrable quadratic variation.", "Since $ \\big [ \\int _0^\\cdot S_{s-} \\mathrm {d}M_s^R \\big ]_t = \\int _0^t \\int _\\mathbb {R}S_{s-}^2 z^2 \\mu ^R(\\mathrm {d}z, \\mathrm {d}s) $ , we have $ \\mathbb { E} \\left( \\big [ \\int _0^\\cdot S_{s-} \\mathrm {d}M_s^R \\big ]_t \\right) =\\mathbb {E} \\left( \\int _0^t S_{s}^2 \\int _\\mathbb {R}z^2 \\eta ^\\mathbf {P}(s, Y_{s-}, \\nu _{s-}\\mathrm {d}z) \\mathrm {d}s\\right) \\le S_0^2 \\tilde{C} \\int _0^t e^{C s} \\mathrm {d}s\\,,$ for every $t\\in [0,T]$ , where ${\\tilde{C}} = \\sup \\big \\lbrace \\int _\\mathbb {R}z^2 \\eta ^\\mathbf {P}(t,e,\\nu ,dz)\\colon t \\in [0,T], \\ e \\in \\mathcal {E}, \\nu \\in [0 , m] \\big \\rbrace $ is finite by Assumption REF .", "[Lemma REF ] Conditions (REF ) and (REF ) imply that $\\tilde{Z}$ is a true martingale, see [42].", "Moreover, $\\beta ( t,Y_{t^-},\\nu _{t^-}, z) > - 1$ , since $\\big ( \\mathrm {d}\\eta ^\\mathbf {P}_t (t,e_i, \\nu ;\\mathrm {d}z)/\\mathrm {d}\\eta ^\\mathbf {Q}_t(\\mathrm {d}z) \\big )(z) >0$ by assumption.", "This implies that $\\widetilde{Z}_T >0$ , and hence the equivalence of $\\mathbf {P}$ and $\\mathbf {Q}$ .", "The Girsanov theorem for random measures (see [16]) shows that under $\\mathbf {P}$ , $\\mu ^{R} (\\mathrm {d}t,\\mathrm {d}z)$ has the predictable compensator $(\\beta (t, Y_{t-}, \\nu _t, z) +1) {\\eta }_t^\\mathbf {Q}(\\mathrm {d}z) \\mathrm {d}t.$ By definition of $\\beta $ this is equal to ${\\eta }^P(t, Y_{t-}, \\nu _t, \\mathrm {d}z) \\mathrm {d}t \\,.$ Moreover, $\\widetilde{Z}$ and $Y$ are orthogonal, since $R$ and $Y$ have no common jumps, so that the law of $Y$ is the same under $\\mathbf {P}$ and under $\\mathbf {Q}$ .", "[Theorem REF ] Our derivation parallels the proof of [7], which deals with the classical case where the observation process is a Brownian motion with drift.", "Recall that for a function $f \\colon \\mathcal {E} \\rightarrow \\mathbb {R}$ the semimartingale decomposition of $f(Y_t)$ is given by $f(Y_t)=f(Y_0)+\\int _0^t \\langle Q\\mathbf {f}, Y_s \\rangle \\mathrm {d}s+M_t^f$ , where $M^f$ is a true $(\\mathbb {F}, \\mathbf {Q})$ -martingale.", "Define the process $\\widetilde{Z}^\\epsilon =(\\widetilde{Z}^\\epsilon _t)_{t\\in [0,T]}$ by $\\widetilde{Z}^\\epsilon _t:=\\frac{\\widetilde{Z}_t}{1+\\epsilon \\widetilde{Z}_t},$ and note that $\\widetilde{Z}^\\epsilon _t < 1/\\epsilon $ for every $t \\in [0,T]$ .", "Now we compute $\\widetilde{Z}^\\epsilon f(Y)$ .", "Notice that $[\\widetilde{Z}^\\epsilon , Y]_t =0$ for every $t \\in [0,T]$ , as $R$ and $Y$ have no common jumps.", "Hence, from Itô's product rule we get $\\mathrm {d}\\big (\\widetilde{Z}^\\epsilon _t f(Y_t)\\big ) &=\\widetilde{Z}^\\epsilon _{t^-} \\langle Q\\mathbf {f}, Y_t \\rangle \\mathrm {d}t + \\widetilde{Z}^\\epsilon _{t^-} \\mathrm {d}M^f_t- f(Y_{t^-}) \\widetilde{Z}^\\epsilon _{t^-} \\int _\\mathbb {R}\\frac{\\beta (t,Y_{t^-},\\nu _{t-},z)}{1 + \\epsilon \\widetilde{Z}_{t^-}} \\eta ^\\mathbf {Q}_t(\\mathrm {d}z) \\mathrm {d}t \\\\& \\quad + f(Y_{t^-}) \\widetilde{Z}^\\epsilon _{t^-}\\int _\\mathbb {R}\\frac{\\beta (t,Y_{t^-},\\nu ,z)}{1+\\epsilon \\widetilde{Z}_{t^-} (1+\\beta (t,Y_{t^-},\\nu _{t-},z))} \\mu ^R(\\mathrm {d}t, \\mathrm {d}z).$ Next we show that $\\mathbb {E}^{\\mathbf {Q}}\\left(\\int _0^t\\widetilde{Z}^\\epsilon _{s^-}\\mathrm {d}M^f_s\\mid \\mathcal {F}^S_t\\right)=0$ .", "By the definition of conditional expectation, this is equivalent to $\\mathbb {E}^{\\mathbf {Q}}\\left(H\\int _0^t\\widetilde{Z}^\\epsilon _{s^-}\\mathrm {d}M^f_s\\right)=0 $ for every bounded, $\\mathcal {F}_t^S$ -measurable random variable $H$ .", "Define an $(\\mathbb {F}^S,\\mathbf {Q})$ -martingale by $H_u=\\mathbb {E}^{\\mathbf {Q}}\\left(H\\mid \\mathcal {F}^S_u\\right)$ , $0\\le u \\le t\\le T$ , and note that $H=H_t$ .", "By the martingale representation theorem for random measures, see, e.g., [39] or [16], we get that there is a bounded $\\mathbb {F}^S$ -predictable random function $\\phi $ such that $H_t=H_0+\\int _0^t\\int _{\\mathbb {R}}\\phi (s,z)(\\mu ^R(\\mathrm {d}s, \\mathrm {d}z)-\\eta ^{\\mathbf {Q}}(\\mathrm {d}z)\\mathrm {d}s), \\quad t \\in [0,T] \\,.", "$ Now, applying the Itô product rule and using that $[M^f, H]_t = [Y,R]_t =0$ for every $t \\in [0,T]$ , we obtain $H_t \\int _0^t\\widetilde{Z}^\\epsilon _{s^-}\\mathrm {d}M^f_s\\!=\\!\\int _0^t H_{s-}\\widetilde{Z}^\\epsilon _{s^-}\\mathrm {d}M^f_s+\\int _0^t\\!\\int _{\\mathbb {R}} \\!\\Big (\\!\\int _0^s\\widetilde{Z}^\\epsilon _{u^-}\\mathrm {d}M^f_u\\Big )\\phi (s,z)\\big (\\mu ^R(\\mathrm {d}s, \\mathrm {d}z)-\\eta ^{\\mathbf {Q}}(\\mathrm {d}z)\\mathrm {d}s\\big ).$ Both integrals on the right hand side of the above representation are martingales.", "This follows from the finite-state property of the Markov chain $Y$ and the boundedness of $\\widetilde{Z}^\\epsilon $ and $H$ .", "Hence, taking the expectation we get that $\\mathbb {E}^{\\mathbf {Q}}\\left(H\\int _0^t\\widetilde{Z}^\\epsilon _{s^-}\\mathrm {d}M^f_s\\right)=0 $ as claimed.", "Now note that for $t \\in [0,T]$ and a generic integrable $\\mathcal {F}_t$ -measurable random variable $U$ it holds that $ \\mathbb {E}^{\\mathbf {Q}}\\left(U\\mid \\mathcal {F}^S_t\\right)=\\mathbb {E}^{\\mathbf {Q}}\\left(U\\mid \\mathcal {F}^S_T\\right);$ this can be shown with similar arguments as in [7].", "Taking the conditional expectation from (REF ) and applying (REF ) and the Fubini theorem we get for every $t \\in [0,T]$ , $\\nonumber &\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}^\\epsilon _t f(Y_t) \\mid \\mathcal {F}^S_t\\right) =\\frac{\\pi _0 (f)}{1+\\epsilon }+\\int _0^t\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}^\\epsilon _{s^-} \\langle Q\\mathbf {f}, Y_s \\rangle \\mid \\mathcal {F}^S_T\\right)\\mathrm {d}s\\\\& \\, + \\int _0^t \\int _\\mathbb {R}\\mathbb {E}^{\\mathbf {Q}}\\left( f(Y_{s^-}) \\widetilde{Z}^\\epsilon _{s^-} \\frac{\\beta (s,Y_{s^-},\\nu _{s^-},z)}{1+\\epsilon \\widetilde{Z}_{s^-} (1+\\beta (s,Y_{s^-},\\nu ,z))}\\mid \\mathcal {F}^S_T\\right)\\mu ^R(\\mathrm {d}s, \\mathrm {d}z) \\\\\\nonumber &\\, -\\int _0^t\\int _\\mathbb {R}\\mathbb {E}^{\\mathbf {Q}}\\left(f(Y_{s^-}) \\widetilde{Z}^\\epsilon _{s^-} \\frac{\\beta (s,Y_{s^-},\\nu _{s^-},z)}{1 + \\epsilon \\widetilde{Z}_{s^-}} \\mid \\mathcal {F}^S_T\\right)\\eta ^\\mathbf {Q}_s(\\mathrm {d}z)\\mathrm {d}s\\,.$ Note that, for every $t \\in [0,T]$ , $\\widetilde{Z}^\\epsilon _t < \\widetilde{Z}_t $ and that $\\widetilde{Z}_t $ is integrable.", "Since $\\beta $ is bounded by assumption, by dominated convergence we get the following three limits $& \\lim _{\\epsilon \\rightarrow 0} \\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}^\\epsilon _t f(Y_t)\\mid \\mathcal {F}^S_t\\right) =\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}_t f(Y_t)\\mid \\mathcal {F}^S_t\\right)\\,, \\\\& \\lim _{\\epsilon \\rightarrow 0} \\int _0^t\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}^\\epsilon _{s^-} \\langle Q\\mathbf {f}, Y_s \\rangle \\mid \\mathcal {F}^S_T\\right)\\mathrm {d}s = \\int _0^t\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}_{s^-} \\langle Q\\mathbf {f}, Y_s \\rangle \\mid \\mathcal {F}^S_T\\right)\\mathrm {d}s\\,,\\\\&\\lim _{\\epsilon \\rightarrow 0} \\int _0^t \\int _\\mathbb {R}\\mathbb {E}^{\\mathbf {Q}}\\left(f(Y_{s^-}) \\widetilde{Z}^\\epsilon _{s^-} \\frac{\\beta (s,Y_{s^-},\\nu _{s^-},z)}{1 + \\epsilon \\widetilde{Z}_{s^-}} \\mid \\mathcal {F}^S_T\\right) \\eta ^\\mathbf {Q}_s(\\mathrm {d}z)\\mathrm {d}s \\\\ & \\qquad =\\int _\\mathbb {R}\\mathbb {E}^{\\mathbf {Q}}\\left(f(Y_{s^-}) \\widetilde{Z}_{s^-} \\beta (s,Y_{s^-},\\nu _{s^-},z) \\mid \\mathcal {F}^S_T\\right) \\eta ^\\mathbf {Q}_s(\\mathrm {d}z)\\mathrm {d}s.$ Finally we consider the integral with respect to $\\mu ^R(\\mathrm {d}s, \\mathrm {d}z)$ in (REF ).", "Let $\\lbrace T_n, Z_n\\rbrace $ be the sequence of jump times and the corresponding jump sizes of the process $R$ .", "Denote by $n(t) $ the number of jumps up to time $t$ , so that $T_{n(t)}$ is the last jump time before $t$ .", "Then $&\\lim _{\\epsilon \\rightarrow 0}\\int _0^t \\int _\\mathbb {R}\\mathbb {E}^{\\mathbf {Q}}\\left( f(Y_{s^-}) \\widetilde{Z}^\\epsilon _{s^-} \\frac{\\beta (s,Y_{s^-},\\nu ,z)}{1+\\epsilon \\widetilde{Z}_{s^-} (1+\\beta (s,Y_{s^-},\\nu ,z))}\\mid \\mathcal {F}^S_T\\right) \\mu ^R(\\mathrm {d}s, \\mathrm {d}z)\\\\&\\qquad = \\lim _{\\epsilon \\rightarrow 0} \\sum _{n=1}^{n(t)} \\mathbb {E}^{\\mathbf {Q}}\\left( f(Y_{T_n^-}) \\widetilde{Z}^\\epsilon _{T_n^-} \\frac{\\beta (T_n,Y_{T_n^-},\\nu ,\\Delta R_{T_n})}{1+\\epsilon \\widetilde{Z}_{T_n^-} (1+\\beta (T_n,Y_{T_n^-},\\nu _{T_n^-},\\Delta R_{T_n}))}\\mid \\mathcal {F}^S_T\\right) \\\\&\\qquad = \\sum _{n=1}^{n(t)} \\mathbb {E}^{\\mathbf {Q}}\\left( f(Y_{T_n^-}) \\widetilde{Z}_{T_n^-} \\beta (T_n,Y_{T_n^-},\\nu _{T_n^-},\\Delta R_{T_n})\\mid \\mathcal {F}^S_T\\right)\\\\&\\qquad = \\int _0^t \\int _\\mathbb {R}\\mathbb {E}^{\\mathbf {Q}}\\left( f(Y_{s^-}) \\widetilde{Z}_{s^-} \\beta (s,Y_{s^-},\\nu _{s^-},z)\\mid \\mathcal {F}^S_T\\right) \\mu ^R(\\mathrm {d}s, \\mathrm {d}z)\\,.$ Assembling the previous results we obtain $&\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}_t f(Y_t)\\mid \\mathcal {F}^S_T\\right)=\\pi _0 (f) +\\int _0^t\\mathbb {E}^{\\mathbf {Q}}\\left(\\widetilde{Z}_{s^-} \\langle Q\\mathbf {f}, Y_s \\rangle \\mid \\mathcal {F}^S_T\\right)\\mathrm {d}s\\\\&\\quad +\\int _0^t\\int _\\mathbb {R}\\mathbb {E}^{\\mathbf {Q}}\\left(f(Y_{s^-}) \\widetilde{Z}_{s^-} \\beta (s,Y_{s^-},\\nu _{s^-},z)\\mid \\mathcal {F}^S_T\\right)\\left(\\mu ^R(\\mathrm {d}s, \\mathrm {d}z)-\\eta ^\\mathbf {Q}_s(\\mathrm {d}z)\\right)\\,,$ and hence the claim of the theorem follows from (REF )." ], [ "Optimization via MDMs: proofs and additional results", "[Lemma REF ] To establish the claim we show that the first derivatives of the vector field $g$ are bounded, uniformly in $\\nu $ .", "The components of $\\frac{\\partial g}{\\partial w}$ and $\\frac{\\partial g}{\\partial s}$ are all 0, and, using Assumption REF , the nonzero components of $\\frac{\\partial g}{\\partial \\pi ^i}$ , $i=1,\\dots , K$ , can be estimated as follows.", "For $i \\ne k$ , $\\left|\\frac{\\partial g^{k+3}}{\\partial \\pi ^i}\\right|\\!&=\\!", "\\left|q^{ik}\\!-\\!\\pi ^k \\!\\!\\int _{\\mathbb {R}}\\!\\!", "u^k(t,\\nu ,\\pi ,z)\\eta ^\\mathbf {P}(t,e_i,\\nu ,\\mathrm {d}z)-\\pi ^k \\sum _{j=1}^K \\pi ^j \\!\\!\\int _{\\mathbb {R}} \\!\\!", "\\frac{\\partial u^k(t,\\nu ,\\pi ,z)}{\\partial \\pi ^i} \\eta ^\\mathbf {P}(t,e_j,\\nu ,\\mathrm {d}z)\\right|\\\\<&\\max _{i,k}q^{ik}+\\pi ^k \\int _{\\mathbb {R}} u^k(t,\\nu ,\\pi ,z)\\eta ^\\mathbf {P}(t,e_i,\\nu ,\\mathrm {d}z)\\\\ &+ \\pi ^k \\sum _{j=1}^K \\pi ^j \\int _{\\mathbb {R}} \\frac{{\\mathrm {d}\\eta ^{\\mathbf {P}}(t,e_i,\\nu )}/{\\mathrm {d}\\eta ^\\mathbf {Q}_t}( z){\\mathrm {d}\\eta ^{\\mathbf {P}}(t,e_k,\\nu )}/{\\mathrm {d}\\eta ^\\mathbf {Q}_t}( z)}{\\left(\\sum _{l=1}^K \\pi ^l {\\mathrm {d}\\eta ^{\\mathbf {P}}(t,e_l,\\nu )}/{\\mathrm {d}\\eta ^\\mathbf {Q}_t}( z)\\right)^2}\\eta ^\\mathbf {P}(t,e_j,\\nu ,\\mathrm {d}z),$ and this is smaller than $\\max _{i,k}q^{ik}+(M^4+M^2)\\lambda ^{\\max }$ .", "For $i=k$ we get $\\left|\\frac{\\partial g^{i+3}}{\\partial \\pi ^i}\\right|=&\\left| q^{ii}-2\\pi ^i \\int _{\\mathbb {R}} u^i(t,\\nu ,\\pi ,z)\\eta ^\\mathbf {P}(t,e_i,\\nu ,\\mathrm {d}z)- \\sum _{j\\ne i}\\pi ^j\\int _{\\mathbb {R}}u^i(t,\\nu ,\\pi ,z)\\eta ^\\mathbf {P}(t,e_j,\\nu ,\\mathrm {d}z) \\right.\\\\&\\left.-\\pi ^i \\sum _{j=1}^K \\pi ^j \\int _{\\mathbb {R}} \\frac{\\partial u^i(t,\\nu ,\\pi ,z)}{\\partial \\pi ^i} \\eta ^\\mathbf {P}(t,e_j,\\nu ,\\mathrm {d}z)\\right|<\\max _{i}q^{ii}(M^4+3 M^2)\\lambda ^{\\max }\\,.$ [Lemma REF ] First we estimate the reward function introduced in (REF ).", "Since $f \\ge 0$ , $e^{- \\rho t} \\le 1$ , and $h(w)\\le w$ , we get that $r(\\widetilde{x}, \\alpha ) \\le s \\int _0^{\\tau ^\\varphi } e^{-\\Lambda ^{\\alpha }_u} \\alpha _u \\mathrm {d}u + s e^{-\\Lambda ^{\\alpha }_{\\tau ^\\varphi }} w_{\\tau ^\\varphi }^\\alpha $ .", "Partial integration gives $ \\int _0^{\\tau ^\\varphi } e^{-\\Lambda ^{\\alpha }_u} \\alpha _u \\mathrm {d}u =\\big [- w_u^\\alpha e^{-\\Lambda ^{\\alpha }_u}\\big ]_0^{\\tau ^\\varphi } - \\int _0^{\\tau ^\\varphi } \\lambda ^\\alpha _u e^{-\\Lambda ^{\\alpha }_u} w_u^\\alpha \\mathrm {d}u \\le w - e^{-\\Lambda ^{\\alpha }_{\\tau ^\\phi }} w_{\\tau ^\\phi }^\\alpha \\, ,$ and hence $r(\\widetilde{x}, \\alpha ) \\le s w$ .", "Next we estimate $Q_L b (\\widetilde{x},\\alpha )$ .", "Recall the definition of $\\bar{\\eta }^\\mathbf {P}$ from (REF ) and let $c_{\\eta } := \\sup \\big \\lbrace \\bar{\\eta }^\\mathbf {P}(t,e,0) \\colon (t,e) \\in [0,T] \\times \\mathcal {E} \\big \\rbrace .$ It holds that $Q_L b (\\widetilde{x},\\alpha ) &= \\int _0^{\\tau ^{\\varphi }} \\hspace{-5.69046pt} e^{\\gamma (T-(u+t))}e^{-\\Lambda ^{\\alpha }_u}\\sum _{j=1}^K \\pi _j s w_u^\\alpha (1 + \\bar{\\eta }^\\mathbf {P}( t+u, e_j, \\alpha _u)) \\mathrm {d}u{\\le } s w e^{\\gamma (T-t)} c_\\eta \\int _0^{\\tau _\\varphi } \\hspace{-5.69046pt} e^{-\\gamma u}\\,\\mathrm {d}u\\,,$ where we have used that $w_u^\\alpha \\le w$ and $e^{-\\Lambda ^{\\alpha }_u}<1$ .", "The last term is bounded by $ b(\\widetilde{x})\\frac{c_{\\eta }}{\\gamma }$ , and the MDM is contracting for $\\gamma >c_\\eta $ .", "The following lemma is needed in the proof of Proposition REF .", "Lemma B.1 Consider a function $v\\in \\mathcal {C}_b$ .", "Then the mapping $(\\widetilde{x}, \\nu ) \\mapsto \\bar{Q} v(\\widetilde{x}, \\nu )$ is continuous on $\\widetilde{\\mathcal {X}} \\times [0, \\nu ^{\\text{max}}]$ .", "It suffices to show that for $j=1,\\dots ,K$ the mapping $ (t,w,s,\\pi ,\\nu ) \\mapsto \\int _\\mathbb {R}v \\left(t,s(1+z), \\pi ^1(1+ u^1(t, \\nu , \\pi ,z), \\dots , \\pi ^K(1+ u^K(t, \\nu , \\pi ,z)\\right) \\eta ^j(t, \\nu , \\mathrm {d}z)$ is continuous on $ \\widetilde{\\mathcal {X}} \\times [0, \\nu ^{\\text{max}}]$ , where $\\eta ^j(t, \\nu , \\mathrm {d}z) := \\eta (t,e_j,\\nu ,\\mathrm {d}z)$ .", "Consider a sequence with elements $ (t_n, \\nu _n, \\pi _n) \\xrightarrow[ n \\rightarrow \\infty ]{} (t, \\nu , \\pi )$ .", "Note that, for sufficiently large $n$ , the set $\\lbrace s^n (1+z)\\colon z \\in \\text{supp}(\\eta ) \\rbrace $ is contained in a compact subset $[\\underline{s}, \\overline{s}] \\subset (0, \\infty )$ .", "Moreover, $v$ is uniformly continuous on the compact set $ [0,T] \\times [0, w_0] \\times [\\underline{s}, \\overline{s}] \\times \\mathcal {S}^K \\times [0, \\nu ^{\\text{max}}]\\,.$ Then, Assumption REF -(2) implies that the sequence $\\lbrace v^n\\rbrace $ with $v^n(z) := v \\left(t_n,s_n(1+z), \\pi _{n}^1(1+ u^1(t_n, \\nu _n, \\pi _n,z), \\dots \\pi ^K_n(1+ u^K_n(t_n, \\nu _n, \\pi _n,z)\\right)$ converges uniformly in $z \\in \\text{supp}(\\eta )$ to $v(z) := v(t,s,\\pi , \\nu , z)$ .", "Hence the following estimate holds: $\\nonumber &\\Big | \\int _{\\text{supp}(\\eta )} \\hspace{-5.69046pt} v^n(z) \\eta ^j(t_n, \\nu _n,\\mathrm {d}z) - \\int _{\\text{supp}(\\eta )} \\hspace{-5.69046pt} v(z) \\eta ^j(t, \\nu ,\\mathrm {d}z) \\Big | \\\\& \\le \\int _{\\text{supp}(\\eta )} \\hspace{-19.91684pt} \\big |v^n (z) - v(z) \\big | \\eta ^j(t_n, \\nu _n , \\mathrm {d}z) +\\Big |\\int _{\\text{supp}(\\eta )} \\hspace{-19.91684pt} v(z) \\eta ^j(t_n, \\nu _n,\\mathrm {d}z) -\\int _{\\text{supp}(\\eta )} \\hspace{-19.91684pt} v(z) \\eta ^j(t, \\nu ,\\mathrm {d}z) \\Big |\\,.$ Finally, the first term in (REF ) can be estimated by $ \\lambda ^{\\text{max}} \\sup \\lbrace |v^n(z) - v(z)| \\colon z \\in \\text{supp}(\\eta ) \\rbrace $ , which converges to zero as $v^n$ converges to $v $ uniformly; the second term in (REF ) converges to zero by Assumption REF -(1) (continuity of the mapping $(t,\\nu ) \\mapsto \\eta ^j(t,\\nu , \\mathrm {d}z) $ in the weak topology)." ] ]
1606.05079
[ [ "A direct proof of F. Riesz representation Theorem" ], [ "Abstract A direct proof of the Riesz representation theorem is provided.", "This theorem characterizes the linear functionals acting on the vector space $C(K)$ of continuous functions defined on a compact subset $K$ of the real numbers $\\mathbb{R}$.", "This proof avoids complicated arguments commonly used in generalizations of Riesz original theorem." ], [ "arrows,backgrounds,decorations.pathmorphing,decorations.pathreplacing,positioning,fit DRAFT 2022/12/09 19:20:14130 65 foot=0.9cm pdftex Abstract A direct proof of the Riesz representation theorem is provided.", "This theorem characterizes the linear functionals acting on the vector space $\\mathcal {C}(K)$ of continuous functions defined on a compact subset $K$ of the real numbers $\\mathbb {R}$ .", "The proof avoids complicated arguments commonly used in generalizations of Riesz original theorem.", "Abstract A direct proof of the Riesz representation theorem is provided.", "This theorem characterizes the linear functionals acting on the vector space $\\mathcal {C}(K)$ of continuous functions defined on a compact subset $K$ of the real numbers $\\mathbb {R}$ .", "The proof avoids complicated arguments commonly used in generalizations of Riesz original theorem.", "section10em1.7em1.2emIntroduction The Riesz representation theorem is a remarkable result which describes the continuous linear functionals acting on the space of continuous functions defined on a set $K$ .", "It is very surprising that all these functionals are just integrals and vice versa.", "In case $K$ is a closed interval of real numbers, any such functional is represented by Riemann-Stieltjes integral, which is a generalization of the usual Riemann integral.", "This was first announced by F. Riesz in 1909 [13].", "In case $K$ is compact set (not necessarily a closed interval), then a more general concept of integral is needed, because the Riemann-Stieltjes integral used by Riesz is defined only for functions on intervals.", "In this work, we prove that there is a short path between the two cases.", "Besides its aesthetic appeal, the above mentioned theorem has far-reaching applications.", "It allows a short proof of the Kolmogoroff consistency theorem, see [3] thm 10.6.2., and can be used to give an elegant proof of the spectral theorem for selfadjoint bounded operators, see section VII.2 of [12].", "Both these theorems are main results in probability and functional analysis respectively.", "Moreover, the entire theory of integration for general spaces can be recovered using the theorem of Riesz.", "See for example [18], where the Lebesgue measure on $\\mathbb {R}^n$ is constructed.", "More generally it can also be used to show the existence of the Haar measure on a group, see [3] chap.", "9.", "In this note we give a short proof of the Riesz representation theorem for the case $K$ is an arbitrary compact set of real numbers, see Theorem REF below.", "This is interesting because in many situations we have a compact set which is not a closed interval.", "To prove the spectral theorem, for example, one considers the set of continuous functions defined on the spectrum of selfadjoint bounded operator, which is a compact set of $\\mathbb {R}$ , but not necessarily a closed interval.", "We get our result starting from the nondecreasing function that appears in the Riemann-Stieltjes integral representation of Riesz original formulation.", "To this function we associate a measure which is used to integrate over general compact sets.", "Then we show how this Lebesgue integral representation can be seen as a Riemann-Stieltjes integral again.", "Our proof is new, its reliance on measure theory make it not completely elementary, but it is very direct and quite simple.", "section10em1.7em1.2emPreliminaries Let us introduce first some definitions and notations we shall use.", "Let $\\mathcal {C}(K):=\\lbrace f:K\\rightarrow \\mathbb {R} : f \\text{ continuous}\\rbrace $ where $K$ is a compact subset of $\\mathbb {R}$ , the real numbers.", "A functional is an assignment $L: \\mathcal {C}(K)\\rightarrow \\mathbb {R}$ .", "The functional is linear if $L(c_1f+c_2g)=c_1L(f)+c_2 L(g)$ for all $f,g\\in \\mathcal {C}(K), c_1,c_2 \\in \\mathbb {R} $ .", "It is continuous if there exists a fixed $M>0$ such that $ |Lf|\\le M \\Vert f\\Vert _\\infty $ for all $f\\in \\mathcal {C}(K)$ , where $\\Vert \\cdot \\Vert _\\infty $ denotes the uniform norm, that is, $\\Vert f\\Vert _\\infty =\\sup \\lbrace |f(x)|: x\\in K\\rbrace $ .", "We define the norm of such functional as $\\Vert L\\Vert _{\\mathcal {C}(K)}=\\Vert L\\Vert = \\sup \\lbrace |L(f)|:f\\in \\mathcal {C}(K) \\text{ and }\\,\\Vert f\\Vert _\\infty \\le 1 \\rbrace ,$ We denote the set of the linear continuous functionals on $\\mathcal {C}(K)$ by $\\mathcal {C}(K)^*$ .", "It is called the dual space.", "In general, the dual of normed linear space $X$ is denoted by $X^*$ .", "A functional $L$ on $\\mathcal {C}(K)$ is said to be a positive if $L(f)\\ge 0$ whenever $f(x)\\ge 0$ for every $x\\in \\mathbb {R}$ .", "We use the notation $C(K)^*_+$ for the set of positive linear functionals on $C(K)$ .", "The function $\\alpha :[a,b]\\longrightarrow \\mathbb {R}$ is said to be normalized, if $\\alpha (a)=0$ and $\\alpha (t)=\\alpha (t+),\\, $ $a< t< b$ , that is, $\\alpha $ is continuous from the right inside the interval (not at a!", "If it were right continuous at $a$ , theorem (REF ) would not hold for the functional $L(f)=f(a)$ ).", "The total variation of a monotone increasing function $\\alpha $ is defined as $V(\\alpha )= \\alpha (b)-\\alpha (a)$ .", "We denote the characteristic function of a set $A\\subset [a,b]$ by $ \\mathbf {1}_{A}$ where $\\mathbf {1}_{A}(x)=1$ if $x \\in A$ and 0 if $x \\in [a,b]\\setminus A$ .", "We formulate the above-mentioned result by F. Riesz as follows: Theorem 0.1 Let $L:\\mathcal {C}[a,b]\\longrightarrow \\mathbb {R}$ be a positive linear functional.", "There exists a unique normalized monotone function $\\alpha :[a,b]\\longrightarrow \\mathbb {R}$ such that $Lf=\\int _a^bf(x)d\\alpha (x).$ The integral is understood in the sense of Riemann-Stieltjes.", "Moreover $\\Vert L \\Vert = V(\\alpha ).$ The Riemann-Stieltjes integral is a generalization of the Riemann integral, where instead of taking the length of the intervals, a $\\alpha $ -weighted length is taken.", "For an interval $I$ the $\\alpha $ -length is given by $\\alpha (I)= \\alpha (y)-\\alpha (x)$ , where $x,y$ are the end points of $I$ and $\\alpha $ is a function of finite variation.", "The integral of a continuous function $f$ on $[a, b]$ is defined as the limit, when it exists, of the sum $\\sum _i f(c_i) \\alpha (I_i)$ where $\\lbrace I_i\\rbrace $ is a finite collection of subintervals whose endpoints form a partition of $[a,b]$ and $c_i\\in I_i$ .", "See [16] p.122.", "There are different proofs of the above theorem, see for example [21].", "Here we will give a sketch of the proof which uses the following result about extensions of functionals known as the Hahn-Banach theorem: Let $X$ a normed linear space, $Y$ a subspace of $X$ , and $\\lambda $ an element of $Y^*$ .", "Then there exists a $\\Lambda \\in X^*$ extending $\\lambda $ with the same norm.", "See [12] for a proof.", "[ Proof of theorem REF ].", "We may assume that $[a,b]=[0,1]$ .", "Since $L\\in \\mathcal {C}[0,1]^*$ we use Hahn-Banach theorem to conclude the existence of $\\Lambda \\in B[0,1]^*$ such that $\\Vert \\Lambda \\Vert =\\Vert L\\Vert $ and $L=\\Lambda $ on $\\mathcal {C}[0,1]$ and where $B[0,1]$ is the set of bounded functions on $[0,1]$ .", "Let us define the functions $\\mathbf {1}_x:= \\mathbf {1}_{[0,x]}$ , that is $\\mathbf {1}_x(t)=1$ when $t\\in [0,x]$ and zero otherwise.", "Set $\\alpha (x) = \\Lambda (\\mathbf {1}_x)$ for all $x\\in [0,1]$ .", "Now for $f\\in \\mathcal {C}[0,1]$ , define $f_n =\\sum _{j=1}^n f(j/n) (\\mathbf {1}_{j/n}-\\mathbf {1}_{(j-1)/n}).$ Since $f$ is continuous, it is uniformly continuous on $[0,1]$ and so $\\Vert f_n-f\\Vert _\\infty \\rightarrow 0$ .", "Thus $ \\lim _n \\Lambda (f_n) =\\Lambda (f)=L(f).$ Using the definition of $\\alpha $ we get $\\Lambda (f_n)=\\sum _{j=1}^n f(j/n) (\\alpha (j/n)-\\alpha ((j-1)/n)).$ This in turn implies $ \\Lambda (f)=\\lim _n \\Lambda (f_n)=\\int _0^1 f \\,d\\alpha .$ Now to see that $\\Vert L \\Vert = V(\\alpha )$ : Let $\\varepsilon >0$ and choose $f\\in \\mathcal {C}[0,1]$ such that $\\Vert f\\Vert _\\infty \\le 1$ and $\\Vert L\\Vert \\le |L(f)|+\\varepsilon $ , we apply (REF ) and we get $\\Vert L\\Vert \\ \\le |L(f)|+\\varepsilon =\\left|\\int _0^1f(x)d\\alpha (x)\\right|\\ + \\varepsilon \\le \\alpha (1)-\\alpha (0)+\\varepsilon =V(\\alpha )+\\varepsilon .", "$ It is possible to normalize $\\alpha $ and in this case we easily have the other inequality, that is, $V(\\alpha )=\\alpha (1)-\\alpha (0)=\\alpha (1)=\\Lambda (\\mathbf {1}_1)\\le \\Vert \\Lambda \\Vert =\\Vert L\\Vert .$ Remarks.", "(1) The standard textbook's proof uses Hahn-Banach's theorem ([9],[21]), but the original proof of F. Riesz does not use it.", "See [16] section 50 and [14],[15].", "(2) E. Helly [7] should have similar results.", "J. Radon extended theorem REF to compact subsets $K\\subset \\mathbb {R}^n $ [11].", "S. Banach and S. Saks extended the result to compact metric spaces, see appendix of [20] and [19].", "The proof by S. Saks is particularly elegant and clean.", "For compact Hausdorff spaces the theorem was proven by S. Kakutani [8] and for normal spaces by A. Markoff [10].", "Nowadays this theorem is also known as Riesz-Markoff or Riesz-Markoff-Kakutani theorem.", "More information on the history of this theorem can be found in [5] p. 231, the references therein, [22] p.238 and [6].", "(3) Positivity of a linear functional $L$ implies continuity of $L$ .", "To see it, we take the function $\\textbf {1}(x)=1$ for all $x\\in K$ , then $\\textbf {1}\\in \\mathcal {C}(K)$ and $|f(x)|\\le \\Vert f\\Vert _{\\infty }\\textbf {1}(x)$ , therefore $\\Vert f\\Vert _{\\infty }\\,\\textbf {1}(x)\\pm f(x)\\ge 0\\qquad \\mbox{implies}\\qquad \\,\\Vert f\\Vert _{\\infty }L(\\textbf {1})\\pm L(f)\\ge 0$ so $|L(f)|\\le L(\\textbf {1})\\Vert f\\Vert _{\\infty }.$ See [5] Prop.", "7.1.", "The standard textbook's proof uses Hahn-Banach's theorem ([9],[21]), but the original proof of F. Riesz does not use it.", "See [16] section 50 and [14],[15].", "E. Helly [7] should have similar results.", "J. Radon extended theorem REF to compact subsets $K\\subset \\mathbb {R}^n $ [11].", "S. Banach and S. Saks extended the result to compact metric spaces, see appendix of [20] and [19].", "The proof by S. Saks is particularly elegant and clean.", "For compact Hausdorff spaces the theorem was proven by S. Kakutani [8] and for normal spaces by A. Markoff [10].", "Nowadays this theorem is also known as Riesz-Markoff or Riesz-Markoff-Kakutani theorem.", "More information on the history of this theorem can be found in [5] p. 231, the references therein, [22] p.238 and [6].", "Positivity of a linear functional $L$ implies continuity of $L$ .", "To see it, we take the function $\\textbf {1}(x)=1$ for all $x\\in K$ , then $\\textbf {1}\\in \\mathcal {C}(K)$ and $|f(x)|\\le \\Vert f\\Vert _{\\infty }\\textbf {1}(x)$ , therefore $\\Vert f\\Vert _{\\infty }\\,\\textbf {1}(x)\\pm f(x)\\ge 0\\qquad \\mbox{implies}\\qquad \\,\\Vert f\\Vert _{\\infty }L(\\textbf {1})\\pm L(f)\\ge 0$ so $|L(f)|\\le L(\\textbf {1})\\Vert f\\Vert _{\\infty }.$ See [5] Prop.", "7.1. section10em1.7em1.2emMain Result Next theorem is our main result.", "It is a generalization of Theorem REF to continuous functions defined on arbitrary compact sets $K\\subset \\mathbb {R}$ .", "Since an ordinary Riemann-Stieltjes integral is not defined for functions on general compact $K$ , we shall introduce the Lebesgue integral which makes sense for such functions.", "In the Appendix, we collect the basic facts and definitions of measure theory we need.", "Theorem 0.2 Let $K$ a compact subset of $\\mathbb {R}$ and let $\\ell :\\mathcal {C}(K)\\rightarrow \\mathbb {R}$ be a positive linear functional.", "Then, there is a unique finite Borel measure $\\mu $ such that $\\mu (K)=\\Vert \\ell \\Vert \\ _{C(K)^*}$ and $ \\ell f=\\int _Kfd\\mu .", "$ Proof.", "The proof proceeds in stages.", "i) Integral representation.", "Let $[a,b]$ be a closed and bounded interval containing $K$ .", "Let $r:\\mathcal {C}[a,b]\\longrightarrow \\mathcal {C}(K)$ be the restriction operator, that is, for every $f\\in \\mathcal {C}[a,b]$ , $r(f)(x)=f(x)$ for $x\\in K$ .", "It is clear that $r$ is a bounded linear operator, so we can define its transpose operator, see [22] p.11, also known as adjoint, see [21].", "Recall $r^t$ is defined as follows $r^t: \\mathcal {C}(K)^* \\rightarrow \\mathcal {C}[a,b]^*$ , $r^t(\\ell )(f)=\\ell (r(f))$ for $f\\in \\mathcal {C}[a,b]$ ; the expression $\\ell (r(f))$ assigns a scalar to each function $f\\in \\mathcal {C}[a,b]$ .", "Let $\\ell $ be a positive linear functional in $C(K)$ and we define $Lf=r^t(\\ell ) (f)=\\ell (rf)$ .", "Since $\\ell $ and $r$ are positive linear functionals, so is $L$ and we can apply theorem REF and (c) in the Appendix to find a monotone increasing function $\\alpha $ and an associated Borel measure $\\mu $ such that $Lf=r^t(\\ell )(f)=\\int _a^bfd\\alpha =\\int _a^bfd\\mu $ for every $f\\in \\mathcal {C}[a,b]$ .", "Denote $K^c := [a,b]\\setminus K$ .", "We will show that $\\mu (K^c)=0$ .", "Let $\\varepsilon >0$ and choose $F_\\varepsilon $ as a closed subset of $K^c$ such that $\\mu (K^c\\setminus F_\\varepsilon )<\\varepsilon ,$ see (a) in the Appendix.", "Let $\\tilde{f}\\in \\mathcal {C}[a,b]$ be a continuous function such that $\\tilde{f}(x)=1$ if $x\\in K$ , $\\tilde{f}(x)=0$ if $x\\in F_\\varepsilon $ and $\\Vert \\tilde{f}\\Vert _\\infty \\le 1$ .", "One can take for instance $ \\tilde{f}(x)=\\frac{d(x,F_\\varepsilon )}{d(x,F_\\varepsilon )+d(x,K)},$ where $d(x,A)=\\inf _{y\\in A}|x-y|$ .", "Note that since $|d(x,A)-d(y,A)|\\le |x-y|$ the function $d(x,A)$ is even uniformly continuous, (cf.", "Urysohn's Lemma.", "[5], 4.15.).", "Therefore $L(\\tilde{f})=\\int _a^b\\tilde{f}d\\mu =\\int _Kd\\mu +\\int _{K^c\\backslash F_\\varepsilon }\\tilde{f}d\\mu +\\int _{F_\\varepsilon }\\tilde{f}d\\mu $ The third integral on the right is equal zero, by definition of $\\tilde{f}$ .", "We can estimate the second integral as follows, $0\\le \\int _{K^c\\backslash F_\\epsilon }\\tilde{f}d\\mu \\le \\int _{K^c\\backslash F_\\varepsilon }d\\mu =\\mu (K^c\\backslash F_\\varepsilon )<\\varepsilon , $ since $\\tilde{f}\\le 1$ and using (REF ).", "Then $ L(\\tilde{f})<\\int _Kd\\mu +\\varepsilon =\\mu (K)+\\varepsilon .$ We have that $\\mu (K)+\\mu (K^c)=\\int _a^bd\\mu =L(\\mathbf {1}_{[a,b]} )=L(\\tilde{f})<\\mu (K)+\\varepsilon ,$ The third equality follows from $r(\\tilde{f})=r(\\mathbf {1}_{[a,b]} )$ .", "Thus $0\\le \\mu (K^c)<\\varepsilon $ , since $\\mu (K)<\\infty $ .", "To conclude, let $f\\in \\mathcal {C}(K)$ and $f^*$ a continuous extension of $f$ to the closed interval $[a,b]$ .", "We can do this extension taking, for example, straight lines as follows: since $K^c$ is an open subset of $[a,b]$ , it is at most a countable union of pairwise disjoint open intervals $(\\alpha _i, \\beta _i)$ intersected with the interval $[a,b]$ , (see Lindeloef's thm., [17] Prop.9.", "p.40).", "For $ x\\in (\\alpha _i,\\beta _i)$ we define $f^*(x) =(1-t) f(\\alpha _i)+t f(\\beta _i)$ if $x= \\alpha _i (1-t)+t \\beta _i$ for $t\\in (0,1).$ The function $f^*$ is continuous on the interval $[a,b]$ since on $K$ coincides with the continuous function $f$ and on $K^c$ consists of straight lines, (cf.", "Tietze's Theorem [5], 4.16).", "Then we have $ \\ell (f)=\\ell (r(f^*))=Lf^*=\\int _a^b f^*d\\alpha =\\int _a^b f^*d\\mu =\\int _Kf^*d\\mu =\\int _Kfd\\mu .$ as was to be shown.", "ii) Conservation of norm.", "Take $f\\in \\mathcal {C}(K)$ such that $\\Vert f\\Vert _\\infty \\le 1$ .", "Since (REF ) holds we have, $|\\ell (f)|= \\left|\\int _Kfd\\mu \\right|\\le \\Vert f\\Vert _\\infty \\,\\mu (K) \\le \\mu (K).$ For the reverse inequality, let $\\textbf {1}(x)=1$ for all $x$ , as defined in remark (3), so $\\Vert \\ell \\Vert \\ge |\\ell (\\textbf {1})|=\\left|\\int _K \\textbf {1}\\, d\\mu \\right| = \\mu (K),$ we can conclude that $\\mu (K) = \\Vert \\ell \\Vert $ .", "iii) Uniqueness.", "Suppose $\\mu $ and $\\nu $ are finite measures that satisfy (REF ).", "Since $\\mu $ and $\\nu $ are regular measures, from (a) in the Appendix, it is enough to show that $\\mu (C) =\\nu (C)$ for any closed set $C$ of $K$ .", "Let $C$ a nonempty closed set of $K$ and set $f_k (x):= \\max \\lbrace 0,1- k d(x,C)\\rbrace $ for all $k$ and $x\\in K$ , where $d(x,C)=\\inf _{y\\in C}|x-y|$ .", "These functions are bounded, by 0 and 1, and continuous.", "Thus $f_k$ belongs to $\\mathcal {C}(K)$ for all $k$ .", "Notice that they form a sequence that decreases to the indicator of $C$ , i.e., $f_k\\downarrow \\textbf {1}_C$ , where $\\textbf {1}_C(x)=1$ if $x\\in C$ and $\\textbf {1}_C(x)=0$ if $x\\notin C$ .", "Thus, for all $k$ we must have that $\\int _K f_k d\\mu =\\int _K f_k d\\nu $ , and so we can use the dominated convergence theorem, see (b) in the Appendix, to conclude that $\\mu (C) = \\lim _k \\int _K f_k d\\mu =\\lim _k \\int _K f_k d\\nu = \\nu (C).", "$ Integral representation.", "Let $[a,b]$ be a closed and bounded interval containing $K$ .", "Let $r:\\mathcal {C}[a,b]\\longrightarrow \\mathcal {C}(K)$ be the restriction operator, that is, for every $f\\in \\mathcal {C}[a,b]$ , $r(f)(x)=f(x)$ for $x\\in K$ .", "It is clear that $r$ is a bounded linear operator, so we can define its transpose operator, see [22] p.11, also known as adjoint, see [21].", "Recall $r^t$ is defined as follows $r^t: \\mathcal {C}(K)^* \\rightarrow \\mathcal {C}[a,b]^*$ , $r^t(\\ell )(f)=\\ell (r(f))$ for $f\\in \\mathcal {C}[a,b]$ ; the expression $\\ell (r(f))$ assigns a scalar to each function $f\\in \\mathcal {C}[a,b]$ .", "Let $\\ell $ be a positive linear functional in $C(K)$ and we define $Lf=r^t(\\ell ) (f)=\\ell (rf)$ .", "Since $\\ell $ and $r$ are positive linear functionals, so is $L$ and we can apply theorem REF and (c) in the Appendix to find a monotone increasing function $\\alpha $ and an associated Borel measure $\\mu $ such that $Lf=r^t(\\ell )(f)=\\int _a^bfd\\alpha =\\int _a^bfd\\mu $ for every $f\\in \\mathcal {C}[a,b]$ .", "Denote $K^c := [a,b]\\setminus K$ .", "We will show that $\\mu (K^c)=0$ .", "Let $\\varepsilon >0$ and choose $F_\\varepsilon $ as a closed subset of $K^c$ such that $\\mu (K^c\\setminus F_\\varepsilon )<\\varepsilon ,$ see (a) in the Appendix.", "Let $\\tilde{f}\\in \\mathcal {C}[a,b]$ be a continuous function such that $\\tilde{f}(x)=1$ if $x\\in K$ , $\\tilde{f}(x)=0$ if $x\\in F_\\varepsilon $ and $\\Vert \\tilde{f}\\Vert _\\infty \\le 1$ .", "One can take for instance $ \\tilde{f}(x)=\\frac{d(x,F_\\varepsilon )}{d(x,F_\\varepsilon )+d(x,K)},$ where $d(x,A)=\\inf _{y\\in A}|x-y|$ .", "Note that since $|d(x,A)-d(y,A)|\\le |x-y|$ the function $d(x,A)$ is even uniformly continuous, (cf.", "Urysohn's Lemma.", "[5], 4.15.).", "Therefore $L(\\tilde{f})=\\int _a^b\\tilde{f}d\\mu =\\int _Kd\\mu +\\int _{K^c\\backslash F_\\varepsilon }\\tilde{f}d\\mu +\\int _{F_\\varepsilon }\\tilde{f}d\\mu $ The third integral on the right is equal zero, by definition of $\\tilde{f}$ .", "We can estimate the second integral as follows, $0\\le \\int _{K^c\\backslash F_\\epsilon }\\tilde{f}d\\mu \\le \\int _{K^c\\backslash F_\\varepsilon }d\\mu =\\mu (K^c\\backslash F_\\varepsilon )<\\varepsilon , $ since $\\tilde{f}\\le 1$ and using (REF ).", "Then $ L(\\tilde{f})<\\int _Kd\\mu +\\varepsilon =\\mu (K)+\\varepsilon .$ We have that $\\mu (K)+\\mu (K^c)=\\int _a^bd\\mu =L(\\mathbf {1}_{[a,b]} )=L(\\tilde{f})<\\mu (K)+\\varepsilon ,$ The third equality follows from $r(\\tilde{f})=r(\\mathbf {1}_{[a,b]} )$ .", "Thus $0\\le \\mu (K^c)<\\varepsilon $ , since $\\mu (K)<\\infty $ .", "To conclude, let $f\\in \\mathcal {C}(K)$ and $f^*$ a continuous extension of $f$ to the closed interval $[a,b]$ .", "We can do this extension taking, for example, straight lines as follows: since $K^c$ is an open subset of $[a,b]$ , it is at most a countable union of pairwise disjoint open intervals $(\\alpha _i, \\beta _i)$ intersected with the interval $[a,b]$ , (see Lindeloef's thm., [17] Prop.9.", "p.40).", "For $ x\\in (\\alpha _i,\\beta _i)$ we define $f^*(x) =(1-t) f(\\alpha _i)+t f(\\beta _i)$ if $x= \\alpha _i (1-t)+t \\beta _i$ for $t\\in (0,1).$ The function $f^*$ is continuous on the interval $[a,b]$ since on $K$ coincides with the continuous function $f$ and on $K^c$ consists of straight lines, (cf.", "Tietze's Theorem [5], 4.16).", "Then we have $ \\ell (f)=\\ell (r(f^*))=Lf^*=\\int _a^b f^*d\\alpha =\\int _a^b f^*d\\mu =\\int _Kf^*d\\mu =\\int _Kfd\\mu .$ as was to be shown.", "Conservation of norm.", "Take $f\\in \\mathcal {C}(K)$ such that $\\Vert f\\Vert _\\infty \\le 1$ .", "Since (REF ) holds we have, $|\\ell (f)|= \\left|\\int _Kfd\\mu \\right|\\le \\Vert f\\Vert _\\infty \\,\\mu (K) \\le \\mu (K).$ For the reverse inequality, let $\\textbf {1}(x)=1$ for all $x$ , as defined in remark (3), so $\\Vert \\ell \\Vert \\ge |\\ell (\\textbf {1})|=\\left|\\int _K \\textbf {1}\\, d\\mu \\right| = \\mu (K),$ we can conclude that $\\mu (K) = \\Vert \\ell \\Vert $ .", "Uniqueness.", "Suppose $\\mu $ and $\\nu $ are finite measures that satisfy (REF ).", "Since $\\mu $ and $\\nu $ are regular measures, from (a) in the Appendix, it is enough to show that $\\mu (C) =\\nu (C)$ for any closed set $C$ of $K$ .", "Let $C$ a nonempty closed set of $K$ and set $f_k (x):= \\max \\lbrace 0,1- k d(x,C)\\rbrace $ for all $k$ and $x\\in K$ , where $d(x,C)=\\inf _{y\\in C}|x-y|$ .", "These functions are bounded, by 0 and 1, and continuous.", "Thus $f_k$ belongs to $\\mathcal {C}(K)$ for all $k$ .", "Notice that they form a sequence that decreases to the indicator of $C$ , i.e., $f_k\\downarrow \\textbf {1}_C$ , where $\\textbf {1}_C(x)=1$ if $x\\in C$ and $\\textbf {1}_C(x)=0$ if $x\\notin C$ .", "Thus, for all $k$ we must have that $\\int _K f_k d\\mu =\\int _K f_k d\\nu $ , and so we can use the dominated convergence theorem, see (b) in the Appendix, to conclude that $\\mu (C) = \\lim _k \\int _K f_k d\\mu =\\lim _k \\int _K f_k d\\nu = \\nu (C).", "$ $\\Box $ Remarks (a) It is possible to represent the linear positive functionals acting on $\\mathcal {C}(K)$ as Riemann-Stieltjes integrals, similar to the original work of F. Riesz.", "This follows immediately from the chain of equalities (REF ).", "The caveat is that we cannot use $f$ directly in order to define the Riemann-Stieltjes integral, but any continuous extension of $f$ works, cf.", "theorem REF below.", "This integral is independent of the extension of $f$ .", "(b) As just seen, the use of compact set $K$ above allows us to extend the continuous functions to the entire interval $[a,b]$ , using an elementary version of the Tietze's theorem.", "This construction is in general not possible if $K$ is an arbitrary subset of the real line.", "It is possible to represent the linear positive functionals acting on $\\mathcal {C}(K)$ as Riemann-Stieltjes integrals, similar to the original work of F. Riesz.", "This follows immediately from the chain of equalities (REF ).", "The caveat is that we cannot use $f$ directly in order to define the Riemann-Stieltjes integral, but any continuous extension of $f$ works, cf.", "theorem REF below.", "This integral is independent of the extension of $f$ .", "As just seen, the use of compact set $K$ above allows us to extend the continuous functions to the entire interval $[a,b]$ , using an elementary version of the Tietze's theorem.", "This construction is in general not possible if $K$ is an arbitrary subset of the real line.", "As a consequence of the previous results, we shall see that two spaces of functionals are practically the same.", "One of the spaces consists of Lebesgue integrals on compact subsets of $[a,b]$ and the other of Riemann-Stieltjes integrals over the whole interval $[a,b]$ .", "In this way we show how the Lebesgue integral representation, that was introduced to represent functionals in the case of general compact sets, can be seen as a Riemann-Stieltjes integral.", "To state this precisely we need to introduce the terms isomorphic and constant in $K^c$.", "A transformation $T$ which preserves the norm, that is $ \\Vert Tx\\Vert =\\Vert x\\Vert $ , is called an isometry.", "Two normed vector spaces $X$ and $Y$ are said to be isomorphic if there is a linear, bijective, isometry $T: X\\rightarrow Y$ .", "Such functions are called isomorphisms.", "Since an isomorphism preserves the linear as well as the metric structure of the spaces, two isomorphic spaces can be considered identical, the isomorphism corresponding just to a labeling of the elements.", "We say that the monotone function $\\alpha \\text{ \\it is constant in } K^c$ if it is constant in each interval of $K^c$ .", "Recall that $\\mathcal {C}(X)^*_+$ denotes the set of positive linear functionals on $\\mathcal {C}(X)$ .", "The result mentioned above can be then stated as follows: Theorem 0.3 The normed spaces $ \\left\\lbrace L_\\alpha \\in \\mathcal {C}[a,b]^*_+ : \\alpha \\text{ is constant in } K^c \\right\\rbrace $ and $ \\mathcal {C}(K)^*_+$ are isomorphic.", "Before we prove this theorem we need two preparatory results.", "Proposition 1 $r^t :\\mathcal {C}(K)^*_+ \\rightarrow \\mathcal {C}[a,b]^*_+$ is an isometry.", "$\\Vert r^t \\ell \\Vert _{\\mathcal {C}[a,b]^*_+}=V(\\alpha )=\\alpha (b)-\\alpha (a)=\\mu ([a,b])=\\mu (K) +\\mu ([a,b]\\setminus K)=\\mu (K)=\\Vert \\ell \\Vert _{\\mathcal {C}(K)^*_+}$ .", "The first equality follows from Theorem REF .", "The function $\\alpha $ depends on $\\ell $ .", "The second is the definition of the total variation of $\\alpha $ and the third is the definition of $\\mu $ .", "The last two equalities follow from the construction of Theorem REF .", "We denote the range of $r^t$ by Rang $r^t =\\left\\lbrace L\\in \\mathcal {C}[a,b]^*_+: \\exists l \\in \\mathcal {C}(K)^*_+ \\text{ s.t. }", "L=r^t l \\right\\rbrace $ Proposition 2 Let $ L_\\alpha $ denote the functional with corresponding monotone function $\\alpha $ as introduced in (REF ).", "Then $\\text{Rang } r^t=\\left\\lbrace L_\\alpha \\in \\mathcal {C}[a,b]^*_+ : \\alpha \\text{ is constant in } K^c \\right\\rbrace $ \"$\\subset $ \" Let $L\\in \\text{Rang } r^t\\subset \\mathcal {C}[a,b]^*_+$ .", "Then there exists $\\ell \\in \\mathcal {C}(K)^*_+$ such that as in (REF ) $r^t(\\ell )(f)=Lf=L_\\alpha f=\\int _a^bfd\\alpha =\\int _a^bfd\\mu $ As was shown in the proof of Theorem REF i), $\\mu (K^c) =0$ .", "Since $K^c$ is a countable union of intervals, these have $\\mu $ measure zero.", "By the relation which is given in (REF ) below, between the measure $\\mu $ and the monotone function $\\alpha $ we conclude that $\\alpha $ is constant in each one of the intervals of $K^c$ . \"", "$\\supset $ \" Let $ L_\\alpha \\in \\mathcal {C}[a,b]^*_+$ with $\\alpha $ constant in each interval of $K^c$ and $\\mu $ be the measure associated with this $\\alpha $ , as in Appendix (c).", "Define $\\ell \\in \\mathcal {C}(K)^*_+$ as $\\ell h=\\int _Kh d\\mu $ .", "We shall show that $r^t(\\ell )(f)=L_\\alpha f$ for every $f\\in \\mathcal {C}[a,b]$ .", "Since $\\alpha $ constant in each interval of $K^c$ this implies, using again (REF ), that $\\mu (K^c) =0$ .", "Then we have $ L_\\alpha f= \\int _a^bfd\\alpha =\\int _a^bfd\\mu = \\int _Kfd\\mu =\\int _Kr(f)d\\mu = \\ell (r(f))= r^t(\\ell )(f) $ where $r(f)$ denotes, as in Theorem (REF ) i) above, the restriction of $f$ to $K$ .", "[Proof of Theorem REF ].", "From Proposition REF and Proposition REF it follows that $r^t$ is a bijective isometry.", "Since $r^t$ is linear as follows from its definition, then it is an isomorphism.", "Acknowledgments We thank C. Bosch and Ma.", "C. Arrillaga for useful comments.", "We are grateful to Ma.", "R. Sanchez for her help in the search of bibliographical information.", "section10em1.7em1.2emAppendix A collection of subsets $\\mathcal {A}$ of $X$ is called an $\\sigma $ -algebra if it is closed under finite (countable) union, complements and $X\\in \\mathcal {A}$ .", "If our space is $\\mathbb {R}$ , the Borel $\\sigma $ -algebra, $\\mathcal {B}_{\\mathbb {R}}$ , is the smallest $\\sigma $ -algebra containing all the open intervals.", "A function $\\mu : \\mathcal {A}\\rightarrow [0,\\infty ]$ , where $\\mathcal {A}$ is a $\\sigma $ -algebra, it is called a measure if it is countable additive, that is $\\mu (\\bigcup A_n) =\\sum \\mu (A_n)$ whenever $\\lbrace A_n\\rbrace $ is a disjoint sequence of elements in $\\mathcal {A}$ , and $\\mu (\\varnothing ) =0$ .", "A Borel measure is a measure defined on $\\mathcal {B}_{\\mathbb {R}}$ .", "We say that a measure is regular if every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.", "A function $f$ from $(X,\\mathcal {A}, \\mu )$ to $(\\mathbb {R}, B_{\\mathbb {R}} )$ is $\\mathcal {A}$ -measurable if $\\lbrace x: f(x)\\le t\\rbrace \\in \\mathcal {A}$ for all $t\\in \\mathbb {R}$ .", "The following results are used in the proof of Theorem REF .", "(a) Every Borel measure in a metric space is regular.", "We will only use inner regularity, that is, for every Borel set $A$ and every $\\varepsilon >0$ there exist a compact set $F_\\varepsilon $ such that $F_\\varepsilon \\subset A$ and $\\mu (A\\setminus F_\\epsilon )<\\epsilon $ .", "[2] Thm 7.1.7. or [3] Lemma 1.5.7.", "(b) (Dominated convergence theorem) Let $(X,\\mathcal {A}, \\mu )$ a measure spaces.", "Let $g$ be a $[0,\\infty ]$ -valued integrable function on $X$ , that is, $\\int g d\\mu <\\infty $ , and let $f, f_1, f_2, \\ldots $ real-valued $\\mathcal {A}$ -measurable functions on $X$ such that $f(x)= \\lim _n f_n (x)$ and $|f_n(x)|\\le g(x)$ .", "Then $f$ and $\\lbrace f_n\\rbrace $ are integrable and $\\int f d\\mu = \\lim _n \\int f_n d\\mu $ .", "(c) Given a normalized monotone function $\\alpha $ in the closed interval $[a,b]$ , there is a unique Borel measure $\\mu $ associated with it.", "This can be seen as follows (see for example [4]): for $a\\le s \\le t \\le b\\,$ let define $\\, \\langle s, t]$ where Every Borel measure in a metric space is regular.", "We will only use inner regularity, that is, for every Borel set $A$ and every $\\varepsilon >0$ there exist a compact set $F_\\varepsilon $ such that $F_\\varepsilon \\subset A$ and $\\mu (A\\setminus F_\\epsilon )<\\epsilon $ .", "[2] Thm 7.1.7. or [3] Lemma 1.5.7.", "(Dominated convergence theorem) Let $(X,\\mathcal {A}, \\mu )$ a measure spaces.", "Let $g$ be a $[0,\\infty ]$ -valued integrable function on $X$ , that is, $\\int g d\\mu <\\infty $ , and let $f, f_1, f_2, \\ldots $ real-valued $\\mathcal {A}$ -measurable functions on $X$ such that $f(x)= \\lim _n f_n (x)$ and $|f_n(x)|\\le g(x)$ .", "Then $f$ and $\\lbrace f_n\\rbrace $ are integrable and $\\int f d\\mu = \\lim _n \\int f_n d\\mu $ .", "Given a normalized monotone function $\\alpha $ in the closed interval $[a,b]$ , there is a unique Borel measure $\\mu $ associated with it.", "This can be seen as follows (see for example [4]): for $a\\le s \\le t \\le b\\,$ let define $\\, \\langle s, t]$ where Let $\\mathcal {F}_0 = \\bigg \\lbrace \\bigcup _{\\text{ finite}}\\langle s_k, t_k]: \\langle s_k, t_k] \\subset [a,b] \\text{ pairwise disjoint}\\bigg \\rbrace $ Then $\\mathcal {F}_0$ is an algebra of subsets of $[a,b]$ and therefore we can define a set function as $\\mu _0 \\left(\\bigcup _{\\text{finite}}\\langle s_k, t_k] \\right)= \\sum _{\\text{finite}} \\alpha (t_k)-\\alpha (s_k).$ Moreover, $\\mu _0$ has a unique extension to a measure in the smallest $\\sigma $ -algebra containing $\\mathcal {F}_0$ (Caratheodory's Theorem).", "See [1] .", "Moreover, for any continuous function $f$ it happens that $\\int _a^bfd\\alpha =\\int _a^bfd\\mu $ where the integral on the left is a Riemann-Stieltjes integral, whereas the integral on the right is an integral in the sense of Lebesgue." ] ]
1606.05026
[ [ "Flare Generated Shock Wave Propagation Through Solar Coronal Arcade\n Loops and Associated Type II Radio Burst" ], [ "Abstract This paper presents multiwavelength observations of a flare-generated type II radio burst.", "The kinematics of the shock derived from the type II closely match a fast EUV wave seen propagating through coronal arcade loops.", "The EUV wave was closely associated with an impulsive M1.0 flare without a related coronal mass ejection, and was triggered at one of the footpoints of the arcade loops in active region NOAA 12035.", "It was initially observed in the AIA 335 \\AA\\ images with a speed of ~800 km/s and accelerated to ~1490 km/s after passing through the arcade loops.", "A fan-spine magnetic topology was revealed at the flare site.", "A small, confined filament eruption (~340 km/s) was also observed moving in the opposite direction to the EUV wave.", "We suggest that breakout reconnection in the fan-spine topology triggered the flare and associated EUV wave that propagated as a fast shock through the arcade loops." ], [ "INTRODUCTION", "Type II radio bursts are considered to be an indirect signature of shocks [42], [61].", "Most of the coronal type II radio bursts are excited by coronal mass ejection (CME) pistons [57], [15], [21].", "Another source of type II radio bursts are shocks generated by blast waves created by a pressure increase at the flare site [58], [24], [25].", "The flare impulsive phase and CME acceleration phase often occur simultaneously [72], therefore, it is difficult to confirm the exact driver of coronal shocks in terms of flare blast or CME piston.", "An argument in favour of the CME driver is that confined flares (even X-class flares) do not produce coronal shocks.", "On the other hand, shocks are only seen when the associated CME is accompanied by a flare [14], [20].", "In support of the flare-driven shock, some studies using SOHO/EIT and LASCO coronagraph images suggest the formation of a type II radio burst during the flare impulsive phase when the CME acceleration phase starts 10-20 min later, or the CME is too slow to drive a shock wave ($<$ 500 km s$^{-1}$ ) [38], [36], [37].", "The shock waves in such events were associated with impulsive/short duration flare events with rise times less than 4 min.", "These studies have used low resolution SOHO/EIT images (12 min cadence) and it was therefore not possible to track the low coronal eruptions (plasmoid/jets) related to short duration impulsive flares and their associated shock waves ($\\sim $ 1000 km $^{-1}$ ) in extreme ultraviolet (EUV) images.", "Now data from the Atmospheric Image Assembly (AIA; [33]) on board the Solar Dynamics Observatory (SDO; [47]) provides an opportunity to study fast EUV waves and associated plasma ejections in the low corona and it is therefore possible to investigate the wave driver and kinematics in more detail.", "[28] provided evidence of a blast wave ($>$ 1000 km s$^{-1}$ ) propagating alongside and ahead of an erupting plasmoid that produced a metric type II radio burst.", "Recently, [29] reported a clear observation of a fast-mode wave propagating along arcade loops and its partial reflection from the other footpoint.", "They also observed a second fast EUV wavefront propagating perpendicular to the arcade loops which triggered a metric type II radio burst.", "The wave was excited by an impulsive C-class flare (without a CME) at one of the footpoints of the arcade loops.", "Solar eruptions are generally associated with EUV waves.", "These waves were discovered by SOHO/EIT [56].", "There has been a long-lasting debate about their nature (true wave or pseudo wave) and driver (in terms of CME or flare) ([61], [59], [64], [66], [46], [35], [65] and references cited therein).", "At present, high resolution observations from SDO/AIA suggest the existence of two wavefronts (i.e., fast and slow) [10], [27], [30].", "The fast wavefront speed (of the order of $\\sim $ 1000 km s$^{-1}$ ) is about three times larger than the slow one as predicted in the numerical simulation by [9].", "The fast wavefront is either a true fast-mode wave or an magnetohydrodynamic (MHD) shock ahead of the slow wavefront.", "The slow wavefront is basically expanding CME loops interpreted as a pseudo wave (non-wave component).", "The fast EUV wavefront triggers transverse oscillations in coronal loops [68], [4], [52], [3], [27] or filament channels when passing through these structures [1], [53] and shows reflection, refraction and transmission when it interacts with neighboring active regions [30] or coronal holes [22], [44].", "On the other hand, the slow wavefront generally stops at the boundary of an active region or at magnetic separatrices [17], therefore, supporting the non-wave interpretation.", "In this paper, we study a type II radio burst associated with a fast EUV wave observed during an impulsive C-class flare (without a CME) on 16 April 2014.", "The EUV wave propagates through arcade loops and triggers transverse oscillations in the loop system.", "We focus on the wave characteristics, its propagation, and excitation.", "In section 2, we present the observation and results.", "In the last section, we discuss the results and draw some conclusions.", "The type II burst was seen in the dynamic radio spectra from e-Callisto (Roswell station, 220-450 MHz) and RSTN (Sagamore hill station, 25-180 MHz) from about 19:57:40 UT to 20:10 UT (Figure REF (a,b)).", "The composite spectrum shows both the fundamental and harmonic of the type II burst.", "The fundamental shows band splitting probably caused by emission from the pre- and post-shock plasma [60], [62], [13].", "The splitting is typical of that seen in other CME-associated type IIs and corresponds to a compression ratio of 1.6 around 120 MHz and 1.4 around 40 MHz.", "We used the Newkirk one and two-fold density model [43] to estimate the height of the type II exciter by selecting a few data points (marked by the + symbol) from the second harmonic of the type II burst.", "The time-height plot is shown in Figure REF (a).", "The average speed (from the linear fit) of the type II emission source is $\\sim $ 670 and $\\sim $ 810 km s$^{-1}$ for the one- and two-fold models, respectively.", "An unusual feature of the radio spectrum is a positive drifting feature from $\\sim $ 220 MHz to $\\sim $ 250 MHz starting simultaneously with the type II, which implies that there were sunward propagating energetic electrons at the same time as the outward moving shock creating the type II.", "To estimate the exciter speed, we selected the data points marked with a green + sign and converted the frequencies into the emission heights in the corona using the Newkirk density models (considering fundamental emission).", "The downward moving source speed (by a linear fit to the emission heights) is $\\sim $ 110 and $\\sim $ 130 km s$^{-1}$ from the Newkirk one-fold and two-fold density models, respectively.", "There was also a type III radio burst at high frequencies (220-450 MHz) but not below 200 MHz suggesting acceleration of sub-relativistic electrons that did not escape into interplanetary space.", "To find the source of the type II and other features of the radio spectrum, we investigated EUV images and magnetic field data from AIA and the Heliospheric and Magnetic Imager (HMI; [51]) on SDO, images and spectra from Hinode's XRT (X-Ray Telescope; [19]) and EIS (EUV Imaging Spectrometer; [16]), hard X-ray images from RHESSI (Reuven Ramaty High Energy Solar Spectroscopic Imager, [34]), and SOHO/LASCO [7], [70] and STEREO-B COR-1 [69], [23] coronagraph images.", "At the time of the type II, there was an M1.0 flare in active region NOAA 12035, when it was located in the southeast (S19E12) with a $\\beta \\gamma $ magnetic configuration.", "The M1.0 flare was an impulsive short duration flare which started at 19:54 UT, peaked at 19:59, and ended at 20:04 UT.", "Figure REF (c) shows the RHESSI X-ray flux in the 6-12 keV, 12-25 keV, and 25-50 keV channels.", "The type III radio burst correlates with the hard X-ray flux in 12-25 and 25-50 keV channels, suggesting the emission from non-thermal electrons.", "The 6-12 keV flux peaked slightly later ($\\sim $ 30 sec), which indicates the contribution from the thermal emission.", "A careful check of the coronagraph images revealed that there was no CME associated with this flare; however the EUV images showed a fast, outward propagating wave and the possible source of the positive drifting feature.", "This study utilizes AIA 94 Å (Fe10, Fe18, $T\\approx $ 1 MK, $T\\approx $ 6.3 MK), 131 Å (Fe8, Fe21, Fe23, i.e., 0.4, 10, 16 MK), 171 Å (Fe9, $T\\approx $ 0.7 MK), 193 Å (Fe12, Fe24, $T\\approx $ 1.5, 20 MK), 304 Å (He2, T$\\approx $ 0.05 MK), 335 Å (Fe16, T$\\approx $ 2.5 MK), and 1600 Å (C4 + continuum, $T\\approx $ 0.1 MK) images.", "Figure: (a,b) Dynamic radio spectrum from e-Callisto (Roswell station, 220-450 MHz) and RSTN (Sagamore hill station, 25-180 MHz).", "F and H denote the fundamental and second harmonic of the type II burst.", "The black pluses mark the frequencies used for obtaining the type II exciter speed.", "The green pluses mark the frequencies for calculating the speed of the positive drift feature.", "(c) RHESSI X-ray flux profiles in 6-12, 12-25, and 25-50 keV channels.", "Two vertical dotted lines represent the onset time of nonthermal particle acceleration in the radio (type III burst) and hard X-ray (12-50 keV) channels.Figure: (a) Time-height plots (R ⊙ _{\\odot }, above the solar surface) estimated from the Newkirk one-(diamond) and two-fold (triangle) density models using the second harmonic of the type II radio burst.", "The estimated shock speed from the linear-fit is ∼\\sim 670 and ∼\\sim 810 km s -1 ^{-1}, respectively, from the Newkirk one- and two-fold density models.", "The distance of the EUV wave from the flare center (Figure ) using AIA 335 (filled diamond) and 193 (filled circle) images is over-plotted in red.", "(b) EUV wave speed profile estimated from the time-distance values (red) taken from panel (a).", "GOES soft X-ray flux in 1-8 Å  channel is also included (dark green).Figure: AIA 335, 193 and 171 Å  running difference images showing the propagating EUV disturbance (marked by arrows) left to the flare site.", "The green box in panel (c) represents the size of the top panels (a,b).", "(An animation of this figure is available)Figure: Time-distance intensity (running difference) plots along the slices S1 (AIA 335 Å), S2, and S3 (AIA 193 Å).", "The red dashed curve in panel (a) shows the GOES soft X-ray flux profile in 1-8 Å  channel.Figure: (a,b) AIA 131 and 94 Å  images showing the flare and arcade loops.", "(c) AIA 131 Å  intensity (running difference) plot along the slice S4 (marked in panel(a)) showing the kink oscillation of the arcade loops.", "(d) Decaying sine function fit (red color) to the kink oscillating loop.", "The period of oscillation is ∼\\sim 210 s. The start time is ∼\\sim 20:04 UT.", "(An animation of this figure is available)Figure: (a,b) Potential field extrapolation of the active region (at ∼\\sim 19:45 UT) showing fan loops at the flare site.", "The closed and open field lines are shown by the black and white color, respectively.", "The fast-mode wave propagated through the closed loops (eastward) and triggered kink oscillation of the arcade loops.", "The red box in panel (a) represents the size of panel (b).", "(c,d) Hinode/XRT image of the active region overlaid by HMI magnetogram contours of positive (green) and negative (yellow) polarities.", "(e,f) AIA 1600 Å  image showing the quasi-circular flare ribbon.", "The contour levels are ±\\pm 500, ±\\pm 1000, and ±\\pm 1500 Gauss.Figure: (a-c) Stack plots of the AIA 94 Å  intensity along the slices S5 and S6.", "The red rectangular box indicates the EIS field of view (refer to the next figure).", "(d-f) AIA 304 Å  images overlaid by RHESSI X-ray contours in 6-12 keV (green) and 12-25 keV (blue) channels.", "The contour levels are 30%\\%, 50%\\%, 70%\\%, and 90%\\% of the peak intensity.", "HMI magnetogram contours of positive (white) and negative (black) polarities are overlaid in panel (e).", "R1 and R2 indicate the flare ribbons.", "(An animation of this figure is available)Figure: (a-c) Hinode/EIS Fe24 line intensity, velocity, and width images.", "(d-f) Fe23, Fe16, and Fe12 line intensity images.", "The start time of the scan is 19:57:13 UT." ], [ "EUV wave", "Figure REF displays running difference images in the AIA 335, 193, and 171 Å channels.", "The time difference ($\\Delta $ t) for the difference images is 1 minute.", "The AIA 335 Å image at 19:56:50 UT shows the EUV disturbance propagating to the left of the flare site.", "The wavefront is simultaneously observed in the AIA 171 and 193 channels as it propagates through the arcade loops connected to the flare site.", "Note that the initial phase (from 19:56:38 UT to 19:58:38 UT) of the EUV wavefront was best observed in the AIA 335 Å channel.", "Later it could not be observed in the AIA 335 Å channel; however, it was observed clearly in the AIA 193 channel from 19:58:42-20:01:35 UT.", "Figure REF (c,d) shows the semi-circular wavefront (F, yellow arrows) moving outward towards the eastern limb after passing through the arcade loops.", "We did not see a clear wavefront in the AIA 171 Å channel, but a long coronal loop can be seen expanding at 19:56:37 UT and shortly afterwards contracting in the 171 Å images of the AIA 171, 131 and 94 intensity movie as the loop was crossed by the wave.", "At 19:59:35 UT, we see brightening in the funnel loops (in the AIA 171 channel) possibly caused by compression generated by the propagating wavefront (Figure REF (e)) and later we see a part of the wavefront at 20:00:23 UT (marked by green arrows).", "The details of the propagating wavefront can be viewed in the AIA 335, 193, and 171 running-difference movie.", "The interesting points to be noted here are: (i) the wavefront is highly directive toward the eastern limb, (ii) the wavefront is not moving in all the directions, as generally seen in the case of an expanding CME bubble, (iii) there is only a single wavefront not multiple wave trains, (iv) transverse loop oscillations are seen throughout the arcade after the passage of the wave through them, (v) the flare occurred at one of the footpoints of the arcade loops, (vi) we do not see any CME loop or flux rope running behind the propagating wavefront in any of the AIA channels.", "To estimate the speed of the propagating disturbance, we selected the straight slices S1, S2, and S3 shown in Figure REF (a,c) in the AIA 335 and 193 Å  channels.", "The flare center is chosen as the origin for all slices.", "The time-distance intensity (running difference) plot is displayed in Figure REF .", "S1 and S2 are selected almost in the same direction and the estimated speeds are the projected speeds of the wavefront.", "From a linear fit, we estimated the speed of the wavefront in the AIA 335 and 193 Å  channels, to be $\\sim $ 800$\\pm $ 40 km s$^{-1}$ and $\\sim $ 1490$\\pm $ 190 km s$^{-1}$ respectively.", "We assumed 5 pixels error in the distance of the wavefront.", "The speed along slice S3 was $\\sim $ 1360 $\\pm $ 210 km s$^{-1}$ .", "The initial speed of the front in the 193 Å  channel (S2) was $\\sim $ 740$\\pm $ 40 km s$^{-1}$ .", "The speed of the semi-circular front (outside the arcade loops) is almost similar in the 193 Å  channel and does not change significantly along S2 and S3.", "To show the kinematics of the wavefront, we derived the speed of the wavefront using time-distance measurements from the AIA 335 and 193 Å  channels (Figure REF (b)).", "We utilized 3-point Lagrangian interpolation method to estimate the speed of the wavefront.", "Initially, the wave accelerated ($\\sim $ 700-900 km s$^{-1}$ ) slowly while passing through the arcade loops.", "The speed of the wavefront jumped ($\\sim $ 2800 km s$^{-1}$ ) after leaving the arcade loops and subsequently showed rapid deceleration ($\\sim $ 2800-1100 km s$^{-1}$ ).", "To see the association of the EUV wave with the M1.0 flare, we over-plotted the GOES soft X-ray flux (dashed curve) from the 1-8 Å  channel in Figure REF (a).", "The EUV wave onset time ($\\sim $ 19:56 UT) closely matched the flare impulsive phase.", "The time-distance measurements of the EUV wave in the AIA 335 and 193 Å  channels have been over-plotted by the red + symbol and filled circles, respectively in Figure REF .", "The speed of the type II exciter from the Newkirk two-fold model is roughly consistent with the EUV wave speed observed in the AIA 335 Å  channel.", "An exact match cannot be expected because the radio gives the outward speed and the EUV the front speed across the disk.", "The previous studies have also suggested the consistency of the Newkirk two-fold model with the observed shock wave in the AIA [27], [29].", "Therefore, the EUV disturbance (shock wave) passing through the arcade loops is the exciter of the type II radio burst." ], [ "Transverse oscillations in the arcade loops", "The AIA 335, 193, and 131 Å  composite movie (running difference) shows transverse oscillations of many loops within the arcade.", "These oscillations were triggered when the fast EUV wave passed through the arcade loops.", "The EUV disturbance seems to be reflected back after reaching the opposite footpoint of the arcade loops.", "Although we observed the loop system from above, it is difficult to say whether it is due to reflection of the disturbance or transverse oscillations of arcade loops.", "It is very likely that the fast wave propagates not only along the arcade loops but also across/perpendicular to the loop system [29].", "It is quite difficult to extract the oscillation of an individual loop due to the orientation and mixing of the complex loop system in the 193 and 171 Å images.", "However, some of the loops have been heated and are visible in the 131 Å channel.", "We can use one of these to illustrate the oscillations.", "Figure REF (a) shows a stack plot (running difference intensity) along the slice S4 marked on the 131 Å  image (Figure REF (a)).", "In the 131 Å  channel, the oscillation starts $\\sim $ 20:03 UT, after the passage of the disturbance through it.", "We can clearly see four peaks.", "First of all, we determined the positions of the oscillating loop from the stack plot.", "We subtracted a second order polynomial profile in order to detrend the oscillation profile.", "We fitted a decaying sine function (y=A.", "cos[2$\\pi $ t/T+$\\phi $ ].e$^{-t/\\tau }$ ) to the oscillation profile (red curve).", "Where, A, $\\phi $ , T, and $\\tau $ are the initial amplitude, phase angle, period, and decay time of the oscillation.", "The estimated period and decay time of the oscillation are 210 s and 506 s, respectively.", "To determine the phase speed of the wave, we need to measure the loop length.", "This active region was lying nearly behind the western limb of STEREO-B.", "Therefore, we need to determine the 3D structure of the loop with a single point observation.", "We used the curvature radius maximization method [2] to determine the loop length.", "The estimated loop length was $\\sim $ 0.17 R$_\\odot $ (Figure REF (b)).", "The calculated phase speed of the wave (for the fundamental mode) is 2L/P$\\approx $ 1140 km s$^{-1}$ , and the Alfvén speed is 840 km s$^{-1}$ ." ], [ "Magnetic configuration", "In the previous sub-sections, we described the wave propagation through the arcade loops toward the eastern limb.", "Figure REF (a) displays the potential field extrapolation of the active region based on the HMI magnetogram at 19:45 UT (before the flare onset) .", "The black and white curved lines represent the closed and open field lines, respectively.", "A small negative polarity region is surrounded by opposite polarity (positive) regions at the flare site, creating a quasi-circular polarity inversion line (PIL).", "The enlarged view of the flare site (marked by the red box) is shown in Figure REF (b).", "We can see the fan loops connecting to the negative polarities from the surrounding opposite polarity field region.", "This morphology is quite similar to the fan-spine topology [45].", "Figure REF (c) shows the Hinode XRT image (Al-poly filter) of the active region before the flare (19:54:12 UT).", "Two sets of loops are observed here: lower small loops and higher loops, which is consistent with the extrapolated field lines.", "Note that the higher loops are most affected by the flare and associated EUV wave.", "Also the fan loops at the flare site are clearly observed in the XRT image (Figure REF (d)).", "These images are overlaid by the HMI magnetogram contours of positive (green) and negative (yellow) polarities to view the connectivity of the coronal loops.", "In Figure REF (e,f), we display the AIA 1600 Å  images during the flare (19:58:16 UT).", "Interestingly, we see the formation of a quasi-circular ribbon at the flare site.", "In addition, there is a remote ribbon toward the eastern side of the flare.", "These ribbons are the precipitation sites of non-thermal electrons accelerated during the flare.", "The formation of a quasi-circular ribbon confirms the fan-spine topology at the flare site [40].", "Therefore, reconnection most likely occurred at the null point of the fan-spine topology producing a quasi-circular ribbon.", "One of the footpoints of the higher loops was connected to the flare site, therefore, accelerated particles propagated along these loops and precipitated to the opposite footpoint to form the remote ribbon." ], [ "Eruption", "There was a small eruption at the time of the flare.", "To investigate the eruption, we analyzed AIA 94 and 304 Å  images.", "The evolution of the plasma at the site of the eruption is illustrated in Figure REF .", "Figure REF shows the stack plots of the AIA 94 Å intensity along the slices S5 and S6 (marked in Figure REF (c)).", "S5 is selected to show the timing along the quasi-circular ribbon (Figure REF (b)).", "The first brightening (ribbon) was at $\\sim $ 19:55:30 UT, followed by plasma ejections.", "The AIA 94 Å movie shows a sequential brightening that propagates along the quasi-circular ribbon in the counterclockwise direction with a speed $\\sim $ 80$\\pm $ 10 km s$^{-1}$ .", "The slice S5 was selected along the direction of the cool plasma ejection.", "Initially there was a small filament along the quasi-circular polarity inversion line (see AIA 304 and 94 composite movie).", "At flare onset (19:55:30 UT), the southern end of the filament, overlaid by hard X-ray contours in Figure REF (c), brightened at all wavelengths and the quasi-circular ribbon appeared, followed by eruption of the filament.", "The speed of the plasma ejection (i.e., small filament) was $\\sim $ 340$\\pm $ 40 km s$^{-1}$ .", "After the first ejection, we see a series of minor ejections below the filament, which look like multiple plasma blobs or small plasmoids.", "The AIA 171, 131, 94 intensity movie reveals the formation of postflare loops below the erupting filament.", "The eruption did not produce a CME.", "It looks as though the filament material was stopped by the overlying arcades.", "Therefore, the filament eruption failed (e.g., [31]).", "In the case of a piston-driven shock, the speed of the shock could be 2-3 times the speed of the driver.", "The wavefront should be initially located ahead of the driver and can decouple (from the driver) later to propagate freely.", "An important thing to note is that in this flare eruption and EUV wave move simultaneously in the opposite directions.", "Also, the speed of the EUV wave is about 3-4 times larger than the speed of the erupting plasma so it is unlikely that the wave was triggered by the filament eruption.", "If the wave was driven by the small filament; we should have seen the EUV wavefront ahead of the filament at least for some duration as reported by [30].", "An alternative mechanism for non-CME type IIs has been recently proposed by [55].", "They suggest that the rapid expansion of loops following reconnection may provide a suitable piston to generate shocks in the lower corona.", "In this event, we saw no evidence for rapidly expanding loops that could have driven the shock.", "Figure REF (d-f) shows the AIA 304 Å images (chromosphere and transition region) overlaid with the RHESSI X-ray contours in 6-12 keV (green) and 12-25 keV (blue) energy channels (refer to Figure REF (c) for the flux profiles).", "To construct the X-ray images, we used the PIXON algorithm [41] with an integration time of 30 s for each image.", "The evolution can be followed in the AIA 304 and 94 Å composite movie.", "The RHESSI 12-25 keV contours (Figure REF (d)) show two footpoint sources.", "If we compare them with the RHESSI flux profile in 12-25 keV, they are associated with the flare impulsive phase (mostly non-thermal emission).", "The erupting filament and two-ribbons (R1 and R2) underneath are marked in Figure REF (e).", "Note that R1 is a part of the quasi-circular ribbon.", "We observed RHESSI footpoint sources (below the filament) in 6-12 and 12-25 keV from 19:57:30-19:59:00 UT.", "These sources are almost co-spatial with the R1 and R2 ribbons.", "The position of the footpoint sources seem to have changed from the impulsive phase to the decay phase probably due to the change in the particle precipitation site.", "We speculate that it may be somehow connected with the sequential brightening observed in the counterclockwise direction which may change the particle precipitation site associated with the rise of the small filament.", "As mentioned before, we observed the formation of a remote ribbon (AIA 1600 Å) and a circular ribbon.", "Most of the particles accelerated during the flare were confined along the field lines (higher arcade loops) in the low corona.", "The electron beam from the reconnection site (null point of the fan-spine topology) precipitates downward into the chromosphere (along the fan loops), forming a circular ribbon.", "In addition, high energy electrons probably follow the higher arcade loops and cause a remote ribbon seen in the 1600 Å images in the eastward direction.", "In addition, we analysed the available Hinode/EIS spectra of the flare.", "The EIS field-of-view is shown in Figure REF (b) by a rectangular box (red color).", "It only covered part of the arcade loops located east of the flare site but not the flare ribbons.", "These are the arcade loops through which the fast EUV wave propagated.", "Figure REF (a-c) displays Fe24 (192.03 Å, log T=7.1) line intensity, velocity, and width profiles starting at 19:57:13 UT (from right to left).", "Note that by that time, the fast EUV wave had already passed through these loops (see Figure REF (a-b)), and therefore, we see the after-wave signatures in the spectra.", "Interestingly, we see strong blue shifts ($\\sim $ 50-100 km s$^{-1}$ ) along the higher loops in the Fe24 line.", "The same loops are also seen in the Fe23 line (263.76 Å, log T =7) (Figure REF (d)) confirming that these are 10 MK loops.", "The Fe24 line is considered the most prominent EIS lines during large flares [71].", "The observed blue shifted upflows may be generated by chromospheric evaporation along the higher loops.", "They coincide with hot arcade loops emanating from the flare site seen in AIA 131/94 Å images.", "The Fe16 (251.063 Å, log T =6.4) intensity image shows the higher coronal parts of the lower arcade loops that are also visible in the Hinode XRT image.", "The Fe12 (193.5 Å, log T=6.1) intensity image is more patchy because it is emitted from the cooler lower parts of the loops, closer to their photospheric footpoints." ], [ "DISCUSSION AND CONCLUSION", "We reported on the direct observation of a fast EUV wave propagating through coronal arcade loops and its associated type II radio burst (without a CME).", "Initially, the wave was best observed in the AIA 335 Å with a speed of $\\sim $ 800 km s$^{-1}$ and later in the AIA 193 and 171 Å channels (1490 km s$^{-1}$ ).", "The fast wave was most likely triggered during breakout reconnection in a fan-spine topology.", "At the flare onset site, a quasi-circular ribbon was seen just before a small, failed-filament eruption ($\\sim $ 340 km s$^{-1}$ ) and, in the opposite direction, launching of a fast EUV wave across the arcade loops.", "We rule out the small filament as a driver of the EUV wave because (i) the speed was $\\sim $ 3-4 times smaller than the speed of the fast EUV wave, (ii) it moves in the opposite direction, (iii) we do not see the EUV wavefront ahead of the filament apex as expected in the case of piston driven shock.", "The propagation of fast EUV waves is generally affected by active regions and coronal holes.", "The EUV waves may suffer reflection, trapping, and transmission when they encounter with nearby AR or coronal holes [22], [44], [30], [29].", "The speed of the EUV wave was initially lower as it passed through the arcade loops.", "Later after it had escaped, it speeded up.", "A similar jump in shock speed was inferred from the dynamic radio spectrum (type II radio burst).", "The jump probably occurred because the arcade loops have higher density than the corona, so the wave could travel faster in the corona.", "A shock wave propagating through the high density loops (at higher heights, $\\sim $ 0.1 Rs) could strengthen due to its lower Alfvénic speed [12].", "The denser overlying loops around the active region may have a relatively weaker field strength, therefore, lower Alfvénic speed [48].", "This scenario is probably similar to the excitation of a type II radio burst by a shock wave passing through a denser coronal streamer (i.e., lower Alfvénic speed regions) [11], [26].", "The shock wave is stronger while passing through the denser loops and can excite the fragmented type II radio burst as suggested by [48].", "In our case, the type II radio burst structure is fragmented during the passage of fast EUV wave through the denser overlying loops.", "The wave showed acceleration while passing through the arcade loops as reported in numerical simulations [48].", "The lower loops (in the AIA 171 Å  channel) in the active region did not show transverse oscillation and were almost unaffected by the EUV wave.", "The higher loops were mainly affected by the EUV wave and exhibit strong transverse oscillation.", "These higher loops are the most likely candidate for exciting the type II radio burst (second harmonic$>$ 300 MHz).", "Fast EUV waves generally follow the path of lower Alfvénic speed [61] and type II radio bursts could be excited by the flare generated fast shock wave passing through the high density loops at the periphery of the active region.", "This could be a reason why the EUV wave does not propagate in all directions from the flare center and followed only the denser loops.", "In addition, we did not observe any slow wavefront behind the observed fast EUV wave.", "The slow wavefronts generally have speeds three times smaller than the fast wave and are usually interpreted as CME stretched loops (i.e., pseudo wave) running behind the fast wave [9], [10], [27].", "Alternatively, a pseudo wave may be explained by the current shell model [18] and the reconnection front model [5].", "In our case, the wave excitation closely matches the flare impulsive phase.", "This observation reveals that a low coronal type II radio burst can be generated by an impulsive flare ignited shock wave (i.e., blast wave).", "We agree that most of the low coronal type II radio bursts are generated by a CME.", "However, our finding suggests that not all the low coronal type II radio bursts may be generated by a CME (piston-driven).", "As the fast EUV wave propagated through the coronal arcade loops, it triggered kink oscillations of the loops.", "This suggests the wave was highly directional.", "According to [25], a high directionality is a common characteristic of flare blast waves that trigger kink oscillations of certain loop-system whereas other nearby loops remain stationary [54], [67].", "For one of the loops the kink oscillation had a period of 210 s and a phase speed of $\\sim $ 1140 km s$^{-1}$ .", "For the first time, we see a positive slowly drifting burst ($\\sim $ 220 MHz) simultaneously with the onset of a type II radio burst and closely associated with magnetic reconnection.", "The emission source drifts downward with a speed of $\\sim $ 100-130 km s$^{-1}$ .", "In the AIA 94 Å channel, we observed a co-temporal brightening propagating counterclockwise along the circular ribbon with a projected speed of $\\sim $ 80 km s$^{-1}$ .", "It is likely that the propagating brightening could be evidence of the rotation/slippage of fan loops around the spine.", "The speed of the radio source is roughly consistent with the rotating brightening.", "Therefore, we suggest that the downward moving radio source could be generated by the slippage of the field lines that may cause the shift in the reconnection point (inclined motion) resulting in particle acceleration at increasing densities along the slipping field lines.", "However, more observational studies are needed to validate this interpretation.", "According to 3D numerical simulation, torsional spine reconnection or slipping reconnection are expected in a fan-spine magnetic configuration [49], [50], [40].", "Alternatively, the narrow band positive drifting structure may also be interpreted by a particle acceleration (downward) at a reconnection outflow generated termination shock [6], [39], [8] as it was observed simultaneously during the type II radio burst.", "[39] explained radio emission features (200-400 MHz) as evidence of particle acceleration at the termination shock generated by reconnection outflow.", "We also observed a two-ribbon flare within the circular flare ribbon.", "This result is consistent with our previous finding of a two-ribbon flare (within a global circular ribbon) generated by untwisting small jets produced during the coalescence of two sheared J-shaped H$\\alpha $ loops [32].", "The magnetic field configuration was quite similar in both cases, i.e., a fan-spine topology with a quasi-circular ribbon [49], [50], [40], [45], [63].", "Here we observed breakout reconnection followed by the failed eruption of a small filament with a speed of $\\sim $ 340 km s$^{-1}$ .", "In summary, breakout reconnection in a fan-spine topology launched a fast-mode MHD shock that propagated perpendicular to the arcade loops generating a type II radio burst.", "We speculate that a specific magnetic configuration may be an important candidate for the flare ignited shock wave.", "However, future studies with high resolution observation will shed more light on this issue.", "We thank the referee for the positive and constructive comments/suggestions that improved the manuscript considerably.", "SDO is a mission for NASA's Living With a Star (LWS) program.", "The SDO data were (partly) provided by the Korean Data Center (KDC) for SDO in cooperation with NASA and SDO/HMI team.", "RHESSI is a NASA Small Explorer.", "Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and UKSA as international partners.", "It is operated by these agencies in co-operation with ESA and NSC (Norway).", "This work was supported by the “Operation of Korea Space Weather Center\" of KASI and the KASI basic research funds." ] ]
1606.05056
[ [ "Plasmon polaritons in cubic lattices of spherical metallic nanoparticles" ], [ "Abstract We theoretically investigate plasmon polaritons in cubic lattices of spherical metallic nanoparticles.", "The nanoparticles, each supporting triply-degenerate localized surface plasmons, couple through the Coulomb dipole-dipole interaction, giving rise to collective plasmons that extend over the whole metamaterial.", "The latter hybridize with photons forming plasmon polaritons, which are the hybrid light-matter eigenmodes of the system.", "We derive general analytical expressions to evaluate both plasmon and plasmon-polariton dispersions, and the corresponding eigenstates.", "These are obtained within a Hamiltonian formalism, which takes into account retardation effects in the dipolar interaction between the nanoparticles and considers the dielectric properties of the nanoparticles as well as their surrounding.", "Within this model we predict polaritonic splittings in the near-infrared to the visible range of the electromagnetic spectrum that depend on polarization, lattice symmetry and wavevector direction.", "Finally, we show that the predictions of our model are in excellent quantitative agreement with conventional finite-difference frequency-domain simulations, but with the advantages of analytical insight and significantly reduced computational cost." ], [ "Introduction", "Plasmonic metamaterials can be exploited to manipulate light at subwavelength scales and may be used to tailor optical properties [1], [2], [3].", "They consist of meta-atoms with possibly complicated subwavelength structures that are arranged in a controlled fashion [4].", "Potential applications of such metamaterials range from optical cloaking over planar hyperlenses to optical data processing [5], [6].", "The study of the optical properties of one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) arrays of metallic particles is a very active field of research [7].", "In the past, most theoretical and experimental research has been focused on 1D and 2D systems, since they are much easier to fabricate with well-established techniques [3], [4].", "However, the development of reliable techniques to control 3D assemblies of plasmonic nanoparticles is presently making substantial advances, and such 3D assemblies can now be achieved by using surface ligands or DNA templates [8], [9], [10], [7], [11].", "It is thus of current interest to also understand systematically the structure-property relationships in 3D crystalline arrangements of meta-atoms, where, beside the shape and the size of the nanoparticles themselves, the spacing and the crystal symmetry can be controlled independently.", "The optical properties of a plasmonic metamaterial are governed in the first instance by those of the individual metallic nanoparticles [4].", "Of primary importance to understand such optical properties are the localized surface plasmons (LSPs), which correspond to collective oscillations of the valence electrons against the ionic background.", "The resonance frequency and polarization of the LSP modes are determined by the size, shape and material of the nanoparticles.", "Classical electrodynamics can be used to understand many of the optical properties of 1D, 2D, and 3D plasmonic metamaterials [2], [7].", "Depending on the distance between the meta-atoms, two qualitatively different regimes emerge [4]: In the first regime, the distance between the meta-atoms is on the order of or larger than the wavelength associated with the LSP resonance of individual nanoparticles, so that diffractive far-field interactions between the meta-atoms of the array can interfere, leading to collective modes termed surface lattice resonances.", "In the second, opposite regime, the meta-atom separation is much smaller than the LSP resonance wavelength so that near-field interactions are predominant, yielding collective plasmons that are extended over the whole metamaterial.", "In the present work we concentrate on the latter regime.", "Early studies on the plasmonic properties of near-field-coupled metallic nanoparticles focused on 1D chains using a nonretarded model of point dipoles [12], [13], [14], [15], followed by fully-retarded classical approaches applied to 1D [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] and 2D systems [26], [27], [28], [29], [30].", "Three-dimensional metastructures were also investigated using more approximate approaches such as the Maxwell-Garnett effective medium theory [7] or Bruggeman effective medium theory [31].", "In addition to the classical, typically fully numerical treatments, an analytically tractable approach based on a Hamiltonian formalism was recently applied to 1D [32], [33], [34], [35], 2D [36], [37], [38] and 3D systems [39].", "In this work we study the less explored 3D plasmonic arrays in the regime of near-field coupling between spherical metallic nanoparticles.", "Spherical particles are chosen in order to focus on the effects of crystal structure on the optical properties only.", "The nature of the modes supported by a plasmonic metamaterial depends crucially on the dimensionality of the lattice.", "For 1D and 2D lattices, the collective plasmons couple to a continuum of photonic modes with different wavevector components along directions where translational symmetry is absent.", "However, as it has been pointed out by Hopfield in the context of exciton polaritons [40], in stark contrast to lower dimensional systems, collective plasmons in 3D lattices only couple to photons which conserve crystal momentum due to the discrete translational symmetry of the system.", "As a result, the true eigenmodes of the metamaterial are coherent superpositions of plasmons and photons, which we call plasmon polaritons.", "We study them by means of an analytically tractable Hamiltonian-based approach, which importantly incorporates retardation effects.", "In what follows we consider 3D lattices of spherical metallic nanoparticles, including simple cubic (sc), face-centered cubic (fcc), and body-centered cubic (bcc) structures.", "In the quasistatic limit [41], each nanoparticle supports a discrete set of multipolar LSP modes.", "However, as we consider small nanoparticles (of some $[10]{nm}$ in radius), we neglect higher-order multipolar modes and focus on the fundamental dipolar LSPs, whose corresponding frequency lies in the visible to ultraviolet range of the spectrum.", "In this regime, quantum-size effects in the optical response of the nanoparticles can be significant [6].", "Due to the spherical symmetry of the nanoparticles, each dipolar LSP is triply-degenerate with three polarization degrees of freedom.", "We work in the Coulomb gauge [42], [43], where the scalar and vector potentials describe the longitudinal and transverse components of the electromagnetic field, respectively.", "The scalar potential, which depends only on the matter degrees of freedom, takes the form of the instantaneous Coulomb interaction between the LSPs.", "This results in collective plasmonic modes, which extend across the whole metamaterial.", "The effects of retardation are then included in the light-matter coupling through the interaction of the LSPs with the transverse vector potential.", "In this way, transverse photons hybridize with the collective plasmons to form plasmon polaritons.", "We also take into account screening effects from the core electrons as well as the dielectric medium surrounding the nanoparticles.", "Here, we decisively extend inspiring work of some of the authors [39].", "Although it is stated in Ref.", "[39] that spherical metallic nanoparticles are used, these nanoparticles were assumed to exhibit only one polarization degree of freedom that was fixed in a given direction.", "This gives rise to a single plasmon band, whose polarization does not depend on the wavevector.", "In fact this model does not correctly describe lattices of spherical nanoparticles, but could be used to study lattices of resonators that have a nondegenerate fundamental eigenmode, such as plasmonic nanorods.", "Our treatment fixes this issue by considering plasmon polaritons which arise from the hybridization of photons with three plasmonic bands with wavevector-dependent polarizations.", "Furthermore we show that the model yields plasmon-polariton properties in excellent agreement with classical electrodynamics simulations at a much reduced computational cost and at the benefit of analytical intuition.", "With our newly developed tools, we demonstrate that these highly symmetric cubic systems exhibit polarization-dependent optical properties such as band splittings in the near-infrared or visible range of the spectrum.", "With the emerging fabrication techniques for 3D metallic nanoparticle lattices, this work is an important step towards accurate predictions of their polaritonic properties, and the model can be readily extended to more complex lattices and nanoparticle shapes.", "The paper is organized as follows: In Sec.", "we describe our theoretical model to study plasmon polaritons.", "The general solution to this model is subsequently presented in Sec.", "REF .", "The resulting dispersion relations of the collective plasmons and plasmon polaritons for sc, fcc and bcc lattices are discussed in Secs.", "REF and REF , respectively.", "In Sec.", ", we compare our predictions to classical electrodynamics simulations.", "We finally summarize our results in Sec. .", "In the Appendix we discuss the form of the dielectric tensor that shows a nonlocal response." ], [ "Model", "We consider sc, fcc, and bcc lattices of spherical metallic nanoparticles separated by a center-to-center distance $a$ between nearest neighbors, as depicted in Figs.", "REF (a)-(c).", "The corresponding first Brillouin zones are shown in Figs.", "REF (d)-(f).", "We describe the nanoparticles with a Drude-like dielectric function $\\epsilon _{\\mathrm {r}}^{\\mathrm {D}}(\\omega ) = \\epsilon _{\\mathrm {d}} -\\frac{\\omega _{\\mathrm {p}}^2 }{\\omega (\\omega +\\mathrm {i} \\gamma ^{\\mathrm {D}})},$ where $\\omega _{\\mathrm {p}}$ is the plasma frequency of the considered (noble) metal and where the dielectric constant $\\epsilon _{\\mathrm {d}}$ takes into account the screening of the conduction electrons by the $d$ -electrons.", "In our model we use $\\gamma ^{\\mathrm {D}}=0$ , but we will explore the effect of a nonvanishing Drude damping in the finite-difference frequency-domain (FDFD) calculations, presented in Sec. .", "The surrounding medium that fills the space between the nanoparticles is characterized by the dielectric constant $\\epsilon _{\\mathrm {m}}$ .", "The magnetic permeabilities of the nanoparticles and the embedding medium are assumed to be equal to the vacuum permeability.", "Each nanoparticle in the lattice supports three degenerate dipolar LSPs polarized in the $x$ , $y$ or $z$ direction.", "They interact with their neighbors through the quasistatic dipole-dipole interaction $V_\\mathrm {dip}(\\mathbf {R}, \\mathbf {R}^{\\prime })=\\frac{9\\epsilon _{\\mathrm {m}}}{\\left(\\epsilon _{\\mathrm {d}}+2\\epsilon _{\\mathrm {m}} \\right)^2}\\frac{\\mathbf {p}\\cdot \\mathbf {p}^{\\prime }-3(\\mathbf {p}\\cdot \\hat{n})(\\mathbf {p}^{\\prime }\\cdot \\hat{n})}{4\\pi \\epsilon _0|\\mathbf {R}-\\mathbf {R}^{\\prime }|^3},$ where $\\mathbf {p}$ and $\\mathbf {p}^{\\prime }$ are the dipole moments associated with the LSPs of the nanoparticles located at the lattice sites $\\mathbf {R}$ and $\\mathbf {R}^{\\prime }$ , respectively, while $\\hat{n}=(\\mathbf {R}-\\mathbf {R}^{\\prime })/|\\mathbf {R}-\\mathbf {R}^{\\prime }|$ , and $\\epsilon _0$ is the vacuum permittivity.", "Here and in what follows, hats denote unit vectors.", "In the expression above, the prefactor takes into account the two dielectric environments and arises from a model in which each point dipole is located inside a sphere with dielectric constant $\\epsilon _\\mathrm {d}$ , and separated by a medium with dielectric constant $\\epsilon _\\mathrm {m}$ [44].", "As we only consider dynamical degrees of freedom relating to the fundamental dipolar LSPs, we are thus neglecting any effects of higher-order multipolar plasmons.", "This approximation has been shown to be valid for center-to-center interparticle separations $a\\gtrsim 3r_{\\mathrm {np}}$ [13], with $r_{\\mathrm {np}}$ the nanoparticle radius (see Fig.", "REF ).", "We demonstrate the validity of this approximation in Sec.", "by comparing our results to FDFD simulations.", "We write the full Hamiltonian of the system as $H =H_{\\mathrm {pl} } + H_{\\mathrm {ph} } + H_{\\mathrm {pl}\\textrm {-}\\mathrm {ph} },$ where $H_{\\mathrm {pl}}$ and $H_{\\mathrm {ph}}$ denote the plasmonic and photonic Hamiltonians, respectively, and where $H_{\\mathrm {pl}\\textrm {-}\\mathrm {ph} }$ is the interaction Hamiltonian between both subsystems.", "In the Coulomb gauge [42], [43], the purely plasmonic Hamiltonian reads [36], [37], [39], [33], [34] Hpl =0 q, bq bq +q,,' fq,'[bq(bq'+b-q')+ h.c. ], with $f_{\\mathbf {q}}^{\\hat{\\sigma },\\hat{\\sigma }^{\\prime }} =\\sum _{{{\\rho }\\\\(a\\leqslant \\rho \\leqslant \\rho _\\mathrm {c})}}\\left(\\frac{a}{\\rho }\\right)^3 \\frac{\\cos {\\left(\\mathbf {q}\\cdot {\\rho } \\right)}}{2}\\left[\\delta _{\\hat{\\sigma }\\hat{\\sigma }^{\\prime }}-3(\\hat{\\sigma }\\cdot \\hat{\\rho })(\\hat{\\sigma }^{\\prime }\\cdot \\hat{\\rho })\\right].$ Here, $\\mathbf {q}=q\\, \\hat{q}$ is the plasmonic wavevector in the first Brillouin zone.", "In Eq.", "(), $b_{\\mathbf {q}}^{\\hat{\\sigma }}=\\mathcal {N}^{-1/2}\\sum _\\mathbf {R}\\exp {(-\\mathrm {i}\\mathbf {q}\\cdot \\mathbf {R})}b_\\mathbf {R}^{\\hat{\\sigma }}$ is defined as the Fourier transform of the bosonic operator $b_\\mathbf {R}^{\\hat{\\sigma }}$ , which annihilates an LSP at lattice site $\\mathbf {R}$ with polarization $\\hat{\\sigma }=\\hat{x}$ , $\\hat{y}$ or $\\hat{z}$ , where $\\mathcal {N}$ is the number of unit cells of the metacrystal.", "The first term on the right-hand side of Eq.", "() describes the uncoupled LSPs with Mie frequency [41] $\\omega _0=\\frac{\\omega _{\\mathrm {p}}}{\\sqrt{\\epsilon _{\\mathrm {d}}+2\\epsilon _{\\mathrm {m}}}},$ while the second one with coupling constant $\\Omega =\\frac{3\\epsilon _{\\mathrm {m}}}{2(\\epsilon _{\\mathrm {d}}+2\\epsilon _{\\mathrm {m}})}\\omega _0\\left(\\frac{r_\\mathrm {np}}{a}\\right)^3$ corresponds to the Coulomb dipole-dipole interaction [cf.", "Eq.", "(REF )] between nanoparticles linked by the separation vector $\\rho $ .", "Crucially, we consider Coulomb interactions up to a large cut-off distance $\\rho _\\mathrm {c}\\gg a$ , beyond the nearest-neighbor approximation that was employed in Ref.", "[39].", "As will be highlighted later, these long-range Coulomb interactions are critical for obtaining the correct plasmonic dispersions.", "As discussed in detail in Ref.", "[45], there is a region of slow convergence of $f_{\\mathbf {q}}^{\\hat{\\sigma },\\hat{\\sigma }^{\\prime }}$ around the $\\Gamma $ point [see Eq.", "(REF )].", "This stems from discontinuities of $f_{\\mathbf {q}}^{\\hat{\\sigma },\\hat{\\sigma }^{\\prime }}$ at $\\mathbf {q}=0$ for $\\rho _\\mathrm {c}\\rightarrow \\infty $ .", "These discontinuities lead to the Gibbs-Wilbraham phenomenon [46], and the summation in Eq.", "(REF ) does not easily converge with increasing cutoff radius $\\rho _\\mathrm {c}$ .", "Thus, for small wavevectors $q< \\alpha \\rho _\\mathrm {c}^{-1}$ , with $\\alpha $ a real positive number, we use the correction $f_{\\mathbf {q}}^{\\hat{\\sigma },\\hat{\\sigma }^{\\prime }}=-2\\pi \\left[\\delta _{\\hat{\\sigma }\\hat{\\sigma }^{\\prime }}-3(\\hat{\\sigma }\\cdot \\hat{q})(\\hat{\\sigma }^{\\prime }\\cdot \\hat{q})\\right]/3\\nu $ for the infinite lattice [45].", "It contains the factor $\\nu $ , which accounts for the different volumes of the primitive cells of the considered lattices and equals $\\nu =1$ for sc, $\\nu =2^{-1/2}\\simeq 0.71$ for fcc, and $\\nu =4/3^{3/2}\\simeq 0.77$ for bcc lattices, respectively.", "In Eq.", "(REF ) the photonic subsystem is described by $H_{\\mathrm {ph} }=\\sum \\limits _{\\mathbf {q},\\hat{\\lambda }_{\\mathbf {q}}}\\hbar \\omega _{\\mathrm {ph},\\mathbf {q}}c_{\\mathbf {q}}^{\\hat{\\lambda }_{\\mathbf {q}}\\dagger }c_{\\mathbf {q}}^{\\hat{\\lambda }_{\\mathbf {q}}}, $ where $c_{\\mathbf {q}}^{\\hat{\\lambda }_{\\mathbf {q}}}$ annihilates and $c_{\\mathbf {q}}^{\\hat{\\lambda }_{\\mathbf {q}}\\dagger }$ creates a photon with wavevector $\\mathbf {q}$ , dispersion $\\omega _{\\mathrm {ph},\\mathbf {q}}=cq/\\sqrt{\\epsilon _{\\mathrm {m}}}$ , and transverse polarization $\\hat{{\\lambda }}_{\\mathbf {q}}$ (with $\\hat{{\\lambda }}_{\\mathbf {q}}\\cdot \\mathbf {q}=0$ ).", "Here $c/\\sqrt{\\epsilon _{\\mathrm {m}}}$ is the speed of light in the embedding medium.", "In the long-wavelength limit $qr_\\mathrm {np}\\ll 1$ , the minimal light-matter coupling Hamiltonian in Eq.", "(REF ) takes the form Hpl-ph = i 0 q,,qq q ( bqcqq + bqc-qq -h.c.) +0q,qq2 (cqqcqq +cqqc-qq + h.c. ), where $\\xi _{\\mathbf {q}}=[2\\Omega \\pi /(\\nu \\omega _{\\mathrm {ph},\\mathbf {q}})]^{1/2}$ .", "Since we consider lattice constants $a$ much smaller than the wavelength associated with the LSP resonances, we neglect Umklapp processes in Eqs.", "(REF ) and ().", "However, the model can be readily extended to include such Umklapp scattering in order to describe metamaterials with larger lattice constants.", "Let us point out that the first term on the right-hand side of Eq.", "() describes, to second-order in perturbation theory, the exchange of virtual photons among the nanoparticles of the lattice [43].", "Such a term therefore incorporates the retardation effects in the dipolar coupling between the LSPs." ], [ "General solution", "The full Hamiltonian (REF ), representing collective plasmons strongly coupled to photons, can be diagonalized by introducing the bosonic operator qq = ( uqq, bq +vqq, b-q ) +q( mqq,q cqq +nqq,q c-qq ), which annihilates a plasmon polariton with wavevector $\\mathbf {q}$ and polarization $\\hat{\\tau }_{\\mathbf {q}}$ , the latter being generally not aligned with the $\\hat{\\sigma }$ -axis.", "Imposing that the operator in Eq.", "(REF ) and its adjoint diagonalize the Hamiltonian (REF ) as $H=\\sum _{\\mathbf {q},\\hat{\\tau }_{\\mathbf {q}}}\\hbar \\omega _{\\mathrm {pp},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\eta _{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}\\dagger } \\eta _{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}} ,$ the Heisenberg equation of motion $[ \\eta _{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}},H]=\\hbar \\omega _{\\mathrm {pp},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\eta _{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ leads to the $10\\times 10$ eigensystem $\\begin{pmatrix}\\omega _0 \\mathbb {1}_{3}+2\\Omega F_\\mathbf {q} & -2\\Omega F_\\mathbf {q} &-\\mathrm {i}\\omega _0\\xi _\\mathbf {q}P_\\mathbf {q} &\\mathrm {i}\\omega _{0}\\xi _\\mathbf {q}P_\\mathbf {q} \\\\[.1cm] 2\\Omega F_\\mathbf {q} & -(\\omega _0 \\mathbb {1}_{3}+2\\Omega F_\\mathbf {q})&\\mathrm {i}\\omega _0\\xi _\\mathbf {q}P_\\mathbf {q} &-\\mathrm {i}\\omega _0\\xi _\\mathbf {q}P_\\mathbf {q} \\\\[.1cm]\\mathrm {i}\\omega _0\\xi _\\mathbf {q}P_\\mathbf {q}^\\top &\\mathrm {i}\\omega _0\\xi _\\mathbf {q}P_\\mathbf {q}^\\top & (\\omega _{\\mathrm {ph},\\mathbf {q}}+2\\omega _0\\xi _\\mathbf {q}^2)\\mathbb {1}_{2} &-2\\omega _0\\xi _\\mathbf {q}^2\\mathbb {1}_{2} \\\\[.1cm]\\mathrm {i}\\omega _0\\xi _\\mathbf {q}P_\\mathbf {q}^\\top &\\mathrm {i}\\omega _0\\xi _\\mathbf {q}P_\\mathbf {q}^\\top &2\\omega _0\\xi _\\mathbf {q}^2\\mathbb {1}_{2} & -(\\omega _{\\mathrm {ph},\\mathbf {q}}+2\\omega _0\\xi _\\mathbf {q}^2)\\mathbb {1}_{2}\\\\\\end{pmatrix}\\begin{pmatrix}\\mathbf {u}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\mathbf {v}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\mathbf {m}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\mathbf {n}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\end{pmatrix}=\\omega _{\\mathrm {pp},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\begin{pmatrix}\\mathbf {u}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\mathbf {v}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\mathbf {m}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\mathbf {n}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}\\\\\\end{pmatrix},$ where the vectors $\\mathbf {u}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ , $\\mathbf {v}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ , $\\mathbf {m}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ , and $\\mathbf {n}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ consist of $u^{\\hat{\\tau }_{\\mathbf {q}},\\hat{\\sigma }}_\\mathbf {q}$ , $v_\\mathbf {q}^{\\hat{\\tau }_{\\mathbf {q}},\\hat{\\sigma }}$ , $m^{\\hat{\\tau }_{\\mathbf {q}},\\hat{\\lambda }_{\\mathbf {q}}}_\\mathbf {q}$ , and $n_\\mathbf {q}^{\\hat{\\tau }_{\\mathbf {q}},\\hat{\\lambda }_{\\mathbf {q}}}$ , respectively, as defined in Eq.", "(REF ).", "In Eq.", "(REF ), $\\mathbb {1}_{n}$ stands for the $n\\times n$ identity matrix, the $3\\times 3$ symmetric matrix $F_{\\mathbf {q}}$ is defined by its elements $f_{\\mathbf {q}}^{\\hat{\\sigma },\\hat{\\sigma }^{\\prime }}$ as given in Eq.", "(REF ), while the $3\\times 2$ matrix $P_\\mathbf {q}$ is introduced as $P_\\mathbf {q}=\\begin{pmatrix}\\hat{x}\\cdot \\hat{\\lambda }_{1, \\mathbf {q}}&\\hspace{2.84544pt}\\hat{x}\\cdot \\hat{\\lambda }_{2, \\mathbf {q}}\\\\[.1cm]\\hat{y}\\cdot \\hat{\\lambda }_{1, \\mathbf {q}}&\\hspace{2.84544pt} \\hat{y}\\cdot \\hat{\\lambda }_{2, \\mathbf {q}}\\\\[.1cm]\\hat{z}\\cdot \\hat{\\lambda }_{1, \\mathbf {q}}&\\hspace{2.84544pt} \\hat{z}\\cdot \\hat{\\lambda }_{2, \\mathbf {q}}\\end{pmatrix},$ and $P_\\mathbf {q}^\\top $ represents its transpose.", "Here, the two photon polarizations can be parameterized, e.g., as $\\hat{\\lambda }_{1, \\mathbf {q}}=\\hat{z}\\times \\hat{q}/|\\hat{z}\\times \\hat{q}|$ and $\\hat{\\lambda }_{2, \\mathbf {q}}=\\hat{q}\\times \\hat{\\lambda }_{1, \\mathbf {q}}/|\\hat{q}\\times \\hat{\\lambda }_{1, \\mathbf {q}}|$ for $\\hat{q}\\nparallel \\hat{z}$ , while for $\\hat{q}=\\hat{z}$ , we choose $\\hat{\\lambda }_{1, \\mathbf {q}}=\\hat{x}$ and $\\hat{\\lambda }_{2, \\mathbf {q}}=\\hat{y}$ .", "We note that the plasmon-polariton eigenfrequencies $\\omega _{\\mathrm {pp},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ arising from the eigensystem (REF ) occur in pairs of positive and negative eigenvalues.", "Below, we will focus on the physically relevant, positive solutions.", "If not stated otherwise, we will use an interparticle distance $a=3r_{\\mathrm {np}}$ , a cutoff radius $\\rho _\\mathrm {c}=150 a$ , and $\\alpha =10$ .", "We have checked that the latter choices provide numerically-converged results for the collective plasmon and plasmon-polariton dispersions, presented in the next subsections." ], [ "Collective plasmons", "Before considering the fully coupled system, represented by the Hamiltonian (REF ), it is instructive to analyze in detail the purely plasmonic problem described by $H_\\mathrm {pl}$ in Eq. ().", "We will therefore set the light-matter coupling to zero in this subsection.", "In this way, plasmon properties are computed in the quasistatic limit, neglecting all retardation effects.", "Setting $\\xi _\\mathbf {q}=0$ , the matrix defined in Eq.", "(REF ) becomes block-diagonal.", "On the one hand, the lower $4\\times 4$ block is diagonal and corresponds to the two degenerate photon branches with dispersion $\\omega _{\\mathrm {ph},\\mathbf {q}}$ for the two positive eigenvalues.", "The three positive eigenvalues of the upper $6\\times 6$ block, on the other hand, yield the collective plasmon dispersion $\\omega _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}} }$ , which is represented in Fig.", "REF as a function of wavevector $\\mathbf {q}$ along the red paths given in Figs.", "REF (d)-(f) for the sc [Fig.", "REF (a)], fcc [Fig.", "REF (b)], and bcc [Fig.", "REF (c)] lattices.", "In the figure we use $\\epsilon _{\\mathrm {d}}= 5.6$ , as determined for silver films [47], [48], and $\\epsilon _{\\mathrm {m}}= 4$ , mimicking an embedding medium made of glass or polymer.", "In Fig.", "REF we also show the collective plasmon-polarization angle $\\phi _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}=\\arccos {(|\\hat{\\tau }_{\\mathbf {q}}\\cdot \\hat{q}|)}$ , where we choose $\\hat{\\tau }_{\\mathbf {q}}=\\hat{{u}}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ .", "Notice that the alternative choice $\\hat{\\tau }_{\\mathbf {q}}=\\hat{{v}}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ leads to the same polarization angle, as the vectors $\\mathbf {u}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ and $\\mathbf {v}_{\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ are proportional for a given wavevector $\\mathbf {q}$ .", "With the above definition of $\\phi _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}$ , longitudinal collective plasmons, which do not couple to light, have a polarization angle $\\phi _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}=0$ (black lines in Fig.", "REF ), while purely transverse modes have a corresponding polarization $\\phi _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}=\\pi /2$ (yellow lines in Fig.", "REF ).", "Figure: Collective plasmon dispersionω pl ,𝐪 τ ^ 𝐪 \\omega _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}} in units of the LSPfrequency ω 0 \\omega _0 along the paths shown in red inFigs.", "(d)-(f) for the (a) sc, (b) fcc, and (c) bcclattices.", "The color code corresponds to the collective plasmon-polarizationangle φ pl ,𝐪 τ ^ 𝐪 \\phi _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\tau }_{\\mathbf {q}}}, which equals 0(π/2\\pi /2) for purely longitudinal (transverse) plasmons.", "In the figure weuse a=3r np a=3r_{\\mathrm {np}}, ρ c =150a\\rho _{\\mathrm {c}}=150a and α=10\\alpha =10 for thecolored thick lines, while in panel (a) we choose ρ c =a\\rho _{\\mathrm {c}}=a andα=0\\alpha =0 for the gray thin lines, corresponding to nearest-neighborinteractions only [cf. Eq. ()].", "In all cases thedielectric constants are set to ϵ d =5.6\\epsilon _{\\mathrm {d}}= 5.6 andϵ m =4\\epsilon _{\\mathrm {m}}= 4.Our results in Fig.", "REF indicate that there are two purely transverse collective plasmons and one purely longitudinal one along the high-symmetry axes in the first Brillouin zone [i.e., axes with 2- to 4-fold rotational symmetry, see Figs.", "REF (d)-(f)].", "For less symmetric axes the collective modes can be of a mixed type [see, e.g., the $XM$ and $MR$ lines in Fig.", "REF (a)].", "Moreover, along 3- and 4-fold symmetry axes, the two transverse modes are degenerate [see, e.g., the $\\Gamma R$ and $\\Gamma X$ lines in Fig.", "REF (a)].", "This is a manifestation of Neumann's principle [49]: For the collective plasmon dispersion this enforces the degeneracy of the transverse modes for the 3- and 4-fold symmetry lines.", "The latter degeneracy is lifted for wavevector directions with lower symmetry.", "We also note that one would expect the longitudinal and transverse plasmon modes to be degenerate at the $\\Gamma $ point since the latter has the full point-group symmetry of the lattice.", "As we will see later, there is a radiative correction from the light-matter interaction Hamiltonian () that enforces this degeneracy in the polariton spectrum.", "Before we move on to the discussion of the fully coupled system, a comment is in order about the importance of the dipole-dipole interaction beyond nearest neighbors for the collective plasmon dispersion.", "In Fig.", "REF (a) we represent by thin gray lines the plasmon dispersion of the sc lattice, including nearest-neighbor interactions only.", "(Note that we do not correct for the Wilbraham-Gibbs phenomenon around the $\\Gamma $ point in this case, i.e., we use $\\alpha =0$ .)", "Under these conditions, the matrix $F_\\mathbf {q}$ is diagonal, and its elements read $f_\\mathbf {q}^{\\hat{\\sigma },\\hat{\\sigma }^{\\prime }}=\\delta _{\\hat{\\sigma }\\hat{\\sigma }^{\\prime }}\\sum _{\\hat{\\sigma }^{\\prime \\prime }=\\hat{x}, \\hat{y}, \\hat{z}}(1-3\\delta _{\\hat{\\sigma }\\hat{\\sigma }^{\\prime \\prime }})\\cos {(a\\hat{\\sigma }^{\\prime \\prime }\\cdot \\mathbf {q})}.$ The plasmonic Hamiltonian () is therefore separable into $\\hat{x}$ , $\\hat{y}$ and $\\hat{z}$ directions and can be diagonalized analytically, yielding $\\omega _{\\mathrm {pl},\\mathbf {q}}^{\\hat{\\sigma }}=\\omega _0\\sqrt{1+4\\frac{\\Omega }{\\omega _0} f_\\mathbf {q}^{\\hat{\\sigma },\\hat{\\sigma }}}.$ This result and the corresponding coefficients of the Bogoliubov transformation (REF ), which we do not report explicitly here, coincide with those found in Ref.", "[39] for LSP polarizations along $\\hat{x}$ , $\\hat{y}$ or $\\hat{z}$ and $\\epsilon _{\\mathrm {m}}=\\epsilon _{\\mathrm {d}}=1$ .", "As can be seen in Fig.", "REF (a), including the dipole-dipole interactions beyond nearest neighbors can have a qualitative effect on the collective plasmon dispersion, most noticeably around the $\\Gamma $ point.", "They further lift the degeneracy between plasmon branches, e.g., along the $\\Gamma M$ and $\\Gamma R$ directions.", "In other regions of the first Brillouin zone the difference between the full dispersion and those from nearest neighbors only is less significant." ], [ "Plasmon polaritons", "We now consider the fully coupled system, represented by the eigensystem (REF ), and numerically solve for its five positive eigenvalues.", "These eigenvalues yield the plasmon-polariton spectrum $\\omega _{\\mathrm {pp},\\mathbf {q}}^{\\hat{\\tau }_\\mathbf {q}}$ , which is shown by solid lines in Fig.", "REF for the sc [Figs.", "REF (a)-(c)], fcc [Figs.", "REF (d)-(f)], and bcc lattices [Figs.", "REF (g)-(i)] along 2-fold [Figs.", "REF (a),(d),(g)], 3-fold [Figs.", "REF (b),(e),(h)], and 4-fold symmetry axes [Figs.", "REF (c),(f),(i)], cf.", "Figs.", "REF (d)-(f).", "Along the high symmetry axes of the first Brillouin zone, the five modes split up into four polaritonic branches (colored solid lines) and one purely longitudinal collective plasmon, which does not couple to transverse photons (black lines).", "The four polaritonic modes result from the coupling of transverse collective plasmons (see Fig.", "REF ) to photons, whose dispersion relation is shown by dashed lines in Fig.", "REF .", "According to the construction of our effective model and the nature of the Coulomb gauge, retardation effects are taken into account for all plasmon-polariton branches, where photons and plasmons interact via Eq. ().", "Figure: Solid lines: plasmon-polariton dispersion ω pp ,𝐪 τ ^ 𝐪 \\omega _{\\mathrm {pp},\\mathbf {q}}^{\\hat{\\tau }_\\mathbf {q}} in units of the LSP frequency ω 0 \\omega _0for the (a)-(c) sc, (d)-(f) fcc, and (g)-(i) bcc lattices along the(a),(d),(g) 2-fold, (b),(e),(h) 3-fold, and (c),(f),(i) 4-fold symmetryaxes, shown in Fig. .", "Dashed gray lines: free photondispersion ω ph ,𝐪 \\omega _{\\mathrm {ph},\\mathbf {q}}.", "The parameters used in thefigure are a=3r np a=3r_{\\mathrm {np}}, ρ c =150a\\rho _\\mathrm {c}=150a, α=10\\alpha =10,ω 0 r np /c=0.237\\omega _0r_{\\mathrm {np}}/c=0.237, ϵ d =5.6\\epsilon _{\\mathrm {d}}= 5.6 andϵ m =4\\epsilon _{\\mathrm {m}}= 4.As can be inferred from Fig.", "REF , there are two high-energy polaritonic branches (orange solid lines) and two low-energy ones (green solid lines).", "The two high-energy branches are nearly degenerate.", "The low-energy polaritonic branches, shown by green solid lines in Fig.", "REF , have the same twofold degeneracy along 3-fold and 4-fold symmetry axes as the collective plasmon dispersion (compare with Fig.", "REF ) and the light-matter interaction does not lift this degeneracy.", "As mentioned previously, there is a radiative correction to the transverse plasmonic modes at the $\\Gamma $ point, which is equal to the longitudinal-transverse splitting observed in the plasmonic spectrum (Fig.", "REF ).", "As a result, one observes that the longitudinal and transverse high-energy polaritonic branches are degenerate at the $\\Gamma $ point (Fig.", "REF ), as required by symmetry.", "For wavevectors close to the edge of the first Brillouin zone the high-energy polaritonic branches (orange solid lines in Fig.", "REF ) asymptotically approach the light cone, while the low-energy ones (green solid lines in the figure) tend to the collective plasmon dispersion.", "For $\\mathbf {q}\\rightarrow 0$ (i.e., close to the $\\Gamma $ point), the states corresponding to the low-energy branches are mostly photon-like, with a renormalized group velocity, which is smaller than $c/\\sqrt{\\epsilon _{\\mathrm {m}}}$ , indicating an effective index of refraction larger than $\\sqrt{\\epsilon _{\\mathrm {m}}}$ .", "However, the high-energy branches do not tend to the values displayed in Fig.", "REF at the $\\Gamma $ point due to the strong coupling between collective plasmons and photons [cf. Eq. ()].", "This results in a splitting between the low- and high-energy polaritonic branches.", "We define this polaritonic splitting $\\Delta _{\\hat{q}}$ as the frequency difference between the minimum of the high-energy polaritonic branches and the maximum of the lower branches over all wavevectors $\\mathbf {q}$ in the first Brillouin zone along a fixed direction $\\hat{q}$ from the $\\Gamma $ point.", "As can be seen in Fig.", "REF for $\\epsilon _{\\mathrm {d}}=5.6$ and $\\epsilon _{\\mathrm {m}}=4$ , the polaritonic splitting reaches values of the order of $[25]{\\%}$ of the LSP resonance frequency $\\omega _0$ .", "For noble-metal nanoparticles the latter typically lies in the visible to ultraviolet range ($\\omega _0\\simeq 2$ –$[4]{eV/\\hbar }$ ), resulting in a splitting of about $\\Delta _{\\hat{q}}\\simeq 0.5$ –$[1.0]{eV/\\hbar }$ .", "The splitting in the polaritonic dispersion has important experimental consequences for the optical properties of the metamaterial.", "Indeed, along a certain direction $\\hat{q}$ in the Brillouin zone, no plasmon polariton can propagate for frequencies within the bandgap, so that the reflectivity of the metacrystal should be perfect.", "We would like to emphasize that the physical origin of these band gaps is entirely different from those emerging in conventional photonic crystals which are the result of Bragg scattering [50].", "In fact, we neglect Umklapp processes and therefore the band gaps emerge as a result of polaritonic hybridization between Mie resonances and photons.", "Interestingly, the polaritonic splitting depends on the polarization for the two-fold symmetry axes of the three cubic lattices [see Figs.", "REF (a),(d),(g)].", "This birefringence is directly related to the polarization dependence of the collective plasmon dispersion, the latter being due to the anisotropic nature of the dipole-dipole interaction between the nanoparticles composing the metamaterial.", "The modulation of the band splitting can be rather significant for the sc and fcc lattices (around $[12]{\\%}$ of $\\omega _0$ ), while for the bcc lattice it is comparatively less (around $[3]{\\%}$ of $\\omega _0$ ).", "In the following, we will refer to the modulation of $\\Delta _{\\hat{q}}$ for different polarizations as $\\delta _{\\hat{q}}$ .", "Figure: Plasmon-polariton dispersions for various values of thedielectric constants ϵ d \\epsilon _{\\mathrm {d}} andϵ m \\epsilon _{\\mathrm {m}}.", "The lines present the results of our Hamiltonianapproach [see Eq.", "()] and the symbols those found assolutions in FDFD simulations.", "Solid and dashed lines: plasmon-polaritondispersions ω pp ,𝐪 τ ^ 𝐪 \\omega _{\\mathrm {pp}, \\mathbf {q}}^{\\hat{\\tau }_\\mathbf {q}} in units ofthe LSP frequency ω 0 \\omega _0 for the sc lattice along the 2-fold symmetryaxis (see Fig. ).", "Dotted gray lines: free photon dispersionsω ph ,𝐪 \\omega _{\\mathrm {ph},\\mathbf {q}}.", "The parameters for the Hamiltonian approach area=3r np a=3r_{\\mathrm {np}}, α=10\\alpha =10, and ρ c =150a\\rho _\\mathrm {c}=150a, whilewe choose r np =[10]nmr_{\\mathrm {np}}=[10]{nm} and a=3r np a=3r_{\\mathrm {np}}in the FDFD calculations.", "In panels (a) and (b) the LSP frequencyω 0 r np /c=0.177\\omega _0r_{\\mathrm {np}}/c=0.177 [see Eq.", "()] is keptconstant, while in panel (c) ω p =[9.6]eV/ℏ\\omega _{\\mathrm {p}}=[9.6]{eV/\\hbar } isconstant.", "All other parameters are indicated in the respective panels.In the FDFD calculations presented in panel (c), we choose a finite Drude dampingγ D \\gamma ^{\\mathrm {D}} , and plot the calculated imaginaryparts of the eigenfrequencies, i.e., damping rates, as error bars.Let us now discuss the dependence of the plasmon-polariton dispersions on the dielectric constants $\\epsilon _{\\mathrm {d}}$ and $\\epsilon _{\\mathrm {m}}$ for the two-fold symmetry axes, as shown in Fig.", "REF .", "To simplify the discussion, we keep the Mie frequency $\\omega _0$ constant in Figs.", "REF (a) and REF (b) by adjusting $\\omega _{\\mathrm {p}}$ , while the Mie frequency is varied in Fig.", "REF (c).", "As indicated in Fig.", "REF (a), an increase in the screening of the core electrons decreases the polaritonic splitting and leads to a corresponding flattening of the longitudinal plasmon branch.", "The smaller splitting can be understood by noting that the coupling constant $\\Omega \\propto 1/(2+\\epsilon _{\\mathrm {d}}/\\epsilon _{\\mathrm {m}})$ in the plasmonic part [see Eq.", "(REF )] decreases with increasing $\\epsilon _{\\mathrm {d}}$ .", "The dependence of the polaritonic dispersion on the dielectric constant of the surrounding medium $\\epsilon _{\\mathrm {m}}$ is more complex, as displayed in Fig.", "REF (b).", "An increasing $\\epsilon _{\\mathrm {m}}$ reduces the effective speed of light in the medium.", "Hence, this reduces the slope of the low-energy polaritonic branches around the $\\Gamma $ -point, while the slope of the high-energy polaritonic branches is modified away from the $\\Gamma $ point.", "Furthermore, with increasing $\\epsilon _{\\mathrm {m}}$ a larger polaritonic splitting $\\Delta _{\\hat{q}}$ as well as increased modulation $\\delta _{\\hat{q}}$ between the low-energy polaritonic branches of different polarization is observed.", "We attribute this to two factors.", "Most importantly the coupling constant $\\Omega $ increases with increasing $\\epsilon _{\\mathrm {m}}$ , and thus the related band splittings get larger.", "An increasing $\\epsilon _{\\mathrm {m}}$ also enhances the plasmon-photon coupling as $\\xi _{\\mathbf {q}}\\propto \\epsilon _{\\mathrm {m}}^{1/4}/(2+\\epsilon _{\\mathrm {d}}/\\epsilon _{\\mathrm {m}})^{1/2}$ [see Eq.", "()], but the effect of $\\xi _{\\mathbf {q}}$ on the polaritonic dispersion is not easily quantified.", "In Fig.", "REF (c) the plasma frequency $\\omega _{\\mathrm {p}}$ is fixed to the value of silver films [47], [48], while the dielectric constant $\\epsilon _{\\mathrm {m}}$ of the medium is varied.", "In this case, we observe similar effects as in Fig.", "REF (b).", "We note that for certain high-symmetry axes it is possible to derive analytic expressions for the components of the dielectric tensor of the metamaterial, as we show in the Appendix.", "Their dependencies on the wavevector and frequency indicate a nonlocal behavior of the metamaterial in space and time.", "The experimental observability of the band splittings $\\Delta _{\\hat{q}}$ and of their polarization-dependent modulation $\\delta _{\\hat{q}}$ , discussed above, may be hindered by damping mechanisms, leading to the decay of the plasmon polaritons.", "The latter are mostly subject to two sources of damping: Ohmic (absorption) losses with decay rate $\\gamma ^\\mathrm {D}$ inherent to any type of metallic nanostructure [see Eq.", "(REF )], and Landau damping with decay rate $\\gamma ^{\\mathrm {L}}$ , i.e., the decay of the plasmon excitation into electron-hole pairs [41], [51].", "Note that radiation damping is irrelevant for the infinite metacrystals considered here since there is no photonic continuum into which the plasmons can decay.", "Ohmic losses were experimentally estimated to be of the order of $\\gamma ^\\mathrm {D}\\simeq [24]{meV/\\hbar }$ for bulk silver [47].", "Moreover, it has been shown that Landau damping only weakly depends on the dipole-dipole interaction [52], [33], [34], so that we estimate it with the Landau damping of a single nanoparticle.", "This yields $\\gamma ^\\mathrm {L}=3 v_{\\mathrm {F}}g/4r_{\\mathrm {np}}$ , where $v_{\\mathrm {F}}$ is the Fermi velocity and $g$ is a numerical factor of the order of 1 [41], [51], [53], [54].", "For Ag nanoparticles, we obtain $\\hbar \\gamma ^\\mathrm {L}\\simeq [690]{meV}/r_{\\mathrm {np}}\\mathrm {[nm]}$ .", "For the nanoparticle radii that we consider (typically of the order of [10]nm), the total linewidth of the plasmon-polariton bandstructure is therefore of the order of $\\gamma ^\\mathrm {D}+\\gamma ^\\mathrm {L}\\simeq [100]{meV}/\\hbar $ .", "For this reason the splittings in the plasmon-polariton dispersion $\\Delta _{\\hat{q}}$ , as well as their polarization dependence $\\delta _{\\hat{q}}$ for certain directions in the first Brillouin zone, should be experimentally accessible." ], [ "Comparison to classical electrodynamics simulations", "To validate the predictions of our Hamiltonian approach presented in the preceding section, we compare them here to calculations based on classical electrodynamics.", "FDFD simulations are carried out with the electromagnetic wave module of the COMSOL Multiphysics package with the eigenfrequency solver.", "We numerically search for solutions to the eigenequation ${\\nabla }\\times \\left[{\\nabla } \\times \\mathbf {E}(\\mathbf {r},\\omega )\\right]-\\left(\\frac{\\omega }{c}\\right)^2\\epsilon _{\\mathrm {r}}(\\mathbf {r},\\omega )\\mathbf {E}(\\mathbf {r},\\omega )=0,$ where $\\mathbf {E}(\\mathbf {r},\\omega )$ corresponds to the electric field at position $\\mathbf {r}$ and frequency $\\omega $ , and where $\\epsilon _{\\mathrm {r}}(\\mathbf {r},\\omega )$ characterizes the dielectric properties of the metamaterial.", "We consider an infinite, sc lattice with a lattice constant of $[30]{nm}$ , which allows us to simplify the numerical calculations by applying Floquet periodicity on the faces of a unit cell for the electric and magnetic fields.", "We choose nanoparticles of radius $[10]{nm}$ and model them using the Drude dielectric function of Eq.", "(REF ), while in the embedding medium $\\epsilon _{\\mathrm {r}}(\\mathbf {r},\\omega )=\\epsilon _{\\mathrm {m}}$ .", "Note that since we use the eigenvalue solver in COMSOL, we do not insert a driving source into the system.", "The meshes on three surfaces of the cubic cell are of a free triangular type.", "They are copied to the opposite side to be compatible with the Floquet periodicity.", "The cubic cell is filled with an automatically-generated tetrahedral mesh, and the parameters utilized for generating the triangular and tetrahedral meshes are listed in Table REF .", "Table: Parameters of the triangular and tetrahedral meshes used in the COMSOL simulations.The results of the FDFD calculations for the low-energy polaritonic branches are summarized with symbols in Fig.", "REF .", "As for the Hamiltonian approach, parameters in Figs.", "REF (a) and REF (b) are adjusted to give the same Mie frequency $\\omega _0=[3.48]{eV}/\\hbar $ for the nanoparticles.", "In Fig.", "REF (c) we keep $\\omega _{\\mathrm {p}}$ and $\\epsilon _{\\mathrm {d}}$ constant, varying $\\epsilon _{\\mathrm {m}}$ and exploring the influence of a finite Drude damping $\\gamma ^{\\mathrm {D}}$ , which is not contained in our Hamiltonian-based model.", "The parameters $\\omega _{\\mathrm {p}}$ and $\\gamma ^{\\mathrm {D}}$ are chosen as specified for silver in Ref. [47].", "We find an excellent agreement of the FDFD simulations with the predictions of our effective model in all cases, confirming its validity.", "To avoid repetition, we refrain from discussing in further detail the results of the FDFD calculations in Figs.", "REF (a) and REF (b), but concentrate on the new aspect due to the inclusion of a finite damping in Fig.", "REF (c).", "There, the imaginary part of the eigenfrequencies, which can be interpreted as the linewidth broadening due to Ohmic losses, is represented by error bars.", "We find a general trend of an increased damping with increasing wavevector.", "Since the broadenings turn out to have nearly no influence on the polaritonic dispersion relations, our model reproduces the dispersions with great accuracy.", "The small red shift of the FDFD calculations with respect to our model can be understood by the fact that we neglect Umklapp scattering and higher-order multipolar bands, which would push the bands downward in energy.", "Even if a wavevector-independent broadening $\\gamma ^{\\mathrm {L}}$ due to Landau damping would be added, which we argued to be actually larger than the broadening due to Ohmic losses (see the discussion in Sec.", "REF ), the polarization-dependent band gap modulation $\\delta _{\\hat{q}}$ should still be observable.", "With the distribution of the electric field available in the COMSOL package, we can check the polarization direction that our Hamiltonian approach predicts for the sc lattice.", "Along the $\\Gamma M$ direction with $\\hat{q}=(\\hat{x}+\\hat{y})/\\sqrt{2}$ [see Figs.", "REF (d) and REF ] we find that the lowest-energy transverse plasmon-polariton branch exhibits a polarization $\\hat{\\tau }_{\\mathbf {q}}$ parallel to the $\\hat{z}$ -axis, while the second lowest-energy one exhibits a polarization $\\hat{\\tau }_{\\mathbf {q}}$ parallel to $\\hat{y} - \\hat{x}$ .", "This is indeed confirmed by the FDFD calculations for all the parameter sets tested in Fig.", "REF .", "An example of the field distributions is given in Figs.", "REF (a) and REF (b).", "For different lengths of the reciprocal wavevectors $q$ , these modes change in details like the field distribution in the middle of the nanoparticle or the calculated field strength, but the polarization directions and the overall dumbbell shape remain the same.", "Figure: (a),(b) Density plots for the distribution of the electric field ofthe low-energy transverse polaritonic modes, shown on a plane cuttingthrough the center of the primitive cell.", "These distributions arecalculated at 𝐪=0.49π(x ^+y ^)/a\\mathbf {q}=0.49\\pi (\\hat{x}+\\hat{y})/a in the direction of thetwofold symmetry axis, using the parameters r np =[10]nmr_{\\mathrm {np}}=[10]{nm},a=3r np a=3r_{\\mathrm {np}}, ϵ d =5.6\\epsilon _{\\mathrm {d}}=5.6,ϵ m =1\\epsilon _{\\mathrm {m}}=1, ω p =[9.6]eV/ℏ\\omega _{\\mathrm {p}}=[9.6]{eV/\\hbar } andγ D =[22.8]meV/ℏ\\gamma ^\\mathrm {D}=[22.8]{meV/\\hbar } .", "The dipolar modesexhibit a polarization oriented in the (a) y ^-x ^\\hat{y} - \\hat{x} and (b) z ^\\hat{z} directions.We would like to highlight some difficulties in obtaining the polariton dispersions using FDFD calculations.", "As we are trying to solve a non-linear equation in 3D, the solver also converges on many unphysical solutions where the electric field is, e.g., concentrated in a single spot or exhibits a random-looking distribution.", "As a result, one has to manually inspect the field profile of the eigenmodes, discarding the artificial solutions and retaining only those with a dipolar-like character, such as the ones shown in Fig.", "REF .", "Also, the starting and linearization points for the eigenfrequency search were varied for the different calculations, and we checked that they had negligible effect on the real part of the eigenfrequency.", "However, we find that the imaginary part is less robust and changes with distance of the real part of the eigenfrequency from the linearization point.", "For this reason, we took care that the linearization points were located close to the respective eigenfrequencies at each wavevector.", "Furthermore, in the discussion above, we have focused on the low-energy polariton branches which show the interesting polarization-dependent band splittings.", "However, at higher frequencies one can also find, in addition to the longitudinal branch, many polaritonic branches which arise from multipolar plasmon modes.", "The excellent agreement of plasmon-polariton dispersions predicted by our Hamiltonian-based model with those of the FDFD simulations shows that higher multipolar modes beyond the considered dipolar interactions as well as intraparticle retardation effects are irrelevant in the studied parameter regime.", "In addition, we want to point out that the computational costs of our Hamiltonian approach are only a fraction of those of the COMSOL simulations and avoid the cumbersome problems related to the convergence to unphysical solutions.", "Our Hamiltonian-based approach is thus an efficient way to quantitatively predict the response of metacrystals in the near-field regime, when meta-atom separations are much smaller than the LSP resonance wavelength, i.e., $\\omega _0 a/c\\ll 1$ ." ], [ "Conclusions", "In this work we have theoretically studied plasmon polaritons in sc, fcc and bcc lattices of spherical metallic nanoparticles.", "We have developed a model based on a quantum-mechanical Hamiltonian, justified for small nanoparticles (i.e., with a radius between ca.", "1 and $[20]{nm}$ ) in the near-field dipolar regime.", "The dipole-dipole interaction between the nanoparticles leads to collective plasmons, which are delocalized over the metacrystal.", "The strong coupling of these collective plasmons to photons results in the formation of plasmon polaritons.", "Our model readily incorporates retardation effects and considers the dielectric properties of the nanoparticles and of the medium in which they are embedded.", "This has enabled us to derive semi-analytical expressions, which determine collective plasmon dispersions, plasmon-polariton dispersions and their corresponding polarization dependence, and we have analyzed these aspects in detail for the three cubic lattices.", "We have discussed the influence of the dielectric screening due to core electrons of the nanoparticles and due to the embedding medium on these optical properties.", "Specifically, we have shown that the polaritonic dispersions present band splittings in the near-infrared to the visible range of the spectrum for all three cubic lattices and for all high-symmetry axes starting from the center of the first Brillouin zone.", "Remarkably, for special directions in the reciprocal space the polaritonic splitting depends on the polarization, suggesting the possibility to realize a birefringent metacrystal, despite the high degree of cubic symmetry of the latter.", "By comparing our model to classical electrodynamics simulations, we have shown that it is in quantitative agreement at much reduced computational costs.", "This robustness emphasizes that the predicted polarization-dependent band dispersions and band splittings should be observable.", "We thank Pierre Gilliot for enlightening discussions and Charles A.", "Downing for his careful reading of the manuscript.", "S.L.", "and F.P.", "acknowledge funding through the Junior Professorship Program of the Ministry of Science, Research and the Arts (MWK) of Baden-Württemberg within the project “Theory of Plasmonic Nanostructures”, through the Carl Zeiss Foundation and the Collaborative Research Center (SFB) 767 of the German Research Foundation (DFG).", "C.-R.M.", "would like to acknowledge financial support from the EPSRC Center for Doctoral Training in Metamaterials (Grant No.", "EP/L015331/1).", "C.-R.M.", "and E.M. acknowledge financial support by the Royal Society (International Exchange Grant No.", "IE140367, Newton Mobility Grant NI160073, Theo Murphy Award TM160190) and by the Leverhulme Trust (Research Project Grant RPG-2015-101).", "G.W.", "is grateful to the French National Research Agency ANR (Project No.", "ANR-14-CE26-0005 Q-MetaMat) and the CNRS PICS program (Contract No.", "6384 APAG) for financial support.", "Part of this work was performed on the computational resource bwUniCluster, funded by the MWK and the universities of the state of Baden-Württemberg within the framework program bwHPC." ], [ "Appendix: Dielectric tensor", "In this Appendix, we show that our model of interacting plasmonic nanoparticles leads to a nonlocal, dispersive response.", "The dielectric tensor of the metamaterial is calculated explicity for a special, analytically tractable case, and is found to depend on both the wavevector and the frequency.", "We consider the sc crystal and assume $\\mathbf {q}=q \\hat{x}$ .", "In this case, the matrix $F_\\mathbf {q}$ is diagonal and $f_{\\mathbf {q}}^{\\hat{x},\\hat{x}}\\ne f_{\\mathbf {q}}^{\\hat{y},\\hat{y}}=f_{\\mathbf {q}}^{\\hat{z},\\hat{z}}$ .", "Furthermore, the choice of $\\mathbf {q}$ results in a sparse matrix $P_\\mathbf {q}$ with the only nonvanishing components being $P_{\\mathbf {q}}^{\\hat{y},\\hat{\\lambda }_{1, \\mathbf {q}}}= P_{\\mathbf {q}}^{\\hat{z},\\hat{\\lambda }_{2, \\mathbf {q}}}$ .", "Hence the matrix on the left-hand side of Eq.", "(REF ), which we now call $M_{\\mathbf {q}}$ , can be reordered into a block-diagonal form with block matrices $M_{x,\\mathbf {q}}\\ne M_{y,\\mathbf {q}}=M_{z,\\mathbf {q}}$ , which read $M_{x,\\mathbf {q}} = \\begin{pmatrix} \\omega _0 +2\\Omega f_{\\mathbf {q}}^{\\hat{x},\\hat{x}} & -2\\Omega f_{\\mathbf {q}}^{\\hat{x},\\hat{x}} \\\\[.1cm]2\\Omega f_{\\mathbf {q}}^{\\hat{x},\\hat{x}} & -\\omega _0 -2\\Omega f_{\\mathbf {q}}^{\\hat{x},\\hat{x}}\\end{pmatrix}$ and $M_{y,\\mathbf {q}} = \\begin{pmatrix} \\omega _0+2\\Omega f_{\\mathbf {q}}^{\\hat{y},\\hat{y}} & -2\\Omega f_{\\mathbf {q}}^{\\hat{y},\\hat{y}} & -\\mathrm {i}\\omega _0\\xi _\\mathbf {q} &\\mathrm {i}\\omega _0\\xi _\\mathbf {q} \\\\[.1cm] 2\\Omega f_{\\mathbf {q}}^{\\hat{y},\\hat{y}} & -\\omega _0 -2\\Omega f_{\\mathbf {q}}^{\\hat{y},\\hat{y}} & \\mathrm {i}\\omega _0\\xi _\\mathbf {q} &-\\mathrm {i}\\omega _0\\xi _\\mathbf {q} \\\\[.1cm] \\mathrm {i}\\omega _0\\xi _\\mathbf {q} &\\mathrm {i}\\omega _0\\xi _\\mathbf {q} & \\omega _{\\mathrm {ph},\\mathbf {q}}+2\\omega _0\\xi _\\mathbf {q}^2 & -2\\omega _0\\xi _\\mathbf {q}^2 \\\\[.1cm]\\mathrm {i}\\omega _0\\xi _\\mathbf {q} & \\mathrm {i}\\omega _0\\xi _\\mathbf {q} & 2\\omega _0\\xi _\\mathbf {q}^2 &-\\omega _{\\mathrm {ph}, \\mathbf {q}}-2\\omega _0\\xi _\\mathbf {q}^2.\\\\\\end{pmatrix}.$ The matrix $M_{x,\\mathbf {q}}$ leads to the longitudinal plasmon, which does not couple to light within our model.", "For this reason, we concentrate on the transverse components.", "We follow Hopfield [40] to find an expression for the transverse components of the dielectric tensor of the metamaterial $\\epsilon _{\\mathrm {meta}}^{\\hat{y},\\hat{y}}(\\mathbf {q},\\omega )=\\epsilon _{\\mathrm {meta}}^{\\hat{z},\\hat{z}}(\\mathbf {q},\\omega )$ .", "For this purpose, we calculate $\\det \\left(M_{y,\\mathbf {q}}-\\omega ^2\\mathbb {1}_{4} \\right)=0$ and substitute the definition of the dielectric function $c^2q^2= \\epsilon _{\\mathrm {meta}}^{\\hat{y}\\hat{y}}(\\mathbf {q},\\omega ) \\omega ^2$ in the resulting expressions.", "Solving for $\\epsilon _{\\mathrm {meta}}^{\\hat{y}\\hat{y}}(\\mathbf {q},\\omega )$ and exploiting the plasmonic dispersion relation $(\\omega _{\\mathrm {pl},\\mathbf {q}}^{\\hat{y}})^2=\\omega _0^2+4 \\Omega \\omega _0 f_{\\mathbf {q}}^{\\hat{y},\\hat{y}}$ finally yields $\\epsilon _{\\mathrm {meta}}^{\\hat{y}\\hat{y}}(\\mathbf {q},\\omega ) =\\epsilon _{\\mathrm {m}} \\left[ 1 + \\frac{8 \\pi \\Omega \\omega _0}{(\\omega _{\\mathrm {pl},\\mathbf {q}}^{\\hat{y}})^2-\\omega ^2}\\right].$ Equation (REF ) is the same expression as Eq.", "(21) in Ref.", "[39] for $\\epsilon _{\\mathrm {m}}=\\epsilon _{\\mathrm {d}}=1$ and for the respective polarization of the collective plasmon, but we consider here the dipole-dipole interaction beyond the nearest-neighbor limit." ] ]
1606.04897
[ [ "HARQ and AMC: Friends or Foes?" ], [ "Abstract To ensure reliable communication in randomly varying and error-prone channels, wireless systems use adaptive modulation and coding (AMC) as well as hybrid ARQ (HARQ).", "In order to elucidate their compatibility and interaction, we compare the throughput provided by AMC, HARQ, and their combination (AMC-HARQ) under two operational conditions: in slow- and fast block-fading channels.", "Considering both, incremental redundancy HARQ (HARQ-IR) and repetition redundancy HARQ (HARQ-RR) we optimize the rate-decision regions for AMC/HARQ and compare them in terms of attainable throughput.", "Under a fairly general model of the channel variation and the decoding functions, we conclude that i) adding HARQ on top of AMC may be counterproductive in the high average signal- to-noise ratio regime for fast fading channels, and ii) HARQ is useful for slow fading channels, but it provides moderate throughput gains.", "We provide explanations for these results which allow us to propose paths to improve AMC-HARQ systems." ], [ "Introduction", "amc and harq are two transmission strategies commonly used in modern wireless systems to communicate over error-prone and time-varying channels.", "The main objective of this work is to compare amc and harq from the throughput point of view, which is a well-established reference criterion particularly suited for transmission of delay-insensitive contents.", "In broad terms, amc consists in adjusting the transmission parameters (such as the modulation type, the coding rate, and/or the transmission power) to the channel conditions [1], [2]; these are most often defined by a predefined set of mcs.", "The receiver selects a suitable mcs and conveys its index to the transmitter via a feedback channel.", "Transmission errors, unavoidable in any practical system, are handled by the retransmission protocol known as arq[3], where the receiver uses a feedback channel to inform the transmitter about a successful decoding—via a ack message—or about a decoding failure—via a nack message.", "Each nack triggers a new transmission round (or a retransmission), and increasing their number improves reliability.", "In this work, we consider the retransmission protocol known as hybrid arq (harq) in which coding is done across the transmission rounds and thus is intimately related to the amc for which coding and modulation are the core elements.", "In more general terms, both amc and harq may be seen as variable rate transmission strategies: the former, due to the explicit variation of the number of bits conveyed over the channel, the latter, due to the variability of the transmission time resulting from variable number of transmission rounds.", "Both are thus naturally coupled and, in this work we want to clarify to what extent this coupling should be preserved or exploited.", "Therefore, although the formalism of the communication layers tend to separate the amc from harq, the practice may call for their holistic view.", "In particular, the lte standard specifies the harq operation as part of the mac layer [4], whereas the channel measurements procedures are defined in the phy [5].", "However, the amc, i.e., the way the mcs should be chosen based on the reported channel measurement is unspecified and left for implementation.", "We will consider the amc and harq as mechanisms of phy.", "The main difficulty in providing a qualitative insight into the relationship between harq and the amc lies in the fact that they are affected by various elements, such as the adopted channel model, the available coding/modulation schemes, or the power adaptation strategies.", "It is thus critical to strike a realistic balance between the general model and the plethora of clumsy technical details of the practical systems or standards.", "With this idea in mind, the channel model must be kept simple and we will consider two extreme cases of the operating conditions with respect to the channel correlation between harq rounds: i) in fast block-fading channels, all transmission rounds are carried out over independent fading realizations, and ii) in slow block-fading channels, all harq transmission rounds (of the same packet) are carried out in the same operating conditions (are thus perfectly correlated).", "Under these models, the relationship between amc and harq becomes tractable while still allowing us to extrapolate the findings to the situations where the channel correlation is only partial – a case which is difficult to tackle analytically and a numerical approach might be necessary.", "Another important point is the choice of a suitable performance criterion.", "The throughput seen by the upper layers provides a simple and fair comparison baseline, but ignores most of the considerations related to delay or packet loss.", "The former is acceptable by assuming a delay-insensitive traffic and a saturated buffer operating mode; the latter requires a few assumptions about the upper llc layer taking care of all residual retransmission errors which then become irrelevant to the analysis (more on this in Sec.", "REF ).", "The numerical examples that will illustrate our conclusions and findings will be based on a simple model of the decoding errors which may be fit to the experimental data obtained with practical decoders.", "We will then use simplifications to characterize the behavior of irharq when different parts of the codeword are transmitted in the different rounds [6], [7], and of rrharq when each round carried the same codeword [8], [9].", "The contributions of this work can be summarized as follows: We provide a general model which leads to a qualitative evaluation of the relationship between amc and harq.", "We provide the throughput expressions valid for fast- and slow-fading channel models, and compare to the formulations appearing in the literature.", "We discuss how the transmission rate should be chosen based on the observed csi, to maximize the throughput of amc and amc-harq for fast- and slow-fading channels.", "We prove that, in the high average snr regime, combining harq with the amc affects negatively the system throughput in fast-fading channel, and we provide an intuitive explanation of this result.", "We also show that harq is always productive for slow-fading channel.", "Finally, we propose en evaluate two strategies to enhance the performance of harq for fast-fading channels which remove the throughput penalty introduced by the conventional harq.", "The rest of this paper is organized as follows.", "The system model is described in Sec.", ", where – somewhat unconventionally – Sec.", "REF positions this work with respect to the literature.", "We found it the most appropriate because amc and harq have been studied under many different modeling assumptions, and it is difficult to discuss them before the model is introduced.", "The throughput analysis of amc and harq is presented in Sec.", ", Sec.", ", and Sec. .", "The directions towards the improvement of AMC-HARQ systems are explored in Sec.", "and we present our conclusions in Sec. .", "We consider the communication over a block flat-fading channel, so the received signal can be written as $y[n]=\\sqrt{\\mathsf {snr}[n]}s[n]+z[n],$ where $z[n]$ is a zero-mean, unit-variance complex Gaussian noise, $s[n]$ is the signal composed of ${\\mathop {N_{\\textnormal {s}}}}$ symbols carrying the encoded message, and $\\mathsf {snr}[n]$ is the value of the snr in the block-time $n$ .", "We assume that the user is granted access to the channel in predefined, non-consecutive, time blocks.", "The time-difference between the beginning of the blocks $s[n], s[n+1],\\ldots $ is given by $(1+q){\\mathop {N_{\\textnormal {s}}}}$ .", "It is convenient to think about $q$ as an integer, which should be interpreted as the number of blocks left between two consecutive transmission.", "We assume also a transmission with a constant power and, thus, the snr $\\mathsf {snr}[n]$ captures the variation of the channel.", "The relationship between the snrs $\\mathsf {snr}[n], \\mathsf {snr}[n+1]\\ldots $ , depends on the channel coherence time $\\tau _\\textrm {coh}$ , and we consider two different cases Fast fading, where $q{\\mathop {N_{\\textnormal {s}}}}\\gg \\tau _\\textrm {coh}$ , and then the snrs $\\mathsf {snr}[n], \\mathsf {snr}[n+1], \\ldots $ may be modeled as independent random variables.", "This corresponds to the case of fast-moving users where channel conditions change quickly from one block to another.", "Slow fading, where $q{\\mathop {N_{\\textnormal {s}}}}\\ll \\tau _\\textrm {coh}$ , and then the transmissions experience the same snr in many subsequent blocks, i.e., $\\mathsf {snr}[n]\\approx \\mathsf {snr}[n+1]\\approx \\mathsf {snr}[n+2]\\approx \\ldots $ ; this model is suitable for slowly-moving users.", "In the numerical examples, we will use the Rayleigh fading so the snr is modeled by a random variable $\\mathsf {SNR}$ following an exponential distribution $p_{\\mathsf {SNR}}(\\mathsf {snr})=(1/\\overline{\\mathsf {snr}})\\exp \\bigl (-\\mathsf {snr}/\\overline{\\mathsf {snr}}\\bigr ),$ where $\\overline{\\mathsf {snr}}$ is the average snr." ], [ "Physical Layer (PHY): AMC and HARQ", "The role of amc consists in encoding the information bits, $b$ obtained from the llc, and transmitting them over the channel.", "We assume that the transmission rates to be adopted are taken from the set $\\mathcal {R}=\\lbrace R_l\\rbrace _{l=1}^L$ (measured in [bits/symbols]); each transmission thus carries an encoded version of $R_lN_s$ bits.", "This corresponds to the reality of current systems which use predefined sets of transmission rates, each corresponding to a particular mcs supported by both the transmitter and the receiver [5].", "This is different from the approach used in [10], where the optimized rates depended on $\\overline{\\mathsf {snr}}$ .", "We note that the performance of the system is, of course, affected by the choice of mcss, nevertheless, the methodology of comparison and the main conclusions we draw are valid independently of a particular choice.", "The transmitter uses the rate $R_{\\hat{l}}$ , where $\\hat{l}\\in \\lbrace 1,\\ldots ,L\\rbrace $ is the mcs index sent by the receiver over the error-free feedback channel, see Fig.", "REF .", "Due to the block-fading model (REF ), it is enough to discretize the snr, $\\mathsf {snr}$ which we also assume to be perfectly estimated at the receiver, i.e.,  $\\hat{l}=\\sum _{l=1}^L l~{I}_{[\\mathsf {snr}\\in \\mathcal {D}_l]},$ where $\\mathcal {D}_l$ is the mcs decision region that can be adjusted in order to maximize the criterion of interest (here, the throughput).", "It will be formally defined later.", "Figure: Model of the transmission with the arq implemented by the llc; amc and harq belong to phy.Another role of amc is to prepare the upcoming harq transmission rounds.", "To this end, amc encodes the packet into $K$ sub-codewords $x_{1},\\ldots , x_{K}$ of equal length.", "The role of harq is to transmit successively these sub-codewords upon reception of nack messages.", "For the moment, we assume that each sub-codeword occupies the whole available block $s[n]$ , thus, the information in each round is transmitted with the rate $R_l$ determined by the amc in the first round.", "This is an important assumption, which allows us to focus the analysis; we will relaxe it later.", "In harq, the information packet $b$ is sent using many channel blocks, so it is convenient to use a packet-oriented notation and denote the respective variables using subindices, e.g., $y_{t}$ and $\\mathsf {snr}_t$ will denote the channel outcome and the respective snr of the $t$ -th round of the packet $b$ .", "The phy-related actions terminate when ack is received or when the final $K$ -th harq round is reached.", "The receiver then discards the channel outcomes and sends a final acknowledgement that is shared with the llc layer which takes over the communication process." ], [ "Logical Link Control Layer", "In our model, the final acknowledgement of harq is shared with the llc layer as shown schematically in Fig.", "REF .", "llc ignores the details of the operations of phy and only relies on the final ack/nack, implementing thus a basic form of arq: upon reception of ack, the packet $b$ is removed from the llc buffer; if nack is received, the contents (bits) of $b$ are kept in the buffer.", "Then, a new packet $b^{\\prime }$ is formed, which contains ${\\mathop {N_{\\textnormal {s}}}}R_{\\hat{l}}$ bits ($\\hat{l}$ is the mcs index obtained by the transmitter at the moment it has to form the packet $b^{\\prime }$ ).", "The new packet $b^{\\prime }$ may contain some (or all) of the bits from the previously “NACKed\" packet $b$ which, in general, used a rate $R_k\\ne R_{\\hat{l}}$ .", "The feedback channel carrying one-bit ack/nack messages at phy and llc is assumed to be error-free as.", "theoretically, these bits can be protected with arbitrary strength, whose overhead may be neglected for sufficiently large ${\\mathop {N_{\\textnormal {s}}}}$ .In practice, the acknowledgement messages for harq are received with a non-zero error probability; e.g., the lte specifies a minimum snr requirement for reception of the ack messages with errors no greater than 1% [11].", "On the other hand, the acknowledgement messages at the llc are grouped for various packets and protected against errors through coding; then they are considered error-free.", "For tractability, we do not include these elements in our model.", "Since the packet is discarded from the llc buffer only after ack is received, there is no loss of data independently of how unreliable phy is.", "We thus implicitly assumed that we deal with delay-tolerant but loss-sensitive applications, which is justified, for example, when files with critical contents are being transmitted.", "Because it might not be immediately obvious, we emphasize that implementing arq at the llc does not change the llc throughput seen by the upper layers.", "In other words, retransmitting the NACKed packets does not degrade the throughput because this criterion is blind to which bits are actually being transmitted – the fact that the same bits are retransmitted is irrelevant in the throughput evaluation.", "In fact, the throughput remains the same independently of the number of allowed arq rounds at llc, i.e., truncated arq does not change the throughput neither.In practice, the addition delay induced by the retransmissions may be a source of the throughput degradation.", "For example, the TCP may interpret the delay as a congestion and respond by decreasing the size of the transmitted packets, which lowers the throughput.", "This is formally stated in [12], and it holds in absence of other system-related considerations such as the communication overhead in the feedback channel, or the buffer overflow probability, which we ignore here.", "In our case, assuming an unlimited number of arq rounds leads to the lossless communication which is particularly useful from the theoretical point of view, as it allows us to make a fair comparison between various phy strategies using a single criterion – the throughput." ], [ "Decoding Errors: AMC", "The probability of error in the first transmission round, $\\mathsf {Err}_1$ , depends on the experienced snr and the adopted rate $R_l$ , i.e., $\\Pr \\left\\lbrace \\mathsf {Err}_1|R_l,\\mathsf {snr}\\right\\rbrace =\\mathsf {PER}_l(\\mathsf {snr})$ and it might be established experimentally for a given encoding and decoding.", "However, for the numerical analysis, it is convenient to use the parametric description of the per function $\\mathsf {PER}_l(\\mathsf {snr})&={\\left\\lbrace \\begin{array}{ll}1 &\\text{if}\\quad \\mathsf {snr}<\\mathsf {snr}_{\\textrm {th},l}\\\\\\exp [-\\tilde{a}_l (\\mathsf {snr}/\\mathsf {snr}_{\\textrm {th},l}-1) ] &\\text{if}\\quad \\mathsf {snr}\\ge \\mathsf {snr}_{\\textrm {th},l}\\end{array}\\right.", "},$ where the thresholds $\\mathsf {snr}_{\\textrm {th},l}$ and the decay parameters $\\tilde{a}_l$ should be found from the empirical/measured data.", "A form similar to (REF ) has been also used in [13], which provided tabulated values of $\\exp (\\tilde{a}_l)$ , $\\mathsf {snr}_{\\textrm {th},l}$ , and $\\tilde{a}_l/\\mathsf {snr}_{\\textrm {th},l}$ for a particular class of encoders/modulators and decoders.", "We found that, a common decay value $\\tilde{a}_l=\\tilde{a}$ allows for a compact description of the snr-per relationship for the same family of encoders/decoders: for convolutional codes, $\\tilde{a}\\approx 4$ fits well the experimental data, while $\\tilde{a}\\approx 15$ should be used for turbo-coded transmissions with large codewords.", "[13] shows the values $a_l=\\exp (\\tilde{a}_l)$ , i.e., $\\tilde{a}_l=\\log {a}_l\\in \\lbrace 5.6, 4.4, 4.2, 3.9, 4.0, 3.5\\rbrace $ .", "It is not clear how these $a_l$ were obtained, but the per curves in [13] may be approximated using a common value $\\tilde{a}_l=4$ .", "Here, instead of matching the value of $\\tilde{a}$ to a particular family of per curves as done in [13], we treat $\\tilde{a}$ as a parameter which defines how quickly the per curve decays as a function of snr.", "Its operational meaning may be defined from (REF ) in terms of snr, $\\Delta \\cdot \\mathsf {snr}_{\\textrm {th},l}$ , necessary to achieve the desired level of the per, $\\mathsf {PER}_l(\\Delta \\cdot \\mathsf {snr}_{\\textrm {th},l})=\\epsilon $ , as $\\Delta ={\\ln (1/\\epsilon )}/{\\tilde{a}}+1.$ For example, setting $\\epsilon =10^{-2}$ , we obtain $\\Delta =3.3{\\textnormal {dB}}$ for $\\tilde{a}=4$ , and $\\Delta =10{\\textnormal {dB}}$ for $\\tilde{a}=0.5$ .", "We also use $I(\\mathsf {snr}_{\\textrm {th},l})=R_l$ , where $I(\\mathsf {snr})=\\mathsf {I}(S;Y)$ is the mutual information between the random variables $S$ and $Y$ modeling the transmitted and the received signals $s[n]$ and $y[n]$ .", "This coincides with the threshold decoding of the capacity achieving codes which can be modelled with $\\tilde{a}=\\infty $ ; it is convenient because, the results are then comparable to those already presented in the literature for amc [10] or harq [7]." ], [ "Decoding Errors: HARQ", "Let $\\mathsf {NACK}_k=\\lbrace \\mathsf {Err}_1,\\ldots , \\mathsf {Err}_k\\rbrace $ be the event of $k$ consecutive decoding errors.", "Each error event $\\mathsf {Err}_t$ , $t=1,\\ldots , k$ , depends on the snrs experienced in $t$ rounds and on the transmission rate $R=R_l$ adopted in the first round.", "As in Sec.", "REF , the probability of this event is given by the per function $\\Pr \\left\\lbrace \\mathsf {Err}_t|R_l\\right\\rbrace =\\mathsf {PER}_l(\\mathsf {snr}_1, \\ldots ,\\mathsf {snr}_t)$ which now depends on $t$ different snrs.", "However, a multidimensional representation of the per is not tractable, and we adopt the simplifying approach of [14], [15] which reduces the multidimensional function to the following scalar representation $\\mathsf {PER}_l(\\mathsf {snr}_1,\\ldots ,\\mathsf {snr}_k)\\approx \\mathsf {PER}_l(\\mathsf {snr}^\\Sigma _k),$ where the “aggregate” snr obtained as $\\mathsf {snr}^\\Sigma _k=h^{-1}\\Big ( \\sum _{t=1}^k h(\\mathsf {snr}_t)\\Big );$ the function $h(\\cdot )$ and its inverse $h^{-1}(\\cdot )$ depend on the encoding; i.e., on the type of harq.", "For rrharq, each codeword is the same, i.e., $x_1=\\ldots =x_K$ , and the receiver applies a mrcrrharq is also known as “Chase combining” harq.", "to combine the received signals $y_1,\\ldots ,y_k$ .", "We thus can use $h(x)=x$ , and then $\\mathsf {snr}^\\Sigma _k=\\sum _{t=1}^k\\mathsf {snr}_k$ is the “accumulated” snr [7].", "In this case, (REF ) is exact, and the approximation sign may be removed.", "In the case of irharq, codewords $x_k$ , $k=1,\\ldots , K$ , are obtained by taking non-overlapping elements of the “mother” codeword $x_\\textrm {o}=[x_1,\\ldots ,x_K]$ .Codewords $x_k$ are punctured versions of $x_\\textrm {o}$ ; for example, $x_k$ may consist of new parity symbols/bits in each harq round.", "As in [14], [16], [17], we will use $h(x)=I(x)$ , which is concave so we guarantee that in the case of irharq $\\mathsf {snr}^\\Sigma _k>\\sum _{t=1}^k\\mathsf {snr}_k$ .", "That is, as expected, irharq yields smaller probabilities of error compared to rrharq.", "For simplicity, we use $I(\\mathsf {snr})=\\log _2(1+\\mathsf {snr})$ ; although this corresponds to a Gaussian model for $S$ , it is done merely to simplify the modelling.", "The final step requires finding the probability of the event $\\mathsf {NACK}_k$ , and we use here the backward error implication assumption, $\\mathsf {Err}_k \\Rightarrow \\mathsf {Err}_{k-1}\\Rightarrow \\ldots \\Rightarrow \\mathsf {Err}_1$ , already proposed in [18], [19], which yields $\\Pr \\left\\lbrace \\mathsf {NACK}_k\\right\\rbrace &=\\Pr \\left\\lbrace \\mathsf {Err}_1,\\ldots ,\\mathsf {Err}_k\\right\\rbrace =\\Pr \\left\\lbrace \\mathsf {Err}_k\\right\\rbrace \\approx \\mathsf {PER}_k(\\mathsf {snr}_k^\\Sigma ).$ We quickly note that, although formally $\\Pr \\left\\lbrace \\mathsf {NACK}_k\\right\\rbrace \\le \\Pr \\left\\lbrace \\mathsf {Err}_k\\right\\rbrace $ , the approximation (REF ) is very accurate and should be preferred to the very imprecise $\\Pr \\left\\lbrace \\mathsf {NACK}_k\\right\\rbrace =\\prod _{l=1}^k\\Pr \\left\\lbrace \\mathsf {Err}_l\\right\\rbrace $ , which also appeared in the literature, e.g., [20], [21]." ], [ "Relation to Previous Works", "The interaction between the amc and harq has raised a considerable interest in the literature.", "For instance, the throughput of harq with the amc was analyzed in [13], [20], [22], [23], [24], [25], and their delay in [26], [27], [28], [29], [30].", "However, the available results do not allow us to draw clear-cut conclusions, mainly because they are based on different assumptions very often connected to a particular coding or channel models.", "The difficulty is to strike a balance between the simplicity of the analysis and the generality of the conclusions.", "To address this challenge, we use arguably the simplest non-trivial decoding error model, and consider extreme assumptions with regard to the channel model (slow and fast fading), while all the remaining cases (e.g., correlated channels) are expected to yield intermediate results.", "Furthermore, we initially make no efforts to optimize the operation of harq, as done for example in [24], providing thus a “canonical\" model for an interaction between the amc and harq.", "However, the main difference with the previous works is that we adopt the throughput at phy as the unique performance criterion, allowing the llc to handle the residual errors of harq or the amc, at the cost of buffering and delay for the individual packets.", "The conclusions (and the analysis) change when packet loss or delay are taken into consideration.", "For example, the authors of [13] analyze the llc with truncated the arq operating on top of the amc: under constraints on the probability of packet loss at llc, increasing the number of the arq rounds allows the amc to select rather aggressively the rates providing hence a higher throughput.", "In our perspective, the loss is irrelevant and the highest throughput is attained by optimizing the amc without any constraints, letting the llc to deal with the errors." ], [ "AMC", "The throughput is a the long-term average number of correctly received bits per transmitted symbol.", "Since the errors in the amc are block-wise memoryless, the throughput is unaffected by the fading type (slow or fast) and we can take expectation for each block $\\eta ^{\\textnormal {amc}}&\\triangleq {E}_{\\mathsf {SNR}}\\Big [R_{\\hat{l}}\\big (1-\\mathsf {PER}_{\\hat{l}}(\\mathsf {SNR})\\big )\\Big ] \\\\&=\\sum _{l=1}^L \\int _{\\mathsf {snr}\\in \\mathcal {D}_{l}}p_{\\mathsf {SNR}}(x) \\eta _l(x) \\,\\mathrm {d}x,$ where we used (REF ) to obtain (REF ), with $\\eta _l(\\mathsf {snr})\\triangleq R_l(1-\\mathsf {PER}_l(\\mathsf {snr}))$ being the “instantaneous” throughput defined for an snr, $\\mathsf {snr}$ .", "The optimal snr decision region is hence given by $\\mathcal {D}^{{\\textnormal {amc}}}_l=\\lbrace \\mathsf {snr}: \\eta _l(\\mathsf {snr}) \\ge \\eta _k(\\mathsf {snr}), \\quad \\forall k\\ne l\\rbrace .$ Proposition 1 If the per function is defined by (REF ), the optimal decision regions are intervals $\\mathcal {D}^{{\\textnormal {amc}}}_l=[\\gamma _l, \\gamma _{l+1})$ .", "It is enough to show that there exists $\\mathsf {snr}_{\\textnormal {o}}\\ge \\mathsf {snr}_{\\textrm {th},l+1}$ such that $\\eta _{l+1}(\\mathsf {snr})<\\eta _{l}(\\mathsf {snr})$ for $\\mathsf {snr}\\le \\mathsf {snr}_{\\textnormal {o}}$ , and $\\eta ^{\\prime }_{l+1}(\\mathsf {snr})>\\eta ^{\\prime }_{l}(\\mathsf {snr})$ for $\\mathsf {snr}\\ge \\mathsf {snr}_{\\textnormal {o}}$ .", "Then, it can be easily shown with basic algebra that there is only one value of $\\mathsf {snr}$ solving $\\eta _{l+1}(\\mathsf {snr})=\\eta _{l}(\\mathsf {snr})$ .", "Indeed, in most of the practically interesting cases, the decision regions of amc are intervals [1] with boundaries $\\gamma _l$ defined by the intersection of $\\eta _l(\\mathsf {snr})$ and $\\eta _{l-1}(\\mathsf {snr})$ $&R_l\\big (1-\\mathsf {PER}_l(\\gamma _l ) \\big )=R_{l-1}\\big (1-\\mathsf {PER}_{l-1}(\\gamma _l ) \\big ),$ where also, for notational convenience, we use $\\gamma _0\\triangleq 0$ and $\\gamma _{L+1}=\\infty $ .", "This expression may be further simplified assuming that the probability of decoding error when transmitting with rate $R_{l-1}$ is very small at the right border of $\\mathcal {D}^{{\\textnormal {amc}}}_{l-1}$ .It is, indeed, the case if $\\gamma _{l-1}\\ll \\gamma _l$ and the per function decays quickly with $\\mathsf {snr}$ as per (REF ).", "Then, (REF ) becomes $\\mathsf {PER}_{l} (\\gamma _l)\\approx 1-\\frac{R_{l-1}}{R_l}, \\quad l=2,\\ldots ,L.$ Here, we see that if $R_l$ is much larger than $R_{l-1}$ , the nominal error rate at the interval border $\\gamma _l$ may be quite high.", "For example, if $R_{l-1}=1$ and $R_{l}=2$ , we obtain $\\mathsf {PER}_{l} (\\gamma _l)=0.5$ .", "This result may be contrasted with the “hard” limits imposed on the per, $\\mathsf {PER}_{l} (\\gamma _l)=\\mathsf {PER}_{\\textnormal {t}}$ , suggested in [13] and specified by the lte as $\\mathsf {PER}_{\\textnormal {t}}=10^{-1}$ [5].", "Finally, we express the throughput compactly as follows: $\\eta ^{\\textnormal {amc}}&=\\sum _{l=1}^L R_l \\cdot (1-f_{1,l})p_l,$ where $f_{1,l}&\\triangleq \\frac{1}{p_{l}}\\int _{\\gamma _l}^{\\gamma _{l+1}} p_{\\mathsf {SNR}}(x) \\mathsf {PER}_l(x) \\,\\mathrm {d}x,\\quad p_{l}\\triangleq \\int _{\\gamma _l}^{\\gamma _{l+1}} p_{\\mathsf {SNR}}(x) \\,\\mathrm {d}x.$ Using (REF ) in (REF ), we can calculate the border of the decision region as $\\gamma _l &=\\mathsf {snr}_{{\\textnormal {th}},l}\\Big (1+ \\frac{1}{\\tilde{a}}\\ln \\frac{R_l}{R_l-R_{l-1}}\\Big ).$ Therefore, not surprisingly, an increased strength of the coding (large $\\tilde{a}$ ) moves the borders of the decision region closer to the decoding threshold, while a weak coding will result in larger $\\gamma _l$ ; this, via (), will also move the throughput curve to the regions of higher snr." ], [ "Discussion", "We note that, according to the observations in Sec.", "REF , the above results hold also when arq is used on top of the amc.", "Since a similar setup is considered in [13], it is instructive to contrast both results.", "The main difference is that, according to our results, the throughput-optimal intervals defined by $\\gamma _l$ do not change with $M$ the number of arq rounds.", "This is exactly the strength of the throughput criterion which does not change when arq is added on top of a particular phy transmission strategy.", "On the other hand, [13] imposed limits on the packet loss $P_{{\\textnormal {loss}}}$ , and the snr thresholds $\\check{\\gamma }_l$ were adjusted by solving the equation $[\\mathsf {PER}_l(\\check{\\gamma }_l)]^M=P_{{\\textnormal {loss}}}$ , or $\\mathsf {PER}_l(\\check{\\gamma }_l)=P_{{\\textnormal {t}}},$ where $P_{\\textnormal {t}}=P^{1/M}_{\\textnormal {loss}}$ is the target per at each interval boundary $\\check{\\gamma }_l$ .", "Hence, for a small $P_{\\textnormal {loss}}$ , with a growing $M$ it is possible to increase $P_{\\textnormal {t}}$ and then $\\check{\\gamma }_l$ decreases approaching the throughput-optimal value $\\gamma _l$ ; this effect was also observed in [13].", "However, a caution is required because $\\check{\\gamma }_l$ should never be smaller than $\\gamma _l$ , which may happen using (REF ) for large $M$ .", "Example 1 To illustrate the importance of the decision regions, we show in Fig.", "REF the throughput of amc for the (optimal) decision intervals based on (REF ) (as they are practically the same as with (REF )), and based on (REF ), for both cases $P_{\\textnormal {t}}=10^{-1}$ and $P_{\\textnormal {t}}=10^{-2}$ .", "We use $L=5$ rates from the set $\\mathcal {R}=\\lbrace R_l\\rbrace _{l=1}^L$ , where $R_l=l\\cdot 0.75$ .", "From (REF ), we find that $\\mathsf {PER}(\\gamma _l)\\approx 0.5, 0.33, 0.25, 0.2$ .", "To get an insight into the importance of the coding strength, we use $\\tilde{a}\\in \\lbrace 4, 0.5\\rbrace $ , where the weaker code throughput curve is right-shifted by $\\sim 6\\textrm {dB}$ due to the increased borders $\\gamma _l$ of the decision regions, cf.", "(REF ).", "Quite clearly, the strict constraints on the packet loss (small $P_{\\textnormal {t}}$ ) move the snr boundaries $\\check{\\gamma }_l$ to the right; this decreases the throughput and is particularly notable for weak codes (here, $\\tilde{a}=0.5$ ).", "Adding more arq rounds (increasing $M$ ) increases $P_{\\textnormal {t}}$ and moves $\\gamma _l$ to the left (cf.", "(REF )), hence improving the throughput which approaches the optimal solution derived in (REF ).", "Figure: The throughput η amc \\eta ^{\\textnormal {amc}} with the decision intervals based on () and (), for P t =10 -1 P_{\\textnormal {t}}=10^{-1} and P t =10 -2 P_{\\textnormal {t}}=10^{-2}." ], [ "HARQ: Slow Fading", "In the case of slow-fading channels, we may assume that the snr, $\\mathsf {snr}$ , remains constant during many blocks $s[n]$ , which we treat together as a “super channel-block”.", "We may then apply the same approach we already used to analyze amc, and calculate the throughput as $\\eta ^{{\\textnormal {harq}}}_{K}&=\\sum _{l=1}^L \\int _{x \\in \\mathcal {D}_{l}}p_{\\mathsf {SNR}}(x) \\eta ^{{{\\textnormal {harq}}}}_{K,l}(x) \\,\\mathrm {d}x,$ where again $\\mathcal {D}_{l}$ is the snr decision region in which we use the transmission rate $R_l$ , and $ \\eta ^{{{\\textnormal {harq}}}}_{K,l}(x)$ is the throughput of harq carried out with the rate $R_l$ over snr $x$ .", "The main questions are: how to calculate $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(x)$ , and how to find the optimal decision regions $\\mathcal {D}_l$ ?", "As to the throughput, $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(x)$ , the number of channel blocks used to transmit the information packet is now variable due to the decoding errors whose probability is captured by the per function in (REF ).", "Modeling harq rounds by the states of a Markov process, the beginning of the harq cycle corresponds to the renewal of the process.", "Thus, we can use the renewal-reward theorem to obtain [31], [7], [20],[32] $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(\\mathsf {snr})=\\frac{{E}[\\mathsf {R}_l]}{{E}[\\mathsf {T}_l]},$ where $\\mathsf {R}_l\\in \\lbrace R_l,0\\rbrace $ is a random “reward” (here, the number of successfully delivered bits normalized by the number of symbols, ${\\mathop {N_{\\textnormal {s}}}}$ ) at the end of the harq cycle, and $\\mathsf {T}_l\\in \\lbrace 1,\\ldots ,K\\rbrace $ is a random inter-renewal time, i.e., the number of harq rounds.", "The expectations in (REF ) are taken with respect to the decoding errors $\\mathsf {Err}_1, \\ldots , \\mathsf {Err}_K$ which, conditioned on $\\mathsf {SNR}=\\mathsf {snr}$ , are independent from one harq cycle to another.", "Thus, the reward $\\mathsf {R}_l$ and the time $\\mathsf {T}_l$ are also independent between cycles.", "This is, in fact, the necessary condition to use the renewal-reward theorem (REF ) [32].", "On the other hand, $\\mathsf {R}_l$ and the time $\\mathsf {T}_l$ are not unconditionally independent, because they depend on the same realization of the snr.", "For this reason, the expectation cannot be taken with respect to $\\mathsf {SNR}$ as it would imply that the snr is independent between the harq cycles; while this approach may be suitable in bursty communication scenarios [33][34], it is inappropriate in our case.A direct consequence of bursty communications assumption is that the transmissions over “better” channels require shorter times than those made over “worse” channels.", "This creates a coupling between the channel model and the transmission protocol [34] which is undesirable in our communication model.", "The throughput can be hence calculated as [7], [20] $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(\\mathsf {snr})=\\frac{R_l(1-f_{K,l}(\\mathsf {snr}))}{1+\\sum _{k=1}^{K-1} f_{k,l}(\\mathsf {snr})},$ where $f_{k,l}(\\mathsf {snr})=\\Pr \\left\\lbrace \\mathsf {Err}_1,\\ldots ,\\mathsf {Err}_k| R_l\\right\\rbrace $ is the probability of $k$ consecutive decoding errors conditioned on having the first harq round carried out with the rate $R_l$ .", "These are calculated using the approach explained in (REF ) $f_{k,l}(\\mathsf {snr})\\approx \\mathsf {PER}_l (\\mathsf {snr}_k^\\Sigma ),$ where $\\mathsf {snr}_k^\\Sigma $ is obtained from (REF ), and thus depends on the type of harq.", "In particular, for rrharq we have $\\mathsf {snr}_k^\\Sigma =k\\mathsf {snr}$ , while for irharq it is obtained through $I(\\mathsf {snr}_k^\\Sigma )=kI(\\mathsf {snr})$ , where $I(x)=\\log _2(1+x)$ (see also Sec.", "REF ).", "For notational convenience, we use $f_{0,l}(\\mathsf {snr})\\triangleq 1$ .", "Before progressing any further, we first address the following simple question: does the throughput of harq increase with then number of allowed transmissions, $K$ ?", "Proposition 2 If $\\frac{f_{k+1,l}(\\mathsf {snr})}{f_{k,l}(\\mathsf {snr})} \\le \\frac{f_{k,l}(\\mathsf {snr})}{f_{k-1,l}(\\mathsf {snr})}, \\quad \\forall k\\ge 1$ then, for $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(\\mathsf {snr})$ given by (REF ), the following holds: $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(\\mathsf {snr})\\le \\eta ^{{{\\textnormal {harq}}}}_{K+1,l}(\\mathsf {snr})$ .", "Moreover, if there exists $k$ for which the inequality in (REF ) is strict, then we obtain a strict inequality $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(\\mathsf {snr})<\\eta ^{{{\\textnormal {harq}}}}_{K+1,l}(\\mathsf {snr})$ .", "Our objective is to derive a sufficient condition, under which the expression $\\eta _K=\\frac{R(1-f_{K})}{1+\\sum _{k=1}^{K-1} f_{k}}$ is increasing with $K$ , where the dependence on $l$ and $\\mathsf {snr}$ is removed from (REF ) for notational brevity.", "That is we require $\\eta _{K}\\le \\eta _{K+1}&\\Rightarrow \\frac{R\\big (1-f_{K}\\big )}{1+\\sum _{k=1}^{K-1} f_{k}} \\le \\frac{R\\big (1-f_{K+1}\\big )}{1+\\sum _{k=1}^{K} f_{k}}\\\\&\\Rightarrow f_{K+1}\\big (1 + \\sum _{k=1}^{K-1} f_{k}\\big ) \\le f_{K}\\sum _{k=1}^{K} f_{k}\\Rightarrow \\sum _{k=1}^{K} f_{K+1} f_{k-1} \\le \\sum _{k=1}^{K} f_{K} f_{k}.$ Obviously, () is a necessary and sufficient condition for the throughput to be increasing with $K$ .", "To remove the dependence on the sum, we strengthen () by imposing the inequality on each term yielding the sufficient (not necessary) condition $f_{K+1} f_{k-1}\\le f_{K} f_{k}\\quad \\forall k\\le K$ , which must hold $\\forall K \\ge 1$ ; this is (REF ).", "Any strict inequality will make the inequality in (REF ) strict as well.", "This terminates the proof.", "The above proposition rectifies the conditions exposed in [7], where the “subgeometric” property $f_{k,l}< f_{1,l}^k$ was conjectured as a sufficient condition for the throughput to be increasing with $K$ .", "In fact, it is not, as we may show by constructing a subgeometric relationship and yet obtaining a non-increasing throughput with $K$ .", "For example: the subgeometric terms $f_{2}=0.5f_{1}^2, f_{3}=0.75f_{1}^3$ yield $\\eta ^{{{\\textnormal {harq}}}}_{2}>\\eta ^{{{\\textnormal {harq}}}}_{3}$ , which invalidates the conjecture of [7].", "The value of Proposition REF is that, if the error probabilities in harq satisfy the conditions (REF ), we can guarantee that harq will improve the throughput of amc provided the same snr decision regions are used, i.e., $\\mathcal {D}_l=\\mathcal {D}^{{\\textnormal {amc}}}_l$ .", "It is easy to show that the conditions in Proposition REF are satisfied using rrharq and irharq under the decoding model of Sec.", "REF .", "Thus, harq will outperform the amc in the slow-fading channels, and the gains may be larger if the decision regions (REF ) take into account the reality of harq and are redefined as follows $\\mathcal {D}^{{{\\textnormal {harq}}}}_l=\\lbrace \\mathsf {snr}: \\eta ^{{{\\textnormal {harq}}}}_{K,l}(\\mathsf {snr}) \\ge \\eta ^{{{\\textnormal {harq}}}}_{K,k}(\\mathsf {snr}), \\quad \\forall k\\ne l\\rbrace .$ Figure: In slow-fading channels, decision regions 𝒟 l harq \\mathcal {D}^{{\\textnormal {harq}}}_l are unions of intervals.", "Here, R 4 =3R_4=3, R 5 =3.75R_5=3.75, and we assume irharq with K=4K=4.", "a ˜=4\\tilde{a}=4.Here, however, we may run into a difficulty because the throughput $\\eta ^{{{\\textnormal {harq}}}}_{K,l}(\\mathsf {snr})$ is not a concave function of $\\mathsf {snr}$ .", "Since then, the unique intersection condition we obtained in Proposition REF cannot be guaranteed, the decision regions $\\mathcal {D}^{{{\\textnormal {harq}}}}_l$ , in general, are non-convex sets and should be represented by the union of intervals as shown in the following example.", "Example 2 With two rates $R_5=3.75$ and $R_4=3$ – as in Example REF , and irharq with $K=4$ , we can see from Fig.", "REF that each decision region $\\mathcal {D}^{{\\textnormal {harq}}}_4$ (shaded) and $\\mathcal {D}^{{\\textnormal {harq}}}_5$ (white) is the union of intervals.", "The boundary of the right-most piece of $\\mathcal {D}^{{\\textnormal {harq}}}_5$ is shown with a black marker, where we can observe it being close to the boundary of $\\mathcal {D}^{\\textnormal {amc}}_5$ (shown with a white marker).", "Moreover, in most of the practical cases the rates are relatively close to each other, then if $R_l>R_{l-1}/2$ is ensured, it means that the boundary of $\\mathcal {D}^{{\\textnormal {harq}}}_l$ will be very close to the boundary of $\\mathcal {D}^{\\textnormal {amc}}_l$ .", "Furthermore, if sufficiently “dense\" set of rates $\\mathcal {R}$ is available, the region of the achievable throughput is well covered.", "While, in theory, the system may operate with arbitrary $\\mathcal {D}^{{{\\textnormal {harq}}}}_l$ , it is much more convenient to use intervals as in the case of amc.", "In fact, using $\\mathcal {D}^{\\textnormal {amc}}_l$ is not a bad idea since Proposition REF confirms that the throughput will increase when using harq on top of the amc.", "Figure: Throughput of amc and amc-harq for slow-fading channels using a) rrharq, and b) irharq.Example 3 In Fig.", "REF , we compare the throughput obtained using the amc and harq with rates from the set $\\mathcal {R}$ as in Example REF .", "Both irharq and rrharq are considered with $K=4$ rounds, where we also consider two cases for the decision regions for harq, namely, $\\mathcal {D}^{\\textnormal {amc}}_l$ and $\\mathcal {D}^{{\\textnormal {harq}}}_l$ .", "Two coding strengths are considered: $\\tilde{a}\\in \\lbrace 0.5, 4\\rbrace $ .", "As expected, the throughput of harq is improved with respect to the amc even if ${\\mathcal {D}}^{{\\textnormal {amc}}}_{l}$ is used.", "The gains of using the optimal decision regions $\\mathcal {D}^{{{\\textnormal {harq}}}}_{l}$ are important only for irharq; they are notable in low snr and are pronounced for weak codes.", "On the other hand, in the case of rrharq, we obtain $\\mathcal {D}^{{{\\textnormal {harq}}}}_{l}\\approx \\mathcal {D}^{{\\textnormal {amc}}}_{l}$ and that is why we do not show two distinct curves in this case." ], [ "HARQ: Fast Fading", "In the case of fast-fading channels, the throughput can again be expressed using the renewal-reward theorem (REF ) $\\eta ^{{\\textnormal {harq}}}=\\frac{{E}[\\mathsf {R}]}{{E}[\\mathsf {T}]},$ where $\\mathsf {R}$ is the random reward.", "Since the snrs vary independently from one transmission round to another (and from one harq cycle to another), the expectations in (REF ) are taken also with respect to all the snrs affecting the transmission (unlike in (REF ), where the expectation over $\\mathsf {SNR}_1$ was taken outside of the fraction).", "Consequently, unlike (REF ), Since the rate is random from one harq cykle to another, the expected reward is given by ${E}[\\mathsf {R}]&=\\sum _{l=1}^L R_l (1-f_{K,l}) p_l,$ where $p_l$ is the probability of choosing the rate $R_l$ in the first harq round, given by $p_l\\triangleq \\Pr \\left\\lbrace \\mathsf {SNR}\\in \\mathcal {D}_l\\right\\rbrace ,$ with $\\mathcal {D}^{{\\textnormal {harq}}}_l$ being the decision regions we have to define.", "Again, $f_{k,l}$ that denotes the probability of $k$ consecutive errors (conditioned on starting the harq cycle with the rate $R_l$ ) is given by $f_{k,l}&\\triangleq \\Pr \\left\\lbrace \\mathsf {NACK}_k | R_l \\right\\rbrace =\\frac{1}{p_l}\\int _{\\mathcal {D}_l}p_{\\mathsf {SNR}_1}(x)f_{k,l}(x)\\,\\mathrm {d}x,\\\\f_{k,l}(x)&={E}_{\\lbrace \\mathsf {SNR}_l\\rbrace _{l=2}^k }\\big [\\mathsf {PER}_l(x,\\mathsf {SNR}_2,\\ldots ,\\mathsf {SNR}_k) \\big ].$ Both (REF ) and () require a one-dimensional integration respectively over $\\mathsf {SNR}_1\\in \\mathcal {D}_l$ and an aggregate snr (merging $\\mathsf {snr}_1=x$ with $\\mathsf {SNR}_2,\\ldots , \\mathsf {SNR}_k$ via (REF )).", "Similarly, the expected number of harq rounds is given by ${E}[\\mathsf {T}]&=\\sum _{l=1}^L \\int _{\\mathcal {D}_l}p_{\\mathsf {SNR}}(x) T_{K,l}(x) \\,\\mathrm {d}x=\\sum _{l=1}^L\\overline{T}_{K,l} p_l,$ where $T_{K,l}(\\mathsf {snr})=1+\\sum _{k=1}^{K-1} f_{k,l}(\\mathsf {snr})$ is the expected number of rounds with the first being carried out over $\\mathsf {snr}\\in \\mathcal {D}^{{\\textnormal {harq}}}_l$ , and $\\overline{T}_{K,l}=1+\\sum _{k=1}^{K-1} f_{k,l}$ .", "The throughput of harq is then given by $\\eta ^{{\\textnormal {harq}}}_K=\\frac{\\sum _{l=1}^L R_l (1-f_{K,l})p_l}{\\sum _{l=1}^L \\overline{T}_{K,l}p_l },$ where both the numerator and the denominator depend on $\\mathcal {D}^{{\\textnormal {harq}}}_l$ , $l=1, \\ldots , L$ .", "For the moment, we do not make any assumption about the coding, i.e., we make no distinction between irharq and rrharq." ], [ "Decision Regions", "The regions $\\mathcal {D}^{{\\textnormal {harq}}}_l$ that maximize (REF ) are in general non-convex and must by represented as a union of intervals.", "Since this would lead to a tedious optimization problem, we restrict our considerations to the amc-like decision regions $\\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_l=[\\gamma _l,\\gamma _{l+1}),$ which, as we will see, become an insightful proxy to the optimal solution despite their suboptimality.", "Denoting (REF ) as $\\eta ^{{\\textnormal {harq}}}_K(\\gamma )$ , where $\\gamma \\triangleq [\\gamma _1,\\ldots , \\gamma _L]$ , we solve the following optimization problem $\\hat{\\eta }^{{\\textnormal {harq}}}_K\\triangleq &\\max _{\\gamma } \\eta ^{{\\textnormal {harq}}}_K(\\gamma )\\quad \\text{s.t.", "}\\quad \\gamma _1\\le \\gamma _2 \\le \\ldots \\le \\gamma _L\\le \\gamma _{L+1},$ where $\\gamma _1=0, \\gamma _{L+1}=\\infty $ are adopted for notational convenience.", "Since the numerator and the denominator of (REF ) depend on $\\gamma $ , we will use the fractional programming approach [35].", "We define the following function $F(\\gamma ,\\lambda )\\triangleq \\sum _{l=1}^L R_l &p_l\\big (1-f_{K,l}\\big )-\\lambda \\big (\\sum _{l=1}^L \\overline{T}_{K,l} p_l\\big ),$ and solve the new optimization problem $\\gamma _{\\lambda }\\triangleq &\\mathop {\\mathrm {argmax}}_{\\gamma } F (\\gamma ,\\lambda )\\quad \\text{s.t.}", "\\quad 0=\\gamma _1\\le \\gamma _2 \\le \\ldots \\le \\gamma _L\\le \\infty .$ Then, the optimal solution of (REF ) is found examining the sign of $F(\\gamma ,\\lambda )$ due to the simple relationships [35] $F(\\gamma _{\\lambda },\\lambda )>0 & \\iff \\lambda <\\hat{\\eta }^{{\\textnormal {harq}}}_K, \\\\F(\\gamma _{\\lambda },\\lambda )<0 & \\iff \\lambda >\\hat{\\eta }^{{\\textnormal {harq}}}_K, \\\\F(\\gamma _{\\lambda },\\lambda )=0 & \\iff \\lambda =\\hat{\\eta }^{{\\textnormal {harq}}}_K .$ Consequently, to solve (REF ), it is enough to solve (REF ) for each $\\lambda $ , and find $\\lambda $ such that $F(x_{\\lambda },\\lambda )=0$ .", "The latter can be done efficiently using simple numerical methods (a bisection search, for instance).", "To solve (REF ), we differentiate (REF ) with respect to $\\gamma _l$ , and obtain the following kkt equations $\\nonumber R_{l-1}\\big (1-{E}_{\\lbrace \\mathsf {SNR}_l\\rbrace _{l=2}^K }\\big [\\mathsf {PER}_{l-1}(\\mathsf {snr}^\\Sigma _K(\\gamma _l)) \\big ]\\big )&-R_{l}\\big (1-{E}_{\\lbrace \\mathsf {SNR}_l\\rbrace _{l=2}^K }\\big [\\mathsf {PER}_l(\\mathsf {snr}^\\Sigma _K(\\gamma _l)) \\big ]\\big )\\\\&\\qquad =\\lambda \\big (T_{K,l-1}(\\gamma _l) -T_{K,l}(\\gamma _l)\\big ),$ that may be solved for each $\\gamma _l$ under the constraints $\\gamma _l\\le \\gamma _{l+1}$ , $l=1,\\ldots , L$ .", "We note that it is possible to obtain $\\gamma _l=\\gamma _{l+1}$ ; in which case the decision region is degenerate $\\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_l=\\varnothing $ .", "Figure: Optimal thresholds γ l \\gamma _l defining the decision regions 𝒟 ˜ l harq \\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_l for fast-fading channels with a) a ˜=4\\tilde{a}=4, and b) a ˜=0.5\\tilde{a}=0.5.", "The filled circles on the 𝗌𝗇𝗋 ¯\\overline{\\mathsf {snr}} axis indicate the value 𝗌𝗇𝗋 ¯ low,l \\overline{\\mathsf {snr}}_{{\\textnormal {low}},l} such that if 𝗌𝗇𝗋 ¯<𝗌𝗇𝗋 ¯ low,l \\overline{\\mathsf {snr}}<\\overline{\\mathsf {snr}}_{{\\textnormal {low}},l}; 𝒟 ˜ l harq \\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_l is not degenerate, i.e., γ l >γ l-1 \\gamma _l>\\gamma _{l-1}.", "We note that in a) we have 𝗌𝗇𝗋 ¯ low,4 =𝗌𝗇𝗋 ¯ low,5 \\overline{\\mathsf {snr}}_{{\\textnormal {low}},4}=\\overline{\\mathsf {snr}}_{{\\textnormal {low}},5}.", "The hollow circle indicates the value 𝗌𝗇𝗋 ¯ high \\overline{\\mathsf {snr}}_{{\\textnormal {high}}} (common to all decision regions) such that if 𝗌𝗇𝗋 ¯>𝗌𝗇𝗋 ¯ high \\overline{\\mathsf {snr}}>\\overline{\\mathsf {snr}}_{{\\textnormal {high}}}; 𝒟 ˜ l harq \\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_l is not degenerate.Example 4 Considering the same set of rates $\\mathcal {R}$ as in previous examples, we show in Fig.", "REF the thresholds $\\lbrace \\gamma _2,\\gamma _3,\\gamma _4,\\gamma _5\\rbrace $ obtained after solving (REF ).", "The optimal thresholds for the amc, which do not depend on $\\overline{\\mathsf {snr}}$ , are shown as a reference.", "The relationship between $\\gamma _l$ and $\\overline{\\mathsf {snr}}$ is quite complex, and it provides an immediate indication on how tedious it will be to find the optimal regions $\\mathcal {D}^{{\\textnormal {harq}}}_l$ (as a union of intervals).", "Some intuition about the potential gains of using harq on top of amc may be obtained from Fig.", "REF a.", "For high (average) snrs, the thresholds of harq are higher compared to the amc, this more “conservative\" choice of rates is meant to avoid errors and, hence, not to use harq.", "On the other hand, for low snrs, the thresholds of harq are lower compared to the amc: the rates are adopted in a more “aggressive\" way indicating that the retransmissions would be beneficial.", "This becomes particularly clear with medium snrs, where the regions degenerate and only $\\tilde{\\mathcal {D}}_5=[0,\\infty )$ ; this occurs with irharq for $\\overline{\\mathsf {snr}}\\in (1~\\!", "{\\textnormal {dB}}, 9~\\!", "{\\textnormal {dB}})$ , and this interval is indicated with the markers on the $\\overline{\\mathsf {snr}}$ axis.", "Similar conclusions may be drawn from Fig.", "REF b, where the relationship is slightly more complex due to a much less steeper per curve of the decoder.", "We also observe that, for high snrs, all decision regions are not degenerate, and we conjecture that it would be always true.", "We adopt the following reasoning: we know that the region $\\mathcal {D}^{{\\textnormal {harq}}}_L$ is not degenerate because, if it was, it would not be possible to attain a throughput that is close to $R_L$ .", "Now, for all $\\mathsf {snr}<\\mathsf {snr}_{{\\textnormal {th}},L}$ , transmitting with the rate $R_L$ provokes an error, so we ask a question: is it more beneficial to avoid errors or to count on retransmissions?", "This can be answered comparing the rewards, $\\mathsf {R}$ , associated with the selections of rates (actions).", "The assignment $\\mathsf {snr}\\in \\mathcal {D}^{{\\textnormal {harq}}}_{L-1}$ yields an expected immediate reward of $\\mathsf {R}=R_{L-1}(1-f_{L-1,1})$ ; on the other hand, if $\\mathsf {snr}\\in \\mathcal {D}^{{\\textnormal {harq}}}_L$ , the reward is at most $\\mathsf {R}=R_L/2$ because it takes at least two harq rounds.", "Thus, for $R_{L-1}\\gg R_L/2$ and a sufficiently small $f_{l,1}$ , we will assign $\\mathsf {snr}$ to $\\mathcal {D}_l$ .", "A similar reasoning holds when $R_{L-1}\\gg R_L/k$ .", "We thus conclude that the region $\\mathcal {D}^{{\\textnormal {harq}}}_{L-1}$ is not degenerate.", "Furthermore, with the same arguments for $\\mathsf {snr}<\\mathsf {snr}_{{\\textnormal {th}},L-1}$ , we can find that the regions $\\mathcal {D}^{{\\textnormal {harq}}}_{L-2}$ is not degenerate." ], [ "Throughput", "Proposition REF answered the important question about the value of using harq on top of the amc in slow-fading channels.", "In fast-fading channels, however, the derivations are more complicated due to the sums appearing in the numerator and the denominator in (REF ).", "To obtain answers in this case, we will consider separately the regimes of high- and low- average snrs.", "From the results in Example REF , we have already obtained an indication about the added-value of harq: retransmissions were more valuable in the low- than in the high-snr regime.", "However, the final conclusions will be drawn by looking at the throughput.", "Example 5 The throughput obtained by adopting the decision regions shown in Example REF is represented in Fig.", "REF .", "The results may seem surprising at the first glance, but they are completely in line with the behavior of the decision regions: the advantage of harq is well pronounced in low $\\overline{\\mathsf {snr}}$ .", "On the other hand, for high $\\overline{\\mathsf {snr}}$ , adding harq on top of the amc is counterproductive.", "The throughput penalty is small but clearly observable, e.g., considering $\\tilde{a}=4$ and rrharq, this effect occurs above $3~\\!", "{\\textnormal {dB}}$ , and above $9~\\!", "{\\textnormal {dB}}$ for irharq; these break-points are indicated with markers on the figure.", "We also note that using the optimized decision regions $\\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_l$ is beneficial for amc-irharq when the break-points move to the high-snrs.", "However, this gain is moderate for strong codes ($\\tilde{a}=4$ ), and the general conclusion holds: there are regions of snr for which it is counterproductive to use harq.", "Finally, Fig.", "REF b shows the throughput with irharq in the range $\\overline{\\mathsf {snr}}\\in (1~\\!", "{\\textnormal {dB}}, 9~\\!", "{\\textnormal {dB}})$ for $\\tilde{a}=4$ , where $\\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_5=[0, \\infty )$ (cf.", "Example REF ) implies that we effectively ignore the csi; meaning that the adaptation to the channel is counterproductive.", "This observation is explained by the fact that amc-harq cannot be considered as a fully adaptive transmission since it ignores the csi after the first round.", "In fact, this is the source of a surprising and disappointing behavior of harq in high-snrs.", "Figure: The throughput of amc and amc-harq over fast-fading channels for a) rrharq and b) irharq.", "The markers on the 𝗌𝗇𝗋 ¯\\overline{\\mathsf {snr}} axis indicate the points where the harq throughput curves cross the throughput curve of amc.", "Filled and hollow markers correspond to harq based on 𝒟 ˜ l harq \\tilde{\\mathcal {D}}^{{\\textnormal {harq}}}_l and 𝒟 l amc \\mathcal {D}^{\\textnormal {amc}}_l, respectively.We will explain now more formally the results we have observed in the examples.", "We will show that, in the low-snr regime, harq is more productive, but at high $\\overline{\\mathsf {snr}}$ , amc outperforms amc-harq under mild conditions on the per curves of the decoder.", "Proposition 3 (Low SNR) Denote by $\\hat{\\eta }^{{\\textnormal {harq}}}_K$ the optimal throughput of harq based on $\\mathcal {D}^{{\\textnormal {harq}}}_l$ .", "There exists $\\overline{\\mathsf {snr}}_{\\textnormal {o}}$ such that, for all $\\overline{\\mathsf {snr}}<\\overline{\\mathsf {snr}}_{\\textnormal {o}}$ , $\\hat{\\eta }^{{\\textnormal {harq}}}_K \\ge \\eta ^{\\textnormal {amc}}.$ We denote by $\\check{\\eta }^{{\\textnormal {harq}}}_K$ the throughput of harq based on $\\mathcal {D}^{\\textnormal {amc}}_l$ where, due to the non-optimized choice of the decision regions, we have $\\check{\\eta }^{{\\textnormal {harq}}}_K\\le \\hat{\\eta }^{{\\textnormal {harq}}}_K$ .", "Since $\\lim _{\\overline{\\mathsf {snr}}\\rightarrow 0}p_l f_{K,l}=0, l>1$ , and $\\lim _{\\overline{\\mathsf {snr}}\\rightarrow 0}f_{K,1}p_1=f_{K,1}$ , we compare the throughputs using the terms appearing in the limits $\\lim _{\\overline{\\mathsf {snr}}\\rightarrow 0}\\check{\\eta }^{{{\\textnormal {harq}}}}_K&= \\frac{R_1(1-f_{K,1})}{\\overline{T}_{K,1}},\\\\\\lim _{\\overline{\\mathsf {snr}}\\rightarrow 0}\\eta ^{{\\textnormal {amc}}}&= R_1(1-f_{1,1}).$ Thus, in the low-snr regime, both harq and the amc may be considered as single-rate transmission schemes (i.e., without rate adaptation).", "So, from Proposition REF we have $\\check{\\eta }^{{{\\textnormal {harq}}}}_K \\ge \\eta ^{{\\textnormal {amc}}}$ .", "For high $\\overline{\\mathsf {snr}}$ , the situation is slightly more involved, and we cannot use (REF ) because all terms $f_{K,l}p_l$ have the same limit, $\\lim _{\\overline{\\mathsf {snr}}\\rightarrow \\infty }f_{K,l}p_l=0$ .", "To compare the throughputs of harq and amc, we will thus define a trp as a hypothetical harq transmission which, in the first round, behaves just like a conventional harq (and is described by the same probability of decoding error), while in the second transmission round, it guarantees that the message is decoded, i.e., $f_{2,l}=0$ .", "The throughput of trp is given by $\\eta ^{{{\\textnormal {2r}}}} &\\triangleq \\frac{\\sum _{l=1}^L R_l p_l}{1 + \\sum _{l=1}^L f_{1,l} p_l } \\nonumber \\\\&=\\frac{\\sum _{l=1}^L R_l p_l}{1 + \\overline{f}_{1} },$ where $\\overline{f}_{1}\\triangleq \\sum _{l=1}^L f_{1,l} p_l$ is the average probability of error in the first round, i.e., the probability that re-transmissions are required, $\\overline{f}_{1}=\\Pr \\left\\lbrace \\mathsf {NACK}_1\\right\\rbrace $ .This should not be confused with the average per.", "Comparing (REF ) and (REF ), we see that the throughput of trp upper bounds the throughput of harq, $\\eta ^{{\\textnormal {harq}}}_K <\\eta ^{{{\\textnormal {2r}}}}$ .", "To focus the discussion, we start with the assumption that the decision regions $\\mathcal {D}^{{\\textnormal {harq}}}_l$ are not degenerate.", "Proposition 4 Let $x_{l}\\triangleq \\lim _{\\overline{\\mathsf {snr}}\\rightarrow \\infty }\\frac{f_{1,l}p_l}{f_{1,L}p_L}$ .", "If $x_l>0$ , $l=1,\\ldots , L-1$ , then there is a $\\overline{\\mathsf {snr}}_{\\textnormal {o}}$ such that, for all $\\overline{\\mathsf {snr}}>\\overline{\\mathsf {snr}}_{\\textnormal {o}}$ , $\\eta ^{{{\\textnormal {harq}}}}_K < \\eta ^{\\textnormal {amc}}.$ Since $\\eta ^{{\\textnormal {harq}}}_K \\le \\eta ^{{{\\textnormal {2r}}}}$ , it is enough to prove that $\\eta ^{{{\\textnormal {2r}}}}<\\eta ^{{\\textnormal {amc}}}$ to obtain a sufficient condition for (REF ).", "We have $\\eta ^{{{\\textnormal {2r}}}}&<\\eta ^{{\\textnormal {amc}}} \\\\\\frac{\\sum _{l=1}^L R_l p_l}{1 + \\overline{f}_{1} }&<\\sum _{l=1}^L R_l (1-f_{1,l})p_l\\\\\\sum _{l=1}^L R_l p_l&<(1 + \\overline{f}_{1})\\sum _{l=1}^L R_l (1-f_{1,l})p_l\\\\\\sum _{l=1}^L R_l f_{1,l}p_l&<\\overline{f}_{1}\\eta ^{\\textnormal {amc}},$ and by taking the limit $\\overline{\\mathsf {snr}}\\rightarrow \\infty $ of both sides of (), we get $\\sum _{l=1}^L R_l x_{l,L} &< \\sum _{l=1}^L R_L x_{l,L},$ which is obviously true if $x_l>0$ .", "Thus, for sufficiently large $\\overline{\\mathsf {snr}}$ , $\\eta ^{{\\textnormal {amc}}}>\\eta ^{{{\\textnormal {2r}}}}>\\eta ^{{\\textnormal {harq}}}_K$ .", "The value of Proposition REF is that it is obtained using solely the model of the decoding errors in the first transmission.", "Hence, we do not need to consider approximations with regard to the way the redundancy is introduced by harq.", "We also quickly note that, using the model of the decoding errors from (REF ) and [13], we obtain $x_{l,L}>0, \\forall l$ , provided that the decision regions $\\mathcal {D}^{{\\textnormal {harq}}}_l$ are not degenerate (i.e., when $p_l>0$ ).", "This brings us to the following question: will we be able to obtain gains by using harq on top of amc if the decision regions $\\mathcal {D}^{{\\textnormal {harq}}}_l$ are optimized?", "The answer is “no”, because Proposition REF does not assume anything about the decision regions $\\mathcal {D}^{{\\textnormal {harq}}}_l$ , except that there are at least two (in which case there are at least two non-zero terms $x_l$ ).", "This condition materializes for the high-snr regime under the conjecture we made in Example REF .", "Another interesting question is whether we can decrease the gap between $\\eta ^{{\\textnormal {amc}}}$ and $\\eta ^{{{\\textnormal {2r}}}}$ by increasing the number of rated in the set $\\mathcal {R}$ ?", "The question may be difficult to answer.", "Nevertheless we consider here a particular and practical case where we cannot use coding rates below $R_1$ which is determined by the minimum implementable encoding rate (e.g., the coding rate of $1/3$ in the popular turbo-codes) and the minimum modulation rate (e.g., bpsk).", "Also, a finite size of the constellation, e.g., 16-qam, implies that there is a maximum rate $R_L$ which cannot be exceeded.", "We can manipulate, however, the granularity of the set $\\mathcal {R}$ , i.e., we can increase the number of rates, $L$ , while keeping $R_1$ and $R_L$ constant.", "Corollary 1 For $\\tilde{a}=\\infty $ , keeping $R_1$ and $R_L$ constant, the difference $ \\eta ^{{\\textnormal {amc}}}-\\eta ^{{{\\textnormal {2r}}}}$ increases with $L$ .", "We first rewrite (REF ) as follows $\\eta ^{{\\textnormal {amc}}}-\\eta ^{{{\\textnormal {2r}}}}= \\frac{\\overline{f}_{1}(\\eta ^{{\\textnormal {amc}}}-R_1)}{1+\\overline{f}_{1}}.$ Then, for $\\tilde{a}=\\infty $ , $\\overline{f}_{1}$ is independent from $L$ because the only source of error is the event $\\lbrace \\mathsf {SNR}\\in (0, \\mathsf {snr}_{{\\textnormal {th}},1} )\\rbrace $ .", "Since $R_1$ is fixed by assumption, and $\\eta ^{{\\textnormal {amc}}}$ is increasing with $L$ , the difference $\\eta ^{{\\textnormal {amc}}}-\\eta ^{{{\\textnormal {2r}}}}$ can only increase with $L$ .", "Although Corollary REF does not talk about the gap $\\eta ^{{\\textnormal {amc}}}-\\eta ^{{{\\textnormal {harq}}}}$ , according to our observations, for high-snr regime, we obtain a $\\eta ^{{{\\textnormal {harq}}}}$ indistinguishable from $\\eta ^{{\\textnormal {2r}}}$ .", "Moreover, setting a practical (finite) value for $\\tilde{a}$ does not change the results, and we indeed observe that the difference $\\eta ^{{\\textnormal {amc}}}-\\eta ^{{{\\textnormal {harq}}}}$ increases with $L$ even if $\\tilde{a}<\\infty $ ." ], [ "Discussion", "The presented results indicate that adding harq on top of the amc may be counterproductive in terms of throughput.", "This finding is not only counterintuitive; but goes against the general belief that both the amc and harq complement each other in their operations.", "We try to provide more intuition about this unusual—at first glance—behavior.", "We do it in two ways: first, by exploring the temporal relationships of csi, and second, by reinterpreting the equation in the proof of Proposition REF ." ], [ "Temporal evolution of the channel states", "Let us start with the relationship between the channel state in two different time instants assuming operation in high $\\overline{\\mathsf {snr}}$ .", "Then, the amc is very likely to choose a high transmission rate, e.g., $R_L$ .", "A small rate, e.g., $R_1$ , may be chosen as well, which happens when the measured snr, $\\mathsf {snr}[n]$ , falls into the interval $(0,\\gamma _2)$ ; this event has a probability $p_1>0$ .", "Let us assume that the error occurs when transmitting with the rate $R_1$ , so the reward in time $n$ is zero, $\\mathsf {R}[n]=0$ .", "We can now compare the expected reward in the block $n+1$ due to the adoption of amc or amc-harq.", "With the amc-harq, the error is handled by the second harq round where the chances for a successful decoding (both for irharq and rrharq), $f_{1,2}$ are high; the expected reward in this second round is thus given by ${E}\\big [\\mathsf {R}[n+1]\\big ]=R_1(1-f_{1,2})\\approx R_1.$ On the other hand, with the amc, the error is handled by the llc, so the new packet is transmitted in time $n+1$ with the rate $R_l$ .", "The expected reward in this case with the amc in time $n+1$ is thus ${E}\\big [\\mathsf {R}[n+1]\\big ]=R_l(1-f_{l,1})>R_{l-1},$ where the loose lower bound is obtained by using (REF ) and the fact that $f_{l,1}<\\mathsf {PER}_l(\\gamma _l)$ .", "We can see that the reward (REF ) due to a small probability of error, $f_{1,2}\\approx 0$ (obtained thanks to harq), is negligible corresponding to the worst-case gain of the amc given by (REF ).", "In other words, even if the selection of the rate $R_1$ occurs with a very small probability $p_1$ , its effect propagates to the following time instants." ], [ "Probabilistic interpretation", "Let us transform () and evaluate its limits $\\frac{\\sum _{l=1}^L R_l f_{1,l}p_l}{\\overline{f}_{1}}=\\frac{\\sum _{l=1}^L R_l \\Pr \\left\\lbrace \\mathsf {NACK}, R_l\\right\\rbrace }{\\Pr \\left\\lbrace \\mathsf {NACK}\\right\\rbrace }&=\\sum _{l=1}^L R_l \\Pr \\left\\lbrace R_l|\\mathsf {NACK}\\right\\rbrace <\\eta ^{{\\textnormal {amc}}},\\\\&\\sum _{l=1}^L R_l \\Pr \\left\\lbrace R_l|\\mathsf {NACK}\\right\\rbrace <R_L,$ where () is obtained by taking the limits of both sides of () for $\\overline{\\mathsf {snr}}\\rightarrow \\infty $ .", "If, for all $l<L$ , $\\Pr \\left\\lbrace R_l|\\mathsf {NACK}\\right\\rbrace >0$ , then () is satisfied.", "This is another version of the conditions required in Proposition REF .", "It can be verbalized as follows: knowing that an error occurred (i.e., $\\mathsf {NACK}$ is observed), the a posteriori probability of choosing a rate $R_l$ , $l=1,\\ldots , L-1$ does not go to zero.", "This means that, for the high-snr regime, the error event $\\mathsf {NACK}$ provides information about the realization of the snr which led to the error.", "The most obvious interpretation of this relationship is obtained for $\\tilde{a}=\\infty $ with the decision regions $\\mathcal {D}^{{\\textnormal {harq}}}_l=\\mathcal {D}^{\\textnormal {amc}}_l$ , where $\\gamma _1=0$ and $\\gamma _l=\\mathsf {snr}_{{\\textnormal {th}},l}$ for $l>1$ .", "Then, the only source of error is the event $\\mathsf {SNR}\\in (0, \\mathsf {snr}_{{\\textnormal {th}},1} )$ .", "Since this event occurs only when the rate $R_1$ is selected, we can write $\\Pr \\left\\lbrace R_1|\\mathsf {NACK}\\right\\rbrace =1$ .", "That is, even if the a priori probability $p_1=\\Pr \\left\\lbrace R=R_1\\right\\rbrace $ can be arbitrarily small, the a posteriori probability can be very large." ], [ "Improving the Throughput via AMC-HARQ Interaction", "With the lessons from Sec.", ", we explore in this section two simple solutions to limit the throughput penalty incurred by the adoption of harq on top of the amc in fast-fading channels.", "We do it in two steps.", "First, we will try to remove the source of the throughput penalty by making harq aware of the csi; next, we will explore the possibility of increasing the throughput with a more advanced coding." ], [ "AMC-Aware HARQ: Packet dropping", "The main point from the previous analysis is that, while the receiver observes the csi, the adaptation has a very limited scope: the parameters of harq are fixed in the first round, and the subsequent ones ($k=2,\\ldots , K$ ) are not affected by the varying csi.", "In Sec.", "REF , we have also identified situations where harq used on top of the amc is counterproductive: suppose we observe in the $k$ -th round the mcs index, $\\hat{l}_k$ .", "Using the amc the reward can be close to $R_{\\hat{l}_k}$ , see (REF ).", "On the other hand, the reward related to harq depends on the first round mcs index, $\\hat{l}_1$ , and may be close to $R_{\\hat{l}_1}$ .", "Therefore, if $R_{\\hat{l}_k}$ is greater than $R_{\\hat{l}_1}$ , the amc should be used; this modifies the protocol for rounds $k>1$ in the following manner $\\text{if }\\hat{l}_k \\le \\hat{l}_1 &\\Rightarrow \\quad \\text{continue HARQ}\\\\\\text{if }\\hat{l}_k > \\hat{l}_1 &\\Rightarrow \\quad \\text{restart HARQ (drop the packet)}.$ In this way, we adapt the number of harq transmission rounds to the observed mcs indices.", "Quite clearly, only the performance of harq in fast-fading channels will be affected because, under slow-fading model, we observe $\\hat{l}_1=\\hat{l}_2=\\ldots =\\hat{l}_K$ , and subsequently (REF ) applies in all rounds.", "Using (), the harq the number of rounds of one cycle may depend on the rate used in the cycle which follows.", "Thus, to calculate the throughput of the pdharq we have just introduced, we cannot directly use the previous formulas based on the expected reward and the expected duration in one harq cycle.", "However, this is not an issue since we are merely interested in the throughput with fixed transmission strategy parameters, and we will also use the predefined decision regions of amc, $\\mathcal {D}^{\\textnormal {amc}}_l$ .", "Therefore, to obtain the results we show in Fig.", "REF , we used a Monte Carlo integration.", "We can appreciate that the packet dropping proposed in () practically eliminates the throughput loss introduced by harq.", "This is an indirect confirmation that the main source of the throughput gap was correctly identified in Sec.", "REF .", "Figure: The throughput obtained using packet-dropping pdharq for a) rrharq, and b) irharq, and using variable-length vlharq, which can be implemented only for irharq." ], [ "Adaptive HARQ: Variable-Length Coding", "Here, we would like to improve the throughput beyond what amc alone can offer, especially in the high-snr regime.", "We follow [36], [37], [38], [39], [40], which proposed to use variable-length coding for harq; the main idea of vlharq is to reduce the number of symbols involved in the transmission of a packet.", "This can be done by decreasing the length of the codewords in rounds $k=2,\\ldots , K$ .", "In order to comply with the constant-length channel block assumptions we used previously, modifications are needed regarding the way the packets are encoded and transmitted in a channel block.", "First, we redefine the operation of the amc.", "Previously, the rate $R_l$ was interpreted as a transmitting of one packet containing $R_l{\\mathop {N_{\\textnormal {s}}}}$ bits per block.", "We assume henceforth that the packet contents is fixed to ${\\mathop {N_{\\textnormal {b}}}}=R_1{\\mathop {N_{\\textnormal {s}}}}$ bits so selecting the rate $R_l$ means that the transmission requires ${\\mathop {N_{{\\textnormal {s}},l}}}={\\mathop {N_{\\textnormal {b}}}}/R_l=\\ell _l{\\mathop {N_{\\textnormal {s}}}}$ symbols, where $\\ell _l=R_1/R_l$ is the normalized length of the codeword.Equivalently, $\\ell _l$ is the fraction of ${\\mathop {N_{\\textnormal {s}}}}$ required to transmit the packet with the rate $R_l$ .", "If we also use $R_l=l R_1$ , $l=1,\\ldots , L$ (as we already did in the numerical examples), then the amc transmission with the rate $R_l$ means that $l$ packets are transmitted in one channel block, as done also in [13], [26], [40].", "Since the relationship between the rate $R_l$ and the length $\\ell _l$ is bijective, it is convenient to reformulate the discussion in terms of packet length.", "The first transmission round of harq is done using codewords $x_1$ with the predefined length taken from the set $\\mathcal {L}=\\lbrace 1,1/2,\\ldots , 1/L\\rbrace $ ; we are thus compatible with the amc.", "On the other hand, in the subsequent harq rounds we are allowed to use the subcodewords $x_k$ with arbitrary length taken from $\\mathcal {L}$ .", "Since the idea is to use shorter subcodewords in order to decrease the channel use, we may also expand the decision space and use auxiliary set of codewords lengths $\\mathcal {L}^{\\prime }={\\ell _{L+1}, \\ldots , \\ell _{L^{\\prime }}}$ , where $\\ell _{k-1}<\\ell _{k}, k=L+1,\\ldots , L^{\\prime }$ .", "Since it is allowed to transmit in the same channel block the “fresh” (not transmitted) packets as well as the redundancy codewords for the nacked packets, the operation of harq needs to be modified.", "To do so, we assume that all packets, which are candidates for the transmission are gathered in the harq buffer and we index them with $h\\in \\lbrace 1,\\ldots , H\\rbrace $ , where $H\\le K\\cdot L $ is the maximum number of packets we need to consider.", "Let $\\lbrace \\ell _h\\rbrace _{h=1}^H$ denote the length we will assign to each packets in the harq buffer, where setting $\\ell _h=0$ means, de facto, that the packet is not “scheduled\" for transmission.", "We will maximize the instantaneous reward for a given snr, $\\mathsf {snr}$ .", "As all the packets carry ${\\mathop {N_{\\textnormal {b}}}}$ bits, we solve the following problem: $\\lbrace \\hat{\\ell }_h\\rbrace &=\\mathop {\\mathrm {argmax}}_{\\lbrace \\ell _h\\rbrace } \\sum _{h=1}^H (1-f(h)),\\quad \\text{s.t.", "}\\quad \\sum _{h=1}^H \\ell _h\\le 1$ where $f(h)$ is the conditional probability of error for the $h$ -th packet, calculated as $f(h)=\\Pr \\left\\lbrace \\mathsf {Err}_{k(h)+1}|\\mathsf {Err}_{k(h)}\\right\\rbrace $ and $k(h)$ is the harq counter of the $h$ -th packet (i.e., the number of time the packet was scheduled for transmission).", "To calculate (REF ) we first express the probability of decoding error after $k(h)$ transmissions with the codewords' lengths $\\ell _{1}(h),\\ldots ,\\ell _{k(h)}(h)\\in \\lbrace \\mathcal {L},\\mathcal {L}^{\\prime }\\rbrace $ using (REF )–(REF ) $\\Pr \\left\\lbrace \\mathsf {Err}_k|\\ell _{1}(h),\\ldots ,\\ell _{k(h)}(h) \\right\\rbrace =\\mathsf {PER}_{l(h)}(\\mathsf {snr}^\\Sigma (h)),$ where, $l(h)$ is the index of the first transmission (i.e., $\\ell _{l(h)}=\\ell _{1}(h)$ ), and for irharq the $\\mathsf {snr}^\\Sigma (h)$ is given by $\\mathsf {snr}^\\Sigma (h)=I^{-1}\\Big ( \\frac{1}{\\ell _{1}(h)}\\sum _{t=1}^{k(h)} \\ell _{t}(h) I(\\mathsf {snr}_t(h))\\Big ),$ where $\\mathsf {snr}_{t}(h)$ is the snr experienced by the $h$ -th packet during the $t$ -th round and (REF ) expresses the fact that the mi accumulation depends on the subcodeword lengths.", "Note that, as expected, (REF ) is equivalent to (REF ) if $\\ell _{1}(h)=\\ldots =\\ell _{k(h)}(h)$ .", "Thus, using (REF ) we calculate (REF ) as $f(h)=\\frac{\\mathsf {PER}_{l(h)}(\\mathsf {snr}^{\\prime }(h))}{\\mathsf {PER}_{l(h)}(\\mathsf {snr}^\\Sigma (h))},$ where $\\mathsf {snr}^{\\prime }(h) &= I^{-1}\\Big (I(\\mathsf {snr}^\\Sigma (h))+\\frac{ \\ell _h}{\\ell _1(h)}\\mathsf {snr}\\Big )$ is the aggregate snr that will be experienced by the $h$ -th packet provided it is scheduled for transmission.", "After erroneous decoding of the $h$ -th packet, signaled by a nack message, $k(h)$ is incremented if $k(h)<K-1$ (otherwise the packet is discarded from the harq buffer), and $\\mathsf {snr}^\\Sigma (h)$ is updated using (REF ) $\\mathsf {snr}^\\Sigma (h)&\\leftarrow \\mathsf {snr}^{\\prime }(h).$ We thus see that it is not necessary to keep track of all lengths used to transmit the packet $h$ : observing the current snr, $\\mathsf {snr}$ , and knowing the triplet $\\big (k(h), \\ell _{1}(h), \\mathsf {snr}^{\\Sigma }(h)\\big )$ , for all the packets $h=1,\\ldots , H$ , we have enough information to calculate the probability of error for each scheduled packet.", "We note that, if all the packets in the buffer are fresh, the optimization problem (REF ) is equivalent to finding the throughput-maximizing rates as done for amc when the decision regions are defined by (REF ).", "On the other hand, in presence of packets with harq counters $k(h)>0$ , the notion of decision regions is less clear; since they become dependent on the parameters of all the packets in the harq buffer via (REF ), the simple one-to-one relationship between the packet and the transmission rate does not exist anymore.", "To make a comparison with the amc fair, we assume that in order to have a decodable first transmission (i.e., $f(h)<1$ ) we must use the length $\\ell \\in \\mathcal {L}$ .", "Again, this is motivated by the practical considerations: for example, using a finite-size constellation, such as 16-qam, any transmission with rate $R>4$ (or $\\ell <R_1/4$ ) fails.", "Therefore, using the instantaneous throughput criterion, the first transmission length must be taken from $\\mathcal {L}$ so we cannot exploit $\\mathcal {L}^{\\prime }$ to improve the throughput of the amc.", "We terminate noting that the memoryless scheduling decisions obtained solving (REF ) are suboptimal in terms of long-term throughput.", "In general, finding the optimal set $\\lbrace \\hat{\\ell }_h\\rbrace _{h=1}^H$ is difficult mainly due to the dependence of the throughput on the future decisions and would have to be solved using a framework of mdp; a problem is compounded by the dimensionality of the observation space.", "The numerical results we show in Fig.", "REF are obtained by Monte Carlo simulations, where the set of lengths $\\mathcal {L}=\\lbrace 1,1/2,1/3,1/4,1/5\\rbrace $ is the same as in all previous examples, and the auxiliary set is given by $\\mathcal {L}^{\\prime } =\\lbrace 1/8, 1/12, 1/16\\rbrace $ .", "The discrete optimization problem defined in (REF ) was solved by exhaustive search over the entire solution space; this is possible here due to a relatively small dimensionality of the sets $\\mathcal {L}$ and $\\mathcal {L}^{\\prime }$ .", "The improvement of the throughput of vlharq comparing to amc-harq is clear for high values of $\\overline{\\mathsf {snr}}$ despite of the sub-optimality of the objective function (REF ).", "We conducted more experiments including smaller length $\\ell $ in $\\mathcal {L}^{\\prime }$ , but changes in the throughput we observed were not significant.", "On the other hand, it is was indeed necessary to use $\\mathcal {L}^{\\prime }$ in order to improve the throughput in high snr.", "This is because, under such operating conditions, the first transmission is very likely to use the shortest codewords from $\\mathcal {L}$ and thus, to decrease the average transmission time, we need to use shorter codewords from $\\mathcal {L}^{\\prime }$ ." ], [ "Conclusions", "In this work, we provided a general communication-theoretic point of view on the benefits of combining amc and harq in point-to-point transmissions.", "Using the throughput as a comparison criterion, the main conclusions are the following: i) in slow-fading channels, the throughput increases thanks to harq but the improvements are very moderate, especially when amc ignores the presences of harq, ii) in fast-fading channels, harq is beneficial only in low snrs and is counterproductive in high snrs, irrespectively of the adaptation efforts of amc.", "Then, since harq provides very small (if any) throughput gains, the error-free operation may be taken in charge by the upper layer (llc).", "Following the identified sources of deficiencies of harq used on top of amc, we also proposed a simple modification to the harq protocol, which terminates the harq cycle on the basis of the observed mcs.", "This modification can be implemented without any additional signaling or change at phy, and removes the observed throughput penalty in fast fading channels.", "We pursued the idea of a closer interaction between phy and harq, and evaluated the possibility of using packet dropping (pdharq) which provide a simply remedy to the throughput penalty.", "At the cost of more complicated implementation, the variable-length coding (vlharq) leads to a moderate—yet notable—improvement of the throughput in the region of high snr.", "The conclusions and observations we make hold for delay-insensitive applications, and do not take into account the overhead necessary to retransmit the packet at the llc.", "Thus, a more realistic evaluation should differentiate between the cost of phy and llc transmissions.", "Other important challenges would be to evaluate the performance of harq and amc in the interference-limited scenario or for a delay-sensitive traffic." ] ]
1606.05177
[ [ "Photoinduced Chern insulating states in semi-Dirac materials" ], [ "Abstract Two-dimensional (2D) semi-Dirac materials are characterized by a quadratic dispersion in one direction and a linear dispersion along the orthogonal direction.", "We study the topological phase transition in such 2D systems in the presence of an electromagnetic field.", "We show that a Chern insulating state emerges in a semi-Dirac system with two gapless Dirac nodes in the presence of light.", "In particular, we show that the intensity of a circularly polarized light can be used as a knob to generate topological states with nonzero Chern number.", "In addition, for fixed intensity and frequency of the light, a semi-Dirac system with two gapped Dirac nodes with trivial band topology can reveal the topological transition as a function of polarization of the light." ], [ "Photoinduced Chern insulating states in semi-Dirac materials Kush Saha Department of Physics and Astronomy, University of California, Irvine, California 92697, USA California Institute for Quantum Emulation, Santa Barbara, California 93106, USA Two-dimensional (2D) semi-Dirac materials are characterized by a quadratic dispersion in one direction and a linear dispersion along the orthogonal direction.", "We study the topological phase transition in such 2D systems in the presence of an electromagnetic field.", "We show that a Chern insulating state emerges in a semi-Dirac system with two gapless Dirac nodes in the presence of light.", "In particular, we show that the intensity of a circularly polarized light can be used as a knob to generate topological states with nonzero Chern number.", "In addition, for fixed intensity and frequency of the light, a semi-Dirac system with two gapped Dirac nodes with trivial band topology can reveal the topological transition as a function of polarization of the light.", "Introduction- Recent years have witnessed unprecedented theoretical and experimental advances in the field of materials whose low-energy excitations are described by Dirac fermions [1], [2], [3], [4], [5], [6].", "Such materials with a bulk gapped spectrum (e.g., 2D HgTe/CdTe quantum wells, 3D Bi$_2$ Se$_3$ ) may exhibit exotic electronic properties such as a quantum spin Hall effect[7], topological magneto-caloric effect[8], etc.", "Likewise, gapless or semi-metallic Dirac materials (e.g., 2D graphene, 3D Weyl metal) may reveal many unconventional properties such as a quantum Hall effect at room temperature (particularly in graphene)[9], chiral anomaly induced negative magnetoresistance[10] (particularly in a Weyl metal), and many more[2], [11].", "Due to their exotic properties, on the one hand, there has been a considerable effort to search for new materials with Dirac-like properties.", "This has led to several proposals to realize a new class of materials with specific 2D Dirac dispersion: parabolic in one direction and linear in the perpendicular direction.", "The materials or systems that can host such semi-Dirac (SD) dispersion are TiO$_2$ /V$_2$ O$_3$[12], BEDT-TTF$_2$ I$_3$ salt under pressure[13], hexagonal lattices in the presence of a magnetic field[14], and photonic system[15].", "Due to their unusual dispersion, they may exhibit exotic properties[16], [17] in contrast to graphene or conventional 2D systems.", "On the other hand, there is a race to engineer new materials with topological properties due to the lack of natural materials with such properties.", "The static controllable parameters by which one can induce topological phases in intrinsically nontopological materials include pressure[18], doping[19], disorder[20], temperature[21], and a few more[22].", "However, most of these tools lack a continuous and high degree of controllability.", "Recently, it has been shown and later verified by experiment that time-dependent perturbations can induce topological phases in an intrinsically nontopological insulating material [23], [24], [26], [25], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36] .", "Also, it has been shown that polarized light can open a gap in semi-metallic systems, allowing them to host topologically protected gapless modes responsible for robust transport properties.", "Indeed, circularly polarized light opens a gap in a gapless Dirac system such as graphene, and may give rise to a photoinduced Hall current without applying any magnetic field[23].", "Thus semi-metallic graphene becomes a Chern insulator (CI) in the presence of electromagnetic radiation, making it potential candidate for spintronics or transistors.", "Figure: a) Schematics of shining polarized light on the surface of a SD material.", "The size of the sample is considered to be smaller than the wavelength of the light.", "b) Energy spectrum of the SD Hamiltonian in Eq.", "() with two nodal points at (±δ 0 /α,0)(\\pm \\sqrt{\\delta _0/\\alpha },0).", "Note that, this dispersion is gapped if δ 0 <0\\delta _0<0.This raises an interesting question if semi-Dirac systems can carry a nonzero Chern number in the presence of polarized light.", "In this article, we have answered this affirmatively.", "For circularly polarized light, we show light can open a gap and induce topological states in a semi-metallic SD system.", "In particular, the strength of the polarized light can be used as a knob to drive a topological transition.", "In addition, we show that a SD system with a trivial insulating phase can mimic the Haldane model as a function of polarization of the light.", "Finally, we investigate photoinduced anomalous Hall conductivity for different strengths of the incident beam.", "Note that a similar system was studied [37] recently in the presence of laser light but overlooked the interesting topological transition as presented here.", "Model- The minimal model Hamiltonian describing low-energy electronic bands of a two-dimensional semi-Dirac material is[16], [17] $H_0(\\bf k)={\\bf d(\\bf k)}\\cdot {\\bf \\sigma },$ where ${\\bf \\sigma }=(\\sigma _x,\\sigma _y,\\sigma _z)$ are the Pauli matrices in pseudospin space, $ {\\bf d (\\bf k)}=(\\alpha k_x^2-\\delta _0,v k_y,0)$ , where ${\\bf k}=(k_x, k_y)$ is the crystal momentum, $\\alpha $ is the inverse of quasiparticle mass along $x$ , $v$ is the Dirac velocity along $y$ , and $\\delta _0$ is the gap parameter.", "The energy eigenvalues are given by $E^{\\pm }_{k_x,k_y}=\\pm \\sqrt{(\\alpha k_x^2-\\delta _0)^2+v^2 k_y^2},$ where $\\pm $ denote the conduction and valence band, respectively.", "For $\\delta _0=0$ , the spectrum is gapless with linear dispersion along $k_y$ .", "For $\\delta _0<0$ , it is a gapped system with trivial insulating phase.", "For $\\delta _0>0$ , we obtain two gapless Dirac points at $(\\pm \\sqrt{\\delta _0/\\alpha },0)$ .", "The corresponding energy spectrum is shown in Fig.", "(REF )b.", "Note that, this model obeys effective time-reversal ($\\Theta =\\mathcal {K}$ ), particle-hole ($\\mathcal {P}=\\sigma _y$ ) and chiral symmetries ($\\mathcal {C}=\\sigma _y \\mathcal {K}$ ), where $\\mathcal {K}$ is the complex conjugation operator.", "It is worth mentioning that, in typical SD systems, the gapless modes are protected by mirror symmetry along some symmetry line[39] and cannot be destroyed without breaking that symmetry.", "Figure: Berry curvature of a gapped semi-Dirac Hamiltonian in the presence of perturbation δH=m z σ z \\delta H=m_z\\sigma _z.", "The integral of the Berry curvature over the BZ turns out to be zero since Ω(-k x ,-k y )=-Ω(k x ,k y )\\Omega (-k_x,-k_y)=-\\Omega (k_x,k_y) as a manifestation of time-reversal symmetry.", "b) Same plot as (a) for 2D linear Dirac dispersion .", "In this case, the perturbation breaks time-reversal symmetry and leads to a nonzero Chern number, as is evident from the Berry curvature.Berry curvature and Chern number- Eq.", "(REF ) represents a generic Hamiltonian of a two-level system.", "Its simple form in terms of “vector field\" ${\\bf d(k)}$ allows us to write the Berry curvature as ${\\bf \\Omega }({\\bf k})=\\frac{1}{2}\\frac{\\bf d(k)}{|\\bf d( k)|^3}.$ Since the Chern number is defined as an integral of Berry curvature over the 2D Brillouin zone (BZ), it can be expressed as $C=\\int d{\\bf k}\\Omega ({\\bf k})={1\\over 2\\pi }\\int d{\\bf k}~ {1\\over 2 |\\bf d|^3} {\\bf d} .", "(\\partial _{k_x}{\\bf d}\\times \\partial _{k_y}{\\bf d}),$ where $\\partial _{x}$ is the partial derivative with respect to $x$ .", "Since the $z$ -component of ${\\bf d}$ in Eq.", "(REF ) is zero, the Berry curvature is zero everywhere except at the gapless Dirac points where it diverges.", "The perturbation $\\delta H=m_z \\sigma _z$ opens a gap and leads to a nonzero local Berry curvature, $\\Omega (k_x,k_y)=\\pm \\frac{2\\alpha v m_z k_x}{(E_{k_x,k_y})^3}.$ Note that $\\Omega (k_x,k_y)$ is asymmetric in $k_x$ for constant $m_z$ .", "Thus, the Chern number $C$ is obtained to be zero, as also apparent from the Berry curvature shown in Fig.", "(REF )a.", "The above perturbation therefore cannot change the band topology in the SD system.", "This is in contrast to half semimetals with 2D linear Dirac bands near the Fermi level, where a mass gap induced by spin-orbit coupling leads to a nonzero Chern number[38], [40].", "It is worth pointing out that, in graphene, the perturbation $m_z\\sigma _z$ , namely the “Semenoff mass\" leads to a trivial insulating phase, while perturbation $m_z\\sigma _z\\tau _z $ , namely the “Haldane mass\" gives rise to non-trivial band topology, where $\\tau _z$ denotes valley the degrees of freedom[41].", "Floquet theory- We now investigate the effect of time-dependent radiation generated by a radio frequency source or a laser on the semi-metallic phase of the SD Hamiltonian in Eq.", "(REF ).", "The light field ${\\bf A}(t)=A_0(\\sin (\\omega t),\\sin (\\omega t+\\phi ) )$ minimally couples to the system via the momentum ${\\bf k}\\rightarrow {\\bf k}+e{\\bf A}(t)$ , where $\\phi $ is the polarization of the light, $e$ is the electric charge, $\\omega $ is the frequency of the light, and $A_0$ is the strength of the applied field.", "Note that we neglect spatial dependence of the light field considering the fact that the wavelength of the light field is large compared to the sample size.", "Then, in the presence of a light field, Eq.", "(REF ) can be read off as $H({\\bf k},t)=H_0({\\bf k})+H_{1}({\\bf k},t)\\sigma _x+H_{2}({\\bf k},t)\\sigma _y,$ where $H_{1}({\\bf k},t)=\\frac{\\alpha e^2 A_0^2}{2}(1-\\cos (2\\omega t))+2\\alpha e A_0 k_x \\sin (\\omega t)$ , and $H_{2}({\\bf k},t)=ve A_0 \\sin (\\omega t+\\phi )$ .", "Under this time-dependent Hamiltonian, the quantum state evolves as $\\Psi (t)=U(t,t_0)\\Psi (t_0)$ , where $U(t,t_0)=\\tau \\exp (-\\frac{i}{\\hbar }\\int _{t_0}^{t} dt^{\\prime } H({\\bf k},t^{\\prime })) $ , where $\\tau $ is the time-ordering operator, and $t_0$ is the initial time of the perturbation.", "Using the Floquet formalism, we define an effective static Hamiltonian, namely Floquet Hamiltonian after a full time period $T$ as $H_F ({\\bf k})=\\frac{i}{\\hbar T} \\ln (U(T)),$ where we have assumed $t_0=0$ .", "Following Ref.", "[anatoli], we obtain the Floquet Hamiltonian up to first order in inverse frequency, $H_F({\\bf k} )\\simeq H_F^{0}+H_{F}^1,$ where $H_{F}^0$ is obtained by time-averaging of $H({\\bf k},t)$ : $H_{F}^0=\\frac{1}{T}\\int _{0}^T H({\\bf k},t )=H_0({\\bf k} )+\\frac{\\alpha e^2 A_0^2 }{2}\\sigma _x,$ and $H_{F}^1&=&\\frac{1}{2T i \\hbar }\\int _{0}^{T}dt_1\\int _{0}^{t_1}[H({\\bf k},t_1),H({\\bf k},t_2)] dt_2\\nonumber \\\\&=&(m_0+\\beta k_x^2+\\gamma k_x+\\eta k_x k_y)\\sigma _z,$ where $m_0=\\frac{eA_0 v}{\\hbar \\omega }(\\alpha e^2 A_0^2-2\\delta _0)\\cos (\\phi )$ , $\\beta =\\frac{eA_0 v}{\\hbar \\omega }2\\alpha \\cos (\\phi )$ , $\\gamma =\\frac{e^2A_0^2 v}{\\hbar \\omega }2\\alpha \\sin (\\phi )$ and $\\eta =-\\frac{4\\alpha eA_0 v}{\\hbar \\omega }$ .", "Note that this is a high-frequency expansion when only one photon is considered for the process.", "We recognize $H_{F}^1$ as a gap-opening mass term in Eq.", "(REF ).", "As already discussed, $H_0({\\bf k})$ represents the Hamiltonian of a trivial insulator in the presence of a gap induced by a constant mass term.", "However, we will see that the momentum dependent term in $H_F^1$ plays a crucial role in revealing the band topology in these systems.", "Combining $H_F^{0}$ and $H_{F}^1$ , Eq.", "(REF ) yields $H_F({\\bf k} )={\\bf n({\\bf k})}.", "{\\bf \\sigma },$ with ${\\bf n({\\bf k})}=(\\alpha k_x^2+\\delta _1,v k_y, m_{\\rm eff} ),$ where $\\delta _1=\\frac{\\alpha e^2 A_0^2}{2}-\\delta _0$ , $m_{\\rm eff}=m_0+\\beta k_x^2+\\gamma k_x+\\eta k_x k_y$ .", "Note that the incident beam shifts the position of the Dirac nodes.", "Notice also, that $m_{\\rm eff}$ breaks the effective time-reversal symmetry as $\\Theta m_{\\rm eff}(k_x,k_y)\\Theta ^{-1}\\ne m_{\\rm eff}(-k_x,-k_y)$ .", "With the new vector field ${\\bf n(k)}$ , we define the Chern number for the Floquet Hamiltonian as $C_{\\rm F}=&{1\\over 2\\pi }\\int d{\\bf k}~ {1\\over 2|\\bf n(k)|^3} {\\bf n (k)} .", "(\\partial _{k_x}{\\bf n(k)}\\times \\partial _{k_y}{\\bf n(k)}),$ where $|n({\\bf k})|=\\pm \\sqrt{(\\alpha k_x^2+\\delta _1)^2+v^2 k_y^2+m_{\\rm eff}^2}$ .", "Figure: a) Schematic phase diagram of the photoinduced Chern number (C F C_F) in the δ 0 -αe 2 A 0 2 /2\\delta _0-\\alpha e^2 A_0^2/2 plane.", "The diagonal lines correspond to δ 0 =αe 2 A 0 2 /2\\delta _0=\\alpha e^2A_0^2/2, where C F C_F is ill-defined as the spectrum is gapless.", "b) Phase diagram of C F C_F in the m-φm-\\phi plane for a gapped semi-Dirac system as discussed in the text.", "The red (m c 1 >0m_{c_1}>0) and blue lines (m c 2 <0m_{c_2}<0) correspond to a gapless spectrum due to the gap closing at only one Dirac point.Chern insulating states in gapless SD systems- Together with Eq.", "(REF ) and the integral part of Eq.", "(REF ), we obtain $\\Omega (k_x,k_y)= \\frac{v}{2|n(\\bf k)|^3}\\left[\\gamma (\\alpha k_{x}^2-\\delta _1)-\\eta k_y(\\alpha k_x^2+\\delta _1)\\right].$ It is apparent that $\\Omega (k_x,k_y) \\ne - \\Omega (-k_x,-k_y)$ as a consequence of broken time-reversal symmetry.", "We would like to point out that the numerator of $\\Omega (k_x,k_y)$ does not contain any term involving $m_0$ .", "However, the presence of such a term in the numerator can have important consequences in revealing the band topology as will be evident below.", "For circularly polarized ($\\phi =\\pm \\pi /2$ ) light, and for $\\delta _0>\\alpha e^2A_0^2/2$ , the gapless Dirac nodes become gapped with their new position at $(\\pm \\sqrt{\\frac{|\\delta _1|}{\\alpha }-\\frac{\\gamma ^2}{2\\alpha ^2}},0)$ .", "Then the total Berry flux for this gapped spectrum is found to be $\\pm 2\\pi {\\rm sgn}(\\gamma )$ , where $\\pm $ correspond to the positive and negative energy spectrum, respectively.", "Note that left ($\\phi =-\\pi /2$ ) or right circularly ($\\phi =+\\pi /2$ ) polarized light determines the sign of $\\gamma $ .", "The $2\\pi $ flux comes from the fact that the energy spectrum has two Dirac-like dispersions, and each of those contributes $\\pm \\pi $ to the total flux in contrast to $2\\pi $ flux in bilayer graphene due to quadratic band touching.", "Consequently, two gapped Dirac nodes make a contribution of $\\pm (1/2+1/2)=\\pm 1$ to $C_F$ .", "The total $C_F$ is obtained by summing over the contribution coming from all Dirac nodes in the BZ.", "For typical semi-Dirac materials, there are four symmetry-related momenta [14], which, in turn, give rise to a total of 8 Dirac nodes, hence a total $C_F=\\pm 4$ .", "However, for simplicity, we restrict ourselves to a system with only two Dirac nodes within the BZ[43].", "For $\\delta _0=\\alpha e^2A_0^2/2$ , i.e., $\\delta _1=0$ , the spectrum is gapless.", "For $\\delta _0<\\alpha e^2A_0^2/2$ , the spectrum becomes gapped again.", "In this case, $C_F$ turns out to be zero.", "Thus varying intensity of the incident light, we can drive topological transition in a gapless SD system (cf.", "Fig.", "REF a).", "Note that, unlike the case of gapped trivial insulating phase induced by constant mass term (as discussed earlier), the Berry curvature $\\Omega (k_x,k_y)$ is not asymmetric either in $k_x$ or $k_y$ .", "We therefore expect to have finite anomalous Hall conductivity at finite doping ($\\epsilon _F\\ne 0$ ).", "This will be discussed shortly.", "Chern insulating states in gapped SD systems- The objective of this section is to explore topological phenomena in a SD system with two gapped Dirac nodes within the BZ.", "The low-energy model Hamiltonian is given by $H_{\\rm g}({\\bf k})=H_0({\\bf k})+m\\sigma _z.$ As mentioned earlier, this Hamiltonian represents a trivial insulator with zero Chern number as a consequence of an asymmetric Berry curvature.", "The presence of polarized light leads to $n_{\\rm g}({\\bf k})=(\\alpha k_x^2+\\delta _1-m\\xi ,v k_y+m\\chi k_x, m+m_{\\rm eff} )$ , where $\\xi =2eA_0 v\\cos (\\phi )/\\hbar \\omega $ and $\\chi =4eA_0 \\alpha /\\hbar \\omega $ .", "Note that the position of the Dirac nodes moves along both $k_x$ and $k_y$ directions as opposed to the case discussed in the preceding section.", "Note also, that the movement depends on the polarization of the light.", "With this $n_{\\rm g}({\\bf k})$ , the numerator of the Berry curvature turns out to be dependent on $m$ , and also on $\\cos (\\phi )$[46], in contrast to the previous case where the term involving $m_0$ vanishes identically in the numerator.", "Consequently, for fixed intensity and frequency of the light, we can mimic the phase diagram of the Haldane model by varying $m$ with the polarization of the light.", "The corresponding plot is shown in Fig.", "(REF )b.", "Along the curve lines ( $m_{c_1} \\simeq m_{c_2}=m_c$ ), one Dirac node closes while the other remains open, hence the spectrum becomes gapless.", "Note that $m_c$ depends on the intensity, frequency, and polarization of the light.", "For circularly polarized light, and for $\\delta _1<0$ , we obtain $m_c(\\phi =\\pi /2)=\\pm 2e^2A_0^2 v\\hbar \\omega \\sqrt{\\alpha |\\delta _1|}/(16 e^2A_0^2\\alpha \\delta _1-\\hbar ^2\\omega ^2)$ .", "For any other polarization, $m_c$ can be obtained numerically for fixed $A_0$ and $\\omega $ .", "When $|m|<|m_c|$ , the mass gaps at two Dirac nodes are opposite in sign, hence $C_F=\\pm {\\rm sgn}(\\phi )$ .", "In contrast, for $|m|>|m_c|$ , the mass gaps at two Dirac nodes turn out to be the same in sign.", "Thus we obtain $C_F=0$ , indicating a transition from a topological to a trivial phase as a function of polarization of the light.", "Before ending this section, we would like to point out that a gapped SD system for $\\delta _0<0$ in Eq.", "(REF ) does not host any interesting topological states in the presence of light.", "Thus the presence of gapless or gapped Dirac nodes (when $\\delta _0>0$ ) in the field-free Hamiltonian of SD materials plays a crucial role in revealing the topological transition in the presence of light.", "In sum, this reflects a non-trivial interplay between light and band spectrum.", "Finally, we comment on the lattice model which is crucial to compute the Chern number correctly.", "In doing so, we consider the lattice Hamiltonian, $H=(1-2\\cos (k_x))\\sigma _x+k_y\\cos (k_x)\\sigma _y$ , which has two Dirac nodes within the 2D BZ.", "Note that, this model Hamiltonian can be obtained from hexagonal lattice geometry with appropriate band parameters[45].", "Then the perturbation $\\delta H=\\gamma \\sin (k_x)\\sigma _z$ reveals the same topological properties as discussed in the preceding sections.", "Figure: a) Anomalous Hall conductivity σ xy \\sigma _{xy} for a trivial insulating phase with nonzero Berry curvature as a function of ϵ F \\epsilon _F.", "It is evident that σ xy \\sigma _{xy} increases with the intensity A 0 A_0.", "However, as soon as ϵ F \\epsilon _F reaches the gap, it drops to zero as a consequence of zero Chern number for δ 1 >0\\delta _1>0.", "b) For δ 1 <0\\delta _1<0, we see the conductivity is quantized in units of e 2 /he^2/h if ϵ F \\epsilon _F lies within the gap.", "Away from the gap, we obtain a very small contribution to conductivity since γ\\gamma is small for weak intensity A 0 A_0.Anomalous Hall conductivity- Unlike quantum Hall conductivity, the anomalous Hall conductivity does not require any external magnetic field or nonzero Chern number.", "The local Berry curvature can contribute to conductivity even at zero temperature for finite Fermi energy, $\\epsilon _F$ .", "The Hall conductivity is defined as $\\sigma _{xy}^{\\pm }=\\pm \\frac{e^2}{\\hbar }\\int \\frac{d{\\bf k}}{(2\\pi )^2} f(E^{\\pm }_{k_x,k_y})\\Omega (k_x,k_y),$ where $f(\\epsilon )$ is the Fermi-Dirac (FD) distribution function, $f(\\epsilon )=(1+e^{(\\epsilon -\\epsilon _F)/K_B T})^{-1}$ , where $T$ is the temperature.", "The total conductivity is the sum of two $\\sigma _{xy}^{\\pm }$ s. Note that Eq.", "(REF ) is best valid for equilibrium dc Hall conductivity.", "In nonequilibrium, the expression for $\\sigma _{xy}$ differs due to the nonuniversal nature of the FD distribution, which, in turn, depends on the details of the systems, its environment, etc [23], [47], [48].", "However, we approximate $\\sigma _{xy}$ by its equilibrium value based on the fact that the energy of the perturbing field exceeds any other energy scales of the system[48].", "At zero temperature, $f(E^{\\pm }_{k_x,k_y})=\\Theta (E^{\\pm }_{k_x,k_y}-\\epsilon _F)$ .", "For a typical semi-Dirac material, we take $v=0.65 eV Å$ , $\\alpha =0.75 eV Å^2$ , $\\delta _0=0.01 eV$ .", "The typical phonon energy that is used in experiment is roughly $0.25 eV$ with $eA_0=0.01-0.2 Å^{-1}$ .", "Using these parameters, we compute the anomalous conductivity, particularly for a gapless SD system when $\\delta _0<\\alpha e^2A_0^2/2$ .", "Fig.", "(REF )a shows the dependence of $\\sigma _{xy}$ on $\\epsilon _F$ and $A_0$ .", "Note that $\\epsilon _F=0$ corresponds to Fermi energy in the middle of the gap.", "As $\\epsilon _F$ approaches to the gap, the conductivity gradually increases, and drops to zero as $\\epsilon _F$ reaches the gap.", "Notice that, a measurable conductivity contribution can be obtained if $\\epsilon _F$ lies close to the gap.", "Also, the magnitude of Hall conductivity varies with the strength of the vector potential provided that $v e A_0<<\\hbar \\omega $ as a valid criterion for Floquet expansion.", "Fig.", "(REF )b shows zero temperature conductivity for $\\delta _0>\\alpha e^2A_0^2/2$ .", "As expected, the conductivity is quantized if $\\epsilon _F$ lies within the gap.", "However, away from the gap the conductivity appears to be small.", "This is because of the low intensity set by the condition $\\delta _0>\\alpha e^2A_0^2/2$ .", "The predicted photoinduced chiral nature of the 2D semi-Dirac materials can be verified by dynamical Hall measurement using a crossed ac electric and magnetic field of circularly polarized light[49].", "In addition, since the intensity of the electromagnetic radiation is easily controllable in experiment, the signature of transition from a topological to a trivial phase can be tested in this transport experiment, provided that $\\delta _0$ in the band spectrum is positive.", "Conclusion- We have shown that light can induce topological states in an intrinsically trivial semi-Dirac materials.", "For circularly polarized light, the intensity of the light can be used to induce topological phases in a semi-metallic semi-Dirac system.", "On the other hand, for fixed intensity and frequency, the polarization of light can drive a topological transition in a gapped semi-Dirac system.", "In addition, the intensity of light can be used to tune the anomalous Hall conductivity to measurable values for Fermi energy lying either in the conduction or valence bands.", "KS thanks M. Kolodrubetz, S. A. Parameswaran and I. Garate for useful discussions.", "This work is funded by NSF Grant DMR-1455366 and the California Institute for Quantum Emulation, supported by a President's Research Catalyst Award (CA-15-327861) from the University of California Office of the President." ] ]
1606.04864
[ [ "cltorch: a Hardware-Agnostic Backend for the Torch Deep Neural Network\n Library, Based on OpenCL" ], [ "Abstract This paper presents cltorch, a hardware-agnostic backend for the Torch neural network framework.", "cltorch enables training of deep neural networks on GPUs from diverse hardware vendors, including AMD, NVIDIA, and Intel.", "cltorch contains sufficient implementation to run models such as AlexNet, VGG, Overfeat, and GoogleNet.", "It is written using the OpenCL language, a portable compute language, governed by the Khronos Group.", "cltorch is the top-ranked hardware-agnostic machine learning framework on Chintala's convnet-benchmarks page.", "This paper presents the technical challenges encountered whilst creating the cltorch backend for Torch, and looks in detail at the challenges related to obtaining a fast hardware-agnostic implementation.", "The convolutional layers are identified as the key area of focus for accelerating hardware-agnostic frameworks.", "Possible approaches to accelerating the convolutional implementation are identified including: implementation of the convolutions using the implicitgemm or winograd algorithm, using a GEMM implementation adapted to the geometries associated with the convolutional algorithm, or using a pluggable hardware-specific convolutional implementation." ], [ "Introduction", "This work presents an OpenCL backend for the Torch network library.", "Torch[12] is a deep learning library.", "Other commonly used deep learning libraries include: caffe[17], MXNet[9], and Tensorflow[4].", "Deep learning became popular with the success of the AlexNet[20] deep learning model on the Imagenet[13] task.", "Prior to AlexNet, image recognition used hand-engineered features such as SIFT[24] and SURF[6], in order to get state of the art performance.", "AlexNet used deep learning to extract features automatically, via the expressive power of a hierarchy of layers.", "A key ingredient for this to work was the massive amount of data available in the ImageNet task, compared to earlier tasks, such as MNIST[22] and CIFAR[19].", "This allowed training complex models without overfitting.", "ImageNet gave sufficient training data for the gradients to propagate across all layers, despite the vanishing of the gradient signal as it traverses from the output back through the multiple layers.", "Given the amount of training data involved, and the length of time to train each batch in a deep network, another key ingredient for the success of AlexNet was: the computational capacity available in GPUs.", "GPUs provided sufficient processing power to be able to run multiple epochs on imagenet, using a deep network, in a reasonable timeframe, ie days or weeks; rather than months or years.", "Recent continued developments of SOTA models on ImageNet often improve on AlexNet by adding more layers, making the network deeper.", "It is probably fair to say that the more processing power we can bring to bear onto neural network models, the easier it will be to create powerful models, to train them in a reasonable timeframe, and to train them on ever larger datasets.", "LSUN dataset[38] for example has 20 million images, a 20-fold increase from AlexNet.", "In order to obtain continued development of the computational capacity of neural network hardware, it could plausibly be useful to encourage competition between hardware vendors to as great an extent possible.", "For training neural networks, GPUs are typically used, but there is no particular reason why other hardware, such as Field Programmable Gate Arrays (“FPGAs”), or dedicated Massively Parallel Processors (MPPs), could not be used instead.", "Even within the domain of GPUs, it is probably fair to say that the vast majority of neural network training is not done on multiple hardware vendor's hardware, but using NVIDIA CUDA hardware.", "One reason for this is that NVIDIA was the first vendor to make GPUs accessible for running numerical algorithms in a straightforward fashion, without needing clever hardware hacks.", "NVIDIA introduced the CUDA toolkit, and they released TESLA GPUs, which were dedicated hardware for running embarrassingly parallel numerical algorithms.", "More recently, other GPU manufacturers, such as Intel and AMD, have made their own GPUs available for numerical compute applications.", "However, looking at the machine learning domain, existing neural network libraries were written to use the NVIDIA CUDA toolkit, and it was not therefore possible to run them on non-CUDA hardware.", "Concretely it is not possible to run Caffe, Tensorflow, Torch, or MXNET on non-CUDA hardware.", "In 2015, this situation started to change.", "Caffe now can be used on OpenCL hardware, thanks to the efforts of Gu et al [15], Tschopp[35], and Engel[14].", "DeepCL[29] provides a dedicated deep learning library for OpenCL.", "cltorch, described in the present paper, provides a hardware-agnostic backend for the `Torch' neural network library, using OpenCL.", "These projects facilitate vendor competition, and therefore contribute to the development of state of the art parallel compute hardware.", "cltorch is built using OpenCL.", "OpenCL is a standard governed by the Khronos Group.", "It aims to make it possible to run a single numerical algorithmic implementation across multiple hardwares, without needing any changes to the implementation.", "Hardware vendors who are members of the OpenCL consortium include: AMD, Qualcomm, NVIDIA, Intel, and Xilinx.", "OpenCL is not the only possible cross-platform approach possible.", "A recent alternative is AMD's HCC, which we will touch on briefly, at the end of this paper." ], [ "Key differences between CUDA and\nOpenCL", "CUDA and OpenCL are not so different, both conceptually, and in the details.", "However there are some key differences, that make porting challenging." ], [ "language word differences", "The names of certain functions are different.", "For the example, to determine the id of a thread within a block of threads, on CUDA this uses properties, such as threadIdx.x.", "In OpenCL this uses functions such as get_local_id(0).", "The meanings of such functions are typically the same, or similar, and these differences can be handled in a straightforward fashion, using search and replace." ], [ "C++ templates", "A more fundamental difference is the use of C++, and specifically C++ templates and functors, in CUDA.", "Torch CUDA framework makes extensive use of these.", "OpenCL 1.2 is based on C99, and does not have templating.", "Gu et al solved this problem by using a recent version of OpenCL, which does support C++ templates.", "Unfortunately, this limits the hardware that such a library can run on.", "For example NVIDIA hardware, and some mobile hardware, only supports OpenCL 1.1 and OpenCL 1.2.", "It can be argued that there is no reason to support NVIDIA hardware, because on NVIDIA hardware one can use CUDA instead.", "However, it seems an interesting goal to be able to run a neural network library across all commonly available GPUs, including those for which an alternative implementation already exists.", "This would enable a single codebase to be used across all devices, reducing redundant effort.", "The C++ templates in Torch CUDA implementation cannot be trivially replaced by hand-written C99 code.", "For example, they are used to pre-bake geometry-specific parameters, such as loop sizes for loop unrolling.", "It is not tempting to write unrolled loops by hand for multiple possible geometries.", "The clBLAS[1] library handles this issue by using Python scripts to directly write out the generated OpenCL kernels and related C code, at compile time.", "This is an effective solution.", "It is fast at runtime, because the generated code is entirely generated at build time.", "However, it means that all possible geometries need to be baked in, at compile time, leading to a combinatorial explosion of possible parameter values.", "Therefore, in cltorch, the kernels are generated at runtime, adapted to the exact geometries required.", "Runtime code generation needs a scripting language to be embedded into the network library.", "Lua was selected.", "It is lightweight (about 46KB), and easy to embed.", "A templating language similar to Jinja2 was used, based on a templater by Nachimwald[26].", "This approach was sufficient to express the C++ CUDA templates used by CUDA torch." ], [ "Use of Thrust library", "The Torch CUDA library makes extensive use of the Thrust[7] library, to handle for example sorting and reduction operations.", "Thrust is CUDA-specific, and therefore these methods need careful consideration.", "For now, this was worked around by using alternative approaches.", "In the case of reduction, Torch CUDA backend has internal implementations, and this was ported, and used in place of some of the calls to Thrust.", "Merry [25] provides performance benchmarks of several libraries that might be useful here." ], [ "cl_mem object offsets", "cl_mem objects cannot themselves carry an implicit offset within the allocated buffer, unlike their CUDA float * counterparts.", "In Caffe library OpenCL backends, this was quite a large engineering challenge, since the float *s are passed around inside the library, and offsets are added to these in very many places.", "In Torch OpenCL backend this was not an issue, since the Torch Tensor structure already incorporates a storageOffset value." ], [ "Runtime compilation", "In order to facilitate being able to run across many hardware platforms, OpenCL kernels are typically compiled on the fly, at runtime.", "CUDA kernels are compiled at build time, with the rest of the C and C++ code.", "Compiling the GPU kernels at build time might provide a very slight boost at runtime, but in practice the cost of compiling the OpenCL kernels at runtime was found to be negligible, for deep learning libraries, compared to the time spent on network training.", "Compiling at runtime has the advantage that one can bake in the exact geometries which are required.", "By comparison, many of the Torch CUDA functions, such as the `Apply' method for example, bake in several hundred possible geometries, most of which will never be used.", "Runtime compilation makes it possible to provide some useful functionalities to the framework user, such as being able to provide custom functions, which will be compiled on the fly into GPU kernels.", "These will run on the GPU, at full speed.", "Zagoruyko has implemented similar functionality for CUDA Torch as an additional module [39], which runs the CUDA compiler at runtime." ], [ "OpenCL Kernel Debuggers", "When using CUDA, there are extensive debugging tools available, to facilitate finding root cause on kernel crashes, or numerical accuracy issues.", "When using OpenCL on CUDA devices, no such debugging tools are available.", "Some vendors provide debugging tools for their specific hardware.", "For example AMD has created CodeXL[2].", "However, no cross-platform tool is available.", "An alternative approach is to use a simulator.", "Oclgrind [30] provides OpenCL kernel debugging, using a simulated OpenCL device, loadable via ICD." ], [ "Hardware geometry\ndifferences", "Each specific hardware has its own set of geometry specifications, including, but not limited to: maximum workgroup size warp size local memory size number of registers A kernel that runs prefectly on NVIDIA hardware might fail to run altogether on eg AMD hardware.", "For example, the kernel might be written on the assumption that at least 512 workgroup threads are available, whereas AMD hardware typically has a maximum of 256 workgroup threads.", "Many kernels won't run across multiple hardware platforms until they have been tested on each platform, and platform-specific issues addressed.", "It is not in general safe to assume that code that works on one hardware platform will run on all the others.", "Moreover, the low-level memory management involved in writing OpenCL code contrasts with the use of caching on CPUs, where the caching is handled automatically by the processor, at runtime.", "It arguably might not be appropriate for GPUs to handle caching dynamically at runtime, because one of the distinguishing features of GPU cores is their simplicity.", "However, it does seem plausible that a language that is slightly higher level than OpenCL could perhaps leave the management of the cache and the memory to an effective optimizing compiler to handle?", "This would free the developer to focus on writing down the algorithms in programmatic form, rather than considering different possible permutations and combinations of memory and loop structures.", "PENCIL [5] might be one approach here." ], [ "Compiler differences", "Each hardware vendor's compiler emits slightly different warnings and errors.", "Something that builds cleanly on one vendor's hardware might fail to build altogether on another's.", "In addition, certain syntax, valid C99, entirely crashed certain compilers, a “segfault”.", "The result is that, in addition to hardware geometry differences, each kernel needs to be built using each supported vendor's OpenCL compiler, in order to resolve vendor-specific build issues.", "It is not safe to assume that if it builds on one vendor's OpenCL compiler, and that the kernel contains only valid C99, that other vendor's OpenCL compilers will build the kernel successfully.", "Therefore, if one wishes to support hardware from three different vendors, one needs to have access to representative hardware from each vendor." ], [ "OpenCL version", "There is a compromise between targeting more recent versions of OpenCL, providing additional functionality and performance, or targeting older versions, allowing the use of more possible target platforms.", "cltorch targets OpenCL 1.2, which is supported by many current GPUs from Intel, NVIDIA, and AMD.", "Nevertheless, some mobile devices in common use continue to use OpenCL 1.1, and would therefore need special treatment in order to work with cltorch." ], [ "Kernel caching", "NVIDIA GPUs cache the kernels automatically, even when building from templates, including where the kernel OpenCL code is supplied as a string, at runtime.", "For training a neural network model, the compilation time is negligible.", "However, for development purposes, being able to start the kernels quickly is convenient.", "Thus, there could be an opportunity for other vendors to provide similar kernel caching functionality." ], [ "Comparison with other\nframeworks", "In this section, we will compare OpenCL Torch with other deep learning frameworks and backends, and specifically with: CUDA Torch DeepCL OpenCL Caffe" ], [ "CUDA Torch", "CUDA Torch provides more functionality than the OpenCL Torch implementation currently.", "For example, the sort function is not currently implemented in OpenCL Torch.", "36ptGiven constant hardware, the CUDA implementation runs faster than the OpenCL implementation.", "There are two principle reasons: Semantically identical OpenCL kernels often run slower than their CUDA counterparts, even on the same hardware there is currently no equivalent of the CUDA CUDNN[10] library available for OpenCL Figure: Per-element timings as function of tensorsizeWith regards to the first reason, Figure REF shows a comparison of carrying out per-element operations on a tensor in CUDA Torch vs OpenCL Torch.", "It shows the bandwidth for per-element operations, as a function of the number of floats per kernel launch, for CUDA and for OpenCL.", "These kernels are relatively simple, and the code is more or less the same, to within a search and replace eg of threadIdx.x with get_local_id(0).", "This experiment was performed on an NVIDIA K520 GPU, using CUDA Torch as of 21 May 2016, and OpenCL Torch as of 21 May 2016.", "The scale is log-log.", "We can see that the shape of the graph is pretty similar for both OpenCL and CUDA kernel launches.", "For both backends, the bandwidth is approximately constant down to tensors of $1e5$ floats, then reduces with tensor size, as the overhead of kernel launch and setup becomes dominant.", "The shape of both graphs, CUDA vs OpenCL, on a log-log scale, looks pretty similar.", "Figure: Per-element timings for tensors of 1e7floatsHowever on a linear scale, there is a noticeable difference between the performance of this kernel using CUDA and using OpenCL.", "Figure REF shows the bandwidths on a linear scale for tensors of $1e7$ floats.", "We can see that the OpenCL backend is about 33% slower than the CUDA backend, for this geometry, even though the hardware is identical, and the kernel code is approximately identical, to within a search and replace for eg threadIdx.x vs get_local_id(0).", "An opportunity exists to look at why this is the case, since both CUDA and OpenCL compile via the same PTX IL." ], [ "DeepCL", "DeepCL[29] provides a deep learning framework dedicated to OpenCL, that runs on Windows, as well as on linux and Mac.", "It provides a commandline tool, and a python interface.", "DeepCL does not provide as many network layer types, or as complete an implementation of each layer type, compared with OpenCL Torch, or OpenCL Caffe." ], [ "OpenCL Caffe", "There are several ports of Caffe[17] to OpenCL: Gu et al[15], Engel[14], and Tschopp[35] Caffe is quite challenging to port to OpenCL, because it is quite tightly coupled with the underlying CUDA implementation, and it is challenging to factorize this so that an OpenCL implementation will not adversely affect subsequent Caffe development.", "There is no easy solution for this task, so merging the OpenCL ports to Caffe core is a painstaking time-consuming task, needing careful negotiation between the various parties involved.", "In terms of performance, the port by Gu et al contains a batched im2col implementation, discussed below.", "Tschopp's Greentea initially used im2col.", "Recently, Tschopp has started to implement implicitgemm.", "These algorithms are discussed below.", "Having discussed some alternative implementations in terms of features and functionality, let's look now at performance." ], [ "Performance", "Figure REF shows the timings for running a single forward-backward iteration through the convolutional layers of six popular convnet models.", "These timings are from Chintala's convnet-benchmarks[11].", "The timings shown are the forward-backward training times for a single batch.", "The implementations are described in Table REF .", "Table: Convolutional ImplementationsThus, we compare between torch implementations, between OpenCL Torch and OpenCL Caffe, and with a state of the art convolutional implementation for CUDA cards neon.", "DeepCL is included in the per-layer timings on the convnet-benchmarks page, but is not included in the full network timings shown here, because it lacks certain key implementation details for these models.", "Looking at Figure REF , we can see that the OpenCL implementations are significantly slower than the CUDA implementations.", "cltorch is marginally faster than Tschopp's Greentea im2col implementation.", "cltorch is around 3-4 times slower than the Torch cutorch implementation.", "neon is about 20 times faster than cltorch, on the Googlenet v1 model.", "Gu et al's OpenCL Caffe is not present on the benchmarks page, because it uses AMD-specific OpenCL extensions, and therefore cannot run on the NVIDIA Titan X used here.", "It could be interesting to obtain benchmarks on hardware from other vendors, for example on AMD GPUs.", "Let's look at performance in more detail, and look at what options we have to improve the performance of convolutions in hardware-agnostic neural network frameworks.", "In this section, we will analyse options for improving the performance of hardware-agnostic neural net frameworks.", "We will see that the convolutional layers dominate the training time.", "We will compare common algorithms for computing the convolution, discuss GEMM, hardware-specific implementations, and HCC." ], [ "Importance of convolution", "The training time for a convolutional network is dominated by the convolutional layers.", "Figure REF shows the per-layer forward time for VGG model `A', on an NVIDIA Titan X GPU, for im2col+clBLAS, im2col+cublas and cudnnv4 convolutional implementations.", "Figure: VGG timings grouped by layertypeThe vast majority of the time is going into the convolutional layers, rather than into pooling, or RELU.", "Figure REF sums the timings by layer type, and confirms this analysis.", "It is clear that the convolutional layer is where we should be focusing our attention.", "These graphs shows timings for three Torch GPU backends, described in Table REF .", "Looking again at Figure REF , we can see that the convolutional performance of cltorch is about three times slower than for the CUDA backends for Torch, and therefore there are opportunities to improve in this area.", "Table: Torch GPU Backends" ], [ "Convolutional Algorithms", "In order to go further with hardware-agnostic deep learning frameworks, it could be interesting to analyse how the CUDA learning frameworks obtain excellent performance.", "48pt There are four algorithms in common use: im2col implicitgemm fft winograd In addition, cuda-convnet[18] provides a very fast direct convolutional implementation for very specific convolutional geometries.", "Let's look at each of these algorithms in turn, starting with cuda-convnet" ], [ "Cuda-convnet", "Cuda-convnet provides a very fast convolutional implementation for very specific convolutional geometries.", "Unfortunately it generalizes poorly to other geometries, or hardware.", "For example [17] found that using im2col gave a speedup of 1.3 times on K20, compared to cuda-convnet, which is optimized for GTX580." ], [ "im2col", "im2col [17], [8] is a very general method for implementing convolutions, off-loading the geometry-specific specializations to the underlying GPU BLAS implementation.", "im2col converts the convolution of a single 3d image tensor with the 4d weights tensor into a matrix multiplication of two 2-dimensional matrices.", "This uses more memory than direct convolution, but it generalizes well to many convolutional geometries.", "It can take advantage of existing optimizations in the existing SGEMM implementations.", "The idea for using matrix multiplications for carrying out convolutions is not new, but the observation that this produces excellent performance on GPUs, that generalizes well to many geometries, was made independently by [17] and [8].", "For im2col in cltorch, the BLAS implementation is handled by clBLAS[1].", "clBLAS is highly optimized for many high-performance computing workloads, but till recently had not been used for deep learning im2col workloads.", "The geometries produced by im2col do not fit within the regime of geometries for which clBLAS has been optimized on previously.", "Therefore, the SGEMM performance for these geometries is relatively less good.", "Gu et al[15] found that the overall convolutional speed of clBLAS for these workloads is not competitive with cublas.", "clBLAS works well for large, square matrices, with dimensions of around 1024 or 4096 along the side, and dimensions a multiple of 8 or 16.", "Gu et al showed that using batched im2col, over multiple images, puts clBLAS into a more favorable regime, giving a performance boost of around 4-5 times, on AMD hardware.", "Therefore it could be interesting to incorporate this technique generally into other OpenCL backends.", "In general, the performance of im2col will be strongly dependent on the performance of the underlying GEMM implementation." ], [ "implicitgemm", "Similar to im2col, implicitgemm[10] lowers the convolutions onto matrix multiplication.", "Whereas im2col fully materializes the lowered matrix in off-chip memory, implicitgemm materializes the lowered matrix lazily, in on-chip memory.", "This reduces memory requirements, and can improve performance.", "implicitgemm is implemented in NVIDIA's CUDNNv1[10].", "An OpenCL implementation of implicitgemm is being developed by Tschopp [35].", "This looks like a promising way forward to improve OpenCL convolutional performance." ], [ "FFT", "fft transforms the convolutional problem from the spatial domain into the frequency domain.", "It is constant-time with respect to the size of the convolutional kernel.", "It typically requires a large amount of memory.", "For models such as Alexnet, which have large kernels in the first layer, fft gives noticeable speed benefits.", "Specifically, in the second layer of AlexNet, fft shows a clear benefit for the stride 1 5x5 kernels.", "However, recently the focus of convolutional networks has turned to smaller kernels.", "VGG uses 3x3 kernels.", "GoogleNet includes 1x1 kernels.", "On these models, the benefits of fft are less clear, compared to other algorithms.", "A competitive implementation of fft is fbfft[36].", "In Figure REF , it can be seen that fbfft is fast on AlexNet, and on Overfeat, but offers no advantage on VGG.", "No timings are available for GoogleNet, which contains 1x1 convolutions, and is far from the favorable regime for fft.", "As for im2col, fft needs a fast GEMM implementation.", "The performance of fft will be strongly dependent on the speed of the underlying GEMM." ], [ "Winograd", "The Winograd FIR algorithms[37] are an algorithmic optimization that reduces the number of mathematical operations required for convolution.", "Winograd outlined the general principles in the 80s.", "Lavin and Gray provided a specific implementation for GPU convolution in [21], and implemented these algorithms in neon[3].", "winograd is an algorithmic improvement, and therefore could be implemented also in OpenCL, plausibly providing a speedup similar to the speedup which it provides to neon.", "Note however that this is not the only optimization in neon.", "We will look later at neon's use of SASS, including for the GEMM implementation, which provides additional speedups.", "As for im2col and fft, winograd transforms the problem, but still relies on GEMM.", "And therefore its performance is strongly related also to the efficiency of the underlying GEMM implementation.", "Having touched on neon, we should look at neon specifically, as the current state of the art for convolutional implementations, on NVIDIA CUDA hardware." ], [ "Neon", "neon[3] uses at least the following optimizations to obtain its very fast, near-optimal convolutional performance: winograd algorithm[37], [21] SASS implementation ... including a SASS implementation of GEMM fully fused kernels As discussed, the winograd algorithm could be implemented in OpenCL, and looks a promising possible way forward to improving OpenCL convolutional performance.", "Note that as for fft and im2col, winograd is also dependent on an underlying GEMM implementation.", "Therefore, whilst writing winograd in OpenCL could plausibly produce an efficient implementation, an efficient implementation of GEMM would also be required.", "The SASS implementations are not directly accessible from OpenCL, since they are not just vendor-specific, but architecture specific.", "SASS written for CUDA Maxwell devices is incompatible with SASS written for Kepler devices, and visa versa.", "For example, the SASS implementations make assumptions about the available registers.", "SPIR-V provides a portable IL language, which could help obtain efficient optimizations.", "However SPIR-V is at the level of PTX, rather than at the level of SASS.", "PTX is higher-level than SASS.", "It is device independent IL, albeit vendor-specific.", "How much contribution does the SASS implementation, the custom GEMM, and the fused kernels provide?", "We can estimate this by running a convolution twice, on the same device, one with the Maxwell optimizations on, and one with them turned off, by modifying the code to incorrectly detect the device as Kepler.", "Figure REF shows the results for this.", "This is for forward propagation of a batch of 128 224x224 images, using 3x3 convolutions, running on an NVIDIA Titan X.", "We can see that the Maxwell optimizations reduce the batch time by about 33%.", "Figure: Neon, effect of MaxwelloptimizationsTherefore, an OpenCL implementation of winograd would be at least around 33% slower than a hardware-optimized low-level implementation, such as neon.", "Taking into account the earlier experiments of running “Apply” in OpenCL and in CUDA, on the same device, which showed an additional performance drop of around 33%, all other things equal, we could expect an OpenCL implementation of winograd to approach around 40% of the execution speed of neon.", "However, note that neon depends also on the speed of the underlying GEMM implementation, as do also im2col and fft, so we should discuss GEMM briefly." ], [ "GEMM", "GEMM is at the heart of convolution.", "It is used by all the convolution algorithms detailed above, with the exception of the direct convolution algorithm in cuda-convnet.", "Thus, it is important that the GEMM implementation should be the most efficient possible.", "Currently, cltorch uses the clBLAS GEMM implementation.", "Gu et al showed that clBLAS is an effective GEMM implementation that can be competitive with the CUDA cublas implementation.", "Gu et al showed that it is important to ensure that the matrix sizes fall close to the optimal regime for clBLAS.", "Gu et al showed that by using batching, lower multiple images into matrix multiplication in a single batch, the clBLAS implementation was around 4-5 times faster, on AMD hardware.", "A possible alternative to clBLAS is ViennaCL, which provides a highly hardware-agnostic implementation of GEMM, working not just on OpenCL, but also on CUDA and OpenMP.", "Figure: Comparison of CLBlast with clBLAS, on RadeonHD7950In terms of performance, Nugteren is developing CLBlast[27], based on CLTune auto-tuner[28].", "CLBlast shows a performance benefit relative to clBLAS for matrices whose sides are not a multiple of 128.", "Specifically matrices with a side of $128m+1$ , for integer $m$ , show a clear benefit, Figure REF , [27].", "Tschopp's implicitgemm implementation is based loosely on the CLBlast GEMM implementation, but using fully fused kernels, rather than factorizing the matrix lowering operation and the GEMM into separate kernels.", "It could be interesting to benchmark OpenCL GEMM implementations, under workloads associated with the convolutional algorithms discussed above.", "Lokhmotov's GEMMbench [23] could potentially be useful for this.", "Going back to neon, assuming that one could obtain a GEMM implementation competitive with cublas for OpenCL, then it looks like it could be possible to write an OpenCL implementation of winograd, that could approach around 20-30% of the speed of neon (ie 70-80% slower than neon).", "This would be a significant improvement on the current im2col implementations.", "However, can we do better?" ], [ "Pluggable Implementations", "It seems challenging to approach the performance of close-to-the-metal SASS from within OpenCL or SPIR-V, because there is a limit to how far one can improve the performance in languages designed to be portable.", "To go further, two possible approaches could be: implementation of highly optimizing compilers, for OpenCL, or create pluggable hardware-specific convolutional implementations The former approach, of creating highly optimizing compilers, is an active area of research, and there are no easy answers.", "AEcute metadata [16], and the more recent PENCIL[5] are two approaches to facilitate generation of hardware-optimized code.", "PENCIL allows expression of algorithms in a higher-level language, which can be used to generate device-optimized OpenCL.", "Rather than attempting to make the convolutional implementations portable, an alternative approach could be to make them pluggable, loadable at runtime, and strongly hardware-specific.", "Thus they could be written in low-level assembler, and make full use of hardware-specific optimizations, such as knowledge of the exact memory dimensions, register layout, and other hardware characteristics The neural network library frameworks could themselves be written in a portable language, such as OpenCL.", "Concretely, this could work in a similar way to the ICD abstraction layer in OpenCL.", "OpenCL's ICD binds with the vendor-provided OpenCL implementation at runtime.", "In the case of convolution, the machine learning framework could simply call a function conv, within a Khronos-provided API.", "The Khronos API would route the call to a convolutional implementation appropriate to the current targeted hardware device.", "This might look something like Figure REF .", "Figure: Proposal for runtime convolution linkingNote that this architecture says nothing about who will write the highly optimized hardware-specific convolutional implementation.", "For example we could imagine a virtual OpenCL platform, that wraps the neon implementation, for CUDA hardware, see Figure REF .", "Figure: Proposal for virtual OpenCL platform" ], [ "HCC", "Discussion of non-CUDA deep learning implementations would not be complete without touching on AMD's HCC.", "HCC is an alternative open standard to OpenCL.", "It is not backed by Khronos, but it is an open standard, backed by one of the major competitors to NVIDIA.", "It is currently implemented only on AMD hardware, but there is no reason why other hardware vendors couldnt support it in the future.", "Since only one vendor owns the specifications currently, it can potentially evolve rapidly.", "Note that HCC will not in itself directly solve the challenge of writing a fast convolutional layer.", "The key advantages of HCC could be agility, and compatibility with CUDA language.", "The AMD implementation of HCC might become faster than the AMD implementation of OpenCL.", "However, it seems unlikely that convolutional implementations written in pure HCC would approach the performance of Gray's SASS implementations in neon.", "Therefore, the use of pluggable architectures, such as that outlined above, might be key to obtaining the fastest cross-platform convolutional performance.", "HCC solves many of the challenges faced during the development of cltorch, such as the use of c++ templates in the torch CUDA code.", "It could therefore significantly facilitate the development of hardware- agnostic deep learning frameworks in the future.", "The extent to which HCC can contribute to hardware-agnostic deep learning framework will depend on the extent to which multiple hardware vendors adopt the standard.", "In previous sections, possible future evolutions for portable neural network compute have been described in detail.", "Summarizing here, several possible future evolutions could improve the performance of portable deep learning libraries: OpenCL implementation of implicitgemm, for example that currently under development by Tschopp OpenCL implementation of Winograd algorithms Improvements to the OpenCL GEMM implementations, for example that currently under development by Nugteren hardware-specific convolutional implementations, loaded at runtime via a pluggable architecture Finally, HCC might facilitate the future development of hardware-agnostic learning frameworks, by facilitating the porting process, and reducing the disparity between CUDA and non-CUDA code-bases." ], [ "Conclusion", "cltorch provides a hardware-agnostic backend for the Torch deep learning library.", "It is written using the portable language `OpenCL'.", "cltorch provides sufficient functionality to train recent models such as: AlexNet, GoogleNet, Overfeat, and VGG.", "Challenges faced during the creation of the cltorch backend were presented.", "Current OpenCL neural network framework implementations face challenges obtaining training speeds competitive with CUDA implementations.", "Possible approaches to improve the speed of the convolutional implementation have been presented.", "Finally, HCC could solve many of the challenges faced during the development and reduce the disparity between CUDA and non-CUDA codebases, but the extent to which it will become a hardware-agnostic standard, implemented by multiple vendors is as yet uncertain." ] ]
1606.04884
[ [ "Common envelope events with low-mass giants: understanding the energy\n budget" ], [ "Abstract Common envelope events are important interactions between two binary stars that lead to the formation of close binary systems.", "We present here a systematic three-dimensional study in which we model common envelope events with low-mass giant donors.", "The results allow us to revise the energy formalism that is usually used to determine common envelope event outcomes.", "We show that the energy budget for this type of system should include the recombination energy, and that it also must take into account that a significant fraction of the released orbital energy is taken away by the ejecta.", "We provide three ways in which our results can be used by binary population synthesis studies: a relation that links the observed post-common envelope binary with the initial binary parameters, a fitting formula for the $\\alpha_{\\rm ce}\\lambda$ parameter of the standard energy formalism, and a revised energy formalism that takes into account both the recombination energy and the energy that is taken away by the ejecta." ], [ "Introduction", "It is believed that the common envelope event (CEE) is the most important phase in the evolution of a wide range of different types of close binary systems.", "It most likely plays a crucial role in the formation of X-ray binaries, Type Ia supernovae progenitors, double degenerate stars, and more [12].", "CEE is a short-lived physical process when two stars orbit inside a single, shared envelope.", "The outcome of a CEE is either a new binary with a reduced orbit, or a merger of the two stars.", "One of the standard ways to predict an outcome of a CEE is by using the energy formalism.", "This method equates the binding energy of the envelope of the donor with the orbital energy before and after the event [26], [27]: $E_{\\rm bind}=\\Delta E_{\\rm orb} \\ ,$ where $E_{\\rm bind}$ is the binding energy of the envelope of the donor, and $\\Delta E_{\\rm orb}$ is the change in the orbital energy.", "Recognizing that not all the available orbital energy can be used to eject the envelope of the donor, [16] proposed a common-envelope efficiency parameter, $\\alpha _{\\rm CE}$ , defined as the fraction of released orbital energy that has been effectively used to eject the envelope of the donor.", "This $\\alpha _{\\rm CE}$ parameter is now widely used in binary population synthesis studies [7].", "A better understanding of the energy budget of a CEE, better than a simple parameterization, could help us to better predict the population of close binaries.", "The systems where the parameters of the CEE can be best constrained are double-white-dwarf (DWD) binaries.", "It is widely believed that the last episode of mass transfer leading to DWD formation was an unstable MT, a CEE, where the donor was a red giant (RG) star [24], [10], [27], [25].", "For RGs, a well-defined relation between their core masses and radii exists.", "From the observations of DWD systems, we know that one of the white dwarfs (WDs) is usually younger, and therefore is the remnant of the pre-CE RG donor.", "However, the mass of the donor can not be uniquely determined, as long as $\\alpha _{\\rm CE}$ is uncertain.", "In this paper, we perform three-dimensional numerical simulations of CEEs between low mass RG stars and WD companions.", "This work is the extension of our preliminary study devoted to the formation of the DWD WD 1101+364 via a CEE [20].", "Here, we consider a wide parameter space based on the mass of the RG donor, the RG core mass and the companion mass.", "Our DWD binaries have a mass ratio $q=M_1/M_2$ between 0.8 and 1.125, where $M_1$ is the mass of the younger WD (formed during a CEE), and $M_2$ is mass of the older WD.", "The main goal of this paper is to understand the energy budget at the end of the CEE.", "We pay particular attention to the usage of the recombination energy, and to the energy that is taken away by the ejecta.", "Those energies are not usually taken into account by the standard energy formalism, and should explain the deviation of $\\alpha _{\\rm CE}$ from 1.", "We describe the initial conditions, the parameter space and the numerical methods in § .", "§  contains the definitions for the energies.", "In § , we give an overview of the final states of the simulations in terms of the mass of the ejecta, energy evolution during a spiral-in, orbital parameters at the end of a CEE, and discuss how the outcomes of the CEE can be parameterized.", "Finally, §  gives a brief discussion on how our results can be used in population synthesis studies, as well as comparison with the observed binaries that are known to have a post-CE WD." ], [ "Parameter space", "We study the progenitor systems that have likely formed the observed DWDs, in terms of the observed WD masses and the orbital separations.", "We adopt that immediately before a CEE, the DWD progenitor binaries consisted of a low-mass RG with a core mass close to the observed new (second-formed) WD, and of an older (first-formed) WD.", "We consider 24 binaries in the parameter space defined by the RG donor mass, the mass of the newly formed WD (RG core), and the mass of the old WD.", "For the RG donor mass, $M_{\\rm d,1}$ , we take $1.2$ , $1.4$ , $1.6$ , and $1.8$ $M_\\odot $ .", "Each low-mass RG is considered at two evolutionary points on its RG branch, namely when their degenerate He core masses $M_{\\rm c,1}$ are $0.32 M_\\odot $ and $0.36 M_\\odot $ cores (we expect that the mass of the new WD will be very similar to the He core mass of the RG donors).", "For the mass of the old WD, $M_{\\rm a,2}$ , we take $0.32$ , $0.36$ , and $0.40$ $M_\\odot $ , for each case of the RG donor.", "Table REF shows the summary of the considered parameter space, and the initial conditions for each binary.", "To create the initial RG donor stars, we use the TWIN/EV stellar code [2], [3].", "This allows us to obtain a realistic initial one-dimensional (1D) stellar profile for each RG donor.", "Stars are evolved until their degenerate He cores have grown close to $0.32 M_\\odot $ , and $0.36M_\\odot $ .", "Table: Complete parameter space with initial conditions.To model a CEE between a RG and a WD, we use STARSMASHER [5], [17], a Smoothed Particle Hydrodynamics (SPH) code.", "Technical details on using this code to model CEEs can be found in [19].", "We reiterate the point made in [20], that when a 1D star is transferred to a 3D code via the relaxation process in STARSMASHER, the core of the RG, $M_{\\rm c,1}$ , must be increased slightly by about $0.01\\,M_\\odot $ so that the resulting profile of the 3D star for pressure, density, internal energy and other quantities would match that of a 1D star.", "The RG envelope is modeled using $10^5$ particles, and the RG core is modeled as a point mass, as is the WD (note that a point mass only interacts gravitationally with normal SPH particles).", "The envelope mass in our three-dimensional star is $M_{\\rm env,1}=M_{\\rm d,1}-M_{\\rm c,1}$ .", "We found that for most RGs with cores close to $0.4 M_\\odot $ , the profiles could not be matched well with 1D stars after the relaxation, and hence RGs with core masses $>0.4M_\\odot $ were excluded from the considered parameter space.", "Improving the match of the profiles in those more evolved donors is computationally unfeasible now, as this requires such a change in the number of SPH particles and in their smoothing length, that the GPU time would be increased by at least 64 timesThe average time that is spent on obtaining one model presented in this paper is about 1 GPU core year.", "The global GPU resource available at Compute/Calcul Canada for all scientists in Canada, in several GPU-equipped clusters, is about 200 GPU core years..", "The photospheric radius of the star in SPH, $R_{\\rm SPH}$ , is defined as $R_{\\rm SPH}=R_{\\rm out}+2h_{\\rm out}$ , where $R_{\\rm out}$ is the position of the outermost particle and $h_{\\rm out}$ is the smoothing length of that particle .", "Defining the photospheric radius this way ensures that all envelope particles are enclosed within $R_{\\rm SPH}$ for the non-synchronized cases.", "The initial orbital separation, $a_{\\rm orb,ini}$ , for the non-synchronized cases is found from the assumption that $R_{\\rm SPH}$ is equal to the Roche Lobe (RL) radius, $R_{\\rm rl}$ , and using the approximation for the RL radius by [4].", "The initial orbital period, $P_{\\rm orb,ini}$ is found assuming a Keplerian orbit.", "For the synchronized case, the initial orbital period and separation are found at the moment when the outermost particles overfill the donor's RL during the scan process [17].", "During the scan process, the envelope's angular momentum is steadily boosted.", "This leads to the expansion of the radius of the donor as compared to the non-rotating case.", "As a result, the orbital separation in a synchronized and a non-synchronized case may not match, however, the difference in these quantities is not large.", "The photospheric radius is $R_{\\rm SPH}=R_{\\rm out}$ for the synchronized case.", "We use the tabulated equation of state (TEOS) incorporated from MESA [21] and implemented as described in [20].", "This TEOS includes recombination energy for H, He, C, N, O, Ne, and Mg.", "The dominant contribution to the recombination energy comes from H, which account for about 59% of the total energy, followed by He with about 38%, and 3% for the rest of the elements, in all our simulations." ], [ "Definitions", "In this Section we declare definitions for the most important quantities.", "Definitions are adopted from [20], unless stated otherwise.", "Energy formalism.", "The energy formalism compares the donor's envelope binding energy $E_{\\rm bind}$ with the orbital energy before the CEE, $E_{\\rm orb,ini}$ , and after the CEE, $E_{\\rm orb,fin}$ [27], [16]: $E_{\\rm bind}=\\alpha _{\\rm bind} (E_{\\rm orb,fin}-E_{\\rm orb,ini})\\equiv \\alpha _{\\rm bind} \\Delta E_{\\rm orb} \\ .$ Here $\\alpha _{\\rm bind}$ is the fraction of the orbital energy effectively used to expel the CE.", "This parameter is equivalent to the commonly used $\\alpha _{\\rm CE}$ , and is usually assumed to be $0\\le \\alpha _{\\rm bind}\\le 1$ .", "The potential energy of the donor's envelope in SPH is $E_{\\rm pot,ini}=\\frac{1}{2}\\sum _i m_i \\phi _i,$ where $m_i$ , and $\\phi _i$ are the mass, and specific gravitational energy, respectively, for each SPH particle $i$ in the initial RG profile, including the core.", "Note that this quantity is computed before the star is placed in the binary configuration.", "In our SPH method, $\\phi _i$ is calculated as in [8].", "The internal energy of the donor's envelope in SPH is $E_{\\rm int,ini}=\\sum _i m_i\\left(\\frac{3}{2}\\frac{kT_i}{\\mu _im_{\\rm H}}+\\frac{aT_i^4}{\\rho _i}\\right),$ where $T_i$ , $\\rho _i$ and $\\mu _i$ are the temperature, density, and mean molecular mass, respectively, for each particle $i$ in the initial RG profile.", "The constants $k$ , $a$ , and $m_{\\rm H}$ are the Boltzmann constant, radiation constant, and hydrogen atom mass.", "The binding energy of the RG, without the recombination energy, is $E_{\\rm bind} = E_{\\rm pot,ini}+E_{\\rm int,ini}.$ This binding energy was historically parameterized using the parameter $\\lambda $ [1], [12], $E_{\\rm bind} = - \\frac{GM_{\\rm d,1}(M_{\\rm d,1}-M_{\\rm c,1})}{\\lambda R}$ This equation, combined with the energy formalism equation REF , provides the most used equation to find CEE outcomes in binary population synthesis studies, where $\\alpha _{\\rm bind} \\lambda $ are used together as one single parameter: $\\Delta E_{\\rm orb} = - \\frac{GM_{\\rm d,1}(M_{\\rm d,1}-M_{\\rm c,1})}{\\alpha _{\\rm bind} \\lambda R}$ The orbital energy of the binary system in SPH takes the following form: $E_{\\rm orb}= \\frac{1}{2}\\mu |V_{12}|^2+\\frac{1}{2}\\sum _i m_i\\phi _i-\\frac{1}{2}\\sum _j m_j\\phi _j^{\\rm RL_1}-\\frac{1}{2}\\sum _k m_k\\phi _k^{\\rm RL_2},$ where $\\mu =M_1M_2/(M_1+M_2)$ is the reduced mass, and $\\vec{V}_{12}=\\vec{V}_1-\\vec{V}_2$ is the relative velocity of the two stars.", "The first, second, third and fourth terms give the orbital kinetic energy, the total gravitational energy of the binary (with the sum being over all particles $i$ in the binary), the self-gravitational energy of the donor (the sum being over all particles $j$ in star 1), and of the WD (the sum being over all particles $k$ in star 2, initially just the one particle representing the WD), respectively.", "Recombination energy.", "The recombination energy is included in the total value of the specific internal energy provided by TEOS, and can be found as $E_{\\rm rec,ini} = \\sum _i m_i\\left(u_i-\\frac{3}{2}\\frac{kT_i}{\\mu _im_{\\rm H}}-\\frac{aT_i^4}{\\rho _i}\\right)\\equiv \\alpha _{\\rm rec}\\Delta E_{\\rm orb}, $ where $u_i$ is the SPH specific internal energy for each particle.", "$\\alpha _{\\rm rec}$ is the ratio between the recombination energy and the released orbital energy.", "Since the recombination energy acts as an additional (to the orbital energy) source of energy, $\\alpha _{\\rm rec}<0$ .", "This energy is not part of the usually considered binding energy, as it is not available immediately, and its release must be triggered [13].", "The amount of stored recombination energy is proportional to the mass of the envelope $E_{\\rm rec,ini}= \\eta (M_{\\rm d,1}-M_{\\rm c,1})$ .", "In a fully ionized gas that consists of only helium (0.3 mass fraction) and hydrogen (0.7 mass fraction), $\\eta \\simeq 1.5\\times 10^{13}\\,{\\rm erg/g}$ .", "Our gas chemical composition is a bit different, and also our TEOS takes ionization of heavier elements into account as well.", "For our donors, we find $\\eta \\simeq 1.6\\times 10^{13}\\, {\\rm erg/g}$ .", "The version of STARSMASHER we use evolves, for each SPH particle, the specific internal energy $u_i$ and density $\\rho _i$ [5].", "The pressure is then found from the internal energy, density, and the adopted equation of state.", "Total energy.", "The total initial energy, $E_{\\rm tot,ini}$ , is $E_{\\rm tot,ini}=E_{\\rm orb,ini}+E_{\\rm bind}+E_{\\rm rec}.$ This quantity is conserved during the evolution of all our models.", "Bound and unbound material.", "For each particle, its total energy is defined as $E_{{\\rm tot,}i}\\equiv 0.5 m_iv_i^2 +m_i\\phi _i+m_iu_i$ , where the first, second and third terms are the kinetic, potential, and internal energies, respectively.", "We classify our particle as in [20]: (i) the ejecta, $m_{\\rm unb}$ – the particles that have positive energy, (ii) the circumbinary material, $m_{\\rm cir}$ – the matter that is bound to the binary, but is located outside of both RLs, and (iii) the binary material, $m_{\\rm bin}$ – the particles that are inside either of the two RLs.", "The total energy of the unbound material at infinity is found when the unbound mass is in a steady state after the CEE.", "It is computed as $E_{\\rm tot,unb}^\\infty = \\sum _i E_{\\rm tot,i}^{\\rm unb}\\equiv -\\alpha _{\\rm unb}^\\infty \\Delta E_{\\rm orb}\\ .", "$ Note that $E_{\\rm tot,i}^{\\rm unb}$ includes the recombination energy of the unbound material.", "$\\alpha _{\\rm unb}^{\\infty }$ is the ratio of the energy taken away by the unbound material to the released orbital energy.", "Final energies.", "The total energy at the end of the simulation is distributed in the “binding” energy of the gas bound to the binary, $E_{\\rm bind,fin}$ , the final orbital energy of the binary, $E_{\\rm orb,fin}$ , and the total energy of the unbound material at infinity, $E_{\\rm tot,unb}^{\\infty }$ : $E_{\\rm tot,fin}=E_{\\rm orb,fin}+E_{\\rm bind,fin}+E_{\\rm tot,unb}^{\\infty },$ where $E_{\\rm tot,unb}^{\\infty }$ is composed of $E_{\\rm kin,unb}^{\\infty }$ , $E_{\\rm int,unb}^{\\infty }$ , and $E_{\\rm pot,unb}^{\\infty }$ – the kinetic, internal and potential energies of the unbound material, respectively.", "Generally, $E_{\\rm bind,fin}$ has a fairly small absolute value at the end of the simulation, and so can be safely disregarded.", "In addition, the particles around the WD may be accreted during a CEE, and hence their presence there may not have any physical meaning.", "The extended energy formalism, following to [20], can then be written as follows: $\\alpha _{\\rm bind}+\\alpha _{\\rm rec}+\\alpha _{\\rm unb}^\\infty \\approx 1,$ If $\\alpha _{\\rm rec}=\\alpha _{\\rm unb}^\\infty =0$ , then Equation REF reduces to the standard energy formalism.", "For additional analysis of the energies at the end of a CEE, we introduce 3 more quantities : $\\alpha _{\\rm pot}\\equiv E_{\\rm pot,unb}^\\infty /E_{\\rm pot,ini}$ – the ratio of the potential energy taken away by the ejecta, to the initial potential energy of the RG envelope, $\\alpha _{\\rm th}\\equiv E_{\\rm int,unb}^\\infty /(E_{\\rm int,ini}+E_{\\rm rec})$ – the ratio of the internal energy (including recombination) taken away by the ejecta, to the sum of the initial internal energy and the recombination energy of the RG envelope, $\\alpha _{\\rm kin}^\\infty \\equiv -E_{\\rm kin,unb}^\\infty /\\Delta E_{\\rm orb}$ – the ratio of the kinetic energy taken away by the ejecta, to the released orbital energy.", "We point out that $\\alpha _{\\rm kin}^\\infty $ is a part of $\\alpha _{\\rm unb}^\\infty $ , however, $\\alpha _{\\rm pot}$ and $\\alpha _{\\rm th}$ are not a part of $\\alpha _{\\rm unb}^\\infty $ as they describe fractions of their corresponding initial energies.", "All our simulations conserved quite well the total angular momentum and the total energy.", "We have checked and found that the error in the energy conservation in all our simulations is less than 0.1% of the initial total energy, while the error in the angular momentum conservations in all our non-synchronized cases is less than 0.001% of the initial total angular momentum value, and the error in the angular momentum conservation in the only synchronized case is 0.4%." ], [ "Overview", "Masses.", "At the end of each simulation, we form a binary consisting of $M_1$ and $M_2$ (see Table REF ).", "We note that $M_1$ and $M_2$ in Table REF differ from the values given for $M_{\\rm c,1}$ , and $M_{\\rm a,2}$ in Table REF , respectively, as a few SPH gas particles remain within the RLs of the DWD binary.", "Ultimately, the ejected material $M_{\\rm unb}$ is at least 99.4% of the initial RG envelope, and there is no circumbinary matter around the newly formed DWD binary.", "Table: Energies and massesFinal Time.", "We stop our simulations no less than 800 orbits after the end of the plunge-in, and typically we stop the simulations after more than 2000 orbits.", "The plunge-in is the fastest phase of the spiral-in, during which the instantaneous separation (distance) between the RG core and the WD changes substantially on the timescale comparable to its inferred orbital period.", "At the moment we stop, the orbital separation is changing by less than $|\\delta a_{\\rm orb} / a_{\\rm orb}| < 0.002$ , where $\\delta a_{\\rm orb}$ is found over one binary orbital period.", "Some simulations were calculated for much longer, e.g.", "the case 1.8G0.32C0.36D is calculated for more than 10 000 orbits after the end of the plunge-in, and the change of the orbital separation over the binary orbital period, at the end of the simulation, is $|\\delta a_{\\rm orb} / a_{\\rm orb}| \\approx 0.0002$ .", "The final parameters provided in Table REF are expected to be time-converged values.", "Figure: The evolution of the total energy (grey solid line), theorbital energy E orb E_{\\rm orb} (green solid line), the energy in thecircumbinary matter (black solid line), and the energy in the ejecta (bluesolid line) for the case 1.2G0.32C0.32D.Energies.", "Figure REF shows how the energies change during the spiral-in phase for the case 1.2G0.32C0.32D.", "After the spiral-in phase is complete and there is no circumbinary matter left, the circumbinary total energy vanishes, while the ejecta energy and binary total energy (which is the orbital energy plus the “binding” energy of the remaining particles) converge to their final values.", "Table REF provides the final distributions of energies for all our simulations.", "Some ostensible deviations can be observed in Table REF .", "For example, the model 1.8G0.32C0.32D has more energetic ejecta than other models.", "We note that the overall energy budget, and the energy that was extracted from the formed binary, are much higher in this model than in any other model.", "Figure: The evolution of the total recombination energy (grey solidline), the recombination energy in SPH particles bound to thebinary (green solid line), the recombination energy in SPHparticles in the circumbinary material (black solid line), and therecombination energy in the ejecta (blue solid line line), shownfor the case 1.2G0.32C0.32D.Table: Orbital parametersFigure: The evolution of the specific energies and the masses for the case 1.2G0.32C0.32D.The bottom plot shows the specific energies per unit mass in the outer envelope (see § 4.2): kinetic energy ε kin \\varepsilon _{\\rm kin}, total energy including recombination ε tot \\varepsilon _{\\rm tot}, and total energy excluding recombination ε tot -ε rec \\varepsilon _{\\rm tot}-\\varepsilon _{\\rm rec}.", "See the text in §4.2 for details on masses." ], [ "Role of the recombination energy. ", "In brief, we found that the circumbinary recombination energy has a maximum during the plunge-in phase.", "At this moment, almost the entire envelope is no longer within the Roche lobe of the binary and becomes circumbinary material (see an example in Figure REF ).", "The ejection of the circumbinary envelope then takes place on its dynamical time-scale, which, e.g.", "for the case 1.2G0.32C0.32D, is about 100 days.", "Let us consider in detail why the dominant energy source that drives the final ejection of the puffed-up envelope is the recombination energy but not the binary orbital energy.", "For that, we will trace the evolution of specific energies during the crucial time at about the plunge-in, when the puffed-up circumbinary envelope formed initially.", "The binary orbital energy can be expected to boost the envelope's ejection by being transferred into kinetic energy of the envelope.", "However, the acceleration of material by the binary's motion, which also can be called a dynamical tide, will only affect the mass that is located approximately within 3 binary orbital separations of the binary's center of mass [22].", "Note that we do not consider here any secular tidal effects, but only an acceleration that is produced during a period of time that is less than a hundred orbital periods of the binary after the plunge-in, i.e.", "comparable to when the envelope is ejected in our simulations.", "We can test whether the mass located far away from the binary is, or is not, accelerated by the binary in our case.", "First, we separate the bound envelope into two sub-envelopes: the inner envelope - the envelope's mass that is within the distance of $3a_{\\rm orb}$ , $m_{\\rm bound,tid}$ – and the outer envelope, $m_{\\rm bound,notid}$ , or the mass that is beyond $3a_{\\rm orb}$ .", "Here $a_{\\rm orb}$ is the current distance between the RG core and the WD.", "The outer envelope, once it is “decoupled” from the binary's tidal effect, is expected to evolve according to its potential, kinetic, internal and recombination energies.", "Let us consider the case 1.2G0.32C0.32D (see Figure REF ).", "In the evolution shown, the plunge-in takes place from days 1750 to 1770, when the orbital separation shrinks by a factor of 10, approaching closely its final value.", "At day 1770, most of the initial envelope is either in the outer envelope, or is ejected.", "During the plunge-in, the definition of the orbital separation $a_{\\rm orb}$ by the energetic principle can not provide a proper result [14], hence we can only use a geometrical distance between the RG core and WD.", "During the plunge-in, the inferred orbital separation is changing rapidly (in a sense, it can also be described as having a very high eccentricity).", "Since the boundary between the outer and inner envelopes, drawn at $3a_{\\rm orb}$ , oscillates as well as the orbital separation itself, the defined masses of the envelopes and their energies oscillate during the plunge-in.", "In Figure REF , we can see that the specific kinetic energy of the outer envelope is settled by the end of the plunge-in.", "This outer envelope is bound by the conventional definition, in which the total energy excluding recombination is negative.", "At the same time, the outer envelope is effectively decoupled from the binary and is not receiving further boosts to its kinetic energy.", "At the moment of the end of the plunge-in, the outer envelope possesses most of the mass that remains bound to the binary.", "The outer envelope has obtained some kinetic energy from its previous interaction with the shrinking binary during the plunge-in.", "That non-zero kinetic energy leads to the envelope expansion, on the dynamical timescale of the expanded envelope, where every SPH particle in the outer envelope can have only a parabolic (bound) trajectory with respect to the binary, if the recombination energy is not released.", "However, once the material expands and cools down enough to start recombination, an SPH particle gains enough energy to become unbound – it can be seen from Figure REF that at the end of the plunge-in the stored potential recombination energy is sufficient to unbind the material of the outer envelope.", "The outer envelope is now flowing away; as more of its material feels a pressure differential between open space above and the remaining envelope below, it expands further, cools down and becomes unbound after recombination.", "This recombination-driven ejection is gradual and non-explosive, vs. the rather explosive, or dynamical, ejection that takes place during the plunge-in, as described in detail by [14].", "The radius at which the released recombination energy can remove a particle out of the potential well is the “recombination” radius, and was derived in [14].", "We can clarify that there is no recombination energy stored in the ejected material at the end of the simulations.", "It is important to mention that the recombination takes place at large optical depths.", "Using our 3D models, we estimate that typical optical depths have values of at least 10, and 1D studies showed that hydrogen recombination can take place at optical depths above 100 [13]." ], [ "Post-CE orbital parameters", "We find the final orbital separation in a geometrical way as $a_{\\rm orb,fin}=(r_{\\rm a}+r_{\\rm p})/2$ , where $r_{\\rm p}$ is the periastron, and $r_{\\rm a}$ is the apastron.", "We ensure that these two quantities, $r_{\\rm p}$ and $r_{\\rm a}$ , are no longer changing with time at the moment when we extract them from the simulations.", "We find the final orbital period of the binary assuming a Keplerian orbit, $P_{\\rm orb,fin}$ .", "Another important orbital parameter is the eccentricity, $e$ , which is found as $e=(r_{\\rm a}-r_{\\rm p})/(2a_{\\rm orb,fin})$ .", "The final orbital parameters are provided in Table REF .", "Note that the final separation found using the orbital energy, $a_{\\rm orb,fin}^{\\rm En}=-GM_1M_2/(2E_{\\rm orb,fin})$ , differs from the final orbital separation found in the geometric way, $a_{\\rm orb,fin}$ .", "This is for two reasons: There is still mass within the RLs of both point masses, as can be seen by the non-zero value of $E_{\\rm bind, fin}$ .", "The presence of these particles, and their not fully stable orbits around the point masses, makes the energy-based way to calculate $a_{\\rm orb,fin}^{\\rm En}$ uncertain.", "Note that these few particles make the stars aspherical and the equation for $a_{\\rm orb,fin}^{\\rm En}$ is formally not valid.", "The distance between the two point masses (WD and RG core) is less than two times their smoothing lengths, which means that there is some extra smoothing in the gravitational potential equation [8].", "The smoothing length of the point masses acts as the softening term defined by [8].", "For details on the definition and how to determine the smoothing length in the case modeled here, see [17].", "As an example, in the model 1.8G0.32C0.32D, the smoothing length for the RG core is $h_{\\rm core}=0.35 R_\\odot $ , and the smoothing length for the WD is $h_{\\rm WD}=0.73 R_\\odot $ .", "The difference in orbital separations between the geometrical way $a_{\\rm orb,fin}$ and the energy way $a_{\\rm orb,fin}^{\\rm En}$ varies from 7.19% (1.2G0.32C0.36D) to 18.11% (1.8G0.32C0.32W), where the separation derived via the geometrical way is always smaller than the separation derived via the energy way.", "There is a very small discrepancy for the initial orbital separations using the two methods, $<0.24$ %.", "The two values for the orbital separation would be closer to each other if the potential in the SPH code were calculated without a softening term.", "Note however that due to the first reason above, which is the presence of SPH particles inside the RLs, the two terms will never be completely the same.", "The discrepancy between the two values due to the softening term is expected to decrease if the smoothing length is decreased, and that can be done if the number of the particles is increased, although it is not intuitive to state whether the separation found by the geometrical way would increase or decrease.", "Only one test was made for the CEE study of the formation of the specific binary, a simulation with 200k particles resulted in a 7 per cent smaller final orbital separation than the same case modelled with 100k particles [20].", "To clarify, the smoothing length was smaller in the case of 200k by 20 per cent compared to the case of 100k, but the relative difference between the final separations derived in the two ways was smaller.", "The models presented in this study might be affected similarly, but it is likely that the relative change in the final results will be small even if the resolution will be doubled.", "Even though the smoothing lengths of the point masses are partially responsible for the discrepancy between the orbital separations found by the two methods, the smoothing length values cannot explain the unbinding of the puffed-up envelope.", "For instance, let us consider the model mentioned above with the maximum discrepancy of 18%, 1.8G0.32C0.32W.", "The smoothing lengths for the RG core and WD are $h_{\\rm core}=0.35 R_\\odot $ , and $h_{\\rm WD}=0.73 R_\\odot $ .", "The softening in this case starts to work when the distance between the RG core and the WD is $2.16R_\\odot $ .", "At that moment, most of the mass is located at an average distance of 14 $R_\\odot $ , except for a few strongly bound particles which remain bound within about 3 $R_\\odot $ from the center of mass.", "The final separation therefore can be dependent on the mass resolution of the particles that were initially strongly bound and were in the close neighborhood of the RG core (where the smoothing length become important).", "However, as was discussed previously in § REF , the binary is decoupled with the puffed-up outer envelope, which is too far from the binary to be able to effectively transfer away its orbital energy, and the envelope ejection only depends on the stored kinetic and recombination energy, and does not affect the final parameters of the binary.", "Table REF shows that the bigger the initial mass of the RG, the tighter is the final orbit, for each fixed companion mass.", "For each initial RG mass and different companion mass, usually, the smaller the mass of the companion, the tighter the final orbital separation.", "However, there are two exceptions: In the case of the 1.2$\\, M_\\odot $ RG with a 0.32$\\,M_\\odot $ core, the largest final orbital period is for the 0.36$\\,M_\\odot $ WD companion, instead of the 0.40$\\,M_\\odot $ WD.", "This could be because the 1.2G0.32C0.36D case deposited the least of the kinetic energy in the ejecta, as compared to the other two cases during the spiral-in phase (see Table REF ).", "In the case of the 1.4$\\, M_\\odot $ RG with a 0.32$\\,M_\\odot $ core, the tighter final orbit is for the 0.36$\\,M_\\odot $ WD companion instead of the 0.32$\\,M_\\odot $ WD.", "This could be because the 1.4G0.32C0.36D case deposited more kinetic energy in the ejecta than the other two cases.", "The final binary in this case also has less remaining bound mass than in the other two cases (see Table REF ).", "We could not identify any other initial condition that could discriminate why the final orbital separation in the two discussed cases did not follow the trend.", "During a spiral-in, we find that in those two cases the velocity at which the companion plunges into the envelope was higher than in other cases, which is consistent with the ejecta taking away more angular momentum.", "However, what causes this deviation in the ejecta's angular momentum is not fully clear.", "Figure REF shows the final orbital periods for all the simulations, as a function of the initial RG mass.", "It can be seen that qualitatively there are two populations, mainly defined by the mass of the RG core, and with a smaller dependence on the mass of the companion.", "In each of these two populations, the final orbital period appears to depend almost linearly on the initial RG mass.", "Figure: The final orbital periods, plotted against the initial RG mass.", "The open circles correspond to simulations with M core ,1 =0.32M ⊙ M_{\\rm core,1}=0.32M_\\odot , while the open triangles correspond to M core ,1 =0.36M ⊙ M_{\\rm core,1}=0.36M_\\odot .", "The small, medium and big symbols are for companions with 0.32M ⊙ 0.32M_\\odot , 0.36M ⊙ 0.36M_\\odot , and 0.40M ⊙ 0.40M_\\odot , respectively.Figure REF shows the final orbital periods as a function of the initial orbital period, the initial RG mass, the mass of the RG core and the mass of the companion.", "This appeared to produce the relationship that can be expressed as follows: $P_{\\rm orb,fin}=10^{-2.46\\pm 0.05}\\left(P_{\\rm orb,ini} \\times \\frac{M_2}{M_{\\rm d,1} M_{1}}\\right)^{1.18\\pm 0.04},$ Here $\\pm $ indicates the standard error for each coefficient.", "The units for the quantities are $M_\\odot $ for all the masses, and days for periods.", "Figure: Final orbital periods.", "This plot contains all non-synchronized simulations.", "The open circles correspond to simulations with M 1 ≈0.32M ⊙ M_{1}\\approx 0.32M_\\odot , the open triangles correspond to M 1 ≈0.36M ⊙ M_{1}\\approx 0.36M_\\odot .", "The small, medium and big symbols are for companions with 0.32M ⊙ 0.32M_\\odot , 0.36M ⊙ 0.36M_\\odot , and 0.40M ⊙ 0.40M_\\odot , respectively.", "The black solid line corresponds to the best fit for all the simulations (see Eq.", ").", "The orbital periods are in days and the masses are in M ⊙ M_\\odot ." ], [ "$\\alpha _{\\rm bind}\\lambda $ formalism", "In population synthesis models, a crucial parameter is $\\alpha _{\\rm bind}\\lambda $ , which can be found from the results of our simulations as follows (see also Equation REF ): $\\alpha _{\\rm bind}\\lambda =-\\frac{GM_{\\rm d,1}(M_{\\rm d,1}-M_{\\rm c,1})}{R_{\\rm rlof}\\Delta E_{\\rm orb}} \\ .$ Note that this quantity does not imply a separate consideration of the recombination energy as it is simply a fit to the standard energy formalism.", "Figure REF shows the behavior of $\\alpha _{\\rm bind}\\lambda $ in our models.", "Our best fit for $\\alpha _{\\rm bind}\\lambda $ with the assumed multi-linear regression model is: $\\alpha _{\\rm bind}\\lambda =0.92+0.55\\frac{M_{\\rm d,1}}{M_\\odot }-0.79\\frac{M_{2}}{M_\\odot }-1.19\\frac{M_{\\rm c,1}}{M_\\odot }.$ This equation accurately represents all our models, and the maximum deviation between this equation and any data point of 0.13 (1.2G0.32C0.4D), and a minimum deviation of 0.002 (1.8G0.36C0.40D).", "Figure: Values for α bind λ\\alpha _{\\rm bind}\\lambda with respect to the initial RG mass.", "The open circles correspond to simulations with M c,1 =0.32M ⊙ M_{\\rm c,1}=0.32M_\\odot , while the open triangles correspond to M c,1 =0.36M ⊙ M_{\\rm c,1}=0.36M_\\odot .", "The small, medium and big symbols are for companions with 0.32M ⊙ 0.32M_\\odot , 0.36M ⊙ 0.36M_\\odot , and 0.40M ⊙ 0.40M_\\odot , respectively." ], [ "Energy carried away by the ejecta ", "The total energy carried by the ejecta is not negligible, and is comparable, within an order of magnitude, to the initial binding energy of the RG star.", "Figure REF shows the ratio of the energy taken away by the unbound material to the released orbital energy, $\\alpha _{\\rm unb}^\\infty $ .", "It can be seen that $\\alpha _{\\rm unb}^\\infty $ decreases with the mass of the RG.", "A multilinear regression that uses all the points from the simulations (assuming that all the variables presented have linear trends with respect to each other) gives the following dependence: $\\alpha _{\\rm unb}^{\\infty }=-\\frac{E_{\\rm tot,unb}^\\infty }{\\Delta E_{\\rm orb}}=-0.16 - 0.30 \\frac{M_{\\rm d,1}}{M_\\odot }+0.49\\frac{M_{2}}{M_\\odot }+2.27\\frac{M_{\\rm c,1}}{M_\\odot }.$ We note that this equation fits all our models.", "The maximum deviation between this equation and any point is found to be 0.07 (1.2G0.32C0.40D), and the minimum is 0.005 (1.4G0.36C0.32D).", "Figure: Values for α unb ∞ \\alpha _{\\rm unb}^\\infty with respect to theinitial RG mass.", "The open circles correspond tosimulations with M core ,1 =0.32M ⊙ M_{\\rm core,1}=0.32M_\\odot , while the opentriangles correspond to M core ,1 =0.36M ⊙ M_{\\rm core,1}=0.36M_\\odot .", "The small,medium and big symbols are for companions with 0.32M ⊙ 0.32M_\\odot ,0.36M ⊙ 0.36M_\\odot , and 0.40M ⊙ 0.40M_\\odot , respectively.Table REF shows $\\alpha _{\\rm kin}^{\\infty }$ , which is defined as the ratio of the kinetic energy taken away by the ejecta to the released orbital energy.", "Figure REF shows a monotonic decrease of $\\alpha _{\\rm kin}^{\\infty }$ with the initial mass of the RG, very similarly to $\\alpha _{\\rm unb}^\\infty $ .", "The multilinear fitting equation takes the following form $\\alpha _{\\rm kin}^{\\infty }=-\\frac{E_{\\rm kin,unb}^\\infty }{\\Delta E_{\\rm orb}}=0.20-0.26 \\frac{M_{\\rm d,1}}{M_\\odot }+0.44\\frac{M_{2}}{M_\\odot }+0.92 \\frac{M_{\\rm c,1}}{M_\\odot },$ where this equation fits all the points presented in Table REF .", "The maximum deviation between this equation and any listed value in Table REF is 0.07 (1.4G0.32C0.32D), while the minimum deviation is 0.0002 (1.2G0.32C0.32D).", "Figure: Values for α kin ∞ \\alpha _{\\rm kin}^\\infty with respect to theinitial RG mass.", "The open circles correspond tosimulations with M c,1 =0.32M ⊙ M_{\\rm c,1}=0.32M_\\odot , while the opentriangles correspond to M c,1 =0.36M ⊙ M_{\\rm c,1}=0.36M_\\odot .", "The small,medium and big symbols are for companions with 0.32M ⊙ 0.32M_\\odot ,0.36M ⊙ 0.36M_\\odot , and 0.40M ⊙ 0.40M_\\odot , respectively.The potential energy of the ejecta, compared to the initial potential energy, is not really significant, as in all cases $\\alpha _{\\rm pot}\\lesssim 0.04$ .", "The thermal energy the ejecta still has at infinity, as compared to the initial thermal energy plus the recombination energy, is several times larger, albeit also limited to $\\alpha _{\\rm th}\\lesssim 0.16$ .", "The thermal energy of the ejecta is comparable to the kinetic energy of the ejecta, therefore, the internal energy still plays a role in supporting the ongoing expansion of the material even after all the material is unbound.", "Figure REF shows how the specific kinetic energy of the ejecta changes with the initial mass of the RG.", "We can see that overall this energy decreases as the RG mass increases, but no clear trend is observed.", "Figure: Values for E kin , unb ∞ /M unb E_{\\rm kin,unb}^\\infty /M_{\\rm unb} with respectto the initial RG mass.", "The open circlescorrespond to simulations with M c,1 =0.32M ⊙ M_{\\rm c,1}=0.32M_\\odot , while theopen triangles correspond to M c,1 =0.36M ⊙ M_{\\rm c,1}=0.36M_\\odot .", "The small,medium and big symbols are for companions with 0.32M ⊙ 0.32M_\\odot ,0.36M ⊙ 0.36M_\\odot , and 0.40M ⊙ 0.40M_\\odot , respectively.Figure REF shows how the sum of the potential and thermal specific energies of the ejecta changes with the mass of the RG.", "We cannot really see a trend, except that for RGs with the initial mass of $1.8\\, M_\\odot $ this quantity is higher than for the rest.", "Note that this quantity is always smaller than the specific kinetic energy of the ejecta.", "Figure: Values for (E int , unb ∞ +E pot , unb ∞ )/M unb (E_{\\rm int,unb}^\\infty +E_{\\rm pot,unb}^\\infty )/M_{\\rm unb} with respect to the initial RGmass.", "The open circles correspond to simulations withM c,1 =0.32M ⊙ M_{\\rm c,1}=0.32M_\\odot , while the open triangles correspond toM c,1 =0.36M ⊙ M_{\\rm c,1}=0.36M_\\odot .", "The small, medium and big symbols arefor companions with 0.32M ⊙ 0.32M_\\odot , 0.36M ⊙ 0.36M_\\odot , and0.40M ⊙ 0.40M_\\odot , respectively.To study the formation of a DWD binary via a CEE, we have simulated 25 three-dimensional hydrodynamical interactions between a low-mass RG and a WD companion.", "We considered for the initial masses of the low-mass RG star 1.2, 1.4, 1.6, or 1.8 $M_\\odot $ , with a He core of 0.32 or 0.36 $M_\\odot $ , and WD companions with masses 0.32, 0.36, or 0.40 $M_\\odot $ .", "We find that in all the cases, a DWD binary is formed, most of the envelope is ejected, and only a few SPH particles remain bound to the binary in some cases (the bound mass is less than 0.06% of the initial envelope mass).", "The envelope is ejected on the dynamical time-scale of the expanded envelope.", "Our results show that the standard energy formalism should be modified to take into account: (i) the energy that is taken away by the ejecta, as it is a substantial fraction of the released orbital energy, and (ii) the recombination energy, which plays a crucial role in ejection of the formed circumbinary envelope.", "The role of the recombination energy for the CEE with a low-mass RG donor is not that it is necessary for the overall energy budget, as none of the considered systems were expected to merge by the standard energy formalism, but because the recombination occurs exactly at the time when the shrunk binary is no longer capable of transfering its orbital energy to the expanded envelope.", "For future population synthesis studies, we provide three ways in which our results can be used.", "First of all, we provide a fitting formula (Equation REF ) that relates the final orbital period, the initial orbital period, the companion mass, the initial RG mass and the RG core.", "The RG radius and its core mass are coupled for each donor mass (these can be found using single stellar evolution tracks).", "The initial RG mass, its radius and the initial orbital period are also related [4].", "Therefore our fitting formula provides the relation between the observed parameters –the post-CE orbital period and the observed masses of both WDs – and the RG mass and radius before the CE.", "And vice versa, a population study could use this fitting formula to obtain the post-CE orbital period from known binary parameters at the start of a CE.", "Figure: Final orbital periods of sdB stars andtheir pre-CE conditions, as inferred fromEquation .", "The solid black line is the relationprovided by Equation .", "The crossinghorizontal lines show the positions for different initial donormasses, using observed older WD masses.", "Solid symbols are for RGdonors (ZAMS masses from 1.0 to 1.8 M ⊙ M_\\odot with an incrementof 0.1M ⊙ 0.1\\,M_\\odot , with the lowest mass on the right.)", "The RG donorsare selected when their cores are 0.47M ⊙ 0.47 M_\\odot .", "Open symbols arefor AGB donors (ZAMS masses 1.8, 1.9 and 2.0 M ⊙ M_\\odot , thesmallest on the right).", "The AGB donors are selected when their cores are0.53M ⊙ 0.53\\,M_\\odot ).Table: sdB binary predictionsLet us consider how this fitting formula can be used to interpret observed post-CE binaries.", "For example, take the WD 1101+364, which has observed parameters best matching the set of models we have calculated: $P_{\\rm orb,fin}=0.145$ day, $M_{\\rm 1}=0.31\\, M_\\odot $ and $M_{\\rm 2}=0.36\\, M_\\odot $ [18].", "To find the radius of the donor at the time when it had a core of the same mass as a younger WD in the observed sample, we used parameterized evolutionary tracks from [9].", "For WD 1101+364, the fitting formula predicts a pre-CE donor mass of $1.5~M_\\odot $ , and pre-CE orbital period of 33 days.", "We note that more detailed studies devoted to the simulations specifically of WD 1101+364 gave a similar pre-CE donor mass, $1.5~M_\\odot $ [20].", "We note that since the pre-CE radius is a strong function of the core mass, uncertainty in the mass of a younger WD leads to a large uncertainty on the pre-CE donor.", "E.g., if the mass is only a bit smaller, $M_{\\rm 1}=0.29\\,M_\\odot $ , and the companion's mass is $M_{\\rm 2}=0.33\\, M_\\odot $ (defined by the observed mass ratio of 0.87), then the donor would rather have an initial mass of $1.3M_\\odot $ and a pre-CE orbital period of 26 days.", "The second observable type of post-CE binaries for which we can test the fitting equation are hot subdwarf B stars (sdBs).", "These stars are hot core helium-burning stars with masses around $0.5 M_\\odot $ .", "We note that the post-CE remnants of this mass are beyond the set of our current simulations (modeling a CEE with a more evolved donor requires substantially more GPU time than is available at existing Compute/Calcul Canada facilities, and therefore is not feasible yet), but we will try to look at the post-CE binaries to see if we can place any constraint on their past.", "We use 12 sdB binaries for which [15] have found orbital periods, and inferred the minimum companion masses in these systems from the assumption of a canonical mass of $0.47\\,M_\\odot $ for the sdB stars.", "In Figure REF we show the predictions of our fitting formula.", "In addition to checking RG donors, we also took into account AGB donors.", "A prediction for a post-CE outcome for an AGB donor can not be fully trusted, as the fitting formula may not work well for them; both $\\lambda $ in the donor's envelope, and the fraction of the recombination energy in the total binding energy of the initial envelope, are not the same as in the case of a low-mass RG.", "However, it is important that at least half of the considered sdB binary systems can be better explained by a RG donor.", "Table REF summarizes the possible progenitors for each sdB binary.", "We note that we list the values for the closest point on the Figure REF , not the exact intersection between the model and the line with constant final orbital period; the donor mass for each case will not change much from the listed value in the Table, only the radius of the donor and its initial orbital period will change.", "We can see from Table REF or Figure REF that the evolutionary stage of the donor star can be associated with the final orbital period.", "For $P_{\\rm orb,fin}\\lesssim 1$ day, the donor star is likely a (relatively more massive) AGB star, while for $P_{\\rm orb,fin}\\gtrsim 1$ day, the donor star is more likely to be a (relatively less massive) RG.", "For a second way to use our results – as in the population synthesis studies that use $\\alpha _{\\rm bind}\\lambda $ -formalism to find the outcome of a CEE – we supply the parameterization that directly provides the $\\alpha _{\\rm bind}\\lambda $ value, as a function of the initial donor mass, its core mass and the companion mass.", "We note that in no case $\\alpha _{\\rm bind}\\lambda >1.3$ , and our maximum $\\alpha _{\\rm bind}<1.03$ (we remind that $\\alpha _{\\rm bind}$ more than 1 implies that an energy additional to the orbital energy was used, in our case it is the recombination energy).", "Some past population synthesis studies have considered $\\alpha _{\\rm bind}\\lambda =2$ [23] for all CEEs leading to DWD binary formation, but the results of our simulations do not confirm that such a very high value is plausible, at least in the case of CEEs with low-mass RG donors.", "And, finally, we give the preferred way to use our results, which is the most trusted method when one wants to extrapolate our result outside of the parameter space we considered.", "We advise population synthesis studies to use the energy conservation equation that accounts for all energy sinks and sources.", "In the energy conservation equation, all initially available recombination energy can be used as an energy source.", "(Note that this statement is not yet fully justified to extend our results for low-mass giants to the case of more massive or more evolved donors, and shall require further studies.)", "The ejected material can take away 20% to 40% of the released orbital energy, both as thermal energy and as kinetic energy, and this is an energy loss.", "It is this energy that powers those Luminous Red Novae which are produced by a CEE [11].", "For these energy losses we provided a fitting formula.", "Then the CEE outcomes can be found using the revised energy formalism as follows: $(E_{\\rm orb,ini}-E_{\\rm orb,fin}) (1-\\alpha _{\\rm unb}^{\\infty }) + E_{\\rm bind}+ \\eta (M_{\\rm d,1}- M_{\\rm c,1}) =0 \\ .$" ], [ "Acknowledgments", "JLAN acknowledges CONACyT for its support.", "NI thanks NSERC Discovery and Canada Research Chairs Program.", "The authors thank Craig Heinke for checking the English in the manuscript.", "This research has been enabled by the use of computing resources provided by WestGrid and Compute/Calcul Canada." ] ]
1606.04922
[ [ "The iridium double perovskite Sr2YIrO6 revisited: A combined structural\n and specific heat study" ], [ "Abstract Recently, the iridate double perovskite Sr$_2$YIrO$_6$ has attracted considerable attention due to the report of unexpected magnetism in this Ir$^{5+}$ (5d$^4$) material, in which according to the J$_{eff}$ model, a non-magnetic ground state is expected.", "However, in recent works on polycrystalline samples of the series Ba$_{2-x}$Sr$_x$YIrO$_6$ no indication of magnetic transitions have been found.", "We present a structural, magnetic and thermodynamic characterization of Sr$_2$YIrO$_6$ single crystals, with emphasis on the temperature and magnetic field dependence of the specific heat.", "Here, we demonstrate the clue role of single crystal X-ray diffraction on the structural characterization of the Sr$_2$YIrO$_6$ double perovskite crystals by reporting the detection of a $\\sqrt{2}a \\times \\sqrt{2}a \\times 1c$ supercell, where $a$, $b$ and $c$ are the unit cell dimensions of the reported monoclinic subcell.", "In agreement with the expected non-magnetic ground state of Ir$^{5+}$ (5d$^4$) in Sr$_2$YIrO$_6$, no magnetic transition is observed down to 430~mK.", "Moreover, our results suggest that the low temperature anomaly observed in the specific heat is not related to the onset of long-range magnetic order.", "Instead, it is identified as a Schottky anomaly caused by paramagnetic impurities present in the sample, of the order of $n \\sim 0.5(2)$ \\%.", "These impurities lead to non-negligible spin correlations, which nonetheless, are not associated with long-range magnetic ordering." ], [ "Introduction", "The effect of spin-orbit coupling (SOC) on strongly correlated electron systems has demonstrated to be the origin of novel phases [1], [2], like spin liquids [3], unconventional magnetism [4], [5], and topological phases [6], among others.", "Also, it has been demonstrated that for an intermediate strength regime, SOC can work together with Coulomb interactions to enhance the electron correlations, leading to spin-orbit driven Mott insulators [7], [8].", "The key point is that small changes in these interactions can lead to an enormous variety in the electronic and magnetic behaviors.", "Specifically in $5d$ transition metals like Ir the SOC is significant, and comparable to the atomic interactions like crystal electric field $\\Delta $ and the on-site Coulomb interaction $U$ .", "Examples for this scenario are the pyrochlore iridates, which show novel phases like Weyl semimetals [9], [10], [11], topological Mott insulators [12], [13], and axion insulators with unusual electromagnetic responses [14], [15].", "The most common oxidation states of iridium are 3+ and 4+.", "On the other hand oxidation states of 5+ and 6+ are rare and sometimes poorly characterized [16].", "The double perovskite structure R$_2$ MM$^{\\prime }$ O$_6$ may host iridium on the M and/or M$^{\\prime }$ position; Ir will be in octahedral coordination in this structure type.", "Hence, the octahedral crystal electric field splits the $5d$ levels in a $t_{2g}$ triplet and a $e_g$ doublet.", "Then, for Ir$^{4+}$ (5d$^5$ electronic configuration) the large SOC acts on the $t_{2g}$ levels, splitting them into an effective $j = 1/2$ doublet and an effective $j = 3/2$ quartet [17], [18].", "According to the strong spin-orbit coupling J$_{eff}$ model, a non-magnetic $j= 0$ ground state is expected for double perovskites with Ir$^{5+}$ (5d$^4$ ) as the formal oxidation state.", "That is the case for the monoclinic Sr$_2$ YIrO$_6$ , synthesized for the first time by Wakeshima et al.", "[19] in 1999 with the aim of studying the magnetic properties of $5d$ transition metal oxides in which the metal ions are in an unusual oxidation state.", "As expected, no transition into a long-range ordered state was observed.", "However, recently a report by Cao et al.", "on this system [20] reports that a strong non-cubic crystal field (due to the flattening of the IrO$_6$ octahedra) together with an “intermediate-strength” spin-orbit coupling, leads to a different ground-state configuration, i.e.", "an antiferromagnetic ground state withT$_N$ = 1.3 K. The emergence of magnetic ordering was evidenced in magnetization studies at 7 T as well as in specific heat measurements in different applied magnetic fields.", "For the latter characterization, a small anomaly in the low temperature region was interpreted as a signature of long-range magnetic ordering.", "Recent first-principles calculations [21] on both Sr$_2$ YIrO$_6$ and Ba$_2$ YIrO$_6$ compounds gave another explanation for this behavior.", "In Ref [21] the authors argue that the breakdown of the J = 0 state in Sr$_2$ YIrO$_6$ would be due to a band structure effect rather than the non-cubic crystal field effect.", "However, a posterior report by Pajskr et al.", "[22] combining several numerical and semianalytical methods to study the band structure of both compounds contradicts the conclusions of Refs.", "[20] and [21].", "They found that monoclinic Sr$_2$ YIrO$_6$ and cubic Ba$_2$ YIrO$_6$ are quite similar, and that both exhibit no tendency towards a magnetic phase transition at low temperatures.", "More recent works on polycrystalline samples of the series Ba$_{2-x}$ Sr$_x$ YIrO$_6$ [23], [24] did not observe any signature for long-range magnetic order in this material in their magnetic susceptibility characterization, in strong contrast with Cao et al [20].", "On the contrary, these reports reinforce the notion of a non-magnetic ground state dominated by strong SOC.", "In both reports [23], [24] the magnetic characteristics of the system do not change significantly across the series, and the effective magnetic moment per Ir is much less than the value expected for a S = 1 system as reported by Cao et al.", "[20], demonstrating a strongly SOC dominated ground state.", "In this work, in order to settle the dispute described above, we present a combined structural, magnetic and thermodynamic characterization of Sr$_2$ YIrO$_6$ single crystals.", "Structure and composition of the crystals were thoroughly characterized by single crystal X-ray diffraction (XRD), synchroton powder XRD, energy dispersive X-ray spectroscopy (EDX) and scanning electron microscopy (SEM).", "The single crystal XRD data show evidence of a cubic supercell.", "We particularly analyze the specific heat of this material, in order to study the anomaly reported in [20].", "In our studies no long-range magnetic order was found even in fields up to 9 T. The magnetic contribution to the specific heat was extracted, finding a Schottky anomaly due to a small amount of magnetic impurities.", "Further analysis suggests non-negligible spin correlations, which nonetheless, are not associated with long-range magnetic ordering.", "Single crystals of Sr$_2$ YIrO$_6$ were flux-grown using SrCO$_3$ (Alfa Aesar, 99.994%), Y$_2$ O$_3$ (Alfa Aesar, 99.99%) and IrO$_2$ (Alpha Aesar, 99.99%) as the precursor materials.", "Anhydrous SrCl$_2$ (Alpha Aesar, 99.5%) was used as the flux.", "The stoichiometric amount of starting materials and the flux were mixed using a nutrient to solvent weight ratio 1:13.", "All constituents were put in a platinum crucible (50 cubic ml) with a platinum lid.", "The crucible edge and lid were squeezed to semi-seal the crucible assembly.", "The set-up was heated to different high temperatures (1200$^{\\circ }$ C, 1250$^{\\circ }$ C and 1300$^{\\circ }$ C), held for 24 h and then cooled to 900$^{\\circ }$ C with different cooling rates (0.5$^{\\circ }$ C/h, 1$^{\\circ }$ C/h and 2$^{\\circ }$ C/h).", "The furnace was switched off afterwards and cooled to room temperature.", "The crystals were found at the bottom of the crucible and were separated by dissolving the flux in water.", "After that, in order to gain a better understanding of the growth, the Sr$_2$ YIrO$_6$ crystals were examined by optical microscopy.", "As depicted in Figure S1 [25], crystals have different morphologies: cubic-like or irregular polyhedral.", "The cubic ones correspond to the pure Sr$_2$ YIrO$_6$ phase and the irregular ones seem to be cubic crystals inter-grown with each other due to additional nucleation sites.", "All our attempts to avoid the co-crystallization of irregular shaped crystals by varying the hold temperatures, cooling rates and material to flux ratio have been failed.", "For the following studies, we used the samples grown by cooling down from 1200$^{\\circ }$ C to 900$^{\\circ }$ C with a rate of 2$^{\\circ }$ C/h and material to flux ratio of 1:13." ], [ "Characterization", "A bunch of single crystals of Sr$_2$ YIrO$_6$ with different morphologies were selected and mounted on the tip of a glass fiber for X-ray Diffraction (XRD).", "Intensity data were collected at room temperature using $\\omega $ scans on a STOE imaging plate diffraction system (IPDS-II) with graphite-monochromatized Mo K$\\alpha $ radiation ($\\lambda $ = 0.71069 Å) operating at 50 kV and 40 mA using a 34 cm diameter imaging plate.", "Individual frames were collected with a 4 min exposure time and a 0.8$^{\\circ } \\omega $ rotation.", "X-AREA, X-RED, and X-SHAPE software packages [26] were used for data collection, integration, and analytical absorption corrections, respectively.", "SHELXL [27] and JANA2006 [28] software packages were used to solve and refine the structure.", "High-resolution synchrotron powder diffraction data were collected at the beamline P02.1 at the storage ring PETRA III (DESY, Hamburg, Germany) using an average wavelength of 0.2067 Å ($\\sim $ 60 keV), with a relative energy bandwidths $\\Delta $ E/E of the order of 10$^{-4}$ .", "The repeatability of the detector translations along the beam was tested in a series of exposures of a LaB$_6$ standard (NIST 660a) filled into a 0.8 mm-diameter capillary.", "During this measurement, the detector was moved forward and backward (with and without backlash) over the entire travel range in steps of 125 mm [29].", "The synchrotron powder XRD pattern was collected on crushed single crystals of Sr$_2$ YIrO$_6$ sealed in a 0.8 mm-diameter capillary.", "Data were fitted by the Rietveld method [30] using Fullprof in the WinPlotR program package [31].", "The microstructural and compositional analysis was performed using scanning electron microscopy (SEM, Zeiss EVOMA15) along with an electron microprobe analyzer for semi-quantitative elemental analysis in the energy dispersive x-ray (EDX) mode (X-MaxN20 detector from Oxford Instruments with a AZtecEnergy Advanced acquisition and EDX analysis software).", "The magnetization as a function of temperature (in the range 0.43 - 300 K) and magnetic field was obtained for randomly oriented single crystals of Sr$_2$ YIrO$_6$ ($\\sim $ 92.5 mg) using a Superconducting Quantum Interference Device (SQUID) magnetometer from Quantum Design, equipped with an iHelium3 option.", "A thorough background subtraction was performed for all the curves.", "Specific heat measurements were performed on 29 single crystals($\\sim $ 5.6 mg) between 0.4 K and 10 K using a heat-pulse relaxation method in a Physical Properties Measurement System (PPMS) from Quantum Design.", "The heat capacity of the sample holder (addenda) was determined prior to the measurements for the purpose of separating the heat capacity contribution of the sample from the total heat capacity." ], [ " Results and discussion", "Sr$_2$ YIrO$_6$ orders in the double perovskite type structure with the general formula R$_2$ MM$^{\\prime }$ O$_6$ , wherein R denotes an alkaline-earth (or rare-earth) metal and M and M$^{\\prime }$ are d-block elements (or other metals).", "Double perovskites crystallize in a cubic, tetragonal, or monoclinic symmetry with interpenetrating M and M$^{\\prime }$ face-centered cubic (FCC) sublattices.", "We started the analysis of Sr$_2$ YIrO$_6$ single crystal XRD data using the reported monoclinic structural model [20], with the space group P2$_1$ /n and lattice parameters:$a$ = 5.7826(5) Å, b = 5.7830(5) Å, c = 8.1746(7) Å, $\\beta $ = 90.036(7)$^{\\circ }$ .", "The single crystal XRD refinement leads to the agreement factors of R$_{obs}$ = 4.17 % (for I $>2\\sigma $ (I)) andR$_{all}$ = 9.19 %.", "Figure: (Color online) Structural details forSr 2 _2YIrO 6 _6 crystals: (a) Perspective view of the doubleperovskite structure of the monoclinic subcell along thecrystallographic b axis.", "(b) Reconstructed precession photographsfor Sr 2 _2YIrO 6 _6 crystal in the hk0hk0 layers show extrareflections, marked with red circles, required to describe a2×2×1\\sqrt{2} \\times \\sqrt{2} \\times 1 supercell.", "(c) Perspectiveview of the double perovskite structure of the 2×2×1\\sqrt{2} \\times \\sqrt{2} \\times 1 cubic supercell along the crystallographic caxis.", "(d) Comparison between bond lengths and bond angles of theIrO 6 _6 and YO 6 _6 octahedra, monoclinic vs cubic.After an anisotropic refinement of all atoms, a difference electron density Fourier map calculated with phases based on the final parameters shows maximum and minimum residual peaks of +2.03 and -2.58 eÅ$^{-3}$ , respectively.", "A detailed examination of reconstructed precession images of the $hk0$ layer (Figure REF (b)) shows strong diffraction maxima due to the small subcell and additional weak peaks forming distinctive groups of four reflections around the allowed peaks.", "All those extra peaks can be indexed with the $\\sqrt{2}a \\times \\sqrt{2}b \\times 1c$ supercell, where $a$ , $b$ and $c$ are the unit cell dimensions of the monoclinic subcell.", "The integrated data with the $\\sqrt{2}a\\times \\sqrt{2}b \\times 1c$ supercell suggests that Sr$_2$ YIrO$_6$ crystallizes in the cubic space group $Fm\\overline{3}m$ with lattice parameter, $a$ = 8.1773(7) Å.", "The crystal structure of the supercell based on our refinement, shown in Figure REF (c), is built up of corner-sharing IrO$_6$ and YO$_6$ octahedra units, which feature an ordered rock-salt like arrangement, and the Sr placed in between.", "Please note that the Ir$^{5+}$ ions in this structure build up a FCC network, a general feature of the double perovskite type structure.", "The single crystal XRD refinement by taking into account the supercell shows excellent agreement factors of R$_{obs}$ = 1.65 % (for I $>2\\sigma $ (I)) and R$_{all}$ = 3.79 % and less electron density difference after an anisotropic refinement of all atoms, i.e.", "maximum and minimum residual peaks +1.06 and -0.98 eÅ$^{-3}$ , respectively.", "The parameters for data collection and the details of the structure refinement are given in Table REF .", "Atomic coordinates, thermal displacement parameters (U$_{eq}$ ) and occupancies of all atoms are given in Table REF .", "Anisotropic displacement parameters (ADPs) and selected bond lengths and angles are given in the supplemental material, Tables S1 and S2 [25].", "Table: Crystal data and structure refinement for Sr 2 _2YIrO 6 _6 (a)subcell and (b) supercell at 293 K.Table: Atomic coordinates (×10 4 \\times 10^4) and equivalent isotropic displacement parameters (Å×10 3 \\times 10^3) of Sr 2 _2YIrO 6 _6 (a) subcell and (b) supercell at293(2) K with estimated standard deviations in parentheses.In the Sr$_2$ YIrO$_6$ subcell, the monoclinic distortion is very small, as presented in Figure REF (d), with $\\beta $ angles very close to 90$^{\\circ }$ ($\\beta = 90.036(7)^{\\circ }$ ) and unequal Ir-O bond lengths with a small difference of  0.03 Å, while $\\angle $ O-Ir-O deviates from 90$^{\\circ }$ by 2.4$^{\\circ }$ , and the IrO$_6$ octahedra are rotated/tilted with $\\angle $ Ir-O-Y of 162.42$^{\\circ }$ .", "For our Sr$_2$ YIrO$_6$ supercell, the IrO$_6$ octahedra are completely regular with equal Ir-O bond lengths, $\\angle $ O-Ir-O= 90$^{\\circ }$ and $\\angle $ Ir-O-Y = 180$^{\\circ }$ .", "Our experimental results on single crystal XRD are in good agreement with the theoretical paper reported by Pajskr et al.", "[22], which claims that the non-cubic crystal field of Sr$_2$ YIrO$_6$ is found to be rather weak and that Sr$_2$ YIrO$_6$ is quite similar in terms of structure and low-energy properties to the cubic Ba$_2$ YIrO$_6$ analog [32].", "The results of our Rietveld refinement, agreement factors, and refined lattice constants of synchrotron powder XRD ($\\lambda $ = 0.2067 Å) patterns of the Sr$_2$ YIrO$_6$ sample studied in this work are shown in the supplemental material [25].", "Figure S2 shows the experimental synchrotron XRD data, the simulated and residual intensities as well as the corresponding Bragg positions using the monoclinic structure model.", "A tiny trace of unreactedY$_2$ O$_3$ ($\\sim $ 1.06 %) is observed.", "The analysis of the structural data reveals that Sr$_2$ YIrO$_6$ crystallizes in monoclinic P2$_1$ /n space group with lattice parameters: $a$ = 5.7870(7) Å,b = 5.7912(6) Å, c = 8.1805(1) Å, $\\beta $ = 90.214(6)$^{\\circ }$ and R$_f$ = 2.11, R$_{Bragg}$ = 2.01 and reduced $\\chi ^2$ = 2.32.", "The result of the Rietveld refinement of the synchrotron XRD data using the cubic structural model discovered by single crystal XRD is depicted in Figure S3.", "Please note that both the monoclinic and the cubic model describe the synchrotron powder XRD data reasonably well with a slightly better fit by the monoclinic model (Bragg R-factor 2.11 for the monoclinic vs 3.46 for the cubic one).", "However, this difference is not significant since the larger number of fit parameters for the monoclinic model will yield better fit results as such.", "Hence, is very difficult to decide which model, monoclinic or cubic, is correct using powder diffraction data only.", "We emphasize that single crystal diffraction data is required to find the cubic $\\sqrt{2}a \\times \\sqrt{2}b \\times 1c$ supercell.", "Figure REF illustrates an SEM image of the Sr$_2$ YIrO$_6$ single crystal, collected in backscattered electron (BSE) mode.", "EDX analysis was performed on various spots and on several crystals.", "The analysis suggests that the crystals are chemically homogeneous with a stoichiometry close to the target composition, within the accuracy of the EDX method.", "The magnetic susceptibility in an external magnetic field of $\\mu _0$ H = 0.5 T is shown in Fig.", "REF in the temperature range 1.8 -300 K, with no evidence of long-range magnetic order.", "The data obeys the Curie-Weiss law $\\chi (T) = \\chi _0 +C/(T-\\Theta )$ in the temperature range 20 - 300 K, which gives a temperature independent susceptibility contribution $\\chi _0 = 5.90\\times 10^{-4}$ emu/mol Oe, an effective magnetic moment $\\mu _{eff} = 0.21 \\mu _B$ /Ir, and a Curie-Weiss temperature$\\Theta = -2.8$ K. Due to the $5d^4$ electronic configuration of Sr$_2$ YIrO$_6$ , a Van Vleck contribution to the susceptibility is expected: $\\chi _0= \\chi _{Dia} + \\chi _{VV}$ , where $\\chi _{Dia}$ comes from the diamagnetic contribution of the core levels and $\\chi _{VV}$ is the Van Vleck paramagnetic susceptibility.", "The diamagnetic susceptibility can be obtained by adding the diamagnetic contribution of all the individual ions, leading to $\\chi _{Dia} =-7.03 \\times 10^{-5}$ emu/mol Oe, and finally $\\chi _{VV} = 6.6\\times 10^{-4}$ emu/mol Oe, which is in agreement with similar $5d^4$ compounds [32], [33].", "It is worth to notice that the obtained effective magnetic moment $\\mu _{eff} = 0.21 \\mu _B$ /Ir is much smaller than the value 2.38$\\mu _B$ /Ir expected for a conventional S = 1 5d-electron system as mentioned by Cao et al.", "[20], but it is in close agreement with the value $\\mu _{eff} = 0.16 \\mu _B$ /Ir reported for polycrystalline Sr$_2$ YIrO$_6$ [23].", "As will be shown below, the paramagnetic signal can be attributed to a small percentage of paramagnetic centers (less than 1%), so it is likely that the Curie-tail in the magnetic susceptibility stems from paramagnetic impurities in the Sr$_2$ YIrO$_6$ sample, only.", "Figure: (color online).", "Temperature dependence of themagnetic susceptibility in the low temperature region fordifferent magnetic fields (ZFC results).", "The inset shows a zoom ofthe 5 T data down to 0.43 K.In view of the antiferromagnetic transition at T$_N$ = 1.3 K in 7 T magnetic field which has been observed in Ref.", "[20], we performed further magnetization studies at low temperatures and at high magnetic fields.", "From our experimental data no magnetic transition is detected down to 0.43 K, even for applied magnetic fields up to 5 T (see Fig.REF ) [34].", "Figure: (color online).", "Isothermal magnetization curve of Sr 2 _2YIrO 6 _6 at T =0.48 K. The red line shows the fit according to the Brillouin function (for details see text).The magnetization of Sr$_2$ YIrO$_6$ as a function of applied magnetic field is shown in Fig.", "REF for T = 0.48 K. In agreement with the temperature dependent measurements, the magnetic behavior at lowest temperatures is predominantly paramagnetic, and the curve can be fitted with the modified Brillouin function: $M(H) = \\chi _0 H + nN_Ag\\mu _BJ\\left\\lbrace \\frac{2J+1}{2J}\\coth \\Bigg (\\frac{2J+1}{2J}\\frac{g\\mu _BJH}{k_BT}\\Bigg )-\\frac{1}{2J}\\coth \\Bigg (\\frac{1}{2J}\\frac{g\\mu _BJH}{k_BT}\\Bigg )\\right\\rbrace .$ In this expression $n$ represents a scaling factor which gives the percentage of paramagnetic Ir in the sample, $N_A$ is the Avogadro constant, $g$ the Landé factor, $\\mu _B$ the Bohr magneton, $J$ the total angular momentum, and $k_B$ the Boltzmann constant.", "From the fit the parameters g = 2.1, and $\\chi _0 = 9.9 \\times 10^{-4}$ emu/mol Oe are found, together with a low percentage of paramagnetic centers ($J = 1/2$ ) of $n \\sim 0.6$ %.", "These paramagnetic centers could be defects in the sample stemming from oxygen vacancies, chemical disorder (Y/Ir site mixing), off-stoichiometry, etc.", "A possible scenario could be the presence of Ir$^{4+}$ or Ir$^{6+}$ ions created by intermixing and/or off-stoichiometry [32], [35], [36].", "The temperature independent susceptibility value $\\chi _0$ is slightly larger than the value obtained from the Curie-Weiss fitting Fig.", "REF .", "This discrepancy could be due to non-negligible spin correlations at low temperature which are not included in Eq.", "REF , or to a temperature-dependent contribution to the Van Vleck susceptibility.", "It is worth to notice that no metamagnetic features arise in the field dependence of the magnetization in contrast to Ref.", "[20], where a sharp metamagnetic transition has been observed at the critical field H$_c\\approx $ 2.6 K at T = 0.5 K. Figure: (color online).", "Temperature dependence of the specific heat of Sr 2 _2YIrO 6 _6for different magnetic fields, shown in the same scale as .", "The inset shows a zoom into thelow-temperature region.The low temperature specific heat in different magnetic fields is shown in Fig.", "REF .", "A small anomaly is observed below T $\\sim $ 5 K in all the curves, including the zero-field data, and which becomes more pronounced in applied magnetic fields of 1, 3, and 5 T. This peak shifts to higher temperature and broadens with increasing magnetic fields, which in principle suggests its magnetic origin.", "However, its magnitude is small, and instead of being a sharp, $\\lambda $ -like anomaly, pointing towards a second-order phase transition, it resembles more a hump-like feature.", "This makes it difficult to identify the anomaly as a real magnetic phase transition.", "In Ref.", "[20], Cao et al.", "detect a similar anomaly for their Sr$_2$ YIrO$_6$ single crystals, in qualitatively and semi-quantitative agreement with the one observed here.", "In [20], the anomaly in the specific heat was interpreted as the onset of long-range magnetic order, due to its proximity to the ordering temperature T$_N$ = 1.3 K observed as a sharp peak in the magnetization as function of temperature.", "In order to settle this open question, we performed additional specific heat measurements on the non-magnetic cubic analog compound Ba$_2$ YIrO$_6$  [32].", "Since we observe striking similarities in our specific heat data for Sr$_2$ YIrO$_6$ and Ba$_2$ YIrO$_6$ [25], i.e.", "a broad anomaly below $T$ $\\sim $ 5 K, it is rather improbable that the specific heat anomaly in Sr$_2$ YIrO$_6$ marks a transition into a magnetically long-range ordered state.", "Figure: (color online).", "Zero-field specific heatof Sr 2 _2YIrO 6 _6.", "The red line represents the lattice specificheat obtained using a power-law fitting in the range of 6 K ≤\\le T ≤\\le 10 K marked by arrows; for details see text.To gain a deeper insight into the origin of this anomaly, the lattice contribution was modelled for 6 K $\\le $ T $\\le $ 10 K using the approximation $C_{p,lattice} = \\beta T^3 + \\delta T^5$ in order to disentangle possible magnetic contributions in Sr$_2$ YIrO$_6$ (Fig.", "REF ).", "The extracted magnetic contribution shows a broad anomaly.", "From the fit parameters $\\beta = 2.52\\times 10^{-4}$ J/mol K$^4$ and $\\delta = 4.36\\times 10^{-7}$ J/mol K$^6$ , the Debye temperature $\\Theta _D = 425.7$ K was calculated.", "This value is comparable with those obtained for other double-perovskite materials [37], [38] and iridium-based compounds [33], [39].", "Considering that the fitting range was small and chosen to follow the best-fitting procedure in combination with sufficient statistics, the extracted magnetic contribution to the specific heat has only a semi-quantitative character, i.e.", "the assumption $\\Delta C_{p,mag}$ = 0 for$T$ $\\ge $ 6 K has been applied.", "Thus, in order to model the magnetic contribution to $C_p$ in the following a subtraction of the zero-field data from the corresponding field data was performed, yielding the field-dependent magnetic specific heat.", "The subtracted data are shown in Fig.", "REF (a).", "The shape of the $\\Delta C_{p,mag}/T$ curves resembles a two-level Schottky anomaly, shifting to higher temperature with increasing magnetic field.", "Figure: (color online).", "(a) Temperaturedependence of the low temperature specific heat divided bytemperature for different magnetic fields, after subtraction ofthe zero-field data.", "(b) Experimental data from (a) representedwith a constant offset of 0.03 J/mol K 2 ^2 for clarity.", "The redlines represent the fits according to a two-level Schottky anomaly(see text).Consequently, the magnetic contribution shown in Fig.", "REF (a) was modelled as the subtraction of two Schottky anomalies, i.e.", "$C_{m}(T,H) = C_{Sch}(T,H) - C_{Sch}(T,0)$ .", "Following this reasoning, the peaks were fitted in the range 0.4 K $\\le $ T $\\le $ 6 K with the following expression: $C_{m}(T,H) = n\\Bigg [\\Bigg (\\frac{\\Delta (H)}{T}\\Bigg )^2\\frac{e^{\\Delta (H)/T}}{(1+e^{\\Delta (H)/T})^2} -\\Bigg (\\frac{\\Delta _0}{T}\\Bigg )^2\\frac{e^{\\Delta _0/T}}{(1+e^{\\Delta _0/T})^2}\\Bigg ],$ where $n$ is the concentration of paramagnetic impurities, $\\Delta _0$ is the energy separation between the two levels in zero magnetic field, and $\\Delta (H) = g\\mu _BH$ represents the Zeeman splitting in an applied magnetic field $H$ .", "The fits are shown in the Fig.", "REF (b).", "According to the fit, a value $\\Delta _0$ = 0.25 K was found for the gap at zero field together with $n \\approx 0.7$ %, indicating that the anomaly is only based on a small amount of paramagnetic impurities, instead of being due to a real long-range magnetic ordering of all iridium atoms in Sr$_2$ YIrO$_6$ .", "While the fits reasonably describe our data for H $\\le $ 5 T, a slightly different behavior is found for H $\\ge $ 7 T with a lacking peak at low temperatures in our experimental data.", "This suggests that even when a Schottky anomaly can explain the observed peaks for H $\\le $ 5 T (low fields), for higher fields the magnetic contribution is altered.", "Following the discussion about the Brillouin fitting in Fig.", "REF , this could point towards field-induced changes, possibly due to non-negligible spin correlations from correlated magnetic impurities which, nonetheless, are not associated with long-range magnetic ordering, but with short-range correlations only.", "The possibility of magnetic correlations due to impurities was already pointed out for polycrystalline Ba$_{2-x}$ Sr$_x$ YIrO$_6$ [23] and Ba$_2$ YIrO$_6$ single crystals [32].", "Figure: (color online).", "Magnetic field dependence ofthe energy gap between the ground state doublet for paramagneticimpurity spins in Sr 2 _2YIrO 6 _6.", "The red line and the gg valueare the fitting results using Δ=gμ B H\\Delta = g\\mu _BH.We emphasize the fact that both the paramagnetic susceptibility and the Schottky two-level anomaly can be ascribed to the presence of correlated impurity spins Ir$^{4+}$ or Ir$^{6+}$ .", "With this in mind, the Zeeman splitting between the two lowest energy levels obtained from the former fits can be plotted as a function of applied magnetic field (see Fig.", "REF ).", "From $\\Delta =g\\mu _BH$ the Landé $g$ factor was found to be $g$ = 2.1(4), which is consistent with the $g$ factor obtained from the Brillouin fit, $g$ = 2.1(0).", "Figure: (color online).", "Zero-field magneticspecific heat plotted as C p,mag _{p,mag}/T vs T (left scale) togetherwith the magnetic entropy (right scale).Finally, we can evaluate the magnetic entropy $S_{mag}$ for our samples by integrating the zero-field specific heat data from Fig.", "REF according to the expression: $S_{mag} = \\int _{0}^{T} \\!", "\\frac{C_{mag}}{T} \\, dT.$ The resulting curves for C$_{p,mag}$ /T and S$_{mag}$ (T) are shown in Fig.", "REF .", "The small value $S_{mag} = 0.021$ J/mol K is of the same order of magnitude as reported by Cao et al.", "[20]($\\sim 0.01$ J/ mol K).", "This non-zero value again points to the presence of paramagnetic impurities, which need to be taken into account.", "For a $S$ = 1/2 ground state of Sr$_2$ YIrO$_6$ as claimed in Ref.", "[20], a total entropy $S_{mag} = R\\ln (2J+1) = R\\ln (2) = 5.76$ J/mol K is expected, which is evidently far from the experimental observations.", "Nevertheless, considering the small and finite number of $n$ impurities in our samples with presumably $J = 1/2$ contributing to the magnetic entropy, we have $S_{mag} = 0.021$ J/mol K = $nR\\ln (2)$ , and therefore $n\\sim 0.4 \\%$ .", "Overall, our magnetic and thermodynamic characterization is consistent with a non-magnetic ground state of Sr$_2$ YIrO$_6$ with the presence of a small amount of correlated $J =1/2$ paramagnetic centers.", "From our results, it is clear that the low temperature specific heat anomalies are not related with the onset of long-range magnetic order." ], [ " Conclusions", "Following a recent matter of debate as to the evolution of magnetism in monoclinic Sr$_2$ YIrO$_6$ , we have grown single crystals of Sr$_2$ YIrO$_6$ using anhydrous SrCl$_2$ flux.", "Our single crystal XRD results shows for the first time the presence of a $\\sqrt{2}a \\times \\sqrt{2}a \\times 1c$ supercell, where $a$ , $b$ and $c$ are the unit cell dimensions of the monoclinic subcell, highlighting the cubic supercell of this compound.", "The magnetic susceptibility revealed a predominantly paramagnetic behavior in the temperature range 0.4 K $<$ T $\\le $ 300 K with an effective magnetic moment $\\mu _{eff} = 0.21 \\mu _B$ /Ir and non-negligible but small spin correlations.", "From the Brillouin fit for the isothermal magnetization curve at T = 0.48 K a low percentage of $J = 1/2$ impurities were identified as the source for the observed paramagnetism.", "No long-range magnetic ordering was observed down to 430 mK.", "A thorough study of the specific heat was carried out, in which a broad anomaly in the low temperature region T $\\le $ 5 K was identified as a Schottky anomaly caused by paramagnetic impurities present in the sample, which are of the same order of magnitude ($n \\sim 0.4 - 0.7$ %) as obtained from magnetization measurements.", "Our results are in strong contrast with the reported ones by Cao et al.", "[20], but in full agreement with recent reports on polycrystalline samples [23], [24], pointing towards a non-magnetic ground state with the presence of a small amount of correlated $J = 1/2$ impurities." ], [ "Acknowledgments", "The authors would like to thank M. Vojta, D. Efremov, J. van den Brink and J. Wosnitza for fruitful discussions.", "This work has been supported by the Deutsche Forschungsgemeinschaft DFG under SFB 1143 and the Emmy-Noether program (Grant No.", "WU595/3-3)." ] ]
1606.05104
[ [ "Bias-Reduction in Variational Regularization" ], [ "Abstract The aim of this paper is to introduce and study a two-step debiasing method for variational regularization.", "After solving the standard variational problem, the key idea is to add a consecutive debiasing step minimizing the data fidelity on an appropriate set, the so-called model manifold.", "The latter is defined by Bregman distances or infimal convolutions thereof, using the (uniquely defined) subgradient appearing in the optimality condition of the variational method.", "For particular settings, such as anisotropic $\\ell^1$ and TV-type regularization, previously used debiasing techniques are shown to be special cases.", "The proposed approach is however easily applicable to a wider range of regularizations.", "The two-step debiasing is shown to be well-defined and to optimally reduce bias in a certain setting.", "In addition to visual and PSNR-based evaluations, different notions of bias and variance decompositions are investigated in numerical studies.", "The improvements offered by the proposed scheme are demonstrated and its performance is shown to be comparable to optimal results obtained with Bregman iterations." ], [ "Introduction", "Variational regularization methods with nonquadratic functionals such as total variation or $\\ell ^1$ -norms have evolved to a standard tool in inverse problems [10], [34], image processing [13], compressed sensing [12], and recently related fields such as learning theory [15].", "The popularity of such approaches stems from superior structural properties compared to other regularization approaches.", "$\\ell ^1$ -regularization for example leads to sparse solutions with very accurate or even exact reconstruction of the support of the true solution.", "On the other hand it is known that such methods suffer from a certain bias due to the necessary increased weighting of the regularization term with increasing noise.", "Two well-known examples are the loss of contrast in total variation regularization [10], [27] or shrinked peak values in $\\ell ^1$ -regularization.", "Accordingly, quantitative values of the solutions have to be taken with care.", "Several approaches to reduce or eliminate the bias of regularization methods have been considered in literature: For $\\ell ^1$ -regularization and similar sparsity-enforcing techniques an ad-hoc approach is to determine the support of the solution by the standard variational methods in a first step, then use a second debiasing step that minimizes the residual (or a general data fidelity) restricted to that support, also known as refitting [20], [22], [23].", "A slightly more advanced approach consists in adding a sign-constraint derived from the solution of the variational regularization method in addition to the support condition.", "This means effectively that the solution of the debiasing step shares an $\\ell ^1$ -subgradient with the solution of the variational regularization method.", "A different and more general approach is to iteratively reduce the bias via Bregman iterations [27] or similar approaches [7], [35].", "Recent results for the inverse scale space method in the case of $\\ell ^1$ -regularization (respectively certain polyhedral regularization functionals [8], [25], [5]) show that the inverse scale space performs some kind of debiasing.", "Even more, under certain conditions, the variational regularization method and the inverse scale space method provide the same subgradient at corresponding settings of the regularization parameters [6].", "Together with a characterization of the solution of the inverse scale space method as a minimizer of the residual on the set of elements with the same subgradient, this implies a surprising equivalence to the approach of performing a debiasing step with sign-constraints.", "Recently, bias and debiasing in image processing problems were discussed in a more systematic way by Deledalle et al.", "[16], [17].", "They distinguish two different types of bias, namely method bias and model bias.", "In particular they suggest a debiasing scheme to reduce the former, which can be applied to some polyhedral one-homogeneous regularizations.", "The key idea of their approach is the definition of suitable spaces, called model subspaces, on which the method bias is minimized.", "The remaining model bias is considered as the unavoidable part of the bias, linked to the choice of regularization and hence the solution space of the variational method.", "The most popular example is the staircasing effect that occurs for total variation regularization due to the assumption of a piecewise constant solution.", "In the setting of $\\ell ^1$ -regularization a natural model subspace is the set of signals with a given support, which yields consistency with the ad-hoc debiasing approach mentioned above.", "Based on this observation, the main motivation of this paper is to further develop the approach in the setting of variational regularization and unify it with the above-mentioned ideas of debiasing for $\\ell ^1$ -regularization, Bregman iterations, and inverse scale space methods.", "Let us fix the basic notations and give a more detailed discussion of the main idea.", "Given a bounded linear operator $A \\colon \\mathcal {X}\\rightarrow \\mathcal {Y}$ between Banach spaces, a convex regularization functional $J \\colon \\mathcal {X}\\rightarrow \\mathbb {R} \\cup \\lbrace \\infty \\rbrace $ and a differentiable data fidelity $H: \\mathcal {Y}\\times \\mathcal {Y}\\rightarrow \\mathbb {R}$ , we consider the solution of the variational method $ u_\\alpha \\in \\arg \\min _{u \\in \\mathcal {X}} \\ H(Au,f) + \\alpha J(u)$ as a first step.", "Here $\\alpha > 0$ is a suitably chosen regularization parameter.", "This problem has a systematic bias, as we further elaborate on below.", "The optimality condition is given by $ A^* \\partial _{u} H (Au_\\alpha ,f) + \\alpha p_\\alpha = 0,\\ p_\\alpha \\in \\partial J(u_\\alpha ),$ where $\\partial _{u} H$ is the derivative of $H$ with respect to the first argument.", "Now we proceed to a second step, where we only keep the subgradient $p_\\alpha $ and minimize $ \\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathcal {X}} \\ H(Au,f) \\text{ s.t. }", "p_\\alpha \\in \\partial J(u).$ Obviously, this problem is only of interest if there is no one-to-one relation between subgradients and primal values $u$ , otherwise we always obtain $\\hat{u}_{\\alpha }=u_\\alpha $ .", "The most interesting case with respect to applications is the one of $J$ being absolutely one-homogeneous, i.e.", "$J(\\lambda u) = |\\lambda | J(u)$ for all $\\lambda \\in \\mathbb {R}$ , where the subdifferential can be multivalued at least at $u=0$ .", "The debiasing step can be reformulated in an equivalent way as $ \\min _{u \\in \\mathcal {X}} \\ H(Au,f) \\text{ s.t. }", "{D}_J^{p_\\alpha }(u,u_\\alpha ) = 0,$ with the (generalized) Bregman distance given by ${D}_J^{p}(u,v) = J(u)-J(v)-\\langle p, u-v \\rangle , \\quad p \\in \\partial J(v).$ We remark that for absolutely one-homogeneous $J$ this simplifies to ${D}_J^{p}(u,v) = J(u)-\\langle p, u\\rangle , \\quad p \\in \\partial J(v).$ The reformulation in terms of a Bregman distance indicates a first connection to Bregman iterations, which we make more precise in the sequel of the paper.", "Summing up, we examine the following two-step method: Compute the (biased) solution $u_\\alpha $ of (REF ) with optimality condition (REF ), Compute the (debiased) solution $\\hat{u}_{\\alpha }$ as the minimizer of (REF ) or equivalently (REF ).", "In order to relate further to the previous approaches of debiasing $\\ell ^1$ -minimizers given only the support and not the sign, as well as the approach with linear model subspaces, we consider another debiasing approach being blind against the sign.", "The natural generalization in the case of an absolutely one-homogeneous functional $J$ is to replace the second step by $ \\min _{u \\in \\mathcal {X}} \\ H(Au,f) \\text{ s.t. }", "\\mathrm {ICB}_J^{p_{\\alpha }}(u,u_\\alpha )= 0,$ where $\\mathrm {ICB}_J^{p_{\\alpha }}(u,u_\\alpha ) := \\big [J^{p_\\alpha }(\\cdot ,u_\\alpha ) \\Box J^{\\text{-}p_\\alpha }(\\cdot ,-u_\\alpha )\\Big ](u)$ denotes the infimal convolution between the Bregman distances $J^{p_\\alpha }(\\cdot ,u_\\alpha )$ and $J^{-p_\\alpha }(\\cdot ,-u_\\alpha )$ , evaluated at $u \\in \\mathcal {X}$ .", "The infimal convolution of two functionals $F$ and $G$ on a Banach space $\\mathcal {X}$ is defined as $(F \\Box G)(u) &= \\inf _{\\begin{array}{c}\\phi , \\psi \\in \\mathcal {X},\\\\ \\phi + \\psi = u\\end{array}} F(\\phi ) + G(\\psi ) \\\\&= \\inf _{z\\in \\mathcal {X}} F(u-z) + G(z).$ For the sake of simplicity we carry out all analysis and numerical experiments in this paper for a least-squares data fidelity (related to i.i.d.", "additive Gaussian noise) $ H(Au,f) = \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2$ for some Hilbert space $\\mathcal {Y}$ , but the basic idea does not seem to change for other data fidelities and noise models.", "We show that the sets characterized by the constraints $D_J^{p_\\alpha }(u,u_\\alpha ) = 0 \\quad \\text{ and } \\quad \\mathrm {ICB}_J^{p_{\\alpha }}(u,u_\\alpha )= 0$ constitute a suitable extension of the model subspaces introduced in [16] to general variational regularization.", "In particular, we use those manifolds to provide a theoretical basis to define the bias of variational methods and investigate the above approach as a method to reduce it.", "Moreover, we discuss its relation to the statistical intuition of bias.", "At this point it is important to notice that choosing a smaller regularization parameter will also decrease bias, but on the other hand strongly increase variance.", "The best we can thus achieve is to reduce the bias at fixed $\\alpha $ by the two-step scheme while introducing only a small amount of variance.", "The remainder of the paper is organized as follows: In Section we motivate our approach by considering bias related to the well-known ROF-model [31] and we review a recent approach on debiasing [16].", "In the next section we introduce our debiasing technique supplemented by some first results.", "Starting with a discussion of the classical definition of bias in statistics, we consider a deterministic characterization of bias in Section .", "We reintroduce the notion of model and method bias as well as model subspaces as proposed in [16] and extend it to the infinite-dimensional variational setting.", "We furthermore draw an experimental comparison between the bias we consider in this paper and the statistical notion of bias.", "Finally, we comment on the relation of the proposed debiasing to Bregman iterations [27] and inverse scale space methods [32], [7].", "We complete the paper with a description of the numerical implementation via a first-order primal-dual method and show numerical results for signal deconvolution and image denoising." ], [ "Motivation", "Let us start with an intuitive approach to bias and debiasing in order to further motivate our method.", "To do so, we recall a standard example for denoising, namely the well-known ROF-model [31], and we rewrite a recent debiasing approach [16] in the setting of our method." ], [ "Bias of total variation regularization", "As already mentioned in the introduction, variational regularization methods suffer from a certain bias.", "This systematic error becomes apparent when the regularization parameter is increased.", "Indeed this causes a shift of the overall energy towards the regularizer, and hence a deviation of the reconstruction from the data in terms of quantitative values.", "Intuitively, this can be observed from the discrete version of the classical ROF-model [31], i.e.", "$u_\\alpha \\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2} \\Vert u - f \\Vert _2^2 + \\alpha \\Vert \\Gamma u\\Vert _1,$ with a discrete gradient operator $\\Gamma \\in \\mathbb {R}^{m \\times n}$ .", "It yields a piecewise constant signal $u_\\alpha $ reconstructed from an observation $f \\in \\mathbb {R}^n$ , which has been corrupted by Gaussian noise (see Figure REF (a)).", "Figure: Illustration of the bias of the ROF model on a 1D signal.", "(a) Original signal, and noisy signal corrupted by additive Gaussian noise.", "(b) Restoration of the noisy signal with TV regularization and Bregman iterations.The TV reconstruction recovers the structure of the signal but suffers from a loss of contrast, which is however well recovered with Bregman iterations.Figure REF (b) shows the solution of (REF ) together with the true, noiseless signal we aimed to reconstruct.", "Even though the structure of the true signal is recovered, the quantitative values of the reconstruction do not match the true signal.", "Instead, jumps in the signal have a smaller height, which is often referred to as a loss of contrast.", "Without any further definition, one could intuitively consider this effect as the bias (or one part of the bias) of the ROF model.", "Hence, the goal of a bias reduction method would be to restore the proper signal height while keeping the (regularized) structure.", "It has been shown in [27], [1] that this can be achieved by the use of Bregman iterations, i.e.", "by iteratively calculating $u_\\alpha ^{k+1} \\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2} \\Vert u - f \\Vert _2^2 + \\alpha D_{J}^{p_\\alpha ^k}(u,u_\\alpha ^k),$ where in our case $J(u) = \\Vert \\Gamma u\\Vert _1$ , and $p_\\alpha ^k \\in \\partial J(u_\\alpha ^k)$ is a subgradient of the last iterate $u_\\alpha ^k$ .", "Since for total variation regularization the subgradient $p_\\alpha ^k$ essentially encodes the edge information of the last iterate, its iterative inclusion allows to keep edges while restoring the correct height of jumps across edges.", "We further elaborate on that in Section .", "Indeed, the reconstruction via (REF ) in Figure REF (b) shows an almost perfect recovery of the true signal even in terms of quantitative values.", "This indicates that Bregman iterations are able to reduce or even eliminate our heuristically defined bias.", "However, a theoretical basis and justification is still missing, i.e.", "a proper definition of the bias of variational methods, a proof that Bregman iterations indeed reduce the bias in that sense, and in particular a link to the statistical definition and understanding of bias.", "With this paper we aim to define a proper basis for this link, and in particular further establish the connection between bias reduction techniques and Bregman distances." ], [ "Recent debiasing and Bregman distances", "In order to further motivate the use of Bregman distances for bias reduction let us recall and review a very recent approach on debiasing and work out its relation to Bregman distances.", "In [16], Deledalle et al.", "introduce a debiasing algorithm for anisotropic TV-type regularized problems $u_\\alpha \\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2} \\Vert Au - f \\Vert _2^2 + \\alpha \\Vert \\Gamma u\\Vert _1,$ with a linear operator $A \\in \\mathbb {R}^{n \\times d}$ , a discrete gradient operator $\\Gamma \\in \\mathbb {R}^{n \\times m}$ and noisy data $f \\in \\mathbb {R}^d$ .", "In [16] the authors argued that the loss of contrast characteristic for this kind of regularization is indeed bias in their sense.", "In order to correct for that error, the proposed debiasing method in [16] consists in looking for a debiased solution $\\hat{u}_{\\alpha }$ such that $\\Gamma \\hat{u}_{\\alpha }$ and $\\Gamma u_\\alpha $ share the same support, but $\\hat{u}_{\\alpha }$ features the right intensities.", "Mathematically, the solution $\\hat{u}_{\\alpha }$ of their debiasing problem is given by $\\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\sup _{z \\in F_{\\cal I}} \\ \\frac{1}{2} \\Vert Au - f\\Vert _2^2 + \\langle \\Gamma u, z \\rangle ,$ where $F_{\\cal I}=\\lbrace z \\in \\mathbb {R}^m ~|~z_{\\cal I} = 0 \\rbrace $ , and ${\\cal I}$ is the set of indices corresponding to nonzero entries of $\\Gamma u_\\alpha $ .", "We can explicitly compute the supremum (the convex conjugate of the indicator function of the set $F_{\\cal I}$ ), which is $\\sup _{z \\in F_{\\cal I}} \\langle \\Gamma u, z \\rangle = {\\left\\lbrace \\begin{array}{ll}\\infty , &(\\Gamma u)_i \\ne 0 \\text{\\small { for some }} i \\notin {\\cal I}, \\\\0, & \\text{ else.}\\end{array}\\right.", "}$ Hence, $\\hat{u}_{\\alpha }$ can only be a minimizer of (REF ) if $\\mathrm {supp}(\\Gamma \\hat{u}_{\\alpha }) \\subset \\mathrm {supp}(\\Gamma u_\\alpha )$ , thus $\\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\ &\\frac{1}{2} \\Vert Au - f\\Vert _2^2 \\nonumber \\\\\\text{ s.t. }", "&\\mathrm {supp}(\\Gamma \\hat{u}_{\\alpha }) \\subset \\mathrm {supp}(\\Gamma u_\\alpha ).$ We can also enforce this support condition using the infimal convolution of two $\\ell ^1$ -Bregman distances.", "Defining $J(u) = \\Vert \\Gamma u\\Vert _1$ , the subdifferential of $J$ at $u_\\alpha $ is given by $\\partial J(u_\\alpha ) &= \\lbrace \\Gamma ^T q_\\alpha \\in \\mathbb {R}^n ~|~ \\Vert q_\\alpha \\Vert _\\infty \\le 1,\\\\(q_\\alpha )_i &= \\text{sign}((\\Gamma u_\\alpha )_i) \\text{ for } (\\Gamma u_\\alpha )_i \\ne 0 \\rbrace .$ In particular $|(q_\\alpha )_i| = 1$ on the support of $\\Gamma u_\\alpha $ .", "Let $q_\\alpha $ be such a subgradient and consider the $\\ell ^1$ -Bregman distances $D_{\\Vert \\cdot \\Vert _1}^{q_\\alpha }(\\cdot , \\Gamma u_\\alpha )$ and $D_{\\Vert \\cdot \\Vert _1}^{-q_\\alpha }(\\cdot , -\\Gamma u_\\alpha )$ .", "According to [24], their infimal convolution evaluated at $\\Gamma u$ is given by: $&\\quad \\mathrm {ICB}_{\\Vert \\cdot \\Vert _1}^{q_{\\alpha }}(\\Gamma u,\\Gamma u_\\alpha ) \\\\&= [ D_{\\Vert \\cdot \\Vert _1}^{q_\\alpha }(\\cdot , \\Gamma u_\\alpha ) \\Box D_{\\Vert \\cdot \\Vert _1}^{-q_\\alpha }(\\cdot , -\\Gamma u_\\alpha )] (\\Gamma u) \\\\&= \\sum _{i=1}^m (1-|(q_{\\alpha })_i|)|(\\Gamma u)_i|.$ We observe that this sum can only be zero if $|(q_\\alpha )_i| = 1$ or $(\\Gamma u)_i = 0$ for all $i$ .", "Assuming that a qualification condition holds, i.e.", "$p_\\alpha = \\Gamma ^T q_\\alpha \\in \\partial J(u_\\alpha )$ with $| (q_\\alpha )_i | < 1 $ for $i \\notin {\\cal I}$ , i.e.", "$| (q_\\alpha )_i | = 1 \\Leftrightarrow (\\Gamma u_\\alpha )_i \\ne 0$ , we can rewrite the above debiasing method (REF ) as $\\min _{u \\in \\mathbb {R}^n} \\ \\frac{1}{2} \\Vert Au - f\\Vert _2^2 \\text{ s.t.", "}\\mathrm {ICB}_{\\Vert \\cdot \\Vert _1}^{q_{\\alpha }}(\\Gamma u,\\Gamma u_\\alpha ) = 0.$ Note that the zero infimal convolution exactly enforces the support condition (REF ) only if $| (q_\\alpha )_i | < 1$ for all $i \\in {\\cal I}$ .", "Intuitively, since the subdifferential is multivalued at $(\\Gamma u_\\alpha )_i = 0$ , this leads to the question of how to choose $q_\\alpha $ properly.", "However, our method does not depend on the choice of a particular $q_\\alpha $ , but instead we use a unique subgradient $p_\\alpha $ coming from the optimality condition of the problem.", "We further comment on this in Section .", "Figure: TV denoising of a one-dimensional noisy signal and debiasing using the proposed approach with zero Bregman distance.Figure: Denoising of a cartoon image.", "First row: original image, noisy image corrupted by Gaussian noise.Second row: TV reconstruction and debiasing using the Bregman distance and its infimal convolution, respectively.The TV reconstruction recovers well the structures of the images but suffers from a loss of contrast, while the debiased solutionsallow for a more accurate dynamic.", "1 ^1 The color image is provided in order to point out that it is indeed a giraffe and not a cow." ], [ "Debiasing", "Inspired by the above observations, let us define the following two-step-method for variational regularization on Banach spaces.", "At first we compute a solution $u_\\alpha $ of the standard variational method $\\begin{alignedat}{5}&\\text{1)} \\; \\; \\; && u_\\alpha &&\\in \\arg \\min _{u \\in \\mathcal {X}} \\ \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2 + \\alpha J(u),\\end{alignedat}$ where $A \\colon \\mathcal {X}\\rightarrow \\mathcal {Y}$ is a linear and bounded operator mapping from a Banach space $\\mathcal {X}$ to a Hilbert space $\\mathcal {Y}$ , $J \\colon \\mathcal {X}\\rightarrow \\mathbb {R} \\cup \\lbrace \\infty \\rbrace $ denotes a convex and one-homogeneous regularization functional and $f \\in \\mathcal {Y}$ .", "We point out that in the following, we will always make the standard identification $\\mathcal {Y}^* = \\mathcal {Y}$ without further notice.", "The first-order optimality condition of (REF ) reads: $p_\\alpha = \\frac{1}{\\alpha } A^* (f - Au_\\alpha ), \\ p_\\alpha \\in \\partial J(u_\\alpha ),$ and it is easy to show that this $p_\\alpha $ is unique (cf.", "Section REF , Thm.", "REF ).", "We use this subgradient to carry over information about $u_\\alpha $ to a second step.", "In the spirit of the previous paragraph the idea is to perform a constrained minimization of the data fidelity term only: $\\begin{alignedat}{4}\\begin{split}&\\text{2 a)} && \\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathcal {X}} \\; \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2\\\\& \\; && \\; \\qquad \\text{ s.t. }", "\\mathrm {ICB}_J^{p_{\\alpha }}(u,u_\\alpha )= 0.\\end{split}\\end{alignedat}$ If we reconsider the ad-hoc idea of $\\ell ^1$ or TV-type debiasing from the introduction, it can be beneficial to add a sign or direction constraint to the minimization, rather than a support condition only.", "This can be achieved by the use of a single Bregman distance.", "Hence it is self-evident to define the following alternative second step: $\\begin{alignedat}{4}\\begin{split}&\\text{2 b)} && \\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathcal {X}} \\; \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2\\\\& \\; && \\; \\qquad \\text{ s.t. }", "{D}^{p_\\alpha }_J(u,u_\\alpha )= 0.\\end{split}\\end{alignedat}$ We would like to point out that until now we only argued heuristically that the above method actually performs some kind of debiasing for specific problems.", "But since we are able to recover the debiasing method of [16] for $J(u) = \\Vert \\Gamma u\\Vert _1$ as a special case, at least for this specific choice of regularization (and a finite-dimensional setting) our method is provably a debiasing in their sense.", "However, our method is much more general.", "Since in contrast to [16] it does not depend on a specific representation of $u_\\alpha $ , it can theoretically be carried out for any suitable regularizer $J$ .", "In particular, the method does not even depend on the specific choice of the data term.", "In order to obtain a unique subgradient $p_\\alpha $ from the optimality condition it is desirable e.g.", "to have a differentiable data fidelity, but if we drop that condition, the data term is theoretically arbitrary.", "Since this generalization requires more technicalities, we focus on a squared Hilbert space norm in this paper in order to work out the basics of the approach.", "Before we actually lay a theoretical foundation for our framework and prove that our method indeed is a debiasing method, we show some motivating numerical results and prove the well-definedness of the method." ], [ "A first illustration", "To give a first glimpse of the proposed method, we revisit the ROF-reconstruction model (REF ) from Section and show some numerical results in one and two dimensions.", "Taking the subgradient $p_\\alpha $ of the TV reconstruction $u_\\alpha $ of the one-dimensional signal and performing our debiasing method, we obtain the results in Figure REF .", "The second step restores the right height of the jumps and yields the same result as the Bregman iterations we performed in Section .", "As a second example we perform denoising on a cartoon image corrupted by Gaussian noise.", "The first row of Figure REF shows the original image and its noisy version.", "The left image in the second row is the denoising result obtained with the ROF-model (REF ).", "We observe that noise has been reduced substantially, but some part of the contrast is lost.", "The second step of our method restores the contrast while keeping the structure of the first solution, yielding the two results depicted in the middle and on the right of the second row." ], [ "Well-definedness of the method", "The aim of this section is to show that the method defined above is well-defined, i.e.", "that there always exists at least one solution to the problem.", "We fix the setup by restricting ourselves to conditions ensuring that the original variational problem (REF ) with quadratic data fidelity has a solution.", "The following result can be established by standard arguments: Theorem 3.1 Let $\\mathcal {Y}$ be a Hilbert space, $\\mathcal {X}$ be the dual space of some Banach space $\\mathcal {Z}$ , such that the weak-star convergence in $\\mathcal {X}$ is metrizable on bounded sets.", "Moreover, let $A: \\mathcal {X}\\rightarrow \\mathcal {Y}$ be the adjoint of a bounded linear operator $B: \\mathcal {Y}\\rightarrow \\mathcal {Z}$ , $J$ be the convex conjugate of a proper functional on the predual space $\\mathcal {Z}$ , and let the map $u \\mapsto \\frac{1}{2}\\Vert A u \\Vert _\\mathcal {Y}^2 + J(u)$ be coercive in $\\mathcal {X}$ .", "Then the variational problem (REF ) with data-fidelity (REF ) has a minimizer $u_\\alpha \\in \\mathcal {X}$ and there exists a subgradient $p_\\alpha \\in \\partial J(u_\\alpha )$ such that the optimality condition $p_\\alpha = \\frac{1}{\\alpha }A^* (f - Au_\\alpha ) = \\frac{1}{\\alpha }B(f-Au_\\alpha )$ holds.", "Moreover, if $u_\\alpha \\ne \\tilde{u}_\\alpha $ are two minimizers, then $A u_\\alpha = A\\tilde{u}_\\alpha $ and the corresponding subgradient is unique, i.e., $p_\\alpha = \\frac{1}{\\alpha }B (f - Au_\\alpha ) = \\frac{1}{\\alpha }B ( f- A\\tilde{u}_\\alpha ) = \\tilde{p}_\\alpha .$ Since the functional $J$ is proper, there exists a nonempty sublevel set of the functional $u \\mapsto \\frac{1}{2} \\Vert A u -f \\Vert _\\mathcal {Y}^2 + \\alpha J(u)$ , and by the coercivity assumption this sublevel set is bounded.", "The Banach-Alaoglu theorem now implies precompactness of the sublevel set in the weak-star topology.", "Since the latter is metrizable on bounded sets, it suffices to show that the objective functional is sequentially weak-star lower semicontinuous in order to obtain existence of a minimizer.", "For the regularization functional $J$ , this follows from a standard argument for convex conjugates of proper functionals along the lines of [18].", "The assumption $A=B^*$ guarantees further that $A$ is continuous from the weak-star topology in $\\mathcal {X}$ to the weak topology of $\\mathcal {Y}$ and the weak lower semicontinuity of the norm also implies the weak-star lower semicontinuity of the data fidelity.", "Those arguments together yield the existence of a minimizer.", "The first equation of the optimality condition for the subgradient $p_\\alpha $ follows from the fact that the data fidelity is Fréchet-differentiable.", "From the argumentation in [4] we see that the assumption $A=B^*$ furthermore implies that $A^*$ indeed maps to the predual space $\\mathcal {Z}$ (and not to the bigger space $\\mathcal {Z}^{**}$ ), such that (REF ) holds true.", "More precisely, this special property of $A^*$ is derived from the fact that $A$ is sequentially continuous from the weak-star topology of $\\mathcal {X}$ to the weak(-star) topology of $\\mathcal {Y}$ , which implies that it posseses an adjoint which maps $\\mathcal {Y}$ into $\\mathcal {Z}$ regarded as a closed subspace of $\\mathcal {Z}^{**}$ (note that the weak and the weak-star topology coincide on the Hilbert space $\\mathcal {Y}$ ).", "Consequently $p_\\alpha \\in \\mathcal {Z}$ .", "Finally, assume that $u_\\alpha $ and $\\tilde{u}_\\alpha $ are two solutions, then we find $p_\\alpha = B w_\\alpha , \\quad w_\\alpha = \\frac{1}{\\alpha }(f-Au_\\alpha ),$ and an analogous identity for $\\tilde{p}_\\alpha $ respectively $\\tilde{u}_\\alpha $ .", "Consequently, we have $(w_\\alpha - \\tilde{w}_\\alpha ) + \\frac{1}{\\alpha }A(u_\\alpha - \\tilde{u}_\\alpha ) = 0.$ Computing the squared norm of the left-hand side, we find $\\Vert w_\\alpha - \\tilde{w}_\\alpha \\Vert _{\\mathcal {Y}}^2 &+ \\frac{2}{\\alpha }\\langle p_\\alpha - \\tilde{p}_\\alpha , u_\\alpha - \\tilde{u}_\\alpha \\rangle \\\\&+ \\frac{1}{\\alpha ^2} \\Vert A(u_\\alpha - \\tilde{u}_\\alpha )\\Vert _{\\mathcal {Y}}^2 \\quad = 0.$ The dual product can be expressed as a symmetric Bregman distance $D^{\\mathrm {sym}}_J(u_\\alpha ,\\tilde{u}_\\alpha ) = D^{\\tilde{p}_\\alpha }_J(u_\\alpha ,\\tilde{u}_\\alpha ) + D^{p_\\alpha }_J(\\tilde{u}_\\alpha , u_\\alpha ).$ Hence all three terms are nonnegative and we find in particular $A u_\\alpha = A\\tilde{u}_\\alpha $ , $w_\\alpha = \\tilde{w}_\\alpha $ and thus $p_\\alpha = \\tilde{p}_\\alpha $ .", "$\\square $ By exploiting that $p_\\alpha $ lies in the range of $B$ we can prove coercivity and subsequently existence for problem (REF ).", "In fact, we can give a more general result.", "Theorem 3.2 Let the conditions of Theorem REF hold and let $p \\in \\partial J(0) \\cap \\mathcal {Z} \\subset \\mathcal {X}^*$ be such that there exists $w$ with $ J^*\\left(\\frac{p - B w}{\\tau }\\right) = 0 $ for some $0 <\\tau < 1$ .", "Then there exists a minimizer of $\\min _{u \\in \\mathcal {X}} \\frac{1}{2}\\Vert Au - f \\Vert ^2_{\\mathcal {Y}} \\text{ s.t. }", "J(u)-\\langle p,u \\rangle =0.$ Let $\\mathcal {A} = \\lbrace u \\in \\mathcal {X}~|~ J(u)-\\langle p,u \\rangle =0 \\rbrace $ be the admissible set.", "Since $0 \\in \\mathcal {A}$ we can look for a minimizer in the sublevel set $S = \\big \\lbrace u \\in \\mathcal {A} ~|~ \\Vert Au -f \\Vert _{\\mathcal {Y}} \\le \\Vert f \\Vert _{\\mathcal {Y}} \\big \\rbrace .$ By the triangle inequality we have $\\Vert Au \\Vert _{\\mathcal {Y}} \\le 2 \\Vert f \\Vert _{\\mathcal {Y}}$ and hence $\\frac{1}{2}\\Vert Au \\Vert ^2_{\\mathcal {Y}} \\le 2 \\Vert f \\Vert ^2_{\\mathcal {Y}}$ on $S$ .", "Accordingly, $u \\mapsto \\frac{1}{2}\\Vert Au\\Vert ^2_{\\mathcal {Y}}$ is bounded on $S$ .", "From the definition of the convex conjugate we know that for all $u \\in \\mathcal {X}, r \\in \\mathcal {X}^*$ we have $\\langle r,u \\rangle \\le J^*(r) + J(u).$ Hence for $u \\in S$ we find $J(u) &= \\langle p,u \\rangle \\\\&= \\langle p - Bw,u \\rangle + \\langle w, Au \\rangle \\\\&\\le \\langle \\frac{p-B w}{\\tau }, \\tau u \\rangle + \\Vert w \\Vert _{\\mathcal {Y}} \\Vert Au\\Vert _{\\mathcal {Y}}\\\\&\\le J^* \\left(\\frac{p-B w}{\\tau } \\right) + J(\\tau u) + \\Vert w \\Vert _{\\mathcal {Y}} \\Vert Au\\Vert _{\\mathcal {Y}}$ which implies by the one-homogeneity of $J$ that $J(u) \\le \\frac{\\Vert w \\Vert _{\\mathcal {Y}} ~ \\Vert Au \\Vert _{\\mathcal {Y}}}{1-\\tau }.$ Thus we obtain the boundedness of $u \\mapsto \\frac{1}{2}\\Vert Au \\Vert ^2_{\\mathcal {Y}} + J(u)$ on $S$ .", "The remaining steps follow the proof of Theorem REF .", "$\\square $ Note that, provided that the operator $A$ fulfills the conditions of Theorem REF , the assumptions of Theorem REF always hold for $p=p_\\alpha $ obtained from (REF ) with $w=\\frac{1}{\\alpha }(f-Au_\\alpha )$ and $\\tau $ arbitrarily small, hence we conclude the existence of a minimizer $\\hat{u}_{\\alpha }$ of (REF ).", "The situation for (REF ) is less clear, since there is no similar way to obtain coercivity.", "As we shall see in Section , (REF ) consists in minimizing a quadratic functional over a linear subspace, which immediately implies the existence of $\\hat{u}_{\\alpha }$ if $\\mathcal {X}$ has finite dimensions.", "In an infinite-dimensional setting we cannot provide an existence result in general, since there is neither a particular reason for the subspace to be closed nor for the quadratic functional to be coercive (in ill-posed problems we typically deal with an operator $A$ with nonclosed range)." ], [ "Optimal debiasing on singular vectors", "In the following we work out the behavior of the debiasing method on singular vectors [1], which represent the extension of the concept of classical singular value decomposition to nonlinear regularization functionals.", "According to [1], $u^\\lambda \\in \\mathcal {X}$ is a singular vector if for some $\\lambda > 0$ $\\lambda A^* A u^\\lambda \\in \\partial J(u^\\lambda )$ holds.", "Without going too much into detail, singular vectors can be considered as generalized “eigenfunctions” of the regularization functional $J$ .", "As such, they describe a class of exact solutions to problem $(\\ref {eq:firstStep})$ in the following sense: Let us consider a multiple $c u^\\lambda $ of such a singular vector for $c > \\lambda \\alpha $ .", "According to [1], the solution $u_\\alpha $ of the variational problem (REF ) for data $f=c Au^\\lambda $ is given by $u_\\alpha = (c- \\alpha \\lambda ) u^\\lambda ,$ and the subgradient from the optimality condition is $p_\\alpha = \\lambda A^* A u^\\lambda \\in \\partial J(u_\\alpha ).$ Hence $u_\\alpha $ recovers $c u^\\lambda $ up to a (known) scalar factor $\\alpha \\lambda $ and shares a subgradient with $u^\\lambda $ .", "This means that the variational method leaves the singular vector basically untouched, which allows for its exact recovery.", "Intuitively, the quantity $-\\lambda \\alpha u^\\lambda $ hence represents the bias of the variational method in this case, which should be removed by our debiasing method (REF ).", "And indeed we obtain $\\hat{u}_{\\alpha }=c u^\\lambda $ as a minimizer of (REF ), since $\\Vert A\\hat{u}_{\\alpha }- f\\Vert _{\\mathcal {Y}} = \\Vert A(\\hat{u}_{\\alpha }- cu^\\lambda )\\Vert _{\\mathcal {Y}} = 0$ and since $\\hat{u}_{\\alpha }$ lies in the admissible set due to the shared subgradient.", "If $A$ has trivial nullspace, $\\hat{u}_{\\alpha }$ is even unique.", "Hence, the debiasing strategy leads to the exact reconstruction of the solution and corrects the bias $-\\lambda \\alpha u^\\lambda $ .", "Note that this is indeed an important result, since if the debiasing method failed for singular vectors it would be doubtful whether the method is reliable in general.", "Since the infimal convolution of Bregman distances is nonnegative and less or equal than either of the Bregman distances, it also vanishes at $\\hat{u}_{\\alpha }=c u^\\lambda $ .", "In particular $\\mathrm {ICB}_J^{p_{\\alpha }}(c u^\\lambda , u_\\alpha ) &\\le D_J^{p_\\alpha }( c u^\\lambda , u_\\alpha ) \\\\&= J(c u^\\lambda ) - \\langle p_\\alpha , c u^\\lambda \\rangle = 0.$ Consequently, $\\hat{u}_{\\alpha }$ is also a solution of (REF )." ], [ "Bias and Model Manifolds", "In the following we provide a more fundamental discussion of bias and decompositions obtained by debiasing methods.", "An obvious point to start is the definition of bias, which is indeed not always coherent in the imaging literature with the one in statistics." ], [ "Definitions of bias", "We first recall the classical definition of bias in statistics.", "Let $f$ be a realization of a random variable modeling a random noise perturbation of clean data $f^*=Au^*$ , such that $\\mathbb {E}[f] = f^*$ .", "If we consider a general canonical estimator $\\hat{U}(f)$ , the standard definition of bias in this setup is given by $\\begin{split}\\mathbb {B}^{\\text{stat}}(\\hat{U}) &= \\mathbb {E}[u^*-\\hat{U}(f)]\\\\&= u^*-\\mathbb {E}[\\hat{U}(f)].\\end{split}$ Unfortunately, this bias is hard to manipulate for nonlinear estimators.", "Hence, we consider a deterministic definition of bias, which relies on the clean data $f^*$ : $\\begin{split}\\mathbb {B}^*(\\hat{U}) &= \\mathbb {E}[u^*-\\hat{U}(f^*)]= u^* - \\hat{U}(f^*)\\\\&= u^*-\\hat{U}(\\mathbb {E}[f]).\\end{split}$ We immediately note the equivalence of the two definitions in the case of linear estimators, but our computational experiments do not show a significant difference between $\\mathbb {B}^{\\text{stat}}$ and $\\mathbb {B}^*$ even for highly nonlinear variational methods.", "In general, the purpose of debiasing is to reduce the quantitative bias $B^d$ , i.e.", "here the error between $u^*$ and $\\hat{U}(f^*)$ in an appropriate distance measure $d$ : $B^d(\\hat{U}(f^*)) = d(\\hat{U}(f^*), u^*).$ Let us consider the specific estimator $u_\\alpha (f^*)$ , i.e.", "the solution of problem (REF ) with clean data $f^*$ .", "As already argued in Section , it suffers from a certain bias due to the chosen regularization.", "Following [16], this bias can be decomposed into two parts.", "The first part is related to the regularization itself, and it occurs if the assumption made by the regularization does not match the true object that we seek to recover.", "For example, trying to recover a piecewise linear object using TV regularization leads to the staircasing effect due to the assumption of a piecewise constant solution.", "This part of the bias is unavoidable since it is inherent to the regularization, and it is referred to as model bias.", "In particular, we cannot hope to correct it.", "However, even if the regularity assumption fits, the solution still suffers from a systematic error due to the weight on the regularization.", "For TV regularization for example, this is the loss of contrast observed in Section .", "This remaining part is referred to as method bias, and this is the part that we aim to correct.", "As we shall see in the remainder of the section, the estimator $u_\\alpha (f^*)$ provides the necessary information to correct this bias.", "Deledalle et al.", "[16] define an appropriate linear model subspace related to that estimator, on which the debiasing takes place.", "It allows to define the model bias as the difference between $u^*$ and its projection onto the model subspace.", "The remaining part of the difference between the reconstructed solution and $u^*$ is then the method bias.", "In the following we reintroduce the notion of model subspaces provided by [16] and further generalize it to the variational setting in infinite dimensions.", "The latter may imply the nonclosedness of the model subspace and hence nonexistence of the projection of $u^*$ onto it.", "Moreover, it seems apparent that in some nonlinear situations it might be more suitable to consider a model manifold instead of a linear space and we hence generalize the definition in this direction.", "We remark that the use of the term manifold is for technical reasons.", "As we shall see, the sets we consider in the course of the paper are for example (linear) subspaces or convex cones.", "The latter are not linear, but can be considered as manifolds with boundaries.", "Therefore we shall use the term model manifold in general, and be more precise for particular instances of model manifolds.", "Let us first assume that we are already given an appropriate model manifold.", "Definition 4.1 Let ${\\cal M}_{f^*}$ be a given model manifold and $d\\colon \\mathcal {X}\\times \\mathcal {X}\\rightarrow [0,\\infty )$ a distance measure.", "An estimator $\\hat{U}(f^*)$ of $u^*$ is a debiasing of $u_\\alpha (f^*)$ if $\\hat{U}(f^*) \\in \\mathcal {M}_{f^*}$ and $d(\\hat{U}(f^*),u^*) < d(u_\\alpha (f^*),u^*).$ If there exists a minimizer $\\hat{u}_{\\alpha }(f^*) \\in \\arg \\min _{v \\in {\\cal M}_{f^*}} d(v,u^*),$ we call it an optimal debiasing.", "In any case, we define the magnitude of the model bias as $B_{\\text{mod}}^d({\\cal M}_{f^*}) = \\inf _{v \\in {\\cal M}_{f^*}} d(v,u^*).$ Obviously the model bias only depends on the model manifold and for a given $u_\\alpha (f^*)$ it is hence, as already indicated, a fixed quantity that we cannot manipulate.", "Instead we want to perform the debiasing on the manifold only, so we consider another bias for elements of ${\\cal M}_{f^*}$ only.", "Since according to the above definition there might exist more than one optimal debiasing, we shall from here on assume that we are given one of them.", "Definition 4.2 For a fixed optimal debiasing $\\hat{u}_{\\alpha }(f^*)$ on ${\\cal M}_{f^*}$ , we define the magnitude of the method bias of $v \\in {\\cal M}_{f^*}$ related to $\\hat{u}_{\\alpha }(f^*)$ as $B_{\\text{meth}}^d(v) = d(v, \\hat{u}_{\\alpha }(f^*)).$ The optimal debiasing $\\hat{u}_{\\alpha }(f^*)$ obviously does not suffer from method bias.", "Note that if the minimizer in (REF ) does not exist, which can happen in particular in ill-posed problems in infinite dimensions, then the magnitude of the method bias is not well-defined or has to be set to $+\\infty $ .", "With these definitions at hand, we now aim to compute an optimal debiasing, i.e.", "the solution of (REF ).", "The remaining questions are how to choose an appropriate model manifold $\\mathcal {M}_{f^*}$ and the distance measure $d$ .", "We start with the latter.", "An easy choice for the distance measure $d$ is a squared Hilbert space norm: If the minimizer of (REF ) exists, e.g.", "if $\\mathcal {M}_{f^*}$ is nonempty, convex and closed, the optimal debiasing $\\hat{u}_{\\alpha }(f^*)$ is the (unique) projection of $u^*$ onto $\\mathcal {M}_{f^*}$ .", "We obtain a decomposition of the bias of any estimator $v \\in \\mathcal {M}_{f^*}$ into method and (constant) model bias: $v - u^* = \\underbrace{ v - \\hat{u}_{\\alpha }(f^*) }_{\\text{method bias}} + \\underbrace{ \\hat{u}_{\\alpha }(f^*) - u^* }_{\\text{model bias}}.$ In case $\\mathcal {M}_{f^*}$ is a closed subspace of $\\mathcal {X}$ , this decomposition is even orthogonal, i.e.", "$B^d(v) &= \\Vert v - u^*\\Vert ^2 \\\\&= \\Vert v - \\hat{u}_{\\alpha }(f^*) \\Vert ^2 + \\Vert \\hat{u}_{\\alpha }(f^*) - u^* \\Vert ^2 \\\\&= B_{\\text{meth}}^d(v) + B_{\\text{mod}}^d({\\cal M}_{f^*}).$ Unfortunately, for general inverse problems with a nontrivial operator we do not know $u^*$ and hence cannot compute its projection onto $\\mathcal {M}_{f^*}$ .", "Instead we have access to the data $f^* = Au^*$ (or rather to one noisy realization $f$ of $f^*$ in practice, which we discuss later).", "In order to make the bias (and the associated debiasing) accessible, we can consider bias through the operator $A$ .", "Hence the optimal debiasing comes down to computing the minimizer of (REF ) with a distance defined over $A (\\mathcal {M}_{f^*})$ , i.e.", "$\\hat{u}_{\\alpha }(f^*) &= \\arg \\min _{v \\in {\\cal M}_{f^*}} \\Vert Au^* - Av\\Vert ^2 \\nonumber \\\\&= \\arg \\min _{v \\in {\\cal M}_{f^*}} \\Vert f^* - Av\\Vert ^2.$ Correspondingly, if such a minimizer $\\hat{u}_{\\alpha }(f^*)$ exists, we measure the magnitude of model and method bias in the output space, rather than in image space, i.e.", "$& B_{\\text{mod}}^d({\\cal M}_{f^*}) = \\inf _{v \\in {\\cal M}_{f^*}} \\Vert Av - f^*\\Vert ^2, \\\\& B_{\\text{meth}}^d(v) = \\Vert A\\hat{u}_{\\alpha }(f^*) - Av\\Vert ^2.$ We can hence at least guarantee that the optimal debiasing has zero method bias in the output space.", "For denoising problems without any operator ($A$ being the identity), or for $A$ invertible on $\\mathcal {M}_{f^*}$ we obtain the equivalence of both approaches.", "In ill-posed inverse problems it is usually rather problematic to measure errors in the output space, since noise can also be small in that norm.", "Notice however that we do not use the output space norm on the whole space, but on the rather small model manifold, on which - if chosen appropriately - the structural components dominate.", "On the latter the output space norm is reasonable.", "The main advantage of this formulation is that we are able to compute a minimizer of (REF ), since it is in fact a constrained least-squares problem with the data fidelity of (REF ).", "Its solution of course requires a proper choice of the underlying model manifold $\\mathcal {M}_{f^*}$ , which we discuss in the following." ], [ "Model manifolds", "In general, a model manifold can be characterized as the space of possible solutions for the debiasing step following the first solution $u_\\alpha (f)$ of the variational problem (REF ).", "As such it contains the properties of $u_\\alpha (f)$ that we want to carry over to the debiased solution.", "In the context of sparsity-enforcing regularization this is basically a support condition on the debiased solution." ], [ "Differential model manifolds", "Deledalle et al.", "[16] use the notion of Fréchet derivative to define their model subspace in a finite-dimensional setting.", "We naturally generalize this concept using the directional derivative instead, and further extend it to infinite dimensions.", "The following definitions can e.g.", "be found in [33].", "Definition 4.3 Let $ and $ W$ be Banach spaces.A mapping $ F W$ is called Fréchet differentiable at $ x if there exists a linear and bounded operator $\\mathrm {D}F(x;\\cdot ) \\colon \\mathcal {W}$ such that $\\lim _{\\Vert g\\Vert _{ \\rightarrow 0} \\dfrac{\\Vert F(x + g) - F(x) + \\mathrm {D}F(x;g) \\Vert _\\mathcal {W}}{\\Vert g\\Vert _{} = 0.", "}}\\begin{definition}A mapping F \\colon \\mathcal {W} is called directionally differentiable in the sense of Gâteaux at x \\in if the limit{\\begin{@align*}{1}{-1}\\mathrm {d}F(x;g) := \\lim _{t \\rightarrow 0^+} \\dfrac{F(x + tg) - F(x)}{t}\\end{@align*}}exits for all g \\in .\\end{definition}We can immediately deduce from the definition that, if the directional derivative \\mathrm {d}F(x;\\cdot ) exits, it is positively one-homogeneous in g, i.e.", "{\\begin{@align*}{1}{-1}\\mathrm {d}F(x;\\lambda g) = \\lambda \\mathrm {d}F(x;g)\\end{@align*}}for all \\lambda \\ge 0 and g \\in .If it is linear in g, we call F Gâteaux differentiable at x.$ Provided a unique and Fréchet differentiable map $f \\mapsto u_\\alpha (f)$ , Deledalle et al.", "[16] introduce the tangent affine subspace $\\mathcal {M}_f^{\\mathrm {F}}= \\big \\lbrace u_\\alpha (f) + \\mathrm {D}u_\\alpha (f;g) ~|~ g \\in \\mathcal {Y}~\\big \\rbrace ,$ where $\\mathrm {D}u_\\alpha (f;\\cdot )\\colon \\mathcal {Y}\\rightarrow \\mathcal {X}$ is the Fréchet derivative of $u_\\alpha (f)$ at $f$ .", "To be less restrictive, the easiest generalization of $\\mathcal {M}_f^{\\mathrm {F}}$ is to consider the directional derivative.", "Definition 4.4 If the map $f \\mapsto u_\\alpha (f)$ is directionally differentiable with derivative $\\mathrm {d}u_\\alpha (f; \\cdot )$ , we define $\\mathcal {M}_f^{\\mathrm {G}}= \\big \\lbrace u_\\alpha (f) + \\mathrm {d}u_\\alpha (f;g) ~|~ g \\in {\\cal Y} ~ \\big \\rbrace .$ Note that if the map is Fréchet differentiable, $\\mathcal {M}_f^{\\mathrm {G}}$ is a linear space and coincides with the model subspace $\\mathcal {M}_f^{\\mathrm {F}}$ .", "We now derive a few illustrative examples that we use throughout the remainder of the paper.", "In order to keep it as simple as possible, the easiest transition from the finite-dimensional vector space setting to infinite dimensions are the $\\ell ^p$ -spaces of $p$ -summable sequences: Definition 4.5 For $1 \\le p < \\infty $ we define the spaces $\\ell ^p$ of $p$ -summable sequences with values in $\\mathbb {R}^d$ by $\\ell ^p(\\mathbb {R}^d) = \\big \\lbrace (x_i)_{i \\in \\mathbb {N}}, x_i \\in \\mathbb {R}^d : \\sum _{i \\in \\mathbb {N}} | x_i|^p < \\infty \\big \\rbrace ,$ where $| \\cdot |$ denotes the Euclidean norm on $\\mathbb {R}^d$ .", "For $p = \\infty $ we define $\\ell ^{\\infty }(\\mathbb {R}^d) = \\big \\lbrace (x_i)_{i \\in \\mathbb {N}}, x_i \\in \\mathbb {R}^d : \\sup _{i \\in \\mathbb {N}} | x_i| < \\infty \\big \\rbrace .$ It is easy to show that $\\ell ^p(\\mathbb {R}^d) \\subset \\ell ^q(\\mathbb {R}^d)$ for $1 \\le p \\le q \\le \\infty $ .", "In particular for $d=1$ we denote by $\\ell ^1$ , $\\ell ^2$ and $\\ell ^{\\infty }$ the spaces of summable, square-summable and bounded scalar-valued sequences.", "Example 4.6 Anisotropic shrinkage.", "Let $f \\in \\ell ^2$ be a square-summable sequence.", "The solution of $u_\\alpha (f) \\in \\arg \\min _{u \\in \\ell ^1} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2}^2 + \\alpha \\Vert u\\Vert _{\\ell ^1}$ for $\\alpha > 0$ is given by $[u_\\alpha (f)]_i = {\\left\\lbrace \\begin{array}{ll}f_i - \\alpha ~ \\mathrm {sign}(f_i), & |f_i| \\ge \\alpha , \\\\0, & |f_i| < \\alpha .\\end{array}\\right.", "}$ Its support is limited to where $|f_i|$ is above the threshold $\\alpha $ .", "The directional derivative $\\mathrm {d}u_\\alpha (f;g)$ of $u_\\alpha (f)$ into the direction $g \\in \\ell ^2$ is given by $[\\mathrm {d}u_\\alpha (f;g)]_i& \\\\ = &{\\left\\lbrace \\begin{array}{ll}g_i, & |f_i| > \\alpha \\\\0, & |f_i| < \\alpha \\\\g_i, & |f_i| = \\alpha , \\mathrm {sign}(f_i) = \\mathrm {sign}(g_i) \\\\0, & |f_i| = \\alpha , \\mathrm {sign}(f_i) \\ne \\mathrm {sign}(g_i).\\end{array}\\right.", "}$ See Appendix REF .", "First, if we exclude the case $|f_i| = \\alpha $ , the directional derivative is linear, hence it is a Gâteaux derivative.", "In fact it is even an infinite-dimensional Fréchet derivative, and the resulting model manifold coincides with the model subspace defined in finite dimensions in [16]: $\\mathcal {M}_f^{\\mathrm {F}}= \\big \\lbrace u \\in \\ell ^2~|~ \\mathrm {supp}(u) \\subset \\mathrm {supp}(u_\\alpha (f)) \\big \\rbrace .$ The model manifold carries over information about the support of the first solution $u_\\alpha (f)$ .", "Note that $\\mathcal {M}_f^{\\mathrm {F}}$ contains all elements of $\\ell ^2$ which share the same support as $u_\\alpha (f)$ , but as well allows for zeros where $u_\\alpha (f) \\ne 0$ .", "In that sense $u_\\alpha (f)$ defines the maximal support of all $u \\in \\mathcal {M}_f^{\\mathrm {F}}$ .", "If we allow $|f_i|$ to be equal to $\\alpha $ , we obtain a larger set which allows for support changes in the direction of $f_i$ on the threshold: $u \\in \\mathcal {M}_f^{\\mathrm {G}}\\Leftrightarrow u_i = {\\left\\lbrace \\begin{array}{ll}\\lambda \\in \\mathbb {R}, & |f_i|> \\alpha , \\\\0, & |f_i|< \\alpha , \\\\\\lambda \\ge 0 , & f_i = \\alpha , \\\\\\lambda \\le 0 , & f_i = - \\alpha .\\end{array}\\right.", "}$ Note that the case $|f_i| > \\alpha $ reveals a remaining shortcoming of the definition via the directional derivative, e.g.", "if $f_i> \\alpha $ it is counter-intuitive to allow for negative elements in $\\mathcal {M}_f^{\\mathrm {G}}$ , while this is not the case for $f_i = \\alpha $ .", "The main reason appears to be the strong deviation of the linearization in such directions from the actual values of $[u_\\alpha (f)]_i$ , which is not controlled by the definition.", "However, minimizing the data term over $\\mathcal {M}_f^{\\mathrm {G}}$ for the debiasing in Eq.", "(REF ) forces the changes to have the right sign and the debiased solution $\\hat{u}_{\\alpha }(f)$ corresponds to hard-thresholding: $[\\hat{u}_{\\alpha }(f)]_i = {\\left\\lbrace \\begin{array}{ll}f_i, & |f_i| \\ge \\alpha , \\\\0, & |f_i| < \\alpha .\\end{array}\\right.", "}$ Note that we as well maintain the signal directly on the threshold.", "We obtain analogous results for isotropic shrinkage, i.e.", "if $f \\in \\ell ^2(\\mathbb {R}^d)$ for $d > 1$ .", "Since the computation of the derivative requires a little more work, we provide the results in Appendix REF .", "A more interesting example is the model manifold related to anisotropic $\\ell ^1$ -regularized general linear inverse problems.", "Example 4.7 Anisotropic $\\ell ^1$ -regularization.", "For $r > 1 $ let $A \\colon \\ell ^r \\rightarrow \\ell ^2$ be a linear and bounded operator and $f \\in \\ell ^2$ .", "Consider the solution $u_\\alpha (f)$ of the $\\ell ^1$ -regularized problem $u_\\alpha (f) \\in \\arg \\min _{u \\in \\ell ^1} \\frac{1}{2} \\Vert Au - f\\Vert _{\\ell ^2}^2 + \\alpha \\Vert u \\Vert _{\\ell ^1},$ where we assume that the solution is unique for data in a neighborhood of $f$ .", "Computing the directional derivative directly is a more tedious task in this case, but computing the model manifold $\\mathcal {M}_f^{\\mathrm {G}}$ is actually easier via a slight detour.", "Let $u_\\alpha (f)$ be the solution for data $f$ and $u_\\alpha (\\tilde{f})$ the solution for data $\\tilde{f}$ .", "First, we derive an estimate on the two subgradients from the optimality conditions $0 &= A^* ( A u_\\alpha (f) - f) + \\alpha p_\\alpha , \\hspace{5.0pt} p_\\alpha \\in \\partial \\Vert u_\\alpha (f) \\Vert _{\\ell ^1}, \\\\0 &= A^* ( A u_\\alpha (\\tilde{f}) - \\tilde{f}) + \\alpha \\tilde{p}_{\\alpha }, \\hspace{5.0pt} \\tilde{p}_{\\alpha } \\in \\partial \\Vert u_\\alpha (\\tilde{f}) \\Vert _{\\ell ^1}.$ Following the ideas of [11], we subtract the two equations and multiply by $u_\\alpha (f) - u_\\alpha (\\tilde{f})$ to arrive at $\\Vert A &u_\\alpha (f) - A u_\\alpha (\\tilde{f}) \\Vert _{\\ell ^2}^2 \\\\&+ \\alpha \\langle p_\\alpha - \\tilde{p}_{\\alpha }, u_\\alpha (f) - u_\\alpha (\\tilde{f}) \\rangle \\\\&= \\langle f - \\tilde{f},A u_\\alpha (f) - A u_\\alpha (\\tilde{f}) \\rangle \\\\&\\le \\frac{1}{2} \\Vert f - \\tilde{f} \\Vert _{\\ell ^2}^2 + \\frac{1}{2} \\Vert A u_\\alpha (f) - A u_\\alpha (\\tilde{f})\\Vert _{\\ell ^2}^2.$ The last line follows from the Fenchel-Young inequality, obtained by applying the inequality (REF ) to $J = \\frac{1}{2} \\Vert \\cdot \\Vert _{\\ell ^2}^2$ .", "The second term on the left hand side is a symmetric Bregman distance, i.e.", "the sum of two Bregman distances (cf.", "[11]), hence positive.", "Leaving it out and rearranging then yields $\\Vert A u_\\alpha (f) &- A u_\\alpha (\\tilde{f}) \\Vert _{\\ell ^2} \\le \\Vert f - \\tilde{f} \\Vert _{\\ell ^2}.$ Since $A^* \\colon \\ell ^2\\rightarrow \\ell ^s$ , where $s^{-1} + r^{-1} = 1$ , $A^*$ is also continuous to $\\ell ^{\\infty }$ , hence we derive the following estimate from the optimality conditions: $&\\Vert p_\\alpha - \\tilde{p}_\\alpha \\Vert _{\\ell ^{\\infty }} \\\\&= \\frac{1}{\\alpha } \\Vert A^*(A u_\\alpha (f) - A u_\\alpha (\\tilde{f})) - A^*(f - \\tilde{f}) \\Vert _{\\ell ^{\\infty }} \\\\&\\le \\frac{\\Vert A^*\\Vert }{\\alpha } \\Vert A u_\\alpha (f) - A u_\\alpha (\\tilde{f})\\Vert _{\\ell ^2} \\\\&+ \\frac{\\Vert A^*\\Vert }{\\alpha } \\Vert f - \\tilde{f} \\Vert _{\\ell ^2} \\\\&\\le \\frac{C}{\\alpha } \\Vert f - \\tilde{f} \\Vert _{\\ell ^2},$ where we used (REF ) for the last inequality and $\\Vert \\cdot \\Vert $ denotes the operator norm.", "Next, we note that since $A^*$ maps to $\\ell ^s$ and $p_\\alpha $ and $\\tilde{p}_\\alpha $ lie in its range, they necessarily have to converge to zero.", "This implies the existence of $N \\in \\mathbb {N}$ such that for all $i \\ge N$ both $|(p_\\alpha )_i|$ and $|(\\tilde{p}_\\alpha )_i|$ are strictly smaller than 1 and hence $u_\\alpha (f)$ and $u_\\alpha (\\tilde{f})$ vanish for all $i \\ge N$ .", "As a consequence it is sufficient to consider a finite dimensional setting for the following reasoning.", "In view of the subdifferential of the $\\ell ^1$ -norm, $\\partial \\Vert u\\Vert _{\\ell ^1} = \\lbrace &p \\in \\ell ^{\\infty }: \\Vert p\\Vert _{\\ell ^{\\infty }} \\le 1, \\\\&p_i = \\mathrm {sign}(u_i) \\text{ for } u_i \\ne 0 \\rbrace ,$ we have to consider several cases.", "If $[u_\\alpha (f)]_i= 0$ and $|(p_\\alpha )_i| < 1$ , we derive from $|(\\tilde{p}_{\\alpha })_i|&\\le |(\\tilde{p}_{\\alpha })_i - (p_\\alpha )_i| + |(p_\\alpha )_i| \\\\&\\le \\frac{C}{\\alpha } \\Vert f - \\tilde{f} \\Vert _{\\ell ^2} + |(p_\\alpha )_i|,$ that if $\\Vert f - \\tilde{f} \\Vert _{\\ell ^2}$ is sufficiently small, then $|(\\tilde{p}_{\\alpha })_i| < 1$ .", "Hence $[u_\\alpha (\\tilde{f})]_i = 0$ , and the derivative related to the perturbed data $\\tilde{f}$ vanishes.", "In case $[u_\\alpha (f)]_i= 0$ and $(p_\\alpha )_i = 1$ , by a similar argument $(\\tilde{p}_{\\alpha })_i \\ne -1$ and thus $[u_\\alpha (\\tilde{f})]_i \\ge 0$ and $[\\mathrm {d}u_\\alpha (f;g)]_i \\ge 0$ .", "Analogously, $[\\mathrm {d}u_\\alpha (f;g)]_i \\le 0$ if $[u_\\alpha (f)]_i= 0$ and $(p_\\alpha )_i = -1$ .", "If $[u_\\alpha (f)]_i\\ne 0$ , the directional derivative is an arbitrary real number depending on the data perturbation.", "Summing up we now know that every directional derivative is an element $v \\in \\ell ^1$ fulfilling $v_i = {\\left\\lbrace \\begin{array}{ll}0, &|(p_\\alpha )_i| < 1,\\\\\\lambda \\ge 0, & (u_\\alpha )_i = 0, (p_\\alpha )_i = 1 , \\\\\\lambda \\le 0, & (u_\\alpha )_i = 0, (p_\\alpha )_i = -1.\\end{array}\\right.", "}$ Note again that $v$ differs from 0 only for a finite number of indices.", "Hence, for $v$ satisfying (REF ), we can pick $t > 0$ sufficiently small such that $p_\\alpha $ is a subgradient of $\\tilde{u} = u_\\alpha (f) + t v$ .", "Indeed, for example if $(u_\\alpha )_i = 0$ and $(p_\\alpha )_i = 1$ , then $v_i \\ge 0$ , so $\\tilde{u} \\ge 0$ , and hence $(p_\\alpha )_i$ fulfills the requirement of a subgradient of $\\tilde{u}$ .", "The other cases follow analogously.", "Then from the optimality condition of $u_\\alpha (f)$ we get: $& \\ A^*(Au_\\alpha (f) -f) )+ \\alpha p_\\alpha = 0\\\\\\Leftrightarrow & \\ A^*(A(\\underbrace{u_\\alpha (f)+tv}_{\\tilde{u}}) - (f+tAv)) + \\alpha p_\\alpha = 0.$ We then deduce that $\\tilde{u}$ is a minimizer of problem (REF ) with data $\\tilde{f} = f + tAv$ .", "Hence, there exists a data perturbation such that $v$ is the directional derivative of $u_\\alpha (f)$ .", "Putting these arguments together we now know that $ u \\in \\mathcal {M}_f^{\\mathrm {G}}$ if and only if $u_i = {\\left\\lbrace \\begin{array}{ll}\\lambda \\in \\mathbb {R}, & [u_\\alpha (f)]_i\\ne 0, \\\\0, & [u_\\alpha (f)]_i= 0, |(p_\\alpha )_i| < 1, \\\\\\lambda \\ge 0, & [u_\\alpha (f)]_i= 0, (p_\\alpha )_i = 1, \\\\\\lambda \\le 0, & [u_\\alpha (f)]_i= 0, (p_\\alpha )_i = -1.\\end{array}\\right.", "}$ It is not surprising that $\\mathcal {M}_f^{\\mathrm {G}}$ has a similar structure as the model manifold for the anisotropic shrinkage in Example REF .", "It allows for arbitrary changes on the support of $u_\\alpha (f)$ and permits only zero values if $[u_\\alpha (f)]_i= 0$ and $|(p_\\alpha )_i| < 1$ .", "If we exclude the case where $|(p_\\alpha )_i| = 1$ even though $[u_\\alpha (f)]_i$ vanishes, debiasing on $\\mathcal {M}_f^{\\mathrm {G}}$ effectively means solving a least-squares problem with a support constraint on the solution.", "But we again find an odd case where changes are allowed outside of the support of the initial solution $u_\\alpha (f)$ .", "It occurs when $|(p_\\alpha )_i| = 1$ even though $[u_\\alpha (f)]_i$ vanishes, which seems to be the indefinite case.", "However, it has been argued in [24] that a subgradient equal to $\\pm 1$ is a good indicator of support, hence it is reasonable to trust the subgradient in that case." ], [ "Variational model manifolds", "As we have shown so far, the appropriate use of a derivative can yield suitable spaces for the debiasing.", "However, for already supposedly easy problems such as the latter example the explicit computation of such spaces or of the derivatives can be difficult or impossible.", "And even if it is possible, there remains the question of how to effectively solve the debiasing on those spaces, both theoretically and numerically.", "On the other hand, the latter example implies that a subgradient of the first solution rather than the solution itself can provide the necessary information for the debiasing.", "This naturally leads us to the idea of Bregman distances in order to use the subgradient in a variational debiasing method.", "And indeed we show that the associated manifolds are closely related, and that they link the concept of model manifolds to the already presented debiasing method from Section .", "Furthermore, this does not only provide a theoretical framework, but also numerical solutions to perform debiasing in practice, even for more challenging problems.", "In the following we introduce related manifolds motivated by the variational problem itself.", "The optimality condition of the variational problem (REF ) defines a unique map $f \\mapsto p_\\alpha \\in \\partial J(u_\\alpha )$ , which allows us to consider the following manifolds.", "We drop the dependence of $u_\\alpha $ on $f$ for the sake of readability.", "Definition 4.8 For $p_\\alpha \\in \\partial J(u_\\alpha )$ defined by (REF ) we define $\\mathcal {M}_f^{\\mathrm {B}}&= \\big \\lbrace u \\in \\mathcal {X}~|~ D_J^{p_\\alpha } (u, u_\\alpha ) = 0 \\big \\rbrace , \\\\\\mathcal {M}_f^{\\mathrm {IC}}&= \\big \\lbrace u \\in \\mathcal {X}~|~ \\mathrm {ICB}_J^{p_{\\alpha }}(u, u_\\alpha ) = 0 \\big \\rbrace .$ In order to assess the idea of the above manifolds, we first revisit the anisotropic shrinkage problem of Example REF .", "Example 4.9 Anisotropic shrinkage.", "The optimality condition of problem (REF ) yields the subgradient $(p_\\alpha )_i = \\dfrac{f_i-(u_\\alpha )_i}{\\alpha } = {\\left\\lbrace \\begin{array}{ll}\\mathrm {sign}(f_i), & |f_i| \\ge \\alpha , \\\\\\frac{f_i}{\\alpha }, & |f_i| < \\alpha ,\\end{array}\\right.", "}$ and for $J = \\Vert \\cdot \\Vert _{\\ell ^1}$ the Bregman distance takes the following form: $D_{\\ell ^1}^{p_\\alpha }(u,u_\\alpha )&= \\Vert u \\Vert _{\\ell ^1} - \\langle p_\\alpha ,u \\rangle \\\\&= \\sum _{i \\in \\mathbb {N}} |u_i| - (p_\\alpha )_i u_i \\\\&= \\sum _{i \\in \\mathbb {N}} (\\mathrm {sign}(u_i) - (p_\\alpha )_i ) u_i.$ A zero Bregman distance thus means that either $u_i = 0$ or $\\mathrm {sign}(u_i) = (p_\\alpha )_i$ .", "Having a closer look at the subgradient (REF ), we observe that if $|f_i| < \\alpha $ , then $|(p_\\alpha )_i| < 1$ .", "Hence the latter condition cannot be fulfilled, so in this case $u_i$ has to be zero.", "We can thus characterize the model manifold related to a zero Bregman distance as: $u \\in \\mathcal {M}_f^{\\mathrm {B}}\\Leftrightarrow u_i = {\\left\\lbrace \\begin{array}{ll}\\lambda ~ \\mathrm {sign}(f_i), \\lambda \\ge 0, & |f_i| \\ge \\alpha , \\\\0, & |f_i| < \\alpha .\\end{array}\\right.", "}$ As for $\\mathcal {M}_f^{\\mathrm {G}}$ , the model manifold $\\mathcal {M}_f^{\\mathrm {B}}$ fixes the maximum support to where $|f_i| \\ge \\alpha $ .", "However, $\\mathcal {M}_f^{\\mathrm {B}}$ only allows for values on the support sharing the same sign as $f_i$ (respectively $(u_\\alpha )_i$ ).", "By adapting the proof of [24], we obtain a similar result for the infimal convolution of Bregman distances, without the restriction on the sign: $\\mathrm {ICB}_{\\ell ^1}^{p_{\\alpha }}(u,u_\\alpha )&= [D_{\\ell ^1}^{p_\\alpha }(\\cdot ,u_\\alpha ) \\Box D_{\\ell ^1}^{-p_\\alpha }(\\cdot ,-u_\\alpha )](u) \\\\&= \\sum _{i \\in \\mathbb {N}} (1 - |(p_\\alpha )_i|)|u_i|.$ For this infimal convolution to be zero we need either $u_i=0$ or $|(p_\\alpha )_i|=1$ .", "By the structure of the subgradient $p_\\alpha $ we thus find $u \\in \\mathcal {M}_f^{\\mathrm {IC}}\\Leftrightarrow u_i = {\\left\\lbrace \\begin{array}{ll}\\lambda \\in \\mathbb {R}, & |f_i| \\ge \\alpha , \\\\0, & |f_i| < \\alpha .\\end{array}\\right.", "}$ Hence we observe the following connection between the manifolds: $\\mathcal {M}_f^{\\mathrm {B}}\\subset \\mathcal {M}_f^{\\mathrm {G}}\\subset \\mathcal {M}_f^{\\mathrm {IC}}.$ Note that the manifold $\\mathcal {M}_f^{\\mathrm {G}}$ related to the directional derivative seems to be the odd one of the three.", "While allowing for arbitrary sign for $|f| > \\alpha $ , it only allows for changes in the direction of $f$ directly on the threshold.", "In that sense, $\\mathcal {M}_f^{\\mathrm {B}}$ and $\\mathcal {M}_f^{\\mathrm {IC}}$ seem to be more suitable in order to either include or exclude the sign-constraint.", "A closer inspection at the manifolds reveals that $\\mathcal {M}_f^{\\mathrm {IC}}$ is a linear space, as we further elaborate on in the next subsection.", "In this case it is actually even the span of $\\mathcal {M}_f^{\\mathrm {B}}$ , which is however not true in general.", "This can e.g.", "be seen from the next example of isotropic TV-type regularization.", "Example 4.10 Isotropic TV-type regularization.", "Let $A \\colon \\ell ^2(\\mathbb {R}^n) \\rightarrow \\ell ^2(\\mathbb {R}^d)$ and $\\Gamma \\colon \\ell ^2(\\mathbb {R}^n) \\rightarrow \\ell ^1(\\mathbb {R}^m)$ be linear and bounded operators and $J(u) = \\Vert \\Gamma u \\Vert _{\\ell ^1(\\mathbb {R}^m)}$ for $d,m,n \\in \\mathbb {N}$ .", "We aim to find the variational model manifolds for the debiasing of the solution $u_\\alpha \\in \\operatornamewithlimits{arg\\,min}_{u \\in \\ell ^2(\\mathbb {R}^n)} \\frac{1}{2} \\Vert Au - f \\Vert _{\\ell ^2(\\mathbb {R}^d)} + \\alpha \\Vert \\Gamma u \\Vert _{\\ell ^1(\\mathbb {R}^m)}.$ Given the (unique) subgradient $p_\\alpha \\in \\partial J(u_\\alpha )$ from the optimality condition, the chain rule for subdifferentials [18] implies the existence of a $q_\\alpha \\in \\partial \\Vert \\cdot \\Vert _{\\ell ^1(\\mathbb {R}^m)} (\\Gamma u_\\alpha )$ such that $p_\\alpha = \\Gamma ^* q_\\alpha $ and $D_J^{p_\\alpha }(u,u_\\alpha ) = D_{\\ell ^1(\\mathbb {R}^m)}^{q_\\alpha }(\\Gamma u, \\Gamma u_\\alpha ).$ If we denote the angle between $(\\Gamma u)_i$ and $(q_\\alpha )_i$ by $\\varphi _i$ , the Bregman distance reads: $D_J^{p_\\alpha }(u,u_\\alpha )&= D_{\\ell ^1(\\mathbb {R}^m)}^{q_\\alpha }(\\Gamma u, \\Gamma u_\\alpha ) \\\\&= \\sum _{i \\in \\mathbb {N}} |(\\Gamma u)_i| - (q_\\alpha )_i \\cdot (\\Gamma u)_i \\\\&= \\sum _{i \\in \\mathbb {N}} |(\\Gamma u)_i| \\big (1 - \\cos (\\varphi _i) |(q_\\alpha )_i| \\big )$ For a zero Bregman distance we can distinguish two cases: If $|(q_\\alpha )_i| < 1$ , then $(\\Gamma u)_i$ has to be zero.", "If $|(q_\\alpha )_i| =1$ , then either $(\\Gamma u)_i = 0$ or $\\cos (\\varphi _i) = 1$ , hence $(\\Gamma u)_i = \\lambda (q_\\alpha )_i$ for $\\lambda \\ge 0$ .", "Hence the model manifold $\\mathcal {M}_f^{\\mathrm {B}}$ is given by $u \\in \\mathcal {M}_f^{\\mathrm {B}}&\\Leftrightarrow \\\\(\\Gamma u)_i &= {\\left\\lbrace \\begin{array}{ll}\\lambda (q_\\alpha )_i, \\lambda \\ge 0, & |(q_\\alpha )_i| = 1, \\\\0, & |(q_\\alpha )_i| < 1.\\end{array}\\right.", "}$ In particular, if $(\\Gamma u_\\alpha )_i \\ne 0$ , then by the structure of the $\\ell ^1(\\mathbb {R}^m)$ -subdifferential we know that $(q_\\alpha )_i = \\frac{(\\Gamma u_\\alpha )_i}{|(\\Gamma u_\\alpha )_i|}$ and thus $(\\Gamma u)_i = \\mu (\\Gamma u_\\alpha )_i$ for some $\\mu \\ge 0$ .", "So provided that $|(q_\\alpha )_i| < 1$ whenever $(\\Gamma u_\\alpha )_i = 0$ we find $u \\in \\mathcal {M}_f^{\\mathrm {B}}&\\Leftrightarrow \\\\(\\Gamma u)_i &= {\\left\\lbrace \\begin{array}{ll}\\mu (\\Gamma u_\\alpha )_i, \\mu \\ge 0, & (\\Gamma u_\\alpha )_i \\ne 0, \\\\0, & (\\Gamma u_\\alpha )_i = 0.\\end{array}\\right.", "}$ Performing the debiasing on the latter manifold hence means minimizing the data term with a support and direction constraint on the gradient of the solution.", "This in particular allows to restore the loss of contrast which we have observed for TV regularization in Section .", "Note that the condition $|(q_\\alpha )_i| < 1 \\Leftrightarrow (\\Gamma u_\\alpha )_i = 0$ excludes the odd case where the subgradient seems to contain more information than the first solution, as already seen in Example REF .", "In the above illustration of the model manifold, the debiasing seems to rely on the choice of $q_\\alpha $ , which is obviously not unique.", "However, in practice we still use the unique subgradient $p_\\alpha $ from the optimality condition which avoids the issue of the choice of a “good” $q_\\alpha $ .", "The computation of $\\mathcal {M}_f^{\\mathrm {IC}}$ is a little more difficult in this case, since we cannot access an explicit representation of the functional $\\mathrm {ICB}_J^{p_{\\alpha }}(\\cdot , u_\\alpha )$ .", "However, since $\\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ) \\le \\mathrm {ICB}_J^{p_{\\alpha }}(u,u_\\alpha )$ (cf.", "Appendix REF , Thm.", "REF ), we can instead use the infimal convolution of two $\\ell ^1(\\mathbb {R}^m)$ -Bregman distances to illustrate the model manifold.", "We have (cf.", "Appendix REF , Thm.", "REF ) $\\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ) = \\sum _{i \\in \\mathbb {N}} G((\\Gamma u)_i, (q_\\alpha )_i)$ with $G \\colon \\mathbb {R}^m \\times \\mathbb {R}^m \\rightarrow \\mathbb {R}$ defined as $&G((\\Gamma u)_i, (q_\\alpha )_i) =\\\\& {\\left\\lbrace \\begin{array}{ll}|(\\Gamma u)_i| (1 - |\\cos (\\varphi _i)| |(q_\\alpha )_i|), \\\\\\hspace{85.0pt} \\text{ if } |(q_\\alpha )_i| < |\\cos (\\varphi _i)|, \\\\|(\\Gamma u)_i| | \\sin (\\varphi _i)| \\sqrt{1 - |(q_\\alpha )_i|^2}, \\\\\\hfill \\text{ if } |(q_\\alpha )_i| \\ge |\\cos (\\varphi _i)|.\\end{array}\\right.", "}$ For $G$ to be zero we once again distinguish two situations.", "If $|(q_\\alpha )_i| < 1$ , in the first case $G$ can only vanish if $(\\Gamma u)_i = 0$ .", "In the second case, since $1 > |(q_\\alpha )_i| \\ge |\\cos (\\varphi _i)|$ , we infer $\\varphi _i \\notin \\lbrace 0,\\pi \\rbrace $ , and hence neither the sinus nor the square root can vanish.", "This means once again that $(\\Gamma u)_i = 0$ .", "If $|(q_\\alpha )_i| = 1$ , we can only be in the second case and $G$ vanishes independently of $(\\Gamma u)_i$ .", "Thus $(\\Gamma u)_i$ can be arbitrary.", "Putting the arguments together, we find $u \\in \\mathcal {M}_f^{\\mathrm {IC}}&\\Rightarrow \\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ) = 0 \\\\&\\Leftrightarrow (\\Gamma u )_i = {\\left\\lbrace \\begin{array}{ll}\\lambda \\in \\mathbb {R}^m, & |(q_\\alpha )_i| = 1, \\\\0, & |(q_\\alpha )_i| < 1.\\end{array}\\right.", "}$ This is indeed not the span of $\\mathcal {M}_f^{\\mathrm {B}}$ , but it instead allows for arbitrary elements if $|(q_\\alpha )_i| = 1$ .", "From this example, we cannot immediately state that $\\mathcal {M}_f^{\\mathrm {B}}\\subset \\mathcal {M}_f^{\\mathrm {IC}}$ , because so far we only know that $\\mathcal {M}_f^{\\mathrm {B}}$ as well as $\\mathcal {M}_f^{\\mathrm {IC}}$ are subsets of the set $\\lbrace u \\in \\mathcal {X}~|~ \\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ) = 0\\rbrace $ .", "However, in the next subsection we see that $\\mathcal {M}_f^{\\mathrm {B}}\\subset \\mathcal {M}_f^{\\mathrm {IC}}$ is indeed true and it is actually a general property of the variational model manifolds.", "Note that we gain the same support condition on the gradient as for $\\mathcal {M}_f^{\\mathrm {B}}$ , but allow for arbitrary gradient directions on the support, which intuitively does not seem restrictive enough.", "However, in practice for the debiasing the direction is not arbitrary, but the data term decides, so we can expect a similar result for debiasing in $\\mathcal {M}_f^{\\mathrm {B}}$ and $\\mathcal {M}_f^{\\mathrm {IC}}$ .", "Indeed the numerical studies in Section confirm these expectations." ], [ "Properties of variational model manifolds", "In the following we discuss some properties of the variational manifolds $\\mathcal {M}_f^{\\mathrm {B}}$ and $\\mathcal {M}_f^{\\mathrm {IC}}$ .", "All results are general and do not depend on the particular choice of a subgradient, so we drop the dependence on $f$ in the notation of the manifolds.", "Let $v \\in \\mathcal {X}$ and $p \\in \\partial J(v)$ .", "We start with a result on the structure of $\\mathcal {M}^{\\mathrm {B}}$ : Theorem 4.11 The set $\\mathcal {M}^{\\mathrm {B}}= \\lbrace u\\in \\mathcal {X}~|~D_J^p(u,v) = 0 \\rbrace $ is a nonempty convex cone.", "The map $u \\mapsto D_J^p(u,v)$ is convex and nonnegative, hence $\\lbrace u~|~D_J^p(u,v) = 0 \\rbrace = \\lbrace u~|~D_J^p(u,v) \\le 0 \\rbrace $ is convex as a sublevel set of a convex functional.", "Moreover, for each $c \\ge 0$ we have $D_J^p(c u,v)=c~D_J^p(u,v),$ i.e.", "if $u$ is an element of the set, then every positive multiple $c u$ is an element, too.", "Hence it is a convex cone.", "Since $D_J^p(v,v)=0$ it is not empty.$\\square $ The structure of $\\mathcal {M}^{\\mathrm {IC}}$ is even simpler; as announced in a special example above it is indeed a linear space: Theorem 4.12 The set $\\mathcal {M}^{\\mathrm {IC}}= \\lbrace u\\in \\mathcal {X}|[D_J^p(\\cdot ,v)\\Box D_J^{-p}(\\cdot ,-v)](u) = 0 \\rbrace $ is a nonempty linear subspace of $\\mathcal {X}$ .", "By analogous arguments as above we deduce the convexity and since $\\mathrm {ICB}_J^p(0,v) &= \\inf _{\\phi + \\psi = 0} D_J^p(\\phi ,v) + D_J^{-p}(\\psi ,-v)\\\\ &\\le D_J^p(v,v) + D_J^{-p}(-v,-v) = 0$ the set is not empty.", "For arbitrary $c \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ we have $&\\mathrm {ICB}_J^p(cu,v)\\\\ &= \\inf _z J(cu-z) + J(z) - \\langle p, cu - 2z \\rangle \\\\&= \\vert c \\vert \\inf _w J(u-w) + J(w) - \\langle p, u -2w \\rangle ,$ where we use the one-to-one transform $z=cw$ for $c > 0$ and $z=c(u-w)$ for $c < 0$ .", "This implies that $\\mathrm {ICB}_J^p(cu,v)=0$ if $\\mathrm {ICB}_J^p(u,v) = 0$ .", "Now let $u_1,u_2 \\in \\mathcal {M}^{\\mathrm {IC}}$ , i.e.", "$\\mathrm {ICB}_J^p(u_1,v) = 0$ and $\\mathrm {ICB}_J^p(u_2,v) = 0$ .", "Then by definition of the infimum there exist sequences $(z_1^n)_{n\\in \\mathbb {N}}, (z_2^n)_{n\\in \\mathbb {N}}$ such that $\\lim _{n \\rightarrow \\infty } J(u_1 - z_1^n) &+ J(z_1^n) &&- \\langle p, u_1 - 2 z_1^n \\rangle &&= 0, \\\\\\lim _{n \\rightarrow \\infty } J(u_2 - z_2^n) &+ J(z_2^n) &&- \\langle p, u_2 - 2 z_2^n \\rangle &&= 0.$ Due to its convexity and absolute one-homogeneity $J$ is a seminorm and thus satisfies the triangle inequality: $&\\mathrm {ICB}_J^p(u_1 + u_2,v) \\\\&= \\inf _z J(u_1 + u_2 - z) + J(z) \\\\&\\hspace{30.0pt} - \\langle p, u_1 +u_2 - 2z \\rangle \\\\&\\le J(u_1 + u_2 - z_1^n - z_2^n) + J(z_1^n + z_2^n) \\\\&\\hspace{30.0pt} - \\langle p, u_1 + u_2 - 2z_1^n - 2z_2^n \\rangle \\\\&\\le J(u_1 - z_1^n) + J(z_1^n) - \\langle p, u_1 - 2 z_1^n \\rangle \\\\& \\hspace{30.0pt} + J(u_2 - z_2^n) + J(z_2^n) - \\langle p, u_2 - 2 z_2^n \\rangle \\\\& \\rightarrow 0, \\text{ for } n \\rightarrow \\infty .$ Hence $ u_1 + u_2 \\in \\mathcal {M}^{\\mathrm {IC}}$ and $\\mathcal {M}^{\\mathrm {IC}}$ is a linear subspace.", "$\\square $ As one may expect from the fact that the infimal convolution is a weaker distance than the original Bregman distance, we obtain an immediate inclusion between the corresponding manifolds: Lemma 4.13 $\\mathcal {M}^{\\mathrm {B}}\\subset \\mathcal {M}^{\\mathrm {IC}}$ .", "Let $u \\in \\mathcal {M}^{\\mathrm {B}}$ , i.e.", "$D_J^p(u,v) = 0$ .", "For $c \\ge 0$ we have $D_J^{-p}(-c u,-v)=c~D_J^p(u,v).$ Thus we deduce $\\mathrm {ICB}_J^p(u,v)&= \\inf _{\\phi + \\psi = u} D_J^p(\\phi ,v) + D_J^{-p}(\\psi ,-v) \\\\&\\le D_J^p(2u,v) + D_J^{-p}(-u,-v) \\\\&= 2 D_J^p(u,v) + D_J^p(u,v)= 0.$ The assertion follows by the nonnegativity of the maps $u \\mapsto D_J^p(u,v)$ and $u \\mapsto D_J^{-p}(u,-v)$ .", "Note that for $p \\ne 0$ the subset is proper, since e.g.", "$-v \\in \\mathcal {M}^{\\mathrm {IC}}$ but $-v \\notin \\mathcal {M}^{\\mathrm {B}}$ .$\\square $ We finally elaborate on the importance of absolute one-homogeneity of $J$ for our approach (respectively also other debiasing approaches as in [16]), such that the subdifferential can be multivalued.", "Otherwise the model manifolds may just contain a single element and debiasing in this manifold cannot produce any other solution.", "This is e.g.", "the case for a strictly convex functional.", "Lemma 4.14 Let $J$ be strictly convex.", "Then $\\mathcal {M}^{\\mathrm {B}}$ is a singleton.", "For strictly convex $J$ the mapping $u \\mapsto D_J^p(u,v)$ is strictly convex as well, hence $D_J^p(u,v) = 0$ if and only if $u = v$ and $\\mathcal {M}^{\\mathrm {B}}= \\lbrace v \\rbrace $ .", "$\\square $ However, one can easily exclude this case since our assumption of $J$ being one-homogeneous guarantees that it is not strictly convex." ], [ "Bias-variance estimates", "Another justification for the deterministic definition of bias as well as our choice for the distance measure in Section REF can be found in the variational model itself.", "In order to derive quantitative bounds for bias and variance in a variational model, we start with the Tikhonov regularization (Ridge regression) model related to the functional $J(u) = \\frac{1}{2} \\Vert u \\Vert _{\\mathcal {X}}^2$ .", "The optimality condition for this problem is given by $A^* (A u_\\alpha (f) -f) + \\alpha u_\\alpha (f) = 0.$ We easily see that there exists $w_\\alpha = \\frac{1}{\\alpha }(f- Au_\\alpha (f))$ such that $u_\\alpha (f) = A^* w_\\alpha $ and $Au_\\alpha (f) - Au^* + \\alpha w_\\alpha = f- Au^*.$ Now let us assume that a source condition $u^* \\in \\mathrm {Im}[A^*]$ holds, i.e.", "$u^* = A^* w^*$ for some $w^*$ .", "In this case we can subtract $\\alpha w^*$ on both sides and take a squared norm to arrive at $&\\Vert Au_\\alpha (f) - Au^* \\Vert _{\\mathcal {Y}}^2 + \\alpha ^2 \\Vert w_\\alpha - w^* \\Vert _{\\mathcal {Y}}^2\\\\& \\hspace{53.30002pt} + 2 \\alpha \\langle Au_\\alpha (f) - Au^* , w_\\alpha - w^* \\rangle \\\\= \\ &\\Vert f- Au^* \\Vert _{\\mathcal {Y}}^2 + \\alpha ^2 \\Vert w^* \\Vert _{\\mathcal {Y}}^2 \\hspace{0.89996pt} - \\hspace{0.89996pt} 2\\alpha \\langle f - Au^*, w^* \\rangle .$ Now taking the expectation on both sides and using $\\mathbb {E}[f] = f^* =Au^*$ we find $&\\mathbb {E}[ \\Vert Au_\\alpha (f) - Au^* \\Vert _{\\mathcal {Y}}^2 ] + \\alpha ^2 \\mathbb {E}[\\Vert w_\\alpha - w^* \\Vert _{\\mathcal {Y}}^2 ] \\nonumber \\\\&\\hspace{80.0pt} + 2 \\alpha \\mathbb {E}[ \\Vert u_\\alpha (f) - u^* \\Vert _{\\mathcal {X}}^2] \\nonumber \\\\ = \\ & \\mathbb {E}[\\Vert f- Au^* \\Vert _{\\mathcal {Y}}^2 ] + \\alpha ^2 \\Vert w^* \\Vert _{\\mathcal {Y}}^2.$ The left-hand side is the sum of three error terms for the solution measured in different norms: in the output space, the space of the source element, and the original space used for regularization.", "All of them can be decomposed in a bias and a variance term, e.g.", "$&\\mathbb {E}[ \\Vert u_\\alpha (f) - u^* \\Vert _{\\mathcal {X}}^2]\\\\ = &\\Vert \\mathbb {E}[u_\\alpha (f)]- u^* \\Vert _{\\mathcal {X}}^2 + \\mathbb {E}[ \\Vert u_\\alpha (f) - \\mathbb {E}[u_\\alpha (f)] \\Vert _{\\mathcal {X}}^2] .$ The term $\\mathbb {E}[\\Vert f- Au^* \\Vert _{\\mathcal {Y}}^2 ]$ in (REF ) is exactly the variance in the data.", "As a consequence $\\alpha \\Vert w^* \\Vert _{\\mathcal {Y}}$ measures the bias in this case.", "Note that in particular for zero variance we obtain a direct estimate of the bias via $\\alpha \\Vert w^* \\Vert _{\\mathcal {Y}}$ .", "Figure: TV denoising and debiasing of the Giraffe and the Parrot images for either noisy data ff or clean data f * f^*,with the same regularization parameter α=0.3\\alpha =0.3.In the case of the variational model (REF ) this can be generalized using recent approaches [5], [9], [11], [29] using the source condition $A^* w^* \\in \\partial J(u^*)$ .", "Now completely analogous computations as above yield $&\\mathbb {E}[ \\Vert Au_\\alpha (f) - Au^* \\Vert _{\\mathcal {Y}}^2 ] + \\alpha ^2 \\mathbb {E}[\\Vert w_\\alpha - w^* \\Vert _{\\mathcal {Y}}^2 ] \\\\&\\hspace{73.50006pt} + 2 \\alpha \\mathbb {E}[ D^{\\mathrm {sym}}_J(u_\\alpha (f),u^*) ] \\\\= \\ &\\mathbb {E}[\\Vert f- Au^* \\Vert _{\\mathcal {Y}}^2 ] + \\alpha ^2 \\Vert w^* \\Vert _{\\mathcal {Y}}^2 ,$ with the only difference that we now use the symmetric Bregman distance $D^{\\mathrm {sym}}_J(u_\\alpha (f),u^*) = \\langle A^*w_\\alpha - A^*w^* , u_\\alpha (f) - u^* \\rangle ,$ with $A^*w_{\\alpha } \\in \\partial J(u_\\alpha (f))$ .", "The bias-variance decomposition on the right-hand side remains the same.", "In the noiseless case it is then natural to consider this (here, deterministic) estimate as a measure of bias: $&\\Vert Au_\\alpha (f^*) - Au^* \\Vert _{\\mathcal {Y}}^2 + \\alpha ^2 \\Vert w_\\alpha - w^* \\Vert _{\\mathcal {Y}}^2 \\\\&\\hspace{83.99994pt} + 2 \\alpha D^{\\mathrm {sym}}_J(u_\\alpha (f^*),u^*) \\\\&= \\ \\alpha ^2 \\Vert w^* \\Vert _{\\mathcal {Y}}^2 ,$ Here, as already discussed in Section REF , we again consider a difference between the exact solution $u^*$ and the estimator for $\\mathbb {E}[f] = f^*$ , i.e.", "the expectation of the noise, rather than the expectations of the estimators $u_\\alpha (f)$ over all realizations of $f$ (which coincide if $J$ is quadratic).", "We observe that there are three natural distances to quantify the error and thus also the bias: a quadratic one in the output space and a predual space (related to $w$ ), and the symmetric Bregman distance related to the functional $J$ .", "The first term $\\Vert Au_\\alpha (f^*) - Au^* \\Vert _{\\mathcal {Y}}^2$ is exactly the one we use as a measure of bias.", "The second term $ \\alpha ^2 \\mathbb {E}[\\Vert w_\\alpha - w^* \\Vert _{\\mathcal {Y}}^2 ]$ is constant on the model manifold $\\mathcal {M}_{f^*}^{\\mathrm {B}}$ , since by definition of the manifold $p_\\alpha = A^* w_\\alpha $ is a subgradient of all the elements in $\\mathcal {M}_{f^*}^{\\mathrm {B}}$ .", "The third term $ D^{\\mathrm {sym}}_J(u_\\alpha (f^*),u^*)$ is not easy to control; if the manifold is appropriate, meaning that $p_\\alpha \\in \\partial J(u^*)$ , then the symmetric Bregman distance vanishes for every element in $\\mathcal {M}_{f^*}^{\\mathrm {B}}$ .", "In any other case, we do not have access to a subgradient $p^* \\in \\partial J(u^*)$ , so we cannot control the Bregman distance for any element of the manifold.", "Hence, with our method we minimize the part of the bias that we can actually control.", "In fact, if the model manifold is right, we even minimize the whole bias." ], [ "Back to the proposed method", "To sum up, the debiasing method we have introduced in Equations (REF ) and (REF ) comes down to debiasing over ${\\cal M}_{f^*}^{\\mathrm {IC}}$ and ${\\cal M}_{f^*}^{\\mathrm {B}}$ , respectively, while the results of Section guarantee the existence of the optimal debiasing $\\hat{u}_{\\alpha }(f^*)$ at least on ${\\cal M}_{f^*}^{\\mathrm {B}}$ .", "However in practice, we do not have access to the clean data $f^*$ , but often only to one noisy realization $f$ , which makes the regularization in (REF ) necessary in the first place.", "Instead of the true model manifold $\\mathcal {M}_{f^*}$ , we hence use an approximation $\\mathcal {M}_{f}$ computed from the noisy data $f$ to perform the debiasing of the reconstruction $u_\\alpha (f)$ for noisy data.", "The following experiments show that $\\mathcal {M}_{f}$ is a good approximation of $\\mathcal {M}_{f^*}$ in terms of the resulting bias and bias reduction.", "They also relate the different definitions of bias that we have considered.", "In particular, we distinguish between the statistical bias of Equation (REF ) which is the expectation over several noisy realizations $f$ and the deterministic bias that we define in Equation (REF ), which instead considers the outcome given the noiseless data $f^*$ .", "Figure REF displays the TV denoising and debiasing (using the Bregman distance model manifold) results obtained with noisy data $f$ (first row) or clean data $f^*$ (second row) with the same regularization parameter $\\alpha =0.3$ .", "We have performed the experiments for both the cartoon Giraffe image and the natural Parrot imagehttp://r0k.us/graphics/kodak/.", "First, for the Giraffe image we observe that the TV denoised solution $u_{\\alpha }(f^*)$ for clean data suffers from a heavy loss of contrast, i.e.", "from method bias.", "The debiased solution $\\hat{u}_{\\alpha }(f^*)$ however is again close to the original data $f^*$ .", "This shows that if the noiseless data is well represented by the choice of regularization (and hence $\\mathcal {M}_{f^*}$ ), i.e.", "if there is no or little model bias, the debiasing procedure allows to recover the original signal almost perfectly.", "On the other hand, the same experiments on the natural Parrot image show the problem of model bias since the choice of regularization does not entirely match the data $f^*$ .", "The debiasing allows to recover the lost contrast, but even the result for noiseless data still suffers from bias, i.e.", "the loss of small structures, which is model bias in that case.", "Besides, if $\\alpha $ is big enough to effectively remove noise during the denoising step, then the TV solutions $u_{\\alpha }(f)$ and $u_{\\alpha }(f^*)$ are close to each other.", "This leads to comparable model manifolds and hence debiased solutions, which confirms that $\\mathcal {M}_{f}$ is indeed a good approximation to $\\mathcal {M}_{f^*}$ .", "Figure: Bias estimation.", "First row: Statistical bias computed on five hundred noisy realizations of the Giraffe cartoon image.Second row: Deterministic bias computed between the clean data and the recovered solution from clean data f * f^*.In the first column, TV denoising leads to bias.In the second column, the debiasing that has been performed has reduced (or suppressed) the method bias.", "The remaining (small) model bias is due to the necessary regularization.In the third column, the difference between u ^ α (f * )\\hat{u}_{\\alpha }(f^*) and u α (f * )u_{\\alpha }(f^*) shows the bias that has been reduced by the debiasing step, hence the method bias.Figure: Bias estimation.", "First row: Statistical bias computed on five hundred noisy realizations of the Parrot natural image.Second row: Deterministic bias computed between the clean data and the recovered solution from clean data f * f^*.On the first column, TV denoising leads to both kinds of bias, model bias and method bias.On the second column, the debiasing that has been performed has reduced (or suppressed) the method bias, and the remaining bias is model bias.On the third column, the difference between u ^ α (f * )\\hat{u}_{\\alpha }(f^*) and u α (f * )u_{\\alpha }(f^*) shows the bias that has been reduced by the debiasing step, hence the method bias.Furthermore, we can assess the bias for both manifolds.", "On $\\mathcal {M}_{f^*}$ we can only use the deterministic definition (REF ) of bias whereas on $\\mathcal {M}_{f}$ we use the statistical definition (REF ).", "Figures REF and REF show the bias estimation on the Giraffe cartoon image and the natural Parrot image.", "The first row shows the estimations of the statistical bias $\\mathbb {B}^{\\text{stat}}$ for the two estimators $u_{\\alpha }(f)$ and $\\hat{u}_{\\alpha }(f)$ for noisy data $f$ .", "In the second row the bias $\\mathbb {B}^*$ for the two estimators $u_{\\alpha }(f^*)$ and $\\hat{u}_{\\alpha }(f^*)$ for clean data $f^*$ is displayed.", "This deterministic bias can also be decomposed into the associated model and method bias, whereas such a decomposition has not been defined for the statistical bias.", "The overall deterministic bias $\\mathbb {B}^*(u_{\\alpha }(f^*)) = u^*-u_{\\alpha }(f^*)$ for TV denoising appears to be really close to the statistical bias on noisy data in the first row.", "The same applies for the bias of the debiased solutions in the second column.", "This confirms that the estimation of the model manifold that we perform with noisy data is indeed a good approximation to the ideal model manifold for clean data, and that the resulting statistical and deterministic bias are closely related.", "Besides, the difference $u^*-\\hat{u}_{\\alpha }(f^*)$ in the second row shows the remaining bias after the debiasing step, which is model bias.", "For the Giraffe image, this bias is small because the cartoon image is well approximated in the model manifold associated to TV regularization.", "The Parrot image however suffers from a heavier model bias, for example the loss of the small structures around the eye.", "Finally, in the third column, the difference $\\hat{u}_{\\alpha }(f^*) - u_{\\alpha }(f^*)$ shows the error that has been removed by the debiasing step, which corresponds to the method bias.", "It is particularly interesting for the Parrot image.", "Here one can see the piecewise constant areas which correspond to the re-establishment of the lost contrast within the piecewise constant model provided by the model manifold." ], [ "Relation to inverse scale space methods", "We finally comment on the relation of the debiasing approaches to Bregman iterations respectively inverse scale space methods, which are rather efficiently reducing bias as demonstrated in many examples [27], [32], [7].", "The Bregman iteration is iteratively constructed by $u^{k+1} &\\in \\text{arg}\\min _{u \\in \\mathcal {X}} \\frac{1}{2} \\Vert A u - f \\Vert _{\\mathcal {Y}}^2 + \\alpha D_J^{p^k}(u,u^k), \\\\p^{k+1} &= p^k + \\frac{1}{\\alpha }A^* (f- Au^{k+1}) \\in \\partial J(u^{k+1}).$ In the limit $\\alpha \\rightarrow \\infty $ we obtain the time continuous inverse scale-space method, which is the differential inclusion $\\partial _t p(t) = A^* (f -Au(t)), \\hfill p(t) \\in \\partial J(u(t)),$ with initial values $u(0)=0$ , $p(0)=0$ .", "A strong relation to our debiasing approach comes from the characterization of the primal solution given $p(t)$ [8], [25], [26] $u(t) \\in \\text{arg}\\min _{u \\in \\mathcal {X}} \\Vert Au -f \\Vert _{\\mathcal {Y}}^2 \\quad \\text{s.t. }", "p(t) \\in \\partial J(u(t)).$ This reconstruction step is exactly the same as the variational debiasing step using the Bregman distance, however with a different preceding construction of the subgradient $p(t)$ (noticing that $t$ corresponds to $\\frac{1}{\\alpha }$ for the variational method).", "From the last observation it becomes apparent that the Bregman debiasing approach with (REF ) and (REF ) is exactly equivalent if the variational method yields the same subgradient as the inverse scale space method, i.e.", "$p_\\alpha =p(\\frac{1}{\\alpha })$ .", "This can indeed happen, as the results for singular vectors demonstrate [1].", "Moreover, in some cases there is full equivalence for arbitrary data, e.g.", "in a finite-dimensional denoising setting investigated in [6].", "It has been shown that for $A$ being the identity and $J(u) = \\Vert \\Gamma u \\Vert _{1}$ with $\\Gamma \\Gamma ^*$ being diagonally dominant the identity $p_\\alpha =p(\\frac{1}{\\alpha })$ holds, which implies that the Bregman debiasing approach and the inverse scale space method yield exactly the same solution.", "For other cases that do not yield a strict equivalence we include the Bregman iteration for comparison in numerical studies discussed below." ], [ "Numerical Implementation", "Primal-Dual Algorithm for Variational Regularization (Step 1) $f$ , $\\alpha > 0$ $\\sigma ,\\tau > 0$ , $u^0 = \\bar{u}^0 = 0, \\; y_1^0 = y_2^0 = 0$ not converged $y_1^{k+1} = \\frac{y_1^k + \\sigma Au^k - \\sigma f}{1 + \\sigma }$ $y_2^{k+1} = \\Pi _{B^\\infty _{\\alpha }}(y_2^k + \\sigma \\Gamma u^k)$ $u^{k+1} = u^k - \\tau (A^* y_1^{k+1} + \\Gamma ^* y_2^{k+1})$ $\\bar{u}^{k+1} = 2u^{k+1} - u^{k}$ $u_\\alpha = u^{k+1}$ , $p_\\alpha = \\frac{1}{\\alpha }A^*(f - Au_\\alpha )$ (c.f.", "(REF )) Primal-Dual Algorithm for Bias-Reduction with $\\mathcal {M}_f^{\\mathrm {B}}$ (Step 2 a)) $f$ , $\\gamma > 0$ , $p_\\alpha $ , which is obtained via Algorithm $\\sigma ,\\tau > 0$ , $u^0 = \\bar{u}^0 = 0$ , $y_1^0 = y_2^0 = 0$ not converged $y_1^{k+1} = \\frac{y_1^k + \\sigma Au^k - \\sigma f}{1 + \\sigma }$ $y_2^{k+1} = \\Pi _{B^\\infty _{\\gamma }}(y_2^k + \\sigma \\Gamma u^k)$ $u^{k+1} = u^k - \\tau (A^* y_1^{k+1} + \\Gamma ^* y_2^{k+1} - \\gamma p_\\alpha )$ $\\bar{u}^{k+1} = 2u^{k+1} - u^{k}$ $\\hat{u}_{\\alpha }= u^{k+1}$ Primal-Dual Algorithm for Bias-Reduction with $\\mathcal {M}_f^{\\mathrm {IC}}$ (Step 2 b)) $f$ , $\\gamma > 0$ and $p_\\alpha $ , which is obtained via Algorithm .", "$\\sigma ,\\tau > 0$ , $u^0 = z^0 = \\bar{u}^0 = \\bar{z}^0 = 0 , \\; y_1^0 = y_2^0 = y_3^0 = 0$ not converged $y_1^{k+1} = \\frac{y_1^k + \\sigma Au^k - \\sigma f}{1 + \\sigma }$ $y_2^{k+1} = \\Pi _{B^\\infty _{\\gamma }}(y_2^k + \\sigma \\Gamma (u^k - z^k))$ $y_3^{k+1} = \\Pi _{B^\\infty _{\\gamma }}(y_3^k + \\sigma \\Gamma z^k)$ $u^{k+1} = u^k - \\tau (A^* y_1^{k+1} + \\Gamma ^* y_2^{k+1} - \\gamma p_\\alpha )$ $z^{k+1} = z^k - \\tau (-\\Gamma ^* y_2^{k+1} + \\Gamma ^* y_3^{k+1} + 2 \\gamma p_\\alpha )$ $\\bar{u}^{k+1} = 2u^{k+1} - u^{k}$ $\\bar{z}^{k+1} = 2z^{k+1} - z^{k}$ $\\hat{u}_{\\alpha }= u^{k+1}$ In Section we have introduced a two-step-method (cf.", "Eq.", "(REF ) – (REF )) in order to compute a variationally regularized reconstruction with reduced method bias in the sense discussed in Section .", "Its solution requires the minimization of the data fidelity over the model manifold defined by a zero Bregman distance or a zero infimal convolution thereof, respectively.", "This constraint is difficult to realize numerically, but can be approximated by a rather standard variational problem.", "We can translate the hard constraint into a soft constraint such that for $\\gamma > 0$ the reformulated problems read: $&\\text{a) } &&\\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathcal {X}} \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2 + \\gamma {D}^{p_\\alpha }_J(u,u_\\alpha ),\\\\&\\text{b) } &&\\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathcal {X}} \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2 + \\gamma \\mathrm {ICB}_J^{p_{\\alpha }}(u,u_\\alpha ).$ For $\\gamma \\rightarrow \\infty $ we obtain the equivalence of the hard and soft constrained formulations.", "However, for the numerical realization already a moderately large $\\gamma $ is enough to enforce the constraint up to a satisfactory level.", "For our simulations we chose $\\gamma = 1000$ , but our tests showed that already for $\\gamma \\ge 500$ the value of the Bregman distance or its infimal convolution stays numerically zero.", "Of course the choice of the parameter $\\gamma $ depends on the specific problem we aim to solve and probably has to be adjusted slightly for different image sizes or involved operators." ], [ "Discretization", "For our numerical experiments we choose the setting $\\mathcal {X}= \\mathbb {R}^n, \\; \\mathcal {Y}= \\mathbb {R}^d$ and $J(u) = \\Vert \\Gamma u \\Vert _{1}$ .", "In general $\\Gamma \\in \\mathbb {R}^{n \\times m}$ denotes a discrete linear operator, for the experiments with total variation regularization we choose a discretization of the gradient with forward finite differences.", "For a general linear forward operator $A \\in \\mathbb {R}^{n \\times d}$ we hence end up with the following discrete optimization problems: $\\begin{alignedat}{5}&\\text{1.}", "\\; && \\quad && u_\\alpha &&\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2}\\Vert Au - f \\Vert _2^2 + \\alpha \\Vert \\Gamma u \\Vert _1,\\end{alignedat}$ $\\begin{alignedat}{5}&\\text{2.}", "\\; && \\text{a)} \\quad && \\hat{u}_{\\alpha }&&\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2} \\Vert Au - f \\Vert _2^2\\\\& \\; && \\quad && && + \\gamma \\left( \\Vert \\Gamma u \\Vert _1 - \\langle p_\\alpha ,u \\rangle \\right),\\\\& \\; && \\text{b)} \\quad && \\hat{u}_{\\alpha }&&\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2} \\Vert Au - f \\Vert _2^2\\\\& \\; && \\quad && && + \\gamma \\min _{z \\in \\mathbb {R}^n} \\Big \\lbrace \\Vert \\Gamma (u - z) \\Vert _1 - \\langle p_\\alpha ,u-z \\rangle \\\\& \\; && \\quad && && +\\Vert \\Gamma z \\Vert _1 + \\langle p_\\alpha ,z \\rangle \\Big \\rbrace ,\\end{alignedat}$ where we leave out the particular spaces for the primal (and dual) variables for the sake of simplicity in the following.", "Taking a closer look at these minimization problems, we observe that we can exactly recover the optimization problem in the first step by means of problem $2.", "\\; \\text{a)}$ if we choose $\\gamma = \\alpha $ and $p_\\alpha = 0$ .", "We therefore concentrate on the minimization problems in the second step.", "Using the notion of convex conjugates [30], the corresponding primal-dual and dual formulations of our problems are given by $\\text{a)} \\quad &\\min _u \\max _{y_1,y_2} ~ \\langle y_1,Au \\rangle + \\langle y_2,\\Gamma u \\rangle - \\gamma ~ \\langle p_\\alpha , u \\rangle \\\\& \\hspace{30.0pt} - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle - \\iota _{B^{\\infty }_\\gamma }(y_2) \\\\= &\\max _{y_1,y_2} ~ - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle - \\iota _{B^{\\infty }_\\gamma }(y_2) \\\\& \\hspace{30.0pt}- \\iota _{\\gamma p_\\alpha }(A^* y_1 + \\Gamma ^* y_2),\\\\\\text{b)} \\quad &\\min _{u,z} \\max _{y_1,y_2,y_3} ~ \\langle y_1,Au \\rangle + \\langle y_2,\\Gamma u - \\Gamma z \\rangle \\\\& \\hspace{30.0pt} + \\langle y_3,\\Gamma z \\rangle - \\gamma \\; \\langle p_\\alpha , u\\rangle + 2\\gamma \\; \\langle p_\\alpha , z \\rangle \\\\& \\hspace{30.0pt} - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle \\\\& \\hspace{30.0pt} - \\iota _{B^{\\infty }_\\gamma }(y_2)- \\iota _{B^{\\infty }_\\gamma }(y_3) \\\\= &\\max _{y_1,y_2,y_3} ~ - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle \\\\& \\hspace{30.0pt} - \\iota _{B^{\\infty }_\\gamma }(y_2)- \\iota _{B^{\\infty }_\\gamma }(y_3) \\\\& \\hspace{30.0pt}- \\iota _{\\gamma p_\\alpha }(A^* y_1 + \\Gamma ^* y_2) \\\\& \\hspace{30.0pt}- \\iota _{-2 \\gamma p_\\alpha }(-\\Gamma ^*y_2 + \\Gamma ^* y_3),$ In order to find a saddle point of the primal-dual formulations, we apply a version of the popular first-order primal-dual algorithms [28], [19], [14].", "The basic idea is to perform gradient descent on the primal and gradient ascent on the dual variables.", "Whenever the involved functionals are not differentiable, here the $\\ell ^1$ -norm, this comes down to computing the corresponding proximal mappings.", "The specific updates needed for our method are summarized in Algorithm for the first regularization problem, and Algorithm and Algorithm for the two different debiasing steps.", "We comment on our choice of the stopping criterion.", "We consider the primal-dual gap of our saddle point problem, which is defined as the difference between the primal and the dual problem for the current values of variables.", "As in the course of iterations the algorithm is approaching the saddle point, this gap converges to zero.", "Hence we consider our algorithm converged if this gap is below a certain threshold $\\epsilon _1 > 0$ .", "We point out that the indicator functions regarding the $\\ell ^\\infty $ -balls are always zero due to the projection of the dual variables in every update.", "Since the constraints with respect to the other indicator functions, for example $A^* y_1 + \\Gamma ^* y_2 - \\gamma p_\\alpha = 0$ in case a), are hard to satisfy exactly numerically, we instead control that the norm of the left-hand side is smaller than a certain threshold $\\epsilon _2$ (respectively $\\epsilon _3$ for case b)).", "All in all we stop the algorithm if the current iterates satisfy: $\\text{a)} \\quad & PD(u,y_1,y_2) = \\big (-\\gamma \\langle p_\\alpha ,u \\rangle \\\\& \\qquad + \\frac{1}{2} \\Vert Au - f \\Vert _2^2 + \\gamma \\Vert \\Gamma u \\Vert _1\\\\& \\qquad + \\frac{1}{2}\\Vert y_1 \\Vert _2^2 + \\langle y_1,f \\rangle \\big ) / n < \\epsilon _1$ and $&\\Vert A^* y_1 + \\Gamma ^* y_2 - \\gamma p_\\alpha \\Vert _1 /n < \\epsilon _2$ $\\text{b)} \\quad & PD(u,z,y_1,y_2) = \\big (-\\gamma \\langle p_\\alpha ,u \\rangle + 2\\gamma \\langle p_\\alpha ,z \\rangle \\\\& \\qquad + \\frac{1}{2} \\Vert Au - f \\Vert _2^2\\\\& \\qquad + \\gamma \\Vert \\Gamma u - \\Gamma z\\Vert _1 + \\gamma \\Vert \\Gamma z \\Vert _1\\\\& \\qquad + \\frac{1}{2}\\Vert y_1 \\Vert _2^2 + \\langle y_1,f \\rangle \\big )/n < \\epsilon _1$ and $&\\Vert A^* y_1 + \\Gamma ^* y_2 - \\gamma p_\\alpha \\Vert _1/n < \\epsilon _2, \\\\&\\Vert -\\Gamma ^* y_2 + \\Gamma ^* y_3 + 2\\gamma p_\\alpha \\Vert _1 /n < \\epsilon _3.$ Note that we normalize the primal-dual gap and the constraints by the number of primal pixels $n$ in order to keep the thresholds $\\epsilon _1, \\epsilon _2$ and $\\epsilon _3$ independent of varying image resolutions.", "We give an example for the specific choice of parameters for our total variation denoising problems in Table REF .", "Table: Choice of parameters for a total variation denoising problem of an imageof size 256x256 with values in [0,1][0,1], corrupted by Gaussian noise with variance 0.050.05.Figure: ℓ 1 \\ell ^1-deconvolution of a 1D signal.Original and noisy convolved signals, and ℓ 1 \\ell ^1-reconstruction, Bregman debiasing and Infimal convolution debiasing.Figure: ℓ 1 \\ell ^1-deconvolution of a 1D signal.Average bias and variance computed over one thousand realizations of the noisy signal forthe noisy, restored and debiased signals." ], [ "Numerical Results", "This section provides further experiments and numerical results that illustrate the proposed debiasing method." ], [ "$\\ell ^1$ -deconvolution", "The first application that we illustrate is the deconvolution of a one-dimensional signal using anisotropic shrinkage (REF ).", "Figure REF displays the original signal, the blurry signal corrupted by additive Gaussian noise with standard deviation $\\sigma =0.05$ , the $\\ell ^1$ -reconstructed signal and the debiased signals computed over the Bregman manifold $\\mathcal {M}_f^{\\mathrm {B}}$ and the infimal convolution subspace $\\mathcal {M}_f^{\\mathrm {IC}}$ .", "The last two completely overlap on these two plots.", "One can see that provided that the $\\ell ^1$ -reconstruction finds the right peak locations, the debiasing method is able to restore the amplitude of the original signal.", "Figure REF displays the evolution of the average bias of the estimated signals as well as the standard deviation of the error.", "They were computed over one thousand noisy realizations for the noisy, $\\ell ^1$ -reconstructed and debiased signals, as a function of the regularization parameter $\\alpha $ .", "These curves illustrate several behaviors: As expected, the residual variance decreases when the regularization parameter increases.", "For a very low value of $\\alpha $ , the debiasing reintroduces some noise so the average variance is higher than for the $\\ell ^1$ -reconstructed signal, revealing the bias-variance trade-off that has to be settled.", "As $\\alpha $ increases, the gap between the variance of the $\\ell ^1$ -reconstructed and debiased signal vanishes.", "On the other hand, the average bias is indeed smaller for the debiased signal than for the $\\ell ^1$ -reconstructed signal.", "Besides, for small values of the regularization parameter the average bias for the debiased signal is stable and close to zero, showing the effective reduction of the method bias.", "Then it increases by steps which correspond to the progressive vanishing of the peaks, related to model bias.", "All in all, these plots show the ability of the proposed approach to reduce the method bias (here, the loss of intensity on the peaks), hence allowing for more efficient noise reduction and reconstruction for a wider range of regularization parameters." ], [ "Anisotropic TV denoising", "In this subsection we study debiasing by means of the discrete ROF-model [31] given by: $ u_\\alpha (f) \\in \\arg \\min _{u\\in \\mathbb {R}^n} \\frac{1}{2}\\Vert u-f\\Vert _{2}^2 + \\alpha \\Vert \\Gamma u \\Vert _1,$ where the 1-norm is anisotropic, i.e.", "$\\Vert \\Gamma u \\Vert _1 = \\sum _{i=1}^{m/2} \\vert (\\Gamma u)_{1,i} \\vert + \\vert (\\Gamma u)_{2,i} \\vert ,$ with $(\\Gamma u)_1$ and $(\\Gamma u)_2$ denoting the discrete gradient images in horizontal and vertical direction, respectively.", "We compare the original denoising result of Problem (REF ) to the proposed debiased solutions obtained with the Bregman manifold $\\mathcal {M}_f^{\\mathrm {B}}$ or the infimal convolution subspace $\\mathcal {M}_f^{\\mathrm {IC}}$ .", "Figure: Evolution of (a) The total variation and the residual for the cartoon Giraffe image,(b) The average PSNR for TV denoising, Bregman debiasing and infimal convolution debiasing for the cartoon Giraffe imageand (c) The average PSNR for TV denoising, Bregman debiasing and infimal convolution debiasing for the natural Parrotimage as a function of the regularization parameter α\\alpha ." ], [ "Cartoon image", "The Giraffe cartoon image has been designed not to have model bias; it is piecewise constant, which makes it suitable for TV denoising and allows us to study the reduction of the method bias only.", "It takes values in $[0,1]$ and has been artificially corrupted with additive Gaussian noise with zero mean and variance $\\sigma ^2=0.05$ , reaching an initial PSNR of about 13dB.", "The original image and a noisy realization are already displayed on the first line of Fig.", "REF in Section .", "Figure REF displays the TV denoising result as well as the debiased solutions computed on the Bregman manifold $\\mathcal {M}_f^{\\mathrm {B}}$ or the infimal convolution subspace $\\mathcal {M}_f^{\\mathrm {IC}}$ for different values of the regularization parameter $\\alpha $ .", "On the first line, $\\alpha =0.15$ is the optimal regularization parameter for TV denoising (in terms of PSNR, see Fig.", "REF -(b)).", "However, when performing the debiasing, noise is strongly amplified.", "On the second line, $\\alpha =0.3$ is the optimal regularization parameter for debiasing, and overall, (in terms of PSNR, see Fig.", "REF -(b)).", "On the third line $\\alpha =0.6$ leads to an oversmoothed solution, but the debiasing step still allows to recover a lot of the lost contrast.", "Since we expect the variational method to systematically underestimate the value of the regularization functional and overestimates the residual (see [1] for a precise computation on singular values), we compare the corresponding quantities when varying $\\alpha $ in Figure REF -(a).", "We observe that for a very large range of values of $\\alpha $ there appears to be an almost constant offset between the values for the solution $u_\\alpha (f)$ and the debiased solution $\\hat{u}_{\\alpha }(f)$ (except for very small values of $\\alpha $ , when noise dominates).", "This seems to be due to the fact that the debiasing step can correct the bias in the regularization functional (here total variation) and residual to a certain extent.", "This corresponds well to the plot of PSNR vs. $\\alpha $ in Fig.", "REF -(b), which confirms that the PSNR after the debiasing step is significantly larger than the one in $u_\\alpha (f)$ for a large range of values of $\\alpha $ , which contains the ones relevant in practice.", "The fact that the PSNR is decreased by the debiasing step for very small $\\alpha $ corresponds to the fact that indeed the noise is amplified in such a case, visible also in the plots for the smallest value of $\\alpha $ in Figure REF .", "Altogether, these results show that the proposed debiasing approach improves the denoising of the cartoon image both visually and quantitatively.", "Figure: Denoising of the Giraffe cartoon image for different values of the regularization parameter α\\alpha .First column: TV denoising.", "Second column: Debiasing on the Bregman manifold.Third column: Debiasing on the infimal convolution subspace." ], [ "Natural image", "The debiasing can also be evaluated on natural images such as the Parrot picture.", "TV denoising on such images leads to both method bias and model bias.", "We expect to reduce the former with the proposed method, while the latter is due to the piecewise constant approximation associated with the ROF-model.", "The Parrot image takes values in $[0,1]$ and has been artificially corrupted with additive Gaussian noise with zero mean and variance $\\sigma =0.05$ , reaching an initial PSNR of about 13dB.", "The original image and a noisy realization are displayed on the first line of Figure REF .", "Analogously to Figure REF , Figure REF also displays the TV denoising result as well as the debiased solutions computed on the Bregman subspace or the infimal convolution subspace for different values of the regularization parameter $\\alpha $ .", "On the second line, $\\alpha =0.15$ is the optimal regularization parameter for TV denoising (in terms of PSNR, see Fig.", "REF -(c)).", "However, when performing the debiasing, the remaining noise is strongly amplified.", "On the third line, $\\alpha =0.3$ is the optimal regularization parameter for debiasing (in terms of PSNR, see Fig.", "REF -(c)).", "On the fourth line $\\alpha =0.6$ leads to an oversmoothed solution but the debiasing step still allows to recover the lost contrast.", "Note that in the Parrot case, the optimal result in terms of PSNR is obtained for the TV denoising, for $\\alpha =0.15$ .", "However, the debiasing obtained with $\\alpha =0.3$ visually provides a smoother result on the background, while preserving the fine structures such as the stripes around the eye.", "Note also that in each case the artifacts of TV denoising such as staircasing remain and even become more apparent.", "This however seems natural as the contrast is increased.", "Since these issues are in fact model bias they are not dealt with by the debiasing method we perform here, but could be reduced by an appropriate choice of regularization such as total generalized variation [3]." ], [ "Statistical behavior", "For both images, the statistical behavior of the proposed debiasing methods can be evaluated by computing the statistical bias $ \\mathbb {E}[u^* - \\hat{U}]$ as well as the variance $\\mathbb {V}\\mathrm {ar}[u^* - \\hat{U}]$ between the true image $u^*$ and an estimator $\\hat{U}$ .", "In our case this is either the solution of the ROF-model (REF ) or the corresponding debiased result.", "Figure REF displays the evolution of the estimated statistical bias and standard deviation of the TV, Bregman debiased and infimal convolution debiased estimators for the cartoon Giraffe and natural Parrot images, as a function of the regularization parameter $\\alpha $ .", "These curves reflect some interesting behaviors: As expected, the residual variance decreases as the regularization parameter increases.", "Besides, the variance is always slightly higher for the debiased solutions, which reflects the bias-variance compromise that has to be settled.", "However, as the regularization parameter increases, the gap between the denoised and debiased variance decreases.", "On the other hand, as the regularization parameter grows, the bias increases for each method, and it always remains higher for the denoised solutions than for the debiased solutions.", "One interesting fact is the behavior of the bias curve for the cartoon Giraffe image: for low values of the regularization parameter (up to $\\alpha \\approx 0.3$ ), the evolution of the bias for the debiased solutions is relatively stable.", "This means that for those values, one can increase the regularization parameter in order to reduce the variance without introducing too much (at this point, method) bias.", "Then, for higher regularization parameters the bias increases in a steeper way, parallel to the evolution of the original bias for the TV denoised image.", "This reflects the evolution of the model bias from this point on, when the high regularization parameter provides a model subspace whose elements are too smooth compared to the true image.", "For the natural Parrot image, the model bias occurs even for small values of the regularization parameter, because the model manifold provided by the TV regularization does not properly fit the image prior.", "Figure: Evolution of the average residual bias and standard deviationcomputed over 500 noisy realizations of (a) Giraffe and(b) Parrot for TV denoising, Bregman debiasing and infimal convolution debiasing.These curves also illustrate the optimal bias-variance balance that can be achieved with or without the debiasing procedure.", "Intuitively, one would expect the optimal bias-variance trade-off to be reached when the bias and the standard deviation curves intersect each other.", "This is indeed confirmed by the PSNR curves from Fig.", "REF -(b) and REF -(c).", "Looking at those intersection points on both curves for the TV denoised solution on the one hand and for the debiased solutions on the other hand, one can see that the optimal compromise for the debiasing is reached for a higher regularization parameter than for the denoising.", "This offers more denoising performance, and it leads to a smaller (for the Giraffe image) or equal (for the Parrot image) average bias and standard deviation." ], [ "Isotropic TV denoising", "Finally, we extend the examples presented in [16] with a few numerical results for isotropic TV denoising: $\\Vert \\Gamma u \\Vert _1 = \\sum _{i=1}^{m/2} \\sqrt{\\vert (\\Gamma u)_{1,i} \\vert ^2 + \\vert (\\Gamma u)_{2,i} \\vert ^2}.$ We then compare the denoising result to the solutions provided by the two alternative second steps of our debiasing method.", "Moreover, we also compare them to the result obtained from Bregman iterations.", "Figure REF displays the optimal (in terms of PSNR) denoising and debiasing for the Giraffe and Parrot images.", "The regularization parameter has been set to $\\alpha =0.2$ for the denoising result and to $\\alpha =0.3$ for the debiasing.", "Similarly to the anisotropic case, the debiasing both visually and quantitatively improves the quality of the cartoon Giraffe image.", "For the natural Parrot image, even though the PSNR is not improved by the debiasing process, one can still observe that the higher regularization parameter offers a better denoising of the background, while the debiasing guarantees that the fine structures around the eye are preserved with a good contrast.", "Besides, the proposed debiasing approach offers similar results to Bregman iterations, displayed in the fourth column.", "However, the interesting aspect of our debiasing approach is that we only apply a two-step procedure, while Bregman iterations have to be performed iteratively with a sufficiently high number of steps.", "Note that our numerical approach to debiasing (see Section ) is actually equivalent to performing one Bregman iteration with zero initialization of the subgradient, then updating the subgradient and solving a second Bregman step with a sufficiently high regularization parameter.", "Figure: Denoising of the Parrot image for different values of the regularization parameter α\\alpha .First column: TV denoising.", "Second column: Debiasing on the Bregman manifold.Third column: Debiasing on the infimal convolution subspace." ], [ "Conclusion", "We have introduced two variational debiasing schemes based on Bregman distances and their infimal convolution, which are applicable for nonsmooth convex regularizations and generalize known debiasing approaches for $\\ell ^1$ and TV-type regularization.", "Based on a recent axiomatic approach to debiasing by Deledalle and coworkers [16], which we further generalized towards infinite-dimensional problems, we were able to provide a theoretical basis of our debiasing approach and work out meaningful model manifolds for variational methods.", "Moreover, we were able to relate the approach to Bregman iterations and inverse scale space methods.", "From the numerical experiments we observe that the debiasing scheme improves the results for a wide range of regularization parameters, which includes the ones providing optimal results.", "Surprisingly, we often find visually optimal choices of the regularization parameters in the range where bias and standard deviation of the debiased solution are approximately of the same size.", "Various questions remain open for future studies: one might study the generalization to other regularization schemes such as total generalized variation [3], spatially adaptive methods that would further reduce the model bias [21] or nonlocal methods for improved results on natural images.", "As already indicated in the introduction, the method is theoretically not restricted to squared Hilbert-space norms.", "Instead, it can be carried out for any suitable data fidelity $H$ and we expect it to improve the results.", "From a theoretical, and in particular from a statistical viewpoint, the question is then how to relate the method to actual bias reduction, and how to properly motivate and define bias in this setting.", "Another further improvement might be achieved by only approximating the model manifold by tuning the parameter $\\gamma $ without letting it tend to infinity.", "We acknowledge a very recent and related work on the topic from another perspective, which has been developed in parallel to this work [17].", "It will be interesting to investigate the connections in future work." ], [ "Appendix", "We have included some examples and proofs in the Appendix in order not to interrupt the flow of the paper.", "These are in particular the proof for shrinkage and the calculation of the corresponding derivatives for isotropic and anisotropic shrinkage in Example REF , and the calculation of the infimal convolution of two $\\ell ^1$ -Bregman distances in Example REF ." ], [ "Shrinkage", "Let $f \\in \\ell ^2(\\mathbb {R}^d)$ be a vector-valued signal for $d \\in \\mathbb {N}$ .", "Then the solution of the isotropic shrinkage problem $u_\\alpha (f) \\in \\operatornamewithlimits{arg\\,min}_{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\alpha \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^d)}$ is given by the isotropic soft-thresholding $[u_\\alpha (f)]_i = {\\left\\lbrace \\begin{array}{ll}(1 - \\frac{\\alpha }{|f_i|}) f_i, & |f_i| > \\alpha , \\\\0, & |f_i| \\le \\alpha .\\end{array}\\right.", "}$ We first point out, that the objective allows to exploit strong duality.", "Following [2], strong duality holds if $\\mathrm {dom}( \\Vert \\cdot \\Vert _{\\ell ^1(\\mathbb {R}^d)} ) \\cap \\mathrm {cont}\\left(\\dfrac{1}{2} \\Vert \\cdot -f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 \\right) \\ne \\emptyset .$ Since the $\\ell ^1(\\mathbb {R}^d)$ -norm has full domain and the $\\ell ^2(\\mathbb {R}^d)$ -norm is continuous everywhere, this is trivially fulfilled.", "Hence, by the dual definition of the $\\ell ^1(\\mathbb {R}^d)$ -norm we find $&\\quad \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\alpha \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^d)} \\\\&= \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\sup _{\\begin{array}{c}r \\in \\ell ^{\\infty }(\\mathbb {R}^d) \\\\ \\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha \\end{array}} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,u \\rangle \\\\&= \\sup _{\\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha } \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,u \\rangle ,$ where we used strong duality to interchange the infimum and the supremum.", "We can explicitely compute the minimizer for $u$ as $u = f-r$ and hence $&\\quad \\sup _{\\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha } \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,u \\rangle \\\\&= \\sup _{\\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha } - \\dfrac{1}{2} \\Vert r\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,f \\rangle .$ This supremum can be computed explicitely pointwise with the corresponding Lagrangian $\\mathcal {L}(r_i, \\lambda ) = - \\dfrac{1}{2} |r_i|^2 + r_i \\cdot f_i + \\lambda (|r_i|^2 - \\alpha ^2)$ with $\\lambda \\le 0$ .", "Note that both the objective function and the constraints are continuously differentiable and that Slater's condition holds.", "Optimality with respect to $r_i$ yields $f_i - r_i + 2 \\lambda r_i = 0$ and hence $r_i = \\dfrac{f_i}{1- 2 \\lambda }.$ We distinguish two cases: If $|r_i| = \\alpha $ , then $\\alpha (1- 2\\lambda ) = |f_i|$ and $u_i = f_i - r_i = f_i - \\frac{f_i}{1-2\\lambda } = (1 - \\frac{\\alpha }{|f_i|}) f_i.$ The nonpositivity of $\\lambda $ implies that $|f_i| \\ge \\alpha $ .", "In case $|r_i| < \\alpha $ , we obtain that $\\lambda = 0$ and hence $r_i = f_i$ and $u_i = 0$ when $|f_i| < \\alpha $ .", "Note that since $f \\in \\ell ^2(\\mathbb {R}^d)$ there exists a finite $N$ such that $|f_i| \\le \\alpha $ for all $i > N$ .", "Hence trivially $u_\\alpha (f) \\in \\ell ^1(\\mathbb {R}^d)$ as $\\sum _{i \\in \\mathbb {N}} \\vert \\left[u_\\alpha (f)\\right]_i \\vert $ is a finite sum.", "This yields the assertion.", "$\\square $ Remark: For $d = 1$ and a square-summable sequence $f \\in \\ell ^2$ we immediately obtain the anisotropic case: The solution to $u_\\alpha \\in \\arg \\min _{u \\in \\ell ^1} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2}^2 + \\alpha \\Vert u\\Vert _{\\ell ^1}$ for $\\alpha > 0$ is given by $[u_\\alpha (f)]_i = {\\left\\lbrace \\begin{array}{ll}f_i - \\alpha ~ \\mathrm {sign}(f_i), & |f_i| \\ge \\alpha \\\\0, & |f_i| < \\alpha .\\end{array}\\right.", "}$ Directional derivative: The computation of the directional derivative requires a little more work.", "At first, let us compute the directional derivative of the function $F \\colon \\mathbb {R}^d\\backslash \\lbrace 0\\rbrace \\rightarrow \\mathbb {R}$ , $x \\mapsto \\frac{1}{|x|}$ into the direction $g \\in \\mathbb {R}^d$ .", "We define $G \\colon \\mathbb {R}^d\\backslash \\lbrace 0\\rbrace \\rightarrow \\mathbb {R}$ , $x \\mapsto \\frac{1}{|x|^2}$ and calculate $\\mathrm {d}G(x;g)&= \\lim _{t \\rightarrow 0^+} \\frac{G(x+tg) - G(x)}{t} \\\\&= \\lim _{t \\rightarrow 0^+} \\frac{1}{t} \\left( \\frac{1}{|x+tg|^2} - \\frac{1}{|x|^2} \\right) \\\\&= \\lim _{t \\rightarrow 0^+} \\frac{1}{t} \\left( \\frac{|x|^2 - |x + tg|^2}{|x|^2 |x + tg|^2} \\right) \\\\&= \\lim _{t \\rightarrow 0^+} \\frac{1}{t} \\left( \\frac{- 2 t x \\cdot g - t^2 |g|^2}{|x|^2 |x + tg|^2} \\right) \\\\&= - 2 \\frac{x \\cdot g}{|x|^4}.$ Then by the chain rule we obtain $\\mathrm {d}F(x;g)&= \\mathrm {d}\\sqrt{G}(x;g) = \\frac{\\mathrm {d}G(x;g)}{2 \\sqrt{G(x)}} \\\\&= -2 \\frac{x \\cdot g}{|x|^4} \\frac{|x|}{2} = - \\frac{x \\cdot g}{|x|^3}.$ Let us further define the projection of a vector $x \\in \\mathbb {R}^d$ onto another vector $y \\in \\mathbb {R}^d\\backslash \\lbrace 0\\rbrace $ as $\\Pi _y (x) = \\frac{y \\cdot x}{|y|^2} y.$ We now have to compute $[\\mathrm {d}u_\\alpha (f;g)]_i = \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i - [u_\\alpha (f) ]_i \\big )$ and we can distinguish four cases: Let at first $|f_i| > \\alpha $ .", "Then for $t$ small enough we have $|f_i + t g_i| > \\alpha $ and hence $\\ &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i - [u_\\alpha (f) ]_i \\big ) \\\\= &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( \\left( 1 - \\frac{\\alpha }{|f_i + t g_i|} \\right) (f_i + t g_i)\\right.\\\\& \\hspace{116.99997pt} \\left.- \\left( 1 - \\frac{\\alpha }{|f_i|} \\right) f_i \\right) \\\\= &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( f_i + t g_i - \\alpha \\frac{f_i + t g_i}{|f_i + t g_i|} - f_i + \\alpha \\frac{f_i}{|f_i|} \\right) \\\\= &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( t g_i - \\frac{\\alpha t g_i}{|f_i + t g_i|} \\right.\\\\& \\hspace{76.00006pt} \\left.", "- \\alpha f_i \\left( \\frac{1}{|f_i + t g_i|} - \\frac{1}{|f_i|} \\right) \\right) \\\\= &\\ g_i - \\alpha \\frac{g_i}{|f_i|} + \\alpha f_i \\frac{f_i \\cdot g_i}{|f_i|^3} \\\\= &\\ g_i + \\frac{\\alpha }{|f_i|} \\left( \\Pi _{f_i}(g_i) - g_i \\right).$ For $|f_i| < \\alpha $ and $t$ small enough we easily find $|f_i + t g_i| < \\alpha $ and hence $[\\mathrm {d}u_\\alpha (f;g)]_i = 0.$ In case $|f_i| = \\alpha $ we need to distinguish whether $|f_i + t g_i| > \\alpha $ or $|f_i + t g_i| \\le \\alpha $ for arbitrarily small $t$ .", "We hence compute $&\\quad && |f_i + t g_i| &&> \\alpha \\\\&\\Leftrightarrow \\ &&|f_i + t g_i|^2 &&> \\alpha ^2 \\\\&\\Leftrightarrow \\ &&|f_i|^2 + 2 t f_i \\cdot g_i + t^2 |g_i|^2 &&> \\alpha ^2 \\\\&\\Leftrightarrow \\ && 2 f_i \\cdot g_i + t |g_i|^2 &&> 0,$ which for arbitrarily small $t$ is true only if $f_i \\cdot g_i \\ge 0$ .", "Analogously we find that $|f_i + t g_i| < \\alpha $ for small $t$ is only true if $f_i \\cdot g_i < 0$ .", "Hence let now $|f_i| = \\alpha $ and $f_i \\cdot g_i \\ge 0$ .", "Then we obtain $[\\mathrm {d}u_\\alpha &(f;g)]_i= \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i \\big ) \\\\&= \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( \\left( 1- \\frac{\\alpha }{|f_i + t g_i|} \\right) (f_i + t g_i) \\right).$ Using $\\alpha = |f_i|$ , we find $& \\lim _{t \\rightarrow 0^+} \\dfrac{|f_i|f_i}{t} \\left( \\frac{1}{|f_i|} \\text{ - } \\frac{1}{|f_i + t g_i|} \\right) + g_i \\text{ - } \\frac{|f_i| g_i}{|f_i + t g_i|} \\\\&= |f_i|f_i \\frac{f_i \\cdot g_i}{|f_i|^3} \\\\&= \\Pi _{f_i}(g_i).$ In the last case $|f_i| = \\alpha $ and $f_i \\cdot g_i < 0$ , we find $[\\mathrm {d}u_\\alpha (f;g)]_i = \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i \\big ) = 0.$ Summing up we have $[\\mathrm {d}u_\\alpha (f;&g)]_i \\\\&= \\left\\lbrace \\begin{array}{lcr}\\multicolumn{2}{l}{g_i + \\frac{\\alpha }{|f_i|} \\left( \\Pi _{f_i}(g_i) - g_i \\right),\\qquad } & |f_i|> \\alpha ,\\\\0, & & |f_i|< \\alpha ,\\\\\\Pi _{f_i}(g_i),\\qquad & \\multicolumn{2}{r}{\\quad |f_i|= \\alpha ,\\ f_i \\cdot g_i > 0,} \\\\0, & \\multicolumn{2}{r}{|f_i|= \\alpha ,\\ f_i \\cdot g_i \\le 0.", "}\\end{array}\\right.$ It remains to show that $\\Big \\Vert \\frac{ u_\\alpha (f + tg) - u_\\alpha (f) }{t} - \\mathrm {d}u_\\alpha (f;g) \\Big \\Vert _{\\ell ^1(\\mathbb {R}^d)} \\rightarrow 0$ for $t \\rightarrow 0^+$ .", "Again, since $f \\in \\ell ^2(\\mathbb {R}^d)$ , there exists $N \\in \\mathbb {N}$ such that $|f_i| < \\alpha $ and hence $[\\mathrm {d}u_\\alpha (f;g)]_i = 0$ for all $i > N$ .", "The difference quotient as well vanishes for all $i > N$ , hence the above $\\ell ^1$ norm is a finite sum and thus we trivially obtain convergence in $\\ell ^1(\\mathbb {R}^d)$ .", "Remark: For $d=1$ and $f \\in \\ell ^2$ we obtain the anisotropic result: $[\\mathrm {d}u_\\alpha (f;g)]_i& \\\\ = &{\\left\\lbrace \\begin{array}{ll}g_i, & |f_i| > \\alpha \\\\0, & |f_i| < \\alpha \\\\g_i, & |f_i| = \\alpha , \\mathrm {sign}(f_i) = \\mathrm {sign}(g_i) \\\\0, & |f_i| = \\alpha , \\mathrm {sign}(f_i) \\ne \\mathrm {sign}(g_i),\\end{array}\\right.", "}$ where we mention that here $\\Pi _{f_i}(g_i) = g_i$ .", "Model manifold: The corresponding (isotropic) model manifold is given by $u \\in \\mathcal {M}_f^{\\mathrm {G}}\\Leftrightarrow u_i = {\\left\\lbrace \\begin{array}{ll}v \\in \\mathbb {R}^d, & |f_i| > \\alpha , \\\\0, & |f_i| < \\alpha , \\\\\\lambda f_i,\\ \\lambda \\ge 0, & |f_i| = \\alpha .\\end{array}\\right.", "}$ Analogously to the anisotropic case discussed in Example REF , the model manifold allows for arbitrary elements, here even including the direction, if the magnitude $|f_i|$ of the signal is strictly above the threshold parameter $\\alpha $ .", "As already discussed in Example REF , $|f_i| = \\alpha $ is the odd case of the three, since in contrast to $|f_i| > \\alpha $ it only allows for changes into the direction of the signal $f_i$ .", "If we exclude that case, we again find a linear derivative, hence a Gâteaux derivative and even a Fréchet derivative.", "Accordingly the isotropic shrinkage is the immediate generalization of the anisotropic shrinkage, which we can find as a special case for $d = 1$ .", "Summing up, the debiasing procedure on this manifold again yields the solution of hard thresholding: $[\\hat{u}(f)]_i = {\\left\\lbrace \\begin{array}{ll}f_i, & |f_i| \\ge \\alpha , \\\\0, & |f_i|< \\alpha .\\end{array}\\right.", "}$ Note that we again maintain the signal directly on the threshold." ], [ "Infimal convolution of $\\ell ^1$ Bregman distances", "Theorem 8.1 Let $\\Gamma \\colon \\ell ^2(\\mathbb {R}^n) \\rightarrow \\ell ^1(\\mathbb {R}^m)$ be linear and bounded and $J(u) = \\Vert \\Gamma u \\Vert _{\\ell ^1(\\mathbb {R}^m)}$ for $m,n \\in \\mathbb {N}$ .", "Let further $q_\\alpha \\in \\partial \\Vert \\cdot \\Vert _{\\ell ^1(\\mathbb {R}^m)} (\\Gamma u_\\alpha )$ such that $p_\\alpha = \\Gamma ^* q_\\alpha $ .", "Then $\\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ) \\le \\mathrm {ICB}_J^{p_{\\alpha }}(u, u_\\alpha ).$ $& \\quad \\mathrm {ICB}_J^{p_{\\alpha }}(u, u_\\alpha ) \\\\&= \\inf _{z \\in \\ell ^2(\\mathbb {R}^n)} ~ D_J^{p_\\alpha }(u-z,u_\\alpha ) + D_J^{-p_\\alpha }(z,-u_\\alpha ) \\\\&= \\inf _{z \\in \\ell ^2(\\mathbb {R}^n)} ~ \\Vert \\Gamma (u-z) \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle p_\\alpha ,u-z \\rangle \\\\&\\quad + \\Vert \\Gamma z\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle p_\\alpha ,z \\rangle \\\\&= \\inf _{z \\in \\ell ^2(\\mathbb {R}^n)} ~ \\Vert \\Gamma (u-z) \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q_\\alpha ,\\Gamma (u-z) \\rangle \\\\&\\quad + \\Vert \\Gamma z\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q_\\alpha ,\\Gamma z \\rangle \\\\&= \\inf _{\\Gamma z \\in \\ell ^1(\\mathbb {R}^m)} \\Vert \\Gamma (u-z) \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q_\\alpha ,\\Gamma (u-z) \\rangle \\\\&\\quad + \\Vert \\Gamma z\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q_\\alpha ,\\Gamma z \\rangle \\\\&\\ge \\inf _{w \\in \\ell ^1(\\mathbb {R}^m)} ~ \\Vert \\Gamma u - w \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q_\\alpha ,\\Gamma u -w \\rangle \\\\&\\quad + \\Vert w\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q_\\alpha ,w \\rangle \\\\&= \\inf _{w \\in \\ell ^1(\\mathbb {R}^m)} ~ D_{\\ell ^1(\\mathbb {R}^m)}^{q_\\alpha }(\\Gamma u - w, \\Gamma u_\\alpha ) \\\\&\\quad + D_{\\ell ^1(\\mathbb {R}^m)}^{-q_\\alpha }(w, -\\Gamma u_\\alpha ) \\\\&= \\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ).$ $\\square $ Note that we get equality for surjective $\\Gamma $ in Theorem REF .", "Theorem 8.2 Let $v,u \\in \\ell ^1(\\mathbb {R}^m)$ and $q \\in \\partial \\Vert v\\Vert _{\\ell ^1(\\mathbb {R}^m)}$ .", "Then $\\textnormal {ICB}_{\\ell ^1(\\mathbb {R}^m)}^q(u,v) = \\sum _{i \\in \\mathbb {N}} G(u_i, q_i)$ with $G \\colon \\mathbb {R}^m \\times \\mathbb {R}^m \\rightarrow \\mathbb {R}$ defined as $&G(u_i, q_i) \\\\&= {\\left\\lbrace \\begin{array}{ll}|u_i| (1 - |\\cos (\\varphi _i)| |q_i|), & |q_i| < |\\cos (\\varphi _i)|, \\\\|u_i| | \\sin (\\varphi _i)| \\sqrt{1 - |q_i|^2}, & |q_i| \\ge |\\cos (\\varphi _i)|.\\end{array}\\right.", "}$ where $\\varphi _i$ denotes the angle between $u_i$ and $q_i$ , i.e.", "$\\cos (\\varphi _i) |u_i| |q_i| = u_i \\cdot q_i$ with $\\varphi _i := 0$ for $q_i = 0$ or $u_i = 0$ .", "Let $f_1(u) &= D_{\\ell ^1(\\mathbb {R}^m)}^q(u,v) &= \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q,u \\rangle , \\\\f_2(u) &= D_{\\ell ^1(\\mathbb {R}^m)}^{-q}(u,-v) &= \\Vert u \\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q, u \\rangle .$ Since $(f_1 \\Box f_2)^* = f_1^* + f_2^*$ and by the definition of the biconjugate, we know that $f_1 \\Box f_2 \\ge (f_1^* + f_2^*)^*.$ (1) We shall first compute the right-hand side.", "We have $f_1^*(w) &= \\iota _{B^{\\infty }(1)}(w +q), \\\\f_2^*(w) &= \\iota _{B^{\\infty }(1)}(w -q),$ where $\\iota _{B^{\\infty }(1)}$ denotes the characteristic function of the $\\ell ^{\\infty }(\\mathbb {R}^m)$ -ball $B^{\\infty }(1) = \\big \\lbrace w \\in \\ell ^{\\infty }(\\mathbb {R}^m) ~|~ \\Vert w \\Vert _{\\ell ^{\\infty }(\\mathbb {R}^m)} \\le 1 \\big \\rbrace .$ Thus $&(f_1^* + f_2^*)^*(u) = \\sup _{w \\in \\ell ^{\\infty }(\\mathbb {R}^m) } \\langle u, w \\rangle \\\\&\\text{ s.t.}", "~ \\Vert w+q\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^m)} \\le 1, \\Vert w-q\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^m)} \\le 1.$ Taking into account the specific form of these constraints, we can carry out the computation pointwise, i.e.", "$\\sup _{w_i \\in \\mathbb {R}^m } u_i \\cdot w_i ~ \\text{ s.t.}", "~ |w_i + q_i| \\le 1, |w_i - q_i| \\le 1.$ From now on we drop the dependence on $i$ for simplicity.", "$\\bullet $ Let us first consider the case $|q| = 1$ .", "We immediately deduce that $w = 0$ and $u \\cdot w = 0$ .", "$\\bullet $ Hence we assume $|q| < 1$ from now on, and set up the corresponding Lagrangian $\\mathcal {L}(w, \\lambda , \\mu ) = - w \\cdot u &+ \\lambda (|w - q|^2 -1) \\nonumber \\\\&+ \\mu (|w + q|^2 -1).$ Both the objective functional and the constraints are differentiable, so every optimal point of (REF ) has to fulfill the four Karush-Kuhn-Tucker conditions, namely $&\\dfrac{\\partial }{\\partial w} \\mathcal {L}(w, \\lambda , \\mu ) = 0, \\quad &\\lambda (|w -q|^2-1) = 0, \\\\&\\lambda , \\mu \\ge 0, & \\mu (|w + q|^2 -1) = 0,$ Slater's condition implies the existence of Lagrange multipliers for a KKT-point of (REF ).", "The first KKT-condition yields $-u + 2 \\lambda (w-q) + 2 \\mu (w + q) = 0.$ $\\ast $ Let us first remark that the case $u=0$ causes the objective function to vanish anyway, hence in the following $u\\ne 0$ .", "$\\ast $ Then let us address the case $q=0$ in which (REF ) yields $u = 2 (\\lambda + \\mu ) w.$ In case $|w| = 1$ we find that $2 (\\lambda + \\mu ) = |u|$ , hence $w = \\frac{u}{|u|}$ .", "We infer $w \\cdot u = \\frac{u \\cdot u}{|u|} = |u|.$ Note that for $|w| < 1$ , we find that $\\lambda = \\mu = 0$ and hence $u = 0$ .", "$\\ast $ If $q \\ne 0$ , we can distinguish four cases: 1st case: $|w -q|^2 <1, |w +q|^2 = 1$ .", "Thus $\\lambda = 0$ and (REF ) yields $u = 2 \\mu (w +q).$ Since $|w + q|^2 = 1$ , we deduce $\\mu = |u|/2$ , so $w = \\dfrac{u}{|u|} - q$ and finally for the value of the objective function $w \\cdot u = \\left( \\dfrac{u}{|u|} - q \\right) \\cdot u = |u| - q \\cdot u.$ 2nd case: $|w +q|^2 <1, |w -q|^2 = 1$ .", "We analogously find $w \\cdot u = |u| + q\\cdot u.$ The first two cases thus occur whenever (insert $w$ into the conditions) $\\left| \\dfrac{u}{|u|} - 2q \\right| < 1 \\text{ or } \\left| \\dfrac{u}{|u|} + 2q \\right| < 1.$ We calculate $& \\quad &&\\left| \\dfrac{u}{|u|} - 2q \\right|^2 &&< 1 \\\\&\\Leftrightarrow && \\hspace{33.50006pt} |q|^2 &&< q \\cdot \\dfrac{u}{|u|} \\\\&\\Leftrightarrow &&\\hspace{33.50006pt} |q| &&< \\cos (\\varphi ).$ Hence $q\\cdot u >0$ and $|u| - q \\cdot u = |u| -|q \\cdot u|.$ In the second case we analogously find $|q| < -\\cos (\\varphi ),$ hence $q \\cdot u <0$ and $|u| + q \\cdot u = |u| -|q \\cdot u|,$ so we may summarize the first two cases as $w \\cdot u = |u| - |q \\cdot u| = |u| (1 - |\\cos (\\varphi )| |q|),$ whenever $|q| < |\\cos (\\varphi )|$ .", "3rd case: $|w -q|^2 =1, |w +q|^2 = 1$ .", "At first we observe that from $|w +q |^2 = |w -q|^2$ we may deduce that $w \\cdot q = 0$ .", "Therefore we have $|w +q|^2 = 1 \\Rightarrow |w| = \\sqrt{1-|q|^2}.$ We multiply the optimality condition (REF ) by $q$ and obtain $&\\qquad &&u \\cdot q &&= 2\\lambda (w -q) \\cdot q + 2\\mu (w +q) \\cdot q \\\\&\\Leftrightarrow && u \\cdot q &&= 2(\\mu - \\lambda )~ |q|^2 \\\\&\\Leftrightarrow &&(\\mu - \\lambda ) &&= \\frac{u}{2} \\cdot \\dfrac{q}{|q|^2}.$ Multiplying (REF ) by $w$ yields $u \\cdot w = 2 (\\lambda + \\mu ) |w|^2$ and another multiplication of (REF ) by $u$ yields $|u|^2 &= 2 (\\lambda +\\mu ) w \\cdot u + 2 (\\mu -\\lambda ) q\\cdot u \\\\&= 4 (\\lambda +\\mu )^2 |w|^2 + \\left( u\\cdot \\dfrac{q}{|q|} \\right)^2,$ where we inserted the previous results in the last two steps.", "We rearrange and find $2 (\\lambda + \\mu ) = \\sqrt{ |u|^2 - \\left( u \\cdot \\dfrac{q}{|q|} \\right)^2} |w|^{-1}.$ Note that $|w| > 0$ since $|q| < 1$ .", "This finally leads us to $u \\cdot w &= 2 (\\lambda + \\mu ) |w|^2 \\\\&= \\sqrt{ |u|^2 - \\left( u \\cdot \\dfrac{q}{|q|} \\right)^2} |w| \\\\&= |u| \\sqrt{\\left(1 - \\left( \\dfrac{u}{|u|} \\cdot \\dfrac{q}{|q|} \\right)^2\\right) \\left(1 - |q|^2\\right)} \\\\&= |u| \\sqrt{\\left(1 - |\\cos (\\varphi )|^2\\right) \\left(1 - |q|^2\\right)} \\\\&= |u| |\\sin (\\varphi )| \\sqrt{\\left(1 - |q|^2\\right)}.$ 4th case: $|w -q|^2 < 1, |w +q|^2 < 1$ .", "Here the first KKT-condition yields $u =0$ , which can only occur if the objective function $w\\cdot u$ vanishes anyway.", "Summing up, we have $(f_1^* + f_2^*)^*(u) = \\sum _{i \\in \\mathbb {N}} G(u_i,q_i) \\le \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^m)}.$ (2) It remains to show that $(f_1 \\Box f_2)(u) &= \\inf _{z \\in \\ell ^1(\\mathbb {R}^m)} \\sum _{i \\in \\mathbb {N}} g_i(z_i) \\\\&\\le (f_1^* + f_2^*)^*(u),$ where $g_i(z_i) = |u_i - z_i| + |z_i| - q_i \\cdot (u_i - 2z_i) \\ge 0.$ Again we need to distinguish four cases.", "1st case: If $|q_i| < \\cos (\\varphi _i)$ , we have $q_i \\cdot u_i > 0$ and we can choose $z_i = 0$ to obtain $g_i(z_i) = |u_i| - q_i \\cdot u_i = |u_i| - |q_i \\cdot u_i|.$ 2nd case: Analogously if $|q_i| < -\\cos (\\varphi _i)$ , we have $q_i \\cdot u_i < 0$ and choose $z_i = u_i$ , thus $g_i(z_i) = |u_i| + q_i \\cdot u_i = |u_i| - |q_i \\cdot u_i|.$ 3rd case: If $|q_i| = 1$ , we compute for $z_i = \\frac{u_i}{2} - \\frac{c}{2}q_i$ , $c > 0$ , $g_i(z_i) &= \\left| \\frac{u_i}{2} + \\frac{c}{2} q_i \\right| + \\left| \\frac{u_i}{2} - \\frac{c}{2} q_i \\right| - c|q_i|^2 \\\\&=\\frac{c}{2} \\left( \\left| q_i + \\frac{u_i}{c} \\right| + \\left| q_i - \\frac{u_i}{c} \\right| - 2 \\right).$ Using a Taylor expansion around $q$ we obtain $\\left| q_i + \\frac{u_i}{c} \\right| &= |q_i| + \\frac{q_i}{|q_i|} \\cdot \\frac{u_i}{c} + O (c^{-2}), \\\\\\left| q_i - \\frac{u_i}{c} \\right| &= |q_i| - \\frac{q_i}{|q_i|} \\cdot \\frac{u_i}{c} + O (c^{-2}).$ Hence with $|q_i| = 1$ we find $g_i(z_i) = \\frac{c}{2} (2 |q_i| + O(c^{-2}) - 2) = O(c^{-1}) \\rightarrow 0$ for $c \\rightarrow \\infty $ .", "Hence for every $\\varepsilon $ there exists a $c_i > 0$ such that $g_i(z_i) \\le \\varepsilon / 2^i$ .", "4th case: Finally, if $|q_i| \\ge |\\cos (\\varphi _i)|$ and $|q_i| < 1$ , we pick $z_i = 2 \\lambda _i (w_i - q_i)$ , with $\\lambda _i$ and $w_i$ being the Lagrange multiplier and the dual variable from the above computation of $(f_1^* + f_2^*)^*$ .", "It is easy to see that $g_i(z_i) = |u_i| | \\sin (\\varphi _i)| \\sqrt{1 - |q_i|^2}.$ Hence we define $z := (z_i)_i$ such that $z_i = {\\left\\lbrace \\begin{array}{ll}0, & \\text{ if } |q_i| < \\cos (\\varphi _i),\\\\u_i, & \\text{ if } |q_i| < -\\cos (\\varphi _i),\\\\\\frac{u_i}{2} - \\frac{c_i}{2}q_i, & \\text{ if } |q_i| = 1, \\\\\\lambda _i (w_i - q_i) & \\text{ if } |q_i| \\ge |\\cos (\\varphi _i)|, \\\\& \\hspace{12.5pt} |q_i| < 1.\\end{array}\\right.", "}$ Let $z^N$ denote $z$ truncated at index $N \\in \\mathbb {N}$ , i.e.", "$z_i^N = {\\left\\lbrace \\begin{array}{ll}z_i, & \\text{ if } i \\le N, \\\\0, & \\text{ else.}\\end{array}\\right.", "}$ Then trivially $z^N \\in \\ell ^1(\\mathbb {R}^m)$ and we compute $&(f_1 \\Box f_2)(u) \\le \\sum _{i \\in \\mathbb {N}} g_i(z_i^N) \\\\\\le & \\sum _{i = 1}^N \\big ( G(u_i, q_i) + \\frac{\\varepsilon }{2^i} \\big ) + \\sum _{i=N+1}^\\infty g_i(0) \\\\= &\\sum _{i = 1}^\\infty G(u_i, q_i) + \\sum _{i = 1}^N \\frac{\\varepsilon }{2^i}\\\\& + \\sum _{i=N+1}^\\infty \\big ( |u_i| - q_i \\cdot u_i - G(u_i,q_i) \\big ) \\\\\\le & \\sum _{i = 1}^\\infty G(u_i, q_i) + \\sum _{i = 1}^N \\frac{\\varepsilon }{2^i} + 3 \\sum _{i=N+1}^\\infty |u_i| \\\\\\rightarrow & \\sum _{i = 1}^\\infty G(u_i, q_i) + \\varepsilon $ as $N \\rightarrow \\infty $ .", "This completes the proof.", "$\\square $" ], [ "Acknowledgements", "This work was supported by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse.", "MB acknowledges support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster, Germany." ], [ "Numerical Implementation", "Primal-Dual Algorithm for Variational Regularization (Step 1) $f$ , $\\alpha > 0$ $\\sigma ,\\tau > 0$ , $u^0 = \\bar{u}^0 = 0, \\; y_1^0 = y_2^0 = 0$ not converged $y_1^{k+1} = \\frac{y_1^k + \\sigma Au^k - \\sigma f}{1 + \\sigma }$ $y_2^{k+1} = \\Pi _{B^\\infty _{\\alpha }}(y_2^k + \\sigma \\Gamma u^k)$ $u^{k+1} = u^k - \\tau (A^* y_1^{k+1} + \\Gamma ^* y_2^{k+1})$ $\\bar{u}^{k+1} = 2u^{k+1} - u^{k}$ $u_\\alpha = u^{k+1}$ , $p_\\alpha = \\frac{1}{\\alpha }A^*(f - Au_\\alpha )$ (c.f.", "(REF )) Primal-Dual Algorithm for Bias-Reduction with $\\mathcal {M}_f^{\\mathrm {B}}$ (Step 2 a)) $f$ , $\\gamma > 0$ , $p_\\alpha $ , which is obtained via Algorithm $\\sigma ,\\tau > 0$ , $u^0 = \\bar{u}^0 = 0$ , $y_1^0 = y_2^0 = 0$ not converged $y_1^{k+1} = \\frac{y_1^k + \\sigma Au^k - \\sigma f}{1 + \\sigma }$ $y_2^{k+1} = \\Pi _{B^\\infty _{\\gamma }}(y_2^k + \\sigma \\Gamma u^k)$ $u^{k+1} = u^k - \\tau (A^* y_1^{k+1} + \\Gamma ^* y_2^{k+1} - \\gamma p_\\alpha )$ $\\bar{u}^{k+1} = 2u^{k+1} - u^{k}$ $\\hat{u}_{\\alpha }= u^{k+1}$ Primal-Dual Algorithm for Bias-Reduction with $\\mathcal {M}_f^{\\mathrm {IC}}$ (Step 2 b)) $f$ , $\\gamma > 0$ and $p_\\alpha $ , which is obtained via Algorithm .", "$\\sigma ,\\tau > 0$ , $u^0 = z^0 = \\bar{u}^0 = \\bar{z}^0 = 0 , \\; y_1^0 = y_2^0 = y_3^0 = 0$ not converged $y_1^{k+1} = \\frac{y_1^k + \\sigma Au^k - \\sigma f}{1 + \\sigma }$ $y_2^{k+1} = \\Pi _{B^\\infty _{\\gamma }}(y_2^k + \\sigma \\Gamma (u^k - z^k))$ $y_3^{k+1} = \\Pi _{B^\\infty _{\\gamma }}(y_3^k + \\sigma \\Gamma z^k)$ $u^{k+1} = u^k - \\tau (A^* y_1^{k+1} + \\Gamma ^* y_2^{k+1} - \\gamma p_\\alpha )$ $z^{k+1} = z^k - \\tau (-\\Gamma ^* y_2^{k+1} + \\Gamma ^* y_3^{k+1} + 2 \\gamma p_\\alpha )$ $\\bar{u}^{k+1} = 2u^{k+1} - u^{k}$ $\\bar{z}^{k+1} = 2z^{k+1} - z^{k}$ $\\hat{u}_{\\alpha }= u^{k+1}$ In Section we have introduced a two-step-method (cf.", "Eq.", "(REF ) – (REF )) in order to compute a variationally regularized reconstruction with reduced method bias in the sense discussed in Section .", "Its solution requires the minimization of the data fidelity over the model manifold defined by a zero Bregman distance or a zero infimal convolution thereof, respectively.", "This constraint is difficult to realize numerically, but can be approximated by a rather standard variational problem.", "We can translate the hard constraint into a soft constraint such that for $\\gamma > 0$ the reformulated problems read: $&\\text{a) } &&\\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathcal {X}} \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2 + \\gamma {D}^{p_\\alpha }_J(u,u_\\alpha ),\\\\&\\text{b) } &&\\hat{u}_{\\alpha }\\in \\arg \\min _{u \\in \\mathcal {X}} \\frac{1}{2} \\Vert A u - f \\Vert _\\mathcal {Y}^2 + \\gamma \\mathrm {ICB}_J^{p_{\\alpha }}(u,u_\\alpha ).$ For $\\gamma \\rightarrow \\infty $ we obtain the equivalence of the hard and soft constrained formulations.", "However, for the numerical realization already a moderately large $\\gamma $ is enough to enforce the constraint up to a satisfactory level.", "For our simulations we chose $\\gamma = 1000$ , but our tests showed that already for $\\gamma \\ge 500$ the value of the Bregman distance or its infimal convolution stays numerically zero.", "Of course the choice of the parameter $\\gamma $ depends on the specific problem we aim to solve and probably has to be adjusted slightly for different image sizes or involved operators." ], [ "Discretization", "For our numerical experiments we choose the setting $\\mathcal {X}= \\mathbb {R}^n, \\; \\mathcal {Y}= \\mathbb {R}^d$ and $J(u) = \\Vert \\Gamma u \\Vert _{1}$ .", "In general $\\Gamma \\in \\mathbb {R}^{n \\times m}$ denotes a discrete linear operator, for the experiments with total variation regularization we choose a discretization of the gradient with forward finite differences.", "For a general linear forward operator $A \\in \\mathbb {R}^{n \\times d}$ we hence end up with the following discrete optimization problems: $\\begin{alignedat}{5}&\\text{1.}", "\\; && \\quad && u_\\alpha &&\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2}\\Vert Au - f \\Vert _2^2 + \\alpha \\Vert \\Gamma u \\Vert _1,\\end{alignedat}$ $\\begin{alignedat}{5}&\\text{2.}", "\\; && \\text{a)} \\quad && \\hat{u}_{\\alpha }&&\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2} \\Vert Au - f \\Vert _2^2\\\\& \\; && \\quad && && + \\gamma \\left( \\Vert \\Gamma u \\Vert _1 - \\langle p_\\alpha ,u \\rangle \\right),\\\\& \\; && \\text{b)} \\quad && \\hat{u}_{\\alpha }&&\\in \\arg \\min _{u \\in \\mathbb {R}^n} \\frac{1}{2} \\Vert Au - f \\Vert _2^2\\\\& \\; && \\quad && && + \\gamma \\min _{z \\in \\mathbb {R}^n} \\Big \\lbrace \\Vert \\Gamma (u - z) \\Vert _1 - \\langle p_\\alpha ,u-z \\rangle \\\\& \\; && \\quad && && +\\Vert \\Gamma z \\Vert _1 + \\langle p_\\alpha ,z \\rangle \\Big \\rbrace ,\\end{alignedat}$ where we leave out the particular spaces for the primal (and dual) variables for the sake of simplicity in the following.", "Taking a closer look at these minimization problems, we observe that we can exactly recover the optimization problem in the first step by means of problem $2.", "\\; \\text{a)}$ if we choose $\\gamma = \\alpha $ and $p_\\alpha = 0$ .", "We therefore concentrate on the minimization problems in the second step.", "Using the notion of convex conjugates [30], the corresponding primal-dual and dual formulations of our problems are given by $\\text{a)} \\quad &\\min _u \\max _{y_1,y_2} ~ \\langle y_1,Au \\rangle + \\langle y_2,\\Gamma u \\rangle - \\gamma ~ \\langle p_\\alpha , u \\rangle \\\\& \\hspace{30.0pt} - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle - \\iota _{B^{\\infty }_\\gamma }(y_2) \\\\= &\\max _{y_1,y_2} ~ - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle - \\iota _{B^{\\infty }_\\gamma }(y_2) \\\\& \\hspace{30.0pt}- \\iota _{\\gamma p_\\alpha }(A^* y_1 + \\Gamma ^* y_2),\\\\\\text{b)} \\quad &\\min _{u,z} \\max _{y_1,y_2,y_3} ~ \\langle y_1,Au \\rangle + \\langle y_2,\\Gamma u - \\Gamma z \\rangle \\\\& \\hspace{30.0pt} + \\langle y_3,\\Gamma z \\rangle - \\gamma \\; \\langle p_\\alpha , u\\rangle + 2\\gamma \\; \\langle p_\\alpha , z \\rangle \\\\& \\hspace{30.0pt} - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle \\\\& \\hspace{30.0pt} - \\iota _{B^{\\infty }_\\gamma }(y_2)- \\iota _{B^{\\infty }_\\gamma }(y_3) \\\\= &\\max _{y_1,y_2,y_3} ~ - \\frac{1}{2}\\Vert y_1 \\Vert _2^2 - \\langle y_1,f \\rangle \\\\& \\hspace{30.0pt} - \\iota _{B^{\\infty }_\\gamma }(y_2)- \\iota _{B^{\\infty }_\\gamma }(y_3) \\\\& \\hspace{30.0pt}- \\iota _{\\gamma p_\\alpha }(A^* y_1 + \\Gamma ^* y_2) \\\\& \\hspace{30.0pt}- \\iota _{-2 \\gamma p_\\alpha }(-\\Gamma ^*y_2 + \\Gamma ^* y_3),$ In order to find a saddle point of the primal-dual formulations, we apply a version of the popular first-order primal-dual algorithms [28], [19], [14].", "The basic idea is to perform gradient descent on the primal and gradient ascent on the dual variables.", "Whenever the involved functionals are not differentiable, here the $\\ell ^1$ -norm, this comes down to computing the corresponding proximal mappings.", "The specific updates needed for our method are summarized in Algorithm for the first regularization problem, and Algorithm and Algorithm for the two different debiasing steps.", "We comment on our choice of the stopping criterion.", "We consider the primal-dual gap of our saddle point problem, which is defined as the difference between the primal and the dual problem for the current values of variables.", "As in the course of iterations the algorithm is approaching the saddle point, this gap converges to zero.", "Hence we consider our algorithm converged if this gap is below a certain threshold $\\epsilon _1 > 0$ .", "We point out that the indicator functions regarding the $\\ell ^\\infty $ -balls are always zero due to the projection of the dual variables in every update.", "Since the constraints with respect to the other indicator functions, for example $A^* y_1 + \\Gamma ^* y_2 - \\gamma p_\\alpha = 0$ in case a), are hard to satisfy exactly numerically, we instead control that the norm of the left-hand side is smaller than a certain threshold $\\epsilon _2$ (respectively $\\epsilon _3$ for case b)).", "All in all we stop the algorithm if the current iterates satisfy: $\\text{a)} \\quad & PD(u,y_1,y_2) = \\big (-\\gamma \\langle p_\\alpha ,u \\rangle \\\\& \\qquad + \\frac{1}{2} \\Vert Au - f \\Vert _2^2 + \\gamma \\Vert \\Gamma u \\Vert _1\\\\& \\qquad + \\frac{1}{2}\\Vert y_1 \\Vert _2^2 + \\langle y_1,f \\rangle \\big ) / n < \\epsilon _1$ and $&\\Vert A^* y_1 + \\Gamma ^* y_2 - \\gamma p_\\alpha \\Vert _1 /n < \\epsilon _2$ $\\text{b)} \\quad & PD(u,z,y_1,y_2) = \\big (-\\gamma \\langle p_\\alpha ,u \\rangle + 2\\gamma \\langle p_\\alpha ,z \\rangle \\\\& \\qquad + \\frac{1}{2} \\Vert Au - f \\Vert _2^2\\\\& \\qquad + \\gamma \\Vert \\Gamma u - \\Gamma z\\Vert _1 + \\gamma \\Vert \\Gamma z \\Vert _1\\\\& \\qquad + \\frac{1}{2}\\Vert y_1 \\Vert _2^2 + \\langle y_1,f \\rangle \\big )/n < \\epsilon _1$ and $&\\Vert A^* y_1 + \\Gamma ^* y_2 - \\gamma p_\\alpha \\Vert _1/n < \\epsilon _2, \\\\&\\Vert -\\Gamma ^* y_2 + \\Gamma ^* y_3 + 2\\gamma p_\\alpha \\Vert _1 /n < \\epsilon _3.$ Note that we normalize the primal-dual gap and the constraints by the number of primal pixels $n$ in order to keep the thresholds $\\epsilon _1, \\epsilon _2$ and $\\epsilon _3$ independent of varying image resolutions.", "We give an example for the specific choice of parameters for our total variation denoising problems in Table REF .", "Table: Choice of parameters for a total variation denoising problem of an imageof size 256x256 with values in [0,1][0,1], corrupted by Gaussian noise with variance 0.050.05.Figure: ℓ 1 \\ell ^1-deconvolution of a 1D signal.Original and noisy convolved signals, and ℓ 1 \\ell ^1-reconstruction, Bregman debiasing and Infimal convolution debiasing.Figure: ℓ 1 \\ell ^1-deconvolution of a 1D signal.Average bias and variance computed over one thousand realizations of the noisy signal forthe noisy, restored and debiased signals." ], [ "Numerical Results", "This section provides further experiments and numerical results that illustrate the proposed debiasing method." ], [ "$\\ell ^1$ -deconvolution", "The first application that we illustrate is the deconvolution of a one-dimensional signal using anisotropic shrinkage (REF ).", "Figure REF displays the original signal, the blurry signal corrupted by additive Gaussian noise with standard deviation $\\sigma =0.05$ , the $\\ell ^1$ -reconstructed signal and the debiased signals computed over the Bregman manifold $\\mathcal {M}_f^{\\mathrm {B}}$ and the infimal convolution subspace $\\mathcal {M}_f^{\\mathrm {IC}}$ .", "The last two completely overlap on these two plots.", "One can see that provided that the $\\ell ^1$ -reconstruction finds the right peak locations, the debiasing method is able to restore the amplitude of the original signal.", "Figure REF displays the evolution of the average bias of the estimated signals as well as the standard deviation of the error.", "They were computed over one thousand noisy realizations for the noisy, $\\ell ^1$ -reconstructed and debiased signals, as a function of the regularization parameter $\\alpha $ .", "These curves illustrate several behaviors: As expected, the residual variance decreases when the regularization parameter increases.", "For a very low value of $\\alpha $ , the debiasing reintroduces some noise so the average variance is higher than for the $\\ell ^1$ -reconstructed signal, revealing the bias-variance trade-off that has to be settled.", "As $\\alpha $ increases, the gap between the variance of the $\\ell ^1$ -reconstructed and debiased signal vanishes.", "On the other hand, the average bias is indeed smaller for the debiased signal than for the $\\ell ^1$ -reconstructed signal.", "Besides, for small values of the regularization parameter the average bias for the debiased signal is stable and close to zero, showing the effective reduction of the method bias.", "Then it increases by steps which correspond to the progressive vanishing of the peaks, related to model bias.", "All in all, these plots show the ability of the proposed approach to reduce the method bias (here, the loss of intensity on the peaks), hence allowing for more efficient noise reduction and reconstruction for a wider range of regularization parameters." ], [ "Anisotropic TV denoising", "In this subsection we study debiasing by means of the discrete ROF-model [31] given by: $ u_\\alpha (f) \\in \\arg \\min _{u\\in \\mathbb {R}^n} \\frac{1}{2}\\Vert u-f\\Vert _{2}^2 + \\alpha \\Vert \\Gamma u \\Vert _1,$ where the 1-norm is anisotropic, i.e.", "$\\Vert \\Gamma u \\Vert _1 = \\sum _{i=1}^{m/2} \\vert (\\Gamma u)_{1,i} \\vert + \\vert (\\Gamma u)_{2,i} \\vert ,$ with $(\\Gamma u)_1$ and $(\\Gamma u)_2$ denoting the discrete gradient images in horizontal and vertical direction, respectively.", "We compare the original denoising result of Problem (REF ) to the proposed debiased solutions obtained with the Bregman manifold $\\mathcal {M}_f^{\\mathrm {B}}$ or the infimal convolution subspace $\\mathcal {M}_f^{\\mathrm {IC}}$ .", "Figure: Evolution of (a) The total variation and the residual for the cartoon Giraffe image,(b) The average PSNR for TV denoising, Bregman debiasing and infimal convolution debiasing for the cartoon Giraffe imageand (c) The average PSNR for TV denoising, Bregman debiasing and infimal convolution debiasing for the natural Parrotimage as a function of the regularization parameter α\\alpha ." ], [ "Cartoon image", "The Giraffe cartoon image has been designed not to have model bias; it is piecewise constant, which makes it suitable for TV denoising and allows us to study the reduction of the method bias only.", "It takes values in $[0,1]$ and has been artificially corrupted with additive Gaussian noise with zero mean and variance $\\sigma ^2=0.05$ , reaching an initial PSNR of about 13dB.", "The original image and a noisy realization are already displayed on the first line of Fig.", "REF in Section .", "Figure REF displays the TV denoising result as well as the debiased solutions computed on the Bregman manifold $\\mathcal {M}_f^{\\mathrm {B}}$ or the infimal convolution subspace $\\mathcal {M}_f^{\\mathrm {IC}}$ for different values of the regularization parameter $\\alpha $ .", "On the first line, $\\alpha =0.15$ is the optimal regularization parameter for TV denoising (in terms of PSNR, see Fig.", "REF -(b)).", "However, when performing the debiasing, noise is strongly amplified.", "On the second line, $\\alpha =0.3$ is the optimal regularization parameter for debiasing, and overall, (in terms of PSNR, see Fig.", "REF -(b)).", "On the third line $\\alpha =0.6$ leads to an oversmoothed solution, but the debiasing step still allows to recover a lot of the lost contrast.", "Since we expect the variational method to systematically underestimate the value of the regularization functional and overestimates the residual (see [1] for a precise computation on singular values), we compare the corresponding quantities when varying $\\alpha $ in Figure REF -(a).", "We observe that for a very large range of values of $\\alpha $ there appears to be an almost constant offset between the values for the solution $u_\\alpha (f)$ and the debiased solution $\\hat{u}_{\\alpha }(f)$ (except for very small values of $\\alpha $ , when noise dominates).", "This seems to be due to the fact that the debiasing step can correct the bias in the regularization functional (here total variation) and residual to a certain extent.", "This corresponds well to the plot of PSNR vs. $\\alpha $ in Fig.", "REF -(b), which confirms that the PSNR after the debiasing step is significantly larger than the one in $u_\\alpha (f)$ for a large range of values of $\\alpha $ , which contains the ones relevant in practice.", "The fact that the PSNR is decreased by the debiasing step for very small $\\alpha $ corresponds to the fact that indeed the noise is amplified in such a case, visible also in the plots for the smallest value of $\\alpha $ in Figure REF .", "Altogether, these results show that the proposed debiasing approach improves the denoising of the cartoon image both visually and quantitatively.", "Figure: Denoising of the Giraffe cartoon image for different values of the regularization parameter α\\alpha .First column: TV denoising.", "Second column: Debiasing on the Bregman manifold.Third column: Debiasing on the infimal convolution subspace." ], [ "Natural image", "The debiasing can also be evaluated on natural images such as the Parrot picture.", "TV denoising on such images leads to both method bias and model bias.", "We expect to reduce the former with the proposed method, while the latter is due to the piecewise constant approximation associated with the ROF-model.", "The Parrot image takes values in $[0,1]$ and has been artificially corrupted with additive Gaussian noise with zero mean and variance $\\sigma =0.05$ , reaching an initial PSNR of about 13dB.", "The original image and a noisy realization are displayed on the first line of Figure REF .", "Analogously to Figure REF , Figure REF also displays the TV denoising result as well as the debiased solutions computed on the Bregman subspace or the infimal convolution subspace for different values of the regularization parameter $\\alpha $ .", "On the second line, $\\alpha =0.15$ is the optimal regularization parameter for TV denoising (in terms of PSNR, see Fig.", "REF -(c)).", "However, when performing the debiasing, the remaining noise is strongly amplified.", "On the third line, $\\alpha =0.3$ is the optimal regularization parameter for debiasing (in terms of PSNR, see Fig.", "REF -(c)).", "On the fourth line $\\alpha =0.6$ leads to an oversmoothed solution but the debiasing step still allows to recover the lost contrast.", "Note that in the Parrot case, the optimal result in terms of PSNR is obtained for the TV denoising, for $\\alpha =0.15$ .", "However, the debiasing obtained with $\\alpha =0.3$ visually provides a smoother result on the background, while preserving the fine structures such as the stripes around the eye.", "Note also that in each case the artifacts of TV denoising such as staircasing remain and even become more apparent.", "This however seems natural as the contrast is increased.", "Since these issues are in fact model bias they are not dealt with by the debiasing method we perform here, but could be reduced by an appropriate choice of regularization such as total generalized variation [3]." ], [ "Statistical behavior", "For both images, the statistical behavior of the proposed debiasing methods can be evaluated by computing the statistical bias $ \\mathbb {E}[u^* - \\hat{U}]$ as well as the variance $\\mathbb {V}\\mathrm {ar}[u^* - \\hat{U}]$ between the true image $u^*$ and an estimator $\\hat{U}$ .", "In our case this is either the solution of the ROF-model (REF ) or the corresponding debiased result.", "Figure REF displays the evolution of the estimated statistical bias and standard deviation of the TV, Bregman debiased and infimal convolution debiased estimators for the cartoon Giraffe and natural Parrot images, as a function of the regularization parameter $\\alpha $ .", "These curves reflect some interesting behaviors: As expected, the residual variance decreases as the regularization parameter increases.", "Besides, the variance is always slightly higher for the debiased solutions, which reflects the bias-variance compromise that has to be settled.", "However, as the regularization parameter increases, the gap between the denoised and debiased variance decreases.", "On the other hand, as the regularization parameter grows, the bias increases for each method, and it always remains higher for the denoised solutions than for the debiased solutions.", "One interesting fact is the behavior of the bias curve for the cartoon Giraffe image: for low values of the regularization parameter (up to $\\alpha \\approx 0.3$ ), the evolution of the bias for the debiased solutions is relatively stable.", "This means that for those values, one can increase the regularization parameter in order to reduce the variance without introducing too much (at this point, method) bias.", "Then, for higher regularization parameters the bias increases in a steeper way, parallel to the evolution of the original bias for the TV denoised image.", "This reflects the evolution of the model bias from this point on, when the high regularization parameter provides a model subspace whose elements are too smooth compared to the true image.", "For the natural Parrot image, the model bias occurs even for small values of the regularization parameter, because the model manifold provided by the TV regularization does not properly fit the image prior.", "Figure: Evolution of the average residual bias and standard deviationcomputed over 500 noisy realizations of (a) Giraffe and(b) Parrot for TV denoising, Bregman debiasing and infimal convolution debiasing.These curves also illustrate the optimal bias-variance balance that can be achieved with or without the debiasing procedure.", "Intuitively, one would expect the optimal bias-variance trade-off to be reached when the bias and the standard deviation curves intersect each other.", "This is indeed confirmed by the PSNR curves from Fig.", "REF -(b) and REF -(c).", "Looking at those intersection points on both curves for the TV denoised solution on the one hand and for the debiased solutions on the other hand, one can see that the optimal compromise for the debiasing is reached for a higher regularization parameter than for the denoising.", "This offers more denoising performance, and it leads to a smaller (for the Giraffe image) or equal (for the Parrot image) average bias and standard deviation." ], [ "Isotropic TV denoising", "Finally, we extend the examples presented in [16] with a few numerical results for isotropic TV denoising: $\\Vert \\Gamma u \\Vert _1 = \\sum _{i=1}^{m/2} \\sqrt{\\vert (\\Gamma u)_{1,i} \\vert ^2 + \\vert (\\Gamma u)_{2,i} \\vert ^2}.$ We then compare the denoising result to the solutions provided by the two alternative second steps of our debiasing method.", "Moreover, we also compare them to the result obtained from Bregman iterations.", "Figure REF displays the optimal (in terms of PSNR) denoising and debiasing for the Giraffe and Parrot images.", "The regularization parameter has been set to $\\alpha =0.2$ for the denoising result and to $\\alpha =0.3$ for the debiasing.", "Similarly to the anisotropic case, the debiasing both visually and quantitatively improves the quality of the cartoon Giraffe image.", "For the natural Parrot image, even though the PSNR is not improved by the debiasing process, one can still observe that the higher regularization parameter offers a better denoising of the background, while the debiasing guarantees that the fine structures around the eye are preserved with a good contrast.", "Besides, the proposed debiasing approach offers similar results to Bregman iterations, displayed in the fourth column.", "However, the interesting aspect of our debiasing approach is that we only apply a two-step procedure, while Bregman iterations have to be performed iteratively with a sufficiently high number of steps.", "Note that our numerical approach to debiasing (see Section ) is actually equivalent to performing one Bregman iteration with zero initialization of the subgradient, then updating the subgradient and solving a second Bregman step with a sufficiently high regularization parameter.", "Figure: Denoising of the Parrot image for different values of the regularization parameter α\\alpha .First column: TV denoising.", "Second column: Debiasing on the Bregman manifold.Third column: Debiasing on the infimal convolution subspace." ], [ "Conclusion", "We have introduced two variational debiasing schemes based on Bregman distances and their infimal convolution, which are applicable for nonsmooth convex regularizations and generalize known debiasing approaches for $\\ell ^1$ and TV-type regularization.", "Based on a recent axiomatic approach to debiasing by Deledalle and coworkers [16], which we further generalized towards infinite-dimensional problems, we were able to provide a theoretical basis of our debiasing approach and work out meaningful model manifolds for variational methods.", "Moreover, we were able to relate the approach to Bregman iterations and inverse scale space methods.", "From the numerical experiments we observe that the debiasing scheme improves the results for a wide range of regularization parameters, which includes the ones providing optimal results.", "Surprisingly, we often find visually optimal choices of the regularization parameters in the range where bias and standard deviation of the debiased solution are approximately of the same size.", "Various questions remain open for future studies: one might study the generalization to other regularization schemes such as total generalized variation [3], spatially adaptive methods that would further reduce the model bias [21] or nonlocal methods for improved results on natural images.", "As already indicated in the introduction, the method is theoretically not restricted to squared Hilbert-space norms.", "Instead, it can be carried out for any suitable data fidelity $H$ and we expect it to improve the results.", "From a theoretical, and in particular from a statistical viewpoint, the question is then how to relate the method to actual bias reduction, and how to properly motivate and define bias in this setting.", "Another further improvement might be achieved by only approximating the model manifold by tuning the parameter $\\gamma $ without letting it tend to infinity.", "We acknowledge a very recent and related work on the topic from another perspective, which has been developed in parallel to this work [17].", "It will be interesting to investigate the connections in future work." ], [ "Appendix", "We have included some examples and proofs in the Appendix in order not to interrupt the flow of the paper.", "These are in particular the proof for shrinkage and the calculation of the corresponding derivatives for isotropic and anisotropic shrinkage in Example REF , and the calculation of the infimal convolution of two $\\ell ^1$ -Bregman distances in Example REF ." ], [ "Shrinkage", "Let $f \\in \\ell ^2(\\mathbb {R}^d)$ be a vector-valued signal for $d \\in \\mathbb {N}$ .", "Then the solution of the isotropic shrinkage problem $u_\\alpha (f) \\in \\operatornamewithlimits{arg\\,min}_{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\alpha \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^d)}$ is given by the isotropic soft-thresholding $[u_\\alpha (f)]_i = {\\left\\lbrace \\begin{array}{ll}(1 - \\frac{\\alpha }{|f_i|}) f_i, & |f_i| > \\alpha , \\\\0, & |f_i| \\le \\alpha .\\end{array}\\right.", "}$ We first point out, that the objective allows to exploit strong duality.", "Following [2], strong duality holds if $\\mathrm {dom}( \\Vert \\cdot \\Vert _{\\ell ^1(\\mathbb {R}^d)} ) \\cap \\mathrm {cont}\\left(\\dfrac{1}{2} \\Vert \\cdot -f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 \\right) \\ne \\emptyset .$ Since the $\\ell ^1(\\mathbb {R}^d)$ -norm has full domain and the $\\ell ^2(\\mathbb {R}^d)$ -norm is continuous everywhere, this is trivially fulfilled.", "Hence, by the dual definition of the $\\ell ^1(\\mathbb {R}^d)$ -norm we find $&\\quad \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\alpha \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^d)} \\\\&= \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\sup _{\\begin{array}{c}r \\in \\ell ^{\\infty }(\\mathbb {R}^d) \\\\ \\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha \\end{array}} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,u \\rangle \\\\&= \\sup _{\\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha } \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,u \\rangle ,$ where we used strong duality to interchange the infimum and the supremum.", "We can explicitely compute the minimizer for $u$ as $u = f-r$ and hence $&\\quad \\sup _{\\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha } \\min _{u \\in \\ell ^1(\\mathbb {R}^d)} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,u \\rangle \\\\&= \\sup _{\\Vert r\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^d)} \\le \\alpha } - \\dfrac{1}{2} \\Vert r\\Vert _{\\ell ^2(\\mathbb {R}^d)}^2 + \\langle r,f \\rangle .$ This supremum can be computed explicitely pointwise with the corresponding Lagrangian $\\mathcal {L}(r_i, \\lambda ) = - \\dfrac{1}{2} |r_i|^2 + r_i \\cdot f_i + \\lambda (|r_i|^2 - \\alpha ^2)$ with $\\lambda \\le 0$ .", "Note that both the objective function and the constraints are continuously differentiable and that Slater's condition holds.", "Optimality with respect to $r_i$ yields $f_i - r_i + 2 \\lambda r_i = 0$ and hence $r_i = \\dfrac{f_i}{1- 2 \\lambda }.$ We distinguish two cases: If $|r_i| = \\alpha $ , then $\\alpha (1- 2\\lambda ) = |f_i|$ and $u_i = f_i - r_i = f_i - \\frac{f_i}{1-2\\lambda } = (1 - \\frac{\\alpha }{|f_i|}) f_i.$ The nonpositivity of $\\lambda $ implies that $|f_i| \\ge \\alpha $ .", "In case $|r_i| < \\alpha $ , we obtain that $\\lambda = 0$ and hence $r_i = f_i$ and $u_i = 0$ when $|f_i| < \\alpha $ .", "Note that since $f \\in \\ell ^2(\\mathbb {R}^d)$ there exists a finite $N$ such that $|f_i| \\le \\alpha $ for all $i > N$ .", "Hence trivially $u_\\alpha (f) \\in \\ell ^1(\\mathbb {R}^d)$ as $\\sum _{i \\in \\mathbb {N}} \\vert \\left[u_\\alpha (f)\\right]_i \\vert $ is a finite sum.", "This yields the assertion.", "$\\square $ Remark: For $d = 1$ and a square-summable sequence $f \\in \\ell ^2$ we immediately obtain the anisotropic case: The solution to $u_\\alpha \\in \\arg \\min _{u \\in \\ell ^1} \\dfrac{1}{2} \\Vert u-f\\Vert _{\\ell ^2}^2 + \\alpha \\Vert u\\Vert _{\\ell ^1}$ for $\\alpha > 0$ is given by $[u_\\alpha (f)]_i = {\\left\\lbrace \\begin{array}{ll}f_i - \\alpha ~ \\mathrm {sign}(f_i), & |f_i| \\ge \\alpha \\\\0, & |f_i| < \\alpha .\\end{array}\\right.", "}$ Directional derivative: The computation of the directional derivative requires a little more work.", "At first, let us compute the directional derivative of the function $F \\colon \\mathbb {R}^d\\backslash \\lbrace 0\\rbrace \\rightarrow \\mathbb {R}$ , $x \\mapsto \\frac{1}{|x|}$ into the direction $g \\in \\mathbb {R}^d$ .", "We define $G \\colon \\mathbb {R}^d\\backslash \\lbrace 0\\rbrace \\rightarrow \\mathbb {R}$ , $x \\mapsto \\frac{1}{|x|^2}$ and calculate $\\mathrm {d}G(x;g)&= \\lim _{t \\rightarrow 0^+} \\frac{G(x+tg) - G(x)}{t} \\\\&= \\lim _{t \\rightarrow 0^+} \\frac{1}{t} \\left( \\frac{1}{|x+tg|^2} - \\frac{1}{|x|^2} \\right) \\\\&= \\lim _{t \\rightarrow 0^+} \\frac{1}{t} \\left( \\frac{|x|^2 - |x + tg|^2}{|x|^2 |x + tg|^2} \\right) \\\\&= \\lim _{t \\rightarrow 0^+} \\frac{1}{t} \\left( \\frac{- 2 t x \\cdot g - t^2 |g|^2}{|x|^2 |x + tg|^2} \\right) \\\\&= - 2 \\frac{x \\cdot g}{|x|^4}.$ Then by the chain rule we obtain $\\mathrm {d}F(x;g)&= \\mathrm {d}\\sqrt{G}(x;g) = \\frac{\\mathrm {d}G(x;g)}{2 \\sqrt{G(x)}} \\\\&= -2 \\frac{x \\cdot g}{|x|^4} \\frac{|x|}{2} = - \\frac{x \\cdot g}{|x|^3}.$ Let us further define the projection of a vector $x \\in \\mathbb {R}^d$ onto another vector $y \\in \\mathbb {R}^d\\backslash \\lbrace 0\\rbrace $ as $\\Pi _y (x) = \\frac{y \\cdot x}{|y|^2} y.$ We now have to compute $[\\mathrm {d}u_\\alpha (f;g)]_i = \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i - [u_\\alpha (f) ]_i \\big )$ and we can distinguish four cases: Let at first $|f_i| > \\alpha $ .", "Then for $t$ small enough we have $|f_i + t g_i| > \\alpha $ and hence $\\ &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i - [u_\\alpha (f) ]_i \\big ) \\\\= &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( \\left( 1 - \\frac{\\alpha }{|f_i + t g_i|} \\right) (f_i + t g_i)\\right.\\\\& \\hspace{116.99997pt} \\left.- \\left( 1 - \\frac{\\alpha }{|f_i|} \\right) f_i \\right) \\\\= &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( f_i + t g_i - \\alpha \\frac{f_i + t g_i}{|f_i + t g_i|} - f_i + \\alpha \\frac{f_i}{|f_i|} \\right) \\\\= &\\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( t g_i - \\frac{\\alpha t g_i}{|f_i + t g_i|} \\right.\\\\& \\hspace{76.00006pt} \\left.", "- \\alpha f_i \\left( \\frac{1}{|f_i + t g_i|} - \\frac{1}{|f_i|} \\right) \\right) \\\\= &\\ g_i - \\alpha \\frac{g_i}{|f_i|} + \\alpha f_i \\frac{f_i \\cdot g_i}{|f_i|^3} \\\\= &\\ g_i + \\frac{\\alpha }{|f_i|} \\left( \\Pi _{f_i}(g_i) - g_i \\right).$ For $|f_i| < \\alpha $ and $t$ small enough we easily find $|f_i + t g_i| < \\alpha $ and hence $[\\mathrm {d}u_\\alpha (f;g)]_i = 0.$ In case $|f_i| = \\alpha $ we need to distinguish whether $|f_i + t g_i| > \\alpha $ or $|f_i + t g_i| \\le \\alpha $ for arbitrarily small $t$ .", "We hence compute $&\\quad && |f_i + t g_i| &&> \\alpha \\\\&\\Leftrightarrow \\ &&|f_i + t g_i|^2 &&> \\alpha ^2 \\\\&\\Leftrightarrow \\ &&|f_i|^2 + 2 t f_i \\cdot g_i + t^2 |g_i|^2 &&> \\alpha ^2 \\\\&\\Leftrightarrow \\ && 2 f_i \\cdot g_i + t |g_i|^2 &&> 0,$ which for arbitrarily small $t$ is true only if $f_i \\cdot g_i \\ge 0$ .", "Analogously we find that $|f_i + t g_i| < \\alpha $ for small $t$ is only true if $f_i \\cdot g_i < 0$ .", "Hence let now $|f_i| = \\alpha $ and $f_i \\cdot g_i \\ge 0$ .", "Then we obtain $[\\mathrm {d}u_\\alpha &(f;g)]_i= \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i \\big ) \\\\&= \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\left( \\left( 1- \\frac{\\alpha }{|f_i + t g_i|} \\right) (f_i + t g_i) \\right).$ Using $\\alpha = |f_i|$ , we find $& \\lim _{t \\rightarrow 0^+} \\dfrac{|f_i|f_i}{t} \\left( \\frac{1}{|f_i|} \\text{ - } \\frac{1}{|f_i + t g_i|} \\right) + g_i \\text{ - } \\frac{|f_i| g_i}{|f_i + t g_i|} \\\\&= |f_i|f_i \\frac{f_i \\cdot g_i}{|f_i|^3} \\\\&= \\Pi _{f_i}(g_i).$ In the last case $|f_i| = \\alpha $ and $f_i \\cdot g_i < 0$ , we find $[\\mathrm {d}u_\\alpha (f;g)]_i = \\lim _{t \\rightarrow 0^+} \\dfrac{1}{t} \\big ( [u_\\alpha (f + tg) ]_i \\big ) = 0.$ Summing up we have $[\\mathrm {d}u_\\alpha (f;&g)]_i \\\\&= \\left\\lbrace \\begin{array}{lcr}\\multicolumn{2}{l}{g_i + \\frac{\\alpha }{|f_i|} \\left( \\Pi _{f_i}(g_i) - g_i \\right),\\qquad } & |f_i|> \\alpha ,\\\\0, & & |f_i|< \\alpha ,\\\\\\Pi _{f_i}(g_i),\\qquad & \\multicolumn{2}{r}{\\quad |f_i|= \\alpha ,\\ f_i \\cdot g_i > 0,} \\\\0, & \\multicolumn{2}{r}{|f_i|= \\alpha ,\\ f_i \\cdot g_i \\le 0.", "}\\end{array}\\right.$ It remains to show that $\\Big \\Vert \\frac{ u_\\alpha (f + tg) - u_\\alpha (f) }{t} - \\mathrm {d}u_\\alpha (f;g) \\Big \\Vert _{\\ell ^1(\\mathbb {R}^d)} \\rightarrow 0$ for $t \\rightarrow 0^+$ .", "Again, since $f \\in \\ell ^2(\\mathbb {R}^d)$ , there exists $N \\in \\mathbb {N}$ such that $|f_i| < \\alpha $ and hence $[\\mathrm {d}u_\\alpha (f;g)]_i = 0$ for all $i > N$ .", "The difference quotient as well vanishes for all $i > N$ , hence the above $\\ell ^1$ norm is a finite sum and thus we trivially obtain convergence in $\\ell ^1(\\mathbb {R}^d)$ .", "Remark: For $d=1$ and $f \\in \\ell ^2$ we obtain the anisotropic result: $[\\mathrm {d}u_\\alpha (f;g)]_i& \\\\ = &{\\left\\lbrace \\begin{array}{ll}g_i, & |f_i| > \\alpha \\\\0, & |f_i| < \\alpha \\\\g_i, & |f_i| = \\alpha , \\mathrm {sign}(f_i) = \\mathrm {sign}(g_i) \\\\0, & |f_i| = \\alpha , \\mathrm {sign}(f_i) \\ne \\mathrm {sign}(g_i),\\end{array}\\right.", "}$ where we mention that here $\\Pi _{f_i}(g_i) = g_i$ .", "Model manifold: The corresponding (isotropic) model manifold is given by $u \\in \\mathcal {M}_f^{\\mathrm {G}}\\Leftrightarrow u_i = {\\left\\lbrace \\begin{array}{ll}v \\in \\mathbb {R}^d, & |f_i| > \\alpha , \\\\0, & |f_i| < \\alpha , \\\\\\lambda f_i,\\ \\lambda \\ge 0, & |f_i| = \\alpha .\\end{array}\\right.", "}$ Analogously to the anisotropic case discussed in Example REF , the model manifold allows for arbitrary elements, here even including the direction, if the magnitude $|f_i|$ of the signal is strictly above the threshold parameter $\\alpha $ .", "As already discussed in Example REF , $|f_i| = \\alpha $ is the odd case of the three, since in contrast to $|f_i| > \\alpha $ it only allows for changes into the direction of the signal $f_i$ .", "If we exclude that case, we again find a linear derivative, hence a Gâteaux derivative and even a Fréchet derivative.", "Accordingly the isotropic shrinkage is the immediate generalization of the anisotropic shrinkage, which we can find as a special case for $d = 1$ .", "Summing up, the debiasing procedure on this manifold again yields the solution of hard thresholding: $[\\hat{u}(f)]_i = {\\left\\lbrace \\begin{array}{ll}f_i, & |f_i| \\ge \\alpha , \\\\0, & |f_i|< \\alpha .\\end{array}\\right.", "}$ Note that we again maintain the signal directly on the threshold." ], [ "Infimal convolution of $\\ell ^1$ Bregman distances", "Theorem 8.1 Let $\\Gamma \\colon \\ell ^2(\\mathbb {R}^n) \\rightarrow \\ell ^1(\\mathbb {R}^m)$ be linear and bounded and $J(u) = \\Vert \\Gamma u \\Vert _{\\ell ^1(\\mathbb {R}^m)}$ for $m,n \\in \\mathbb {N}$ .", "Let further $q_\\alpha \\in \\partial \\Vert \\cdot \\Vert _{\\ell ^1(\\mathbb {R}^m)} (\\Gamma u_\\alpha )$ such that $p_\\alpha = \\Gamma ^* q_\\alpha $ .", "Then $\\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ) \\le \\mathrm {ICB}_J^{p_{\\alpha }}(u, u_\\alpha ).$ $& \\quad \\mathrm {ICB}_J^{p_{\\alpha }}(u, u_\\alpha ) \\\\&= \\inf _{z \\in \\ell ^2(\\mathbb {R}^n)} ~ D_J^{p_\\alpha }(u-z,u_\\alpha ) + D_J^{-p_\\alpha }(z,-u_\\alpha ) \\\\&= \\inf _{z \\in \\ell ^2(\\mathbb {R}^n)} ~ \\Vert \\Gamma (u-z) \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle p_\\alpha ,u-z \\rangle \\\\&\\quad + \\Vert \\Gamma z\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle p_\\alpha ,z \\rangle \\\\&= \\inf _{z \\in \\ell ^2(\\mathbb {R}^n)} ~ \\Vert \\Gamma (u-z) \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q_\\alpha ,\\Gamma (u-z) \\rangle \\\\&\\quad + \\Vert \\Gamma z\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q_\\alpha ,\\Gamma z \\rangle \\\\&= \\inf _{\\Gamma z \\in \\ell ^1(\\mathbb {R}^m)} \\Vert \\Gamma (u-z) \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q_\\alpha ,\\Gamma (u-z) \\rangle \\\\&\\quad + \\Vert \\Gamma z\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q_\\alpha ,\\Gamma z \\rangle \\\\&\\ge \\inf _{w \\in \\ell ^1(\\mathbb {R}^m)} ~ \\Vert \\Gamma u - w \\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q_\\alpha ,\\Gamma u -w \\rangle \\\\&\\quad + \\Vert w\\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q_\\alpha ,w \\rangle \\\\&= \\inf _{w \\in \\ell ^1(\\mathbb {R}^m)} ~ D_{\\ell ^1(\\mathbb {R}^m)}^{q_\\alpha }(\\Gamma u - w, \\Gamma u_\\alpha ) \\\\&\\quad + D_{\\ell ^1(\\mathbb {R}^m)}^{-q_\\alpha }(w, -\\Gamma u_\\alpha ) \\\\&= \\mathrm {ICB}_{\\ell ^1(\\mathbb {R}^m)}^{q_{\\alpha }}(\\Gamma u, \\Gamma u_\\alpha ).$ $\\square $ Note that we get equality for surjective $\\Gamma $ in Theorem REF .", "Theorem 8.2 Let $v,u \\in \\ell ^1(\\mathbb {R}^m)$ and $q \\in \\partial \\Vert v\\Vert _{\\ell ^1(\\mathbb {R}^m)}$ .", "Then $\\textnormal {ICB}_{\\ell ^1(\\mathbb {R}^m)}^q(u,v) = \\sum _{i \\in \\mathbb {N}} G(u_i, q_i)$ with $G \\colon \\mathbb {R}^m \\times \\mathbb {R}^m \\rightarrow \\mathbb {R}$ defined as $&G(u_i, q_i) \\\\&= {\\left\\lbrace \\begin{array}{ll}|u_i| (1 - |\\cos (\\varphi _i)| |q_i|), & |q_i| < |\\cos (\\varphi _i)|, \\\\|u_i| | \\sin (\\varphi _i)| \\sqrt{1 - |q_i|^2}, & |q_i| \\ge |\\cos (\\varphi _i)|.\\end{array}\\right.", "}$ where $\\varphi _i$ denotes the angle between $u_i$ and $q_i$ , i.e.", "$\\cos (\\varphi _i) |u_i| |q_i| = u_i \\cdot q_i$ with $\\varphi _i := 0$ for $q_i = 0$ or $u_i = 0$ .", "Let $f_1(u) &= D_{\\ell ^1(\\mathbb {R}^m)}^q(u,v) &= \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^m)} - \\langle q,u \\rangle , \\\\f_2(u) &= D_{\\ell ^1(\\mathbb {R}^m)}^{-q}(u,-v) &= \\Vert u \\Vert _{\\ell ^1(\\mathbb {R}^m)} + \\langle q, u \\rangle .$ Since $(f_1 \\Box f_2)^* = f_1^* + f_2^*$ and by the definition of the biconjugate, we know that $f_1 \\Box f_2 \\ge (f_1^* + f_2^*)^*.$ (1) We shall first compute the right-hand side.", "We have $f_1^*(w) &= \\iota _{B^{\\infty }(1)}(w +q), \\\\f_2^*(w) &= \\iota _{B^{\\infty }(1)}(w -q),$ where $\\iota _{B^{\\infty }(1)}$ denotes the characteristic function of the $\\ell ^{\\infty }(\\mathbb {R}^m)$ -ball $B^{\\infty }(1) = \\big \\lbrace w \\in \\ell ^{\\infty }(\\mathbb {R}^m) ~|~ \\Vert w \\Vert _{\\ell ^{\\infty }(\\mathbb {R}^m)} \\le 1 \\big \\rbrace .$ Thus $&(f_1^* + f_2^*)^*(u) = \\sup _{w \\in \\ell ^{\\infty }(\\mathbb {R}^m) } \\langle u, w \\rangle \\\\&\\text{ s.t.}", "~ \\Vert w+q\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^m)} \\le 1, \\Vert w-q\\Vert _{\\ell ^{\\infty }(\\mathbb {R}^m)} \\le 1.$ Taking into account the specific form of these constraints, we can carry out the computation pointwise, i.e.", "$\\sup _{w_i \\in \\mathbb {R}^m } u_i \\cdot w_i ~ \\text{ s.t.}", "~ |w_i + q_i| \\le 1, |w_i - q_i| \\le 1.$ From now on we drop the dependence on $i$ for simplicity.", "$\\bullet $ Let us first consider the case $|q| = 1$ .", "We immediately deduce that $w = 0$ and $u \\cdot w = 0$ .", "$\\bullet $ Hence we assume $|q| < 1$ from now on, and set up the corresponding Lagrangian $\\mathcal {L}(w, \\lambda , \\mu ) = - w \\cdot u &+ \\lambda (|w - q|^2 -1) \\nonumber \\\\&+ \\mu (|w + q|^2 -1).$ Both the objective functional and the constraints are differentiable, so every optimal point of (REF ) has to fulfill the four Karush-Kuhn-Tucker conditions, namely $&\\dfrac{\\partial }{\\partial w} \\mathcal {L}(w, \\lambda , \\mu ) = 0, \\quad &\\lambda (|w -q|^2-1) = 0, \\\\&\\lambda , \\mu \\ge 0, & \\mu (|w + q|^2 -1) = 0,$ Slater's condition implies the existence of Lagrange multipliers for a KKT-point of (REF ).", "The first KKT-condition yields $-u + 2 \\lambda (w-q) + 2 \\mu (w + q) = 0.$ $\\ast $ Let us first remark that the case $u=0$ causes the objective function to vanish anyway, hence in the following $u\\ne 0$ .", "$\\ast $ Then let us address the case $q=0$ in which (REF ) yields $u = 2 (\\lambda + \\mu ) w.$ In case $|w| = 1$ we find that $2 (\\lambda + \\mu ) = |u|$ , hence $w = \\frac{u}{|u|}$ .", "We infer $w \\cdot u = \\frac{u \\cdot u}{|u|} = |u|.$ Note that for $|w| < 1$ , we find that $\\lambda = \\mu = 0$ and hence $u = 0$ .", "$\\ast $ If $q \\ne 0$ , we can distinguish four cases: 1st case: $|w -q|^2 <1, |w +q|^2 = 1$ .", "Thus $\\lambda = 0$ and (REF ) yields $u = 2 \\mu (w +q).$ Since $|w + q|^2 = 1$ , we deduce $\\mu = |u|/2$ , so $w = \\dfrac{u}{|u|} - q$ and finally for the value of the objective function $w \\cdot u = \\left( \\dfrac{u}{|u|} - q \\right) \\cdot u = |u| - q \\cdot u.$ 2nd case: $|w +q|^2 <1, |w -q|^2 = 1$ .", "We analogously find $w \\cdot u = |u| + q\\cdot u.$ The first two cases thus occur whenever (insert $w$ into the conditions) $\\left| \\dfrac{u}{|u|} - 2q \\right| < 1 \\text{ or } \\left| \\dfrac{u}{|u|} + 2q \\right| < 1.$ We calculate $& \\quad &&\\left| \\dfrac{u}{|u|} - 2q \\right|^2 &&< 1 \\\\&\\Leftrightarrow && \\hspace{33.50006pt} |q|^2 &&< q \\cdot \\dfrac{u}{|u|} \\\\&\\Leftrightarrow &&\\hspace{33.50006pt} |q| &&< \\cos (\\varphi ).$ Hence $q\\cdot u >0$ and $|u| - q \\cdot u = |u| -|q \\cdot u|.$ In the second case we analogously find $|q| < -\\cos (\\varphi ),$ hence $q \\cdot u <0$ and $|u| + q \\cdot u = |u| -|q \\cdot u|,$ so we may summarize the first two cases as $w \\cdot u = |u| - |q \\cdot u| = |u| (1 - |\\cos (\\varphi )| |q|),$ whenever $|q| < |\\cos (\\varphi )|$ .", "3rd case: $|w -q|^2 =1, |w +q|^2 = 1$ .", "At first we observe that from $|w +q |^2 = |w -q|^2$ we may deduce that $w \\cdot q = 0$ .", "Therefore we have $|w +q|^2 = 1 \\Rightarrow |w| = \\sqrt{1-|q|^2}.$ We multiply the optimality condition (REF ) by $q$ and obtain $&\\qquad &&u \\cdot q &&= 2\\lambda (w -q) \\cdot q + 2\\mu (w +q) \\cdot q \\\\&\\Leftrightarrow && u \\cdot q &&= 2(\\mu - \\lambda )~ |q|^2 \\\\&\\Leftrightarrow &&(\\mu - \\lambda ) &&= \\frac{u}{2} \\cdot \\dfrac{q}{|q|^2}.$ Multiplying (REF ) by $w$ yields $u \\cdot w = 2 (\\lambda + \\mu ) |w|^2$ and another multiplication of (REF ) by $u$ yields $|u|^2 &= 2 (\\lambda +\\mu ) w \\cdot u + 2 (\\mu -\\lambda ) q\\cdot u \\\\&= 4 (\\lambda +\\mu )^2 |w|^2 + \\left( u\\cdot \\dfrac{q}{|q|} \\right)^2,$ where we inserted the previous results in the last two steps.", "We rearrange and find $2 (\\lambda + \\mu ) = \\sqrt{ |u|^2 - \\left( u \\cdot \\dfrac{q}{|q|} \\right)^2} |w|^{-1}.$ Note that $|w| > 0$ since $|q| < 1$ .", "This finally leads us to $u \\cdot w &= 2 (\\lambda + \\mu ) |w|^2 \\\\&= \\sqrt{ |u|^2 - \\left( u \\cdot \\dfrac{q}{|q|} \\right)^2} |w| \\\\&= |u| \\sqrt{\\left(1 - \\left( \\dfrac{u}{|u|} \\cdot \\dfrac{q}{|q|} \\right)^2\\right) \\left(1 - |q|^2\\right)} \\\\&= |u| \\sqrt{\\left(1 - |\\cos (\\varphi )|^2\\right) \\left(1 - |q|^2\\right)} \\\\&= |u| |\\sin (\\varphi )| \\sqrt{\\left(1 - |q|^2\\right)}.$ 4th case: $|w -q|^2 < 1, |w +q|^2 < 1$ .", "Here the first KKT-condition yields $u =0$ , which can only occur if the objective function $w\\cdot u$ vanishes anyway.", "Summing up, we have $(f_1^* + f_2^*)^*(u) = \\sum _{i \\in \\mathbb {N}} G(u_i,q_i) \\le \\Vert u\\Vert _{\\ell ^1(\\mathbb {R}^m)}.$ (2) It remains to show that $(f_1 \\Box f_2)(u) &= \\inf _{z \\in \\ell ^1(\\mathbb {R}^m)} \\sum _{i \\in \\mathbb {N}} g_i(z_i) \\\\&\\le (f_1^* + f_2^*)^*(u),$ where $g_i(z_i) = |u_i - z_i| + |z_i| - q_i \\cdot (u_i - 2z_i) \\ge 0.$ Again we need to distinguish four cases.", "1st case: If $|q_i| < \\cos (\\varphi _i)$ , we have $q_i \\cdot u_i > 0$ and we can choose $z_i = 0$ to obtain $g_i(z_i) = |u_i| - q_i \\cdot u_i = |u_i| - |q_i \\cdot u_i|.$ 2nd case: Analogously if $|q_i| < -\\cos (\\varphi _i)$ , we have $q_i \\cdot u_i < 0$ and choose $z_i = u_i$ , thus $g_i(z_i) = |u_i| + q_i \\cdot u_i = |u_i| - |q_i \\cdot u_i|.$ 3rd case: If $|q_i| = 1$ , we compute for $z_i = \\frac{u_i}{2} - \\frac{c}{2}q_i$ , $c > 0$ , $g_i(z_i) &= \\left| \\frac{u_i}{2} + \\frac{c}{2} q_i \\right| + \\left| \\frac{u_i}{2} - \\frac{c}{2} q_i \\right| - c|q_i|^2 \\\\&=\\frac{c}{2} \\left( \\left| q_i + \\frac{u_i}{c} \\right| + \\left| q_i - \\frac{u_i}{c} \\right| - 2 \\right).$ Using a Taylor expansion around $q$ we obtain $\\left| q_i + \\frac{u_i}{c} \\right| &= |q_i| + \\frac{q_i}{|q_i|} \\cdot \\frac{u_i}{c} + O (c^{-2}), \\\\\\left| q_i - \\frac{u_i}{c} \\right| &= |q_i| - \\frac{q_i}{|q_i|} \\cdot \\frac{u_i}{c} + O (c^{-2}).$ Hence with $|q_i| = 1$ we find $g_i(z_i) = \\frac{c}{2} (2 |q_i| + O(c^{-2}) - 2) = O(c^{-1}) \\rightarrow 0$ for $c \\rightarrow \\infty $ .", "Hence for every $\\varepsilon $ there exists a $c_i > 0$ such that $g_i(z_i) \\le \\varepsilon / 2^i$ .", "4th case: Finally, if $|q_i| \\ge |\\cos (\\varphi _i)|$ and $|q_i| < 1$ , we pick $z_i = 2 \\lambda _i (w_i - q_i)$ , with $\\lambda _i$ and $w_i$ being the Lagrange multiplier and the dual variable from the above computation of $(f_1^* + f_2^*)^*$ .", "It is easy to see that $g_i(z_i) = |u_i| | \\sin (\\varphi _i)| \\sqrt{1 - |q_i|^2}.$ Hence we define $z := (z_i)_i$ such that $z_i = {\\left\\lbrace \\begin{array}{ll}0, & \\text{ if } |q_i| < \\cos (\\varphi _i),\\\\u_i, & \\text{ if } |q_i| < -\\cos (\\varphi _i),\\\\\\frac{u_i}{2} - \\frac{c_i}{2}q_i, & \\text{ if } |q_i| = 1, \\\\\\lambda _i (w_i - q_i) & \\text{ if } |q_i| \\ge |\\cos (\\varphi _i)|, \\\\& \\hspace{12.5pt} |q_i| < 1.\\end{array}\\right.", "}$ Let $z^N$ denote $z$ truncated at index $N \\in \\mathbb {N}$ , i.e.", "$z_i^N = {\\left\\lbrace \\begin{array}{ll}z_i, & \\text{ if } i \\le N, \\\\0, & \\text{ else.}\\end{array}\\right.", "}$ Then trivially $z^N \\in \\ell ^1(\\mathbb {R}^m)$ and we compute $&(f_1 \\Box f_2)(u) \\le \\sum _{i \\in \\mathbb {N}} g_i(z_i^N) \\\\\\le & \\sum _{i = 1}^N \\big ( G(u_i, q_i) + \\frac{\\varepsilon }{2^i} \\big ) + \\sum _{i=N+1}^\\infty g_i(0) \\\\= &\\sum _{i = 1}^\\infty G(u_i, q_i) + \\sum _{i = 1}^N \\frac{\\varepsilon }{2^i}\\\\& + \\sum _{i=N+1}^\\infty \\big ( |u_i| - q_i \\cdot u_i - G(u_i,q_i) \\big ) \\\\\\le & \\sum _{i = 1}^\\infty G(u_i, q_i) + \\sum _{i = 1}^N \\frac{\\varepsilon }{2^i} + 3 \\sum _{i=N+1}^\\infty |u_i| \\\\\\rightarrow & \\sum _{i = 1}^\\infty G(u_i, q_i) + \\varepsilon $ as $N \\rightarrow \\infty $ .", "This completes the proof.", "$\\square $" ], [ "Acknowledgements", "This work was supported by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse.", "MB acknowledges support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster, Germany." ] ]
1606.05113
[ [ "On a neutral particle with a magnetic quadrupole moment in a uniform\n effective magnetic field" ], [ "Abstract Quantum effects on a Landau-type system associated with a moving atom with a magnetic quadrupole moment subject to confining potentials are analysed.", "It is shown that the spectrum of energy of the Landau-type system can be modified, where the degeneracy of the energy levels can be broken.", "In three particular cases, it is shown that the analogue of the cyclotron frequency is modified, and the possible values of this angular frequency of the system are determined by the quantum numbers associated with the radial modes and the angular momentum and by the parameters associated with confining potentials in order that bound states solutions can be achieved." ], [ "Introduction", "It is well-known in the literature that the Landau quantization [1] takes place when the motion of a charged particle in a plane perpendicular to a uniform magnetic field acquires distinct orbits and the energy spectrum of this system becomes discrete and infinitely degenerate.", "It is important in studies of two-dimensional surfaces [2], [3], [4], the quantum Hall effect [5] and Bose-Einstein condensation [6], [7].", "With the aim of building a quantum system where the quantum Hall effect for neutral particles could be observed, it has been proposed in Ref.", "[8] a model where the electric field that interacts with the permanent magnetic dipole moment of the neutral particle must satisfy specific conditions: the absence of torque on the magnetic dipole moment of the neutral particle, the electric field must satisfy the electrostatic conditions, and there exists the presence of a uniform effective magnetic field given by $\\vec{B}_{\\mathrm {eff}}=\\vec{\\nabla }\\times \\vec{A}_{\\mathrm {eff}}$ , where $\\vec{A}_{\\mathrm {eff}}=\\vec{\\sigma }\\times \\vec{E}$ corresponds to an effective vector potential, $\\vec{E}$ is the electric field and $\\vec{\\sigma }$ are the Pauli matrices.", "Therefore, by choosing an electric field that satisfies the above conditions, it has been shown in Ref.", "[8] that the motion of the neutral particle acquires discrete orbits, where the energy levels correspond to the analogue of the Landau levels.", "This analogue of the Landau levels corresponds to the Landau-Aharonov-Casher quantization [8].", "Another model of the Landau quantization for neutral particles has been proposed in Ref.", "[9] based on the dual effect of the Aharonov-Casher effect [10], [11], [12], where the neutral particle has a permanent electric dipole moment.", "In this model, there exists an effective vector potential given by $\\vec{A}_{\\mathrm {eff}}=\\vec{\\sigma }\\times \\vec{B}$ , where $\\vec{B}$ is a magnetic field that interacts with the permanent electric dipole moment of the neutral particle.", "This Landau-like quantization is called as the Landau-He-McKellar-Wilkens quantization [9].", "Recently, the Landau quantization for an atom with electric quadrupole moment has been proposed in Ref.", "[13] by imposing that the electric quadrupole tensor must be symmetric and traceless and there exists the presence of a uniform effective magnetic field given by an effective vector potential defined as $\\vec{A}_{\\mathrm {eff}}=\\vec{Q}\\times \\vec{B}$ , where $\\vec{Q}$ is a vector associated with the electric quadrupole tensor [14], [13] and $\\vec{B}$ is the magnetic field in the laboratory frame.", "The aim of this work is to analyse quantum effects on a Landau-type system associated with an atom with a magnetic quadrupole moment subject to some confining potentials.", "A great deal of work can be found in the literature with respect to studies of quadrupole moments of atoms and molecules, for instance, in single crystals [15], refractive index [16], nuclear quadrupole interactions [17], [18], [19], [20], [21], molecules [22], [23], [24], [25], [26], atoms [27], [28], superposition of chiral states [29], geometric quantum phases [14] and noncommutative quantum mechanics [30].", "In particular, quantum particles with a magnetic quadrupole moment have attracted interests in atomic systems [31], [32], molecules [33], [34], [35], chiral anomaly [36], with $P$ - and $T$ -odds effects in atoms [37], [39], [38], Coulomb-type interactions [40], [41], Landau-type quantization [42], [43] and analogue of the quantum Hall effect [44].", "Other interesting studies have been made in Refs.", "[47], [48], [46], [45], [49].", "In this work, we consider the single particle approximation used in Refs.", "[45], [46], [40], [41] and then we deal with a system that consists in a moving atom with a magnetic quadrupole moment that interacts with external fields.", "Then, we introduce the Landau quantization associated with a moving atom that possesses a magnetic quadrupole moment, and thus we analyse the confinement of the Landau-type system to a hard-wall confining potential, a Coulomb-type potential, a linear confining potential and a Coulomb-type plus a linear confining potential.", "The structure of this paper is as follows: in section II, we introduce the Landau-type system associated with an atom with a magnetic quadrupole moment by using the single particle approximation of Refs.", "[45], [46]; in section III, we confine the Landau-type system to a hard-wall confining potential and analyse the bound states solutions; in section IV, we discuss the Landau-type system subject to a Coulomb-type confining potential; in section V, we discuss the Landau-type system subject to a linear confining potential; in section VI, we discuss the Landau-type system subject to a Coulomb-type and a linear confining potentials; in section VII, we present our conclusions." ], [ "Landau-type system", "In this section, we make a brief introduction of the Landau quantization associated with neutral particle (atom or molecule) with a magnetic quadrupole moment.", "First of all, by following Refs.", "[46], [45], then, the single particle approximation that describes an atom with a magnetic quadrupole moment interacting with a magnetic field is given by a potential energy $U_{m}=-\\sum _{i,j}M_{ij}\\,\\partial _{i}\\,B_{j}$ in the rest frame of the particle, where $\\vec{B}$ is the magnetic field and $M_{ij}$ is the magnetic quadrupole moment tensor, whose characteristic is that it is a symmetric and a traceless tensor.", "Recently, by considering a moving atom, it has been shown in Ref.", "[40] that the magnetic quadrupole moment of the atom interacts with a magnetic field given by $\\vec{B}^{\\prime }=\\vec{B}-\\frac{1}{c^{2}}\\,\\vec{v}\\times \\vec{E}$ for $v\\ll c$ [50], where $c$ is the velocity of light.", "Thereby, the quantum dynamics of a moving atom with magnetic quadrupole moment can be described by the Schrödinger equation (with SI units) $i\\hbar \\frac{\\partial \\psi }{\\partial t}=\\frac{1}{2m}\\left[\\hat{p}-\\frac{1}{c^{2}}(\\vec{M}\\times \\vec{E})\\right]^2\\,\\psi -\\vec{M}\\cdot \\vec{B}\\,\\psi ,$ where vector $\\vec{M}$ has the components determined by $M_{i}=\\sum _{j}M_{ij}\\,\\partial _{j}$ , the fields $\\vec{E}$ and $\\vec{B}$ given in Eq.", "(REF ) are the electric and magnetic fields in the laboratory frame, respectively [40], [50].", "An analogue of the Landau quantization for a moving atom that possesses a magnetic quadrupole moment was proposed in Refs.", "[42], [44] based on the properties of the magnetic quadrupole tensor, and the field configuration in the laboratory frame that interacts with the magnetic quadrupole moment of the atom must produce a uniform effective magnetic field perpendicular to the plane of motion of the particle which is given by $\\vec{B}_{\\mathrm {eff}}=\\vec{\\nabla }\\times \\left[\\vec{M}\\times \\vec{E}\\right],$ where the vector $\\vec{E}$ is the the electric field in the laboratory frame and satisfies the electrostatic conditions.", "From this perspective, it has been shown in Ref.", "[42] that an analogue of the Landau quantization can be obtained by considering the magnetic quadrupole moment tensor to be defined by the components: $M_{\\rho z}=M_{z\\rho }=M,$ where $M$ is a constant $\\left(M>0\\right)$ and with all other components of $M_{ij}$ as being zero, and thus there exists an electric field given by $\\vec{E}=\\frac{\\lambda \\,\\rho ^{2}}{2}\\,\\hat{\\rho },$ where $\\lambda $ is a constant associated with a non-uniform distribution of electric charges inside a non-conductor cylinder.", "In this particular case, we have that the magnetic quadrupole moment defined in Eq.", "(REF ) is a symmetric and traceless matrix, and we also have an effective vector potential given by $\\vec{A}_{\\mathrm {eff}}=\\vec{M}\\times \\vec{E}=\\lambda \\,M\\,\\rho \\,\\hat{\\varphi }$ and, consequently, the effective magnetic field (REF ) is uniform in the $z$ -direction $\\vec{B}_{\\mathrm {eff}}=\\lambda \\,M\\,\\hat{z},$ that is, it is perpendicular to the plane of motion of the quantum particle.", "Therefore, the conditions for achieving the Landau quantization associated with a moving atom that possesses a magnetic quadrupole moment are satisfied.", "Henceforth, we are able to analyse the quantum effects of confining potentials on the Landau-type system described above." ], [ "Landau-type system subject to a hard-wall confining potential", "In this section, we analyse the effects of the confinement of the Landau-type system established in the previous section to a hard-wall confining potential.", "By substituting Eqs.", "(REF ) and (REF ) into the Schrödinger equation (REF ), we have (with the units $\\hbar =c=1)$ $i\\frac{\\partial \\psi }{\\partial t}=-\\frac{1}{2m}\\left[\\frac{\\partial ^{2}}{\\partial \\rho ^{2}}+\\frac{1}{\\rho }\\,\\frac{\\partial }{\\partial \\rho }+\\frac{1}{\\rho ^{2}}\\,\\frac{\\partial ^{2}}{\\partial \\varphi ^{2}}+\\frac{\\partial ^{2}}{\\partial z}\\right]\\psi -i\\frac{M\\,\\lambda }{m}\\,\\frac{\\partial \\psi }{\\partial \\varphi }+\\frac{M^{2}\\,\\lambda ^{2}}{2m}\\,\\rho ^{2}\\,\\psi .$ A particular solution to Eq.", "(REF ) is given in terms of the eigenvalues of the operators $\\hat{p}_{z}=-i\\partial _{z}$ and $\\hat{L}_{z}=-i\\partial _{\\varphi }$ as $\\psi \\left(t,\\,\\rho ,\\,\\varphi ,\\,z\\right)=e^{-i\\mathcal {E}t}\\,e^{i\\,l\\,\\varphi }\\,e^{ikz}\\,R\\left(\\rho \\right),$ where $l=0,\\pm 1,\\pm 2,\\ldots $ and $k$ is a constant, since these quantum operators commute with the Hamiltonian of the right-hand side of Eq.", "(REF ).", "By performing a change of variables $\\xi =M\\,\\lambda \\,\\rho ^{2}$ and, after some calculations, we have that the radial wave function is given by [42] $R\\left(\\xi \\right)=\\xi ^{\\frac{\\left|l\\right|}{2}}\\,e^{-\\frac{\\xi }{2}}\\,M\\left(\\frac{\\left|l\\right|}{2}+\\frac{1}{2}-\\frac{m\\mathcal {E}}{2M\\lambda }+\\frac{l}{2},\\,\\left|l\\right|+1,\\,\\xi \\right),$ where $M\\left(\\frac{\\left|l\\right|}{2}+\\frac{1}{2}-\\frac{m\\mathcal {E}}{2M\\lambda },\\,\\left|l\\right|+1,\\,\\xi \\right)$ is the Kummer function of first kind or the confluent hypergeometric function [51].", "In condensed matter physics, a hard-wall confining potential is used with the purpose of describing a more realistic geometry of quantum dots and quantum rings as shown in Refs.", "[52], [53], [54], [55], [56], [57].", "For this purpose, let us now assume that the wave function of the atom is well-behaved at the origin, and thus vanishes at a fixed radius $\\xi _{0}$ , therefore we have $R\\left(\\xi _{0}=M\\,\\lambda \\,\\rho ^{2}_{0}\\right)=0.$ Note that the parameter $\\xi $ is defined in the range $0\\,<\\,\\xi \\,<\\,\\infty $ since the radial coordinate is defined in this range.", "However, by confining the system to a hard-wall confining potential, then, the parameter $\\xi $ becomes defined in the range $0\\,<\\,\\xi \\,<\\,\\xi _{0}$ .", "Therefore, we cannot impose that the confluent hypergeometric series becomes a polynomial of degree $n$ as in made in Ref.", "[42] in order that the Landau quantization could be achieved because we now need to obtain a normalized wave function in the range $0\\,<\\,\\xi \\,<\\,\\xi _{0}$ .", "A particular solution to Eq.", "(REF ) is obtained by considering a fixed value for the parameter $b=\\left|l\\right|+1$ of the confluent hypergeometric function and the intensity of the electric field (REF ) to be small, that is, we consider the parameter $\\lambda $ in Eq.", "(REF ) to be small.", "As a consequence, we can consider the product $M\\lambda $ to be quite small, and hence the parameter $a=\\frac{\\left|l\\right|}{2}+\\frac{1}{2}-\\frac{m\\mathcal {E}}{2M\\lambda }+\\frac{l}{2}$ of the confluent hypergeometric function can be considered to be large, without loss of generality.", "Thereby, for a fixed $\\xi _{0}$ , the confluent hypergeometric function can be written in the form [51]: $M\\left(a,b,\\xi _{0}=M\\,\\lambda \\,\\rho ^{2}_{0}\\right)&\\approx &\\frac{\\Gamma \\left(b\\right)}{\\sqrt{\\pi }}\\,e^{\\frac{\\xi _{0}}{2}}\\left(\\frac{b\\xi _{0}}{2}-a\\xi _{0}\\right)^{\\frac{1-b}{2}}\\times \\nonumber \\\\[-2mm]\\\\[-2mm]&\\times &\\cos \\left(\\sqrt{2b\\xi _{0}-4a\\xi _{0}}-\\frac{b\\pi }{2}+\\frac{\\pi }{4}\\right),\\nonumber $ where $\\Gamma \\left(b\\right)$ is the gamma function.", "Next, by applying the boundary condition established in Eq.", "(REF ), we obtain the following expression for the energy levels: $\\mathcal {E}_{n,\\,l}\\approx \\frac{1}{2m\\rho _{0}^{2}}\\left[n\\pi +\\frac{\\left|l\\right|}{2}\\pi +\\frac{3\\pi }{4}\\right]^{2}+\\frac{M\\,\\lambda \\,l}{m}.$ Thereby, Eq.", "(REF ) is the spectrum of energy of the Landau-type system of an atom with a magnetic quadrupole moment subject to a hard-wall confining potential.", "Despite the presence of an effective uniform magnetic field characteristic of the Landau quantization, the influence of the hard-wall confining potential yields a spectrum of energy that differs from that of the analogue of the Landau quantization [42], where we have that the energy levels (REF ) are parabolic with respect to the quantum number $n$ associated with radial modes and the analogue of the Landau levels are nonparabolic with respect to $n$ ." ], [ "Landau-type system subject to a Coulomb-type confinement", "In this section, our focus is on the effects of a Coulomb-type interaction on the Landau-type system discussed in section II.", "Coulomb-type potentials have been reported as being in the interests of condensed matter physics, for instance, studies that have worked with 1-dimensional systems [58], [59], [60], [61], [62], molecules [63], [64], [65], position-dependent mass systems [66], [67], [68] and the Kratzer potential [69], [70], [71].", "Furthermore, it is worth mentioning studies that have dealt with Coulomb-type potential in the propagation of gravitational waves [72], quark models [73] and relativistic quantum mechanics [74], [75], [76].", "For this purpose, let us consider the Landau-type system to be subject to the confining potential: $V\\left(\\rho \\right)=\\frac{\\alpha }{\\rho },$ where $\\alpha $ is a constant that characterizes the Coulomb-type interaction.", "Since molecules can have a magnetic quadrupole moment [33], [34], [35], then, a particular interest in the Coulomb-type potential (REF ) comes from the studies of molecular interactions described by the Kratzer potential [69], [70], [71].", "From Eqs.", "(REF ) and (REF ), the radial equation with the Coulomb-type scalar potential (REF ) is given by $\\left[2m\\mathcal {E}+2\\,M\\,\\lambda \\,l\\right]R=-R^{\\prime \\prime }-\\frac{1}{\\rho }\\,R^{\\prime }+\\frac{l^{2}}{\\rho ^{2}}\\,R+M^{2}\\,\\lambda ^{2}\\,\\rho ^{2}\\,R+\\frac{2m\\alpha }{\\rho }\\,R,$ where we have also taken $k=0$ .", "Now, we perform a change of variable given by $r=\\sqrt{M\\,\\lambda }\\,\\,\\rho $ , and then, rewrite Eq.", "(REF ) as $R^{\\prime \\prime }+\\frac{1}{r}\\,R^{\\prime }-\\frac{l^{2}}{\\rho ^{2}}\\,R-r^{2}\\,R-\\frac{\\nu }{r}\\,R+\\beta \\,R=0,$ where $\\nu =\\frac{2\\,m\\,\\alpha }{\\sqrt{M\\,\\lambda }};\\,\\,\\,\\,\\,\\,\\beta =\\frac{1}{M\\,\\lambda }\\left[2m\\mathcal {E}+2\\,M\\,\\lambda \\,l\\right].$ Observe that the asymptotic behaviour is determined for $r\\rightarrow 0$ and $r\\rightarrow \\infty $ , then, we can write the function $R\\left(r\\right)$ in terms of an unknown function $G\\left(r\\right)$ as follows [78], [77], [79], [40]: $R\\left(r\\right)=e^{-\\frac{r^{2}}{2}}\\,r^{\\left|l\\right|}\\,G\\left(r\\right).$ By substituting the function (REF ) into Eq.", "(REF ), we obtain the following equation for $G\\left(r\\right)$ : $G^{\\prime \\prime }+\\left[\\frac{2\\left|l\\right|+1}{r}-2r\\right]\\,G^{\\prime }+\\left[\\beta -2-2\\left|l\\right|-\\frac{\\nu }{r}\\right]G=0,$ which is called as the biconfluent Heun equation [79], and the function $G\\left(r\\right)$ is the biconfluent Heun function [79]: $G\\left(r\\right)=H_{B}\\left(2\\left|l\\right|,\\,0,\\,\\beta ,\\,2\\nu ,\\,r\\right)$ .", "We proceed with using the Frobenius method [80], then, we can write the solution to Eq.", "(REF ) as a power series expansion around the origin: $G\\left(r\\right)=\\sum _{k=0}^{\\infty }a_{k}\\,r^{k}$ .", "By substituting this series into Eq.", "(REF ), we obtain the recurrence relation: $a_{k+2}=\\frac{\\nu }{\\left(k+2\\right)\\left(k+2+2\\left|l\\right|\\right)}\\,a_{k+1}-\\frac{\\left(\\beta -2-2\\left|l\\right|-2k\\right)}{\\left(k+2\\right)\\left(k+2+2\\left|l\\right|\\right)}\\,a_{k},$ Let us start with $a_{0}=1$ , then, from Eq.", "(REF ), we can obtain other coefficients of the power series expansion, for instance, the coefficients $a_{1}$ and $a_{2}$ : $a_{1}&=&\\frac{\\nu }{\\left(1+2\\left|l\\right|\\right)};\\nonumber \\\\[-2mm]\\\\[-2mm]a_{2}&=&\\frac{\\nu ^{2}}{2\\,\\left(2+2\\left|l\\right|\\right)\\left(1+2\\left|l\\right|\\right)}-\\frac{\\left(\\beta -2-2\\left|l\\right|\\right)}{2\\left(2+2\\left|l\\right|\\right)}.\\nonumber $ By focusing on achieving bound states solutions, then, we need to impose that the biconfluent Heun series becomes a polynomial of degree $n$ .", "From the recurrence relation (REF ), we have that the biconfluent Heun series becomes a polynomial of degree $n$ by imposing that [75], [40]: $\\beta -2-2\\left|l\\right|=2n;\\,\\,\\,\\,\\,a_{n+1}=0,$ where $n=1,2,3,\\ldots $ .", "By analysing the first condition given in Eq.", "(REF ), we obtain $\\mathcal {E}_{n,\\,l}=\\varpi \\left[n+\\left|l\\right|-l+1\\right],$ where $n=1,2,3,\\ldots $ is the quantum number associated with the radial modes, $l=0,\\pm 1,\\pm 2,\\ldots $ is the angular momentum quantum number and the angular frequency of the system becomes $\\varpi =\\frac{M\\,\\lambda }{m}.$ Note that the the angular frequency is modified in contrast to the cyclotron frequency $\\omega =\\frac{2M\\lambda }{m}$ obtained in Ref.", "[42].", "Next, we analyse the condition $a_{n+1}=0$ given in Eq.", "(REF ) for the ground state of the system ($n=1$ ).", "Let us assume that the angular frequency $\\varpi $ can be adjusted in such a way that the condition $a_{n+1}=0$ , therefore, we shall label $\\varpi =\\varpi _{n,\\,l}$ from now on.", "As a result of this assumption, we have that both conditions imposed in Eq.", "(REF ) are satisfied and a polynomial solution to the function $G\\left(r\\right)$ is obtained.", "Hence, for $n=1$ we have that $a_{n+1}=a_{2}=0$ , and thus the possible values of the angular frequency $\\varpi $ associated with the ground state of the system are given by $\\varpi _{1,\\,l}=\\frac{2m\\alpha ^{2}}{\\left(1+2\\left|l\\right|\\right)},$ which shows us that only some specific values of the angular frequency of the harmonic oscillator are allowed in the system in order that bound state solutions can be obtained.", "Besides, from Eq.", "(REF ), we have that the possible values of the angular frequency are determined by the quantum numbers $\\left\\lbrace n,\\,l\\right\\rbrace $ of the system and the parameter associated with Coulomb-type interaction.", "By substituting (REF ) into Eq.", "(REF ), we have that the energy of the ground state of the system is given by $\\mathcal {E}_{1,\\,l}=\\frac{2m\\alpha ^{2}}{\\left(1+2\\left|l\\right|\\right)}\\,\\left[\\left|l\\right|-l+2\\right].$ Hence, the general expression for the energy levels (REF ) can be written as $\\mathcal {E}_{n,\\,l}=\\varpi _{n,\\,l}\\left[n+\\left|l\\right|-l+1\\right].$ In contrast to Ref.", "[42], we have that the energy levels of the Landau-type system are modified due to the influence of the Coulomb-type interaction.", "Moreover, the angular frequency is also modified by the effects of the Coulomb-type interaction, where the possible values of the angular frequency $\\varpi $ are determined by the quantum numbers of the system $\\left\\lbrace n,\\,l\\right\\rbrace $ and the parameter associated with Coulomb-type interaction.", "The ground state of the system becomes determined by the quantum number $n=1$ instead of the quantum number $n=0$ given in Ref.", "[42], and the degeneracy of the analogue of the Landau levels is broken." ], [ "Landau-type system subject to a linear confining potential", "Several works have dealt with a linear scalar potential in molecular and atomic physics through the perturbation theory [81], [82], [83], [85], [84], [86].", "In the context of the quantum field theory, the interest in the linear scalar potential comes from the studies of confinement of quarks [87], [73], [88], [89], [90], [91].", "Besides, relativistic quantum systems [92], [93], [94], [95], [96], [97], [75], [98], [99], [77], [78] have shown a great interest in the linear scalar potential.", "Now, we focus on the effects of a linear confining potential on the Landau-type system for an atom/molecule with a magnetic quadrupole moment by searching for analytical solutions.", "In this way, we introduce the following scalar potential into the Schrödinger equation (REF ): $V\\left(\\rho \\right)=\\eta \\,\\rho ,$ where $\\eta $ is a constant that characterizes the linear interaction.", "Then, by following the steps from Eq.", "(REF ) to Eq.", "(REF ), then, the radial equation with the linear confining potential (REF ) becomes $\\left[2m\\mathcal {E}+2\\,M\\,\\lambda \\,l\\right]R=-R^{\\prime \\prime }-\\frac{1}{\\rho }\\,R^{\\prime }+\\frac{l^{2}}{\\rho ^{2}}\\,R+M^{2}\\,\\lambda ^{2}\\,\\rho ^{2}\\,R+2m\\,\\eta \\,\\rho \\,R.$ Let us also take $k=0$ and perform the same change of variable made in the previous section, i.e., $r=\\sqrt{M\\,\\lambda }\\,\\,\\rho $ ; thus, we have $R^{\\prime \\prime }+\\frac{1}{r}\\,R^{\\prime }-\\frac{l^{2}}{\\rho ^{2}}\\,R-r^{2}\\,R-\\theta \\,r\\,R+\\beta \\,R=0,$ where the parameter $\\beta $ has been defined in Eq.", "(REF ) and the parameter $\\theta $ is defined as $\\theta =\\frac{2\\,m\\,\\eta }{\\left(M\\,\\lambda \\right)^{3/2}}.$ By analysing the asymptotic behaviour as in the previous section, we can write the function $R\\left(r\\right)$ in terms of an unknown function $H\\left(r\\right)$ as follows [78], [77], [79], [40]: $R\\left(r\\right)=e^{-\\frac{r^{2}}{2}}\\,e^{-\\frac{\\theta \\,r}{2}}\\,r^{\\left|l\\right|}\\,H\\left(r\\right).$ By substituting the function (REF ) into Eq.", "(REF ), we obtain the following equation for $H\\left(r\\right)$ : $H^{\\prime \\prime }+\\left[\\frac{2\\left|l\\right|+1}{r}-\\theta -2r\\right]\\,H^{\\prime }+\\left[\\beta +\\frac{\\theta ^{2}}{4}-2-2\\left|l\\right|-\\frac{\\theta \\left(2\\left|l\\right|+1\\right)}{2r}\\right]H=0,$ which is the biconfluent Heun equation [79], and the function $H\\left(r\\right)$ is the biconfluent Heun function [79]: $H\\left(r\\right)=H_{B}\\left(2\\left|l\\right|,\\,\\theta ,\\,\\beta +\\frac{\\theta ^{2}}{4},\\,0,\\,r\\right)$ .", "Further, by following the steps from Eq.", "(REF ) to Eq.", "(REF ), we obtain a recurrence relation given by $a_{k+2}=\\frac{\\theta \\left(2k+3+2\\left|l\\right|\\right)}{2\\left(k+2\\right)\\left(k+2+2\\left|l\\right|\\right)}\\,a_{k+1}-\\frac{\\left(4\\beta +\\theta ^{2}-8-8\\left|l\\right|-8k\\right)}{4\\left(k+2\\right)\\left(k+2+2\\left|l\\right|\\right)}\\,a_{k},$ and with $a_{0}=1$ , we have that the coefficients $a_{1}$ and $a_{2}$ are given by $a_{1}&=&\\frac{\\theta }{2};\\nonumber \\\\[-2mm]\\\\[-2mm]a_{2}&=&\\frac{\\theta ^{2}\\left(2\\left|l\\right|+3\\right)}{8\\,\\left(2+2\\left|l\\right|\\right)}-\\frac{\\left(4\\beta +\\theta ^{2}-8-8\\left|l\\right|\\right)}{8\\left(2+2\\left|l\\right|\\right)}.\\nonumber $ By imposing that the biconfluent Heun series becomes a polynomial of degree $n$ , then, we have from Eq.", "(REF ), we have that the biconfluent Heun series becomes a polynomial of degree $n$ when: $4\\beta +\\theta ^{2}-8-8\\left|l\\right|=8n;\\,\\,\\,\\,\\,a_{n+1}=0,$ where $n=1,2,3,\\ldots $ .", "The first condition given in Eq.", "(REF ) yields $\\mathcal {E}_{n,\\,l}=\\varpi \\left[n+\\left|l\\right|-l+1\\right]-\\frac{\\eta ^{2}}{2m\\varpi ^{2}},$ where $n=1,2,3,\\ldots $ is the quantum number associated with the radial modes, $l=0,\\pm 1,\\pm 2,\\ldots $ is the angular momentum quantum number and $\\varpi $ is the angular frequency of the system given in Eq.", "(REF ).", "Further, let us analyse the second condition $a_{n+1}=0$ given in Eq.", "(REF ) by considering the ground state of the system as in the previous section.", "Again, we assume that the angular frequency $\\varpi $ is a parameter that can be adjusted in order to satisfy the condition $a_{n+1}=0$ .", "For $n=1$ , we obtain $\\varpi _{1,\\,l}=\\left[\\frac{\\eta ^{2}}{2m}\\left(2\\left|l\\right|+3\\right)\\right]^{1/3},$ where we also have that the possible values of the angular frequency are determined by the quantum numbers of the system $\\left\\lbrace n,\\,l\\right\\rbrace $ and, in this case, by the parameter associated with the linear interaction.", "Besides, from Eqs.", "(REF ) and (REF ), the energy level associated with the ground state is $\\mathcal {E}_{1,\\,l}=\\left[\\frac{\\eta ^{2}}{2m}\\left(2\\left|l\\right|+3\\right)\\right]^{1/3}\\times \\left[\\left|l\\right|-l+2\\right]-\\frac{\\eta ^{2}}{2m}\\left(\\frac{2m}{\\eta ^{2}\\left[2\\left|l\\right|+3\\right]}\\right)^{2/3}.$ Hence, the general form the energy levels of the Landau-type system under the influence of a linear confining potential can be written as $\\mathcal {E}_{n,\\,l,\\,s}=\\varpi _{n,\\,l}\\left[n+\\left|l\\right|-l+1\\right]-\\frac{\\eta ^{2}}{2m\\,\\varpi _{n,\\,l}^{2}}.$ By comparing the spectrum of energy (REF ) with the analogue of the Landau levels, we have that the energy levels are modified by the linear confining potential.", "The effects of the linear interaction yields a new contribution to the energy levels given by the last term of Eqs.", "(REF ) and (REF ).", "The angular frequency is also modified due to the linear interaction in contrast to the cyclotron frequency $\\omega =\\frac{2M\\lambda }{m}$ obtained in Ref.", "[42], but it has the same form of that obtained in the previous section for the Coulomb-type interaction case given in Eq.", "(REF ).", "Furthermore, in this case, the possible values of the angular frequency $\\varpi $ are determined by the quantum numbers of the system $\\left\\lbrace n,\\,l\\right\\rbrace $ and by the parameter associated with the linear interaction.", "The ground state of the system also becomes determined by the quantum number $n=1$ instead of the quantum number $n=0$ of the Landau-like levels.", "Finally, from Eq.", "(REF ) to Eq.", "(REF ), we have that the degeneracy of the analogue of the Landau levels is broken." ], [ "Landau-type system subject to a Coulomb-type plus linear confining potential", "Linear plus Coulomb-type potential has been studied in the context the perturbation theory [100], with the WKB approximation [101] and the Bohr-Sommerfeld quantization [102].", "Analytical solutions to the Schrödinger equation in the presence of a linear and Coulomb-type potentials have been discussed in the context of the quark systems [103] and in mesoscopic systems.", "In this section, we consider the Landau-type system for an atom/molecule with a magnetic quadrupole moment to be subject to the Coulomb-type plus linear confining potential and search for analytical solutions.", "Thereby, let us introduce a scalar potential into the Schrödinger equation (REF ) given by $V\\left(\\rho \\right)=\\frac{\\alpha }{\\rho }+\\eta \\,\\rho ,$ where $\\alpha $ and $\\eta $ are constants as we have established previously.", "From Eqs.", "(REF ) and (REF ), the radial equation becomes $R^{\\prime \\prime }+\\frac{1}{r}\\,R^{\\prime }-\\frac{l^{2}}{\\rho ^{2}}\\,R-r^{2}\\,R-\\theta \\,r\\,R-\\frac{\\nu }{r}\\,R+\\beta \\,R=0,$ where that parameters $\\nu $ , $\\beta $ and $\\theta $ have been defined in Eqs.", "(REF ) and (REF ), respectively.", "By analysing the asymptotic behaviour, we can write the radial wave function $R\\left(r\\right)$ in the same way as given in Eq.", "(REF ); thus, substituting the radial wave function (REF ) into Eq.", "(REF ), we obtain the following second order differential equation By substituting the function (REF ) into Eq.", "(REF ), we obtain the following equation for $H\\left(r\\right)$ : $H^{\\prime \\prime }+\\left[\\frac{2\\left|l\\right|+1}{r}-\\theta -2r\\right]\\,H^{\\prime }+\\left[\\beta +\\frac{\\theta ^{2}}{4}-2-2\\left|l\\right|-\\frac{\\theta \\left(2\\left|l\\right|+1\\right)+2\\nu }{2r}\\right]H=0,$ which is also the biconfluent Heun equation [79], and the function $H\\left(r\\right)$ is the biconfluent Heun function [79]: $H\\left(r\\right)=H_{B}\\left(2\\left|l\\right|,\\,\\theta ,\\,\\beta +\\frac{\\theta ^{2}}{4},\\,2\\nu ,\\,r\\right)$ .", "Further, by following the steps from Eq.", "(REF ) to Eq.", "(REF ), we obtain a recurrence relation given by $a_{k+2}=\\frac{\\theta \\left(2k+3+2\\left|l\\right|\\right)+2\\nu }{2\\left(k+2\\right)\\left(k+2+2\\left|l\\right|\\right)}\\,a_{k+1}-\\frac{\\left(4\\beta +\\theta ^{2}-8-8\\left|l\\right|-8k\\right)}{4\\left(k+2\\right)\\left(k+2+2\\left|l\\right|\\right)}\\,a_{k},$ and with $a_{0}=1$ , we have that the coefficients $a_{1}$ and $a_{2}$ are given by $a_{1}&=&\\frac{\\theta }{2}+\\frac{\\nu }{\\left(1+2\\left|l\\right|\\right)};\\nonumber \\\\[-2mm]\\\\[-2mm]a_{2}&=&\\frac{\\theta ^{2}\\left(3+2\\left|l\\right|\\right)}{8\\left(2+2\\left|l\\right|\\right)}+\\frac{\\theta \\,\\nu \\left(1+\\left|l\\right|\\right)}{\\left(2+2\\left|l\\right|\\right)\\left(1+2\\left|l\\right|\\right)}+\\frac{\\nu ^{2}}{2\\left(2+2\\left|l\\right|\\right)\\left(1+2\\left|l\\right|\\right)}\\nonumber \\\\&-&\\frac{\\left(4\\beta +\\theta ^{2}-8-8\\left|l\\right|\\right)}{8\\left(2+2\\left|l\\right|\\right)}.\\nonumber $ Again, we must impose that the biconfluent Heun series becomes a polynomial of degree $n$ , then, from Eq.", "(REF ), we have that the biconfluent Heun series becomes a polynomial of degree $n$ with the following conditions: $4\\beta +\\theta ^{2}-8-8\\left|l\\right|=8n;\\,\\,\\,\\,\\,a_{n+1}=0,$ where $n=1,2,3,\\ldots $ .", "The first condition of Eq.", "(REF ) yields a general expression for the energy levels: $\\mathcal {E}_{n,\\,l}=\\varpi \\left[n+\\left|l\\right|-l+1\\right]-\\frac{\\eta ^{2}}{2m\\varpi ^{2}},$ where $n=1,2,3,\\ldots $ is the quantum number associated with the radial modes, $l=0,\\pm 1,\\pm 2,\\ldots $ is the angular momentum quantum number and $\\varpi $ is the angular frequency of the system given in Eq.", "(REF ).", "From the second condition $a_{n+1}=0$ given in Eq.", "(REF ), where we also assume that the angular frequency $\\varpi $ is a parameter that can be adjusted in order to satisfy the condition $a_{n+1}=0$ , we have for the ground state of the system ($n=1$ ) that the possible values of the angular frequency associated with the ground state is determined by the following third-degree algebraic equation [75], [41]: $\\varpi ^{3}_{1,\\,l}-\\frac{2m\\alpha ^{2}}{\\left(1+2\\left|l\\right|\\right)}\\,\\varpi ^{2}_{1,\\,l}-\\frac{4\\left(1+\\left|l\\right|\\right)}{\\left(1+2\\left|l\\right|\\right)}\\,\\varpi _{1,\\,l}-\\frac{\\eta ^{2}}{2m}\\,\\left(3+2\\left|l\\right|\\right)=0.$ Despite Eq.", "(REF ) has at least one real solution, we do not write it because its expression is very long.", "Besides, the general expression for the energy levels (REF ) should be written in the same form of Eq.", "(REF ).", "Hence, we have obtained analytical solutions to the Landau-type system for an atom/molecule with a magnetic quadrupole moment to be subject to the Coulomb-type and linear confining potentials.", "Note that the possible values of the angular frequency $\\varpi $ are determined by the quantum numbers of the system $\\left\\lbrace n,\\,l\\right\\rbrace $ and by the parameters associated with the linear and Coulomb-type confining potentials in order that a polynomial solution to the function $H\\left(r\\right)$ can be obtained.", "The ground state of the system is also determined by the quantum number $n=1$ instead of the quantum number $n=0$ of the Landau-like levels.", "Finally, from Eq.", "(REF ) to Eq.", "(REF ), we have that the degeneracy of the analogue of the Landau levels is broken." ], [ "conclusions", "We have investigated the behaviour of a neutral particle (atom or molecule) with a magnetic quadrupole moment in a region with a uniform effective magnetic field subject to confining potentials.", "We have analysed the confinement of the Landau-type system to a hard-wall confining potential, a Coulomb-type potential, a linear confining potential and a linear plus Coulomb-type potential.", "In the confinement to a hard-wall confining potential, we have seen that the spectrum of energy is modified in contrast to the Landau-type levels, where the energy levels are parabolic with respect to the quantum number associated with the radial modes.", "On the other hand, with respect to the confinement to a Coulomb-type potential, a linear confining potential and a linear plus Coulomb-type potential, we have obtained different spectrum of energies.", "In these three cases, the ground state of the system becomes determined by the quantum number $n=1$ instead of the quantum number $n=0$ obtained in the Landau-like levels, and the degeneracy of the analogue of the Landau levels is broken.", "Moreover, the cyclotron frequency of the Landau-type system is modified by the influence of the confining potentials, where the possible values of the angular frequency of the system are determined by the quantum numbers $\\left\\lbrace n,\\,l\\right\\rbrace $ and by the parameters associated with confining potentials.", "In particular, in the case of the confinement to the linear plus Coulomb-type potential, the possible values of the angular frequency associated with the ground state is determined by a third-degree algebraic equation.", "The authors would like to thank the Brazilian agencies CNPq and CAPES for financial support." ] ]
1606.05154
[ [ "Some properties and applications of odd-colorable $r$-hypergraphs" ], [ "Abstract Let $r\\geq2$ and $r$ be even.", "An $r$-hypergraph $G$ on $n$ vertices is called odd-colorable if there exists a map $\\varphi:[n]\\rightarrow\\lbrack r]$ such that for any edge $\\{j_{1},j_{2},\\cdots,j_{r}\\}$ of $G$, we have $\\varphi(j_{1})+\\varphi(j_{2})+\\cdot\\cdot\\cdot+\\varphi(j_{r})\\equiv r/2(\\operatorname{mod}r).$ In this paper, we first determine that, if $r=2^{q}(2t+1)$ and $n\\ge 2^{q}(2^{q}-1)r$, then the maximum chromatic number in the class of the odd-colorable $r$-hypergraphs on $n$ vertices is $2^q$, which answers a question raised by V. Nikiforov recently in [V. Nikiforov, Hypergraphs and hypermatrices with symmetric spectrum.", "Prinprint available in arXiv:1605.00709v2, 10 May, 2016].", "We also study some applications of the symmetric spectral property of the odd-colorable $r$-graphs given in that same paper by V. Nikiforov.", "We show that the Laplacian spectrum and the signless Laplacian spectrum of an $r$-hypergraph $G$ are equal if and only if $G$ is odd-colorable, and then study some further applications of these spectral properties." ], [ "Introduction", "Denote the set $\\lbrace 1,2,\\cdot \\cdot \\cdot ,n\\rbrace $ by $[n]$ .", "An $r$ -hypergraph $G$ $=(V(G),E(G))$ on $n$ vertices is an $r$ -uniform hypergraph each of whose edges contains exactly $r$ vertices ([1]).", "In this paper, $r$ -hypergraph is simply called $r$ -graph for convenience.", "A 2-graph is just an ordinary graph.", "The definition of odd-coloring for tensors (it is called $r$ -matrices in [10]) was introduced in [10], we just focus on its version for $r$ -graph as follows.", "Definition 1 Let $r\\ge 2$ and $r$ be even.", "An $r$ -graph $G$ with $V(G)=[n]$ is called odd-colorable if there exists a map $\\varphi :[n]\\rightarrow [r]$ such that for any edge $\\lbrace j_{1},j_{2},\\cdots ,j_{r}\\rbrace $ of $G$ , we have $\\varphi (j_{1})+\\cdot \\cdot \\cdot +\\varphi (j_{r})\\equiv r/2(\\operatorname{mod}r).$ The function $\\varphi $ is called an odd-coloring of $G.$ The following concept of odd-bipartite $r$ -graphs was taken from [6], and this concept acts as generalizations of the ordinary bipartite graphs.", "Definition 2 [6] An $r$ -graph $G = (V,E)$ is called odd-bipartite, if $r$ is even and there exists some proper subset $V_{1}$ of $V$ such that each edge of $G$ contains exactly odd number of vertices in $V_{1}$ .", "The odd-bipartite $r$ -graphs were also called odd-transversal $r$ -graphs in literature (see [1], [3], or [10]).", "The connection between odd-bipartiteness and spectra of $r$ -graphs was studied in [6], [7], [9] and [14].", "In [10], it was proved that an odd-bipartite graph is always odd-colorable (see Proposition 11 in [10]), and furthermore, in the case $r\\equiv 2($ mod$4)$ , then $G$ is odd-colorable if and only if $G$ is odd-bipartite (see Proposition 12 in [10]).", "An $r$ -graph $G$ is called $k$ -chromatic if its vertices can be partitioned into $k$ sets so that each edge intersects at least two sets.", "The chromatic number $\\chi (G)$ of $G$ is the smallest $k$ for which $G$ is $k$ -chromatic.", "The chromatic number of an odd-colorable $r$ -graph is also considered in [10].", "Clearly, each nontrivial odd-bipartite graph has chromatic number 2.", "A family of 3-chromatic odd-colorable $4k$ -graphs on $n$ vertices is constructed in [10].", "Notice that odd-colorable $r$ -graphs are defined only for even $r$ .", "For further information about the chromatic number of odd-colorable graph, the following question is raised in [10].", "Question 3 Let $r\\equiv 0(\\operatorname{mod}4).$ What is the maximum chromatic number of an odd-colorable $r$ -graph on $n$ vertices?", "In section 2, we will determine that, if $r$ is even, $r=2^{q}(2t+1)$ for some integers $q,t$ and $n\\ge 2^{q}(2^{q}-1)r$ , then the maximum chromatic number in the class of the odd-colorable $r$ -graphs on $n$ vertices is $2^q$ .", "This result provides an answer to the above Question REF .", "Definition 4 [7] [12] Let $G=(V(G),E(G))$ be an $r$ -graph on $n$ vertices.", "The adjacency tensor of $G$ is defined as the order $r$ dimension $n$ tensor $\\mathcal {A}(G)$ whose $(j_{1}\\cdots j_{r})$ -entry is: $(\\mathcal {A}(G))_{j_{1}j_{2}\\cdots j_{r}}={\\left\\lbrace \\begin{array}{ll}\\frac{1}{(r-1)!}", "& \\text{if $\\lbrace j_{1},j_{2},\\cdots ,j_{r}\\rbrace \\in E(G),$}\\\\0 & \\text{otherwise}.\\end{array}\\right.", "}$ Let $\\mathcal {D}(G)$ be an order $r$ dimension $n$ diagonal tensor, with its diagonal entry $\\mathcal {D}_{jj\\cdots j}$ being the degree of vertex $j$ , for all $j\\in [n]$ .", "Then $\\mathcal {L(}G\\mathcal {)}=\\mathcal {D(}G\\mathcal {)}-\\mathcal {A(}G\\mathcal {)}$ is called the Laplacian tensor of $r$ -graph $G$ , and $\\mathcal {Q(}G\\mathcal {)}=\\mathcal {D(}G\\mathcal {)}+\\mathcal {A(}G\\mathcal {)}$ is called the signless Laplacian tensor of $G,$ .", "The following general product of tensors, was defined in [13], which is a generalization of the matrix case.", "Let $\\mathcal {A}$ and $\\mathcal {B}$ be dimension $n$ and order $m\\ge 2$ and $k\\ge 1$ tensors, respectively.", "The product $\\mathcal {AB}$ is the following tensor $\\mathcal {C}$ of dimension $n$ and order $(m-1)(k-1)+1$ with entries: $\\mathcal {C}_{i\\alpha _{1}\\cdots \\alpha _{m-1}}=\\sum _{i_{2},\\cdots ,i_{m}\\in [n_{2}]}\\mathcal {A}_{ii_{2}\\cdots i_{m}}\\mathcal {B}_{i_{2}\\alpha _{1}}\\cdots \\mathcal {B}_{i_{m}\\alpha _{m-1}}, $ where $i\\in [n],\\alpha _{1},\\cdots ,\\alpha _{m-1}\\in [n]^{k-1}$ .", "Let $\\mathcal {T}$ be an order $r$ dimension $n$ tensor, let $x=(x_{1},\\cdot \\cdot \\cdot ,x_{n})^{T}\\in \\mathbb {C}^{n}$ be a column vector of dimension $n$ .", "Then by (1) $\\mathcal {T}x$ is a vector in $\\mathbb {C}^{n}$ whose $j$ th component is as the following $(\\mathcal {T}x)_{j}=\\sum _{j_{2},\\cdots ,j_{r}=1}^{n}\\mathcal {T}_{jj_{2}\\cdots j_{r}}x_{j_{2}}\\cdots x_{j_{r}}.", "$ Let $x^{[r]}=(x_{1}^{r},\\cdots ,x_{n}^{r})^{T}$ .", "Then (see [2] [12]) a number $\\lambda \\in \\mathbb {C}$ is called an eigenvalue of the tensor $\\mathcal {T}$ of order $r$ if there exists a nonzero vector $x\\in \\mathbb {C}^{n}$ satisfying the following eigenequations $\\mathcal {T}x=\\lambda x^{[r-1]}, $ and in this case, $x$ is called an eigenvector of $\\mathcal {T}$ corresponding to eigenvalue $\\lambda $ .", "The spectral radius of $\\mathcal {T}$ is defined as $\\rho (\\mathcal {T})=max\\lbrace |\\mu |:\\mu \\text{ is an eigenvalue of }\\mathcal {T}\\rbrace .$ In order to define the spectra of tensors, we first need to define the determinants of tensors.", "Originally the determinants of tensors were defined as the resultants of some corresponding system of homogeneous equations on $n$ variables.", "Here we give the following equivalent definition of the determinants of tensors.", "Definition 5 Let $\\mathcal {A}$ be an order $m$ dimension $n$ tensor with $m\\ge 2$ .", "Then its determinant $det(\\mathcal {A})$ is defined to be the unique polynomial on the entries of $\\mathcal {A}$ satisfying the following three conditions: (1) $det(\\mathcal {A})=0$ if and only if the system of homogeneous equations $\\mathcal {A}x=0$ has a nonzero solution.", "(2) $det(\\mathcal {A})=1$ , when $\\mathcal {A}=\\mathbb {I}$ is the unit tensor.", "(3) $det(\\mathcal {A})$ is an irreducible polynomial on the entries of $\\mathcal {A}$ , when the entries of $\\mathcal {A}$ are viewed as distinct independent variables.", "Definition 6 Let $\\mathcal {A}$ be an order $m\\ge 2$ dimension $n$ tensor.", "Then the characteristic polynomial of $\\mathcal {A}$ is defined to be the determinant $det(\\lambda \\mathcal {I}- \\mathcal {A})$ .", "The (multi)-set of roots of the characteristic polynomial of $\\mathcal {A}$ (counting multiplicities) is called the spectrum of $\\mathcal {A}$ , denoted by $Spec(\\mathcal {A})$ .", "If the tensor $\\mathcal {T}$ and $-\\mathcal {T}$ have the same spectrum (i.e., the spectrum of $\\mathcal {T}$ is symmetric about the origin), then the spectrum of $\\mathcal {T}$ is said to be symmetric in this paper.", "In [10], Nikiforov studied some symmetric spectral property of the odd-colorable $r$ -graphs.", "He proved that for an $r$ -graph $G$ , $Spec(\\mathcal {A}(G))=-Spec(\\mathcal {A}(G))$ if and only if $r$ is even and $G$ is odd-colorable.", "This result solves a problem in [11] about $r$ -graphs with symmetric spectrum and disproves a conjecture in [18].", "In Section 3, we will give some applications and consequences of these symmetric spectral property of the odd-colorable $r$ -graphs given in [10].", "In particular, we obtain (in Theorem REF ) some further symmetric spectral property of the odd-colorable $r$ -graphs related to the Laplacian and signless Laplacian spectrum of an $r$ -graph $G$ .", "The proof of the disconnected case of this result need to use the Perron-Frobenius Theorem on nonnegative weakly irreducible tensors, the relation between the (Laplacian and signless Laplacian) spectra of an disconnected $r$ -graph $G$ with that of all the connected components of $G$ , and so on.", "We also use these results to study the Question REF proposed in [14] about the relations between H-spectra of $\\mathcal {L}(G)$ and $\\mathcal {Q}(G)$ with the spectra of $\\mathcal {L}(G)$ and $\\mathcal {Q}(G)$ , and obtain an affirmative answer to Question REF for the remaining unsolved case $r\\equiv 2 \\ (mod \\ 4)$ in Theorem REF ." ], [ " Maximum chromatic number of an odd-colorable $r$ -graph", "Let $r$ be even, then there uniquely exist two integers $q,t$ such that $r=2^{q}(2t+1).$ In this section, we will determine that, if $n\\ge 2^{q}(2^{q}-1)r$ , then the maximum chromatic number in the class of the odd-colorable $r$ -graphs on $n$ vertices is $2^q$ .", "This result also provides an answer to Question REF in §1.", "First we prove the following upper bound on the chromatic number of the odd-colorable $r$ -graphs.", "Theorem 7 Let $q\\ge 1,t\\ge 0$ be two integers and $r=2^{q}(2t+1)$ , and $G$ be an odd-colorable $r$ -graph.", "Then $\\chi (G)\\le 2^{q}.$ Proof Suppose $|V(G)|=n$ and let the function $\\varphi :[n]\\rightarrow [r]$ be an odd-coloring of $G.$ For $0\\le i\\le 2^{q}-1,$ set $V_{i}:=\\lbrace j:j\\in [n],\\text{ }\\varphi (j)\\equiv i(\\operatorname{mod}2^{q})\\rbrace ,$ and some $V_{i}$ may be empty.", "Then the vertices set $V(G)$ can be partitioned as $V(G)=V_{0}\\cup \\cdot \\cdot \\cdot \\cup V_{2^{q}-1}.$ We claim that each set $V_{i}$ contains no edge of $G.$ Suppose not, let $e=\\lbrace j_{1},j_{2},\\cdots ,j_{r}\\rbrace $ be an edge in some $V_{i}.$ Since $j_{l}$ is in $V_{i},$ we have $\\varphi (j_{l})=c_{l}2^{q}+i,$ where $c_{l}$ is a nonnegative integer $(l=1,\\cdots ,r)$ .", "Then we have $& \\varphi (j_{1})+\\cdot \\cdot \\cdot +\\varphi (j_{r})\\\\& =(c_{1}2^{q}+i)+\\cdot \\cdot \\cdot +(c_{r}2^{q}+i)\\\\& =2^{q}({\\displaystyle \\sum \\limits _{l=1}^{r}}c_{l})+ir\\\\& =2^{q}({\\displaystyle \\sum \\limits _{l=1}^{r}}c_{l})+i2^{q}(2t+1).$ On the other hand, by the definition of odd-coloring, there exists some integer $c$ such that $& \\varphi (j_{1})+\\cdot \\cdot \\cdot +\\varphi (j_{r})\\\\& =cr+r/2\\\\& =c2^{q}(2t+1)+2^{q-1}(2t+1).$ Hence we have $2^{q}({\\displaystyle \\sum \\limits _{l=1}^{r}}c_{l})+i2^{q}(2t+1)=c2^{q}(2t+1)+2^{q-1}(2t+1),$ which implies that $2[{\\displaystyle \\sum \\limits _{l=1}^{r}}c_{l}+(i-c)(2t+1)]=2t+1,$ and this is a contradiction, since it has different parity of two sides.", "So each set $V_{i}$ does not span any edge and so $\\chi (G)\\le 2^{q}.$ $\\Box $ Now we construct a family of odd-colorable $r$ -graphs on $n$ vertices to show that the upper bound given in Theorem REF is sharp for all $n\\ge 2^{q}(2^{q}-1)r$ .", "Lemma 8 Let $q\\ge 1,t\\ge 0$ be two integers and $r=2^{q}(2t+1).$ If $n\\ge 2^{q}(2^{q}-1)r$ , then there exists a family of $2^{q}$ -chromatic odd-colorable $r$ -graphs on $n$ vertices.", "Proof For any $1\\le i<j\\le 2^{q},$ we may write $j-i=2^{p_{i,j}}(2a_{i,j}+1),\\quad \\mbox{and} \\quad b_{i,j}=2^{q-p_{i,j}-1}(2t+1),$ where $0\\le p_{i,j}<q,$ $a_{i,j}\\ge 0$ are integers.", "By definition of $b_{i,j}$ , we may see that $b_{i,j}=\\frac{2^{q}(2t+1)}{2^{p_{i,j}+1}}=\\frac{r}{2^{p_{i,j}+1}}\\le \\frac{r}{2}.$ Now we start to construct the desired $r$ -graph $G$ .", "First we take $V(G)=[n]$ .", "In order to define the edge set $E(G)$ , we first take any subsets $V_1,\\cdots , V_{2^{q}}$ of $V$ such that $V_{i}^{\\prime }s$ are pairwisely disjoint, and $[n]=V_{1}\\bigcup V_{2}\\bigcup \\cdot \\cdot \\cdot \\bigcup V_{2^{q}}$ , and $|V_{i}|\\ge r(2^{q}-1)$ , $1\\le i\\le 2^{q}.$ Using these subsets $V_1,\\cdots , V_{2^{q}}$ , we can define the following $2^{q-1}(2^{q}-1)$ families $E_{i,j}$ of $r$ -subset of $[n]$ .", "For any $1\\le i<j\\le 2^{q},$ set $E_{i,j}:=\\lbrace e:e\\subset [n],|e\\cap V_{i}|=r-b_{i,j}, \\text{ and}|e\\cap V_{j}|=b_{i,j} \\rbrace .$ Finally, we define the desired $r$ -graph $G$ as $V(G):=[n]$ and $E(G):=E_{1,2}\\cup E_{1,3}\\cup \\cdot \\cdot \\cdot \\cup E_{(2^{q}-1),2^{q}}=\\bigcup _{1\\le i<j\\le 2^{q}}E_{i,j}.$ From the definition we can see that, if there exists a vertex subset $C\\subseteq V$ with $|V_{i}\\cap C|\\ge r$ and $|V_{j}\\cap C|\\ge r$ for some $1\\le i<j\\le 2^{q}$ , then $C$ contains some edge from the set $E_{i,j}$ .", "First we will show that $G$ is odd-colorable.", "Define the map $\\varphi :[n]\\rightarrow [r]$ by letting $\\varphi (v)=i,\\text{ if }v\\in V_{i}.$ We shall check that the function $\\varphi $ is an odd-coloring of $G$ .", "Let $e\\in E(G).$ If $e\\in E_{i,j}$ for some $1\\le i<j\\le 2^{q}$ , then ${\\displaystyle \\sum \\limits _{l\\in e}}\\varphi (l) & =(r-b_{i,j})i+b_{i,j}j\\\\& =ri+b_{i,j}(j-i)\\\\& =ri+2^{q-1}(2t+1)(2a_{i,j}+1)\\\\& =ri+2^{q}a_{i,j}(2t+1)+2^{q-1}(2t+1)\\\\& =ri+ra_{i,j}+r/2\\\\& \\equiv r/2(\\operatorname{mod}r).$ Hence if $\\lbrace j_{1},j_{2},\\cdots ,j_{r}\\rbrace $ is an edge of $G,$ then $\\varphi (j_{1})+\\varphi (j_{2})+\\cdot \\cdot \\cdot +\\varphi (j_{r})\\equiv r/2(\\operatorname{mod}r).$ So $G$ is odd-colorable.", "Now we will show $\\chi (G)=2^{q}.$ First from Theorem REF we know that $\\chi (G)\\le 2^{q},$ since $G$ is odd-colorable.", "Next we show that $\\chi (G)\\ge 2^{q}$ .", "Suppose not, assume that we have a partition $V(G)=C_{1}\\cup \\cdot \\cdot \\cdot \\cup C_{2^{q}-1}$ such that $C_{i}^{\\prime }s$ are pairwisely disjoint and there is no edge in each $C_{i}$ for $1\\le i\\le 2^{q}-1,$ we will use the pigeonhole principle to get a contradiction.", "We first define an auxiliary matrix $T=(t_{ij})$ with $2^q$ rows and $2^q-1$ columns such that $t_{ij}=|V_{i}\\cap C_{j}| \\qquad (i=1,\\cdots , 2^q; \\ j=1,\\cdots , 2^q-1)$ Then the $i$ -th row sum of the matrix $T$ is $\\sum _{j=1}^{2^{q}-1}t_{ij}=\\sum _{j=1}^{2^{q}-1}|V_{i}\\cap C_{j}|=|V_{i}|\\ge r(2^{q}-1)$ This implies that for each $i=1,\\cdots , 2^q$ , there exists some $j\\in \\lbrace 1,\\cdots , 2^q-1\\rbrace $ such that $t_{ij}\\ge r$ .", "An entry $t_{ij}$ of the matrix $T$ is called good if $t_{ij}\\ge r$ .", "Then the above arguments shows that every row of $T$ contains at least one good entry, so altogether $T$ contains at least $2^q$ good entries since $T$ has $2^q$ rows.", "On the other hand, $T$ has $2^q-1$ columns.", "So by the pigeonhole principle, there exists some column of $T$ containing at least two good entries, say $t_{ik}\\ge r$ and $t_{jk}\\ge r$ are good entries $(i<j)$ .", "This implies that the class $C_{k}$ contains an edge from the set $E_{i,j}$ , a contradiction.", "$\\Box $ Combining Theorem REF and Lemma REF , we can obtain the following: Theorem 9 If $r=2^{q}(2t+1)$ is even and $n\\ge 2^{q}(2^{q}-1)r$ , then the maximum chromatic number in the class of the odd-colorable $r$ -graphs on $n$ vertices is $2^q$ .", "Obviously, the special case $q\\ge 2$ of Theorem REF also provides an answer to Question REF in §1." ], [ "Some applications of the symmetric spectral property of the odd-colorable $r$ -graphs", "In [10], Nikiforov studied some symmetric spectral property of the odd-colorable $r$ -graphs.", "He proved the following result of the odd-colorable $r$ -graphs.", "Theorem 10 [10] Let $G$ be an $r$ -graph.", "Then $Spec(\\mathcal {A}(G))=-Spec(\\mathcal {A}(G))$ if and only if $r$ is even and $G$ is odd-colorable.", "In this section, we will give some applications and consequences of this symmetric spectral property of the odd-colorable $r$ -graphs.", "In particular, we obtain some further symmetric spectral property of the odd-colorable $r$ -graphs related to the Laplacian and signless Laplacian spectrum of an $r$ -graph $G$ (see Theorem REF below).", "We also use these results to study the Question REF proposed in [14] about the relations between H-spectra of $\\mathcal {L}(G)$ and $\\mathcal {Q}(G)$ with the spectra of $\\mathcal {L}(G)$ and $\\mathcal {Q}(G)$ , and obtain an affirmative answer to Question REF for the remaining unsolved case $r\\equiv 2 \\ (mod \\ 4)$ in Theorem REF .", "Recall that in Theorem 3.2 of [4], Fan et al.", "proved that in the case of the non-odd-bipartite connected $r$ -graphs, then the following Lemma REF holds.", "Combining this with the Theorems 2.2 and 2.3 in [14] for the odd-bipartite connected case, we have the following result.", "Lemma 11 Let $G$ be a connected $r$ -graph.", "Then $Spec(\\mathcal {A}(G))=-Spec(\\mathcal {A}(G))$ if and only if $ Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G)).$ Proof If $G$ is not odd-bipartite, the result follows from Theorem 3.2 of [4].", "If $G$ is odd-bipartite, the result follows from Theorems 2.2 and 2.3 of [14].", "$\\Box $ Combining Theorem REF and Lemma REF , we can obtain that, for a connected $r$ -graph $G$ , its Laplacian spectrum and signless Laplacian spectrum are equal if and only if $r$ is even and $G$ is odd-colorable.", "In order to extend this result to the disconnected case, we need the following result which is a consequence of Corollary 4.2 of [15].", "Lemma 12 [15] Let $G$ be an $r$ -graph of order $n$ , $G_{1},G_{2},\\cdot \\cdot \\cdot ,G_{k}$ be all the connected components of $G$ , with orders $n_1,\\cdots , n_k$ , respectively.", "Then $Spec(\\mathcal {A}(G))=\\bigcup _{i=1}^k Spec(\\mathcal {A}(G_i))^{(r-1)^{n-n_i}},$ $Spec(\\mathcal {L}(G))=\\bigcup _{i=1}^k Spec(\\mathcal {L}(G_i))^{(r-1)^{n-n_i}},$ $Spec(\\mathcal {Q}(G))=\\bigcup _{i=1}^k Spec(\\mathcal {Q}(G_i))^{(r-1)^{n-n_i}},$ where the notation $S^t$ means the repetition of $t$ times of the multi-set $S$ .", "We also need some more preliminaries for the study of disconnected case of the symmetric spectral property of the Laplacian and signless Laplacian spectrum of odd-colorable $r$ -graphs.", "Lemma 13 [14] Let $G$ be a connected $r$ -graph.", "Then $\\rho (\\mathcal {L}(G))=\\rho (\\mathcal {Q}(G))$ if and only if $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))$ .", "In [5], the weak irreducibility of nonnegative tensors was defined as follows.", "Definition 14 [5] Let $\\mathcal {A}$ be an order $r$ dimension $n$ tensor.", "If there exists a proper subset $I$ of the set $[n]$ such that $a_{i_1i_2\\cdots i_r}=0 \\quad (\\forall \\ i_1\\in I, \\ \\mbox{and at least one of the} \\ i_2,\\cdots , i_r\\notin I).$ Then $\\mathcal {A}$ is called weakly reducible (or sometimes $I$ -weakly reducible).", "If $\\mathcal {A}$ is not weakly reducible, then $\\mathcal {A}$ is called weakly irreducible.", "It was proved in [5] and [17] that an $r$ -graph $G$ is connected if and only if its adjacency tensor $\\mathcal {A(}G\\mathcal {)}$ (and so $\\mathcal {Q(}G\\mathcal {)}$ ) is weakly irreducible.", "Lemma 15 [17] Let $\\mathcal {A}$ and $\\mathcal {B}$ be order $r$ dimension $n$ tensors satisfying $|\\mathcal {B}|\\le \\mathcal {A}$ , $\\mathcal {A}$ is weakly irreducible.", "Then (1).", "$\\rho (\\mathcal {B})\\le \\rho (\\mathcal {A}).$ (2).", "If $\\rho (\\mathcal {A})e^{i\\theta }$ is an eigenvalue of $\\mathcal {B}.$ Then $\\mathcal {B=}$ $e^{i\\theta }U^{-(r-1)}\\mathcal {A}U$ for some nonsingular diagonal matrix $U$ all of whose diagonal entries have absolute value 1.", "Now we can obtain the following result.", "Theorem 16 Let $G$ be an $r$ -graph.", "Then $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))$ if and only if $r$ is even and $G$ is odd-colorable.", "Proof First consider the case that $G$ is connected.", "Then by Theorem REF and Lemma REF we have $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))\\Longleftrightarrow Spec(\\mathcal {A}(G))=-Spec(\\mathcal {A}(G))\\Longleftrightarrow G \\ \\mbox{is odd-colorable and $r$ is even}.$ Now we consider the case that $G$ is disconnected.", "Let $G_{1},G_{2},\\cdot \\cdot \\cdot ,G_{t}$ be all the connected components of $G$ , and the number of vertices of $G_1$ be $n_1$ .", "First we prove the sufficiency part.", "We have $G \\ \\mbox{is odd-colorable}& \\Longrightarrow \\mbox{Every} \\ G_i \\ \\mbox{is odd-colorable} \\ (\\forall i=1,\\cdots ,t)\\\\& \\Longrightarrow Spec(\\mathcal {L}(G_i))=Spec(\\mathcal {Q}(G_i)) \\ (\\forall i=1,\\cdots ,t) \\quad \\mbox{(by the proof of connected case) } \\\\& \\Longrightarrow Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G)) \\quad \\mbox{(by Lemma \\ref {spectrum of union}})$ Now we prove the necessity part of the disconnected case.", "We will use induction on $t$ (the number of connected components of $G$ ).", "Set $\\rho :=\\rho (\\mathcal {Q(}G\\mathcal {)})=\\rho (\\mathcal {L(}G\\mathcal {)}).$ Then $\\rho $ is equal to some $\\rho (\\mathcal {L(}G_{j}\\mathcal {))},$ say $\\rho $ $=$ $\\rho (\\mathcal {L(}G_{1}\\mathcal {))}.$ Since $|\\mathcal {L}(G_1)|=\\mathcal {Q}(G_1)$ and $\\mathcal {Q}(G_1)$ is nonnegative weakly irreducible, by Lemma REF we have $\\rho =\\rho (\\mathcal {Q(}G\\mathcal {)})\\ge \\rho (\\mathcal {Q(}G_{1}\\mathcal {))\\ge }\\rho (\\mathcal {L(}G_{1}\\mathcal {))=\\rho }$ .", "Thus we also have $\\rho $ $=\\rho (\\mathcal {Q(}G_{1}\\mathcal {))}$ .", "So for the connected $r$ -graph $G_{1},$ we have $\\rho (\\mathcal {L(}G_{1}\\mathcal {))=}\\rho (\\mathcal {Q(}G_{1}\\mathcal {))}.$ Then by Lemma REF , we have $Spec(\\mathcal {L}(G_1))=Spec(\\mathcal {Q}(G_1))$ , so $G_{1}$ is odd-colorable by the above arguments for the connected case.", "Now consider the $r$ -graph $G^{\\prime }=G_{2}\\cup \\cdot \\cdot \\cdot \\cup G_{t}.$ Since $G=G_1\\cup G^{\\prime }$ , by Lemma REF we have $ Spec(\\mathcal {L}(G))=Spec(\\mathcal {L}(G_1))^{(r-1)^{n-n_1}}\\bigcup Spec(\\mathcal {L}(G^{\\prime }))^{(r-1)^{n_1}},$ $ Spec(\\mathcal {Q}(G))=Spec(\\mathcal {Q}(G_1))^{(r-1)^{n-n_1}}\\bigcup Spec(\\mathcal {Q}(G^{\\prime }))^{(r-1)^{n_1}},$ Thus $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))$ and $Spec(\\mathcal {L}(G_1))=Spec(\\mathcal {Q}(G_1))$ imply that $Spec(\\mathcal {L}(G^{\\prime }))=Spec(\\mathcal {Q}(G^{\\prime }))$ .", "So by induction on $t$ we obtain that $G^{\\prime }$ is also odd-colorable.", "Therefore we conclude that $G$ is also odd-colorable, since $G=G_1\\cup G^{\\prime }$ and both $G_1$ and $G^{\\prime }$ are odd-colorable.", "$\\Box $ As applications of Theorem REF , we can further obtain the following two results (Theorem REF and Theorem REF ).", "Theorem 17 Let $G$ be an $r$ -graph with $r$ even, and $G_1,\\cdots , G_k$ be all the connected components of $G$ .", "Then $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))$ if and only if $Spec(\\mathcal {L}(G_i))=Spec(\\mathcal {Q}(G_i))$ for every connected component $G_i \\ (i=1,\\cdots , k)$ of $G$ .", "Proof Sufficiency follows from Lemma REF .", "Now we prove the necessary part.", "Since $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))$ , we see by Theorem REF that $G$ is odd-colorable.", "Thus $G_i$ is also odd-colorable, and so by Theorem REF again we have $Spec(\\mathcal {L}(G_i))=Spec(\\mathcal {Q}(G_i)) \\ (i=1,\\cdots , k)$ .", "$\\Box $ An eigenvalue of a tensor $\\mathcal {A}$ is called an H-eigenvalue, if there exists a real eigenvector corresponding to it.", "The H-spectrum of a real tensor $\\mathcal {A}$ , denoted by $Hspec (\\mathcal {A})$ , is defined to be the set of distinct H-eigenvalues of $\\mathcal {A}$ .", "Namely, $Hspec (\\mathcal {A})=\\lbrace \\lambda \\in \\mathbb {R} \\ | \\ \\lambda \\ \\mbox{is an H-eigenvalue of $\\mathcal {A}.$} \\ \\rbrace $ In [14](Theorem 2.2), it was proved that when $r$ is even and the $r$ -graph $G$ is connected, then $Hspec(\\mathcal {L}(G))= Hspec (\\mathcal {Q}(G)) \\Longrightarrow Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G)) $ Also the following question was asked in [14]: Question 18 When $r$ is even, whether the reverse implication of (REF ) is true or not?", "In [4], Fan et al.", "showed that the reverse implication of (REF ) is not true in the case $r\\equiv 0 \\ (mod \\ 4)$ by taking the generalized power hypergraphs $G^{r,r/2}$ ($G$ is a non-bipartite ordinary graph) as counterexamples.", "Now by using Theorem REF , we can show in the following theorem that the reverse implication of (REF ) is true in the case $r\\equiv 2 \\ (mod \\ 4)$ , even when $G$ is not connected, thus provide an affirmative answer to Question REF for the remaining unsolved case.", "Theorem 19 Let $G$ be an $r$ -graph with $r\\equiv 2 \\ (mod \\ 4)$ , and $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))$ .", "Then we have $Hspec(\\mathcal {L}(G))= Hspec (\\mathcal {Q}(G)))$ .", "Proof Let $G_1,\\cdots , G_k$ be all the connected components of $G$ .", "Since $Spec(\\mathcal {L}(G))=Spec(\\mathcal {Q}(G))$ , we obtain by Theorem REF that $G$ is odd-colorable.", "By Proposition 12 of [10] we deduce that $G$ is odd-bipartite since $r\\equiv 2 \\ (mod \\ 4)$ .", "Thus every connected component $G_i$ of $G$ is also odd-bipartite.", "Now by Theorem 2.2 of [14], we obtain that $\\mathcal {L}(G_i)$ and $\\mathcal {Q}(G_i)$ have the same H-spectra for all connected components $G_i \\ (i=1,\\cdots , k)$ of $G$ .", "Therefore we conclude that $\\mathcal {L}(G)$ and $\\mathcal {Q}(G)$ have the same H-spectra.", "$\\Box $ Combining Theorem REF and Lemma REF we have Corollary 20 Let $G$ be a connected $r$ -graph.", "Then $\\rho (\\mathcal {L}(G))=\\rho (\\mathcal {Q}(G))$ if and only if $r$ is even and $G$ is odd-colorable." ] ]
1606.05045
[ [ "Stability of mass transfer from massive giants: double black-hole binary\n formation and ultra-luminous X-ray sources" ], [ "Abstract The mass transfer in binaries with massive donors and compact companions, when the donors rapidly evolve after their main sequence, is one of the dominant formation channels of merging double stellar-mass black hole binaries.", "This mass transfer was previously postulated to be unstable and was expected to lead to a common envelope event.", "The common envelope event then would end with either double black hole formation, or with the merger of the two stars.", "We re-visit the stability of this mass transfer, and find that for a large range of the binary orbital separations this mass transfer is stable.", "This newly found stability allows us to reconcile the theoretical rate for double black hole binary mergers predicted by population synthesis studies, and the empirical rate obtained by LIGO.", "Futhermore, the stability of the mass transfer leads to the formation of ultra-luminous X-ray sources.", "The theoretically predicted formation rates of ultra-luminous X-ray sources powered by a stellar-mass BH, as well as the range of produced X-ray luminosity, can explain the observed bright ultra-luminous X-ray sources." ], [ "Introduction", "Understanding and verification of the proposed formation channels of close black hole-black hole (BH-BH) binaries is important for understanding the nature of the reported gravitational wave signal, the nature of gravitational wave signals that could be detected in the near future, as well as for the overall rate of double compact object mergers [1].", "There are two theoretically dominant formation channels that can form a close double BH binary that can merge in a Hubble time in isolation.", "In the first scenario, the formation depends crucially on the development of at least one common envelope phase during the evolution of a double black hole progenitor binary [6].", "The stability of mass transfer (MT) from massive giants may alter the fate of a binary, and in particular it may prohibit double black hole formation.", "In the second formation scenario, a double BH can be formed from an initially tight massive binary with fast rotating stars.", "If both stars in that binary remain fully mixed, neither star will ever become a giant.", "The chemically homogeneous evolution of rapidly rotating stars was studied in , and this formation scenario was proposed in and .", "In this work, we will reevaluate the plausibility of the first formation channel.", "In this scenario, the episode of the MT that most affects BH-BH formation rates takes place when the initially more massive companion has already become a compact object.", "The second companion, now more massive, starts to evolve from its main sequence, and overfills its Roche lobe.", "At the moment the MT starts, the mass ratio (of donor star to compact object) in this system with one compact object is large.", "By the conventional MT stability criterion for either convective or radiative donors, the MT is deemed to be dynamically unstable and must result in a common envelope event.", "Depending on the energy balance, the outcome of the common envelope event is either the merger of the two components, or the ejection of the common envelope.", "In the latter case, the close binary thus produced may evolve into a double BH and merge within a Hubble time, thus becoming a source of detectable gravitational waves.", "For BH-BH progenitor binaries the decisive episode of MT that is to initiate a common envelope phase was found to predominantly take place when the massive donor is at its Hertzsprung gap (HG), before the star is deemed to appear as a well-developed convective giant [4], [6].", "In this paper we will refer to stars being on the HG in the same manner as [25], [14], even if those stars may already have the next stages of core and/or shell burning.", "A donor in this case does not necessarily have a well developed density contrast between the core and the envelope.", "While the outcome of a common envelope event in the case of a well-developed giant donor is not yet well understood and is commonly parameterized with the energy budget formalism [27], the applicability of the energy formalism for a donor with a poorly-developed density contrast is even less justified.", "The uncertainty over whether this unstable MT results in a merger or creates a close binary was most recently investigated with the StarTrack population synthesis code [14], [15], [16], [9].", "It was found that the merger rate for double BHs changes by more than an order of magnitude depending on whether, assuming that the energy formalism can be applied, a binary survives common envelope, or if a merger takes place.", "In this paper we propose a third outcome: we find that the MT is stable for a large range of donor radii and mass ratios.", "We discuss here why the MT is stable and present the detailed MT calculations covering a limited parameter space.", "This allows us to derive the parameterized criterion for stability, suitable for future studies using population synthesis codes.", "We show that this newly identified MT stability between a massive donor during HG or early core helium burning and a BH: (i) does not lead to the formation of close BH-BH binaries, but (ii) does lead to the appearance of the binary systems as ultra-luminous X-ray sources (ULXs)." ], [ "Understanding the stability of the MT from HG donors", "Population synthesis codes, to treat the MT, require parametrized stability thresholds.", "These thresholds can be based on whether the donor is convective or radiative, the mass ratio of the binary companions at the onset of the Roche lobe overflow, the mode of MT (which is the assumption on how conservative the MT is), and sometimes on additional features of the donor and/or of the accretor.", "A common mechanism underlies all types of runaway MT instabilities – insufficient shrinkage of the donor upon the mass loss compared to the change of the radius of the Roche lobe.", "These responses to the MT, by both the donor and the Roche lobe, are known as the mass-radius exponent, abbreviated as $\\zeta _{\\rm {don}}$ and $\\zeta _{\\rm {RL}}$ , respectively.", "They are defined as the logarithmic derivatives of the donor radius $R_{\\rm {don}}$ and of the Roche lobe radius $R_{\\rm {RL}}$ with respect to the donor mass $M_{\\rm {don}}$ : $\\zeta _{\\rm {don}} = \\frac{\\partial \\log R_{\\rm {don}}}{\\partial \\log M_{\\rm {don}} };\\zeta _{\\rm {RL}} = \\frac{\\partial \\log R_{\\rm {RL}}}{\\partial \\log M_{\\rm {don}} }.$ It was suggested in the past that the donor's response $\\zeta _{\\rm don}$ is low if not negative for stars with convective envelopes – indeed, the simplified polytropic stars were found to expand upon mass loss [23].", "That led to a classical understanding that if the donor has a convective envelope at the moment of Roche lobe overflow, the mass ratio is above the critical value, $\\sim 0.78$ , and the MT is fully conservative, then the ensuing MT will be dynamically unstable [23], .", "Later, when the reaction of realistic stellar models was studied, it was shown that the convective envelopes do not necessarily expand upon the mass loss, and the critical mass ratio for stable MT can even be larger than 1 .Note that the mass ratio as is usually defined in MT stability studies – donor mass to accretor mass – differs from that adopted in some population synthesis studies or from the definition used by LIGO (the ratio of the more massive star's mass to that of the less massive star; thus the mass ratio is always $\\ge 1$ ).", "In radiative donors, $\\zeta _{\\rm {don}}$ is positive, and MT in systems with as large a mass ratio as 3.5 is known to be initially dynamically stable [26], [20].", "We clarify that the StarTrack population synthesis code uses mass-radius exponents to determine the stability of the MT [5].", "The difference in the response between radiative and convective donors is bound to their entropy profiles.", "The flat entropy profile of the convective donors leads to insignificant shrinkage or expansion, while the increase with mass of the entropy profile of radiative donors provides the shrinkage as a result of mass removal.", "One of the consequences is that if a radiative donor started initially stable MT, and during the MT the mass layer with an initially flat entropy profile is exposed, a delayed dynamical instability takes place [26], [20].", "Recently, a new MT framework was developed .", "This approach adopts that the stellar material, that currently has expanded outside the Roche lobe of its donor, cannot be immediately relocated into the accretor's Roche lobe, as the MT rate via the $L_{1}$ neighborhood is finite.", "The framework can follow the MT beyond the $L_{1}$ overflow by calculating the current RL overflow MT rate until $L_{2} / L_{3}$ overflow occurs, if the latter does happen.", "A model that has overfilled its outer Lagrangian point can still be simulated, but the MT through this point is not taken into account.", "Usually by the time this happens the MT itself is already dynamically unstable and the Euler term is comparable to the centrifugal term, i.e.", "the Roche lobe approximation itself breaks down.", "As an immediate application, it was found that for donors with deep convective envelopes, the critical mass ratio is about twice what was previously thought, i.e.", "above 1.6.", "This allows for more massive donors to have MT without initiating a common envelope event.", "If a giant donor has a shallow (in mass) convective envelope, it may respond almost like a radiative donor, i.e.", "the MT could be stable for mass ratios up to 3.5 .", "While no specific study for radiative donors has been made, it is likely that treating arbitrary, including very large, $L_{1}$ overflow in a self-consistent way, provides an overall increase of stability of the MT for all donor types.", "This drastic change of the critical mass ratio that separates stable and unstable binary systems is especially important for HG donors that play a role in BH-BH binary formation.", "We note however that a massive donor's structure can be in general quite complex and include both formally radiative layers with only slightly increasing specific entropy in mass, and formally convective layers in which the entropy decreases quite fast.", "The response of the donor is not solely defined by whether a convective or radiative layer is being removed during MT, but is a complex function of the donor's structure as a whole, which can be only determined with detailed simulations.", "A donor with a shallow or absent convective envelope, may still contain a shell with a relatively flat entropy profile.", "When, during the MT, the donor's outer layers are steadily eaten to expose the shell with a sufficiently flat entropy profile, $\\zeta _{\\rm don}$ decreases dramatically, which might lead in some cases to the delayed dynamical instability, if the mass ratio at this time has not been decreased enough.", "Therefore, the simplified prescription that was supplied in for convective donors less massive than $30~M_\\odot $ is not useful for the case of very massive donors at their HG.", "In this paper we will perform detailed simulations for such massive donors using the MT framework from .", "The aim is to find the range of donor radii at the moment of RLOF, such that the MT, for a given mass ratio, will not be affected by either of the following instabilities: Expansion instability that appears in donors which at the moment of RLOF are experiencing a period of fast thermal-timescale expansion.", "In some donors this almost immediately (within a few thousand years) leads to extremely fast MT and dynamical instability.", "We define the radius that a donor should reach before RLOF to avoid this instability as $R_{\\rm S}$ : donors that are larger than $R_{\\rm S}$ at the start of the MT, will experience stable MT.", "Convection instability that appears in donors with a sufficiently developed convective envelope.", "We define the minimum critical radius, which a donor should reach before RLOF to experience this instability, as $R_{\\rm U}$ : donors that are larger than $R_{\\rm U}$ at the start of the MT, will experience unstable MT." ], [ "Detailed MT Simulations", "To calculate the detailed evolution of the giants and their behavior during the Roche lobe overflow, we use the MT framework described in .", "This MT code is a custom extension to the MESA/binary module that uses the MESA/star stellar code for the evolution of single stars , , .", "For donors, we consider stars of several initial masses – 20, 30, 40, 60 and 80 $\\rm {M_{\\odot }}$ – with solar metallicity taken as $Z=0.02$ and with the metallicity $0.1Z_{}$ .", "The stars are evolved employing the Vink wind prescription .", "Luminous blue variable winds and erruptions are not taken into account.", "It is known that high-mass stars are very sensitive to input parameters, and the evolutionary tracks produced by MESA and other stellar codes can differ vastly unless fine-tuned for their input parameters, especially overshooting and wind loss [12].", "Since for our stability it is the state of the donor at the start of the MT that is important, not the adopted prescription of the overshooting or wind loss (donors of different masses can reach that point depending on the adopted prescriptions), we provide the descriptions of the donor stars at the start of the MT rather than simply relying on giving the initial states (see Table REF ).", "We place each star in a binary containing a BH, varying the BH mass from $7~\\rm {M_{\\odot }}$ to $14\\rm {M_{\\odot }}$ .", "We considered the range of the initial binary separation at the start of MT from a few tenths of $\\rm {R_{\\odot }}$ (this corresponds to the Roche lobe overflow right after the end of the donor's main sequence) to a few thousands of $\\rm {R_{\\odot }}$ (when the donor starts to develop a deep convective envelope).", "After the start of the MT, and during the continuing MT, we examine whether or not the MT is dynamically unstable.", "For this we employ the modified criterion outlined in , namely, ${\\dot{M}}P/M > 2\\%$ , where $P$ is the orbital period and $M$ is the donor's mass.", "This way, we can detect either immediate or delayed dynamical instability, whichever takes place in a given system.", "As the initial MT rate in the systems which are close to the stability/instability region is expected to be very high, we test the stability border in the non-conservative (above the Eddington accretion limit) MT regime with isotropic re-emission for angular momentum loss, where the lost material carries away the specific angular momentum of the accretor.", "The third type of instability – delayed dynamical instability – might take place in some donors that overfill their Roche lobe when their radius is inside the stability region – between $R_{\\rm S}$ and $R_{\\rm U}$ .", "The method we use detects whether the MT becomes unstable, and, if it is, we check whether the reason could be related to delayed dynamical instability.", "In all simulations that were done for this study, the delayed dynamical instability was not detected.", "The radii that border the stability region, $R_{\\rm S}$ and $R_{\\rm U}$ , are thus subject to the same uncertainty as the evolution of high-mass stars in general.", "The plausible reasons include, but are not limited to, the adopted prescription for mixing; the inapplicability of the (simplified) chosen atmospheric conditions; winds, instabilities and eruptions; treatment of convection in the outer layers of massive stars; rotation of the envelope; core overshooting, and more (see discussion of uncertainties and how they can affect population studies in [7], and some details on how MESA outcomes in particular can vary depending on the input can be found in [12]).", "The complete results for solar and sub-solar metallicities are presented in Table REF .", "The values for the stability borders are provided as two evolutionary points, where one point is for a model that is certainly stable, and another point for the model that is certainly unstable.", "The actual stability border should be located between these two points.", "Frequently, the behavior between the two points cannot be classified in terms of certain stability or instability.", "The mapping provided in Table REF between the donor's radius, the mass at the moment of Roche lobe overflow, and the stability of the ensuing MT may be directly used by the population synthesis codes.", "Below we provide analyses of the results: Table: Crytical radii for the MT stability.Figure: Development of the convective envelope (shaded area),which leads to the convection instability at R U R_{\\rm U} in a 30M ⊙ 30~\\rm {M_{\\odot }} giant, Z=0.1Z ⊙ _{\\odot }.This star will have stable MT with a 7 M ⊙ M_\\odot BH companionif it is smaller than 1004 R ⊙ R_\\odot at the moment of RLOF.The MT with a 7 M ⊙ M_\\odot BH companion is immediately unstable ifthe donor is larger than 1111 R ⊙ R_\\odot at the moment RLOF." ], [ "Expansion instability", "The first critical point, $R_{\\rm S}$ , happens during the fast thermal-timescale expansion of the star during the HG.", "Most stars that reach RLOF between the end of the main sequence and this critical point experience unstable MT almost immediately, in a few thousand years after the RLOF.", "However, MT in some binaries is always stable after the end of the donor's main sequence.", "For example, for a binary with a BH of $7M_\\odot $ and a donor with an initial mass $30~\\rm {M_{\\odot }}$ and $Z = 0.1 Z_{\\odot }$ , $R_{\\rm S}$ is located between the radii of 44 and $51~\\rm {R_{\\odot }}$ at the moment of RLOF.", "$44~\\rm {R_{\\odot }}$ corresponds to immediately unstable MT, and $51~\\rm {R_{\\odot }}$ to stable MT (see Table REF ).", "At the same time, a binary with a donor with the initial mass of $20 M_\\odot $ of any metallicity always shows stable MT after the end of the donor's main sequence.", "Also, all solar metallicity donors produce stable MT if they reach RLOF after the end of the donor's main sequence.", "The stability increases as the initial mass ratio decreases.", "This is a very well-known effect , which is caused by the increase of $\\zeta _{\\rm {RL}}$ .", "We have analysed whether the rate of radial expansion after the main sequence can in a direct and simple way explain the instability.", "However, we found that the higher rate of expansion is not always associated with this type of instability, and it is likely a hidden function of the envelope density and entropy gradient at the start of the MT." ], [ "Convective instability", "The second critical point, $R_{\\rm U}$ , corresponds to the moment when the donor expands large enough on the giant branch to develop a deep outer convective envelope.", "The emergence of an initial convective layer in the donor's outer layers does not initiate the convective instability.", "For example, in a binary with a 7 $M_{\\rm BH}$ and a $30~\\rm {M_{\\odot }}$ red giant of $Z=0.1 Z_{\\odot }$ , $R_{\\rm U}$ is located between $1004~\\rm {R_{\\odot }}$ and $1112~\\rm {R_{\\odot }}$ at RLOF (see Table REF ).", "If the radius of the donor at the start of the MT is less than $1004~\\rm {R_{\\odot }}$ , the MT is stable even though the donor's outer convective zone is already 1.6$\\rm {M_{\\odot }}$ in mass.", "If the $30 M_\\odot $ donor's radius is $\\ge R_{\\rm U} = 1112~\\rm {R_{\\odot }}$ at RLOF, the MT is unstable.", "The $L_1$ MT rate reaches about 0.27$~\\rm {M_{\\odot }}$  yr$^{-1}$ ; at this fast MT the condition ${\\dot{M}}P/M < 2\\%$ is not satisfied anymore, and we flag it as dynamical-timescale MT.", "The mass of the donor at this moment is reduced to 26.4 M$_\\odot $ .", "The dynamical MT in this case leads to a common envelope which will start with a less-massive envelope, and with different initial orbital parameters at the MT onset, than if the whole envelope were present.", "We observe that this type of instability requires a sufficiently developed convective envelope to be present in the donor.", "For example, the binary with a 10 $M_{\\rm BH}$ and a $80~\\rm {M_{\\odot }}$ red giant of $Z=0.1 Z_{\\odot }$ , is stable until the convective envelope has increased to $18.2M_\\odot $ despite having a mass ratio of 7.5 (see Table REF ).", "One can notice from the data in Table REF that the boundary of this type of instability quite often, albeit not always, is located when the convective envelope is still formed of convective layers which did not yet merge (see Figure REF for a visual depiction, and Table REF where the models with a layered convective envelope are indicated with asterisks).", "Solar-metallicity donors have stronger winds than donors with $Z=0.1Z_\\odot $ .", "These strong winds increase further the stability of the MT for the 80 $\\rm {M_{\\odot }}$ donors: solar metallicity 80 $\\rm {M_{\\odot }}$ donors never reach radii higher than 100 $\\rm {R_{\\odot }}$ and never develop a convective envelope.", "For that reason the second type of instability is not applicable to them, and binaries with 80 $\\rm {M_{\\odot }}$ donors will have stable MT even with a $10~\\rm {M_{\\odot }}$ BH." ], [ "Behavior in the stability region", "Stars that experience RLOF between the first and the second critical point proceed with dynamically stable MT.", "We note that the donors between two critical points can experience $L_{2}/L_{3}$ overflow, which can be detected by our framework, but is not treated (we do not calculate the mass loss through the $L_2/L_3$ nozzle).", "This overflow is not likely to lead to dynamically unstable MT because the outer layers of these stars are quite rarefied and the corresponding mass loss rates are too low to cause any dynamical instability.", "To warrant that the systems are stable, we test binaries applying fully conservative MT evolution – in real binaries, the fraction of the transferred material that is accreted can be anywhere between 0 and 1, but if the MT is stable in the fully conservative case, it will be also be stable in the non-conservative case, assuming isotropic re-emission for angular momentum loss.", "We also consider the case of non-conservative MT.", "We checked the stability of the MT by considering donors with five initial masses of 20, 30, 40, 60 and 80 $\\rm {M_{\\odot }}$ , and metallicity 0.1 Z$_{\\odot }$ .", "For each mass, the donors were taken to be distributed uniformly in the logarithm of radius at RLOF between $R_{\\rm {S}}$ (if applicable, otherwise, the radius at the end of the main sequence) and $R_{\\rm {U}}$ .", "We use 10 $\\rm {M_{\\odot }}$ BHs as accretors for the 60 and 80 $\\rm {M_{\\odot }}$ donors, and 7 $\\rm {M_{\\odot }}$ BHs for the 20 and 30 $\\rm {M_{\\odot }}$ donors.", "All sequences were found to be stable.", "We did not notice a significant difference between the conservative and non-conservative MT sequences in terms of the MT duration or the MT rate, while the final orbital separations in the case of the non-conservative MT are smaller than in the conservative case.", "For example, let us consider a binary with a 7 $M_{\\rm BH}$ and a $30 M_\\odot $ donor of $Z=0.1 Z_{\\odot }$ , with a radius of $750~\\rm {R_{\\odot }}$ at the moment of RLOF.", "At the start of RLOF, the outer envelope of the star is radiative, and the star has a convective core with a mass $\\approx 5~\\rm {M_{\\odot }}$ and an inner convective layer stretching in mass coordinate from $\\approx 8$ to $14~\\rm {M_{\\odot }}$ .", "The rest of the star is radiative.", "A dynamical timescale MT in this donor can be estimated as 1 M$_\\odot $  yr$^{-1}$ , while the maximum attained MT rate does not exceed even one per cent of that.", "The relative RLOF (by what fraction the donor exceeds its Roche lobe radius, in units of the Roche lobe radius) reaches $37\\%$ , but the mass of the star outside of the Roche lobe is less than $0.5\\%$ of the donor mass.", "The binary experiences a brief period of $L_2/L_3$ overflow during which the MT rate reaches $0.005~M_{\\odot }$ yr$^{-1}$ .", "In total, 4.1 M$_\\odot $ was transferred via the $L_{1}$ point during the $L_2/L_3$ overflow that lasted $\\sim $ 1000 years.", "Note, that the stream of matter via the $L_2/L_3$ nozzles (lost from the system) is negligible in mass when compared to the stream flowing via $L_1$ .", "Similar equatorial outflows have been observed in SS 433, an X-ray binary that has a very high MT rate [10].", "After the giant's $\\zeta _{\\textrm {don}}$ becomes larger than $\\zeta _{\\textrm {RL}}$ , and the degree of the donor's Roche lobe overflow becomes small, the MT rate falls to $\\approx 10^{-5}~M_{\\odot }$  yr$^{-1}$ .", "In this state the system remains for about $2\\times 10^5$ years.", "By the time the donor detaches, $\\approx 11.6 M_\\odot $ has been removed, leaving a core of $\\sim 10~M_\\odot $ and an envelope of $7.5 M_\\odot $ .", "The envelope is predominantly radiative with a convective layer located between $10.5~M_\\odot $ to $12.3~M_\\odot $ .", "The mass ratio of the donor to the BH, by the time the MT stops, is $\\sim 2.5$ .", "The binary has shrunk, and the donor radius at the end is only about 107 compared to 750 R $_\\odot $ at the moment of Roche lobe overflow.", "We note that a strong shrinkage is a common property for the binaries we consider.", "However, for most systems, their final separation does not enable future formation of a double BH that can merge within a Hubble time.", "Figure: History of the MT in a binary system with a 20 M ⊙ \\rm {M_{\\odot }} giant and a 7 M ⊙ \\rm {M_{\\odot }} BH.At the RLOF the donor has a radius of 144 R ⊙ \\rm {R_{\\odot }}, and has Z=0.1Z ⊙ _{\\odot }.Interruptions of the line indicate that the donor's radius is smallerthan the Roche lobe volume radius (e.g., the donor detaches).It does not mean that the mass loss rate is zero, because the donor produces a stellar wind.We conclude that if the binary systems start the MT when their donor's radii are between $R_{\\rm S}$ and $R_{\\rm U}$ , they do not experience unstable MT.", "It means that they are unlikely to experience the common envelope phase, unless the donor reattaches again to the Roche lobe in the future and loses enough mass to reach the DDI (a theoretical possibility that we did not encounter in our models, but cannot exclude completely).", "This rules out the possibility that such binaries can produce double BH mergers.", "The range of the radii, and therefore of the initial orbital periods at which that MT can be initiated, covers almost the entire parameter space between the end of the main sequence and a “convective” giant stage." ], [ "ULX sources", "ULXs are sources with X-ray luminosities above $10^{39}$ erg/s, where this chosen threshold implies exceeding the Eddington limit for a $\\sim 7\\rm {M_{\\odot }}$ BH accreting material with hydrogen abundance $X=0.7$ , and calculated assuming Thompson scattering opacities: $L_{\\rm Edd} = \\frac{2.6\\times 10^{38}}{1+X} \\frac{M_{\\rm acc}}{M_\\odot }$ Here $M_{\\rm acc}$ is the mass of the accretor.", "The are two dominant ways to explain ULXs.", "One is to assume that the accretor is an intermediate mass BH (100 $M_\\odot $ or more) that accretes at (or below) its Eddington limit [13].", "Alternatively, it can be a binary with a NS or a stellar mass BH where the MT rate exceeds the Eddington limit by a factor of up to a few dozens, and the radiation is beamed , .", "If the MT rate is very high, such as produced by thermal-time-scale MT, then relativistic beaming is not required to explain the observed ULX luminosities.", "Strong relativistic beaming is also not expected for BH accretors when the MT rates exceed strongly their Eddington limit, unlike the case with neutron stars accretors , .", "The binary systems considered in this paper can produce the last mentioned type of ULXs, with a stellar mass BH accretor and very high MT rates, where the novelty is that the mass ratio is very large.", "There are at least two important observational examples of an ULX with an accretor having a mass of only a few solar masses, where the observed luminosity exceeds the accretor's Eddington limit, and where the mass ratio between the donor and the accretor is large (note that our study should not be used to explain all ULXs observed to date).", "The first example is SS 433, which is a well known X-ray binary in the Milky Way that has most likely a stellar mass BH accretor [18], [11].", "This system contains a massive donor of $12.3\\pm 3.3 M_\\odot $ and a BH of $4.3\\pm 0.8M_\\odot $ [22].", "SS 433 is a non-conservative system, and the rate of mass outflow exceeds the Eddington accretion limit by a factor of several hundred to a few thousands , .", "This system is our local “misaligned” ULX, and the beamed emission is coming out of a cone with a half opening angle of about $20^{\\circ }$ [3], .", "The second example is ULX M82 X-2, discovered by in the nearby galaxy M82.", "The observed luminosity of ULX M82 X-2 is $1.8 \\times 10^{40}$  erg/s.", "It has been shown recently, via the discovery of X-ray pulsations, that the accretor in M82 X-2 is a neutron star [2], thus the luminosity in this system exceeds the Eddington limit for a 1.4 $\\rm {M_{\\odot }}$ neutron star by a factor of 100.", "Interestingly, the population studies of ULXs, until recently, mostly involved the consideration of an intermediate mass BH with a stellar mass donor , , [24], , , as the MT in systems with very massive donors and stellar mass BH accretors was thought to be dynamically unstable.", "However, observations showed that ULX M82 X-2 consists of a massive giant donor and a neutron star accretor orbiting each other with a period of 2.5 days and a minimum companion mass of $5.2 M_\\odot $ , with the mass ratio exceeding 3.7 [2].", "[19] have shown that it is possible to obtain such systems assuming non-conservative MT from a hydrogen-rich giant donor that fills its Roche lobe, to a neutron star accretor.", "They used an implicit MT scheme that is built such that the donor is always kept within its Roche lobe.", "Our study is somewhat similar albeit we have less technical constraints on which MT systems can be modelled through the MT: our MT framework can handle MT rates which are up to a few per cent of the dynamical timescale MT rate – up to a dozen of $M_\\odot $ yr$^{-1}$ , while [19] had to assume that any system where the MT rate exceeded $10^{-2}\\rm {M_{\\odot }}yr^{-1}$ is bound to start a common envelope.", "our MT framework allows for high degrees of Roche lobe overflow, all the way to $L_{2}/L_{3}$ (outer Lagrangian point of the donor) overflow.", "Figure: The time-averaged distribution of ULXs formed in our simulations (through conservative MT)is shown with the red histogram.", "The distribution of X-ray luminosities of 33 ULXs in nearby ( 5 Mpc)galaxies, as taken from , is shown with the hatched area.", "L X L_X is X-ray luminosity.For modelled ULXs, L X L_X is calculated using MT rate via L 1 L_1,M ˙ L1 , \\dot{M}_{L1}, and Equation () with f=1f=1.It indicates thereforethe upper range of possible L X L_X.We analyzed the MT sequences obtained in §REF , and compared the expected X-ray luminosities of these systems to those of the known ULXs studied in [21].", "The comparison is presented in Figure REF .", "To convert from the MT rate to X-ray luminosity we use: $L_x = \\frac{ \\dot{M} c^2}{ 2 f }$ here $f$ is a factor indicating the “inefficiency” of conversion of the accreted material to energy.", "The material may not emit all its energy as it falls down to the Schwarzschild radius, but may instead plunge from the radius of marginal stability across the horizon without emitting any further energy .", "Then the X-ray luminosity produced per unit accreted mass is smaller, especially for a non-rotating BH, for which $f$ is about 6 .", "With $f=1$ , ULX-thershold luminosity of $10^{39}$  erg $s^{-1}$ can be provided by as low accretion rate on a BH as $3.5\\times 10^{-8}$ $M_\\odot $  yr$^{-1}$ .", "We find that the average time our models spend in the ULX state is $\\sim 10^5$ years.", "We regularly obtain MT rates wich exceed the Eddington luminosity by a factor of $\\sim $ 1000 (see, e.g., the history of MT shown in Figure REF ).", "The time-averaged distribution of X-ray luminosities obtained from the simulations can explain the luminosities of the observed ULXs even if the accreting BHs are non-rotating.", "The donors which are initially more massive than $30 M_\\odot $ dominate above $L_x\\ge 10^{41}$  ergs s$^{-1}$ (this value is smaller by a factor of 6 if the BH accretors are non-rotating).", "These systems with massive donors can nicely explain, for instance, ULX1 located in NGC 5643.", "That ULX has a luminosity of $4\\times 10^{40}$ erg s$^{-1}$ and was argued to have a BH accretor .", "We note, though, that our study shows that the accretor does not require a BH as massive as $30 M_\\odot $ emitting at 10 times the Eddington limit, but this sytem may only have a $10 M_\\odot $ BH, while having a much higher MT rate.", "Let us estimate the rates at which we can produce detectable ULXs via our channel.", "Our ULXs are produced in place of the common envelope events during a HG leading to merging BH-BH, and therefore we will be using those rates for our estimate.", "[14] have provided a suite of 16 population synthesis models, that are also available online syntheticuniverse.org.", "For each model, two submodels were presented.", "In one subset, it was assumed that common envelope initiated by a HG donor is allowed, and energy balance was applied to check if a given system has survived or merged (submodels A).", "In the other subset, it was assumed that each common envelope initiated by a HG donor leads to a merger of two stars, aborting binary evolution (submodels B).", "Not every Roche Lobe overflow with a HG donor in the [14] simulations leads to a common envelope: some MTs with HG donors are stable events in both submodels.", "For submodels A, [14] found that the formation rate of BH-BH mergers can be up to 26 Myr$^{-1}$ per star formation unit of 1 $M_\\odot $ yr$^{-1}$ for metallicity $Z=0.002$ , and up to 8.5 Myr$^{-1}$ per star formation unit of 1 $M_\\odot $ yr$^{-1}$ for solar-type metallicity of $Z=0.02$ .", "These rates assume that if the MT starts when a massive donor crosses the HG, a common envelope event can take place, and that it may potentially result in the formation of a double BH.", "In models where this formation channel is inhibited (submodels B), the resulting merger rate drops dramatically, indicating that the BH-BH merger rates are dominated by common envelopes with HG donors.", "For example, the BH-BH merger rates for submodels B are up to 8.2 Myr$^{-1}$ per star formation unit of 1 $M_\\odot $ yr$^{-1}$ for metallicity $Z=0.002$ , and up to 1.2 Myr$^{-1}$ per star formation unit of 1 $M_\\odot $ yr$^{-1}$ for metallicity $Z=0.02$ [14].", "Not all MTs that are initiated during the HG and result in a common envelope event lead to BH-BH formation, but the difference in merger rates between submodels A and B (quoted above) gives an approximate estimate for the number of MT events that took place, and therefore provides the lower limit on the theoretically expected rate of MTs during the HG.", "On the other hand, we have used the upper limits on BH-BH merger rates from [14] to limit how large the rates inferred using submodels A may reach.", "Our ULXs are therefore formed at the rate 18 Myr$^{-1}$ per unit of star formation for sub-solar metallicity $Z=0.002$ , while the rate drops down to 7.3 Myr$^{-1}$ per unit of star formation for solar-like metallicity $Z=0.02$ .", "With the average lifetime of $\\sim 10^5$ years, and assuming, as in [14], that the star formation rate in the Milky Way is 3.5 $M_\\odot $ yr$^{-1}$ , the theoretical rates above result in $\\sim 2.6$ ULX systems can be present in a galaxy similar to the Milky Way, at a solar metallicity.", "Note that in Milky Way two misaligned ULX systems were tentatively identified (SS 433 and GRS 1915+105), where one observed system, SS 433, resembles our model of a massive donor and a substantially less massive BH.", "A synthetic galaxy similar to the Milky Way with the same star formation rate, but with sub-solar metallicity of $Z=0.002$ would contain $\\sim 6.3$ ULX systems.", "Summing up, the theoretically expected numbers of ULXs per star formation unit of 1 $M_\\odot $ yr$^{-1}$ , are $\\sim 1.8$ ULXs for $Z=0.002$ and $\\sim 0.7$ for $Z=0.02$ .", "A more accurate estimate can be made only with a future full scale population synthesis study.", "Observationally, ULXs are detected at numbers in range $0.2$ –2 per star formation rate of 1 $M_\\odot $ yr$^{-1}$ , , where the larger value is determined from the sample of all galaxies within 14.5 Mpc, and the smaller value is determined from the sample of 17 luminous infrared galaxies within 60 Mpc (these galaxies presumably have a higher metallicity).", "A part of the observed ULXs can potentially be interpreted as systems with intermediate mass BHs, especially for nearby galaxies where more constraints on the spectrum of ULXs were obtained [21].", "More importantly, a fraction of ULXs can be powered by accreting neutron stars, like ULX M82 X-2.", "Adopting that half of the observed ULXs have a stellar mass BH accretor, observationally, ULXs are detected at about 1 per star formation rate of 1 $M_\\odot $ yr$^{-1}$ , similar to our theoretically predicted rates.", "We mainly produce ULXs at the higher end of the ULX luminosity function ($L_X10^{41}$ erg s$^{-1}$ ).", "Even after taking into account the inefficiency $f$ , or considering that some matter can outflow from the system, as observed in SS 433, and thus cannot be converted into radiation, our systems likely will produce predominantly $L_X10^{40}$ erg s$^{-1}$ .", "The observed formation rate of similarly bright ULXs is about 5 times less than the overall ULX formation rate, or 0.4 per star formation rate of 1 $M_\\odot $ yr$^{-1}$ , or even less, if the observed sample contains also ULXs powered by intermediate mass BHs or NSs.", "It is important that some geometrical collimation of the emission cone, for BHs that accrete at rates that exceed vastly their Eddington limits, is expected.", "The fraction of the sky covered by the beamed radiation can be as low as 0.1 .", "Therefore it is possible that observationally, only 10 per cent of the existing bright ULXs are detected.", "Then the intrinsic formation rate of bright ULXs can be as high as 4 ULXs per star formation rate of 1 $M_\\odot $ yr$^{-1}$ .", "We conclude that our theoretical formation rate of bright stellar-mass BH ULXs is similar to the observed one.", "The theoretical rate might even be increased if all MT events that take place during the HG are taken into account, not just those that were assumed to result in the formation of BH-BH mergers from the HG common envelope channel.", "Additionally, it is also very likely that other channels (not involving HG donors) may produce high luminosity ULXs .", "We find that as metallicity increases, the number of produced ULXs drops, which may explain the deficit of ULXs per star formation unit in luminous infrared galaxies found in ." ], [ "Conclusions", "We analyzed the MT in binary systems with massive donors ($20-80~M_\\odot $ ) and stellar-mass BHs ($7-14~M_\\odot $ ) for two metallicities: solar ($Z_\\odot =0.02$ ) and sub-solar ($0.1 Z_\\odot $ ).", "The considered binary systems have high mass ratios, and the mass transfer in these systems was previously considered to be unstable.", "We found the regions where the MT is stable, including systems with an initial (onset of mass transfer) mass ratio as high as $7.5$ .", "The stability regions are bordered by two key instabilities.", "First is the instability that develops during fast post-main sequence expansion; this instability is avoided in a number of donors, especially of solar metallicity.", "Second is the instability that takes place when a sufficiently deep outer convective zone is developed; this instability starts well after the initial development of the outer convective envelope.", "However, in most of the parameter space where donor stars are on the HG (as adopted for this study, this includes early core helium burning stars), the binaries are found to evolve through fast but stable mass transfer.", "Unstable MT takes place only in very expanded giants with well developed envelopes.", "In those cases, by the onset of the common envelope, a substantial part of the envelope had been transferred via the initial stage of the mass transfer.", "The effect of the donor's envelope “reduction” on the common envelope outcomes, in principle, can be tested in population synthesis with common envelope energy prescription parameters, as it might be important.", "A typical case that leads to the formation of a BH-BH merger resembling GW150914 in an isolated binary, without the requirement of homogeneous evolution, involves a core helium-burning giant with a mass of $82.2 M_\\odot $ which enters CE phase at a radius of $1665~\\rm {R_{\\odot }}$ (see Fig.", "1 of [8]).", "This typical binary simulated with the population synthesis code StarTrack is outside the range of parameters tested in our study; it originates at a very low metallicity $Z=0.0006$ , and at the onset of CE the BH mass is $35.1 M_\\odot $ .", "The CE leads to significant orbital decay (the orbital separation decreases from $a=3780~\\rm {R_{\\odot }}$ to $a=43.8~\\rm {R_{\\odot }}$ ) as the large envelope (mass of $45.4 M_\\odot $ ) is ejected (with the adopted $\\alpha =1.0$ and the estimated $\\lambda =0.15$ from ).", "This binary survives and forms a massive BH-BH merger (two BHs with mass $\\sim 30 M_\\odot $ ) at the end of its evolution.", "In the past, and in the majority of present studies, it is assumed that post-main sequence stars (including HG stars) may evolve through, and survive, a common envelope , , , , , [17].", "Since a significant fraction of stellar expansion is encountered during the HG, the most likely formation scenarios of BH-BH mergers in population synthesis studies involve HG donors in the common envelope phase.", "With our findings, the BH-BH merger rates obtained in simulations that allow for common envelopes with HG donors are subject to drastic reduction.", "This effect was approximately quantified with the use of the StarTrack population synthesis code.", "For solar-like metallicity the effect is overwhelming, producing a factor of 750 decrease in the predicted BH-BH merger rates.", "For sub-solar metallicity, this produces however only a factor of 14 decrease (see Tab.", "1 of [6]).", "Most BH-BH mergers are estimated to originate from low-metallicity environments, and a population synthesis calculation that takes into account the forbidden common envelope development predicts a merger rate of 220 Gpc$^{-3}$ yr$^{-1}$ , and the most likely detection of BH-BH mergers with a total mass of 20–$80 M_\\odot $ [8].", "In the same study, the model in which common envelope is allowed for HG donors generates BH-BH merger rates of $\\gtrsim 1000$ Gpc$^{-3}$ yr$^{-1}$ .", "The LIGO empirical estimate of the local Universe BH-BH merger rate is in the range 9–240 Gpc$^{-3}$ yr$^{-1}$ , and the three BH-BH mergers are found with total (the sum of the two merging black hole masses) masses in the range 22–$65 M_\\odot $, .", "The LIGO results indicate therefore that a population synthesis approach in which HG donors will initiate a common envelope is not valid.", "Here we provide a previously missing theoretical foundation that allows us to reconcile the theory and observations.", "We also demonstrate that the MT binaries with HG donors that avoid the common envelope event will become ULXs.", "High thermal timescale MT rates lead to very high X-ray luminosities, which may even exceed $10^{42}$ erg s$^{-1}$ (see also ).", "The theoretically expected formation rate of ULXs that are powered by accretion onto a stellar-mass BH is found to be $0.7$ –$1.8$ per star formation unit of 1 $M_\\odot $ yr$^{-1}$ , for the metallicity range $Z=0.02$ –$0.002$ .", "The binaries that produce bright ULXs consist of donors that are initially 20–$80M_\\odot $ , and significantly less massive BHs.", "This rate may explain the observed formation rate of bright ULXs, which is 0.4-4 per star formation unit of 1 $M_\\odot $ yr$^{-1}$ ." ], [ "Acknowledgements", "NI thanks NSERC Discovery and Canada Research Chairs Program.", "The authors thank C. Heinke for checking the English in the manuscript.", "This research has been enabled by the use of computing resources provided by WestGrid and Compute/Calcul Canada." ] ]
1606.04921
[ [ "Signaling equilibria for dynamic LQG games with asymmetric information" ], [ "Abstract We consider a finite horizon dynamic game with two players who observe their types privately and take actions, which are publicly observed.", "Players' types evolve as independent, controlled linear Gaussian processes and players incur quadratic instantaneous costs.", "This forms a dynamic linear quadratic Gaussian (LQG) game with asymmetric information.", "We show that under certain conditions, players' strategies that are linear in their private types, together with Gaussian beliefs form a perfect Bayesian equilibrium (PBE) of the game.", "Furthermore, it is shown that this is a signaling equilibrium due to the fact that future beliefs on players' types are affected by the equilibrium strategies.", "We provide a backward-forward algorithm to find the PBE.", "Each step of the backward algorithm reduces to solving an algebraic matrix equation for every possible realization of the state estimate covariance matrix.", "The forward algorithm consists of Kalman filter recursions, where state estimate covariance matrices depend on equilibrium strategies." ], [ "Introduction", "Linear quadratic Gaussian (LQG) team problems have been studied extensively under the framework of classical stochastic control with single controller and perfect recall [1].", "In such a system, the state evolves linearly and the controller makes a noisy observation of the state which is also linear in the state and noise.", "The controller incurs a quadratic instantaneous cost.", "With all basic random variables being independent and Gaussian, the problem is modeled as a partially observed Markov decision process (POMDP).", "The belief state process under any control law happens to be Gaussian and thus can be sufficiently described by the corresponding mean and covariance processes, which can be updated by the Kalman filter equations.", "Moreover, the covariance can be computed offline and thus the mean (state estimate) is a sufficient statistic for control.", "Finally, due to the quadratic nature of the costs, the optimal control strategy is linear in the state.", "Thus, unlike most POMDP problems, the LQG stochastic control problem can be solved analytically and admits an easy-to-implement optimal strategy.", "LQG team problems have also been studied under non-classical information structure such as in multi-agent decentralized team problems where two controllers with different information sets minimize the same objective.", "Such systems with asymmetric information structure are of special interest today because of the emergence of large scale networks such as social or power networks, where there are multiple decision makers with local or partial information about the system.", "It is well known that for decentralized LQG team problems, linear control policies are not optimal in general [2].", "However there exist special information structures, such as partially nested [3] and stochastically nested [4], where linear control is shown to be optimal.", "Furthermore, due to their strong appeal for ease of implementation, linear strategies have been studied on their own for decentralized teams even at the possibility of being suboptimal (see [5] and references therein).", "When controllers (or players) are strategic, the problem is classified as a dynamic game and an appropriate solution concept is some notion of equilibrium.", "When players have different information sets, such games are called games with asymmetric information.", "There are several notions of equilibrium for such games, including perfect Bayesian equilibrium (PBE), sequential equilibrium, trembling hand equilibrium [6], [7].", "Each of these notions of equilibrium consists of a strategy and a belief profile of all players where the equilibrium strategies are optimal given the beliefs and the beliefs are derived from the equilibrium strategy profile and using Bayes' rule (whenever possible), with some equilibrium concepts requiring further refinements.", "Due to this circular argument of beliefs being consistent with strategies which are in turn optimal given the beliefs, finding such equilibria is a difficult task.", "To date, there is no known sequential decomposition methodology to find such equilibria for general dynamic games with asymmetric information.", "Authors in [8] studied a discrete-time dynamic LQG game with one step delayed sharing of observations.", "Authors in [9] studied a class of dynamic games with asymmetric information under the assumption that player's posterior beliefs about the system state conditioned on their common information are independent of the strategies used by the players in the past.", "Due to this independence of beliefs and past strategies, the authors of [9] were able to provide a backward recursive algorithm similar to dynamic programming to find Markov perfect equilibria [11] of a transformed game which are equivalently a class of Nash equilibria of the original game.", "The same authors specialized their results in [12] to find non-signaling equilibria of dynamic LQG games with asymmetric information.", "Recently, we considered a general class of dynamic games with asymmetric information and independent private types in [13] and provided a sequential decomposition methodology to find a class of PBE of the game considered.", "In our model, beliefs depend on the players' strategies, so our methodology allows the possibility of finding signaling equilibria.", "In this paper, we build on this methodology to find signaling equilibria for two-player dynamic LQG games with asymmetric information.", "We show that players' strategies that are linear in their private types in conjunction with consistent Gaussian beliefs form a PBE of the game.", "Our contributions are: (a) Under strategies that are linear in players' private types, we show that the belief updates are Gaussian and the corresponding mean and covariance are updated through Kalman filtering equations which depend on the players' strategies, unlike the case in classical stochastic control and the model considered in [12].", "Thus there is signaling [14], [15].", "(b) We sequentially decompose the problem by specializing the forward-backward algorithm presented in [13] for the dynamic LQG model.", "The backward algorithm requires, at each step, solving a fixed point equation in `partial' strategies of the players for all possible beliefs.", "We show that in this setting, solving this fixed point equation reduces to solving a matrix algebraic equation for each realization of the state estimate covariance matrices.", "(c) The cost-to-go value functions are shown to be quadratic in the private type and state estimates, which together with quadratic instantaneous costs and mean updates being linear in the control action, implies that at every time $t$ player $i$ faces an optimization problem which is quadratic in her control.", "Thus linear control strategies are shown to satisfy the optimality conditions in [13].", "(d) For the special case of scalar actions, we provide sufficient algorithmic conditions for existence of a solution of the algebraic matrix equation.", "Finally, we present numerical results on the steady state solution for specific parameters of the problem.", "The paper is structured as follows.", "In Section , we define the model.", "In Section , we introduce the solution concept and summarize the general methodology in [13].", "In Section , we present our main results where we construct equilibrium strategies and belief through a forward-backward recursion.", "In Section  we discuss existence issues and present numerical steady state solutions.", "We conclude in Section .", "We use uppercase letters for random variables and lowercase for their realizations.", "We use bold upper case letters for matrices.", "For any variable, subscripts represent time indices and superscripts represent player identities.", "We use notation $ -i$ to represent the player other than player $i$ .", "We use notation $a_{t:t^{\\prime }}$ to represent vector $(a_t, a_{t+1}, \\ldots a_{t^{\\prime }})$ when $t^{\\prime }\\ge t$ or an empty vector if $t^{\\prime }< t$ .", "We remove superscripts or subscripts if we want to represent the whole vector, for example $ a_t$ represents $(a_t^1, a_t^2) $ .", "We use $\\delta (\\cdot )$ for the Dirac delta function.", "We use the notation $X\\sim F$ to denote that the random variable $X$ has distribution $F$ .", "For any Euclidean set $\\mathcal {S}$ , $\\mathcal {P}(\\mathcal {S})$ represents the space of probability measures on $\\mathcal {S}$ with respect to the Borel sigma algebra.", "We denote by $P^g$ (or $\\mathbb {E}^g$ ) the probability measure generated by (or expectation with respect to) strategy profile $g$ .", "We denote the set of real numbers by $\\mathbb {R}$ .", "For any random vector $X$ and event A, we use the notation $sm(\\cdot |\\cdot )$ to denote the conditional second moment, $sm(X|A) : = \\mathbb {E}[XX^{\\dagger }|A]$ .", "For any matrices $\\mathbf {A}$ and $\\mathbf {B}$ , we will also use the notation $quad(\\cdot ;\\cdot )$ to denote the quadratic function, $quad(\\mathbf {A};\\mathbf {B}) :=\\mathbf {B}^{\\dagger }\\mathbf {A}\\mathbf {B}$ .", "We denote trace of a matrix $\\mathbf {A}$ by $tr(\\mathbf {A})$ .", "$N(\\hat{x},\\mathbf {\\Sigma })$ represents the vector Gaussian distribution with mean vector $\\hat{x}$ and covariance matrix $\\mathbf {\\Sigma }$ .", "All equalities and inequalities involving random variables are to be interpreted in a.s. sense and inequalities in matrices are to be interpreted in the sense of positive definitedness.", "All matrix inverses are interpreted as pseudo-inverses." ], [ "Model", "We consider a discrete-time dynamical system with 2 strategic players over a finite time horizon $\\mathcal {T} := \\lbrace 1, 2, \\ldots T\\rbrace $ and with perfect recall.", "There is a dynamic state of the system $x_t := (x_t^1, x_t^2)$ , where $x_t^i \\in \\mathcal {X}^i :=\\mathbb {R}^{n_i}$ is private type of player $i$ at time $t$ which is perfectly observed by her.", "Player $i$ at time $t$ takes action $u_t^i \\in \\mathcal {U}^i := \\mathbb {R}^{m_i}$ after observing $u_{1:t-1}$ , which is common information between the players, and $x_{1:t}^i$ , which it observes privately.", "Thus at any time $t\\in \\mathcal {T}$ , player $i$ 's information is $u_{1:t-1},x_{1:t}^i$ .", "Players' types evolve linearly as $x_{t+1}^i = \\mathbf {A}_t^ix_t^i + \\mathbf {B}_t^i u_t + w_t^i, $ where $\\mathbf {A}_t^i, \\mathbf {B}_t^i$ are known matrices.", "$(X^1_1,X_1^2,(W_t^i)_{t\\in \\mathcal {T}})$ are basic random variables of the system which are assumed to be independent and Gaussian such that $X_1^i\\sim N(0,\\mathbf {\\Sigma }^i_1)$ and $W_t^i \\sim N(0,\\mathbf {Q}^i)$ .", "As a consequence, types evolve as conditionally independent, controlled Markov processes, $P(x_{t+1}|u_{1:t},x_{1:t}) &= P(x_{t+1}|u_{t},x_{t} )=\\prod _{i=1}^2 Q^i(x_{t+1}^i|u_{t},x_{t}^i ) .$ where $Q^i(x_{t+1}^i|u_{t},x_{t}^i ):= P(w_t^i =x_{t+1}^i -\\mathbf {A}_t^ix_t^i - \\mathbf {B}_t^i u_t )$ .", "At the end of interval $t$ , player $i$ incurs an instantaneous cost $R^i(x_t,u_t)$ , $R^i(x_t,u_t) &= u_t^{\\dagger }\\mathbf {T}^iu_t + x_t^{\\dagger }\\mathbf {P}^ix_t + 2u_t^{\\dagger }\\mathbf {S}^ix_t \\nonumber \\\\&= \\mathbf {u_t^{\\dagger } & x_t^{\\dagger }} \\mathbf {\\mathbf {T}^i &\\mathbf {S}^i \\\\ \\mathbf {S}^{i\\dagger } & \\mathbf {P}^i}\\mathbf {u_t \\\\ x_t},$ where $\\mathbf {T}^i, \\mathbf {P}^i, \\mathbf {S}^i$ are real matrices of appropriate dimensions and $\\mathbf {T}^i,\\mathbf {P}^i$ are symmetric.", "We define the instantaneous cost matrix $\\mathbf {R}^i$ as $\\mathbf {R}^i := \\mathbf {\\mathbf {T}^i &\\mathbf {S}^i \\\\ \\mathbf {S}^{i\\dagger } & \\mathbf {P}^i}$ .", "Let $g^i = ( g^i_t)_{t \\in \\mathcal {T}}$ be a probabilistic strategy of player $i$ , where $g^i_t : (\\mathcal {U}^i)^{t-1}\\times (\\mathcal {X}^i)^t \\rightarrow \\mathcal {P}(\\mathcal {U}^i)$ such that player $i$ plays action $u_t^i$ according to distribution $ g^i_t(\\cdot |u_{1:t-1},x_{1:t}^i)$ .", "Let $ g :=(g^i)_{i=1,2}$ be a strategy profile of both players.", "The distribution of the basic random variables and their independence structure together with the system evolution in (REF ) and players strategy profile $g$ define a joint distribution on all random variables involved in the dynamical process.", "The objective of player $i$ is to maximize her total expected cost $J^{i,g} := \\mathbb {E}^g \\left\\lbrace \\sum _{t=1}^T R^i(X_t,U_t) \\right\\rbrace .", "$ With both players being strategic, this problem is modeled as a dynamic LQG game with asymmetric information and with simultaneous moves." ], [ "Preliminaries", "In this section we introduce the equilibrium concept for dynamic games with asymmetric information and summarize the general methodology developed in [13] to find a class of such equilibria." ], [ "Solution concept", "Any history of this game at which players take action is of the form $h_t = (u_{1:t-1},x_{1:t})$ .", "Let $\\mathcal {H}_t$ be the set of such histories at time $t$ and $\\mathcal {H}:= \\cup _{t=0}^T \\mathcal {H}_t $ be the set of all possible such histories.", "At any time $t$ player $i$ observes $h^i_t = (u_{1:t-1},x_{1:t}^{i})$ and both players together have $h^c_t = u_{1:t-1}$ as common history.", "Let $\\mathcal {H}^i_t$ be the set of observed histories of player $i$ at time $t$ and $\\mathcal {H}^c_t$ be the set of common histories at time $t$ .", "An appropriate concept of equilibrium for such games is the PBE [7] which consists of a pair $(\\beta ^*,\\mu ^*)$ of strategy profile $\\beta ^* = (\\beta _t^{*,i})_{t \\in \\mathcal {T},i=1,2}$ where $\\beta _t^{*,i} : \\mathcal {H}_t^i \\rightarrow \\mathcal {P}(\\mathcal {U}^i)$ and a belief profile $\\mu ^* = (\\mu _t^{*,i})_{t \\in \\mathcal {T},i=1,2}$ where $\\mu _t^{*,i}: \\mathcal {H}^i_t \\rightarrow \\mathcal {P}(\\mathcal {H}_t)$ that satisfy sequential rationality so that for $ i =1,2, \\forall t \\in \\mathcal {T}, h^{i}_t \\in \\mathcal {H}^i_t, {\\beta ^{i}}$ $\\mathbb {E}^{(\\beta ^{*,i} \\beta ^{*,-i},\\, \\mu ^*)}\\left\\lbrace \\sum _{n=t}^T R^i(X_n, U_n)\\big \\vert h^i_t\\right\\rbrace \\le \\mathbb {E}^{({\\beta }^{i} \\beta ^{*,-i},\\, \\mu ^*)}\\left\\lbrace \\sum _{n=t}^T R^i(X_n, U_n)\\big \\vert h^i_t\\right\\rbrace , \\;\\; \\;\\; $ and the beliefs satisfy consistency conditions as described in [7]." ], [ "Structured perfect Bayesian equilibria", "A general class of dynamic games with asymmetric information was considered in [13] by the authors where players' types evolve as conditionally independent controlled Markov processes.", "A backward-forward algorithm was provided to find a class of PBE of the game called structured perfect Bayesian equilibria (SPBE).", "In these equilibria, player $i$ 's strategy is of the form $U_t^i \\sim m_t^i(\\cdot |\\pi ^1_t, \\pi ^2_t ,x_t^i)$ where $m_t^i: \\mathcal {P}(\\mathcal {X}^1) \\times \\mathcal {P}(\\mathcal {X}^2) \\times \\mathcal {X}^i \\rightarrow \\mathcal {P}(\\mathcal {U}^i)$ .", "Specifically, player $i$ 's action at time $t$ depends on her private history $x_{1:t}^i$ only through $x_t^i$ .", "Furthermore, it depends on the common information $u_{1:t-1}$ through a common belief vector $\\underline{\\pi }_t:=(\\pi _t^1,\\pi _t^2)$ where $\\pi _t^i\\in \\mathcal {P}(\\mathcal {X}^i)$ is belief on player $i$ 's current type $x_t^i$ conditioned on common information $u_{1:t-1}$ , i.e.", "$\\pi _t^i(x_t^i):= P^g(X_t^i = x_t^i|u_{1:t-1})$ .", "The common information $u_{1:t-1}$ was summarized into the belief vector $(\\pi _t^1,\\pi _t^2)$ following the common agent approach used for dynamic decentralized team problems [16].", "Using this approach, player $i$ 's strategy can be equivalently described as follows: player $i$ at time $t$ observes $u_{1:t-1}$ and takes action $\\gamma _t^i$ , where $\\gamma _t^i : \\mathcal {X}^i \\rightarrow \\mathcal {P}(\\mathcal {U}^i)$ is a partial (stochastic) function from her private information $x_t^i$ to $u_t^i$ of the form $U_t^i \\sim \\gamma _t^i(\\cdot |x_t^i)$ .", "These actions are generated through some policy $\\psi ^i = (\\psi ^i_t)_{t \\in \\mathcal {T}}$ , $\\psi ^i_t : (\\mathcal {U}^i)^{t-1} \\rightarrow \\left\\lbrace \\mathcal {X}^i \\rightarrow \\mathcal {P}(\\mathcal {U}^i) \\right\\rbrace $ , that operates on the common information $u_{1:t-1}$ so that $\\gamma _t^i = \\psi _t^i[u_{1:t-1}]$ .", "Then any policy of the player $i$ of the form $U_t^i \\sim g_t^i(\\cdot |u_{1:t-1},x_t^{i})$ is equivalent to $U_t^i \\sim \\psi ^i_t[u_{1:t-1}] (\\cdot |x_t^i)$ [16].", "The common belief $\\pi _t^i$ is shown in Lemma 2 of [13] to be updated as $\\pi _{t+1}^i(x_{t+1}^i) =\\frac{ \\int _{x^i_t} \\pi _t^{{i}}(x_t^i) \\gamma _t^i(u_t^i|x_t^i)Q_t^i(x_{t+1}^i|x_t^i ,u_t) dx_t^i}{ \\int _{\\tilde{x}_t^i}\\pi _t^{{i}}(\\tilde{x}_t^i) \\gamma _t^i(u_t^i|\\tilde{x}_t^i)d\\tilde{x}_t^i},$ if the denominator is not 0, and as $\\pi _{t+1}^i(x_{t+1}^i) = \\int _{x^i_t} \\pi _t^{{i}}(x_t^i) Q_t^i(x_{t+1}^i|x_t^i ,u_t) dx_t^i,$ if the denominator is 0.", "The belief update can be summarized as, $\\pi ^{i}_{t+1} = {\\bar{F}}(\\pi ^{i}_t,\\gamma ^i_t,u_t),$ where ${\\bar{F}}$ is independent of players' strategy profile $g$ .", "The SPBE of the game can be found through a two-step backward-forward algorithm.", "In the backward recursive part, an equilibrium generating function $\\theta $ is defined based on which a strategy and belief profile ($\\beta ^*,\\mu ^*$ ) are defined through a forward recursion.", "In the following we summarize the algorithm and results of [13]." ], [ "Backward Recursion", "An equilibrium generating function $\\theta =(\\theta ^i_t)_{i=1,2,t\\in \\mathcal {T}}$ and a sequence of value functions $(V_t^i)_{i=1,2, t\\in \\lbrace 1,2, \\ldots T+1\\rbrace }$ are defined through backward recursion, where $\\theta ^i_t : \\mathcal {P}(\\mathcal {X}^1)\\times \\mathcal {P}(\\mathcal {X}^2) \\rightarrow \\left\\lbrace \\mathcal {X}^i \\rightarrow \\mathcal {P}(\\mathcal {U}^i) \\right\\rbrace $ , $V_t^i : \\mathcal {P}(\\mathcal {X}^1)\\times \\mathcal {P}(\\mathcal {X}^2) \\times \\mathcal {X}^i \\rightarrow \\mathbb {R}$ , as follows.", "(a) Initialize $\\forall \\underline{\\pi }_{T+1}\\in \\mathcal {P}(\\mathcal {X}^1)\\times \\mathcal {P}(\\mathcal {X}^2) , x_{T+1}^i\\in \\mathcal {X}^i$ , $V^i_{T+1}(\\underline{\\pi }_{T+1},x_{T+1}^i) := 0.", "$ (b) For $t = T,T-1, \\ldots 1$ , $ \\forall \\underline{\\pi }_t \\in \\mathcal {P}(\\mathcal {X}^1)\\times \\mathcal {P}(\\mathcal {X}^2) $ , let $\\theta _t[\\underline{\\pi }_t] $ be generated as follows.", "Set $\\tilde{\\gamma }_t = \\theta _t[\\underline{\\pi }_t]$ where $\\tilde{\\gamma }_t$ is the solution of the following fixed point equation, $\\forall i \\in \\mathcal {N},x_t^i\\in \\mathcal {X}^i$ , $\\tilde{\\gamma }^{i}_t(\\cdot |x_t^i) \\in &\\arg \\min _{\\gamma ^i_t(\\cdot |x_t^i)} \\mathbb {E}^{\\gamma ^i_t(\\cdot |x_t^i) \\tilde{\\gamma }^{-i}_t} \\left\\lbrace R^i(X_t,U_t) + V_{t+1}^i (F(\\underline{\\pi }_t, \\tilde{\\gamma }_t, U_t), X_{t+1}^i) \\big \\vert \\underline{\\pi }_t ,x_t^i \\right\\rbrace , $ where expectation in (REF ) is with respect to random variables $(x_t^{-i},u_t ,x_{t+1}^i)$ through the measure ${\\pi }_t^{-i}(x_t^{-i})\\gamma ^i_t(u^i_t|x_t^i) \\tilde{\\gamma }^{-i}_t(u^{-i}_t|x_t^{-i})Q^i(x_{t+1}^i|x_t^i,u_t)$ and $F(\\underline{\\pi }_t,\\gamma _t,u_t):=(\\bar{F}(\\pi _t^1,\\gamma _t^1,u_t),\\bar{F}(\\pi _t^2,\\gamma _t^2,u_t))$ .", "Also define $V^i_{t}(\\underline{\\pi }_t,x_t^i) &:= \\;\\; \\mathbb {E}^{\\tilde{\\gamma }^{i}_t(\\cdot |x_t^i) \\tilde{\\gamma }^{-i}_t}\\left\\lbrace {R}^i (X_t,U_t) +V_{t+1}^i (F(\\underline{\\pi }_t, \\tilde{\\gamma }_t, U_t), X_{t+1}^i)\\big \\vert \\underline{\\pi }_t, x_t^i \\right\\rbrace .", "$ From the equilibrium generating function $\\theta $ defined though this backward recursion, the equilibrium strategy and belief profile ($\\beta ^*,\\mu ^*$ ) are defined as follows." ], [ "Forward Recursion", "(a) Initialize at time $t=1$ , $\\mu ^{*}_1[\\phi ](x_1) &:= \\prod _{i=1}^N Q^i(x_1^i).", "$ (b) For $t =1,2 \\ldots T, \\forall i =1,2, u_{1:t}\\in \\mathcal {H}_{t+1}^c, x_{1:t}^i \\in (\\mathcal {X}^i)^t$ $\\beta _{t}^{*,i}(u_{t}^i|u_{1:t-1}x_{1:t}^i) &:= \\theta _{t}^i[\\mu _{t}^*[u_{1:t-1}]](u^i_{t}|x_{t}^i) $ and $\\mu ^{*,i}_{t+1}[u_{1:t}] &:= \\bar{F}(\\mu _t^{*,i}[u_{1:t-1}], \\theta _t^i[\\mu _t^*[u_{1:t-1}]], u_t) \\\\\\mu ^{*}_{t+1}[u_{1:t}](x_t^1,x_t^2)&:= \\prod _{i=1}^2\\mu ^{*,i}_{t+1}[u_{1:t}](x_t^i).", "$ The strategy and belief profile $(\\beta ^{*}, \\mu ^*)$ thus constructed form an SPBE of the game [13]." ], [ "SPBE of the dynamic LQG game", "In this section, we apply the general methodology for finding SPBE described in the previous section, on the specific dynamic LQG game model described in Section .", "We show that players' strategies that are linear in their private types in conjunction with Gaussian beliefs, form an SPBE of the game.", "We prove this result by constructing an equilibrium generating function $\\theta $ using backward recursion such that for all Gaussian belief vectors $\\underline{\\pi }_t$ , $\\tilde{\\gamma }_t=\\theta _t[\\underline{\\pi }_t]$ , $\\tilde{\\gamma }_t^i$ is of the form $\\tilde{\\gamma }_t^i(u_t^i | x_t^i) =\\delta (u_t^i -\\mathbf {\\tilde{L}}_t^i x^i - \\tilde{m}_t^i) $ and satisfies (REF ).", "Based on $\\theta $ , we construct an equilibrium belief and strategy profile.", "The following lemma shows that common beliefs remain Gaussian under linear deterministic $\\gamma _t$ of the form $\\gamma _t^i(u_t^i | x_t^i) =\\delta (u_t^i - \\mathbf {L}_t^i x_t^i - m_t^i) $ .", "Lemma 1 If $\\pi ^i_t$ is a Gaussian distribution with mean $\\hat{x}_{t}^i$ and covariance $\\mathbf {\\Sigma }_{t}^i$ , and $\\gamma _t^i(u_t^i | x_t^i)=\\delta (u_t^i - \\mathbf {L}_t^i x_t^i - m_t^i) $ then $\\pi _{t+1}^i$ , given by (REF ), is also Gaussian distribution with mean $\\hat{x}_{t+1}^i$ and covariance $\\mathbf {\\Sigma }_{t+1}^i$ , where $\\hat{x}_{t+1}^i &= \\mathbf {A}_t^i\\hat{x}_t^i + \\mathbf {B}_t^i u_t + \\mathbf {A}_t^i\\mathbf {G}_t^i (u_t^i - \\mathbf {L}_t^i\\hat{x}_t^i-m_t^i)\\\\\\mathbf {\\Sigma }_{t+1}^i &= \\mathbf {A}_t^i(\\mathbf {I}- \\mathbf {G}_t^i\\mathbf {L}_t^i)^{\\dagger }\\mathbf {\\Sigma }_t^i(\\mathbf {I}- \\mathbf {G}_t^i\\mathbf {L}_t^i)\\mathbf {A}_t^{i\\dagger }+ \\mathbf {Q}^i, $ where $\\mathbf {G}_t^i = \\mathbf {\\Sigma }_t^i\\mathbf {L}_t^{i\\dagger }(\\mathbf {L}_t^i\\mathbf {\\Sigma }_t^i\\mathbf {L}_t^{i\\dagger })^{-1}.", "$ See Appendix  Based on previous lemma, we define $\\phi _x^i,\\phi _s^i $ as update functions of mean and covariance matrix, respectively, as defined in (REF ), such that $\\hat{x}_{t+1}^i &= \\phi _x^{i}(\\hat{x}_t^i,\\mathbf {\\Sigma }_t^i,\\mathbf {L}_t^i,m_t^i, u_t)\\\\\\mathbf {\\Sigma }_{t+1}^i &= \\phi _s^{i}(\\mathbf {\\Sigma }_t^i,\\mathbf {L}_t^i) .$ We also say, $\\hat{x}_{t+1} &= \\phi _x(\\hat{x}_t,\\mathbf {\\Sigma }_t,\\mathbf {L}_t,m_t, u_t)\\\\\\mathbf {\\Sigma }_{t+1} &= \\phi _s(\\mathbf {\\Sigma }_t,\\mathbf {L}_t) .$ The previous lemma shows that with linear deterministic $\\gamma ^i_t$ , the next update of the mean of the common belief, $\\hat{x}_{t+1}^i$ is linear in $\\hat{x}_t^i$ and the control action $u_t^i$ .", "Furthermore, these updates are given by appropriate Kalman filter equations.", "It should be noted however that the covariance update in () depends on the strategy through $\\gamma _t^i$ and specifically through the matrix $\\mathbf {L}_t^i$ .", "This specifically shows how belief updates depend on strategies on the players which leads to signaling, unlike the case in classical stochastic control and the model considered in [12].", "Now we will construct an equilibrium generating function $\\theta $ using the backward recursion in (REF )–(REF ).", "The $\\theta $ function generates linear deterministic partial functions $\\gamma _t$ , which, from Lemma REF and the fact that initial beliefs (or priors) are Gaussian, generates only Gaussian belief vectors $(\\pi _t^1,\\pi _t^2)_{t\\in \\mathcal {T}}$ for the whole time horizon.", "These beliefs can be sufficiently described by their mean and covariance processes $(\\hat{x}_t^1,\\mathbf {\\Sigma }_t^1)_{t\\in \\mathcal {T}}$ and $(\\hat{x}_t^2,\\mathbf {\\Sigma }_t^2)_{t\\in \\mathcal {T}}$ which are updated using (REF ).", "For $t = T+1,T,\\ldots , 1 $ , we define the vectors $e_t^i := \\mathbf { x_t^i\\\\ \\hat{x}_{t}^1\\\\ \\hat{x}_{t}^2} \\quad z_t^i := \\mathbf {u_t^i\\\\ x_t^i\\\\ \\hat{x}_{t}^1\\\\ \\hat{x}_{t}^2} \\quad y_t^i := \\mathbf {u_t^1\\\\u_t^2\\\\x_t^1\\\\x_t^2\\\\x_{t+1}^i\\\\ \\hat{x}_{t+1}^1\\\\ \\hat{x}_{t+1}^2}.", "$ Theorem 1 The backward recursion (REF )–(REF ) admitsUnder certain conditions, stated in the proof.", "a solution of the form $\\theta _t[\\pi _t] = \\theta _t[\\hat{x}_t,\\mathbf {\\Sigma }_t] = \\tilde{\\gamma }_t$ where ${ \\tilde{\\gamma }^{i}_t}(u_t^{i}|x^{i}_t) = \\delta (u_t^i - \\mathbf {\\tilde{L}}_t^{i}x_t^{i} - \\tilde{m}_t^{i})$ and $\\mathbf {\\tilde{L}}_t^i, \\tilde{m}_t^i$ are appropriately defined matrices and vectors, respectively.", "Furthermore, the value function reduces to $V_t^i(\\underline{\\pi }_t,x_t^i) &= V_t^i(\\hat{x}_t,\\mathbf {\\Sigma }_t,x_t^i) \\\\&= quad(\\mathbf {V}_t^i(\\mathbf {\\Sigma }_t); e_t^i) + \\rho _t^i(\\mathbf {\\Sigma }_t).$ with $\\mathbf {V}_t^i(\\mathbf {\\Sigma }_t)$ and $\\rho _t^i(\\mathbf {\\Sigma }_t)$ as appropriately defined matrix and scalar quantities, respectively.", "We construct such a $\\theta $ function through the backward recursive construction and prove the properties of the corresponding value functions inductively.", "(a) For $i=1,2,\\forall \\ \\mathbf {\\Sigma }_{T+1},$ let $\\mathbf {V}^i_{T+1}(\\mathbf {\\Sigma }_{T+1}) := \\mathbf {0}, \\rho _{T+1}^i(\\mathbf {\\Sigma }_{T+1}) := 0$ .", "Then $\\forall \\ \\hat{x}_{T+1}^1, \\hat{x}_{T+1}^2$ , $ \\mathbf {\\Sigma }_{T+1}^1,\\mathbf {\\Sigma }_{T+1}^2, x_{T+1}^i$ and for $ \\underline{\\pi }_{t}= (\\pi _{t}^1,\\pi _{t}^2)$ , where $\\pi _{t}^i$ is $N(\\hat{x}_{t}^i,\\mathbf {\\Sigma }_{t}^i)$ , $V^i_{T+1}(\\underline{\\pi }_{T+1},x_{T+1}^i) &:=0\\\\&= V_{T+1}^i(\\hat{x}_{T+1},\\mathbf {\\Sigma }_{T+1},x_{T+1}^i)\\\\&=quad(\\mathbf {V}_{T+1}^i(\\mathbf {\\Sigma }_{T+1}),e_{T+1}^i) + \\rho _{T+1}^i(\\mathbf {\\Sigma }_{T+1}).$ (b) For all $t\\in \\lbrace T,T-1,\\ldots , 1 \\rbrace , i=1,2$ , Suppose $V^i_{t+1}(\\underline{\\pi }_{t+1},x_{t+1}^i) = quad(\\mathbf {V}_{t+1}^i(\\mathbf {\\Sigma }_{t+1}),e_{t+1}^i) + \\rho _{t+1}^i(\\mathbf {\\Sigma }_{t+1})$ (from induction hypothesis) where $\\mathbf {V}^i_{t+1}$ is a symmetric matrix defined recursively.", "Define $\\bar{\\mathbf {V}}_t^i$ as $\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {L}_t) &:= \\mathbf {\\mathbf {T}^i&\\mathbf {S}^{i} & \\mathbf {0}\\\\ \\mathbf {S}^{i\\dagger }&\\mathbf {P}^i & \\mathbf {0}\\\\ \\mathbf {0}& \\mathbf {0}& \\mathbf {V}^i_{t+1}(\\phi _s(\\mathbf {\\Sigma }_t,\\mathbf {L}_t)) }.", "$ Since $\\mathbf {T}^i,\\mathbf {P}^i$ are symmetric by assumption, $\\bar{\\mathbf {V}}_t^i$ is also symmetric.", "For ease of exposition, we will assume $i=1$ and for player 2, a similar argument holds.", "At time $t$ , the quantity that is minimized for player $i=1$ in (REF ) can be written as $&\\mathbb {E}^{\\gamma ^1_t(\\cdot |x_t^1)}\\left[ \\mathbb {E}^{ \\tilde{\\gamma }^{2}_t} \\left[ R^1(X_t,U_t) + V_{t+1}^1 (F(\\underline{\\pi }_t, \\tilde{\\gamma }_t, U_t), X_{t+1}^1) \\big \\vert \\underline{\\pi }_t ,x_t^1,u_t^1 \\right] \\big \\vert \\underline{\\pi }_t ,x_t^1\\right].$ The inner expectation can be written as follows, where ${ \\tilde{\\gamma }^{2}_t}(u_t^{2}|x^{2}_t) =\\delta (u_t^2- \\mathbf {\\tilde{L}}_t^{2}x_t^{2} - \\tilde{m}_t^{2})$ , $&\\mathbb {E}^{ \\tilde{\\gamma }^{2}_t} \\left[quad\\left(\\mathbf {\\mathbf {T}^1&\\mathbf {S}^{1}\\\\\\mathbf {S}^{1\\dagger }&\\mathbf {P}^1};z_t^i\\right) + quad\\left(\\mathbf {V}_{t+1}^1(\\phi _s(\\mathbf {\\Sigma }_{t},\\mathbf {\\tilde{L}}_t));e_{t+1}^i\\right) + \\rho _{t+1}^1(\\phi _s(\\mathbf {\\Sigma }_{t},\\mathbf {\\tilde{L}}_t)) \\big \\vert \\pi _t,x_t^1, u_t^i\\right]\\\\&= \\mathbb {E}^{ \\tilde{\\gamma }^{2}_t} \\left[ quad\\left(\\bar{\\mathbf {V}}_t^1(\\mathbf {\\Sigma }_{t},\\mathbf {\\tilde{L}}_t); y_t^1\\right) + \\rho _{t+1}^1(\\phi _s(\\mathbf {\\Sigma }_{t},\\mathbf {\\tilde{L}}_t)) \\big \\vert \\pi _t,x_t^1, u_t^1\\right] \\\\&= quad\\left(\\bar{\\mathbf {V}}_t^1(\\mathbf {\\Sigma }_{t},\\mathbf {\\tilde{L}}_t); \\mathbf {D}_t^1z_t^1+ \\mathbf {C}_t^1 \\mathbf {m_t^1\\\\ \\tilde{m}_t^2} \\right) + \\rho _{t}^1(\\mathbf {\\Sigma }_t), $ where $\\bar{\\mathbf {V}}_t^i$ is defined in (REF ) and function $\\phi _s$ is defined in (); $y_t^i, z_t^i $ are defined in (REF ); $\\rho _t^i$ is given by $\\rho _t^i(\\mathbf {\\Sigma }_t) &= tr\\left(\\mathbf {\\Sigma }_t^{-i} quad\\left(\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t); \\mathbf {J}_t^i \\right)\\right) +tr(\\mathbf {Q}^iV_{11,t+1}^i(\\phi _s(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t))) + \\rho _{t+1}^i(\\phi _s(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)), $ where $V_{11,t+1}^i$ is the matrix corresponding to $x_{t+1}^i$ in $V_{t+1}^i$ i.e.", "in the first row and first column of the matrix $V_{t+1}^i$ ; and matrices $\\mathbf {D}_t^i, \\mathbf {C}_t^i, \\mathbf {J}_t^i$ are as follows, $&\\mathbf {D}_t^1:= \\mathbf {\\mathbf {I}&\\mathbf {0}&\\mathbf {0}&\\mathbf {0}\\\\\\mathbf {0}& \\mathbf {0}&\\mathbf {0}&\\mathbf {\\tilde{L}}_t^2 \\\\\\mathbf {0}&\\mathbf {I}&\\mathbf {0}&\\mathbf {0}\\\\ \\mathbf {0}&\\mathbf {0}&\\mathbf {0}& \\mathbf {I}\\\\\\mathbf {B}^1_{1,t} &\\mathbf {A}_t^1 & \\mathbf {0}&\\mathbf {B}_{2,t}^1\\mathbf {\\tilde{L}}_t^2 \\\\\\mathbf {A}_t^1\\mathbf {G}_t^1+\\mathbf {B}_{1,t}^1 &\\mathbf {0}&\\mathbf {A}_t^1(\\mathbf {I}- \\mathbf {G}_t^1\\mathbf {L}_t^1) & \\mathbf {B}_{2,t}^1\\mathbf {\\tilde{L}}_t^2\\\\\\mathbf {B}_{1,t}^2 &\\mathbf {0}&\\mathbf {0}&\\mathbf {A}_t^2 + \\mathbf {B}_{2,t}^2\\mathbf {\\tilde{L}}_t^2 }$ $& \\mathbf {D}_t^2:= \\mathbf {\\mathbf {0}&\\mathbf {0}&\\mathbf {\\tilde{L}}_t^1 &\\mathbf {0}\\\\\\mathbf {I}& \\mathbf {0}&\\mathbf {0}& \\mathbf {0}\\\\ \\mathbf {0}&\\mathbf {0}&\\mathbf {I}&\\mathbf {0}\\\\ \\mathbf {0}&\\mathbf {I}&\\mathbf {0}& \\mathbf {0}\\\\\\mathbf {B}^2_{2,t} &\\mathbf {A}_t^2 &\\mathbf {B}_{1,t}^2\\mathbf {\\tilde{L}}_t^1&\\mathbf {0}\\\\\\mathbf {B}_{2,t}^1 &\\mathbf {0}&\\mathbf {A}_t^1+ \\mathbf {B}_{1,t}^1\\mathbf {\\tilde{L}}_t^1&\\mathbf {0}\\\\\\mathbf {A}_t^2\\mathbf {G}_t^2+\\mathbf {B}_{2,t}^2 &\\mathbf {0}& \\mathbf {B}_{1,t}^2\\mathbf {\\tilde{L}}_t^1 &\\mathbf {A}_t^2 (\\mathbf {I}- \\mathbf {G}_t^2\\mathbf {L}_t^2) } $ $\\mathbf {C}_t^1&:= \\mathbf {\\mathbf {0}& \\mathbf {0}\\\\\\mathbf {0}& \\mathbf {I}\\\\\\mathbf {0}& \\mathbf {0}\\\\\\mathbf {0}& \\mathbf {0}\\\\ \\mathbf {0}& \\mathbf {B}_{2,t}^1 \\\\-\\mathbf {A}_t^1\\mathbf {G}_t^1& \\mathbf {B}_{2,t}^1 \\\\\\mathbf {0}& \\mathbf {B}_{2,t}^2 } \\quad \\mathbf {C}_t^2 := \\mathbf {\\mathbf {I}& \\mathbf {0}\\\\ \\mathbf {0}& \\mathbf {0}\\\\\\mathbf {0}& \\mathbf {0}\\\\\\mathbf {0}& \\mathbf {0}\\\\\\mathbf {B}_{1,t}^2& \\mathbf {0}\\\\\\mathbf {B}_{1,t}^1 & \\mathbf {0}\\\\ \\mathbf {B}_{1,t}^2& -\\mathbf {A}_t^2\\mathbf {G}_t^2 }$ $& \\mathbf {J}_t^{1\\dagger }:= \\mathbf {\\mathbf {0}& \\mathbf {L}_t^2 & \\mathbf {0}& \\mathbf {I}& \\mathbf {B}_{2,t}^1\\mathbf {L}_t^2 & \\mathbf {B}_{2,t}^1\\mathbf {L}_t^2 & (\\mathbf {B}_{2,t}^2+ \\mathbf {A}_t^2\\mathbf {G}_t^2)\\mathbf {L}_t^2}^{\\dagger }\\nonumber \\\\& \\mathbf {J}_t^{2\\dagger }:=\\mathbf {\\mathbf {L}_t^1& \\mathbf {0}& \\mathbf {I}& \\mathbf {0}& \\mathbf {B}_{1,t}^2\\mathbf {L}_t^1 & (\\mathbf {B}_{1,t}^1 + \\mathbf {A}_t^1\\mathbf {G}_t^1)\\mathbf {L}_t^1 & \\mathbf {B}_{1,t}^2\\mathbf {L}_t^1}^{\\dagger }$ where $\\mathbf {B}_t^i =: \\mathbf {\\mathbf {B}_{1,t}^i & \\mathbf {B}_{2,t}^i}$ , $\\mathbf {B}_{1,t}^i, \\mathbf {B}_{2,t}^i$ are the parts of the matrix $\\mathbf {B}_{t}^i$ that corresponds to $u_t^1,u_t^2$ respectively.", "Let $\\mathbf {D}_t^1 =:\\mathbf {\\mathbf {D}_t^{u1} &\\mathbf {D}_t^{e1}}$ where $\\mathbf {D}_t^{u1}$ is the first column matrix of $\\mathbf {D}_t^1$ corresponding to $u_t^1$ and $\\mathbf {D}_t^{e1}$ is the matrix composed of remaining three column matrices of $\\mathbf {D}_t^1$ corresponding to $e_t^1$ .", "The expression in (REF ) is averaged with respect to $u_t^1$ using the measure $\\gamma _t^1(\\cdot |x_t^1)$ and minimized in (REF ) over $\\gamma _t^1(\\cdot |x_t^1)$ .", "This minimization can be performed component wise leading to a deterministic policy $\\tilde{\\gamma }_t^1(u_t^1|x_t^1) = \\delta (u_t^1 - \\mathbf {\\tilde{L}}_t^1x_t^1 - \\tilde{m}_t^1)=\\delta (u_t^1 - u_t^{1*})$ , assuming that the matrix $\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}\\mathbf {\\tilde{D}}_t^{u1}$ is positive definiteThis condition is true if the instantaneous cost matrix $\\mathbf {R}^i =\\mathbf {\\mathbf {T}^i &\\mathbf {S}^i \\\\ \\mathbf {S}^{i\\dagger } & \\mathbf {P}^i}$ is positive definite and can be proved inductively in the proof by showing that $\\mathbf {V}_t^{i}$ and $\\bar{\\mathbf {V}}_t^{i}$ are positive definite..", "In that case, the unique minimizer $u_t^{1*} = \\mathbf {\\tilde{L}}_t^1 x_t^1 + \\tilde{m}_t^1$ can be found by differentiating (REF ) w.r.t.", "$u_t^{1\\dagger }$ and equating it to $\\mathbf {0}$ , resulting in the equation, $\\mathbf {0}&=2\\mathbf {\\mathbf {I}&\\mathbf {0}&\\mathbf {0}&\\mathbf {0}}\\mathbf {\\tilde{D}}_t^{1\\dagger }\\bar{\\mathbf {V}}_t^{1}(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)\\left(\\mathbf {\\tilde{D}}_t^1z_t^1+\\mathbf {\\tilde{C}}_t^1\\tilde{m}_t \\right)\\\\\\mathbf {0}&=\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)\\left(\\mathbf {\\tilde{D}}_t^{u1} u_t^{1*} + \\mathbf {\\tilde{D}}_t^{e1}e_t^1 + \\mathbf {\\tilde{C}}_t^1\\tilde{m}_t\\right)\\\\\\mathbf {0}&=\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)\\left(\\mathbf {\\tilde{D}}_t^{u1} (\\mathbf {\\tilde{L}}_t^1 x_t^1 + \\tilde{m}_t^1) + [\\mathbf {\\tilde{D}}_t^{e1}]_1x_t^1 + [\\mathbf {\\tilde{D}}_t^{e1}]_{23}\\hat{x}_t + \\mathbf {\\tilde{C}}_t^1\\tilde{m}_t\\right),$ where $[\\mathbf {D}^{ei}]_{1}$ is the first matrix column of $\\mathbf {D}^{ei}$ , $[\\mathbf {D}^{ei}]_{23}$ is the matrix composed of the second and third column matrices of $\\mathbf {D}^{ei}$ .", "Matrices $\\mathbf {\\tilde{D}}_t^i, \\mathbf {\\tilde{C}}_t^i$ are obtained by substituting $\\mathbf {\\tilde{L}}_t^i,\\mathbf {\\tilde{G}}_t^i$ in place of $\\mathbf {L}_t^i,\\mathbf {G}_t^i$ in the definition of $\\mathbf {\\tilde{D}}_t^i, \\mathbf {\\tilde{C}}_t^i$ in (REF ), respectively, and $\\mathbf {\\tilde{G}}_t^i$ is the matrix obtained by substituting $\\mathbf {\\tilde{L}}_t^i$ in place of $\\mathbf {L}_t^i$ in (REF ).", "Thus (REF ) is equivalent to (REF ) and with a similar analysis for player 2, it implies that $\\mathbf {\\tilde{L}}_t^i$ is solution of the following algebraic fixed point equation, $&\\left(\\mathbf {\\tilde{D}}_t^{ui\\dagger }\\bar{\\mathbf {V}}_t^{i}(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)\\mathbf {\\tilde{D}}_t^{ui} \\right)\\mathbf {\\tilde{L}}_t^i = -\\mathbf {\\tilde{D}}_t^{ui\\dagger }\\bar{\\mathbf {V}}_t^{i}(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t) [\\mathbf {\\tilde{D}}_t^{ei}]_1.$ For player 1, it reduces to, $&\\hspace{-10.0pt}\\left[\\mathbf {T}_{11}^{1} + \\mathbf {\\mathbf {B}_{1,t}^1 \\\\ \\mathbf {A}_t^1\\mathbf {G}_t^1+ \\mathbf {B}_{1,t}^1\\\\ \\mathbf {B}_{1,t}^2 }^{\\dagger }\\mathbf {V}_{t+1}^{1}(\\phi _s(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)) \\mathbf { \\mathbf {B}_{1,t}^1 \\\\ \\mathbf {A}_t^1\\mathbf {G}_t^1+ \\mathbf {B}_{1,t}^1 \\\\\\mathbf {B}_{1,t}^2 } \\right] \\mathbf {\\tilde{L}}_t^1 \\nonumber \\\\&= - \\left[ \\mathbf {S}_{11}^{1\\dagger } + \\mathbf {\\mathbf {B}_{1,t}^1 \\\\ \\mathbf {A}_t^1\\mathbf {G}_t^1+ \\mathbf {B}_{1,t}^1\\\\ \\mathbf {B}_{1,t}^2 }^{\\dagger }\\mathbf {V}_{t+1}^{1}(\\phi _s(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)) \\mathbf {\\mathbf {A}_t^1\\\\0\\\\0}\\right],$ and a similar expression holds for player 2.", "In addition, $\\tilde{m}_t$ can be found from (REF ) as $&\\mathbf {\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}\\mathbf {\\tilde{D}}_t^{u1} & \\mathbf {0}\\\\ \\mathbf {0}&\\mathbf {\\tilde{D}}_t^{u2\\dagger }\\bar{\\mathbf {V}}_t^{2}\\mathbf {\\tilde{D}}_t^{u2} }\\tilde{m}_t= -\\mathbf {\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}[\\mathbf {\\tilde{D}}_t^{e1} ]_{23}\\\\ \\mathbf {\\tilde{D}}_t^{u2\\dagger }\\bar{\\mathbf {V}}_t^{2}[ \\mathbf {\\tilde{D}}_t^{e2}]_{23}} \\hat{x}_t-\\mathbf {\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1} \\mathbf {\\tilde{C}}_t^1\\\\\\mathbf {\\tilde{D}}_t^{u2\\dagger }\\bar{\\mathbf {V}}_t^{2} \\mathbf {\\tilde{C}}_t^2} \\tilde{m}_t \\\\\\tilde{m}_t &= -\\left[ \\mathbf {\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}\\mathbf {\\tilde{D}}_t^{u1} & \\mathbf {0}\\\\ \\mathbf {0}&\\mathbf {\\tilde{D}}_t^{u2\\dagger }\\bar{\\mathbf {V}}_t^{2}\\mathbf {\\tilde{D}}_t^{u2} } +\\mathbf {\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1} \\mathbf {\\tilde{C}}_t^1\\\\ \\mathbf {\\tilde{D}}_t^{u2\\dagger }\\bar{\\mathbf {V}}_t^{2} \\mathbf {\\tilde{C}}_t^2} \\right]^{-1}\\mathbf {\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}[\\mathbf {\\tilde{D}}_t^{e1}]_{23}\\\\ \\mathbf {\\tilde{D}}_t^{u2\\dagger }\\bar{\\mathbf {V}}_t^{2} [\\mathbf {\\tilde{D}}_t^{e2}]_{23}} \\hat{x}_t\\\\&=:\\mathbf {\\tilde{M}}_t \\hat{x}_t =: \\mathbf {\\mathbf {\\tilde{M}}_t^1\\\\\\mathbf {\\tilde{M}}_t^2} \\hat{x}_t, $ Finally, the resulting cost for player $i$ is, $V^i_{t}(\\underline{\\pi }_t,x_t^i) &= V^i_{t}(\\hat{x}_t,\\mathbf {\\Sigma }_t,x_t^i) \\\\&:=quad\\left(\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t); \\mathbf {\\mathbf {\\tilde{D}}_t^{ui} & \\mathbf {\\tilde{D}}_t^{ei}} \\mathbf { \\mathbf {\\tilde{L}}_t^i x_t^i + \\mathbf {\\tilde{M}}_t^i\\hat{x}_t\\\\ e_t^i }+ \\mathbf {\\tilde{C}}_t^i\\mathbf {\\tilde{M}}_t\\hat{x}_t \\right) + \\rho _t^i(\\mathbf {\\Sigma }_t)\\\\&=quad\\left(\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t); \\mathbf {\\tilde{D}}_t^{ui}(\\mathbf {\\tilde{L}}_t^i x_t^i + \\mathbf {\\tilde{M}}_t^i\\hat{x}_t )+ \\mathbf {\\tilde{D}}_t^{e1}e_t^i + \\mathbf {\\tilde{C}}_t^i \\mathbf {\\tilde{M}}_t \\hat{x}_t \\right) + \\rho _t^i(\\mathbf {\\Sigma }_t)\\\\&=quad\\left(\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t); \\left(\\mathbf {\\mathbf {\\tilde{D}}_t^{ui}\\mathbf {\\tilde{L}}_t^i & \\mathbf {\\tilde{D}}_t^{ui}\\mathbf {\\tilde{M}}_t^i+ \\mathbf {\\tilde{C}}_t^i\\mathbf {\\tilde{M}}_t} + \\mathbf {\\tilde{D}}_t^{ei}\\right)e_t^i \\right) + \\rho _t^i(\\mathbf {\\Sigma }_t)\\\\&=quad\\left(\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t);\\mathbf {\\tilde{F}}_t^ie_t^i \\right) + \\rho _t^i(\\mathbf {\\Sigma }_t)\\\\&=quad\\left(\\mathbf {\\tilde{F}}_t^{i\\dagger }\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)\\mathbf {\\tilde{F}}_t^i; e_t^i \\right) + \\rho _t^i(\\mathbf {\\Sigma }_t)\\\\&=quad\\left(\\mathbf {V}_t^i(\\mathbf {\\Sigma }_{t}); e_t^i \\right) + \\rho _t^i(\\mathbf {\\Sigma }_t),$ where, $\\mathbf {\\tilde{F}}_t^i &:=\\mathbf {\\mathbf {\\tilde{D}}_t^{ui}\\mathbf {\\tilde{L}}_t^i & \\mathbf {\\tilde{D}}_t^{ui}\\mathbf {\\tilde{M}}_t^i+ \\mathbf {\\tilde{C}}_t^i\\mathbf {\\tilde{M}}_t} + \\mathbf {\\tilde{D}}_t^{ei}\\\\\\mathbf {V}_t^i(\\mathbf {\\Sigma }_{t}) &:= \\mathbf {\\tilde{F}}_t^{i\\dagger }\\bar{\\mathbf {V}}_t^i(\\mathbf {\\Sigma }_t,\\mathbf {\\tilde{L}}_t)\\mathbf {\\tilde{F}}_t^i.", "$ Since $\\bar{\\mathbf {V}}_t^i$ is symmetric, so is $\\mathbf {V}_t^i$ .", "Thus the induction step is completed.", "Taking motivation from the previous theorem and with slight abuse of notation, we define $\\tilde{\\gamma }_t &= \\theta _t[\\underline{\\pi }_t] = \\theta _t[\\hat{x}_t,\\mathbf {\\Sigma }_t],$ and since $\\tilde{\\gamma }_t^i(u_t^i|x_t^i) = \\delta (u_t^i- \\mathbf {\\tilde{L}}_t^i x_t^i - \\tilde{m}_t^i)$ , we define a reduced mapping $(\\theta ^L,\\theta ^m)$ as $\\theta _t^{Li}[\\hat{x}_t,\\mathbf {\\Sigma }_t] = \\theta _t^{Li}[\\mathbf {\\Sigma }_t] := \\mathbf {\\tilde{L}}_t^i \\;\\; \\text{ and } \\;\\; \\theta _t^{mi}[\\hat{x}_t,\\mathbf {\\Sigma }_t] := \\tilde{m}_t^i,$ where $\\mathbf {\\tilde{L}}_t^i$ does not depend on $\\hat{x}_t$ and $\\tilde{m}_t^i$ is linear in $\\hat{x}_t$ and is of the form $\\tilde{m}_t^i = \\mathbf {\\tilde{M}}_t^i\\hat{x}_t$ .", "Now we construct the equilibrium strategy and belief profile $(\\beta ^*,\\mu ^*)$ through the forward recursion in (REF )–(REF ), using the equilibrium generating function $\\theta \\equiv (\\theta ^L,\\theta ^m)$ .", "(a) Let $\\mu ^{*,i}_1[\\phi ](x_1^i)= N(0,\\mathbf {\\Sigma }_1^i).$ (b) For $t =1,2 \\ldots T-1, \\forall u_{1:t} \\in \\mathcal {H}_{t+1}^c$ , if $\\mu _t^{*,i}[u_{1:t-1}] = N(\\hat{x}_{t}^i, \\mathbf {\\Sigma }_t^i)$ , let $\\mathbf {\\tilde{L}}_t^i = \\theta _t^{Li}[\\mathbf {\\Sigma }_t], \\tilde{m}_t^i = \\theta _t^{mi}[\\hat{x}_t,\\mathbf {\\Sigma }_t] = \\mathbf {\\tilde{M}}_t^i\\hat{x}$ .", "Then $ \\forall x_{1:t}^i \\in (\\mathcal {X}^i)^t$ $\\beta _{t}^{*,i}(u_{t}^i|u_{1:t-1}x_{1:t}^i) &:= \\delta (u_t^i - \\mathbf {\\tilde{L}}_t^ix_t^i - \\mathbf {\\tilde{M}}_t^i\\hat{x}_t)\\\\\\mu _{t+1}^{*,i}[u_{1:t}] &:= N(\\hat{x}_{t+1}^i ,\\mathbf {\\Sigma }_{t+1}^i) \\\\\\mu ^{*}_{t+1}[u_{1:t}](x_t^1,x_t^2)&:= \\prod _{i=1}^2\\mu ^{*,i}_{t+1}[u_{1:t}](x_t^i),$ where $\\hat{x}_{t+1}^i = \\phi _x^{i}(\\hat{x}_t^i, \\mathbf {\\tilde{L}}_t^i, \\tilde{m}_t^i, u_t)$ and $\\mathbf {\\Sigma }_{t+1}^i = \\phi _s^{i}(\\mathbf {\\Sigma }_t^i,\\mathbf {\\tilde{L}}_t^i)$ .", "Theorem 2 $(\\beta ^*,\\mu ^*)$ constructed above is a PBE of the dynamic LQG game.", "The strategy and belief profile $(\\beta ^*,\\mu ^*)$ is constructed using the forward recursion steps (REF )–(REF ) on equilibrium generating function $\\theta $ , which is defined through backward recursion steps (REF )–(REF ) implemented in the proof Theorem REF .", "Thus the result is directly implied by Theorem 1 in [13]." ], [ "Existence", "In the proof of Theorem REF , $\\mathbf {\\tilde{D}}_t^{u1\\dagger }\\bar{\\mathbf {V}}_t^{1}\\mathbf {\\tilde{D}}_t^{u1}$ is assumed to be positive definite.", "This can be achieved if $\\mathbf {R}^i$ is positive definite, through which it can be easily shown inductively in the proof of Theorem REF that the matrices $\\mathbf {V}_t^1,\\bar{\\mathbf {V}}_t^1$ are also positive definite.", "Constructing the equilibrium generating function $\\theta $ involves solving the algebraic fixed point equation in (REF ) for $\\mathbf {\\tilde{L}}_t$ for all $\\mathbf {\\Sigma }_t$ .", "In general, the existence is not guaranteed, as is the case for existence of $\\tilde{\\gamma }_t$ in (REF ) for general dynamic games with asymmetric information.", "At this point, we don't have a general proof for existence.", "However, in the following lemma, we provide sufficient conditions on the matrices $\\mathbf {A}_t^i, \\mathbf {B}_t^i, \\mathbf {T}^i, \\mathbf {S}^i, \\mathbf {P}^i,\\mathbf {V}_{t+1}^i$ and for the case $m^i=1$ , for a solution to exist.", "Lemma 2 For $m^1=m^2=1$ , there exists a solution to (REF ) if and only if for $i=1,2$ , $\\exists \\ l^i\\ \\in \\mathbb {R}^{n^i}$ such that $l^{i\\dagger }\\mathbf {\\Delta }^i(l^1,l^2) l^i\\ge 0$ , or sufficiently $\\mathbf {\\Delta }^i(l^1,l^2) + \\mathbf {\\Delta }^{i,\\dagger }(l^1,l^2)$ is positive definite, where $\\mathbf {\\Delta }^i, i=1,2$ are defined in Appendix .", "See Appendix ." ], [ "Steady state", "In Section , we presented the backward/forward methodology to find SPBE for finite time-horizon dynamic games, and specialized that methodology in this chapter, in Section , to find SPBE for dynamic LQG games with asymmetric information, where equilibrium strategies are linear in players' types.", "It requires further investigation to find the conditions for which the backward-forward methodology could be extended to infinite time-horizon dynamic games, with either expected discounted or time-average cost criteria.", "Such a methodology for infinite time-horizon could be useful to characterize steady state behavior of the games.", "Specifically, for time homogenous dynamic LQG games with asymmetric information (where matrices $\\mathbf {A}^i, \\mathbf {B}^i$ are time independent), under the required technical conditions for which such a methodology is applicable, the steady state behavior can be characterized by the fixed point equation in matrices $(\\mathbf {L}^i, \\mathbf {\\Sigma }^i ,\\mathbf {V}^i)_{i=1,2}$ through (), (REF ) and (REF ), where the time index is dropped in these equations, i.e.", "for $i=1,2,$ 1.", "$\\mathbf {\\Sigma }= \\phi _s(\\mathbf {\\Sigma },\\mathbf {L}) $ 2.", "$\\left(\\mathbf {D}^{ui\\dagger }\\bar{\\mathbf {V}}^{i}\\mathbf {D}^{ui} \\right)\\mathbf {L}^i = -\\mathbf {D}^{ui\\dagger }\\bar{\\mathbf {V}}^{i} [\\mathbf {D}^{ei}]_1 $ 3.", "$\\mathbf {V}^i = \\mathbf {F}^{i\\dagger }\\bar{\\mathbf {V}}^i\\mathbf {F}^i,$ where $\\bar{\\mathbf {V}}^i = \\mathbf {\\mathbf {T}^i&\\mathbf {S}^{i} & \\mathbf {0}\\\\ \\mathbf {S}^{i\\dagger }&\\mathbf {P}^i & \\mathbf {0}\\\\ \\mathbf {0}& \\mathbf {0}& \\mathbf {V}^i }$ .", "Observe that in the above equations the matrices $\\mathbf {V}^i$ and $\\bar{\\mathbf {V}}^i$ do not appear as functions of $\\mathbf {\\Sigma }$ , as in the finite horizon case described in (REF ), (REF ), in the proof of Theorem REF .", "The reason for that is as follows.", "The steady state behavior for a general dynamic game with asymmetric information and independent types, if it exists, would involve fixed point equation in value functions $(V^i(\\cdot ))_i$ .", "However, for the LQG case, it reduces to a fixed point equation in $(V^i(\\mathbf {\\Sigma }))_i$ , i.e.", "value functions evaluated at a specific value of $\\mathbf {\\Sigma }$ .", "This is so because the functions $V^i$ are evaluated at $\\mathbf {\\Sigma }$ and $\\phi (\\mathbf {\\Sigma },\\mathbf {L})$ , which at steady state are exactly the same (see (REF )).", "As a result, the fixed point equation reduces to the three algebraic equations as shown above with variables the matrices $\\mathbf {\\Sigma }$ , $\\mathbf {L}$ , $\\bar{\\mathbf {V}}$ and $\\mathbf {V}$ , which represents an enormous reduction in complexity." ], [ "Numerical examples", "In this section, we present numerically found solutions for steady state, assuming that our methodology extends to the infinite horizon problem for the model considered.", "We assume $\\mathbf {B}^i=0$ which implies that the state process $(X_t^i)_{t\\in \\mathcal {T}}$ is uncontrolled.", "1.", "For $i=1,2$ , $ m^i=1, n^i=2, \\mathbf {A}^i = 0.9\\mathbf {I}, \\mathbf {B}^i=\\mathbf {0}, \\mathbf {Q}^i = \\mathbf {I}$ , $\\mathbf {T}^1 = \\mathbf {\\mathbf {I}& \\frac{1}{4}\\mathbf {I}\\\\\\frac{1}{4}\\mathbf {I}&\\mathbf {0}}, \\quad \\mathbf {T}^2 = \\mathbf {\\mathbf {0}& \\frac{1}{4}\\mathbf {I}\\\\\\frac{1}{4}\\mathbf {I}&\\mathbf {I}}, \\quad \\mathbf {P}^1 = \\mathbf {\\mathbf {I}&\\mathbf {0}\\\\ \\mathbf {0}& \\mathbf {0}}, \\nonumber \\\\\\mathbf {P}^2 = \\mathbf {\\mathbf {0}&\\mathbf {0}\\\\ \\mathbf {0}& \\mathbf {I}}, \\quad \\mathbf {S}^1 = \\mathbf { \\mathbf {1}&\\mathbf {0}\\\\ \\mathbf {0}& \\mathbf {0}}, \\quad \\mathbf {S}^2 = \\mathbf { \\mathbf {0}&\\mathbf {0}\\\\\\mathbf {0}& \\mathbf {1}},$ there exists a symmetric solution as, for $i=1,2,$ $\\mathbf {L}^i = -\\mathbf {1.062 & 1.062} , \\mathbf {\\Sigma }^i = \\mathbf {3.132 & -2.132 \\\\ -2.132 & 3.132}.", "$ 2.", "For $i=1,2$ , $ m^i=2, n^i=2, \\mathbf {A}^1 = \\mathbf {0.9 & 0 \\\\ 0 & 0.8}, \\mathbf {A}^2 =0.9\\mathbf {I}, $ and $\\mathbf {B}^i,\\mathbf {T}^i,\\mathbf {P}^i,\\mathbf {S}^i$ used as before with appropriate dimensions, there exists a solution, $\\mathbf {L}^1 =-\\mathbf { 1.680 &1.600 \\\\ 0.191 &0.286 },\\quad \\mathbf {L}^2 = -\\mathbf { 1.363 &1.363 \\\\ 1.363 &1.363 } \\nonumber \\\\\\hspace{-28.45274pt}\\mathbf {\\Sigma }^1 =\\mathbf {I}, \\; \\quad \\mathbf {\\Sigma }^2 = \\mathbf { 3.132 &-2.132 \\\\ -2.132 &3.132 }.$ It is interesting to note that for player 1, where $\\mathbf {A}^1$ does not weigh the two components equally, the corresponding $\\mathbf {L}^1$ is full rank, and thus reveals her complete private information.", "Whereas for player 2, where $\\mathbf {A}^2$ has equal weight components, the corresponding $\\mathbf {L}^2$ is rank deficient, which implies, at equilibrium player 2 does not completely reveal her private information.", "Also it is easy to check from () that with full rank $\\mathbf {L}^i$ matrices, steady state $\\mathbf {\\Sigma }^i = \\mathbf {Q}^i$ ." ], [ "Conclusion", "In this paper, we study a two-player dynamic LQG game with asymmetric information and perfect recall where players' private types evolve as independent controlled Markov processes.", "We show that under certain conditions, there exist strategies that are linear in players' private types which, together with Gaussian beliefs, form a PBE of the game.", "We show this by specializing the general methodology developed in [13] to our model.", "Specifically, we prove that (a) the common beliefs remain Gaussian under the strategies that are linear in players' types where we find update equations for the corresponding mean and covariance processes; (b) using the backward recursive approach of [13], we compute an equilibrium generating function $\\theta $ by solving a fixed point equation in linear deterministic partial strategies $\\gamma _t$ for all possible common beliefs and all time epochs.", "Solving this fixed point equation reduces to solving a matrix algebraic equation for each realization of the state estimate covariance matrices.", "Also, the cost-to-go value functions are shown to be quadratic in private type and state estimates.", "This result is one of the very few results available on finding signaling perfect Bayesian equilibria of a truly dynamic game with asymmetric information." ], [ "This lemma could be interpreted as Theorem 2.30 in [1] with appropriate matrix substitution where specifically, their observation matrix $C_k$ should be substituted by our $L_k$ .", "We provide an alternate proof here for convenience.", "$\\pi _{t+1}^i$ is updated from $\\pi _t^i$ through (REF ).", "Since $\\pi _t^i$ is Gaussian, $\\gamma _t^i(u_t^i|x_t^i) = \\delta (u_t^i - L_t^ix_t^i - m_t^i)$ is a linear deterministic constraint and kernel $Q^i$ is Gaussian, thus $\\pi _{t+1}^i $ is also Gaussian.", "We find its mean and covariance as follows.", "We know that $x_{t+1}^i=\\mathbf {A}_t^i x_t^i + \\mathbf {B}_t^i u_t + w_t^i$ .", "Then, $&\\mathbb {E}[X_{t+1}^i| \\pi ^i_t,\\gamma _t^i, u_t] \\nonumber \\\\&= \\mathbb {E}[\\mathbf {A}_t^i X_t^i + \\mathbf {B}_t^i U_t + W_t^i| \\pi ^i_t,\\gamma _t^i, u_t] \\\\&= \\mathbf {A}_t^i\\mathbb {E}[X_t^i | \\pi ^i_t,\\gamma _t^i, u_t] + \\mathbf {B}_t^i u_t\\\\&= \\mathbf {A}_t^i \\mathbb {E}[X_t^i | \\mathbf {L}_t^iX_t^i = u_t^i-m_t^i ] + \\mathbf {B}_t^i u_t $ where (REF ) follows because $W_t^i$ has mean zero.", "Suppose there exists a matrix $\\mathbf {G}_t^i$ such that $X_t^i-\\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i$ and $\\mathbf {L}_t^iX_t^i$ are independent.", "Then $&\\mathbb {E}[X_t^i\\big |\\mathbf {L}_t^iX_t^i = u_t^i-m_t^i] \\\\&= \\mathbb {E}[X_t^i - \\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i + \\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i \\big | \\mathbf {L}_t^iX_t^i = u_t^i-m_t^i] \\\\&= \\mathbb {E}[X_t^i-\\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i] + \\mathbf {G}_t^i(u_t^i-m_t^i)\\\\&= \\hat{x}_t^i + \\mathbf {G}_t^i(u_t^i- \\mathbf {L}_t^i\\hat{x}_t^i - m_t^i),$ where $\\mathbf {G}_t^i$ satisfies $\\mathbb {E}[(X_t^i-\\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i)&(\\mathbf {L}_t^iX_t^i)^{\\dagger }]\\nonumber \\\\&\\hspace{-28.45274pt}= \\mathbb {E}[(X_t^i-\\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i)]\\mathbb {E}[(\\mathbf {L}_t^iX_t^i)^{\\dagger }]\\\\(\\mathbf {I}-\\mathbf {G}_t^i\\mathbf {L}_t^i)\\mathbb {E}[X_t^iX_t^{i\\dagger }]\\mathbf {L}_t^{i\\dagger } &= (\\mathbf {I}-\\mathbf {G}_t^i\\mathbf {L}_t^i)\\mathbb {E}[X_t^i]\\mathbb {E}[X_t^{i\\dagger }]\\mathbf {L}_t^{i\\dagger }\\\\(\\mathbf {I}-\\mathbf {G}_t^i\\mathbf {L}_t^i)(\\mathbf {\\Sigma }_t^i + \\hat{x}_t^i\\hat{x}_t^{i\\dagger })\\mathbf {L}_t^{i\\dagger } &= (\\mathbf {I}-\\mathbf {G}_t^i\\mathbf {L}_t^i) \\hat{x}_t^i\\hat{x}_t^{i\\dagger } \\mathbf {L}_t^{i\\dagger }\\\\\\mathbf {G}_t^i & = \\mathbf {\\Sigma }_t^i \\mathbf {L}_t^{i\\dagger }(\\mathbf {L}_t^i\\mathbf {\\Sigma }_t^i \\mathbf {L}_t^{i\\dagger })^{-1}.$ $&\\mathbf {\\Sigma }_{t+1}^i = sm\\left(\\mathbf {A}_t^iX_t^i-\\mathbb {E}[\\mathbf {A}_t^iX_t^i|\\mathbf {L}_t^iX_t^i = u_t^i-m_t^i] |\\mathbf {L}_t^iX_t^i = u_t^i-m_t^i\\right) + \\mathbf {Q}^i$ Now $&sm\\left(X_t^i-\\mathbb {E}[X_t^i|\\mathbf {L}_t^iX_t^i = u_t^i-m_t^i] |\\mathbf {L}_t^iX_t^i = u_t^i-m_t^i\\right) \\\\&= sm\\left((X_t^i-\\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i)-(\\mathbb {E}[X_t^i-\\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i|\\mathbf {L}_t^iX_t^i = u_t^i-m_t^i] ) | \\mathbf {L}_t^iX_t^i = u_t^i-m_t^i\\right)\\\\&= sm\\left((X_t^i-\\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i) -(\\mathbb {E}[X_t^i - \\mathbf {G}_t^i\\mathbf {L}_t^iX_t^i] )\\right) \\\\&= sm\\left((\\mathbf {I}-\\mathbf {G}_t^i\\mathbf {L}_t^i)(X_t^i-\\mathbb {E}[X_t^i])\\right) \\\\&= (\\mathbf {I}-\\mathbf {G}_t^i\\mathbf {L}_t^i)\\mathbf {\\Sigma }_t^i(\\mathbf {I}-\\mathbf {G}_t^i\\mathbf {L}_t^i)^{\\dagger }$" ], [ "We prove the lemma for player 1 and the result follows for player 2 by similar arguments.", "For the scope of this appendix, we define $\\mathbf {\\bar{B}}_t^1 = \\mathbf {\\mathbf {B}_{1,t}^1 \\\\ \\mathbf {B}_{1,t}^1\\\\ \\mathbf {B}_{1,t}^2 }$ and for any matrix $\\mathbf {V}$ , we define $\\mathbf {V}_{*i}, \\mathbf {V}_{i*}$ as the $i^{th}$ column and the $i^{th}$ row of $\\mathbf {V}$ , respectively.", "Then the fixed point equation (REF ) can be written as, $&0 =\\left[\\mathbf {T}_{11}^{1} + (\\mathbf {A}_t^1\\mathbf {G}_t^1)^{\\dagger }\\mathbf {V}_{22,t+1}^{1} ( \\mathbf {A}_t^1\\mathbf {G}_t^1) + \\right.", "\\nonumber \\\\&\\left.\\hspace{-5.0pt} \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{*2,t+1}^1\\mathbf {A}_t^1\\mathbf {G}_t^1 +(\\mathbf {A}_t^1\\mathbf {G}_t^1)^{\\dagger }\\mathbf {V}_{2*,t+1}^1\\mathbf {\\bar{B}}_t^1 + \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{t+1}^1\\mathbf {\\bar{B}}_t^1\\right] \\mathbf {L}_t^1 \\nonumber \\\\&+ \\left[ \\mathbf {S}_{11}^{1\\dagger } + (\\mathbf {A}_t^1\\mathbf {G}_t^1)^{\\dagger }\\mathbf {V}_{21,t+1}^{1} \\mathbf {A}_t^1 + \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{*1,t+1}^1\\mathbf {A}_t^1\\right].$ It should be noted that $\\mathbf {V}^i_{t+1}$ is a function of $\\mathbf {\\Sigma }_{t+1}$ , which is updated through $\\mathbf {\\Sigma }_t$ and $\\mathbf {L}_t$ as $\\mathbf {\\Sigma }_{t+1} = \\phi _s(\\mathbf {\\Sigma }_{t},\\mathbf {L}_t)$ (we drop this dependence here for ease of exposition).", "Substituting $\\mathbf {G}_t^1 = \\mathbf {\\Sigma }_t^1\\mathbf {L}_t^{1\\dagger }(\\mathbf {L}_t^1\\mathbf {\\Sigma }_t^1\\mathbf {L}_t^{1\\dagger })^{-1}$ and multiplying (REF ) by $(\\mathbf {L}_t^1\\mathbf {\\Sigma }_t^1\\mathbf {L}_t^{1\\dagger })$ from left and $(\\mathbf {\\Sigma }_t^1\\mathbf {L}_t^{1\\dagger })$ from right, we get $0&=\\mathbf {L}_t^1\\mathbf {\\Sigma }_t^1\\left[ \\mathbf {L}_t^{1\\dagger }(\\mathbf {T}_{11}^{1} + \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{t+1}^1\\mathbf {\\bar{B}}_t^1 ) \\mathbf {L}_t^{1} + \\mathbf {A}_t^{1\\dagger }\\mathbf {V}_{22,t+1}^{1} \\mathbf {A}_t^1\\right.", "\\nonumber \\\\& + \\mathbf {L}_t^{1\\dagger }(\\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{*2,t+1}^1\\mathbf {A}_t^1+ \\mathbf {S}_{11}^{1\\dagger } + \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{*1,t+1}^1\\mathbf {A}_t^1 ) \\nonumber \\\\&\\left.+ (\\mathbf {A}_t^{1\\dagger }\\mathbf {V}_{2*,t+1}^1\\mathbf {\\bar{B}}_t^1+ \\mathbf {A}_t^{1\\dagger }\\mathbf {V}_{21,t+1}^{1} \\mathbf {A}_t^1)\\mathbf {L}_t^{1} \\right] \\mathbf {\\Sigma }_t^1\\mathbf {L}_t^{1\\dagger }$ Let $\\mathbf {\\bar{L}}_t^i = \\mathbf {L}_t^i (\\mathbf {\\Sigma }_t^i)^{1/2}, \\mathbf {\\bar{A}}_t^i = \\mathbf {A}_t^i(\\mathbf {\\Sigma }_t^i)^{1/2}$ , $\\mathbf {\\Lambda }^1_a(\\mathbf {L}_t) & :=\\mathbf {T}_{11}^{1} + \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{t+1}^1\\mathbf {\\bar{B}}_t^1 \\\\\\mathbf {\\Lambda }^1_b(\\mathbf {L}_t) &:=\\mathbf {\\bar{A}}_t^{1\\dagger }\\mathbf {V}_{22,t+1}^{1} \\mathbf {\\bar{A}}_t^1 \\\\\\mathbf {\\Lambda }^1_c(\\mathbf {L}_t) & := \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{*2,t+1}^1\\mathbf {\\bar{A}}_t^1+ \\mathbf {S}_{11}^{1\\dagger }(\\mathbf {\\Sigma }_t^1)^{1/2} + \\mathbf {\\bar{B}}_t^{1\\dagger } \\mathbf {V}_{*1,t+1}^1\\mathbf {\\bar{A}}_t^1 \\\\\\mathbf {\\Lambda }^1_d(\\mathbf {L}_t) &:= \\mathbf {\\bar{A}}_t^{1\\dagger }\\mathbf {V}_{2*,t+1}^1\\mathbf {\\bar{B}}_t^1+ \\mathbf {\\bar{A}}_t^{1\\dagger }\\mathbf {V}_{21,t+1}^{1} \\mathbf {\\bar{A}}_t^1.$ Then, $0&= \\mathbf {\\bar{L}}_t^1\\mathbf {\\bar{L}}_t^{1\\dagger }\\mathbf {\\Lambda }^1_a(\\mathbf {L}_t) \\mathbf {\\bar{L}}_t^{1}\\mathbf {\\bar{L}}_t^{1\\dagger } + \\mathbf {\\bar{L}}_t^1\\mathbf {\\Lambda }^1_b(\\mathbf {L}_t) \\mathbf {\\bar{L}}_t^{1\\dagger }+ \\mathbf {\\bar{L}}_t^1\\mathbf {\\bar{L}}_t^{1\\dagger }\\mathbf {\\Lambda }^1_c(\\mathbf {L}_t) \\mathbf {\\bar{L}}_t^{1\\dagger } + \\mathbf {\\bar{L}}_t^1\\mathbf {\\Lambda }^1_d(\\mathbf {L}_t) \\mathbf {\\bar{L}}_t^{1} \\mathbf {\\bar{L}}_t^{1\\dagger } $ Since m=1, $\\Lambda ^1_a$ is a scalar.", "Let $\\mathbf {\\bar{L}}_t^i=\\lambda ^i l^{i\\dagger }$ , where $\\lambda ^i= || \\mathbf {\\bar{L}}_t^i ||_2$ and $l^i$ is a normalized vector and $t^1 =T_{11}$ .", "Moreover, since the update of $\\mathbf {\\Sigma }_t$ in () is scaling invariant, $\\mathbf {V}^1_{t+1}$ only depends on the directions $l= (l^1,l^2)$ .", "Then, (REF ) reduces to the following quadratic equation in $\\lambda ^1$ $(\\lambda ^1)^2 \\Lambda ^1_a(l) + \\lambda ^1 (\\mathbf {\\Lambda }^1_c(l) l^1 + l^{1\\dagger }\\mathbf {\\Lambda }^1_d(l)) + l^{1\\dagger }\\mathbf {\\Lambda }^1_b(l) l^1 = 0.$ There exists a real-valued solutionNote that a negative sign of $\\lambda ^1$ can be absorbed in $l^1$ .", "of this quadratic equation in $\\lambda ^1$ if and only if $(\\mathbf {\\Lambda }_c(l) l^1 + l^{1\\dagger }\\mathbf {\\Lambda }^1_d(l))^2 \\ge 4 \\Lambda ^1_a(l) l^{1\\dagger }\\mathbf {\\Lambda }^1_b(l) l^1 \\\\l^{1\\dagger }(\\mathbf {\\Lambda }^{1\\dagger }_c (l)\\mathbf {\\Lambda }^1_c(l) + \\mathbf {\\Lambda }^1_d(l) \\mathbf {\\Lambda }^{1\\dagger }_d(l) + 2\\mathbf {\\Lambda }^1_d(l)\\mathbf {\\Lambda }^1_c(l) - 4 \\Lambda ^1_a(l) \\mathbf {\\Lambda }^1_b(l)) l^1 \\ge 0.", "$ $\\text{Let } \\mathbf {\\Delta }^1(l):= (\\mathbf {\\Lambda }^{1\\dagger }_c(l) \\mathbf {\\Lambda }^1_c(l) + \\mathbf {\\Lambda }^1_d(l) \\mathbf {\\Lambda }^{1\\dagger }_d(l) + 2\\mathbf {\\Lambda }^1_d(l)\\mathbf {\\Lambda }^1_c(l) - 4 \\Lambda ^1_a(l) \\mathbf {\\Lambda }^1_b(l)).$ There exists a solution to the fixed point equation (REF ) if and only if $\\exists l^1,l^2\\in \\mathbb {R}^n$ such that $l^{1\\dagger }\\mathbf {\\Delta }^1(l) l^1\\ge 0$ , or sufficiently $\\mathbf {\\Delta }^1(l) + \\mathbf {\\Delta }^{1\\dagger }(l)$ is positive definite." ] ]
1606.04960
[ [ "Systems of sets of lengths: Transfer Krull monoids versus weakly Krull\n monoids" ], [ "Abstract Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid.", "We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far).", "Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths.", "In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid." ], [ "Introduction", "By an atomic monoid we mean a cancellative semigroup with unit element such that every nonunit can be written as a finite product of irreducible elements.", "Let $H$ be an atomic monoid.", "If $a \\in H$ is a nonunit and $a=u_1 \\cdot \\ldots \\cdot u_k$ is a factorization of $a$ into $k$ irreducible elements, then $k$ is called a factorization length and the set $\\mathsf {L} (a) \\subset \\mathbb {N}$ of all possible factorization lengths is called the set of lengths of $a$ .", "Then $\\mathcal {L} (H) = \\lbrace \\mathsf {L} (a) \\mid a \\in H \\rbrace $ is the system of sets of lengths of $H$ .", "Under a variety of noetherian conditions on $H$ (e.g., $H$ is the monoid of nonzero elements of a commutative noetherian domain) all sets of lengths are finite.", "Furthermore, if there is some element $a \\in H$ with $|\\mathsf {L} (a)| > 1$ , then $|\\mathsf {L}(a^N)|>N$ for all $N \\in \\mathbb {N}$ .", "Sets of lengths (together with invariants controlling their structure, such as elasticities and sets of distances) are a well-studied means of describing the arithmetic structure of monoids ([20], [11]).", "Let $H$ be a transfer Krull monoid.", "Then, by definition, there is a weak transfer homomorphism $\\theta \\colon H \\rightarrow \\mathcal {B} (G_0)$ , where $\\mathcal {B} (G_0)$ denotes the monoid of zero-sum sequences over a subset $G_0$ of an abelian group, and hence $\\mathcal {L} (H) = \\mathcal {L} \\big ( \\mathcal {B} (G_0) \\big )$ .", "A special emphasis has always been on the case where $G_0$ is a finite abelian group.", "Thus let $G$ be a finite abelian group and we use the abbreviation $\\mathcal {L} (G) = \\mathcal {L} \\big ( \\mathcal {B} (G) \\big )$ .", "It is well-known that sets of lengths in $\\mathcal {L} (G)$ are highly structured (Proposition REF ), and the standing conjecture is that the system $\\mathcal {L} (G)$ is characteristic for the group $G$ .", "More precisely, if $G^{\\prime }$ is a finite abelian group such that $\\mathcal {L} (G)= \\mathcal {L} (G^{\\prime })$ , then $G$ and $G^{\\prime }$ are isomorphic (apart from two well-known trivial pairings; see Conjecture REF ).", "This conjecture holds true, among others, for groups $G$ having rank at most two, and its proof uses deep results from additive combinatorics which are not available for general groups.", "Thus there is a need for studying $\\mathcal {L} (G)$ with a new approach.", "In Section , we unveil a couple of properties of the system $\\mathcal {L} (G)$ which are first steps on a new way towards Conjecture REF .", "In spite of all abstract work on systems $\\mathcal {L} (G)$ , they have been written down explicitly only for groups $G$ having Davenport constant $\\mathsf {D} (G)\\le 4$ , and this is not difficult to do (recall that a group $G$ has Davenport constant $\\mathsf {D}(G)\\le 4$ if and only if either $|G|\\le 4$ or $G$ is an elementary 2-group of rank three).", "In Section we determine the systems $\\mathcal {L} (G)$ for all groups $G$ having Davenport constant $\\mathsf {D} (G)=5$ .", "Commutative Krull monoids are the classic examples of transfer Krull monoids.", "In recent years a wide range of monoids and domains has been found which are transfer Krull but which are not commutative Krull monoids.", "Thus the question arose which monoids $H$ have systems $\\mathcal {L} (H)$ which are different from systems of sets of lengths of transfer Krull monoids.", "Commutative $v$ -noetherian weakly Krull monoids and domains are the best investigated class of monoids beyond commutative Krull monoids (numerical monoids as well as one-dimensional noetherian domains are $v$ -noetherian weakly Krull).", "Clearly, weakly Krull monoids can be half-factorial and half-factorial monoids are transfer Krull monoids.", "Similarly, it can happen both for weakly Krull monoids as well as for transfer Krull monoids that all sets of lengths are arithmetical progressions with difference 1.", "Apart from such extremal cases, we show in Section that systems of sets of lengths of a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid." ], [ "Background on sets of lengths", "We denote by $\\mathbb {N}$ the set of positive integers, and for real numbers $a, b \\in \\mathbb {R}$ , we denote by $[a,b] = \\lbrace x \\in \\mathbb {Z}\\mid a \\le x \\le b\\rbrace $ the discrete interval between $a$ and $b$ , and by an interval we always mean a finite discrete interval of integers.", "Let $A, B \\subset \\mathbb {Z}$ be subsets of the integers.", "Then $A+B = \\lbrace a+b \\mid a \\in A, b \\in B\\rbrace $ is the sumset of $A$ and $B$ .", "We set $-A= \\lbrace -a \\mid a \\in A\\rbrace $ and for an integer $m \\in \\mathbb {Z}$ , $m+A = \\lbrace m\\rbrace +A$ is the shift of $A$ by $m$ .", "For $m \\in \\mathbb {N}$ , we denote by $mA = A+ \\ldots + A$ the $m$ -fold subset of $A$ and by $m \\cdot A = \\lbrace ma \\mid a \\in A\\rbrace $ the dilation of $A$ by $m$ .", "If $A \\subset \\mathbb {N}$ , we denote by $\\rho (A) = \\sup A/\\min A \\in \\mathbb {Q}_{\\ge 1} \\cup \\lbrace \\infty \\rbrace $ the elasticity of $A$ and we set $\\rho ( \\lbrace 0\\rbrace )=1$ .", "A positive integer $d \\in \\mathbb {N}$ is called a distance of $A$ if there are $a, b \\in A$ with $b-a=d$ and the interval $[a,b]$ contains no further elements of $A$ .", "We denote by $\\Delta (A)$ the set of distances of $A$ .", "Clearly, $\\Delta (A)=\\emptyset $ if and only if $|A|\\le 1$ , and $A$ is an arithmetical progression if and only if $|\\Delta (A)|\\le 1$ .", "Let $G$ be an additive abelian group.", "An (ordered) family $(e_i)_{i \\in I}$ of elements of $G$ is said to be independent if $e_i \\ne 0$ for all $i \\in I$ and, for every family $(m_i)_{i \\in I} \\in \\mathbb {Z}^{(I)}$ , $\\sum _{i \\in I} m_ie_i =0 \\qquad \\text{implies} \\qquad m_i e_i =0 \\quad \\text{for all} \\quad i \\in I\\,.$ A family $(e_i)_{i \\in I}$ is called a basis for $G$ if $e_i \\ne 0$ for all $i \\in I$ and $G = \\bigoplus _{i \\in I} \\langle e_i \\rangle $ .", "A subset $G_0 \\subset G$ is said to be independent if the tuple $(g)_{g \\in G_0}$ is independent.", "For every prime $p \\in \\mathbb {P}$ , we denote by $\\mathsf {r}_p (G)$ the $p$ -rank of $G$ .", "Sets of Lengths.", "We say that a semigroup $S$ is cancellative if for all elements $a,b,c \\in S$ , the equation $ab=ac$ implies $b=c$ and the equation $ba=ca$ implies $b=c$ .", "Throughout this manuscript, a monoid means a cancellative semigroup with unit element, and we will use multiplicative notation.", "Let $H$ be a monoid.", "An element $a \\in H$ is said to be invertible if there exists an element $a^{\\prime }\\in H$ such that $aa^{\\prime }=a^{\\prime }a=1$ .", "The set of invertible elements of $H$ will be denoted by $H^{\\times }$ , and we say that $H$ is reduced if $H^{\\times }=\\lbrace 1\\rbrace $ .", "For a set $P$ , we denote by $\\mathcal {F} (P)$ the free abelian monoid with basis $P$ .", "Then every $a \\in \\mathcal {F} (P)$ has a unique representation in the form $a = \\prod _{p \\in P} p^{\\mathsf {v}_p (a)} \\,,$ where $\\mathsf {v}_p \\colon \\mathcal {F} (P) \\rightarrow \\mathbb {N}_0$ denotes the $p$ -adic exponent.", "An element $a \\in H$ is called irreducible (or an atom) if $a \\notin H^{\\times }$ and if, for all $u, v \\in H$ , $a = u v$ implies that $u \\in H^{\\times }$ or $v \\in H^{\\times }$ .", "We denote by $\\mathcal {A} (H)$ the set of atoms of $H$ .", "The monoid $H$ is said to be atomic if every $a \\in H \\setminus H^{\\times }$ is a product of finitely many atoms of $H$ .", "If $a \\in H$ and $a = u_1 \\cdot \\ldots \\cdot u_k$ , where $k \\in \\mathbb {N}$ and $u_1, \\ldots , u_k \\in \\mathcal {A}(H)$ , then we say that $k$ is the length of the factorization.", "For $a \\in H \\setminus H^{\\times }$ , we call $\\mathsf {L}_H (a) = \\mathsf {L} (a) = \\lbrace k \\in \\mathbb {N} \\mid a \\ \\text{has a factorization of length} \\ k \\rbrace \\subset \\mathbb {N}$ the set of lengths of $a$ .", "For convenience, we set $\\mathsf {L}(a) = \\lbrace 0\\rbrace $ for all $a \\in H^{\\times }$ .", "By definition, $H$ is atomic if and only if $\\mathsf {L} (a) \\ne \\emptyset $ for all $a \\in H$ .", "Furthermore, $\\mathsf {L} (a) = \\lbrace 1\\rbrace $ if and only if $a \\in \\mathcal {A} (H)$ if and only if $1 \\in \\mathsf {L} (a)$ .", "If $a, b \\in H$ , then $\\mathsf {L} (a) + \\mathsf {L} (b) \\subset \\mathsf {L} (a b)$ .", "We call $\\mathcal {L} (H) = \\lbrace \\mathsf {L} (a) \\mid a \\in H \\rbrace $ the system of sets of lengths of $H$ .", "We say that $H$ is half-factorial if $|L|=1$ for every $L \\in \\mathcal {L} (H)$ .", "If $H$ is atomic, then $H$ is either half-factorial or for every $N \\in \\mathbb {N}$ there is an element $a_N \\in H$ such that $|\\mathsf {L} (a_N)| >N$ ([17]).", "We say that $H$ is a BF-monoid if it is atomic and all sets of lengths are finite.", "Let $\\Delta (H) = \\bigcup _{L \\in \\mathcal {L} (H)} \\Delta (L) \\ \\subset \\ \\mathbb {N}$ denote the set of distances of $H$ , and if $\\Delta (H)\\ne \\emptyset $ , then $\\min \\Delta (H) = \\gcd \\Delta (H)$ .", "We denote by $\\Delta _1 (H)$ the set of all $d \\in \\mathbb {N}$ with the following property: For every $k \\in \\mathbb {N}$ there exists an $L \\in \\mathcal {L} (H)$ of the form $L = L^{\\prime } \\cup \\lbrace y, y+d, \\ldots , y+kd\\rbrace \\cup L^{\\prime \\prime }$ where $y \\in \\mathbb {N}$ and $L^{\\prime }, L^{\\prime \\prime } \\subset \\mathbb {Z}$ with $\\max L^{\\prime } < y$ and $y+kd < \\min L^{\\prime \\prime }$ .", "By definition, $\\Delta _1 (H)$ is a subset of $\\Delta (H)$ .", "For every $k \\in \\mathbb {N}$ we define the $k$ th elasticity of $H$ .", "If $H=H^{\\times }$ , then we set $\\rho _k (H) = k$ , and if $H \\ne H^{\\times }$ , then $\\rho _k (H) = \\sup \\lbrace \\sup L \\mid k \\in L \\in \\mathcal {L} (H) \\rbrace \\in \\mathbb {N}\\cup \\lbrace \\infty \\rbrace \\,.$ The invariant $\\rho (H) = \\sup \\lbrace \\rho (L) \\mid L \\in \\mathcal {L} (H) \\rbrace = \\lim _{k \\rightarrow \\infty }\\frac{\\rho _k(H)}{k} \\in \\mathbb {R}_{\\ge 1} \\cup \\lbrace \\infty \\rbrace $ is called the elasticity of $H$ (see [17]).", "Sets of lengths of all monoids, which are in the focus of the present paper, are highly structured (see Proposition REF and Theorems REF – REF ).", "To summarize the relevant concepts, let $d \\in \\mathbb {N}$ , $M \\in \\mathbb {N}_0$ and $\\lbrace 0,d\\rbrace \\subset \\mathcal {D}\\subset [0,d]$ .", "A subset $L \\subset \\mathbb {Z}$ is called an almost arithmetical multiprogression (AAMP for short) with difference $d$ , period $\\mathcal {D}$ , and bound $M$ , if $L = y + (L^{\\prime } \\cup L^* \\cup L^{\\prime \\prime }) \\, \\subset \\, y + \\mathcal {D} + d \\mathbb {Z}$ where $y \\in \\mathbb {Z}$ is a shift parameter, $L^*$ is finite nonempty with $\\min L^* = 0$ and $L^* =(\\mathcal {D} + d \\mathbb {Z}) \\cap [0, \\max L^*]$ , and $L^{\\prime } \\subset [-M, -1]$ and $L^{\\prime \\prime } \\subset \\max L^* + [1,M]$ .", "We say that the Structure Theorem for Sets of Lengths holds for a monoid $H$ if $H$ is atomic and there exist some $M \\in \\mathbb {N}_0$ and a finite nonempty set $\\Delta \\subset \\mathbb {N}$ such that every $L \\in \\mathcal {L}(H)$ is an AAMP with some difference $d\\in \\Delta $ and bound $M$ .", "Monoids of zero-sum sequences.", "We discuss a monoid having a combinatorial flavor whose universal role in the study of sets of lengths will become evident at the beginning of the next section.", "Let $G$ be an additive abelian group and $G_0 \\subset G$ a subset.", "Then $\\langle G_0 \\rangle $ denotes the subgroup generated by $G_0$ , and we set $G_0^{\\bullet }= G_0 \\setminus \\lbrace 0\\rbrace $ .", "In additive combinatorics, a sequence (over $G_0$ ) means a finite sequence of terms from $G_0$ where repetition is allowed and the order of the elements is disregarded, and (as usual) we consider sequences as elements of the free abelian monoid with basis $G_0$ .", "Let $S = g_1 \\cdot \\ldots \\cdot g_{\\ell } = \\prod _{g \\in G_0} g^{\\mathsf {v}_g (S)} \\in \\mathcal {F} (G_0)$ be a sequence over $G_0$ .", "We set $-S = (-g_1) \\cdot \\ldots \\cdot (-g_{\\ell })$ , and we call $ \\operatorname{supp}(S) = \\lbrace g \\in G \\mid \\mathsf {v}_g (S) > 0 \\rbrace \\subset G \\ \\text{the \\ {\\it support} \\ of \\ $S$}\\,$ , $ |S| = \\ell = \\sum _{g \\in G} \\mathsf {v}_g (S) \\in \\mathbb {N}_0 \\ \\text{the \\ {\\it length} \\ of \\ $S$} \\,$ , $ \\sigma (S) = \\sum _{i = 1}^l g_i \\ \\text{the \\ {\\it sum} \\ of \\ $S$} \\,$ , $\\Sigma (S) = \\Big \\lbrace \\sum _{i \\in I} g_i \\mid \\emptyset \\ne I \\subset [1,\\ell ] \\Big \\rbrace \\ \\text{ the \\ {\\it set ofsubsequence sums} \\ of \\ $S$} \\,$ , $\\mathsf {k} (S) = \\sum _{i = 1}^l \\frac{1}{\\operatorname{ord}(g_i)} \\ \\text{the \\ {\\it cross number} \\ of \\ $S$} \\,$ .", "The sequence $S$ is said to be zero-sum free if $0 \\notin \\Sigma (S)$ , a zero-sum sequence if $\\sigma (S) = 0$ , a minimal zero-sum sequence if it is a nontrivial zero-sum sequence and every proper subsequence is zero-sum free.", "The set of zero-sum sequences $\\mathcal {B} (G_0) = \\lbrace S \\in \\mathcal {F} (G_0) \\mid \\sigma (S)=0\\rbrace \\subset \\mathcal {F} (G_0)$ is a submonoid, and the set of minimal zero-sum sequences is the set of atoms of $\\mathcal {B} (G_0)$ .", "For any arithmetical invariant $*(H)$ defined for a monoid $H$ , we write $*(G_0)$ instead of $*(\\mathcal {B} (G_0))$ .", "In particular, $\\mathcal {A} (G_0) = \\mathcal {A} (\\mathcal {B} (G_0))$ is the set of atoms of $\\mathcal {B} (G_0)$ , $\\mathcal {L} (G_0)=\\mathcal {L} (B(G_0))$ is the system of sets of lengths of $\\mathcal {B} (G_0)$ , and so on.", "Furthermore, we say that $G_0$ is half-factorial if the monoid $\\mathcal {B} (G_0)$ is half-factorial.", "We denote by $\\mathsf {D} (G_0) = \\sup \\lbrace |S| \\mid S \\in \\mathcal {A} (G_0) \\rbrace \\in \\mathbb {N}_0 \\cup \\lbrace \\infty \\rbrace $ the Davenport constant of $G_0$ .", "If $G_0$ is finite, then $\\mathsf {D} (G_0)$ is finite.", "Suppose that $G$ is finite, say $G \\cong C_{n_1} \\oplus \\ldots \\oplus C_{n_r}$ , with $r \\in \\mathbb {N}_0$ , $1 < n_1 \\ldots n_r$ , then $r = \\mathsf {r} (G)$ is the rank of $G$ , and we have $ 1 + \\sum _{i=1}^r (n_i-1) \\le \\mathsf {D} (G) \\le |G| \\,.$ If $G$ is a $p$ -group or $\\mathsf {r} (G) \\le 2$ , then $1 + \\sum _{i=1}^r (n_i-1) = \\mathsf {D} (G)$ .", "Suppose that $|G| \\ge 3$ .", "We will use that $\\Delta (G)$ is an interval with $\\min \\Delta (G)=1$ ([26]), and that, for all $k \\in \\mathbb {N}$ , $ \\begin{aligned}\\rho _{2k} (G) & = k \\mathsf {D} (G), \\\\k \\mathsf {D} (G) +1 \\le \\rho _{2k+1} (G) & \\le k \\mathsf {D} (G) + \\lfloor \\mathsf {D} (G)/2 \\rfloor , \\quad \\text{and} \\\\\\rho (G) & = \\mathsf {D} (G)/2 \\,,\\end{aligned}$ ([20])." ], [ "Sets of lengths of transfer Krull monoids", "Weak transfer homomorphisms play a critical role in factorization theory, in particular in all studies of sets of lengths.", "We refer to [20] for a detailed presentation of transfer homomorphisms in the commutative setting.", "Weak transfer homomorphisms (as defined below) were introduced in [5] and transfer Krull monoids were introduced in [17].", "Definition 3.1 Let $H$ be a monoid.", "A monoid homomorphism $\\theta \\colon H \\rightarrow B$ to an atomic monoid $B$ is called a weak transfer homomorphism if it has the following two properties: W(T1) $B = B^{\\times } \\theta (H) B^{\\times }$ and $\\theta ^{-1} (B^{\\times })=H^{\\times }$ .", "(WT2) If $a \\in H$ , $n \\in \\mathbb {N}$ , $v_1, \\ldots , v_n \\in \\mathcal {A} (B)$ and $\\theta (a) = v_1 \\cdot \\ldots \\cdot v_n$ , then there exist $u_1, \\ldots , u_n \\in \\mathcal {A} (H)$ and a permutation $\\tau \\in \\mathfrak {S}_n$ such that $a = u_1 \\cdot \\ldots \\cdot u_n$ and $\\theta (u_i) \\in B^{\\times } v_{\\tau (i)} B^{\\times }$ for each $i \\in [1,n]$ .", "$H$ is said to be a transfer Krull monoid (over $G_0$ ) it there exists a weak transfer homomorphism $\\theta \\colon H \\rightarrow \\mathcal {B} (G_0)$ for a subset $G_0$ of an abelian group $G$ .", "If $G_0$ is finite, then we say that $H$ is a transfer Krull monoid of finite type.", "If $R$ is a domain and $R^{\\bullet }$ its monoid of cancellative elements, then we say that $R$ is a transfer Krull domain (of finite type) if $R^{\\bullet }$ is a transfer Krull monoid (of finite type).", "Let $\\theta \\colon H \\rightarrow B$ be a weak transfer homomorphism between atomic monoids.", "It is easy to show that for all $a \\in H$ we have $\\mathsf {L}_H (a) = \\mathsf {L}_B ( \\theta (a))$ and hence $\\mathcal {L} (H) = \\mathcal {L} (B)$ .", "Since monoids of zero-sum sequences are BF-monoids, the same is true for transfer Krull monoids.", "Let $H^*$ be a commutative Krull monoid (i.e., $H^*$ is commutative, completely integrally closed, and $v$ -noetherian).", "Then there is a weak transfer homomorphism $\\beta \\colon H^* \\rightarrow \\mathcal {B} (G_0)$ where $G_0$ is a subset of the class group of $H^*$ .", "Since monoids of zero-sum sequences are commutative Krull monoids and since the composition of weak transfer homomorphisms is a weak transfer homomorphism again, a monoid is a transfer Krull monoid if and only if it allows a weak transfer homomorphism to a commutative Krull monoid.", "In particular, commutative Krull monoids are transfer Krull monoids.", "However, a transfer Krull monoid need neither be commutative nor $v$ -noetherian nor completely integrally closed.", "To give a noncommutative example, consider a bounded HNP (hereditary noetherian prime) ring $R$ .", "If every stably free left $R$ -ideal is free, then its multiplicative monoid of cancellative elements is a transfer Krull monoid ([34]).", "A class of commutative weakly Krull domains which are transfer Krull but not Krull will be given in Theorem REF .", "Extended lists of commutative Krull monoids and of transfer Krull monoids, which are not commutative Krull, are given in [17].", "The next proposition summarizes some key results on the structure of sets of lengths of transfer Krull monoids.", "Proposition 3.2   Every transfer Krull monoid of finite type satisfies the Structure Theorem for Sets of Lengths.", "For every $M \\in \\mathbb {N}_0$ and every finite nonempty set $\\Delta \\subset \\mathbb {N}$ , there is a finite abelian group $G$ such that the following holds : for every AAMP $L$ with difference $d\\in \\Delta $ and bound $M$ there is some $y_{L} \\in \\mathbb {N}$ such that $y+L \\in \\mathcal {L} (G) \\quad \\text{ for all } \\quad y \\ge y_{L} \\,.$ If $G$ is an infinite abelian group, then $\\mathcal {L} (G) = \\lbrace L \\subset \\mathbb {N}_{\\ge 2} \\mid L \\ \\text{is finite andnonempty} \\ \\rbrace \\ \\cup \\ \\lbrace \\lbrace 0\\rbrace , \\lbrace 1\\rbrace \\rbrace .$ 1.", "Let $H$ be a transfer Krull monoid and $\\theta \\colon H \\rightarrow \\mathcal {B} (G_0)$ be a weak transfer homomorphism where $G_0$ is a finite subset of an abelian group.", "Then $\\mathcal {L} (H) = \\mathcal {L} (G_0)$ , and $\\mathcal {B} (G_0)$ satisfies the Structure Theorem by [20].", "For 2. we refer to [33], and for 3. see [31] and [20].", "The inequalities in (REF ) and the subsequent remarks show that a finite abelian group $G$ has Davenport constant $\\mathsf {D} (G) \\le 4$ if and only if $G$ is cyclic of order $|G|\\le 4$ or if it is isomorphic to $C_2 \\oplus C_2$ or to $C_2^3$ .", "For these groups an explicit description of their systems of sets of lengths has been given, and we gather this in the next proposition (in Section we will determine the systems $\\mathcal {L} (G)$ for all groups $G$ with $\\mathsf {D} (G)=5$ ).", "Proposition 3.3   If $G$ is an abelian group, then $\\mathcal {L}(G) = \\lbrace y +L \\mid y \\in \\mathbb {N}_0, \\ L \\in \\mathcal {L}(G^\\bullet ) \\rbrace \\supset \\bigl \\lbrace \\lbrace y\\rbrace \\,\\bigm |\\, y \\in \\mathbb {N}_0\\bigr \\rbrace $ , and equality holds if and only if $|G| \\le 2$ .", "$\\mathcal {L} (C_3) = \\mathcal {L} (C_2 \\oplus C_2) = \\bigl \\lbrace y+ 2k + [0, k] \\, \\bigm | \\, y,\\, k \\in \\mathbb {N}_0 \\bigr \\rbrace $ .", "$\\mathcal {L} (C_4) = \\bigl \\lbrace y + k+1 + [0,k] \\, \\bigm |\\, y,\\,k \\in \\mathbb {N}_0 \\bigr \\rbrace \\,\\cup \\, \\bigl \\lbrace y + 2k + 2 \\cdot [0,k] \\, \\bigm |\\, y,\\, k \\in \\mathbb {N}_0 \\bigr \\rbrace $ .", "$\\mathcal {L} (C_2^3) = \\bigl \\lbrace y + (k+1) + [0,k] \\,\\bigm |\\, y\\in \\mathbb {N}_0, \\ k \\in [0,2] \\bigr \\rbrace $ $\\quad \\text{\\, } \\ \\qquad $    $\\cup \\ \\bigl \\lbrace y + k + [0,k] \\, \\bigm |\\, y \\in \\mathbb {N}_0, \\ k \\ge 3 \\bigr \\rbrace \\cup \\bigl \\lbrace y + 2k+ 2 \\cdot [0,k] \\, \\bigm |\\, y ,\\, k \\in \\mathbb {N}_0 \\bigr \\rbrace $ .", "See [20].", "Let $G$ and $G^{\\prime }$ be abelian groups.", "Then their monoids of zero-sum sequences $\\mathcal {B} (G)$ and $\\mathcal {B} (G^{\\prime })$ are isomorphic if and only if the groups $G$ and $G^{\\prime }$ are isomorphic ([20]).", "The standing conjecture states that the systems of sets of lengths $\\mathcal {L} (G)$ and $\\mathcal {L} (G^{\\prime })$ of finite groups coincide if and only $G$ and $G^{\\prime }$ are isomorphic (apart from the trivial cases listed in Proposition REF ).", "Here is the precise formulation of the conjecture (it was first stated in [17]).", "Conjecture 3.4 Let $G$ be a finite abelian group with $\\mathsf {D} (G) \\ge 4$ .", "If $G^{\\prime }$ is an abelian group with $\\mathcal {L} (G) = \\mathcal {L} (G^{\\prime })$ , then $G$ and $G^{\\prime }$ are isomorphic.", "The conjecture holds true for groups $G$ having rank $\\mathsf {r} (G) \\le 2$ , for groups of the form $G=C_n^r$ (if $r$ is small with respect to $n$ ), and others ([24], [27], [36]).", "But it is far open in general, and the goal of this section is to develop new viewpoints of looking at this conjecture.", "Let $G$ be a finite abelian group with $\\mathsf {D} (G) \\ge 4$ .", "If $G^{\\prime }$ is a finite abelian group with $\\mathcal {L} (G)=\\mathcal {L} (G^{\\prime })$ , then (REF ) shows that $\\begin{aligned}\\mathsf {D} (G) & = \\rho _2 (G) = \\sup \\lbrace \\sup L \\mid 2 \\in L \\in \\mathcal {L} (G) \\rbrace \\\\& = \\sup \\lbrace \\sup L \\mid 2 \\in L \\in \\mathcal {L} (G^{\\prime }) \\rbrace = \\rho _2 (G^{\\prime }) = \\mathsf {D} (G^{\\prime }) \\,.\\end{aligned}$ We see from the inequalities in (REF ) that there are (up to isomorphism) only finitely many finite abelian groups $G^{\\prime }$ with given Davenport constant, and hence there are only finitely many finite abelian groups $G^{\\prime }$ with $\\mathcal {L} (G)=\\mathcal {L} (G^{\\prime })$ .", "Thus Conjecture REF is equivalent to the statement that for each $m \\ge 4$ and for each two non-isomorphic finite abelian groups $G$ and $G^{\\prime }$ having Davenport constant $\\mathsf {D} (G)=\\mathsf {D}(G^{\\prime })=m$ the systems $\\mathcal {L} (G)$ and $\\mathcal {L} (G^{\\prime })$ are distinct.", "Therefore we have to study the set $\\Omega _m = \\lbrace \\mathcal {L} (G) \\mid G \\ \\text{is a finite abelian group with} \\ \\mathsf {D} (G)=m\\rbrace $ of all systems of sets of lengths stemming from groups having Davenport constant equal to $m$ .", "If a group $G^{\\prime }$ is a proper subgroup of $G$ , then $\\mathsf {D} (G^{\\prime }) < \\mathsf {D} (G)$ ([20]) and hence $\\mathcal {L} (G^{\\prime }) \\subsetneq \\mathcal {L} (G)$ .", "Thus if $\\mathsf {D} (G)=\\mathsf {D} (G^{\\prime })$ for some group $G^{\\prime }$ , then none of the groups is isomorphic to a proper subgroup of the other one.", "Conversely, if $G^{\\prime }$ is a finite abelian group with $\\mathcal {L} (G^{\\prime }) \\subset \\mathcal {L} (G)$ , then $\\mathsf {D} (G^{\\prime }) = \\rho _2 (G^{\\prime }) \\le \\rho _2 (G) = \\mathsf {D} (G)$ .", "However, it may happen that $\\mathcal {L} (G^{\\prime }) \\subsetneq \\mathcal {L} (G)$ but $\\mathsf {D} (G^{\\prime })= \\mathsf {D} (G)$ .", "Indeed, Proposition REF shows that $\\mathcal {L} (C_4) \\subsetneq \\mathcal {L} (C_2^3)$ , and we will observe this phenomenon again in Section .", "Theorem 3.5 For $m \\in \\mathbb {N}$ , let $\\Omega _m = \\lbrace \\mathcal {L} (G) \\mid G \\ \\text{is a finite abelian group with} \\ \\mathsf {D} (G)=m\\rbrace .$ Then $\\mathcal {L} (C_2^{m-1})$ is a maximal element and $\\mathcal {L} (C_m)$ is a minimal element in $\\Omega _m$ (with respect to set-theoretical inclusion).", "Furthermore, if $G$ is an abelian group with $\\mathsf {D} (G)=m$ and $\\mathcal {L} (G) \\subset \\mathcal {L} (C_2^{m-1})$ , then $G \\cong C_m $ or $G \\cong C_2^{m-1}$ .", "If $m \\in [1,2]$ , then $|\\Omega _m|=1$ and hence all assertions hold.", "Since $C_3$ and $C_2 \\oplus C_2$ are the only groups (up to isomorphism) with Davenport constant three, and since $\\mathcal {L} (C_3)= \\mathcal {L} (C_2^2)$ by Proposition REF , the assertions follow.", "We suppose that $m \\ge 4$ and proceed in two steps.", "1.", "To show that $\\mathcal {L} (C_2^{m-1})$ is maximal, we study, for a finite abelian group $G$ , the set $\\Delta _1 (G)$ .", "We define $\\Delta ^* (G) = \\lbrace \\min \\Delta (G_0) \\mid G_0 \\subset G \\ \\text{with} \\ \\Delta (G_0) \\ne \\emptyset \\rbrace \\,,$ and recall that (see [20]) $\\Delta ^* (G) \\subset \\Delta _1 (G) \\subset \\lbrace d_1 \\in \\Delta (G) \\mid d_1 \\ \\text{divides some} \\ d \\in \\Delta ^* (G) \\rbrace \\,.$ Thus $\\max \\Delta _1 (G) = \\max \\Delta ^* (G)$ , and [29] implies that $\\max \\Delta ^* (G) = \\max \\lbrace \\exp (G)-2, \\mathsf {r} (G)-1\\rbrace $ .", "Assume to the contrary that there is a finite abelian group $G$ with $\\mathsf {D} (G) =m\\ge 4$ that is not an elementary 2-group such that $\\mathcal {L} (C_2^{m-1} ) \\subset \\mathcal {L} (G)$ .", "Then $\\begin{split}m-2& =\\max \\Delta ^*(C_2^{m-1}) = \\max \\Delta _1 (C_2^{m-1}) \\le \\max \\Delta _1 (G) \\\\& =\\max \\Delta ^* (G) = \\max \\lbrace \\exp (G)-2, \\mathsf {r} (G)-1\\rbrace \\,.\\end{split}$ If $\\mathsf {r} (G) \\ge m-1$ , then $\\mathsf {D} (G)=m$ implies that $G \\cong C_2^{m-1}$ , a contradiction.", "Thus $\\exp (G) \\ge m$ , and since $\\mathsf {D} (G)=m$ we infer that that $G \\cong C_m$ .", "If $m=4$ , then Proposition REF .4 shows that $\\mathcal {L} (C_2^3) \\lnot \\subset \\mathcal {L} (C_4)$ , a contradiction.", "Suppose that $m \\ge 5$ .", "Then $\\Delta ^*(C_2^{m-1}) = \\Delta _1 (C_2^{m-1}) = \\Delta (C_2^{m-1}) = [1, m-2]$ by [20].", "For cyclic groups we have $\\max \\Delta ^* (C_m) = m-2$ and $\\max ( \\Delta ^* (C_m) \\setminus \\lbrace m-2\\rbrace ) = \\lfloor m/2 \\rfloor - 1$ by [20].", "Therefore $\\mathcal {L} (C_2^{m-1} ) \\subset \\mathcal {L} (C_m)$ implies that $[1, m-2] = \\Delta _1 (C_2^{m-1}) \\subset \\Delta _1 (C_m) \\,,$ a contradiction to $m-3 \\notin \\Delta _1 (C_m)$ .", "2.", "We recall some facts.", "Let $G$ be a group with $\\mathsf {D} (G)=m$ .", "If $U \\in \\mathcal {A} (G)$ with $|U|=\\mathsf {D} (G)$ , then $\\lbrace 2, \\mathsf {D} (G) \\rbrace \\subset \\mathsf {L} \\big ( U(-U) \\big )$ .", "Cyclic groups and elementary 2-groups are the only groups $G$ with the following property: if $L \\in \\mathsf {L} (G)$ with $\\lbrace 2, \\mathsf {D} (G)\\rbrace \\subset L$ , then $L = \\lbrace 2, \\mathsf {D} (G)\\rbrace $ ([20]).", "Now assume to the contrary that there is a finite abelian group $G$ with $\\mathsf {D} (G)=m$ such that $\\mathcal {L} (G) \\subset \\mathcal {L} (C_m)$ .", "Let $L \\in \\mathcal {L} (G)$ with $\\lbrace 2, \\mathsf {D} (G) \\rbrace \\subset L$ .", "Then $L \\in \\mathcal {L} (C_m)$ whence $L = \\lbrace 2, \\mathsf {D} (G)\\rbrace $ which implies that $G$ is cyclic or an elementary 2-group.", "By 1., $G$ is not an elementary 2-group whence $G$ is cyclic which implies $G \\cong C_m$ and hence $\\mathcal {L} (G) = \\mathcal {L} (C_m)$ .", "The furthermore assertion on groups $G$ with $\\mathsf {D} (G)=m$ and $\\mathcal {L} (G) \\subset \\mathcal {L} (C_2^{m-1})$ follows as above by considering sets of lengths $L$ with $\\lbrace 2, \\mathsf {D} (G)\\rbrace \\subset L$ .", "In Section we will see that $\\mathcal {L} (C_2^{m-1})$ need not be the largest element in $\\Omega _m$ , and that indeed $\\mathcal {L} (C_m) \\subset \\mathcal {L} (C_2^{m-1})$ for $m \\in [2,5]$ , where the inclusion is strict for $m \\ge 4$ .", "On the other hand, it is shown in [28] that $\\mathcal {L} (C_m) \\lnot \\subset \\mathcal {L} (C_2^{m-1})$ for infinitely many $m \\in \\mathbb {N}$ .", "Theorem 3.6 We have $\\bigcap \\mathcal {L} (G) = \\bigl \\lbrace y+ 2k + [0, k] \\, \\bigm | \\, y,\\, k \\in \\mathbb {N}_0 \\bigr \\rbrace \\,,$ where the intersection is taken over all finite abelian groups $G$ with $|G| \\ge 3$ .", "By Proposition REF .2, the intersection on the left hand side is contained in the set on the right hand side.", "Let $G$ be a finite abelian group with $|G| \\ge 3$ .", "If $L \\in \\mathcal {L} (G)$ , then $y+L \\in \\mathcal {L} (G)$ .", "Thus it is sufficient to show that $[2k,3k] \\in \\mathcal {L} (G)$ for every $k \\in \\mathbb {N}$ .", "If $G$ contains two independent elements of order 2 or an element of order 4, then the claim follows by Proposition REF .", "Thus, it remains to consider the case when $G$ contains an element $g$ with $\\operatorname{ord}(g)= p$ for some odd prime $p \\in \\mathbb {N}$ .", "Let $k \\in \\mathbb {N}$ and $B_k = ((2g)^p g^p)^k$ .", "We assert that $\\mathsf {L} (B_k)=[2k,3k]$ .", "We set $U_1= g^p$ , $U_2 = (2g)^p$ , $V_1 = (2g)^{(p-1)/2}g$ , and $V_2=(2g)g^{p-2}$ .", "Since $U_1U_2=V_1^2V_2$ and $B_k = (U_1U_2)^k = (U_1 U_2)^{k-\\nu } (V_1^2V_2)^{\\nu } \\quad \\text{for all} \\quad \\nu \\in [0,k] \\,,$ it follows that $[2k,3k] \\subset \\mathsf {L} (B_k)$ .", "In order to show there are no other factorization lengths, we recall the concept of the $g$ -norm of sequences.", "If $S = (n_1g) \\cdot \\ldots (n_{\\ell }g) \\in \\mathcal {B} (\\langle g \\rangle )$ , where $\\ell \\in \\mathbb {N}_0$ and $n_1, \\ldots , n_{\\ell } \\in [1, \\operatorname{ord}(g)]$ , then $||S||_g = \\frac{n_1+ \\ldots + n_{\\ell }}{\\operatorname{ord}(g)} \\in \\mathbb {N}$ is the $g$ -norm of $S$ .", "Clearly, if $S = S_1 \\cdot \\ldots \\cdot S_m$ with $S_1, \\ldots , S_m \\in \\mathcal {A} (G)$ , then $||S||_g = ||S_1||_g + \\ldots + ||S_m||_g$ .", "Note that $U_2=(2g)^p$ is the only atom in $\\mathcal {A} (\\lbrace g,2g\\rbrace )$ with $g$ -norm 2, and all other atoms in $\\mathcal {A} (\\lbrace g,2g\\rbrace )$ have $g$ -norm 1.", "Let $B_k = W_1 \\cdot \\ldots \\cdot W_{\\ell }$ be a factorization of $B_k$ , and let $\\ell ^{\\prime }$ be the number of $i \\in [1,\\ell ] $ such that $W_i = (2g)^p$ .", "We have $\\Vert B_k\\Vert _g= 3k$ and thus $3k = 2\\ell ^{\\prime } + (\\ell - \\ell ^{\\prime })= \\ell ^{\\prime } + \\ell $ .", "Since $\\ell ^{\\prime } \\in [0,k]$ , it follows that $\\ell =3k-\\ell ^{\\prime } \\in [2k , 3k] $ .", "Theorem 3.7 Let $L \\subset \\mathbb {N}_{\\ge 2}$ be a finite nonempty subset.", "Then there are only finitely many pairwise non-isomorphic finite abelian groups $G$ such that $L \\notin \\mathcal {L} (G)$ .", "We start with the following two assertions.", "There is an integer $n_L \\in \\mathbb {N}$ such that $L \\in \\mathcal {L} (C_n)$ for every $n \\ge n_L$ .", "For every $p \\in \\mathbb {P}$ there is an integer $r_{p,L} \\in \\mathbb {N}$ such that $L \\in \\mathcal {L} (C_p^r)$ for every $r \\ge r_{p,L}$ .", "Proof of  A1.", "By Proposition REF .3, there is some $B = \\prod _{i=1}^k m_k \\prod _{j=1}^{\\ell } (-n_j) \\in \\mathcal {B} ( \\mathbb {Z})$ such that $\\mathsf {L} (B)=L$ , where $k,\\ell , m_1, \\ldots , m_k \\in \\mathbb {N}$ and $n_1, \\ldots , n_{\\ell } \\in \\mathbb {N}_0$ .", "We set $n_L = n_1+ \\ldots + n_{\\ell }$ and choose some $n \\in \\mathbb {N}$ with $n \\ge n_L$ .", "If $S \\in \\mathcal {F} (\\mathbb {Z})$ with $S B$ and $f \\colon \\mathbb {Z}\\rightarrow \\mathbb {Z}/n\\mathbb {Z}$ denotes the canonical epimorphism, then $S$ has sum zero if and only if $f (S)$ has sum zero.", "This implies that $\\mathsf {L}_{\\mathcal {B} (\\mathbb {Z}/n\\mathbb {Z})} ( f (B)) = \\mathsf {L}_{\\mathcal {B} ( \\mathbb {Z})} (B) = L$ .", "$\\Box $ [Proof of A1] Proof of  A2.", "Let $p \\in \\mathbb {P}$ be a prime and let $G_p$ be an infinite dimensional $\\mathbb {F}_p$ -vector space.", "By Proposition REF .3, there is some $B_p \\in \\mathcal {B} (G_p)$ such that $\\mathsf {L} (B_p)=L$ .", "If $r_{p,L}$ is the rank of $\\langle \\operatorname{supp}(B_p) \\rangle \\subset G_p$ , then $L = \\mathsf {L} (B_p) \\in \\mathcal {L} ( \\langle \\operatorname{supp}(B_p) \\rangle ) \\subset \\mathcal {L} (C_p^r) \\quad \\text{for } \\quad r \\ge r_{p,L} \\,.", "\\qquad \\Box [\\text{Proof of} {\\bf A2}]$ Now let $G$ be a finite abelian group such that $L \\notin \\mathcal {L} (G)$ .", "Then A1 implies that $\\exp (G) < n_L$ , and A2 implies that $\\mathsf {r}_p (G) < r_{p,L}$ for all primes $p$ with $p \\exp (G)$ .", "Thus the assertion follows." ], [ "Sets of lengths of transfer Krull monoids over small groups", "Since the very beginning of factorization theory, invariants controlling the structure of sets of lengths (such as elasticities and sets of distances) have been in the center of interest.", "Nevertheless, (apart from a couple of trivial cases) the full system of sets of lengths has been written down explicitly only for the following classes of monoids: Numerical monoids generated by arithmetical progressions: see [1].", "Self-idealizations of principal ideal domains: see [10], [4].", "The ring of integer-valued polynomials over $\\mathbb {Z}$ : see [15].", "The systems $\\mathcal {L} (G)$ for infinite abelian groups $G$ and for abelian groups $G$ with $\\mathsf {D} (G) \\le 4$ : see Propositions REF and REF .", "The goal of this section is to determine $\\mathcal {L} (G)$ for abelian groups $G$ having Davenport constant $\\mathsf {D} (G) = 5$ .", "By inequality (REF ) and the subsequent remarks, a finite abelian group $G$ has Davenport constant five if and only if it is isomorphic to one of the following groups: $C_3 \\oplus C_3, \\quad C_5, \\quad C_2 \\oplus C_4, \\quad C_2^4 \\,.$ Their systems of sets of lengths are given in Theorems REF , REF , REF , and REF .", "We start with a brief analysis of these explicit descriptions (note that they will be needed again in Section ; confer the proof of Theorem REF ).", "By Theorem REF , we know that $\\mathcal {L} (C_2^4)$ is maximal in $\\Omega _5 = \\lbrace \\mathcal {L} (C_5), \\mathcal {L} (C_2 \\oplus C_4), \\mathcal {L} (C_3\\oplus C_3), \\mathcal {L} (C_2^4) \\rbrace $ .", "Theorems REF , REF , REF , and REF unveil that $\\mathcal {L} (C_3 \\oplus C_3)$ , $\\mathcal {L} (C_2\\oplus C_4)$ , and $\\mathcal {L} (C_2^4)$ are maximal in $\\Omega _5$ , and that $\\mathcal {L} (C_5)$ is contained in $\\mathcal {L} (C_2^4)$ , but it is neither contained in $\\mathcal {L} (C_3 \\oplus C_3)$ nor in $\\mathcal {L} (C_2 \\oplus C_4)$ .", "Furthermore, Theorems REF , REF , and REF show that $\\mathcal {L} (C_m) \\subset \\mathcal {L} (C_2^{m-1})$ for $m \\in [2,5]$ .", "It is well-known that, for all $m \\ge 4$ , $\\mathcal {L} (C_m) \\ne \\mathcal {L} (C_2^{m-1})$ ([16]), and the standing conjecture is that $\\mathcal {L} (C_m) \\lnot \\subset \\mathcal {L} (C_2^{m-1})$ holds true for almost all $m \\in \\mathbb {N}_{\\ge 2}$ (see [28]).", "The group $C_3 \\oplus C_3$ has been handled in [24].", "Theorem 4.1 $\\mathcal {L} (C_3^2) = \\lbrace y+ [2k, 5k] \\mid y,k \\in \\mathbb {N}_0\\rbrace \\ \\cup \\ \\lbrace y + [2k+1, 5k+2] \\mid y \\in \\mathbb {N}_0,\\, k \\in \\mathbb {N}\\rbrace \\rbrace $ .", "Remark.", "An equivalent way to describe $\\mathcal {L} (C_3^2)$ is $\\lbrace y+\\left\\lceil \\frac{2k}{3}\\right\\rceil +[0,k]\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}_{\\ge 2} \\rbrace \\cup \\lbrace \\lbrace y\\rbrace , y+2+[0,1]\\mid y\\in \\mathbb {N}_0\\rbrace $ .", "The fact that all sets of lengths are intervals is a consequence of the fact $\\Delta (C_3^2)=\\lbrace 1\\rbrace $ .", "Of course, each set of lengths $L$ has to fulfill $\\rho (L) \\le 5/2 = \\rho (C_3^2)$ .", "We observe that the description shows that this is the only condition, provided $\\min L \\ge 2$ .", "The following lemma is frequently helpful in the remainder of this section.", "Lemma 4.2 Let $G$ be a finite abelian group, and let $A\\in \\mathcal {B}(G)$ .", "If $\\operatorname{supp}(A)\\cup \\lbrace 0\\rbrace $ is a group, then $\\mathsf {L}(A)$ is an interval.", "If $A_1$ is an atom dividing $A$ with $|A_1|=2$ , then $\\max \\mathsf {L}(A)=1+\\max \\mathsf {L}(AA_1^{-1})$ .", "If $A$ is a product of atoms of length 2 and if every atom $A_1$ dividing $A$ has length $|A_1|=2$ or $|A_1|=4$ , then $\\max \\mathsf {L}(A)-1\\notin \\mathsf {L}(A)$ .", "1.", "See [20].", "2.", "Let $\\ell =\\max \\mathsf {L}(A)$ and $A=U_1\\cdot \\ldots \\cdot U_{\\ell }$ , where $U_1,\\ldots , U_{\\ell }\\in \\mathcal {A}(G)$ .", "Let $A_1=g_1g_2$ , where $g_1,g_2\\in G$ .", "If there exists $i\\in [1,\\ell ]$ such that $A_1=U_i$ , then $\\max \\mathsf {L}(A)=1+\\max \\mathsf {L}(AA_1^{-1})$ .", "Otherwise there exist distinct $i,j\\in [1,\\ell ]$ such that $g_1U_i$ and $g_2U_j$ .", "Thus $A_1$ divides $U_iU_j$ and hence $1+\\max \\mathsf {L}(AA_1^{-1})\\ge \\ell $ which implies that $\\max \\mathsf {L}(A)=1+\\max \\mathsf {L}(AA_1^{-1})$ by the maximality of $\\ell $ .", "3.", "If $\\max \\mathsf {L}(A)-1\\in \\mathsf {L}(A)$ , then $A=V_1\\cdot \\ldots \\cdot V_{\\max \\mathsf {L}(A)-1}$ with $|V_1|=4$ and $|V_2|=\\ldots =|V_{\\max \\mathsf {L}(A)-1}|=2$ .", "Thus $V_1$ can only be a product two atoms of length 2, a contradiction.", "We now consider the groups $C_5$ , $C_2 \\oplus C_4$ , and $C_2^4$ , each one in its own subsection.", "In the proofs of the forthcoming theorems we will use Proposition REF and Theorem REF without further mention." ], [ "The system of sets of lengths of $C_5$", "The goal of this subsection is to prove the following result.", "Theorem 4.3 $\\quad \\mathcal {L}(C_5)=\\mathcal {L}_1\\cup \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5\\cup \\mathcal {L}_6\\,,$ where $& \\mathcal {L}_1=\\lbrace \\lbrace y\\rbrace \\mid y\\in \\mathbb {N}_0\\rbrace \\,, \\\\&\\mathcal {L}_2=\\lbrace y+2+\\lbrace 0,2\\rbrace \\mid y\\in \\mathbb {N}_0\\rbrace \\,, \\\\&\\mathcal {L}_3= \\lbrace y+3+\\lbrace 0,1,3\\rbrace \\mid y\\in \\mathbb {N}_0\\rbrace \\,, \\\\&\\mathcal {L}_4=\\lbrace y+2k+3 \\cdot [0,k]\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\rbrace \\,,\\\\&\\mathcal {L}_5=\\lbrace y+2\\left\\lceil \\frac{k}{3}\\right\\rceil +[0,k]\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\setminus \\lbrace 3\\rbrace \\rbrace \\cup \\lbrace y+[3,6]\\mid y\\in \\mathbb {N}_0\\rbrace \\, , \\text{ and } \\\\& \\mathcal {L}_6= \\lbrace y+2k+3+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]\\mid y, k \\in \\mathbb {N}_0 \\rbrace \\,.$ We observe that all sets of lengths with many elements are arithmetic multiprogressions with difference 1 or 3.", "Yet, there are none with difference 2.", "This is because $\\Delta ^{*}(C_5) = \\lbrace 1,3\\rbrace $ .", "Moreover, we point out that the condition for an interval to be a set of lengths is different from that of the other groups with Davenport constant 5.", "This is related to the fact that $\\rho _{2k+1}(C_5) = 5k+1$ , while $\\rho _{2k+1}(G) = 5k+2$ for the other groups with Davenport constant 5.", "Before we start the actual proof, we collect some results on sets of lengths over $C_5$ .", "Lemma 4.4 Let $G$ be cyclic of order five, and let $A \\in \\mathcal {B}(G)$ .", "If $g \\in G^{\\bullet }$ and $k \\in \\mathbb {N}_0$ , then $\\mathsf {L} \\big (g^{5(k+1)} (-g)^{5(k+1)} (2g)g^3 \\big )=2k+3+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k] \\,.$ If $2\\in \\Delta (\\mathsf {L} (A))\\subset [1,2]$ , then $\\mathsf {L}(A)\\in \\lbrace \\lbrace y,y+2\\rbrace \\mid y\\ge 2\\rbrace \\cup \\lbrace \\lbrace y,y+1,y+3\\rbrace \\mid y\\ge 3\\rbrace $ or $\\mathsf {L}(A)=3+\\lbrace 0,2,3\\rbrace +\\mathsf {L}(A^{\\prime })$ where $A^{\\prime }\\in \\mathcal {B}(G)$ and $\\mathsf {L}(A^{\\prime })$ is an arithmetical progression of difference 3.", "$\\Delta (G)=[1,3]$ , and if $3\\in \\Delta (\\mathsf {L} (A))$ , then $\\Delta (\\mathsf {L} (A))=\\lbrace 3\\rbrace $ .", "$\\rho _{2k+1} (G) = 5k+1$ for all $k \\in \\mathbb {N}$ .", "1. and 2. follow from the proof of [24].", "3.", "See [20] and [12].", "4.", "See [16].", "Let $G$ be cyclic of order five and let $g\\in G^{\\bullet }$ .", "We first show that all the specified sets occur as sets of lengths, and then we show that no other sets occur.", "Step 1.", "We prove that for every $L\\in \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5\\cup \\mathcal {L}_6$ , there exists an $A\\in \\mathcal {B}(G)$ such that $L=\\mathsf {L}(A)$ .", "We distinguish five cases.", "If $L=\\lbrace y,y+2\\rbrace \\in \\mathcal {L}_2$ with $y\\ge 2$ , then we set $A=0^{y-2} g^5 (-g)^3(-2g)$ and obtain that $\\mathsf {L}(A)=y-2+\\lbrace 2,4\\rbrace =L$ .", "If $L=\\lbrace y,y+1,y+3\\rbrace \\in \\mathcal {L}_3$ with $y\\ge 3$ , then we set $A=0^{y-3} g^5 (-g)^5 g^2(-2g)$ and obtain that $\\mathsf {L}(A)=y-3+\\lbrace 3,4,6\\rbrace =\\lbrace y,y+1,y+3\\rbrace =L$ .", "If $L=y+2k+3 \\cdot [0,k]\\in \\mathcal {L}_4$ with $k\\in \\mathbb {N}$ and $y\\in \\mathbb {N}_0$ , then we set $A=g^{5k}(-g)^{5k}0^y\\in \\mathcal {B}(G)$ and hence $\\mathsf {L}(A)=y+2k+3 \\cdot [0,k]=L$ .", "If $L=y+2k+3+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]\\in \\mathcal {L}_6$ with $k\\in \\mathbb {N}_0$ and $y\\in \\mathbb {N}_0$ , then we set $A=0^y g^{5(k+1)} (-g)^{5(k+1)} (2g)g^3$ and hence $\\mathsf {L}(A)=y+2k+3+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]=L$ by Lemma REF .1.", "Now we suppose that $L\\in \\mathcal {L}_5$ , and we distinguish two subcases.", "First, if $L=y+[3,6]$ with $y\\in \\mathbb {N}_0$ , then we set $A=0^y (2g(-2g)) g^{5} (-g)^{5}$ and hence $\\mathsf {L}(A)=y+[3, 6]=L$ .", "Second, we assume that $L=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]$ with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}\\setminus \\lbrace 3\\rbrace $ .", "If $k\\in \\mathbb {N}$ with $k\\equiv 0\\pmod {3}$ , then $k\\ge 6$ and by Lemma REF .1 we obtain that $\\mathsf {L} \\big ( 0^y (2g)^5 (-2g)^5 g^{5t} (-g)^{5t} \\big ) = y+[2t+2, 5t+5]=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]=L\\,,$ where $k=3t+3$ .", "If $k\\in \\mathbb {N}$ with $k\\equiv 1\\pmod {3}$ , then by Lemma REF .1 we obtain that $\\mathsf {L} \\big (0^y (2g(-g)^2) (g^2(-2g)) g^{5t} (-g)^{5t}\\big ) = y+[2t+2, 5t+3]=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]=L\\,,$ where $k=3t+1$ .", "If $k\\in \\mathbb {N}$ with $k\\equiv 2\\pmod {3}$ , then by Lemma REF .1 we obtain that $\\mathsf {L}\\big (0^y (g^3(2g)) ((-g)^3(-2g)) g^{5t} (-g)^{5t}\\big ) = y+[2t+2, 5t+4]=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]=L\\,,$ where $k=3t+2$ .", "Step 2.", "We prove that for every $A\\in \\mathcal {B}(G^{\\bullet })$ , $\\mathsf {L}(A)\\in \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5\\cup \\mathcal {L}_6$ .", "Let $A \\in \\mathcal {B} (G^{\\bullet })$ .", "We may suppose that $\\Delta (\\mathsf {L} (A)) \\ne \\emptyset $ .", "By Lemma REF .3 we distinguish three cases according to the form of the set of distances $\\Delta (\\mathsf {L} (A))$ .", "CASE 1:   $\\Delta (\\mathsf {L} (A))=\\lbrace 1\\rbrace $ .", "Then $\\mathsf {L}(A)$ is an interval and hence we assume that $\\mathsf {L}(A)=[y,y+k]=y+[0,k]$ where $y\\ge 2$ and $k\\ge 1$ .", "If $k=3$ and $y=2$ , then $\\mathsf {L}(A)=[2,5]$ and hence $\\mathsf {L}(A)=\\mathsf {L}(g^5(-g)^5)=\\lbrace 2,5\\rbrace $ , a contradiction.", "Thus $k=3$ implies that $y\\ge 3$ and hence $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "If $k\\le 2$ , then we obviously have that $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "Suppose that $k\\ge 4$ .", "If $y=2t$ with $t\\ge 2$ , then $y+k\\le 5t$ and hence $y=2t\\ge 2\\lceil \\frac{k}{3}\\rceil $ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "If $y=2t+1$ with $t\\ge 1$ , then $y+k\\le 5t+1$ and hence $y=2t+1\\ge 1+ 2\\lceil \\frac{k}{3}\\rceil $ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "CASE 2:   $\\Delta (\\mathsf {L} (A))=\\lbrace 3\\rbrace $ .", "Then $\\mathsf {L}(A)=y+3 \\cdot [0,k]$ where $y\\ge 2$ and $k\\ge 1$ .", "If $y=2t\\ge 2$ , then $y+3k\\le 5t$ and hence $y=2t\\ge 2k$ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_4$ .", "If $y=2t+1\\ge 3$ , then $y+3k\\le 5t+1$ and hence $y=2t+1\\ge 1+ 2k$ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_4$ .", "CASE 3:   $2\\in \\Delta (\\mathsf {L} (A))\\subset [1,2]$ .", "By Lemma REF .2, we infer that either $\\mathsf {L}(A)\\in \\mathcal {L}_2\\cup \\mathcal {L}_3$ or that $\\mathsf {L}(A)=3+\\lbrace 0,2,3\\rbrace +\\mathsf {L}(A^{\\prime })$ , where $A^{\\prime }\\in \\mathcal {B}(G)$ and $\\mathsf {L}(A^{\\prime })$ is an arithmetical progression of difference 3.", "In the latter case we obtain that $\\mathsf {L}(A^{\\prime })=y+2k+3 \\cdot [0,k]$ , with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}_0$ , and hence $\\mathsf {L}(A)=y+2k+3+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]\\in \\mathcal {L}_6$ ." ], [ "The system of sets of lengths of $C_2 \\oplus C_4$", "We establish the following result, giving a complete description of the system of sets of lengths of $C_2 \\oplus C_4$ .", "Theorem 4.5 $\\quad \\mathcal {L}(C_2\\oplus C_4)=\\mathcal {L}_1\\cup \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5\\,,$ where $\\mathcal {L}_1&=\\lbrace \\lbrace y\\rbrace \\mid y\\in \\mathbb {N}_0 \\rbrace \\,,\\\\\\mathcal {L}_2&=\\lbrace y+2\\left\\lceil \\frac{k}{3}\\right\\rceil +[0,k]\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\setminus \\lbrace 3\\rbrace \\rbrace \\cup \\\\ & \\quad \\quad \\lbrace y+[3,6]\\mid y\\in \\mathbb {N}_0,\\rbrace \\cup \\lbrace [2t+1, 5t+2]\\mid t\\in \\mathbb {N}\\rbrace \\\\&=\\lbrace y+\\left\\lceil \\frac{2k}{3}\\right\\rceil +[0,k]\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\setminus \\lbrace 1,3\\rbrace \\rbrace \\cup \\\\& \\quad \\quad \\lbrace y+3+[0,3], y+2+[0,1]\\mid y\\in \\mathbb {N}_0\\rbrace \\,,\\\\\\mathcal {L}_3&=\\lbrace y+2k+2 \\cdot [0,k]\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\rbrace \\,,\\\\\\mathcal {L}_4&= \\lbrace y+k+1+(\\lbrace 0\\rbrace \\cup [2,k+2])\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\text{\\rm \\ odd}\\rbrace \\,, \\text{ and} \\\\\\mathcal {L}_5&= \\lbrace y+k+2+([0,k]\\cup \\lbrace k+2\\rbrace )\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\rbrace \\,.", "\\\\$ We note that all sets of lengths are arithmetical progressions with difference 2 or almost arithmetical progressions with difference 1 and bound 2.", "This is related to the fact that $\\Delta ( C_2 \\oplus C_4 )= \\Delta ^{*} ( C_2 \\oplus C_4 ) =\\lbrace 1,2\\rbrace $ .", "We start with a lemma determining all minimal zero-sum sequences over $C_2 \\oplus C_4$ .", "Lemma 4.6 Let $(e,g)$ be a basis of $G=C_2\\oplus C_4$ with $\\operatorname{ord}(e)=2$ and $\\operatorname{ord}(g)=4$ .", "Then the minimal zero-sum sequences over $G^{\\bullet }$ are given by the following list.", "The minimal zero sum sequences of length 2 are : $&S_{2}^1=\\lbrace e^2, (e+2g)^2\\rbrace ,\\\\&S_2^2=\\lbrace (2g)^2\\rbrace ,\\\\&S_2^3=\\lbrace g(-g), (e+g)(e-g)\\rbrace $ The minimal zero sum sequences of length 3 are : $&S_3^1=\\lbrace e(2g)(e+2g)\\rbrace \\,,\\\\& S_3^2=\\lbrace g^2(2g),(-g)^2(2g), (e+g)^2(2g), (e-g)^2(2g)\\rbrace \\,,\\\\&S_3^3=\\lbrace eg(e-g), e(-g)(e+g), (e+2g)g(e+g), (e+2g)(-g)(e-g)\\rbrace \\,.\\\\$ The minimal zero sum sequences of length 4 are : $&S_4^1=\\lbrace g^4, (-g)^4, (e+g)^4, (e-g)^4\\rbrace \\,,\\\\&S_4^2=\\lbrace g^2(e+g)^2, (-g)^2(e-g)^2, g^2(e-g)^2, (-g)^2(e+g)^2\\rbrace \\,,\\\\& S_4^3=\\lbrace eg^2(e+2g),e(e+g)^2(e+2g), e(-g)^2(e+2g), e(e-g)^2(e+2g)\\rbrace \\,,\\\\& S_4^4=\\lbrace eg(2g)(e+g),e(-g)(2g)(e-g), (e+2g)g(2g)(e-g), (e+2g)(-g)(2g)(e+g)\\rbrace \\,.\\\\$ The minimal zero sum sequences of length 5 are : $S_5=\\lbrace &eg^3(e+g), e(-g)^3(e-g), e(e+g)^3g, e(e-g)^3(-g)\\\\&(e+2g)g^3(e-g), (e+2g)(-g)^3(e+g), \\\\ & (e+2g)(e+g)^3(-g), (e+2g)(e-g)^3g\\rbrace \\,,\\\\$ Moreover, for each two atoms $W_1,W_2$ in any one of the above sets, there exists a group isomorphism $\\phi \\colon G \\rightarrow G$ such that $\\phi (W_1)=W_2$ .", "We give a sketch of the proof.", "Since a minimal zero-sum sequence of length two is of the form $h(-h)$ for some non-zero element $h\\in G$ , the list given in 1. follows.", "A minimal zero-sum sequence of length three contains either two elements of order four or no element of order four.", "If there are two elements of order four, we can have one element of order four with multiplicity two (see $S_3^2$ ) or two distinct elements of order four that are not the inverse of each other (see $S_3^3$ ).", "If there is no element of order four, the sequence consists of three distinct elements of order two (see $S_3^1$ ).", "A minimal zero-sum sequence of length four contains either four elements of order four or two elements of order four.", "If there are two elements of order four, the sequence can contain one element with multiplicity two (see $S_4^3$ ) or any two distinct elements that are not each other's inverse with multiplicity one (see $S_4^4$ ).", "If there are four elements of order four, the sequence can contain one element with multiplicity four (see $S_4^1$ ) or two elements with multiplicity two (see $S_4^2$ ).", "Since every minimal zero-sum sequence of length five contains an element with multiplicity three, the list given in 4. follows (for details see [20]).", "The existence of the required isomorphism follows immediately from the given description of the sequences.", "The next lemma collects some basic results on $\\mathcal {L} (C_2 \\oplus C_4)$ that will be essential for the proof of Theorem REF .", "Lemma 4.7 Let $G=C_2\\oplus C_4$ , and let $A \\in \\mathcal {B} (G)$ .", "$\\Delta (G)=[1,2]$ , and if $\\lbrace 2,5\\rbrace \\subset \\mathsf {L}(A)$ , then $\\mathsf {L}(A)=\\lbrace 2,4,5\\rbrace $ .", "$\\rho _{2k+1} (G) = 5k+2$ for all $k \\in \\mathbb {N}$ .", "If $(e,g)$ is a basis of $G$ with $\\operatorname{ord}(e)=2$ and $\\operatorname{ord}(g)=4$ , then $\\lbrace 0,g,2g,e+g,e+2g\\rbrace $ and $\\lbrace 0,g,2g,e,e-g\\rbrace $ are half-factorial sets.", "Furthermore, if $\\operatorname{supp}(A)\\subset \\lbrace e,g,2g,e+g,e+2g\\rbrace $ and $\\mathsf {v}_e(A)=1$ , then $|\\mathsf {L}(A)|=1$ .", "1.", "The first assertion follows from [20].", "Let $A \\in \\mathcal {B} (G)$ with $\\lbrace 2,5\\rbrace \\subset \\mathsf {L}(A)$ .", "Then there is an $U \\in \\mathcal {A} (G)$ of length $|U|=5$ such that $A = (-U)U$ .", "By Lemma REF there is a basis $(e,g)$ of $G$ with $\\operatorname{ord}(e)=2$ and $\\operatorname{ord}(g)=4$ such that $U = eg^3(e+g)$ .", "This implies that $\\mathsf {L}(A)=\\lbrace 2,4,5\\rbrace $ .", "2.", "See [19].", "3.", "See [20] for the first statement.", "Suppose that $\\operatorname{supp}(A)\\subset \\lbrace e,g,2g,e+g,e+2g\\rbrace $ and $\\mathsf {v}_e(A)=1$ .", "Then for every atom $W$ dividing $A$ with $eW$ , we have that $\\mathsf {k}(W)=\\frac{3}{2}$ .", "Since $\\operatorname{supp}(AW^{-1})$ is half-factorial, we obtain that $\\mathsf {L}(AW^{-1})=\\lbrace \\mathsf {k}(A)-3/2\\rbrace $ by [20] which implies that $\\mathsf {L}(A)=\\lbrace 1+\\mathsf {k}(A)-3/2\\rbrace =\\lbrace \\mathsf {k}(A)-1/2\\rbrace $ .", "Let $(e,g)$ be a basis of $G=C_2\\oplus C_4$ with $\\operatorname{ord}(e)=2$ and $\\operatorname{ord}(g)=4$ .", "We start by collecting some basic constructions that will be useful.", "Then, we show that all the sets in the result actually are sets of lengths.", "Finally, we show there are no other sets of lengths.", "Step 0.", "Some elementary constructions.", "Let $U_1=eg^3(e+g)$ , $U_2=(e+2g)(e+g)^3(-g)$ , $U_3=e(e-g)^3(-g)$ , $U_4=(-g)^2(e+g)^2$ , and $U_5=e(e+2g)g^2$ .", "Then it is not hard to check that $&\\mathsf {L}(U_1(-U_1))=\\mathsf {L}(U_2(-U_2))=\\lbrace 2,4,5\\rbrace ,&&\\nonumber \\\\&\\mathsf {L}(U_1U_3))=[2,4], \\quad &&\\mathsf {L}(U_1(-U_4))=[2,3]\\,,\\nonumber \\\\&\\mathsf {L}(U_1U_3U_4)=[3,7], \\quad &&\\mathsf {L}(U_1(-U_1)U_2(-U_2))=[4,10]\\,,\\nonumber \\\\&\\mathsf {L}(U_5^2(-g)^4)=\\lbrace 3,4,6\\rbrace , \\quad &&\\mathsf {L}(U_5(-U_5)g^4(-g)^4)=\\lbrace 4,5,6,8\\rbrace \\,,\\nonumber \\\\&\\mathsf {L}(U_1(-U_1)(e+2g)^2)=[3,6]\\,.&&$ Based on these results, we can obtain the sets of lengths of more complex zero-sum sequences.", "Let $k \\in \\mathbb {N}$ .", "Since $[2k+2,4k+5]\\supset \\mathsf {L}(U_1(-U_1)g^{4k}(-g)^{4k})\\supset \\mathsf {L}(U_1(-U_1))+\\mathsf {L}(g^{4k}(-g)^{4k})=2k+2+(\\lbrace 0\\rbrace \\cup [2,2k+3])$ and $2k+3\\notin \\mathsf {L}(U_1(-U_1)g^{4k}(-g)^{4k})$ , we obtain that $\\mathsf {L}(U_1(-U_1)g^{4k}(-g)^{4k})=2k+2+(\\lbrace 0\\rbrace \\cup [2,2k+3])\\,.$ Since $[2(k+1), 5(k+1)]\\supset \\mathsf {L}(U_1(-U_1)U_2^k(-U_2)^k)\\supset \\mathsf {L}(U_1(-U_1)U_2(-U_2))+\\mathsf {L}(U_2^{k-1}(-U_2)^{k-1})=[2(k+1), 5(k+1)]$ , we obtain that $ \\mathsf {L}(U_1(-U_1)U_2^k(-U_2)^k)=[2(k+1), 5(k+1)]\\,.\\\\$ Since $[2(k+1), 5(k+1)-1]\\supset \\mathsf {L}(U_1U_3U_2^k(-U_2)^k)\\supset \\mathsf {L}(U_1U_3)+\\mathsf {L}(U_2^{k}(-U_2)^{k})=[2(k+1), 5(k+1)-1]$ , we obtain that $ \\mathsf {L}(U_1U_3U_2^k(-U_2)^k)=[2(k+1), 5(k+1)-1]\\,.\\\\$ Since $[2(k+1), 5(k+1)-2]\\supset \\mathsf {L}(U_1(-U_4)U_2^k(-U_2)^k)\\supset \\mathsf {L}(U_1(-U_4))+\\mathsf {L}(U_2^{k}(-U_2)^{k})$ and $\\mathsf {L}(U_1(-U_4))+\\mathsf {L}(U_2^{k}(-U_2)^{k})=[2(k+1), 5(k+1)-2]$ , we obtain that $ \\mathsf {L}(U_1(-U_4)U_2^k(-U_2)^k)=[2(k+1), 5(k+1)-2]\\,.\\\\$ Since $ [2k+1, 5k+2]\\supset \\mathsf {L}(U_1U_3U_4U_2^{k-1}(-U_2)^{k-1})\\supset \\mathsf {L}(U_1U_3U_4)+\\mathsf {L}(U_2^{k-1}(-U_2)^{k-1})$ and $\\mathsf {L}(U_1U_3U_4)+\\mathsf {L}(U_2^{k-1}(-U_2)^{k-1})=[2k+1, 5k+2]$ , we obtain that $ \\mathsf {L}(U_1U_3U_4U_2^{k-1}(-U_2)^{k-1})=[2k+1, 5k+2]\\,.\\\\$ Since $[2k+1, 4k+2]\\supset \\mathsf {L}(U_5^2(-g)^4g^{4k-4}(-g)^{4k-4})\\supset \\mathsf {L}(U_5^2(-g)^4)+\\mathsf {L}(g^{4k-4}(-g)^{4k-4}),$ $\\mathsf {L}(U_5^2(-g)^4)+\\mathsf {L}(g^{4k-4}(-g)^{4k-4})=[2k+1, 4k]\\cup \\lbrace 4k+2\\rbrace $ and $4k+1\\notin \\mathsf {L}(U_5^2(-g)^4g^{4k-4}(-g)^{4k-4})$ by Lemma REF .3, we obtain that $ \\mathsf {L}(U_5^2(-g)^{4}g^{4k-4}(-g)^{4k-4})=[2k+1, 4k]\\cup \\lbrace 4k+2\\rbrace \\,.", "\\\\$ Suppose that $k \\ge 2$ .", "Since $[2k, 4k]\\supset \\mathsf {L}(U_5(-U_5)g^{4k-4}(-g)^{4k-4})\\supset \\mathsf {L}(U_5(-U_5)g^4(-g)^4)+\\mathsf {L}(g^{4k-8}(-g)^{4k-8})\\, ,$ $\\mathsf {L}(U_5(-U_5)g^4(-g)^4)+\\mathsf {L}(g^{4k-8}(-g)^{4k-8})=[2k, 4k-2]\\cup \\lbrace 4k\\rbrace $ , and $4k-1\\notin \\mathsf {L}(U_5(-U_5)g^{4k-4}(-g)^{4k-4})$ by Lemma REF .3, we obtain that $ \\mathsf {L}(U_5(-U_5)g^{4k-4}(-g)^{4k-4})=[2k, 4k-2]\\cup \\lbrace 4k\\rbrace \\,.$ Step 1.", "We prove that for every $L\\in \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5$ there exists an $A\\in \\mathcal {B}(G)$ such that $L=\\mathsf {L}(A)$ .", "We distinguish four cases.", "First we suppose that $L\\in \\mathcal {L}_2$ , and we distinguish several subcases.", "If $L=y+[3,6]$ with $y\\in \\mathbb {N}_0$ , then we set $A=0^yU_1(-U_1)(e+2g)^2$ and hence $\\mathsf {L}(A)=y+[3,6]=L$ by Equation (REF ).", "If $L=[2k+1,5k+2]$ with $k\\in \\mathbb {N}$ , then we set $A=U_1U_3U_4U_2^{k-1}(-U_2)^{k-1}$ and hence $\\mathsf {L}(A)=L$ by Equation (REF ).", "Now we assume that $L=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]$ with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}\\setminus \\lbrace 3\\rbrace $ .", "If $k\\equiv 0\\pmod {3}$ , then $k\\ge 6$ and by Equation (REF ) we infer that $\\mathsf {L}\\big (0^yU_1(-U_1)U_2^t(-U_2)^t\\big )=y+[2t+2, 5t+5]=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]=L\\,, \\text{ where $k=3t+3$}\\,.$ If $k\\equiv 1\\pmod {3}$ , then by Equation (REF ) we infer that $\\mathsf {L}\\big (0^{y}U_1(-U_4)U_2^{t}(-U_2)^{t}\\big )=y+[2t+2, 5t+3]=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]=L\\,, \\text{ where $k=3t+1$}\\,.$ If $k\\equiv 2\\pmod {3}$ , then by Equation (REF ) we infer that $\\mathsf {L}\\big (0^yU_1U_3U_2^t(-U_2)^t\\big )=y+[2t+2, 5t+4]=y+2\\lceil \\frac{k}{3}\\rceil +[0,k]=L\\,, \\text{ where $k=3t+2$}\\,.$ If $L=y+2k+2 \\cdot [0,k]\\in \\mathcal {L}_3$ with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}$ , then we set $A=0^yg^{4k}(-g)^{4k}$ and hence $\\mathsf {L}(A)=L$ .", "If $L=y+2t+2+(\\lbrace 0\\rbrace \\cup [2,2t+3])\\in \\mathcal {L}_4$ with $y,t\\in \\mathbb {N}_0$ , then we set $A=0^y U_1 (-U_1) g^{4t} (-g)^{4t}$ and obtain that $\\mathsf {L}(A)=y+2t+2+(\\lbrace 0\\rbrace \\cup [2,2t+3])=L$ by Equation (REF ).", "Finally we suppose that $L=y+k+([0,k-2]\\cup \\lbrace k\\rbrace )\\in \\mathcal {L}_5$ with $k\\ge 3$ and $y\\in \\mathbb {N}_0$ , and we distinguish two subcases.", "If $k=2t$ with $t\\ge 2$ , then we set $A=0^y U_5 (-U_5) g^{4t-4} (-g)^{4t-4}$ and hence $\\mathsf {L}(A)=y+k+([0,k-2]\\cup \\lbrace k\\rbrace )=L$ by Equation (REF ).", "If $k=2t+1$ with $t\\ge 1$ , then we set $A=0^y U_5^2 (-g)^4 g^{4t-4} (-g)^{4t-4}$ and hence $\\mathsf {L}(A)=y+k+([0,k-2]\\cup \\lbrace k\\rbrace )=L$ by Equation (REF ).", "Step 2.", "We prove that for every $A\\in \\mathcal {B}(G^{\\bullet })$ , $\\mathsf {L}(A)\\in \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5$ .", "Let $A \\in \\mathcal {B} (G^{\\bullet })$ .", "We may suppose that $\\Delta (\\mathsf {L} (A)) \\ne \\emptyset $ .", "By Lemma REF .1 we have to distinguish two cases.", "CASE 1:   $\\Delta (\\mathsf {L} (A))=\\lbrace 1\\rbrace $ .", "Then $\\mathsf {L}(A)$ is an interval, say $\\mathsf {L}(A)=[y,y+k]=y+[0,k]$ with $y\\ge 2$ and $k\\ge 1$ .", "If $k=3$ and $y=2$ , then $\\mathsf {L}(A)=[2,5]$ , a contradiction to Lemma REF .1.", "Thus $k=3$ implies that $y\\ge 3$ and hence $\\mathsf {L}(A)\\in \\mathcal {L}_2$ .", "If $k\\le 2$ , then obviously $\\mathsf {L}(A)\\in \\mathcal {L}_2$ .", "Suppose that $k\\ge 4$ .", "If $y=2t$ with $t\\ge 2$ , then $y+k\\le 5t$ and hence $y=2t\\ge 2\\lceil \\frac{k}{3}\\rceil $ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_2$ .", "Suppose that $y=2t+1$ with $t\\in \\mathbb {N}$ .", "If $y+k\\le 5t+1$ , then $y=2t+1\\ge 1+ 2\\lceil \\frac{k}{3}\\rceil $ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_2$ .", "Otherwise $y+k=5t+2$ and hence $\\mathsf {L}(A)=[2t+1,5t+2]\\in \\mathcal {L}_2$ .", "CASE 2:   $2\\in \\Delta (\\mathsf {L} (A))\\subset [1,2]$ .", "We freely use the classification of minimal zero-sum sequence given in Lemma REF .", "Since $2 \\in \\Delta (\\mathsf {L} (A))$ , there are $k \\in \\mathbb {N}$ and $U_1,\\ldots , U_k, V_1,\\ldots , V_{k+2} \\in \\mathcal {A} (G)$ with $|U_1|\\ge |U_2|\\ge \\ldots \\ge |U_k|$ such that $A=U_1\\cdot \\ldots \\cdot U_k=V_1\\cdot \\ldots \\cdot V_{k+2} \\quad \\text{and} \\quad k+1\\notin \\mathsf {L}(A) \\,,$ and we may suppose that $k$ is minimal with this property.", "Then $[\\min \\mathsf {L}(A), k]\\in \\mathsf {L}(A)$ and there exists $k_0\\in [2,k]$ such that $|U_i|\\ge 3$ for every $i\\in [1,{k_0}]$ and $|U_i|=2$ for every $i\\in [k_0+1,k]$ .", "We continue with two simple assertions.", "For each two distinct $i,j\\in [1,k_0]$ , we have that $3\\notin \\mathsf {L}(U_iU_j)$ .", "$|\\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})|\\ge 2$ .", "Proof of  A1.", "Assume to the contrary that there exist distinct $i,j\\in [1,k_0]$ such that $3\\in \\mathsf {L}(U_iU_j)$ .", "This implies that $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "$\\Box $ [Proof of A1] Proof of  A2.", "Assume to the contrary that $|\\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})|=1$ .", "Then Lemma REF .2 implies that $\\max \\mathsf {L}(A)=\\max \\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})+k-k_0=k$ , a contradiction.", "$\\Box $ [Proof of A2] We use A1 and A2 without further mention and freely use Lemma REF together with all its notation.", "We distinguish six subcases.", "CASE 2.1:   $U_1\\in S_5$ .", "Without loss of generality, we may assume that $U_1=eg^3(e+g)$ .", "We choose $j\\in [2,k_0]$ and start with some preliminary observations.", "If $|U_j|=5$ , then the fact that $3\\notin \\mathsf {L}(U_1U_j)$ implies that $U_j=-U_1$ .", "If $|U_j|=4$ , then $3\\notin \\mathsf {L}(U_1U_j)$ implies that $U_j\\in \\lbrace g^2(e+g)^2, g^4, (-g)^4, (e+g)^4\\rbrace $ .", "If $|U_j|=3$ , then $3\\notin \\mathsf {L}(U_1U_j)$ implies that $U_j\\in \\lbrace (e+2g)g(e+g), g^2(2g), (e+g)^2(2g)\\rbrace $ .", "Now we distinguish three cases.", "Suppose that $|U_2|=5$ .", "Then $U_2=-U_1$ and by symmetry we obtain that $U_j\\in \\lbrace g^4,(-g)^4\\rbrace $ for every $j\\in [3,k_0]$ .", "Let $i\\in [k_0+1, k]$ .", "If $U_i\\ne e^2$ , then $4\\in U_1U_2U_i$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Therefore we obtain that $A=U_1(-U_1)(g^4)^{k_1}((-g)^4)^{k_2}(e^2)^{k_3} \\quad \\text{ where} \\quad k_1,k_2,k_3\\in \\mathbb {N}_0\\,,$ and without loss of generality we may assume that $k_1\\ge k_2$ .", "Then it follows that $\\mathsf {L}(A)$ is equal to $k_1-k_2+k_3+\\mathsf {L}(U_1(-U_1)(g^4)^{k_2}((-g)^4)^{k_2})=k_3+k_1-k_2+2k_2+2+(\\lbrace 0\\rbrace \\cup [2,2k_2+3]),$ which is an element of $\\mathcal {L}_4$ .", "Suppose that $|U_2|=4$ and there exists $j\\in [2,k_0]$ such that $U_j=(-g)^4$ , say $j=2$ .", "Let $i\\in [3,k_0]$ .", "If $U_i\\in \\lbrace g^2(e+g)^2, g^2(2g)\\rbrace $ , then $3\\in \\mathsf {L}(U_2U_i)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "If $U_i\\in \\lbrace (e+g)^4, (e+g)^2(2g), (e+2g)g(e+g)\\rbrace $ , then $4\\in \\mathsf {L}(U_1U_2U_i)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Therefore $U_i\\in \\lbrace g^4,(-g)^4\\rbrace $ .", "Let $\\tau \\in [k_0+1,k]$ .", "If $U_{\\tau }\\in \\lbrace (e+2g)^2, (2g)^2, (e+g)(e-g)\\rbrace $ , then $4\\in \\mathsf {L}(U_1U_2U_{\\tau })$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Therefore $U_{\\tau }\\in \\lbrace e^2, g(-g)\\rbrace $ .", "Therefore we obtain that $A=U_1(g^4)^{k_1}((-g)^4)^{k_2}(g(-g))^{k_3}(e^2)^{k_4} \\quad \\text{ where} \\quad k_1,k_3,k_4\\in \\mathbb {N}_0 \\text{ and }k_2\\in \\mathbb {N}$ and hence $\\mathsf {L}(A)$ is equal to $\\mathsf {L}((g^4)^{k_1+1}((-g)^4)^{k_2}(g(-g))^{k_3}(e^2)^{k_4})=k_4+\\mathsf {L}(g^{4k_1+4+k_3}(-g)^{4k_2+k_3})\\,$ which is in $\\mathcal {L}_3$ Suppose that $|U_2|\\le 4$ and for every $j\\in [2,k_0]$ , we have $U_j\\ne (-g)^4$ .", "Then $U_j\\in \\lbrace g^2(e+g)^2, g^4, (e+g)^4, (e+2g)g(e+g), g^2(2g), (e+g)^2(2g) \\rbrace $ .", "Since $\\operatorname{supp}(U_1\\cdot \\ldots \\cdot U_{k_0})\\subset \\lbrace e,g,2g, e+g,e+2g\\rbrace $ and $\\mathsf {v}_e(U_1\\cdot \\ldots \\cdot U_{k_0})=1$ , Lemma REF .3 implies that $|\\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})|=1$ , a contradiction.", "CASE 2.2:   $U_1\\in S_4^4$ .", "Without loss of generality, we may assume that $U_1=eg(2g)(e+g)$ .", "Let $j\\in [2,k_0]$ .", "Suppose that $|U_j|=4$ .", "Since $3\\notin \\mathsf {L}(U_1U_j)$ , we obtain that $U_j\\in \\lbrace g^2(e+g)^2, g^4,(e+g)^4\\rbrace $ .", "Thus $U_1U_j=W_1W_2$ with $|W_1|=5$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.1.", "Suppose that $|U_j|=3$ .", "Since $3\\notin \\mathsf {L}(U_1U_j)$ , we obtain that $U_j\\in \\lbrace (e+2g)g(e+g), g^2(2g), (e+g)^2(2g)\\rbrace $ .", "If $U_j\\in \\lbrace g^2(2g), (e+g)^2(2g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $|W_1|=5$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.1.", "Thus it remains to consider the case where $U_j=(e+2g)g(e+g)$ .", "Therefore we have $U_1\\cdot \\ldots \\cdot U_{k_0}=U_1\\big ((e+2g)g(e+g)\\big )^{k_1} \\quad \\text{ where} \\quad k_1\\in \\mathbb {N}\\,.$ Since $\\operatorname{supp}(U_1\\cdot \\ldots \\cdot U_{k_0})\\subset \\lbrace e,g,2g, e+g,e+2g\\rbrace $ and $\\mathsf {v}_e(U_1\\cdot \\ldots \\cdot U_{k_0})=1$ , Lemma REF .3 implies that $|\\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})|=1$ , a contradiction.", "CASE 2.3:   $U_1\\in S_4^3$ and for every $i\\in [2,k_0]$ , we have $U_i\\notin S_4^4$ .", "Without loss of generality, we may assume that $U_1=eg^2(e+2g)$ .", "Let $j\\in [2,k_0]$ .", "Suppose that $|U_j|=4$ .", "Since $3\\notin \\mathsf {L}(U_1U_j)$ , we obtain that $U_j\\in \\lbrace -U_1, g^2(e+g)^2, g^2(e-g)^2,(e+g)^4, (e-g)^4, g^4\\rbrace $ .", "If $U_j\\in \\lbrace g^2(e+g)^2, g^2(e-g)^2,(e+g)^4, (e-g)^4\\rbrace $ , then $U_1U_j=W_1W_2$ with $|W_1|=5$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.1.", "Thus it remains to consider the cases where $U_j=-U_1$ or $U_j=g^4$ .", "Suppose that $|U_j|=3$ .", "Since $3\\notin \\mathsf {L}(U_1U_j)$ , we obtain that $U_j\\in \\lbrace eg(e-g), (e+2g)g(e+g), g^2(2g), (e+g)^2(2g), (e-g)^2(2g)\\rbrace $ .", "If $U_j\\in \\lbrace eg(e-g), (e+2g)g(e+g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $|W_1|=5$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.1.", "If $U_j\\in \\lbrace (e+g)^2(2g), (e-g)^2(2g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $W_1\\in S_4^4$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.2.", "Thus it remains to consider the case where $U_j=g^2(2g)$ .", "If $U_i\\ne -U_1$ for every $i\\in [2,k_0]$ , then $U_1\\cdot \\ldots \\cdot U_{k_0}=U_1(g^4)^{k_1}(g^2(2g))^{k_2}$ where $k_1,k_2\\in \\mathbb {N}_0$ .", "Since $\\operatorname{supp}(U_1\\cdot \\ldots \\cdot U_{k_0})\\subset \\lbrace e,g,2g, e+g,e+2g\\rbrace $ and $\\mathsf {v}_e(U_1\\cdot \\ldots \\cdot U_{k_0})=1$ , Lemma REF .3 implies that $|\\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})|=1$ , a contradiction.", "Thus there exists some $i\\in [2,k_0]$ , say $i=2$ , such that $U_2=-U_1$ .", "By symmetry we obtain that $k_0=2$ .", "Let $\\tau \\in [3,k]$ .", "If $U_{\\tau }\\in \\lbrace (2g)^2, (e+g)(e-g)\\rbrace $ , then $4\\in \\mathsf {L}(U_1U_2U_{\\tau })$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Therefore $A=U_1(-U_1)(e^2)^{k_1}((e+2g)^2)^{k_2}(g(-g))^{k_3}$ where $k_1,k_2,k_3\\in \\mathbb {N}_0$ .", "Since $[\\min \\mathsf {L}(A), 2+k_1+k_2+k_3]\\subset \\mathsf {L}(A)$ , we obtain that $\\mathsf {L}(A)=[\\min \\mathsf {L}(A), 2+y]\\cup \\lbrace 4+y\\rbrace $ where $y=k_1+k_2+k_3\\in \\mathbb {N}_0$ .", "For every atom $V$ dividing $A$ , we have that $|V|=2$ or $|V|=4$ .", "Thus $\\min \\mathsf {L}(A)\\ge 2+\\frac{y}{2}$ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "CASE 2.4:   $U_1\\in S_4^2$ and for every $i\\in [2,k_0]$ , we have $U_i\\notin S_4^4\\cup S_4^3$ .", "Without loss of generality, we may assume that $U_1=g^2(e+g)^2$ .", "Let $j\\in [2,k_0]$ .", "Suppose that $|U_j|=4$ .", "If $U_j\\in \\lbrace g^2(e-g)^2, (-g)^2(e+g)^2, (-g)^4, (e-g)^4\\rbrace $ , then $3\\in \\mathsf {L}(U_1U_j)$ , a contradiction.", "Thus $U_j\\in \\lbrace U_1, -U_1, g^4, (e+g)^4\\rbrace $ .", "Suppose that $|U_j|=3$ .", "If $U_j\\in \\lbrace (e+2g)(-g)(e-g), (-g)^2(2g), (e-g)^2(2g)\\rbrace $ , then $3\\in \\mathsf {L}(U_1U_j)$ , a contradiction.", "If $U_j\\in \\lbrace eg(e-g), e(-g)(e+g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $|W_1|=5$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.1.", "If $U_j=e(2g)(e+2g)$ , then $U_1U_j=\\big (e(e+g)g(2g)\\big ) \\big (g(e+g)(e+2g)\\big )$ and $e(e+g)g(e+2g)\\in S_4^4$ , going back to CASE 2.2.", "Thus it remains to consider the case where $U_j=g^2(2g)$ or $U_j= (e+g)^2(2g)$ .", "If $U_i\\ne -U_1$ for every $i\\in [2,k_0]$ , then $\\operatorname{supp}(U_1\\cdot \\ldots \\cdot U_{k_0})\\subset \\lbrace g,2g, e+g,e+2g\\rbrace $ is half-factorial by Lemma REF .3, a contradiction.", "Thus there exists some $i\\in [2,k_0]$ , say $i=2$ , such that $U_2=-U_1$ .", "By symmetry we obtain that $\\lbrace U_1, \\ldots , U_{k_0}\\rbrace =\\lbrace U_1,-U_1\\rbrace $ .", "Let $\\tau \\in [k_0+1, k]$ .", "If $U_{\\tau }\\in \\lbrace e^2, (2g)^2, (e+2g)^2\\rbrace $ , then $4\\in \\mathsf {L}(U_1U_2U_{\\tau })$ and $k+1\\in \\mathsf {L}(U_1U_2U_{\\tau })$ , a contradiction.", "Therefore $A=U_1^{k_1}(-U_1)^{k_2}(g(-g))^{k_3}((e+g)(e-g))^{k_4}$ where $k_1,k_2\\in \\mathbb {N}$ and $k_3,k_4\\in \\mathbb {N}_0$ .", "If $k_1+k_2\\ge 3$ , by symmetry we assume that $k_1\\ge 2$ , then $U_1^2(-U_1)=g^4 (-g)^2(e+g)^2 (e+g)(e-g) (e+g)(e-g)$ and hence $4\\in \\mathsf {L}(U_1^2(-U_1))$ which implies that $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Thus $k_1=k_2=1$ and hence $A=U_1(-U_1)(g(-g))^{k_3}((e+g)(e-g))^{k_4}$ where $k_3,k_4\\in \\mathbb {N}_0$ .", "Since $[\\min \\mathsf {L}(A), 2+k_3+k_4]\\in \\mathsf {L}(A)$ , we obtain that $\\mathsf {L}(A)=[\\min \\mathsf {L}(A), 2+y]\\cup \\lbrace 4+y\\rbrace $ where $y=k_3+k_4\\in \\mathbb {N}_0$ .", "For every atom $V$ dividing $A$ , we have that $|V|=2$ or $|V|=4$ .", "Thus $\\min \\mathsf {L}(A)\\ge 2+\\frac{y}{2}$ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "CASE 2.5:   $U_1\\in S_4^1$ and for every $i\\in [2,k_0]$ , we have $U_i\\notin S_4^4\\cup S_4^3\\cup S_4^2$ .", "Without loss of generality, we may assume that $U_1=g^4$ .", "Let $j\\in [2,k_0]$ .", "Suppose that $|U_j|=4$ .", "If $U_j\\in \\lbrace (e+g)^4, (e-g)^4\\rbrace $ , then $U_1U_j=W_1W_2$ with $W_1\\in S_4^2$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.4.", "Thus it remains to consider the case where $U_j=U_1$ or $U_j=-U_1$ .", "Suppose that $|U_j|=3$ .", "If $U_j\\in \\lbrace (-g)^2(2g)\\rbrace $ , then $3\\in \\mathsf {L}(U_1U_j)$ , a contradiction.", "If $U_j\\in \\lbrace e(-g)(e+g), (e+2g)(-g)(e-g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $|W_1|=5$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.1.", "If $U_j\\in \\lbrace (e+g)^2(2g), (e-g)^2(2g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $W_1\\in S_4^2$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.4.", "If $U_j=e(2g)(e+2g)$ , then $U_1U_j=W_1W_2$ with $W_1\\in S_4^3$ , where $W_1,W_2$ are atoms and hence we are back to CASE 2.3.", "Thus it remains to consider the case where $U_j=g^2(2g)$ , or $U_j=eg(e-g)$ , or $U_j=(e+2g)g(e+g)$ .", "First, suppose that $U_i\\ne -U_1$ for every $i\\in [2,k_0]$ .", "Then $U_1\\cdot \\ldots \\cdot U_{k_0}=U_1^{k_1}(eg(e-g))^{k_2}((e+2g)g(e+g))^{k_3}(g^2(2g))^{k_4}\\,,$ where $ k_1\\in \\mathbb {N}$ and $k_2,k_3, k_4\\in \\mathbb {N}_0$ .", "If $k_2\\ge 1$ and $k_3\\ge 1$ , then $eg(e-g) (e+2g)g(e+g)=eg^2(e+2g) (e+g)(e-g)$ , $eg^2(e+2g)\\in S_4^3$ and hence we are back to CASE 2.3.", "Thus we may assume that $k_2=0$ or $k_3=0$ .", "Since $\\lbrace g,2g, e+g, e+2g\\rbrace $ and $\\lbrace g, 2g, e, e-g\\rbrace $ are both half-factorial by Lemma REF .3, we obtain that $|\\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})|=1$ , a contradiction.", "Second, suppose that there exists some $i\\in [2,k_0]$ , say $i=2$ , such that $U_2=-U_1$ .", "By symmetry we obtain that $\\lbrace U_1,\\ldots , U_{k_0}\\rbrace =\\lbrace U_1,-U_1\\rbrace $ .", "Since $4\\in \\mathsf {L}(U_1\\cdot U_2\\cdot (2g)^2)$ , $5\\in \\mathsf {L}(U_1 U_2 e^2 (e-g)(e+g))$ , and $5\\in \\mathsf {L}(U_1 U_2 (e+2g)^2 (e-g)(e+g))$ , we obtain that $\\lbrace U_{k_0+1}, \\ldots ,U_k\\rbrace \\subset \\lbrace (e+g)(e-g), g(-g)\\rbrace \\quad \\text{ or} $ $ \\lbrace U_{k_0+1}, \\ldots ,U_k\\rbrace \\subset \\lbrace e^2, (e+2g)^2, g(-g)\\rbrace \\,.$ This implies that $A=(g^4)^{k_1}((-g)^4)^{k_2}((e+g)(e-g))^{k_3}(g(-g))^{k_4} \\quad \\text{ or}$ $ A=(g^4)^{k_1}((-g)^4)^{k_2}(e^2)^{k_3}((e+2g)^2)^{k_4}(g(-g))^{k_5} \\,,$ where $k_1,k_2\\in \\mathbb {N}$ and $k_3,k_4,k_5\\in \\mathbb {N}_0$ .", "Suppose that $A=(g^4)^{k_1}((-g)^4)^{k_2}((e+g)(e-g))^{k_3}(g(-g))^{k_4}$ , where $k_1,k_2\\in \\mathbb {N}$ and $k_3,k_4,k_5\\in \\mathbb {N}_0$ .", "If $k_1\\ge 2$ and $k_3\\ge 2$ , then $g^4\\ g^4 (-g)^4 (e+g)(e-g) (e+g)(e-g)=\\big (g(-g)\\big )^4 g^2(e+g)^2 g^2(e-g)^2$ and hence $6\\in \\mathsf {L}(g^4 g^4 (-g)^4 (e+g)(e-g) (e+g)(e-g))$ .", "Thus $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Therefore by symmetry $k_3=1$ or $k_1=k_2=1$ .", "If $k_3=1$ , then $\\mathsf {L}(A)=1+\\mathsf {L}((g^4)^{k_1}((-g)^4)^{k_2}(g(-g))^{k_4})\\in \\mathcal {L}_3$ .", "If $k_1=k_2=1$ , then $\\mathsf {L}(A)=[\\min \\mathsf {L}(A), 2+y]\\cup \\lbrace 4+y\\rbrace $ where $y=k_3+k_4\\in \\mathbb {N}_0$ .", "For every atom $V$ dividing $A$ , we have that $|V|=2$ or $|V|=4$ .", "Thus $\\min \\mathsf {L}(A)\\ge 2+\\frac{y}{2}$ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "Suppose that $A=(g^4)^{k_1}((-g)^4)^{k_2}(e^2)^{k_3}((e+2g)^2)^{k_4}(g(-g))^{k_5}$ , where $k_1,k_2\\in \\mathbb {N}$ and $k_3,k_4,k_5\\in \\mathbb {N}_0$ .", "If $k_1\\ge 2$ , $k_3\\ge 1$ , and $k_4\\ge 1$ , then $g^4 g^4 (-g)^4 e^2 (e+2g)^2=\\big (g(-g)\\big )^4 \\big (e(e+2g)g^2\\big )^2$ and hence $6\\in \\mathsf {L}(g^4 g^4 (-g)^4 e^2 (e+2g)^2)$ .", "Thus $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Therefore by symmetry $k_3=0$ , or $k_4=0$ , or $k_1=k_2=1$ .", "If $k_3=0$ or $k_4=0$ , then $\\mathsf {L}(A)=k_3+k_4+\\mathsf {L}((g^4)^{k_1}((-g)^4)^{k_2}(g(-g))^{k_5})\\in \\mathcal {L}_3$ .", "If $k_1=k_2=1$ , then $\\mathsf {L}(A)=[\\min \\mathsf {L}(A), 2+y]\\cup \\lbrace 4+y\\rbrace $ where $y=k_3+k_4+k_5\\in \\mathbb {N}_0$ .", "For every atom $V$ dividing $A$ , we have that $|V|=2$ or 4.", "Thus $\\min \\mathsf {L}(A)\\ge 2+\\frac{y}{2}$ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "CASE 2.6:   $|U_1|=3$ .", "Let $j\\in [2,k_0]$ .", "We distinguish three subcases.", "First, we suppose that $U_1\\in S_3^3$ , and without restriction we may assume that $U_1=eg(e-g)$ .", "If $U_j=-U_1$ , then $3\\in \\mathsf {L}(U_1U_j)$ , a contradiction.", "If $U_j\\in \\lbrace (-g)^2(2g), (e+g)^2(2g), e(2g)(e+2g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $W_1 \\in S_4^4$ where $W_1,W_2$ are atoms and hence we are back to CASE 2.2.", "If $U_j\\in \\lbrace (e+2g)g(e+g), (e+2g)(-g)(e-g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $W_1 \\in S_4^3$ where $W_1,W_2$ are atoms and hence we are back to CASE 2.3.", "If $U_j=U_1$ , then $U_1U_j=W_1W_2$ with $W_1 \\in S_4^2$ where $W_1,W_2$ are atoms and hence we are back to CASE 2.4.", "Thus it remains to consider the case where $U_j=g^2(2g)$ or $(e-g)^2(2g)$ .", "Then $U_1\\cdot \\ldots \\cdot U_{k_0}=U_1(g^2(2g))^{k_1}((e-g)^2(2g))^{k_2}$ where $k_1,k_2\\in \\mathbb {N}_0$ .", "Since $\\lbrace e,g,2g,e-g\\rbrace $ is half-factorial by Lemma REF .3, we obtain that $|\\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})|=1$ , a contradiction.", "Second, we suppose that $U_1\\in S_3^2$ , and without restriction we may assume that $U_1=g^2(2g)$ and $U_j\\notin S_3^3$ .", "If $U_j=-U_1$ , then $3\\in \\mathsf {L}(U_1U_j)$ .", "If $U_j=U_1$ , then $U_1U_j=W_1W_2$ with $W_1 \\in S_4^1$ where $W_1,W_2$ are atoms and hence we are back to CASE 2.5.", "If $U_j\\in \\lbrace (e+g)^2(2g), (e-g)^2(2g)\\rbrace $ , then $U_1U_j=W_1W_2$ with $W_1 \\in S_4^2$ where $W_1,W_2$ are atoms and hence we are back to CASE 2.4.", "If $U_j=e(2g)(e+2g)$ , then $U_1U_j=W_1W_2$ with $W_1 \\in S_4^3$ where $W_1,W_2$ are atoms and hence we are back to CASE 2.3.", "Third, we suppose that $U_1\\in S_3^1$ , and without restriction we assume that $U_j\\in S_3^1$ .", "Thus $3\\in \\mathsf {L}(U_1U_j)$ , a contradiction." ], [ "The system of sets of lengths of $C_2^4$", "Now we give a complete description of the system of sets of lengths of $C_2^4$ .", "Theorem 4.8 $\\quad \\mathcal {L} (C_2^4) = \\mathcal {L}_1\\cup \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5\\cup \\mathcal {L}_6\\cup \\mathcal {L}_7\\cup \\mathcal {L}_8$ , where $\\mathcal {L}_1&=\\big \\lbrace \\lbrace y\\rbrace \\mid y \\in \\mathbb {N}_0\\big \\rbrace , \\\\\\mathcal {L}_2&= \\big \\lbrace y+2k+3 \\cdot [0,k] \\mid y, k \\in \\mathbb {N}_0 \\big \\rbrace , \\\\\\mathcal {L}_3&= \\big \\lbrace y+[2,3], y+[2,4], y+[3,6], y+[3,7],y+[4,9] \\mid y \\in \\mathbb {N}_0\\big \\rbrace \\cup \\\\& \\quad \\ \\big \\lbrace y+[m,m+k] \\mid y \\in \\mathbb {N}_0, k \\ge 6, m \\ \\text{\\rm minimal with} \\ m+k\\le 5m/2 \\big \\rbrace \\\\&=\\big \\lbrace y+\\left\\lceil \\frac{2k}{3}\\right\\rceil +[0,k]\\mid y\\in \\mathbb {N}_0, k \\in \\mathbb {N}\\setminus \\lbrace 1,3\\rbrace \\big \\rbrace \\cup \\\\& \\quad \\ \\lbrace y+3 + [0,3], y+2+[0,1]\\mid y\\in \\mathbb {N}_0\\rbrace , \\\\\\mathcal {L}_4&=\\bigl \\lbrace y + 2k+ 2 \\cdot [0, k] \\, \\bigm |\\, y ,\\, k \\in \\mathbb {N}_0 \\bigr \\rbrace , \\\\\\mathcal {L}_5&= \\lbrace y+k+2+([0,k]\\cup \\lbrace k+2\\rbrace )\\mid y\\in \\mathbb {N}_0, k\\in \\mathbb {N}\\rbrace , \\\\\\mathcal {L}_6&=\\lbrace y+2\\left\\lceil \\frac{k}{3}\\right\\rceil +2+(\\lbrace 0\\rbrace \\cup [2,k+2])\\mid y\\in \\mathbb {N}_0, k\\ge 5 \\text{ or }k=3\\rbrace , \\\\\\mathcal {L}_7&=\\lbrace y+2k+3+\\lbrace 0,1,3\\rbrace +3 \\cdot [0,k]\\mid y, k\\in \\mathbb {N}_0\\rbrace \\ \\cup \\\\& \\quad \\ \\lbrace y+2k+4+\\lbrace 0,1,3\\rbrace +3 \\cdot [0,k]\\cup \\lbrace y+5k+8\\rbrace \\mid y, k\\in \\mathbb {N}_0\\rbrace , \\text{ and } \\\\\\mathcal {L}_8&=\\lbrace y+2k+3+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]\\mid y, k\\in \\mathbb {N}_0\\rbrace \\ \\cup \\\\& \\quad \\ \\lbrace y+2k+4+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]\\cup \\lbrace y+5k+9\\rbrace \\mid y, k\\in \\mathbb {N}_0\\rbrace .", "\\\\ $ We note that the system of sets of lengths of $C_2^4$ is richer than that of the other groups we considered.", "A reason for this is that the set $\\Delta ^{\\ast }(C_2^4)$ is largest, namely $\\lbrace 1,2,3\\rbrace $ (this fact was also crucial in the proof of Theorem REF ).", "We recall some useful facts in the lemma below.", "Lemma 4.9 Let $G=C_2^4$ , and let $A \\in \\mathcal {B} (G)$ .", "$\\Delta (G) = [1,3]$ , and if $3 \\in \\Delta ( \\mathsf {L} (A))$ , then $\\Delta ( \\mathsf {L} (A)) = \\lbrace 3\\rbrace $ and there is a basis $(e_1, \\ldots , e_4)$ of $G$ such that $\\operatorname{supp}(A) \\setminus \\lbrace 0\\rbrace = \\lbrace e_1, \\ldots , e_4, e_1+ \\ldots +e_4\\rbrace $ .", "$\\rho _{2k+1} (G) = 5k+2$ for all $k \\in \\mathbb {N}$ .", "1.", "The first statement follows from [20], and the second statement from [25].", "2.", "See [20].", "In the following result we characterize which intervals are sets of lengths for $C_2^4$ .", "It turns out that, with a single exception, the sole restriction is the one implied by elasticity.", "Proposition 4.10 Let $G=C_2^4$ and let $2 \\le l_1 \\le l_2$ be integers.", "Then $[l_1,l_2] \\in \\mathcal {L} (G)$ if and only if $l_2/l_1 \\le 5/2$ and $(l_1, l_2 ) \\ne (2,5)$ .", "Suppose that $[l_1,l_2] \\in \\mathcal {L} (G)$ with integers $2 \\le l_1 \\le l_2$ .", "Then (REF ) implies that $l_2/l_1 \\le \\rho (G) = 5/2 $ .", "Moreover, $[2,5]=[2, \\mathsf {D} (G)] \\notin \\mathcal {L} (G)$ by [20].", "Conversely, we need to show that for integers $2 \\le l_1 \\le l_2$ with $(l_1, l_2) \\ne (2,5)$ and $l_2/l_1 \\le 5/2$ , we have $[l_1,l_2] \\in \\mathcal {L} (G)$ .", "We start with an observation that reduces the problem to constructing these sets of intervals for extremal choices of the endpoints.", "Let $k \\in \\mathbb {N}$ .", "If $m \\in \\mathbb {N}$ such that $[m,m+k] \\in \\mathcal {L} (G)$ , then $y + [m,m+k] \\in \\mathcal {L} (G)$ for all $y \\in \\mathbb {N}_0$ .", "Thus let $m_k=\\max \\lbrace 2,\\lceil \\frac{2k}{3}\\rceil \\rbrace $ if $k\\in \\mathbb {N}\\setminus \\lbrace 3\\rbrace $ and $m_3=3$ .", "Therefore we only need to prove that $[m_k,m_k+k] \\in \\mathcal {L} (G)$ .", "For $k\\in [1,5]$ we are going to realize sets $[m_k,m_k+k]$ as sets of lengths.", "Then we handle the case $k \\ge 6$ .", "If $k \\in \\lbrace 1,3\\rbrace $ , then the sets $[2,3], [3,6] \\in \\mathcal {L} (C_2^3) \\subset \\mathcal {L} (G)$ .", "To handle the case $k=2$ , we have to show that $[2,4] \\in \\mathcal {L} (G)$ .", "Let $(e_1,\\ldots , e_4)$ be a basis of $G$ and $e_0=e_1+\\ldots +e_4$ .", "If $U_1= e_0 \\cdot \\ldots \\cdot e_4 \\quad \\text{and} \\quad U_2 = e_1e_2(e_1+e_3)(e_2+e_4)(e_3+e_4),$ then $\\max \\mathsf {L} (U_1U_2) < 5$ , and $\\begin{aligned}U_1 U_2 & = \\Big (e_0e_1e_2(e_3+e_4)\\Big ) \\Big ((e_1+e_3)e_1e_3\\Big ) \\Big ((e_2+e_4)e_2e_4\\Big ) \\\\& = \\Big ( e_0(e_1+e_3)(e_2+e_4)\\Big ) \\Big ( e_1^2\\Big ) \\Big (e_2^2\\Big ) \\Big ( (e_3+e_4)e_3e_4\\Big ) \\,,\\end{aligned}$ shows that $\\mathsf {L} (U_1U_2) = [2,4]$ .", "It remains to verify the following assertions.", "$[3,7] \\in \\mathcal {L} (G)$ (this settles the case $k=4$ ).", "$[4,9] \\in \\mathcal {L} (G)$ (this settles the case $k=5$ ).", "Let $k \\ge 6$ .", "Then $[\\lceil \\frac{2k}{3}\\rceil ,\\ \\lceil \\frac{2k}{3}\\rceil +k] \\in \\mathcal {L} (G)$ .", "Proof of  A1.", "Clearly, $\\begin{aligned}U_1 & = e_0 \\cdot \\ldots \\cdot e_4, \\ U_2 = e_1e_2(e_1+e_3)(e_2+e_4)(e_3+e_4), \\quad \\text{and} \\\\U_3 & = (e_1+e_3)(e_2+e_4)e_3e_4(e_1+e_2)\\end{aligned}$ are minimal zero-sum sequences of lengths 5.", "Since $\\begin{aligned}U_1U_2U_3 & = \\Big ( e_0 (e_1+e_2)(e_3+e_4) \\Big ) \\Big ( e_1^2 \\Big ) \\Big ( e_2^2 \\Big ) \\Big ( e_3^2 \\Big )\\Big ( e_4^2 \\Big ) \\Big ( (e_1+e_3)^2\\Big ) \\Big ( (e_2+e_4)^2 \\Big ) \\\\& = \\Big ( e_0(e_1+e_2)(e_3+e_4) \\Big ) \\Big ( (e_1+e_3)e_1e_3 \\Big )^2 \\Big ( (e_2+e_4)^2 \\Big ) \\Big ( e_2^2 \\Big ) \\Big ( e_4^2 \\Big ) \\\\& = \\Big ( e_0 (e_1+e_2)(e_3+e_4) \\Big ) \\Big ( (e_1+e_3)e_1e_3 \\Big )^2 \\Big ( (e_2+e_4)e_2e_4 \\Big )^2 \\\\& = U_2 \\Big ( e_0(e_1+e_2)(e_1+e_3)e_1e_4 \\Big ) \\Big ( (e_2+e_4)e_2e_4 \\Big ) \\Big ( e_3^2 \\Big ) \\,,\\end{aligned}$ it follows that $\\mathsf {L} (U_1U_2U_3) = [3,7]$ .", "Proof of  A2.", "We use the same notation as in A1, set $U_4 = (e_1+e_2)(e_1+e_3)(e_2+e_4)(e_3+e_4)$ , and assert that $\\mathsf {L} (U_1^2 U_2 U_4) = [4,9]$ .", "Clearly, $4 \\in \\mathsf {L} (U_1^2 U_2 U_4)$ and $\\max \\mathsf {L} (U_1^2 U_2 U_4) < 10$ .", "Since $\\begin{aligned}U_1^2U_2U_4 & = \\Big (e_0e_1e_2(e_3+e_4)\\Big ) \\Big ((e_1+e_3)e_1e_3\\Big ) \\Big ((e_2+e_4)e_2e_4\\Big ) U_1U_4 \\\\& = \\Big (e_0(e_1+e_3)(e_2+e_4)\\Big ) \\Big ( e_1^2 \\Big ) \\Big ( e_2^2 \\Big ) \\Big ((e_3+e_4)e_3e_4\\Big ) U_1U_4 \\\\& = \\prod _{\\nu =0}^4 \\big ( e_{\\nu }^2 \\big ) U_2 U_4 \\\\& = \\Big ( (e_1+e_3)^2\\Big ) \\Big ( (e_2+e_4)^2\\Big ) \\Big ( (e_3+e_4)e_3e_4 \\Big )^2 \\Big (e_0^2\\Big ) \\Big ( e_1^2\\Big ) \\Big (e_2^2\\Big ) \\Big ( (e_1+e_2)e_1e_2 \\Big ) \\\\& = \\Big ( (e_1+e_3)^2\\Big ) \\Big ( (e_2+e_4)^2\\Big ) \\Big ( (e_3+e_4)^2 \\Big ) \\Big ( e_3^2 \\Big ) \\Big ( e_4^2 \\Big ) \\Big (e_0^2\\Big ) \\Big ( e_1^2\\Big ) \\Big (e_2^2\\Big ) \\Big ( (e_1+e_2)e_1e_2 \\Big )\\,,\\end{aligned}$ the assertion follows.", "Proof of  A3.", "We proceed by induction on $k$ .", "For $k=6$ , we have to verify that $[4,10] \\in \\mathcal {L} (G)$ .", "We use the same notation as in A1, and assert that $\\mathsf {L} (U_1^2 U_2^2) = [4,10]$ .", "Clearly, $\\lbrace 4, 10\\rbrace \\subset \\mathsf {L} (U_1^2 U_2^2) \\subset [4,10]$ .", "Since $\\begin{aligned}U_1^2 U_2^2 & = \\Big (e_0e_1e_2(e_3+e_4)\\Big ) \\Big ((e_1+e_3)e_1e_3\\Big ) \\Big ((e_2+e_4)e_2e_4\\Big ) U_1U_2 \\\\& = \\Big (e_0e_1e_2(e_3+e_4)\\Big )^2 \\Big ((e_1+e_3)e_1e_3\\Big )^2 \\Big ((e_2+e_4)e_2e_4\\Big )^2 \\\\& = \\prod _{\\nu =0}^4 \\big ( e_{\\nu }^2 \\big ) U_2^2 \\\\& = \\Big ( e_0(e_1+e_3)(e_2+e_4)\\Big )^2 \\Big ( e_1^2\\Big )^2 \\Big (e_2^2\\Big )^2 \\Big ( (e_3+e_4)e_3e_4\\Big )^2 \\\\& = \\Big ( (e_1+e_3)^2\\Big ) \\Big ( (e_2+e_4)^2\\Big ) \\Big ( (e_3+e_4)e_3e_4 \\Big )^2 \\Big (e_0^2\\Big ) \\Big ( e_1^2\\Big )^2 \\Big (e_2^2\\Big )^2\\end{aligned}$ it follows that $[5,9] \\subset \\mathsf {L} (U_1^2 U_2^2)$ , and hence $\\mathsf {L} (U_1^2 U_2^2) = [4,10]$ .", "If $k=7$ , then $[5,12]\\supset \\mathsf {L}(U_1^3U_2U_3)\\supset \\mathsf {L}(U_1U_2U_3)+\\mathsf {L}(U_1^2) = [3,7]+\\lbrace 2,5\\rbrace =[5,12]$ which implies that $[5,12] \\in \\mathcal {L} (G)$ .", "If $k=8$ , then $[6,14]\\supset \\mathsf {L}(U_1^4U_2U_4)\\supset \\mathsf {L}(U_1^2U_2U_4)+\\mathsf {L}(U_1^2) = [4,9]+\\lbrace 2,5\\rbrace =[6,14]$ which implies that $[6,14] \\in \\mathcal {L} (G)$ .", "Suppose that $k \\ge 9$ , and that the assertion holds for all $k^{\\prime } \\in [6, k-1]$ .", "Then the set $[\\lceil \\frac{2(k-3)}{3}\\rceil , \\lceil \\frac{2(k-3)}{3}\\rceil +k-3] \\in \\mathcal {L} (G)$ .", "This implies that $[\\lceil \\frac{2k}{3}\\rceil , \\lceil \\frac{2k}{3}\\rceil +k]=[\\lceil \\frac{2(k-3)}{3}\\rceil , \\lceil \\frac{2(k-3)}{3}\\rceil +k-3]+\\lbrace 2,5\\rbrace \\in \\mathcal {L} (G)$ .", "We now proceed to prove Theorem REF .", "Let $(e_1, e_2,e_3,e_4)$ be a basis of $G=C_2^4$ .", "We set $e_0 = e_1+e_2+e_3+e_4$ , $U =e_0e_1e_2e_3e_4$ , and $V=e_1e_2e_3(e_1+e_2+e_3)$ .", "Step 0.", "Some elementary constructions.", "Let $t_1\\ge 2$ , $t_2\\ge 2$ , $t=t_1+t_2$ , and $L_{t_1,t_2}=\\left\\lbrace \\begin{aligned}&\\lbrace t\\rbrace \\cup [t+2, 5\\lfloor t_1/2\\rfloor +4(t/2-\\lfloor t_1/2\\rfloor )] &&\\text{ if $t$ is even}\\,,\\\\&\\lbrace t\\rbrace \\cup [t+2, 5\\lfloor t_1/2\\rfloor +4((t-1)/2-\\lfloor t_1/2\\rfloor )+1] &&\\text{ if $t$ is odd}\\,.\\end{aligned}\\right.$ Since $\\mathsf {L}(U^2V^2)=\\lbrace 4\\rbrace \\cup [6,9]$ , we have that $\\mathsf {L}(U^{t_1}V^{t_2})\\supset \\mathsf {L}(U^2V^2)+\\mathsf {L}(U^{t_1-2}V^{t_2-2})=L_{t_1,t_2}$ .", "Note that for every atom $W$ dividing $U^{t_1}V^{t_2}$ , we have $W=\\left\\lbrace \\begin{aligned}&U &\\text{ if } |W|=5\\,,\\\\&V &\\text{ if } |W|=4\\,,\\\\&e_0e_4(e_1+e_2+e_3) &\\text{ if } |W|=3\\,.\\end{aligned}\\right.$ Assume to the contrary that $t+1\\in \\mathsf {L}(U^{t_1}V^{t_2})$ .", "Then there exist $t_3,t_4,t_5\\in \\mathbb {N}_0$ and atoms $W_1,\\ldots , W_{t_3+t_4+1}$ such that $U^{t_3}V^{t_4}=W_1\\ldots W_{t_3+t_4+1}$ with $t_3+t_4\\ge 2$ , $t_5\\le \\min \\lbrace t_3,t_4\\rbrace $ , $|W_i|=3$ for $i\\in [1, t_5]$ , and $|W_i|=2$ for $i\\in [t_5+1, t_3+t_4+1]$ .", "It follows that $5t_3+4t_4=3t_5+2(t_3+t_4+1-t_5)\\le 3t_3+2t_4+2$ and hence $t_3+t_4\\le 1$ , a contradiction.", "Therefore $t+1\\notin \\mathsf {L}(U^{t_1}V^{t_2})$ and $\\mathsf {L}(U^{t_1}V^{t_2})=L_{t_1,t_2}.$ Note that for every atom $W$ dividing $U^rV$ with $r\\ge 2$ and $e_1+e_2+e_3W$ , we have $W=V$ or $W=e_0e_4(e_1+e_2+e_3)$ .", "It follows that for all $r \\ge 2$ $&\\mathsf {L}(U^rV)\\nonumber \\\\=&\\big (1+\\mathsf {L}(U^r)\\big )\\cup \\big (1+\\mathsf {L}(e_1^2e_2^2e_3^2U^{r-1})\\big )\\\\ \\nonumber =&\\left\\lbrace \\begin{aligned}&r+1+\\lbrace 0,2,3\\rbrace +3 \\cdot [0, r/2-1], &&\\text{ if $r$ is even}\\,,\\\\&r+1+\\lbrace 0,2,3\\rbrace +3 \\cdot [0, (r-1)/2-1]\\cup \\lbrace r+1+(3r-3)/2+2\\rbrace , &&\\text{ if $r$ is odd}\\,.\\end{aligned}\\right.$ Note that for every atom $W$ dividing $U^rVe_4^2e_0^2$ with $r\\ge 2$ and $e_1+e_2+e_3W$ , we have $W=V$ or $W=e_0e_4(e_1+e_2+e_3)$ .", "It follows that for all $r \\ge 2$ $&\\mathsf {L}(U^rVe_4^2e_0^2)\\nonumber \\\\=&\\big (1+\\mathsf {L}(U^re_4^2e_0^2)\\big )\\cup \\big (1+\\mathsf {L}(U^{r+1})\\big )\\\\ \\nonumber =&\\left\\lbrace \\begin{aligned}&r+2+\\lbrace 0,1,3\\rbrace +3 \\cdot [0, (r+1)/2-1], &&\\text{ if $r$ is odd}\\,,\\\\ \\nonumber &r+2+\\lbrace 0,1,3\\rbrace +3 \\cdot [0, r/2-1]\\cup \\lbrace r+2+3r/2+1\\rbrace , &&\\text{ if $r$ is even}\\,.\\end{aligned}\\right.$ .", "Step 1.", "We prove that for every $L\\in \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5\\cup \\mathcal {L}_6 \\cup \\mathcal {L}_7 \\cup \\mathcal {L}_8$ , there exists an $A\\in \\mathcal {B}(G)$ such that $L=\\mathsf {L}(A)$ .", "We distinguish seven cases.", "If $L=y+2k+3\\cdot [0,k]\\in \\mathcal {L}_2$ with $y, k \\in \\mathbb {N}_0$ , then $L = \\mathsf {L} ( 0^y U^{2k} ) \\in \\mathcal {L} (G)$ .", "If $L \\in \\mathcal {L}_3$ , then the claim follows from Proposition REF .", "If $L= y + 2k + 2 \\cdot [0, k] \\in \\mathcal {L}_4$ with $y, k \\in \\mathbb {N}_0$ , then Proposition REF .4 implies that $L \\in \\mathcal {L} (C_2^3) \\subset \\mathcal {L} (G)$ .", "Suppose that $L=y+k+2+([0,k]\\cup \\lbrace k+2\\rbrace )\\in \\mathcal {L}_5$ with $k\\in \\mathbb {N}$ and $y\\in \\mathbb {N}_0$ .", "Note that $\\mathsf {L}(V^2 (e_1+e_4)^{2}(e_2+e_4)^2(e_3+e_4)^2(e_1+e_2+e_3+e_4)^2)=[4,6]\\cup \\lbrace 8\\rbrace $ .", "If $k$ is even, then we set $A=0^y V^2 (e_1+e_4)^{k}(e_2+e_4)^k(e_3+e_4)^k(e_1+e_2+e_3+e_4)^k$ and obtain that $\\mathsf {L}(A)=L$ by Lemma REF .3.", "If $k$ is odd, then we set $A=0^y V^2 (e_1+e_4)^{k+1}(e_2+e_4)^{k+1}(e_3+e_4)^{k-1}(e_1+e_2+e_3+e_4)^{k-1}$ and obtain that $\\mathsf {L}(A)=L$ by Lemma REF .3.", "Suppose that $L=y+2\\lceil \\frac{k}{3}\\rceil +2+(\\lbrace 0\\rbrace \\cup [2,k+2])\\in \\mathcal {L}_6$ with $\\big (k\\ge 5$ or $k=3\\big )$ and $y\\in \\mathbb {N}_0$ .", "If $k\\equiv 0\\mod {3}$ , then we set $A=0^y U^{2k/3} V^2$ and hence $\\mathsf {L}(A)=L$ by Equation (REF ).", "If $k\\equiv 2\\mod {3}$ , then we set $A=0^y U^{(2k-4)/3} V^4$ and hence $\\mathsf {L}(A)=L$ by (REF ).", "If $k\\equiv 1\\mod {3}$ , then we set $A=0^y U^{(2k-8)/3} V^6$ and obtain that $\\mathsf {L}(A)=L$ by Equation (REF ).", "Suppose that $L\\in \\mathcal {L}_7$ .", "If $L=y+2k+3+\\lbrace 0,1,3\\rbrace +3 \\cdot [0,k]$ with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}_0$ , then we set $A=0^y U^{2k+1} V e_4^2 (e_1+e_2+e_3+e_4)^2$ and obtain that $\\mathsf {L}(A)=L$ by Equation (REF ).", "If $L=y+2k+4+\\lbrace 0,1,3\\rbrace +3 \\cdot [0,k]\\cup \\lbrace y+5k+8\\rbrace $ with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}_0$ , then we set $A=0^y U^{2k+2} V e_4^2 (e_1+e_2+e_3+e_4)^2$ and obtain that $\\mathsf {L}(A)=L$ by Equation (REF ).", "Suppose that $L\\in \\mathcal {L}_8$ .", "If $L=y+2k+3+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]$ with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}_0$ , then we set $A=0^y U^{2k+2} V$ and hence $\\mathsf {L}(A)=L$ by Equation (REF ).", "If $L=y+2k+4+\\lbrace 0,2,3\\rbrace +3 \\cdot [0,k]\\cup \\lbrace y+5k+9\\rbrace $ with $y\\in \\mathbb {N}_0$ and $k\\in \\mathbb {N}_0$ , then we set $A=0^y U^{2k+3} V e_4^2 (e_1+e_2+e_3+e_4)^2$ and obtain that $\\mathsf {L}(A)=L$ by Equation (REF ).", "Step 2.", "We prove that for every $A\\in \\mathcal {B}(G^{\\bullet })$ , $\\mathsf {L}(A)\\in \\mathcal {L}_2\\cup \\mathcal {L}_3\\cup \\mathcal {L}_4\\cup \\mathcal {L}_5\\cup \\mathcal {L}_6 \\cup \\mathcal {L}_7 \\cup \\mathcal {L}_8$ .", "Let $A \\in \\mathcal {B} (G^{\\bullet })$ .", "We may suppose that $\\Delta (\\mathsf {L} (A)) \\ne \\emptyset $ .", "By Lemma REF .1 we have to distinguish four cases.", "CASE 1:   $\\Delta ( \\mathsf {L} (A) ) = \\lbrace 3\\rbrace $ .", "By Lemma REF , there is a basis of $G$ , say $(e_1, e_2, e_3, e_4)$ , such that $\\operatorname{supp}(A) = \\lbrace e_1, \\ldots , e_4, e_0\\rbrace $ .", "Let $n \\in \\mathbb {N}_0$ be maximal such that $U^{2n} A$ .", "Then there exist a proper subset $I \\subset [0,4]$ , a tuple $(m_i)_{i \\in I} \\in \\mathbb {N}_0^{(I)}$ , and $\\epsilon \\in \\lbrace 0,1\\rbrace $ such that $A = U^{\\epsilon } U^{2n} \\prod _{i \\in I} (e_i^2)^{m_i} \\,.$ Using [25], we infer that $\\mathsf {L} (A) = \\epsilon + \\sum _{i \\in I} m_i + \\mathsf {L} (U^{2n}) = \\epsilon + \\sum _{i \\in I} m_i + (2n + 3 \\cdot [0, n])\\in \\mathcal {L}_2 \\,.$ CASE 2:   $\\Delta ( \\mathsf {L} (A)) = \\lbrace 1\\rbrace $ .", "Then $\\mathsf {L} (A)$ is an interval, and it is a direct consequence of Proposition REF that $\\mathsf {L} (A) \\in \\mathcal {L}_3$ .", "CASE 3:   $\\Delta ( \\mathsf {L}(A)) = \\lbrace 2\\rbrace $ .", "The following reformulation turns out to be convenient.", "Clearly, we have to show that for every $L \\in \\mathcal {L}(G)$ with $\\Delta (L) = \\lbrace 2\\rbrace $ there exist $y^{\\prime } \\in \\mathbb {N}_0$ and $k^{\\prime } \\in \\mathbb {N}$ such that $L=y^{\\prime } + 2 \\cdot [k^{\\prime },2k^{\\prime }]$ , which is equivalent to $\\rho (L)= \\max L / \\min L \\le 2$ .", "Assume to the contrary that there is an $L \\in \\mathcal {L} (G)$ with $\\Delta (L)=\\lbrace 2\\rbrace $ such that $\\max L \\ge 2 \\min L + 1$ .", "We choose one such $L \\in \\mathcal {L} (G)$ with $ \\min L$ being minimal, and we choose a $B \\in \\mathcal {B} (G)$ with $\\mathsf {L}(B)= L$ .", "Since $\\min L$ is minimal, we obtain that $0 \\nmid B$ .", "Consequently, $|B| \\ge 2 \\max L \\ge 4 \\min L +2 $ .", "Since $\\mathsf {D} (G) = 5$ , it follows that a factorization of minimal length of $B$ contains at least two (possibly equal) minimal zero-sum sequences $U_1, U_2$ with $|U_1|=|U_2|=5$ , say $U_1 = e_0 \\cdot \\ldots \\cdot e_4$ .", "If $U_1= U_2$ , then $5 \\in \\mathsf {L} (U_1U_2)$ and thus $\\min L + 3 \\in L$ , contradicting the fact that $\\Delta (L)=\\lbrace 2\\rbrace $ .", "Thus $U_1 \\ne U_2$ .", "We assert that $3 \\in \\mathsf {L} (U_1U_2)$ , and thus obtain again a contradiction to the fact that $\\Delta (L)=\\lbrace 2\\rbrace $ .", "Let $g \\in G$ with $g U_2$ but $g \\nmid U_1$ .", "Then $g$ is the sum of two elements from $U_1$ , say $g= e_1+ e_2$ .", "Therefore $g(e_1e_2)^{-1}U_1$ is a minimal zero-sum sequence, whereas the sequence $(e_1e_2)g^{-1}U_2$ cannot be a minimal zero-sum sequence because it has length 6.", "Since $g^{-1}U_2$ is zero-sum free, every minimal zero-sum sequence dividing $(e_1e_2)g^{-1}U_2$ must contain $e_1$ or $e_2$ .", "This shows that $\\mathsf {L}((e_1e_2)g^{-1}U_2)= \\lbrace 2\\rbrace $ and thus $3 \\in \\mathsf {L} (U_1U_2)$ .", "CASE 4:   $\\Delta ( \\mathsf {L} (A)) = \\lbrace 1,2\\rbrace $ .", "Let $k \\in \\mathsf {L} (A)$ be minimal such that $A$ has a factorization of the form $A=U_1\\cdot \\ldots \\cdot U_k=V_1\\cdot \\ldots \\cdot V_{k+2}$ , where $k+1\\notin \\mathsf {L}(A)$ and $U_1,\\ldots , U_k, V_1,\\ldots , V_{k+2} \\in \\mathcal {A} (G)$ with $|U_1|\\ge |U_2|\\ge \\ldots \\ge |U_k|$ .", "Without restriction we may suppose that the tuple $ (|\\lbrace i\\in [1,k]\\mid |U_i|=5\\rbrace |,|\\lbrace i\\in [1,k]\\mid |U_i|=4\\rbrace |,|\\lbrace i\\in [1,k]\\mid |U_i|=3\\rbrace |) \\in \\mathbb {N}_0^3$ is maximal (with respect to the lexicographic order) among all factorizations of $A$ of length $k$ .", "By definition of $k$ , we have $[\\min \\mathsf {L}(A), k]\\in \\mathsf {L}(A)$ .", "Let $k_0\\in [2,k]$ such that $|U_i|\\ge 3$ for every $i\\in [1,{k_0}]$ and $|U_i|=2$ for every $i\\in [k_0+1,k]$ .", "We start with the following assertion.", "A.", "For each two distinct $i,j\\in [1,k_0]$ , we have $3\\notin \\mathsf {L}(U_iU_j)$ .", "For each two distinct $i,j\\in [1,k_0]$ with $|U_i|=|U_j|=5$ , we have $U_i=U_j$ .", "For each two distinct $i,j\\in [1,k_0]$ with $|U_i|=5$ and $|U_j|=4$ , we have $| \\gcd (U_i, U_j)|=3$ .", "Let $i,j\\in [1,k_0]$ be distinct with $|U_i|=|U_j|=4$ , say $U_i=f_1f_2f_3(f_1+f_2+f_3)$ where $(f_1,f_2,f_3,f_4)$ a basis of $G$ .", "Then $U_j=U_i$ , or $U_j=(f_1+f_4)(f_2+f_4)(f_3+f_4)(f_1+f_2+f_3+f_4)$ , or $U_j=f_4(f_1+f_2+f_4)(f_2+f_3+f_4)(f_1+f_3+f_4)$ .", "Furthermore, if $U_i\\ne U_j$ , then for all $t\\in [1, k_0]\\setminus \\lbrace i,j\\rbrace $ , we have $|U_t|\\ne 4$ .", "Let $i,j\\in [1,k_0]$ be distinct with $|U_i|=5$ and $|U_j|=3$ .", "Then there exist $g_1,g_2,g_3\\in G$ such that $g_1g_2g_3U_i$ and $U_j=(g_1+g_2)(g_2+g_3)(g_3+g_1)$ .", "Furthermore, for all $t\\in [1, k_0]\\setminus \\lbrace i,j\\rbrace $ , we have $|U_t|=3$ .", "Let $i,j\\in [1,k_0]$ be distinct with $|U_i|=4$ and $|U_j|=3$ .", "Then $|\\gcd (U_i, U_j)|=0$ , and there exist $g,g_1,g_2\\in G$ such that $gU_j$ , $g_1g_2U_i$ and $g=g_1+g_2$ .", "Furthermore, for all $t\\in [1, k_0]\\setminus \\lbrace i,j\\rbrace $ , we have $|U_t|=3$ .", "For each two distinct $i,j\\in [1,k_0]$ with $|U_i|=|U_j|=3$ , we have $|\\gcd (U_i, U_j)|=0$ .", "Proof of A.", "1.", "If there exist distinct $i,j\\in [1,k_0]$ such that $3\\in \\mathsf {L}(U_iU_j)$ , then $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "2.", "Since $|U_i|=5$ and $U_j\\ne U_i$ , there exist $g,g_1,g_2\\in G$ with $gU_j$ and $g_1g_2U_i$ such that $g=g_1+g_2$ .", "Thus $U_i(g_1g_2)^{-1}g$ is an atom and $U_jg^{-1}g_1g_1$ is a product of two atoms which implies that $3\\in \\mathsf {L}(U_iU_j)$ , a contradiction.", "3.", "Since $|U_i|=5$ and $U_j\\ne U_i$ , there exist $g,g_1,g_2\\in G$ with $gU_j$ and $g_1g_2U_i$ such that $g=g_1+g_2$ .", "Thus $gg_1g_2$ is an atom and $U_iU_j(gg_1g_2)^{-1}$ is a sequence of length 6.", "By 1., $2\\notin \\mathsf {L}(U_iU_j(gg_1g_2)^{-1})$ which implies that $\\mathsf {L}(U_iU_j(gg_1g_2)^{-1})=\\lbrace 3\\rbrace $ and hence $|\\gcd (U_i, U_j)|=3$ .", "4.", "We set $G_1=\\langle f_1,f_2,f_3\\rangle $ and distinguish three cases.", "Case (i): $U_j\\in \\mathcal {B}(G_1)$ .", "Since $3\\notin \\mathsf {L}(U_iU_j)$ , we obtain that $U_j=U_i$ .", "Case (ii): $U_j=(g_1+f_4)(g_2+f_4)g_3g_4$ with $g_1g_2g_3g_4\\in \\mathcal {B}(G_1)$ .", "If $g_3,g_4\\in \\lbrace f_1,f_2,f_3,f_1+f_2+f_3\\rbrace $ , then $3\\in \\mathsf {L}(U_iU_j)$ , a contradiction.", "Thus, without loss of generality, we may assume that $g_3=f_1+f_2\\notin \\lbrace f_1,f_2,f_3,f_1+f_2+f_3\\rbrace $ .", "Thus $g_3f_3(f_1+f_2+f_3)$ is an atom and $(g_1+f_4)(g_2+f_4)f_1f_2g_4$ is a zero-sum sequence of length 5.", "Since $3\\notin \\mathsf {L}(U_iU_j)$ , we have that $(g_1+f_4)(g_2+f_4)f_1f_2g_4$ is an atom of length 5, a contradiction to the maximality condition in Equation (REF ).", "Case (iii): $U_j=(g_1+f_4)(g_2+f_4)(g_3+f_4)(g_4+f_4)$ with $g_1g_2g_3g_4\\in \\mathcal {B}(G_1)$ .", "First, suppose that $g_1g_2g_3g_4$ is an atom.", "If $g_1g_2g_3g_4\\ne U_i$ , then there exist an element $h\\in \\lbrace f_1,f_2,f_3,f_1+f_2+f_3\\rbrace $ and distinct $t_1,t_2\\in [1,4]$ , say $t_1=1, t_2=2$ , such that $h=g_1+g_2=(g_1+f_4)+(g_2+f_4)$ .", "Thus $U_ih^{-1}(g_1+f_4)(g_2+f_4)$ is a zero-sum sequence of length 5 and $h(g_3+f_4)(g_4+f_4)$ is an atom.", "It follows that $U_ih^{-1}(g_1+f_4)(g_2f_4)$ is atom of length 5 since $3\\notin \\mathsf {L}(U_iU_j)$ , a contradiction to the maximality condition in Equation (REF ).", "Therefore $g_1g_2g_3g_4=U_i$ which implies that $U_j=(f_1+f_4)(f_2+f_4)(f_3+f_4)(f_1+f_2+f_3+f_4)$ .", "Second, suppose that $g_1g_2g_3g_4$ is not an atom.", "Without loss of generality, we may assume that $g_1=0$ and $g_2g_3g_4$ is an atom.", "If $\\lbrace g_2,g_3,g_4\\rbrace \\cap \\lbrace f_1,f_2,f_3,f_1+f_2+f_3\\rbrace \\ne \\emptyset $ , say $g_2\\in \\lbrace f_1,f_2,f_3,f_1+f_2+f_3\\rbrace $ , then $g_2(g_3+f_4)(g_4+f_4)$ is an atom and $U_ig_2^{-1}f_4(g_2+f_4)$ is a zero-sum sequence of length 5.", "It follows that $U_ig_2^{-1}f_4(g_2+f_4)$ is atom of length 5 because $3\\notin \\mathsf {L}(U_iU_j)$ , a contradiction to the maximality condition in Equation (REF ).", "Therefore $\\lbrace g_2,g_3,g_4\\rbrace \\cap \\lbrace f_1,f_2,f_3,f_1+f_2+f_3\\rbrace =\\emptyset $ which implies that $g_2g_3g_4=(f_1+f_2)(f_2+f_3)(f_1+f_3)$ and hence $U_j=f_4(f_1+f_2+f_4)(f_2+f_3+f_4)(f_1+f_3+f_4)$ .", "Now suppose that $U_i\\ne U_j$ , and assume to the contrary there exists a $t\\in [1,k_0]\\setminus \\lbrace i,j\\rbrace $ such that $|U_t|=4$ .", "If $U_t\\notin \\lbrace U_i,U_j\\rbrace $ , then $U_iU_jU_t=\\big (f_1f_2f_3(f_1+f_2+f_3)\\big )\\big ( (f_1+f_4)(f_2+f_4)(f_3+f_4)(f_1+f_2+f_3+f_4)\\big )\\big ( f_4(f_1+f_2+f_4)(f_2+f_3+f_4)(f_1+f_3+f_4)\\big )=\\big (f_1(f_2+f_4)(f_1+f_2+f_4)\\big )\\big ( f_2(f_3+f_4)(f_2+f_3+f_4)\\big )\\big ( f_3(f_1+f_4)(f_1+f_3+f_4) \\big )\\big ( f_4(f_1+f_2+f_3)(f_1+f_2+f_3+f_4)\\big )$ .", "Thus $4\\in \\mathsf {L}(U_iU_jU_t)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "If $U_t\\in \\lbrace U_i,U_j\\rbrace $ , then we still have that $4\\in \\mathsf {L}(U_iU_jU_t)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "5.", "Since $3\\notin \\mathsf {L}(U_iU_j)$ , we obtain that $|\\gcd (U_i, U_j)|=0$ .", "Every $h\\in \\operatorname{supp}(U_j)$ is the sum of two distinct elements from $\\operatorname{supp}(U_i)$ .", "Thus there exist $g_1,g_2,g_3\\in G$ with $g_1g_2g_3U_i$ such that $U_j=(g_1+g_2)(g_2+g_3)(g_3+g_1)$ .", "Now we choose an element $t\\in [1,k_0]\\setminus \\lbrace i,j\\rbrace $ , and have to show that $|U_t|=3$ .", "If $|U_t|=5$ , then $U_t=U_i$ by 2. and hence $4\\in \\mathsf {L}(U_iU_tU_j)$ which implies that $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Suppose that $|U_t|=4$ and let $U_i=g_1g_2g_3g_4g_5$ , where $g_4,g_5\\in G$ .", "Then $|\\gcd (U_i,U_t)|=3$ by 3. and by symmetry we only need to consider $\\operatorname{supp}(U_t)\\setminus \\operatorname{supp}(U_i)\\subset \\lbrace g_1+g_2, g_1+g_4,g_4+g_5\\rbrace $ .", "All the three cases imply that $4\\in \\mathsf {L}(U_iU_tU_j)$ .", "It follows that that $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "6.", "If $| \\gcd (U_i, U_j)|=2$ , then $3\\in \\mathsf {L}(U_iU_j)$ , a contradiction.", "If $|\\gcd (U_i, U_j)|=1$ , then $U_1U_2=W_1W_2$ with $W_1,W_2\\in \\mathcal {A}(G)$ and $|W_2|=5$ , a contradiction to the maximality condition in Equation (REF ).", "Thus we obtain that $|\\gcd (U_i, U_j)|=0$ .", "Let $(f_1,f_2,f_3,f_4)$ be a basis and $U_i=f_1f_2f_3(f_1+f_2+f_3)$ .", "Since $|U_j|=3$ , there exists a $g \\in \\operatorname{supp}( U_j)$ such that $g\\in \\langle f_1,f_2,f_3\\rangle $ .", "Since $|\\gcd (U_i, U_j)|=0$ , there exist $g_1,g_2\\in G$ such that $g_1g_2U_i$ and $g=g_1+g_2$ .", "Now we choose an element $t\\in [1,k_0]\\setminus \\lbrace i,j\\rbrace $ and have to show that $|U_t|=3$ .", "Note that 5. implies that $|U_t|\\ne 5$ , and we assume to the contrary that $|U_t|=4$ .", "Without restriction we may assume that $g=f_1+f_2$ , and by 4., we distinguish three cases.", "If $U_t=U_i$ , then $f_1^2,f_2^2, gU_i(f_1f_2)^{-1}, U_t(f_1f_2)^{-1}U_jg^{-1}$ are atoms and hence $4\\in \\mathsf {L}(U_iU_tU_j)$ which implies that $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "If $U_t=(f_1+f_4)(f_2+f_4)(f_3+f_4)(f_1+f_2+f_3+f_4)$ , then $g(f_1+f_2+f_3)(f_1+f_2+f_3+f_4)(f_1+f_4)f_2$ is an atom of length 5 dividing $U_iU_jU_t$ and $U_iU_jU_t(g(f_1+f_2+f_3)(f_1+f_2+f_3+f_4)(f_1+f_4)f_2)^{-1}$ is a product of two atoms, a contradiction to the maximality condition in Equation (REF ).", "If $U_t=f_4(f_1+f_2+f_4)(f_2+f_3+f_4)(f_1+f_3+f_4)$ , then $gf_2f_3f_4(f_1+f_3+f_4)$ is an atom of length 5 dividing $U_iU_jU_t$ and $U_iU_jU_t(gf_2f_3f_4(f_1+f_3+f_4))^{-1}$ is a product of two atoms, a contradiction to the maximality condition in Equation (REF ).", "7.", "If $|\\gcd (U_i, U_j)|\\ge 2$ , then $U_i=U_j$ and hence $3\\in \\mathsf {L}(U_iU_j)$ which implies that $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "If $|\\gcd (U_i, U_j)|=1$ , then $U_iU_j=W_1W_2$ with $W_1,W_2 \\in \\mathcal {A} (G)$ , $|W_1|=2$ , and $|W_2|=4$ , a contradiction to the maximality condition in Equation (REF ).", "Therefore $|\\gcd (U_i, U_j)|=0$ .", "This completes the proof of A.", "$\\Box $ Note that A.5 implies that $\\lbrace |U_i|\\mid i\\in [1,k_0]\\rbrace \\ne \\lbrace 3,4,5\\rbrace $ .", "Thus it remains to discuss the following six subcases.", "CASE 4.1.", "$\\lbrace |U_i|\\mid i\\in [1,k_0]\\rbrace =\\lbrace 3, 5\\rbrace $ .", "By A.5 and A.7, we obtain that $|U_1|=5$ , $|U_2|=\\ldots =|U_{k_0}|=3$ , and that $U_1\\cdot \\ldots \\cdot U_{k_0}$ is square-free.", "This implies that $\\max \\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})=k_0$ , and hence $\\max \\mathsf {L}(A)=\\max \\mathsf {L}(U_0\\cdot \\ldots \\cdot U_{k_0})+k-k_0=k$ , a contradiction.", "CASE 4.2.", "$\\lbrace |U_i|\\mid i\\in [1,k_0]\\rbrace =\\lbrace 3,4\\rbrace $ .", "By A.6 and A.7, we obtain that $|U_1|=4$ , $|U_2|=\\ldots =|U_{k_0}|=3$ , and that $U_1\\cdot \\ldots \\cdot U_{k_0}$ is square-free.", "This implies that $\\max \\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})=k_0$ , and hence $\\max \\mathsf {L}(A)=\\max \\mathsf {L}(U_0\\cdot \\ldots \\cdot U_{k_0})+k-k_0=k$ , a contradiction.", "CASE 4.3.", "$\\lbrace |U_i|\\mid i\\in [1,k_0]\\rbrace =\\lbrace 3\\rbrace $ .", "By A.7, we obtain that $U_1\\cdot \\ldots \\cdot U_{k_0}$ is square-free.", "This implies that $\\max \\mathsf {L}(U_1\\cdot \\ldots \\cdot U_{k_0})=k_0$ , and hence $\\max \\mathsf {L}(A)=\\max \\mathsf {L}(U_0\\cdot \\ldots \\cdot U_{k_0})+k-k_0=k$ , a contradiction.", "CASE 4.4.", "$\\lbrace |U_i|\\mid i\\in [1,k_0]\\rbrace =\\lbrace 5\\rbrace $ .", "By A.2, it follows that $A=U_1^{k_0} U_{k_0+1}\\cdot \\ldots \\cdot U_k$ .", "If $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_k)\\subset \\operatorname{supp}(U_1)$ , then $\\Delta (\\mathsf {L} (A))=\\lbrace 3\\rbrace $ , a contradiction.", "Thus there exists $j\\in [k_0+1,k]$ such that $U_j=g^2$ for some $g\\notin \\operatorname{supp}(U_1)$ .", "Then there exist $g_1,g_2\\in G$ such that $g_1g_2U_1$ and $g=g_1+g_2$ .", "It follows that $U_1^2U_j=g_1^2 g_2^2 (U_1(g_1g_2)^{-1}g)^2$ , where $g_1^2, g_2^2, U_1(g_1g_2)^{-1}g$ are atoms.", "Therefore $4\\in \\mathsf {L}(U_1^2U_j)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "CASE 4.5.", "$\\lbrace |U_i|\\mid i\\in [1,k_0]\\rbrace =\\lbrace 4\\rbrace $ .", "Assume to the contrary, that $k_0\\ge 3$ .", "Then A.4 implies that $U_1\\cdot \\ldots \\cdot U_{k_0}=U_1^{k_0}$ , and we set $G_1=\\langle \\operatorname{supp}(U_1)\\rangle $ .", "If there exists $g\\in \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_k)$ such that $g\\in G_1\\setminus \\operatorname{supp}(U_1)$ , then $4\\in \\mathsf {L}(U_1^2g^2)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "If there exist distinct $g_1,g_2\\in \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_k)$ such that $g_1\\notin G_1$ and $g_2\\notin G_1$ , then $g_1+g_2\\in G_1$ .", "Since $g_1+g_2\\in \\operatorname{supp}(U_1)$ implies that $5\\in \\mathsf {L}(U_1^2g_1^2g_2^2)$ and $k+1\\in \\mathsf {L}(A)$ , we obtain that $g_1+g_2\\in G_1\\setminus \\operatorname{supp}(U_1)$ .", "Then $U_1^2g_1^2g_2^2=W_1^2W_2W_3$ where $W_1,W_2,W_3\\in \\mathcal {A}(G)$ with $|W_1|=4$ , $W_1\\ne U_1$ , and $|W_2|=|W_3|=2$ .", "Say $U_{k_0+1}=g_1^2$ and $U_{k_0+2}=g_2^2$ .", "Then $W_1^2U_3\\cdot \\ldots \\cdot U_{k_0} W_1W_2U_{k_0+3}\\cdot \\ldots \\cdot U_k$ is a factorization of $A$ of length $k$ satisfying the maximality condition of Equation (REF ) and hence applying A.4 to this factorization, we obtain a contradiction.", "Therefore $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_k)\\subset \\operatorname{supp}(U_1)\\cup \\lbrace g\\rbrace $ where $g$ is independent from $\\operatorname{supp}(U_1)$ and hence $\\operatorname{supp}(A)\\subset \\operatorname{supp}(U_1)\\cup \\lbrace g\\rbrace $ which implies that $\\Delta (\\mathsf {L} (A))=\\lbrace 2\\rbrace $ , a contradiction.", "Therefore it follows that $k_0=2$ .", "Then $U_1=U_2$ (since otherwise we would have $\\max \\mathsf {L}(A)=k$ by $U_1U_2$ is square-free), and we obtain that $\\mathsf {L}(A)=[\\min \\mathsf {L}(A), k]\\cup \\lbrace k+2\\rbrace $ .", "Assume to the contrary that there exists a $W \\in \\mathcal {A} (G)$ such that $WA$ and $|W|=5$ .", "Then there exist $g,g_1,g_2\\in G$ such that $gU_1$ , $g_1g_2W$ , and $g=g_1+g_2$ , and hence $|\\lbrace g_1,g_2\\rbrace \\cap \\operatorname{supp}(U_1)|\\le 1$ .", "If $\\lbrace g_1,g_2\\rbrace \\cap \\operatorname{supp}(U_1)=\\emptyset $ , then there exist distinct $t_1,t_2\\in [k_0+1,k]$ such that $U_{t_1}=g_1^2$ and $U_{t_2}=g_2^2$ .", "Thus $5\\in \\mathsf {L}(U_1U_2U_{t_1}U_{t_2})$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Suppose that $|\\lbrace g_1,g_2\\rbrace \\cap \\operatorname{supp}(U_1)|= 1$ , say $g_1\\notin \\operatorname{supp}(U_1)$ and $g_2\\in \\operatorname{supp}(U_1)$ .", "Then there exists $t\\in [k_0+1,k]$ such that $U_t=g_1^2$ .", "Therefore $4\\in \\mathsf {L}(U_1U_2U_t)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Thus every atom $W$ with $WA$ has length $|W| < 5$ .", "It follows that $\\min \\mathsf {L}(A)\\ge \\lceil \\frac{2\\max \\mathsf {L}(A)}{4}\\rceil =\\lceil \\frac{\\max \\mathsf {L}(A)}{2}\\rceil $ and hence $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "CASE 4.6.", "$\\lbrace |U_i|\\mid i\\in [1,k_0]\\rbrace =\\lbrace 4,5\\rbrace $ .", "By A.2, A.3, and A.4, we obtain that $|\\lbrace U_1,\\ldots , U_{k_0}\\rbrace |=2$ .", "Without restriction we may assume that $U_1\\cdot \\ldots \\cdot U_{k_0}=U^{k_1}V^{k_2} $ where $k_1,k_2\\in \\mathbb {N}$ with $k_0=k_1+k_2$ and $V=e_1e_2e_3(e_1+e_2+e_3)$ (recall that $(e_1, \\ldots , e_4)$ is a basis of $G$ , $e_0=e_1+e_2+e_3+e_4$ , and $U=e_1e_2e_3e_4 e_0$ ).", "We claim that $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})\\subset \\operatorname{supp}(UV)$ .", "If $k_1\\ge 2$ , then $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})\\subset \\operatorname{supp}(U)$ , and if $k_2\\ge 2$ , then $\\lbrace e_4,e_0\\rbrace \\lnot \\subset \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})$ .", "Indeed, assume to the contrary that $g\\in \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})\\setminus \\operatorname{supp}(UV)$ .", "By symmetry, we only need to consider $g=e_1+e_2$ and $g=e_1+e_4$ and both cases imply that $4\\in \\mathsf {L}(UVg^2)$ , a contradiction to $k+1\\notin \\mathsf {L}(A)$ .", "If $k_1\\ge 2$ and $g=e_1+e_2+e_3\\in \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})$ , then $4\\in \\mathsf {L}(U^2g^2)$ and $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Thus if $k_1\\ge 2$ , then $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})\\subset \\operatorname{supp}(U)$ .", "If $k_2\\ge 2$ and $\\lbrace e_4,e_0\\rbrace \\subset \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})$ , then $5\\in \\mathsf {L}(V^2e_4^2e_0^2)$ and hence $k+1\\in \\mathsf {L}(A)$ , a contradiction.", "Thus all three claims are proved, and we distinguish three subcases.", "CASE 4.6.1.", "$k_1=1$ .", "If $\\lbrace e_4,e_0\\rbrace \\lnot \\subset \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k}) $ , then $\\mathsf {L}(A)=\\mathsf {L}(UV^{k_2})+k-k_0=\\mathsf {L}(V^{k_0})+k-k_0$ and hence $\\Delta (\\mathsf {L} (A))=\\lbrace 2\\rbrace $ , a contradiction.", "If $\\lbrace e_4,e_0\\rbrace \\subset \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})$ , then $k_2=1$ and we may assume that $U_{k_0+1}=e_4^2$ and that $U_{k_0+2}=e_0^2$ .", "Then $\\mathsf {L}(A)=\\mathsf {L}(UVU_{k_0+1}U_{k_0+2})+k-k_0-2=\\lbrace k-1,k,k+2\\rbrace $ with $k\\ge 4$ , and hence $\\mathsf {L}(A)\\in \\mathcal {L}_5$ .", "CASE 4.6.2.", "$k_1\\ge 2$ and $k_2\\ge 2$ .", "Thus $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})$ is independent and it follows that $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})\\subset \\lbrace e_1,e_2,e_3,e_4\\rbrace $ or $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})\\subset \\lbrace e_1,e_2,e_3,e_0\\rbrace $ .", "Then we have $\\mathsf {L}(A)=\\mathsf {L}(U^{k_1}V^{k_2})+k-k_0$ .", "By Equation (REF ), $\\mathsf {L}(U^{k_1}V^{k_2})$ is equal to $\\left\\lbrace \\begin{aligned}&\\lbrace k_0\\rbrace \\cup [k_0+2, 5\\lfloor k_1/2\\rfloor +4(k_0/2-\\lfloor k_1/2\\rfloor )] &&\\text{ if $k_0=k_1+k_2$ is even}\\,,\\\\&\\lbrace k_0\\rbrace \\cup [k_0+2, 5\\lfloor k_1/2\\rfloor +4((k_0-1)/2-\\lfloor k_1/2\\rfloor )+1] &&\\text{ if $k_0=k_1+k_2$ is odd}\\,.\\end{aligned}\\right.$ Let $\\ell =\\max \\mathsf {L}(U^{k_1}V^{k_2})-k_0-2$ and hence $\\ell =\\left\\lbrace \\begin{aligned}&k_0+\\lfloor \\frac{k_1}{2}\\rfloor -2 &\\text{ if $k_0\\ge 4$ is even}\\,,\\\\&k_0+\\lfloor \\frac{k_1}{2}\\rfloor -3 &\\text{ if $k_0\\ge 5$ is odd}\\,.\\end{aligned}\\right.$ Since $k_1\\ge 2$ and $k_2\\ge 2$ , we obtain that $\\ell \\ge 3$ and $\\ell \\ne 4$ .", "We also have that $\\ell \\le \\left\\lbrace \\begin{aligned}&k_0+\\lfloor \\frac{k_0-2}{2}\\rfloor -2=\\frac{3k_0}{2}-3 &\\text{ if $k_0$ is even}\\,,\\\\&k_0+\\lfloor \\frac{k_0-2}{2}\\rfloor -3=\\frac{3k_0-9}{2} &\\text{ if $k_0$ is odd}\\,.\\end{aligned}\\right.$ Therefore $k_0\\ge \\left\\lbrace \\begin{aligned}&\\frac{2\\ell }{3}+2 &\\text{ if $k_0$ is even}\\,,\\\\&\\frac{2\\ell }{3}+3 &\\text{ if $k_0$ is odd}\\,,\\end{aligned}\\right.$ and hence $k_0\\ge \\left\\lbrace \\begin{aligned}&2\\lceil \\frac{\\ell }{3}\\rceil +2 &\\text{ if $k_0$ is even}\\,,\\\\&2\\lceil \\frac{\\ell }{3}\\rceil +2 &\\text{ if $k_0$ is odd}\\,.\\end{aligned}\\right.$ It follows that $\\mathsf {L}(U^{k_1}V^{k_2})\\in \\mathcal {L}_6$ which implies that $\\mathsf {L}(A)\\in \\mathcal {L}_6$ .", "CASE 4.6.3.", "$k_1\\ge 2$ and $k_2=1$ .", "Then $\\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})\\subset \\lbrace e_1,e_2,e_3,e_4, e_0\\rbrace $ .", "If $\\lbrace e_4,e_0\\rbrace \\lnot \\subset \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k}) $ , then $\\mathsf {L}(A)=\\mathsf {L}(U^{k_1}V)+k-k_0$ is equal to $\\left\\lbrace \\begin{aligned}&k+\\lbrace 0,2,3\\rbrace +3 \\cdot [0, k_1/2-1], &&\\text{ if $k_1$ is even}\\,,\\\\&k+\\lbrace 0,2,3\\rbrace +3 \\cdot [0, (k_1-1)/2-1]\\cup \\lbrace k+(3k_1-3)/2+2\\rbrace , &&\\text{ if $k_1$ is odd}\\,\\end{aligned}\\right.$ by Equation (REF ).", "Therefore $\\mathsf {L}(A)\\in \\mathcal {L}_8$ .", "If $\\lbrace e_4,e_0\\rbrace \\subset \\operatorname{supp}(U_{k_0+1}\\cdot \\ldots \\cdot U_{k})$ , then we may assume that $U_{k_0+1}=e_4^2$ and that $U_{k_0+2}=e_0^2$ .", "Thus $\\mathsf {L}(A)&=\\mathsf {L}(U^{k_1}VU_{k_0+1}U_{k_0+2})+k-k_0-2\\\\&=\\left\\lbrace \\begin{aligned}&k-1+\\lbrace 0,1,3\\rbrace +3 \\cdot [0, (k_1+1)/2-1], &&\\text{ if $k_1$ is odd}\\,,\\\\&k-1+\\lbrace 0,1,3\\rbrace +3 \\cdot [0, k_1/2-1]\\cup \\lbrace k+3k_1/2+1\\rbrace , &&\\text{ if $k_1$ is even}\\,,\\end{aligned}\\right.$ by Equation (REF ) and hence $\\mathsf {L}(A)\\in \\mathcal {L}_7$ ." ], [ "Sets of lengths of weakly Krull monoids", "It is well-known that – under reasonable algebraic finiteness conditions – the Structure Theorem for Sets of Lengths holds for weakly Krull monoids (as it is true for transfer Krull monoids of finite type, see Proposition REF ).", "In spite of this common feature we will demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid (apart from well-described exceptional cases; see Theorems REF to REF ).", "Since half-factorial monoids are transfer Krull monoids, and since there are half-factorial weakly Krull monoids, half-factoriality is such a natural exceptional case.", "So far there are only a couple of results in this direction.", "In [15], Frisch showed that $\\operatorname{Int}(\\mathbb {Z})$ , the ring of integer-valued polynomials over $\\mathbb {Z}$ , is not a transfer Krull domain (nevertheless, the system of sets of lengths of $\\operatorname{Int}(\\mathbb {Z})^{\\bullet }$ coincides with $\\mathcal {L} (G)$ for an infinite abelian group $G$ ).", "To mention a result by Smertnig, let $\\mathcal {O}$ be the ring of integers of an algebraic number field $K$ , $A$ a central simple algebra over $K$ , and $R$ a classical maximal $\\mathcal {O}$ -order of $A$ .", "Then $R$ is a non-commutative Dedekind domain and in particular an HNP ring (see [32]).", "Furthermore, $R$ is a transfer Krull domain if and only if every stably free left $R$ -ideal is free ([35]).", "We gather basic concepts and properties of weakly Krull monoids and domains (Propositions REF and REF ).", "In the remainder of this section, all monoids and domains are supposed to be commutative.", "Let $H$ be a monoid (hence commutative, cancellative, and with unit element).", "We denote by $\\mathsf {q} (H)$ the quotient group of $H$ , by $H_{{\\text{\\rm red}}}=H/H^{\\times }$ the associated reduced monoid of $H$ , by $\\mathfrak {X} (H)$ the set of minimal nonempty prime $s$ -ideals of $H$ , and by $\\mathfrak {m} = H \\setminus H^{\\times }$ the maximal $s$ -ideal.", "Let $\\mathcal {I}_v^* (H)$ denote the monoid of $v$ -invertible $v$ -ideals of $H$ (with $v$ -multiplication).", "Then $\\mathcal {F}_v (H)^{\\times } = \\mathsf {q} ( \\mathcal {I}_v^* (H))$ is the quotient group of fractional $v$ -invertible $v$ -ideals, and $\\mathcal {C}_v (H) = \\mathcal {F}_v (H)^{\\times }/\\lbrace xH \\mid x \\in \\mathsf {q} (H)\\rbrace $ is the $v$ -class group of $H$ (detailed presentations of ideal theory in commutative monoids can be found in [30], [20]).", "We denote by $\\widehat{H} \\subset \\mathsf {q} (H)$ the complete integral closure of $H$ , and by $(H : \\widehat{H}) = \\lbrace x \\in \\mathsf {q} (H) \\mid x \\widehat{H} \\subset H \\rbrace \\subset H$ the conductor of $H$ .", "A submonoid $S \\subset H$ is said to be saturated if $S = \\mathsf {q} (S) \\cap H$ .", "For the definition and discussion of the concepts of being faithfully saturated or being locally tame we refer to [20].", "To start with the local case, we recall that $H$ is said to be primary if $\\mathfrak {m} \\ne \\emptyset $ and for all $a, b \\in \\mathfrak {m}$ there is an $n \\in \\mathbb {N}$ such that $b^n \\subset aH$ .", "strongly primary if $\\mathfrak {m} \\ne \\emptyset $ and for every $a \\in \\mathfrak {m}$ there is an $n \\in \\mathbb {N}$ such that $\\mathfrak {m}^{n} \\subset aH$ .", "We denote by $\\mathcal {M} (a)$ the smallest $n$ having this property.", "a discrete valuation monoid if it is primary and contains a prime element (equivalently, $H_{{\\text{\\rm red}}} \\cong (\\mathbb {N}_0,+)$ ).", "Furthermore, $H$ is said to be weakly Krull ([30]) if $H = \\bigcap _{{\\mathfrak {p}} \\in \\mathfrak {X} (H)} H_{\\mathfrak {p}} \\quad \\text{and} \\quad \\lbrace {\\mathfrak {p}} \\in \\mathfrak {X} (H) \\mid a \\in {\\mathfrak {p}}\\rbrace \\quad \\text{is finite for all} \\ a \\in H \\,.$ weakly factorial if one of the following equivalent conditions is satisfied ([30]) : Every non-unit is a finite product of primary elements.", "$H$ is a weakly Krull monoid with trivial $t$ -class group.", "Clearly, every localization $H_{\\mathfrak {p}}$ of $H$ at a minimal prime ideal $\\mathfrak {p} \\in \\mathfrak {X} (H)$ is primary, and a weakly Krull monoid $H$ is $v$ -noetherian if and only if $H_{\\mathfrak {p}}$ is $v$ -noetherian for each $\\mathfrak {p} \\in \\mathfrak {X} (H)$ .", "Every $v$ -noetherian primary monoid $H$ is strongly primary and $v$ -local, and if $(H : \\widehat{H})\\ne \\emptyset $ , then $H$ is locally tame ([21]).", "Every strongly primary monoid is a primary BF-monoid ([20]).", "Therefore the coproduct of a family of strongly primary monoids is a BF-monoid, and every coproduct of a family of primary monoids is weakly factorial.", "A $v$ -noetherian weakly Krull monoid $H$ is weakly factorial if and only if $\\mathcal {C}_v (H)=0$ if and only if $H_{{\\text{\\rm red}}} \\cong \\mathcal {I}_v^* (H)$ .", "By a numerical monoid $H$ we mean an additive submonoid of $(\\mathbb {N}_0, +)$ such that $\\mathbb {N}_0 \\setminus H$ is finite.", "Clearly, every numerical monoid is $v$ -noetherian primary, and hence it is strongly primary.", "Note that a numerical monoid is half-factorial if and only if it is equal to $(\\mathbb {N}_0,+)$ .", "Let $R$ be a domain.", "Then $R^{\\bullet } = R \\setminus \\lbrace 0\\rbrace $ is a monoid, and all arithmetic and ideal theoretic concepts introduced for monoids will be used for domains in the obvious way.", "The domain $R$ is weakly Krull (resp.", "weakly factorial) if and only if its multiplicative monoid $R^{\\bullet }$ is weakly Krull (resp.", "weakly factorial).", "Weakly Krull domains were introduced by Anderson, Anderson, Mott, and Zafrullah ([2], [3]).", "We recall some most basic facts and refer to an extended list of weakly Krull domains and monoids in [22].", "The monoid $R^{\\bullet }$ is primary if and only if $R$ is one-dimensional and local.", "If $R$ is one-dimensional local Mori, then $R^{\\bullet }$ is strongly primary and locally tame ([23]).", "Furthermore, every one-dimensional semilocal Mori domain with nontrivial conductor is weakly factorial and the same holds true for generalized Cohen-Kaplansky domains.", "It can be seen from the definition that one-dimensional noetherian domains are $v$ -noetherian weakly Krull domains.", "Proposition REF summarizes the main algebraic properties of $v$ -noetherian weakly Krull monoids and Proposition REF recalls that their arithmetic can be studied via weak transfer homomorphisms to weakly Krull monoids of very special form.", "Proposition 5.1 Let $H$ be a $v$ -noetherian weakly Krull monoid.", "The monoid $\\mathcal {I}_v^* (H)$ is isomorphic to the coproduct of $(H_{\\mathfrak {p}})_{{\\text{\\rm red}}}$ over all $\\mathfrak {p} \\in \\mathfrak {X} (H)$ .", "In particular, $\\mathcal {I}_v^* (H)$ is weakly factorial and $v$ -noetherian.", "Suppose that $\\mathfrak {f} = (H : \\widehat{H}) \\ne \\emptyset $ .", "We set $\\mathcal {P}^* = \\lbrace \\mathfrak {p} \\in \\mathfrak {X} (H) \\mid \\mathfrak {p} \\supset \\mathfrak {f} \\rbrace $ , and $\\mathcal {P} = \\mathfrak {X} (H) \\setminus \\mathcal {P}^*$ .", "Then $\\widehat{H}$ is Krull, $\\mathcal {P}^*$ is finite, and $H_{\\mathfrak {p}}$ is a discrete valuation monoid for each $\\mathfrak {p} \\in \\mathcal {P}$ .", "In particular, $\\mathcal {I}_v^* (H)$ is isomorphic to $\\mathcal {F} (\\mathcal {P}) \\times \\prod _{\\mathfrak {p} \\in \\mathcal {P}^*} (H_{\\mathfrak {p}})_{{\\text{\\rm red}}}$ .", "If $\\mathcal {H} = \\lbrace aH \\mid a \\in H \\rbrace $ is the monoid of principal ideals of $H$ , then $\\mathcal {H} \\subset \\mathcal {I}_v^* (H)$ is saturated.", "Moreover, if $H$ is the multiplicative monoid of a domain, then all monoids $H_{\\mathfrak {p}}$ are locally tame and $\\mathcal {H} \\subset \\mathcal {I}_v^* (H)$ is faithfully saturated.", "1.", "See [22].", "2.", "For (a) we refer to [20] and for (b) we refer to [20].", "Proposition 5.2 Let $D = \\mathcal {F} (\\mathcal {P}) \\times \\prod _{i=1}^n D_i$ be a monoid, where $\\mathcal {P} \\subset D$ is a set of primes, $n \\in \\mathbb {N}_0$ , and $D_1, \\ldots , D_n$ are reduced primary monoids.", "Let $H \\subset D$ be a saturated submonoid, $G = \\mathsf {q} (D)/\\mathsf {q} (H)$ , and $G_\\mathcal {P} = \\lbrace \\, p \\, \\mathsf {q} (H) \\mid p \\in \\mathcal {P}\\rbrace \\subset G$ the set of classes containing primes.", "There is a saturated submonoid $B \\subset F=\\mathcal {F} (G_{\\mathcal {P}}) \\times \\prod _{i=1}^n D_i$ and a weak transfer homomorphism $\\theta \\colon H \\rightarrow B$ .", "Moreover, if $G$ is a torsion group, then there is a monomorphism $\\mathsf {q} (F)/\\mathsf {q} (B) \\rightarrow G$ .", "If $G$ is a torsion group, then $H$ is weakly Krull.", "1.", "See [20].", "2.", "See [22].", "Note that, under the assumption of REF .2, the embedding $\\mathcal {H} \\hookrightarrow \\mathcal {I}_v^* (H)$ fulfills the assumptions imposed on the embedding $H \\hookrightarrow D$ in Proposition REF .", "Thus Proposition REF applies to $v$ -noetherian weakly Krull monoids.", "For simplicity and in order to avoid repetitions, we formulate the next results (including Theorem REF ) in the abstract setting of Proposition REF .", "However, $v$ -noetherian weakly Krull domains and their monoids of $v$ -invertible $v$ -ideals are in the center of our interest.", "If (in the setting of Proposition REF ) $G_{\\mathcal {P}}$ is finite, then $F=\\mathcal {F} (G_P)\\times \\prod _{i=1}^n D_i$ is a finite product of primary monoids and $B \\subset F$ is a saturated submonoid.", "We formulate the main structural result for sets of lengths in $v$ -noetherian weakly Krull monoids in this abstract setting.", "Proposition 5.3 Let $D_1, \\ldots , D_n$ be locally tame strongly primary monoids and $H \\subset D=D_1 \\times \\ldots \\times D_n$ a faithfully saturated submonoid such that $\\mathsf {q} (D)/\\mathsf {q} (H)$ is finite.", "The monoid $H$ satisfies the Structure Theorem for Sets of Lengths.", "There is a finite abelian group $G$ such that for every $L \\in \\mathcal {L} (H)$ there is a $y \\in \\mathbb {N}$ such that $y+L \\in \\mathcal {L} (G)$ .", "1. follows from [20], and 2. follows from 1. and from Proposition REF .2.", "The next lemma on zero-sum sequences will be crucial in order to distinguish between sets of lengths in weakly Krull monoids and sets of lengths in transfer Krull monoids.", "Lemma 5.4 Let $G$ be an abelian group and $G_0 \\subset G$ a non-half-factorial subset.", "There exists a half-factorial subset $G_1\\subset G_0$ with $\\mathcal {B} (G_1) \\ne \\lbrace 1\\rbrace $ .", "There are $M \\in \\mathbb {N}$ and zero-sum sequences $B_k \\in \\mathcal {B} (G_0)$ for every $k \\in \\mathbb {N}$ such that $2 \\le |\\mathsf {L} (B_k)| \\le M$ but $\\min \\mathsf {L} (B_k) \\rightarrow \\infty $ as $k \\rightarrow \\infty $ .", "1.", "Since $G_0$ is not half-factorial, there is a $B\\in \\mathcal {B}(G_0)$ such that $|\\mathsf {L}(B)|>1$ .", "Thus $\\operatorname{supp}(B)$ is finite and not half-factorial, say $\\operatorname{supp}(B) = \\lbrace g_1, \\ldots , g_{\\ell }\\rbrace $ with $\\ell \\ge 2$ .", "Without restriction we may suppose that every proper subset of $\\lbrace g_1, \\ldots , g_{\\ell } \\rbrace $ is half-factorial.", "Assume to the contrary that for every subset $G_1\\subsetneq \\lbrace g_1, \\ldots , g_{\\ell }\\rbrace $ we have $\\mathcal {B}(G_1)=\\lbrace 1\\rbrace $ .", "Since $\\lbrace g_1, \\ldots , g_{\\ell }\\rbrace $ is minimal non-half-factorial, there is an atom $A_1 \\in \\mathcal {A}(\\lbrace g_1, \\ldots , g_{\\ell }\\rbrace )$ such that $\\mathsf {v}_{g_i}(A_1)>0$ for every $i \\in [1, \\ell ]$ .", "Since $\\lbrace g_1, \\ldots , g_{\\ell }\\rbrace $ is not half-factorial, there is an atom $A_2 \\in \\mathcal {A}(\\lbrace g_1, \\ldots , g_{\\ell }\\rbrace )$ distinct from $A_1$ , say $A_1=g_1^{k_1}\\cdot \\ldots \\cdot g_{\\ell }^{k_{\\ell }} \\quad \\text{and} \\quad A_2=g_1^{t_1}\\cdot \\ldots \\cdot g_{\\ell }^{t_{\\ell }}$ where $k_i \\in \\mathbb {N}$ and $ t_i \\in \\mathbb {N}_0$ for every $i\\in [1,\\ell ]$ .", "Let $\\tau \\in [1, \\ell ]$ such that $\\frac{t_{\\tau }}{k_{\\tau }}= \\max \\lbrace \\frac{t_j}{k_j} \\mid j\\in [1,\\ell ] \\rbrace $ .", "Then $k_jt_{\\tau }-t_jk_{\\tau }\\ge 0$ for every $j\\in [1,\\ell ]$ whence $W=A_2^{t_{\\tau }} A_1^{-k_{\\tau }} \\in \\mathcal {B}( \\lbrace g_1, \\ldots , g_{\\ell }\\rbrace \\setminus \\lbrace g_{\\tau }\\rbrace )\\,,$ which implies that $W=1$ .", "Therefore $\\frac{t_{\\tau }}{k_{\\tau }}=\\frac{t_j}{k_j}$ for every $j\\in [1,\\ell ]$ and hence $A_1A_2$ or $A_2A_1$ , a contradiction.", "2.", "Let $B\\in \\mathcal {B}(G_0)$ with $|\\mathsf {L}(B)|>1$ .", "By 1., there exists a half-factorial subset $G_1\\subsetneq G_0$ such that $\\mathcal {B}(G_1)\\ne \\lbrace 1\\rbrace $ .", "Let $A\\in \\mathcal {A}(G_1)$ and $B_k=A^k B$ for every $k\\in \\mathbb {N}$ .", "Obviously there exists $k_0\\in \\mathbb {N}$ such that $\\mathsf {L}(B_k)=\\mathsf {L}(A^{k-k_0})+\\mathsf {L}(B_{k_0})=k-k_0+\\mathsf {L}(B_{k_0})$ for every $k\\ge k_0$ .", "Thus $|\\mathsf {L}(B_k)|\\le \\max \\mathsf {L}(B_{k_0})-\\min \\mathsf {L}(B_{k_0})$ and $\\min \\mathsf {L} (B_k) = k-k_0 + \\min \\mathsf {L} (B_{k_0})$ .", "Now we consider strongly primary monoids and work out a feature of their systems of sets of lengths which does not occur in the system of sets of lengths of any transfer Krull monoid.", "To do so we study the set $\\lbrace \\rho (L) \\mid L \\in \\mathcal {L} (H) \\rbrace $ of elasticities of all sets of lengths.", "This set was studied first by Chapman et al.", "in a series of papers (see [6], [13], [7], [8]).", "Among others they showed that in an atomic monoid $H$ , which has a prime element and an element $a \\in H$ with $\\rho ( \\mathsf {L} (a) ) = \\rho (H)$ , every rational number $q$ with $1 \\le q \\le \\rho (H)$ can be realized as the elasticity of some $L \\in \\mathcal {L} (H)$ ([6]).", "Primary monoids, which are not discrete valuation monoids, have no prime elements and their set of elasticities is different, as we will see in the next theorem.", "Statement 1. of Theorem REF was proved for numerical monoids in [13].", "Theorem 5.5 Let $H$ be a strongly primary monoid that is not half-factorial.", "There is a $\\beta \\in \\mathbb {Q}_{>1}$ such that $\\rho (L) \\ge \\beta $ for all $L \\in \\mathcal {L} (H)$ with $\\rho (L) \\ne 1$ .", "$\\mathcal {L} (H) \\ne \\mathcal {L} (G_0)$ for any subset $G_0$ of any abelian group.", "In particular, $H$ is not a transfer Krull monoid.", "If one of the following two conditions holds, then $H$ is locally tame.", "$\\sup \\lbrace \\min \\mathsf {L}(c) \\mid c \\in H \\rbrace < \\infty $ .", "There exists some $u \\in H \\setminus H^\\times $ such that $\\rho _{\\mathcal {M}(u)} (H) < \\infty $ .", "If $H$ is locally tame, then $\\Delta (H)$ is finite, and there is an $M \\in \\mathbb {N}_0$ such that every $L \\in \\mathcal {L} (H)$ is an AAMP with period $\\lbrace 0, \\min \\Delta (H)\\rbrace $ and bound $M$ .", "Remark.", "If $H$ is the multiplicative monoid of a one-dimensional local Mori domain $R$ with nonzero conductor $(R : \\widehat{R}) \\ne \\lbrace 0\\rbrace $ , then one of the conditions in 3. is satisfied (see [20]).", "However, there are strongly primary monoids for which none of the conditions holds and which are not locally tame ([21]).", "1.", "Let $b\\in H$ such that $|\\mathsf {L}(b)|\\ge 2$ and let $u\\in \\mathcal {A}(H)$ .", "Since $H$ is a strongly primary monoid, we have $(H\\setminus H^{\\times })^{\\mathcal {M}(b)}\\in bH$ and $(H\\setminus H^{\\times })^{\\mathcal {M}(u)}\\in uH$ .", "Thus $bu^{\\mathcal {M}(b)}$ and hence $|\\mathsf {L}(u^{\\mathcal {M}(b)})|\\ge 2$ .", "We define $\\beta _1=\\frac{\\mathcal {M}(b)+\\mathcal {M}(u)+1}{\\mathcal {M}(b)+\\mathcal {M}(u)}\\,, \\quad \\quad \\beta _2=\\frac{\\max \\mathsf {L}(u^{\\mathcal {M}(b)})+\\mathcal {M}(b)+\\mathcal {M}(u)}{\\min \\mathsf {L}(u^{\\mathcal {M}(b)})+\\mathcal {M}(b)+\\mathcal {M}(u)}\\,,$ and observe that $\\beta = \\min \\lbrace \\beta _1, \\beta _2\\rbrace >1$ .", "Let $a\\in H$ with $\\rho (\\mathsf {L}(a)) \\ne 1$ .", "We show that $\\rho (\\mathsf {L}(a)) \\ge \\beta $ .", "Let $k\\in \\mathbb {N}_0$ be maximal such that $u^k \\mid a$ , say $a=u^k u^{\\prime }$ with $u^{\\prime }\\in H$ .", "Thus $u\\nmid u^{\\prime }$ and thus $\\max \\mathsf {L}(u^{\\prime })<\\mathcal {M}(u)$ .", "If $k<\\mathcal {M}(b)$ , then $\\min \\mathsf {L}(a)\\le \\min \\mathsf {L}(u^k)+\\min \\mathsf {L}(u^{\\prime })\\le \\mathcal {M}(b)+\\mathcal {M}(u)$ , and hence $\\rho (\\mathsf {L}(a))&=\\frac{\\max \\mathsf {L}(a)}{\\min \\mathsf {L}(a)}\\ge \\frac{\\min \\mathsf {L}(a)+1}{\\min \\mathsf {L}(a)}\\ge \\frac{\\mathcal {M}(b)+\\mathcal {M}(u)+1}{\\mathcal {M}(b)+\\mathcal {M}(u)}=\\beta _1\\ge \\beta \\,.$ If $k\\ge \\mathcal {M}(b)$ , then there exist $t\\in \\mathbb {N}$ and $t_0\\in [0, \\mathcal {M}(b)-1]$ such that $k=t\\mathcal {M}(b)+t_0$ , and hence $\\rho (\\mathsf {L}(a))&=\\frac{\\max \\mathsf {L}(a)}{\\min \\mathsf {L}(a)}\\ge \\frac{\\max \\mathsf {L}(u^k)+\\max \\mathsf {L}(u^{\\prime })}{\\min \\mathsf {L}(u^k)+\\min \\mathsf {L}(u^{\\prime })} \\\\& \\ge \\frac{t\\max \\mathsf {L}(u^{\\mathcal {M}(b)})+\\max \\mathsf {L}(u^{t_0})+\\max \\mathsf {L}(u^{\\prime })}{t\\min \\mathsf {L}(u^{\\mathcal {M}(b)})+\\min \\mathsf {L}(u^{t_0})+\\min \\mathsf {L}(u^{\\prime })}\\\\&\\ge \\frac{t\\max \\mathsf {L}(u^{\\mathcal {M}(b)})+t_0+\\max \\mathsf {L}(u^{\\prime })}{t\\min \\mathsf {L}(u^{\\mathcal {M}(b)})+t_0+\\max \\mathsf {L}(u^{\\prime })} \\\\& \\ge \\frac{t\\max \\mathsf {L}(u^{\\mathcal {M}(b)})+\\mathcal {M} (b) + \\mathcal {M} (u)}{t\\min \\mathsf {L}(u^{\\mathcal {M}(b)})+\\mathcal {M} (b) + \\mathcal {M} (u)} \\ge \\beta _2\\ge \\beta \\,.$ 2.", "Assume to the contrary that there are an abelian group $G$ and a subset $G_0 \\subset G$ such that $\\mathcal {L} (H) = \\mathcal {L} (G_0)$ .", "Since $H$ is not half-factorial, $G_0$ is not half-factorial.", "By 1., there exists $\\beta \\in \\mathbb {Q}$ with $\\beta >1$ such that $\\rho (L)\\ge \\beta $ for every $L \\in \\mathcal {L} (H)$ .", "Lemma REF .2 implies that there are zero-sum sequences $B_k \\in \\mathcal {B} (G_0)$ such that $\\rho ( \\mathsf {L} (B_k) ) \\rightarrow 1$ as $k \\rightarrow \\infty $ , a contradiction.", "3.", "This follows from [20].", "Sets of lengths of numerical monoids have found wide attention in the literature (see, among others, [9], [1], [14]).", "As can be seen from Theorem REF .3, the structure of their sets of lengths is simpler than the structure of sets of lengths of transfer Krull monoids over finite abelian groups.", "Thus it is no surprise that there are infinitely many non-isomorphic numerical monoids whose systems of sets of lengths coincide, and that an analog of Conjecture REF for numerical monoids does not hold true ([1]).", "It is open whether for every $d \\in \\mathbb {N}$ and every $M \\in \\mathbb {N}_0$ there is a strongly primary monoid $D$ such that every AAMP with period $\\lbrace 0,d\\rbrace $ and bound $M$ can (up to a shift) be realized as a set of lengths in $D$ (this would be the analog to the realization theorem given in Proposition REF .2).", "However, for every finite set $L \\subset \\mathbb {N}_{\\ge 2}$ there is a $v$ -noetherian primary monoid $D$ and an element $a \\in D$ such that $L = \\mathsf {L} (a)$ ([21]).", "By Theorem REF and Proposition REF .3, we know that $\\lbrace k, k+1\\rbrace \\in \\mathcal {L} (G)$ for every $k \\ge 2$ and every abelian group $G$ with $|G| \\ge 3$ .", "Furthermore, Theorem REF is in sharp contrast to Theorem REF .1.", "Theorem 5.6 Let $D = D_1 \\times \\ldots \\times D_n$ be the direct product of strongly primary monoids $D_1, \\ldots , D_n$ , which are not half-factorial.", "For every finite nonempty set $L\\subset \\mathbb {N}$ , there is a $y_L \\in \\mathbb {N}_0$ such that $y+L \\notin \\mathcal {L} (D)$ for any $y \\ge y_L$ .", "We have $\\mathcal {L} (D) \\ne \\mathcal {L} (G_0)$ for any subset $G_0$ of any abelian group, and hence $D$ is not a transfer Krull monoid.", "If $D_1, \\ldots , D_n$ are locally tame, then $D$ satisfies the Structure Theorem for Sets of Lengths.", "For every $i \\in [1,n]$ we choose an element $a_i \\in D_i$ such that $|\\mathsf {L} (a_i)| > 1$ .", "1.", "Let $L\\subset \\mathbb {N}$ be a finite nonempty set and let $y_L = |L|(\\mathcal {M} (a_1)+ \\ldots + \\mathcal {M} (a_n))$ .", "Assume to the contrary that there are $y \\ge y_L$ and an element $b=b_1 \\cdot \\ldots \\cdot b_n \\in D$ such that $\\mathsf {L} (b) = y+L$ .", "Then there is an $i \\in [1,n]$ such that $\\min \\mathsf {L} (b_i) \\ge |L| \\mathcal {M} (a_i)$ .", "Then $b_i \\in (D_i \\setminus D_i^{\\times })^{\\min \\mathsf {L} (b_i)} \\subset (D_i \\setminus D_i^{\\times })^{|L| \\mathcal {M} (a_i)} \\subset a_i^{|L|} D_i$ .", "Thus there is a $c_i \\in D_i$ such that $a_i^{|L|}c_i = b_i$ .", "This implies that $|L|\\mathsf {L} (a_i) + \\mathsf {L} (c_i) \\subset \\mathsf {L} (b_i)$ .", "Since $|\\mathsf {L} (a_i)| \\ge 2$ , we infer that $|\\mathsf {L} (b_i)| \\ge |L|+1$ and hence $|L|=|y+L|=|\\mathsf {L} (b)| \\ge |\\mathsf {L} (b_i)| \\ge |L|+1$ , a contradiction.", "2.", "By 1. and Lemma REF .2, the first conclusion follows.", "If $D_1, \\ldots , D_n$ are locally tame, then $D$ satisfies the Structure Theorem by Proposition REF .1.", "Theorem 5.7 Let $D = \\mathcal {F} (\\mathcal {P}) \\times D_1$ be the direct product of a free abelian monoid with nonempty basis $\\mathcal {P}$ and of a locally tame strongly primary monoid $D_1$ , and let $G$ be an abelian group.", "Then $D$ satisfies the Structure Theorem for Sets of Lengths, and the following statements are equivalent : $\\mathcal {L} (D)= \\mathcal {L} (G)$ .", "One of the following cases holds : (b1) $|G|\\le 2$ and $\\rho (D)=1$ .", "(b2) $G$ is isomorphic either to $C_3$ or to $C_2\\oplus C_2$ , $[2,3] \\in \\mathcal {L} (D)$ , $\\rho (D)=3/2$ , and $\\Delta (D)=\\lbrace 1\\rbrace $ .", "(b3) $G$ is isomorphic to $C_3 \\oplus C_3$ , $[2,5] \\in \\mathcal {L} (D)$ , $\\rho (D)=5/2$ , and $\\Delta (D)=\\lbrace 1\\rbrace $ .", "Remark.", "Let $H$ be a $v$ -noetherian weakly Krull monoid.", "If the conductor $(H : \\widehat{H}) \\in v$ -$\\max (H)$ , then by Proposition REF , $\\mathcal {I}_v^* (H)$ is isomorphic to a monoid $D$ as given in Theorem REF .", "Since $\\mathcal {P}$ is nonempty, $\\mathcal {L} (D) = \\lbrace y+L \\mid y \\in \\mathbb {N}_0, L \\in \\mathcal {L} (D_1) \\rbrace $ whence $\\Delta (D)=\\Delta (D_1)$ and $\\rho (D)=\\rho (D_1)$ .", "In particular, $D$ is half-factorial if and only if $D_1$ is half-factorial.", "Since $D_1$ satisfies the Structure Theorem of Sets of Lengths by Theorem REF .3, the same is true for $D$ .", "If $D$ is half-factorial and $\\mathcal {L} (D)=\\mathcal {L} (G)$ , then $\\rho (D)=\\rho (D_1)=1$ and $G$ is half-factorial whence $|G| \\le 2$ by Proposition REF .", "Conversely, if $|G|\\le 2$ and $\\rho (D)=1$ , then $G$ and $D$ are half-factorial and $\\mathcal {L} (G)=\\mathcal {L} (D)$ .", "Thus from now on we suppose that $D_1$ is not half-factorial and that (b1) does not hold.", "Then $\\Delta (D) \\ne \\emptyset $ and we set $\\min \\Delta (D)=d$ .", "(a) $\\Rightarrow $ (b) Theorem REF .3 and Proposition REF .3 imply that $G$ is finite.", "Since $G$ is not half-factorial, it follows that $|G| \\ge 3$ .", "Theorem REF .3 shows that $\\Delta _1 (D) = \\lbrace d\\rbrace $ , and since $1 \\in \\Delta _1 (G) = \\Delta _1 (D)$ , we infer that $d=1$ .", "Corollary 4.3.16 in [20] and [29] imply that $\\max \\lbrace \\exp (G)-2, \\mathsf {r} (G)-1\\rbrace = \\max \\Delta _1 (G) = \\max \\Delta _1 (D) = 1 \\,.$ Therefore $G$ is isomorphic to one of the following groups: $C_2\\oplus C_2$ , $C_3$ , $C_3 \\oplus C_3$ .", "We distinguish two cases.", "CASE 1:   $G$ is isomorphic to $C_2 \\oplus C_2$ or to $C_3$ .", "By Proposition REF , we have $\\mathcal {L} (D) = \\mathcal {L} (C_2 \\oplus C_2) = \\mathcal {L} (C_3) = \\lbrace y + 2k+[0,k] \\mid y,k \\in \\mathbb {N}_0\\rbrace \\,.$ In particular, we have $3/2 = \\rho (G)= \\rho (D)$ and $\\lbrace 1\\rbrace =\\Delta (G)=\\Delta (D)$ .", "CASE 2:   $G$ is isomorphic to $C_3 \\oplus C_3$ .", "By Theorem REF , just using different notation, we have $\\begin{aligned}\\mathcal {L} (D) = \\mathcal {L} (C_3^2) & = \\lbrace [2k, \\ell ] \\mid k \\in \\mathbb {N}_0, \\ell \\in [2k, 5k]\\rbrace \\\\& \\quad \\cup \\ \\lbrace [2k+1, \\ell ] \\mid k \\in \\mathbb {N}, \\ell \\in [2k+1, 5k+2] \\rbrace \\cup \\lbrace \\lbrace 1\\rbrace \\rbrace \\,.\\end{aligned}$ In particular, we have $5/2=\\rho (G)=\\rho (D)$ and $\\lbrace 1\\rbrace =\\Delta (G)=\\Delta (D)$ .", "(b) $\\Rightarrow $ (a) First suppose that Case (b2) holds.", "We show that $\\mathcal {L} (D) = \\bigl \\lbrace y + 2k + [0, k] \\, \\bigm | \\, y,\\, k \\in \\mathbb {N}_0 \\bigr \\rbrace \\,.$ Then $\\mathcal {L} (D)=\\mathcal {L} (G)$ by Proposition REF .", "Since $\\rho (D)=3/2$ and $\\Delta (D)=\\lbrace 1\\rbrace $ , it follows that $\\mathcal {L} (D)$ is contained in the above family of sets.", "Thus we have to verify that for every $y, k \\in \\mathbb {N}_0$ , the set $y+ [2k,3k] \\in \\mathcal {L} (D)$ .", "Since $\\mathcal {P}$ is nonempty, $D$ contains a prime element and hence it suffices to show that $[2k,3k] \\in \\mathcal {L} (D)$ for all $k \\in \\mathbb {N}$ .", "Let $a \\in D$ with $\\mathsf {L} (a) = \\lbrace 2,3\\rbrace $ , and let $k \\in \\mathbb {N}$ .", "Then $\\min \\mathsf {L} (a^k) \\le 2k$ and $\\max \\mathsf {L} (a^k) \\ge 3k$ .", "Since $\\rho ( \\mathsf {L} (a^k)) \\le \\rho (D) =3/2$ , it follows that $\\min \\mathsf {L} (a^k)=2k$ and $\\max \\mathsf {L} (a^k)=3k$ .", "Since $\\Delta (D)=\\lbrace 1\\rbrace $ , we finally obtain that $\\mathsf {L} (a^k)=[2k,3k]$ .", "Now suppose that Case (b3) holds.", "We show that $\\mathcal {L} (D) $ is equal to $\\lbrace [2k, \\ell ] \\mid k \\in \\mathbb {N}_0, \\ell \\in [2k, 5k]\\rbrace \\ \\cup \\ \\lbrace [2k+1, \\ell ] \\mid k \\in \\mathbb {N}, \\ell \\in [2k+1, 5k+2] \\rbrace \\cup \\lbrace \\lbrace 1\\rbrace \\rbrace \\,.$ Then $\\mathcal {L} (D)=\\mathcal {L} (G)$ by Theorem REF .", "Since $\\rho (D)=5/2$ and $\\Delta (D)=\\lbrace 1\\rbrace $ , it follows that $\\mathcal {L} (D)$ is contained in the above family of sets.", "Now the proof runs along the same lines as the proof in Case (b2).", "We show that the Cases (b2) and (b3) in Theorem REF can actually occur.", "Recall that numerical monoids are locally tame and strongly primary.", "Let $D_1$ be a numerical monoid distinct from $(\\mathbb {N}_0,+)$ , say $\\mathcal {A} (D_1)=\\lbrace n_1, \\ldots , n_t\\rbrace $ where $t \\in \\mathbb {N}_{\\ge 2}$ and $1 < n_1 < \\ldots < n_t$ .", "Then, by [13] and [9], $\\rho (D_1) = \\frac{n_t}{n_1} \\quad \\text{and} \\quad \\min \\Delta (D_1) = \\gcd (n_2-n_1, \\ldots , n_t - n_{t-1}) \\,.$ Suppose that $\\rho (D_1) = m/2$ with $m \\in \\lbrace 3,5\\rbrace $ and $\\Delta (D_1) =\\lbrace 1\\rbrace $ .", "Then there is an $a \\in D_1$ with $\\mathsf {L} (a) = [2,m] \\in \\mathcal {L} (D_1)$ .", "Clearly, there are non-isomorphic numerical monoids with elasticity $m/2$ and set of distances equal to $\\lbrace 1\\rbrace $ .", "Theorem 5.8 Let $R$ be a $v$ -noetherian weakly Krull domain with conductor $\\lbrace 0\\rbrace \\subsetneq \\mathfrak {f} = (R : \\widehat{R}) \\subsetneq R$ , and let $\\pi \\colon \\mathfrak {X} ( \\widehat{R}) \\rightarrow \\mathfrak {X} (R)$ be the natural map defined by $\\pi ( \\mathfrak {P}) = \\mathfrak {P} \\cap R$ for all $\\mathfrak {P} \\in \\mathfrak {X} (\\widehat{R})$ .", "$\\mathcal {I}_v^* (H)$ is locally tame with finite set of distances, and it satisfies the Structure Theorem for Sets of Lengths.", "If $\\pi $ is not bijective, then $\\mathcal {L} \\big ( \\mathcal {I}_v^* (H) \\big ) \\ne \\mathcal {L} (G_0)$ for any finite subset $G_0$ of any abelian group and for any subset $G_0$ of an infinite cyclic group.", "In particular, $\\mathcal {I}_v^* (H)$ is not a transfer Krull monoid of finite type.", "If $R$ is seminormal, then the following statements are equivalent : $\\pi $ is bijective.", "$\\mathcal {I}_v^* (H)$ is a transfer Krull monoid of finite type.", "$\\mathcal {I}_v^* (H)$ is half-factorial.", "Suppose that the class group $\\mathcal {C}_v (R)$ is finite.", "The monoid $R^{\\bullet }$ of nonzero elements of $R$ is locally tame with finite set of distances, and it satisfies the Structure Theorem for Sets of Lengths.", "If $\\pi $ is not bijective, then $\\mathcal {L} (R^{\\bullet }) \\ne \\mathcal {L} (G_0)$ for any finite subset $G_0$ of any abelian group and for any subset $G_0$ of an infinite cyclic group.", "In particular, $R$ is not a transfer Krull domain of finite type.", "If $\\pi $ is bijective, $R$ is seminormal, every class of $\\mathcal {C}_v (R)$ contains a $\\mathfrak {p} \\in \\mathfrak {X} (R)$ with $\\mathfrak {p} \\lnot \\supset \\mathfrak {f}$ , and the natural epimorphism $\\delta \\colon \\mathcal {C}_v (R) \\rightarrow \\mathcal {C}_v(\\widehat{R})$ is an isomorphism, then there is a weak transfer homomorphism $\\theta \\colon R^{\\bullet } \\rightarrow \\mathcal {B} ( \\mathcal {C}_v(R))$ .", "In particular, $R$ is a transfer Krull domain of finite type.", "Since $\\mathfrak {f} \\ne R$ , it follows that $R \\ne \\widehat{R}$ and that $R$ is not a Krull domain.", "We use the structural description of $\\mathcal {I}_v^* (H)$ as given in Proposition REF .", "1.", "(a) and 2.", "(a) Both monoids, $R^{\\bullet }$ and $\\mathcal {I}_v^* (H)$ , are locally tame with finite set of distances by [20].", "Furthermore, they both satisfy the Structure Theorem for Sets of Lengths by Proposition REF (use Propositions REF and REF ).", "1.", "(b) and 2.", "(b) Suppose that $\\pi $ is not bijective.", "Then $\\rho \\big ( \\mathcal {I}_v^* (H) \\big ) = \\rho (R^{\\bullet }) = \\infty $ by [20].", "Let $G_0$ be a finite subset of an abelian group $G$ .", "Then $\\mathcal {B} (G_0)$ is finitely generated, the Davenport constant $\\mathsf {D} (G_0)$ is finite whence the set of distances $\\Delta (G_0)$ and the elasticity $\\rho (G_0)$ are both finite (see [20]).", "Thus $\\mathcal {L} \\big ( \\mathcal {I}_v^* (H) \\big ) \\ne \\mathcal {L} (G_0)$ and $\\mathcal {L} (R^{\\bullet }) \\ne \\mathcal {L} (G_0)$ .", "If $G_0$ is a subset of an infinite cyclic group, then the set of distances is finite if and only if the elasticity is finite by [18], and hence the assertion follows again.", "1.", "(c) Suppose that $R$ is seminormal.", "By 1.", "(b) and since half-factorial monoids are transfer Krull monoids of finite type, it remains to show that $\\pi $ is bijective if and only if $\\mathcal {I}_v^* (H)$ is half-factorial.", "Since $R$ is seminormal, all localizations $R_{\\mathfrak {p}}$ with $\\mathfrak {p} \\in \\mathfrak {X} (H)$ are seminormal.", "Thus $\\mathcal {I}_v^* (H)$ is isomorphic to a monoid of the form $\\mathcal {F} ( \\mathcal {P}) \\times D_1 \\times \\ldots \\times D_n$ , where $n \\in \\mathbb {N}$ and $D_1, \\ldots , D_n$ are seminormal finitely primary monoids, and this monoid is half-factorial if and only if each monoid $D_1, \\ldots , D_n$ is half-factorial.", "By [22], $D_i$ is half-factorial if and only if it has rank one for each $i \\in [1,n]$ , and this is equivalent to $\\pi $ being bijective (see [20]).", "2.", "(c) This follows from [22].", "Note that every order $R$ in an algebraic number field is a $v$ -noetherian weakly Krull domain with finite class group $\\mathcal {C}_v (R)$ such that every class contains a $\\mathfrak {p} \\in \\mathfrak {X} (R)$ with $\\mathfrak {p} \\lnot \\supset \\mathfrak {f}$ .", "If $R$ is a $v$ -noetherian weakly Krull domain as above, then Theorems REF , REF , and REF provide further instances of when $R$ is not a transfer Krull domain, but a characterization of the general case remains open.", "We formulate the following problem (see also [17]).", "Problem 5.9 Let $H$ be a $v$ -noetherian weakly Krull monoid with nonempty conductor $(H : \\widehat{H})$ and finite class group $\\mathcal {C}_v (H)$ .", "Characterize when $H$ and when the monoid $\\mathcal {I}_v^* (H)$ are transfer Krull monoids resp.", "transfer Krull monoids of finite type." ] ]
1606.05063
[ [ "Lift force due to odd (Hall) viscosity" ], [ "Abstract We study the problem of flow of a neutral gas past an infinite cylinder at right angle to its axis at low Reynolds number when the fluid is characterized by broken time-reversal invariance, and hence by odd viscosity in addition to the normal even one.", "We solve the Oseen approximation to Navier-Stokes equation and calculate the lift force which appears due to the odd viscosity." ], [ "This line only printed with preprint option Lift force due to odd (Hall) viscosity E. Kogan Jack and Pearl Resnick Institute, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Max-Planck-Institut fur Physik komplexer Systeme, Dresden 01187, Germany Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea We study the problem of flow of a neutral gas past an infinite cylinder at right angle to its axis at low Reynolds number when the fluid is characterized by broken time-reversal invariance, and hence by odd viscosity in addition to the normal even one.", "We solve the Oseen approximation to Navier-Stokes equation and calculate the lift force which appears due to the odd viscosity.", "47.10.ad;47.15.G- Viscosity, i.e.", "the resistance to a flow in which adjacent parts of a fluid move with different velocities, is a basic property of all classical and quantum liquids, and becomes relevant for electron liquids as well, when disorder and coupling to the lattice are not too strong.", "In rotationally-invariant systems, when the time-reversal symmetry is not broken, the viscosity tensor is entirely described by two scalar transport coefficients, the shear and the bulk viscosities (denoted by $\\eta $ and $\\zeta $ respectively), which are both dissipative.", "The Hall viscosity, also known as the Hall viscosity and Lorentz shear modulus, is an off-diagonal viscosity term that is dissipationless and produces forces perpendicular to the direction of the fluid flow.", "It can have a quantum mechanical origin in, for example, systems exhibiting the quantum Hall effect [1], [3], [2], [6], [7], [8], [9], [10], [11], [4], [5], [12] or a classical origin in plasmas at finite-temperature [13].", "More generally, Hall viscosity presents a link between quantum Hall systems, plasmas and liquid crystals [14].", "We will not focus on the microscopic origin of the Hall viscosity coefficient, but only assume it to be non-vanishing in conjunction with the usual viscosity coefficients.", "Despite the extensive theoretical discussions on its properties, the question of how to measure the odd viscosity still poses a challenge.", "We will show that odd viscosity of a neutral gas can be determined from the measurement of the lift force which acts due to flow at right angle to the axis of infinite cylinder Euler's equation can be written in the form $\\frac{\\partial }{\\partial t}(\\rho v_i)=-\\frac{\\partial \\Pi _{ik}}{\\partial x_k},$ where $\\rho $ is the fluid density, $v_i$ is the fluid velocity, and $\\Pi _{ik}$ is the momentum flux density tensor.", "The equation of motion of a viscous fluid may be obtained by subtracting from the \"ideal\" momentum flux a viscous stress tensor $\\sigma ^{\\prime }_{ik}$ which gives the viscous transfer of momentum in the fluid [15] $\\Pi _{ik}=p\\delta _{ik}+\\rho v_iv_k-\\sigma ^{\\prime }_{ik}$ .", "Considering the system with the symmetry, equivalent to those which characterizes isotropic system with magnetic field along the $z$ -axis, we can write down the components of the stress tensor as [16] $\\sigma ^{\\prime }_{xx}&=&-\\eta _0\\left(V_{zz}-\\frac{1}{3}\\text{div}\\;{\\bf V}\\right)+\\eta _1\\left(V_{xx}-V_{yy}\\right)\\nonumber \\\\&+&2\\eta _3V_{xy}+\\zeta \\; \\text{div}\\;{\\bf V}+\\zeta _1V_{zz}\\nonumber \\\\\\sigma ^{\\prime }_{yy}&=&-\\eta _0\\left(V_{zz}-\\frac{1}{3}\\text{div}\\;{\\bf V}\\right)+\\eta _1\\left(V_{xx}-V_{yy}\\right)\\nonumber \\\\&-&2\\eta _3V_{xy}+\\zeta \\;\\text{div}\\;{\\bf V}+\\zeta _1V_{zz}\\nonumber \\\\\\sigma ^{\\prime }_{zz}&=&2\\eta _0\\left(V_{zz}-\\frac{1}{3}\\text{div}\\;{\\bf V}\\right)+\\zeta \\;\\text{div}\\;{\\bf V}+\\zeta _1(V_{zz}+\\text{div}\\;{\\bf V})\\nonumber \\\\\\sigma ^{\\prime }_{xy}&=&2\\eta _1V_{xy}-\\eta _3\\left(V_{xx}-V_{yy}\\right)\\nonumber \\\\\\sigma ^{\\prime }_{xz}&=&2\\eta _2V_{xz}+2\\eta _4V_{yz}\\nonumber \\\\\\sigma ^{\\prime }_{yz}&=&2\\eta _2V_{yz}-2\\eta _4V_{xz},$ where $V_{ij}=\\frac{1}{2}\\left(\\frac{\\partial v_i}{\\partial x_j}+\\frac{\\partial v_j}{\\partial x_i}\\right).$ Here the terms with the coefficients $\\eta _3$ and $\\eta _4$ describe odd viscosity.", "As a side note, we would like to mention that Eq.", "(REF ) would be also valid for magnitoactive plasma.", "In this case the equations are even somewhat simplified by the vanishing of two second viscosity coefficients $\\zeta $ and $\\zeta _1$ [16].", "For the reader to get a feeling of the physics behind the viscosity coefficients introduced above, we would like to reproduce the results obtained in the framework of the fluid dynamics in a magnitoactive plasma.", "In this case in the strong magnetic field one gets [16] $\\eta _1&=&\\frac{\\eta _2}{4}=\\frac{2\\pi ^{1/2}(ze)^4L_iN_i^2}{5(MT)^{1/2}\\omega _{Bi}^2} \\\\\\eta _3&=&\\frac{\\eta _4}{2}=\\frac{N_iT}{2\\omega _{Bi}},$ where $T$ is the temperature, $M$ is the ion mass, $ze$ is the ion charge, $N_i$ is the ion concentration, $\\omega _{Bi}=zeB/Mc$ , $B$ is the magnetic field, and $L_i$ is the Coulomb logarithm.", "However, for the case of plasma, if we assume, as it is normally done, that magnetic field is the source of the odd viscosity, we should include Lorentz force into Eq.", "(REF ).", "The results for this case will be reported separately.", "Eq.", "(REF ) is simplified if we consider two-dimensional (in the $xy$ plane) fluid motion, and additionally assume the fluid incompressible (${\\bf \\nabla \\cdot v} = 0$ ).", "Combining this equation with Eq.", "(REF ) we obtain modified Navier-Stokes equation, which can be written in vector form [3] $\\rho \\left[\\partial _t{\\bf v} + ({\\bf v\\cdot \\nabla }){\\bf v}\\right] =-{\\bf \\nabla }p + \\eta _1\\Delta {\\bf v}+\\eta _3\\Delta {\\bf v}^*,$ where the vector ${\\bf v}$ is two-dimensional, and the dual is defined, as usual, by $v_i^*=\\epsilon _{ij}v_j.$ Here probably it is appropriate to mention another aspect of Hall viscosity $\\eta _3$ relevant for modern studies of the quantum Hall systems.", "The Hall viscosity is an instance of a class of \"anomalous transport coefficients\" - of which the Hall conductivity is the best known example - which are given by the imaginary part of an off-diagonal linear response function [12], in this case [4] $\\eta _3=\\lim _{\\omega \\rightarrow 0}\\text{Im}\\frac{<<P_{xx};P_{xy}>>_{\\omega }}{\\omega },$ where $<<P_{xx};P_{xy}>>_{\\omega }$ is a shorthand for the off-diagonal stress-stress response function.", "Returning to Eq.", "(REF ) we realize that equation ${\\bf \\nabla \\cdot v} = 0$ allows to introduce the stream function $\\psi $ $v_x=\\frac{\\partial \\psi }{\\partial y},\\;\\;\\;v_y=-\\frac{\\partial \\psi }{\\partial x}.$ Noticing that $v^*=-\\text{grad}\\;\\psi ,$ we can rewrite Eq.", "(REF ) in the form [3] $\\rho \\left[\\partial _t{\\bf v} + ({\\bf v\\cdot \\nabla }){\\bf v}\\right] =-{\\bf \\nabla }(p+\\eta _3\\Delta \\psi ) + \\eta _1\\Delta {\\bf v}.$ The pressure can be eliminated from Eq.", "(REF ) by taking the curl of both sides.", "We get the well known equation $\\partial _t(\\text{curl}\\;{\\bf v}) + ({\\bf v\\cdot \\nabla })(\\text{curl}\\;{\\bf v})-(\\text{curl\\;}{\\bf v}\\cdot {\\bf \\nabla }){\\bf v}\\nonumber \\\\= \\nu \\Delta (\\text{curl}\\;{\\bf v}).$ When the velocity distribution is known, the pressure distribution in the fluid can be found from Eq.", "(REF ).", "Substituting $\\psi $ for ${\\bf v}$ we get [15] $\\frac{\\partial }{\\partial t}\\Delta \\psi -\\frac{\\partial \\psi }{\\partial x}\\frac{\\partial \\Delta \\psi }{\\partial y}+\\frac{\\partial \\psi }{\\partial y}\\frac{\\partial \\Delta \\psi }{\\partial x}-\\nu \\Delta \\Delta \\psi =0.$ It is evident from Eq.", "(REF ) that typically odd viscosity does not influence the distribution on velocity, but does influence distribution of pressure.", "Additional pressure appearing in the system due to odd viscosity can be presented as $\\Delta p=\\eta _3\\left(\\text{curl}\\;{\\bf v}\\right)_z.$ Thus odd viscosity, at least in principle, can be measure by measuring forces the fluid apply to the bodies it flows past.", "To warm up, consider two elementary problems.", "Consider first, steady flow between two fixed parallel lines in the presence of a pressure gradient.", "We choose the $x$ -axis in the direction of the motion of the fluid, and $y$ -axis in the perpendicular direction, so the lines are given by equations $y=0$ and $y=h$ .", "Since the velocity clearly is in the $x$ direction and does not depend upon $x$ , we can look for $\\psi $ in the form $\\psi =\\psi (y)$ , and Eq.", "(REF ) gives $\\frac{d^4 \\psi }{dy^4}=0.$ The two boundary conditions are $\\left.\\frac{d\\psi }{dy}\\right|_{y=0}=\\left.\\frac{d\\psi }{dy}\\right|_{y=h}=0.$ The solution of Eq.", "(REF ) is $\\psi =Ay^2\\left(\\frac{3}{2}h-y\\right)$ (the constant term in $\\psi $ is irrelevant).", "Thus for velocity we obtain $v_x=3Ay(h-y)$ and $p=-6A\\left(\\eta _1 x-{\\eta _3}y\\right)+\\text{const}.$ To consider our next elementary example, motion of a fluid between two coaxial circles rotating with radii $R_1$ , $R_2$ ($R_2>R_1$ ), rotating about their axis with angular velocities $\\Omega _1$ , $\\Omega _2$ , we need to rewrite Eqs.", "(REF ), (REF ) and (REF ) in polar coordinates.", "The dual becomes $v^*_{\\theta }=-v_r,\\;\\;\\;v_r=v_{\\theta },$ Eq.", "(REF ) becomes $v_\\theta =\\frac{\\partial \\psi }{\\partial r},\\;\\;\\;v_r=-\\frac{1}{r}\\frac{\\partial \\psi }{\\partial \\theta }.$ and Eq.", "(REF ) becomes $\\frac{\\partial }{\\partial t}\\Delta \\psi -\\frac{1}{r}\\left(\\frac{\\partial \\psi }{\\partial r}\\frac{\\partial \\Delta \\psi }{\\partial \\theta }+\\frac{\\partial \\psi }{\\partial \\theta }\\frac{\\partial \\Delta \\psi }{\\partial r}\\right)-\\nu \\Delta \\Delta \\psi =0.$ From the symmetry we have $\\psi =\\psi (r)$ , and Eq.", "(REF ) gives $\\frac{1}{r}\\frac{d}{dr}\\left\\lbrace r\\frac{d}{dr}\\left[\\frac{1}{r}\\frac{d}{dr}\\left(r\\frac{d\\psi }{dr}\\right)\\right]\\right\\rbrace =0$ The two boundary conditions are $\\left.\\frac{d\\psi }{dr}\\right|_{r=R_1}=\\Omega _1R_1^2,\\;\\;\\;\\left.\\frac{d\\psi }{dr}\\right|_{r=R_2}=\\Omega _2R_1^2.$ The solution of Eq.", "(REF ) is $\\psi =\\frac{\\Omega _2R_2^2-\\Omega _1R_1^2}{R_2^2-R_1^2}\\frac{r^2}{2}+\\frac{(\\Omega _1-\\Omega _2)R_1^2R_2^2}{R_2^2-R_1^2}\\ln r.$ Hence, for velocity we obtain [15] $v_{\\phi }=\\frac{\\Omega _2R_2^2-\\Omega _1R_1^2}{R_2^2-R_1^2}r+\\frac{(\\Omega _1-\\Omega _2)R_1^2R_2^2}{R_2^2-R_1^2}\\frac{1}{r}.$ The pressure is $p=\\rho \\frac{v_{\\phi }^2}{r}-\\eta _3\\Delta \\psi +\\text{const}.$ Thus we see that odd viscosity does not influence ether velocity distribution or the applied torque, but just pressure on the circles, which are proportional to the rate of rotation (this fact can be seen directly from Eq.", "(REF )).", "Now consider the lift force on the cylinder of radius $a$ moving in a fluid with velocity ${\\bf U}$ , corresponding to low Reynolds number.", "(Because the problem is essentially two-dimensional we'll use Eq.", "(REF ).)", "The natural desire would be to ignore the term $({\\bf v\\cdot \\nabla }){\\bf v}$ in this equation.", "However, it is well known that (in the absence of odd viscosity) for the problem considered, such amputated equation does not have a solution [17], [15].", "The way out was discovered by Oseen and Lamb [17].", "It consists in approximating Navier-Stokes equation $\\rho \\left[\\partial _t{\\bf v} + ({\\bf v\\cdot \\nabla }){\\bf v}\\right] =-{\\bf \\nabla }p + \\eta _1\\Delta {\\bf v}$ by the equation $\\rho ({\\bf U\\cdot \\nabla }){\\bf v} =-{\\bf \\nabla }p + \\eta _1\\Delta {\\bf v}.$ Hence, in our case one has to solve equation $\\rho ({\\bf U\\cdot \\nabla }){\\bf v} =-{\\bf \\nabla }p + \\eta _1\\Delta {\\bf v}+\\eta _3\\Delta {\\bf v}^*,$ or alternatively $\\rho ({\\bf U\\cdot \\nabla }){\\bf v} =-{\\bf \\nabla }(p+\\eta _3\\Delta \\psi ) + \\eta _1\\Delta {\\bf v}.$ Let us recall classical results due to Lamb (in the absence of odd viscosity) [17].", "Going to the frame of reference moving with the circle, one obtains that Eq.", "(REF ) is solved by the substitutions ($x$ axis is chosen in the direction of ${\\bf U}$ ): $v_x&=&-\\frac{\\partial \\phi }{\\partial x}+\\frac{1}{2k}\\frac{\\partial \\chi }{\\partial x}-\\chi \\\\v_y&=&-\\frac{\\partial \\phi }{\\partial y}+\\frac{1}{2k}\\frac{\\partial \\chi }{\\partial y}\\\\p&=&\\rho U \\frac{\\partial \\phi }{\\partial x}$ where $k=U/2\\nu $ , provided that functions $\\phi $ and $\\chi $ satisfy equations $\\Delta \\phi &=&0\\\\\\left(\\Delta -2k\\frac{\\partial }{\\partial x}\\right)\\chi &=&0.$ The solution of Eq.", "(REF ) is $\\chi =Ce^{kx}K_0(kr).$ where $K_0$ is the modified Bessel function.", "For small values of $kr$ we have $\\chi =-C(1+kx)\\left(\\gamma + \\ln \\left(\\frac{1}{2}ka\\right)\\right),$ where $\\gamma $ is Euler' constant.", "Hence for these values $&&\\frac{1}{2k}\\frac{\\partial \\chi }{\\partial x}-\\chi =-\\frac{C}{2k}\\left\\lbrace k\\left[\\frac{1}{2}-\\gamma -\\ln \\left(\\frac{1}{2}ka\\right)\\right]\\right.\\nonumber \\\\&&+\\left.\\frac{\\partial }{\\partial x}\\ln r-\\frac{1}{2}kr^2\\frac{\\partial ^2}{\\partial x^2}\\ln r+\\dots \\right\\rbrace \\\\&&\\frac{1}{2k}\\frac{\\partial \\chi }{\\partial y}=-\\frac{C}{2k}\\left\\lbrace \\frac{\\partial }{\\partial y}\\ln r-\\frac{1}{2}kr^2\\frac{\\partial ^2}{\\partial x\\partial y}\\ln r+\\dots \\right\\rbrace .\\nonumber \\\\$ If we set $\\phi =A_0\\ln r +A_1\\frac{\\partial }{\\partial x}\\ln r+\\dots ,$ than we'll find that conditions $v_x=-U$ , $v_y=0$ at $r=a$ will be satisfied if we set approximately $C&=&\\frac{2U}{\\frac{1}{2}-\\gamma -\\ln \\left(\\frac{1}{2}ka\\right)}\\\\A_0&=&-\\frac{C}{2k}\\\\A_1&=&\\frac{1}{4}Ca^2$ The vorticity is thus given by the equation [17] $\\left(\\text{curl}\\;{\\bf v}\\right)_z=-kC\\frac{y}{r}e^{kx}K_1(kr),$ Taking into account that $ka\\ll 1$ we obtain at the surface of the cylinder $\\left(\\text{curl}\\;{\\bf v}\\right)_z=-\\frac{C}{a}\\sin \\theta .$ Thus in the approximation considered $\\Delta F_x&=&0\\\\\\Delta F_y&=&-\\pi \\eta _3C.$ Notice, that the drag force on the unit of length of the cylinder [17] $F_x=2\\pi \\eta _1 C.$ To conclude we would like to emphasize again that the calculated lift force appears exclusively due to the non-dissipative odd viscosity, and hence can exist only if the time reversal invariance in the system is broken, either explicitly or implicitly.", "We also want to mention that the obtained results can be applied, in addition to neutral gas, mentioned in the beginning of the paper, to other systems, like, for example, nematic liquid crystals.", "Discussions with M. Sherafati, which actually initiated the present work, are gratefully acknowledged.", "The authors also cordially thanks for the hospitality extended to him during his stay: Max-Planck-Institut fur Physik komplexer Systeme, where the work was initiated, and Center for Theoretical Physics of Complex Systems, where the work continued." ] ]
1606.05082
[ [ "Multiple Outflows in the Giant Eruption of a Massive Star" ], [ "Abstract The supernova impostor PSN J09132750+7627410 in NGC 2748 reached a maximum luminosity of approximately -14 mag.", "It was quickly realized that its was not a true supernova, but another example of a non-terminal giant eruption.", "PSN J09132750+7627410 is distinguished by multiple P Cygni absorption minima in the Balmer emission lines that correspond to outflow velocities of -400, -1100, and -1600 km/s.", "Multiple outflows have been observed in only a few other objects.", "In this paper we describe the evolution of the spectrum and the P Cygni profiles for three months past maximum, the post-maximum formation of a cool, dense wind, and the identification of a possible progenitor.", "One of the possible progenitors is an infrared source.", "Its pre-eruption spectral energy distribution suggests a bolometric luminosity of -8.3 mag and a dust temperature of 780 degrees K. If it is the progenitor it is above the AGB limit unlike the intermediate luminosity red transients.", "The three P Cygni profiles could be due to ejecta from the current eruption, the wind of the progenitor, or previous mass loss events.", "We suggest that they were all formed as part of the same high mass loss event and are due to material ejected at different velocities or energies.", "We also suggest that multiple outflows during giant eruptions may be more common than reported." ], [ "Introduction", "Transient surveys are finding an increasing number of what appear to be non-terminal giant eruptions.", "Many of these giant eruptions are spectroscopically similar to Type IIn supernovae and thus receive a supernova (SN) designation, but are later recognized as sub-luminous or their spectra and light curves do not develop like true supernovae.", "Consequently, they are often referred to as “supernova impostors” [28].", "These impostors or giant eruptions are examples of high mass loss episodes apparently from evolved massive stars (see Van Dyk & Matheson 2012 for a review and references therein).", "Authors often refer to them as Luminous Blue Variables (LBVs), but these giant eruptions are distinctly different from LBV/S Doradus variability in which the star does not increase in luminosity and the eruption or maximum light can last for several years.", "PSN J09132750+7627410 did not receive a supernova designation.", "It was posted on the CBAT Transient Objects Confirmation Page on 2015 February 10 by K. Itagaki as a possible supernova (PSN) in NGC 2748.", "The first reported magnitudes of 17.7 to 18.1 suggested a luminosity of $\\approx $ -14 mag based on membership in NGC 2748 at a mean distance of 20.97 Mpc via NED, and based on Tully-Fisher and kinematic distances.", "Thus, it was likely not a true supernova, but another example of a giant eruption.", "A spectrum obtained only a day later, on 2015 February 11 reported by [23] showed narrow emission lines.", "They also measured an apparent V magnitude of 18.7 and a luminosity of -13 mag.", "Based on the spectral appearance and the luminosity, they suggested that this PSN was a SN impostor.", "Our first spectrum of PSN J09132750+7627410 [9] observed on 2015 February 16, showed multiple P Cygni absorption components in the prominent Balmer emission lines.", "These features could be due to ejecta from the current eruption, from previous mass loss events, or the wind of the progenitor.", "Evidence for multiple outflows have been previously observed in the spectra of only a few objects; two impostor eruptions (SN2000ch, SN Hunt 248), the peculiar SN2009ip, and a Type IIn supernova (SN2005gj), see §4.", "For that reason we obtained additional spectra and photometry.", "In this brief Paper we present our observations, describe the spectrum and our measurements, discuss a possible progenitor, and the origin of the multiple P Cygni absorption features.", "Moderate resolution spectra of PSN J09132750+7627410 were observed with the MODS1 spectrograph on the Large Binocular Telescope (LBT) in February, April and May, 2015.", "The MODS1 uses a dichroic to obtain blue and red spectra simultaneously with the G400L and G750L gratings, respectively.", "The total wavelength coverage is from 3200Å to more than 1$\\mu $ m. We used a 1slit yielding a resolution of 1500 in the blue and 2000 in the red.", "This gives a velocity resolution of 150 km s$^{-1}$ at H$\\alpha $ .", "The two dimensional spectra were initially reduced using the modsCCDred pipeline for bias subtraction and flat fielding.", "The spectra were then extracted, and wavelength and flux calibrated using the standard IRAF twodspec and onedspec packages.", "The extracted spectra cover the wavelength ranges 3600 – 5600 and 6000 – 9000 Å.", "The MODS1 uses standard lamps exposed during the afternoon for wavelength calibration, but despite a flexure model, there remain wavelength calibration uncertainties.", "For this reason we used the night sky lines for the wavelength calibration in the red.", "This may introduce a small velocity offset between the blue and red spectra.", "But, for this object in NGC 2748 (velocity +1476 km sec$^{-1}$ ), the difference is not significant for our discussion.", "The journal of observations is given in Table 1." ], [ "Photometry During the Eruption", "Broadband CCD photometry with ASTRODON Johnson-Cousins filters was obtained with the 20-inch telescope at the Barber Observatory with an Apogee U42 CCD camera using a back-illuminated E2V CCD42-40 chip.", "All images were flat-fielded and bias and dark subtracted.", "Aperture photometry was measured using VPHOT (https://www.aavso.org/vphot) developed primarily by Geir Klingenberg (coding and design) and Arne Henden (photometry).", "Photometry was measured in circular apertures with a diameter 3.0 times the FWHM of the stellar profiles.", "Sky background was sampled from an annulus centered on each star with an inner radius of 11.5$$ and 3$$ wide.", "No effort was made to subtract the underlying contribution from the galaxy.", "The galaxy contribution is estimated from images without the target present to be V $=$ 21 mag arcsec$^{-2}$ (about V= 19 mag in a typical aperture).", "The brightness of the target is an unweighted average measured with respect to 9 reference stars in the magnitude range V = 14.0 to 16.2 and within 10 arcminutes of the galaxy (but well separated from it) selected from the UCAC4 catalog (Zacharias et al.", "2013) using the prescription of Toone (2005).", "Photometric error for the target was calculated as the quadrature sum of the CCD equation noise for the target, plus the standard deviation of the reference stars.", "We used the standard photometric transformations for the optical system.", "The photometric observations are summarized in Table 2 which also includes the discovery magnitudes from the CBAT “Transient Objects Followup Reports”.", "The earlest photometry observed during the first 24hrs shows a decline of 0.6 mag which could be due to a difference in calibration.", "We note though that the early I band magnitude is consistent with later photometry.", "Additional post-maximum magnitudes were recorded in the Pan-STARRS Survey for Transients.", "The target is PS15jf.", "The r(P1) and i(P1) magnitudes in Table 2 are on the Pan-STARRS filter system (Tonry et al.", "2012).", "Although the photometric record is sparse, and the onset of the eruption is uncertain, the available data suggest that this eruption was relatively brief.", "PSN J09132750+7627410 was at or near maximum light for only about a month.", "The photometry shows that a decline had begun by about 30 days post-maximum.", "Interestingly, our data point from 2015 May 19 suggests that the object may have brightened somewhat.", "This is supported by the flux-calibrated spectrum from May 20 observed under good conditions.", "Short-term oscillations in brightness are common in these objects as is evident from the erratic behavior of SN2009ip [19], [15], [16].", "To check on variability and a possible recovery, we observed it again in 2016 January and February.", "The target remained below our limit of detectability defined as the median value in the aperture that was not statistically different from the median value in the background annulus.", "We note that the seeing and observing conditions were excellent for the February measurement." ], [ "Pre-Eruption Images and a Possible Progenitor", "NGC 2748 is a well-studied spiral galaxy at a distance of 20.97 Mpc, modulus 31.60 mag, from NED.", "It is the site of a super-massive black hole in its nucleus and two previous supernovae, SN1985A (Ia) and SN2013ff (Ic).", "Consequently, there are numerous space-based images in the Mikulski Archive for Space Telescopes (MAST), but PSN J09132750+7627410 is in the outer parts of the galaxy, and was just off the frame in most of the HST images.", "We identified the target field on four HST/WFPC2 images of NGC 2748; two with F450W and two with F814W.", "(GO-9042,PI:Smartt) obtained 2001 July 6.", "The exposure times were 230s each and the target was on the WF3 chip.", "We processed the images using DOLPHOT [4] which successfully fit a PSF to six objects within 2of the target position with sigma threshold 2.5, the default.", "We only considered DOLPHOT detections consistent with a point source (class 1 and 2) using the default WFPC2 DOLPHOT settings.", "Only two objects were above the sky background in all four images.", "The resulting photometry in VEGAMAGs for these two objects is in Table 3.", "The F450W and F814W filters are comparable to the Johnson B and Cousins I bands, respectively, yielding an instrumental F450W -F814W or b-i color.", "For comparison with the target's published position, which we confirmed with astrometry from our CCD imagesWe measured an average postion of RA 09:13:27.55, Dec +76:27:41.1 in ICRS J2000 coordinates with a standard deviation of 0$$ 2 from six separate images using GSC 2.3 stars in the field with astrometry.net [14].", "This agrees very well with the discovery position., we determined the offsets of their WCS image centers relative to ICRS J2000 coordinates using four GSC 2.3 stars on the same frame, shifted in X and Y and with rotation with a standard deviation of 0$$ 4 in RA and Dec.", "The combined uncertainty of the target's position and of the WCS in the HST image is then 0$$ 45.", "Their ICRS positions are included in Table 3.", "The two summed HST frames are shown in Figure 1, centered on the impostor's position and corrected for the offset.", "Star A is within the small circle, radius 0$$ 2, centered on the target's position.", "The fainter star B, further from the center, is just outside a circle with the 0$$ 45 radius.", "It is apparently quite red, but with a large uncertainty.", "Based on its position, star A is the more likely progenitor.", "Its absolute blue magnitude (F450W) would be $\\approx $ -7.3. with a foreground galactic extinction A$_{B}$ of 0.097 [20].", "Its extinction-corrected b-i color of 0.58 mag implies an A-type or early F-type supergiant.", "NGC 2748 has prominent dust lanes, although none are apparent close to the target, so there may be some additional internal extinction and star A would be both brighter and bluer.", "There is also an extended area of low level emission around star A visible in the F814W image which could be due to unresolved stars.", "Figure: The summed HST/WFPC2 F450W and F814W images including the regionaround PSN J09132750+7627410.The smaller dashed circle 02 in radius is centered on the positionof the impostor.", "The cosmic rays have not been removed.We have also identified a source in the Spitzer Science Archive in three separate epochs at 3.6$\\mu $ m and 4.5$\\mu $ m at nearly the same position.", "Since the background is high and variable, we initially used a profile fitting routine to determine the flux distribution across the image and the appropriate radius for aperture photometry.", "We then used the Astropy-affiliated package “photutils” to measure aperture photometry on the Level 2 PBCD mosaic images.", "Since there is a significant gradient in the infrared background, particularly in the 4.5 images, the sky is modeled as a two-dimensional polynomial surface across the source and subtracted from a 1.8radius aperture.", "In each image, subpixel centroids are calculated as the center of mass determined from image moments.", "The magnitudes are included in Table 3 with errors calculated from the uncertai nty maps provided by the Spitzer Science Archive for each field.", "The IRAC images from 2014 have the best signal to noise and are shown with the HST/WFPC2 F814W image for comparison in Figure 2.", "We found no systematic offset between the the IRAC position and the ICRS J2000 using the same reference stars, but with a standard deviation of 0$$ 64 and 0$$ 46 in RA and Dec respectively.", "The WFPC2 and IRAC frames are thus on the same coordinate system.", "The IRAC position is included in Table 3.", "We were not able to identify the source in the WISE survey, because it is not resolvable from the background galaxy.", "Figure: The HST/WFPC2 F814W image and IRAC 3.6μ\\mu m and 4.5μ\\mu m imagesfrom 2014 of the region around PSN J09132750+7627410.", "The WCS coordinateswere corrected to the ICRS J2000 using the same reference stars so that theWFPC2 and IRAC frames are on the same coordinate system..The spectral energy distributions (SED) are shown in Figure 3 for stars A and B and the IRAC source.", "The IRAC source is very red with a rising energy distribution to longer wavelengths similar to obscured AGB stars and OH/IR stars [10] with extensive circumstellar dust.", "Without longer wavelength data it is not possible to know where the SED peaks.", "We show Planck curves fits to the data points from 2014 with a color temperature of 780K.", "The source could be both cooler and more luminous.", "Planck curves are also shown fit to the optical photometry for stars A and B.", "Figure: The SEDs for stars A and B and the infrared source near the positionof PSN J09132750+7627410.", "Planck curve fits are shown with color temperaturesof 8200K and 4600K for stars A , filled circles, and B,open circles, and 780K for the infrared source.", "No corrections for possible interstellar extinction have been applied.We have two possibilities for an optical progenitor; star A or B, alone or in combination with the infrared source.", "The photometry for star A is incompatible with the IR SED.", "If A is the progenitor, it would be an intermediate-type supergiant of relatively low luminosity (9 $\\times $ 10$^{4}$ L$_{\\odot }$ , M$_{Bol} = -7.6$ mag) and initial mass $\\approx $ 15 – 20 M$_{\\odot }$ .", "Star B has a luminosity of 4.4 $\\times $ 10$^{4}$ L$_{\\odot }$ and based on its somewhat uncertain color, it could be an evolved star of approximately 12 – 15 M$_{\\odot }$ .", "Based on these limited SEDs, it is possible that the infrared source could be associated with star B.", "If so, then star B's luminosity would be dominated by the infrared radiation.", "Given the lower spatial resolution with IRAC, another possibility is that the infrared source, possibly with no optical counterpart, is the progenitor.", "The postions of the impostor and the IRAC source marginally agree at the 1 sigma level and are coincident within two sigma of their positional uncertainties.", "If this is the case, PSN J09132750+7627410 could be similar to the dusty Intermediate Luminosity Red Transients (ILRTs) like SN 2008S [21] and the 2008 NGC 300 OT [3], [7].", "Assuming that the infrared source is similar to an AGB, we use its 3.6$\\mu $ m $-$ 4.5$\\mu $ m color index in 2014 with the bolometric calibration from [2] to derive a bolometric magnitude of -8.3 mag (1.6 $\\times $ 10$^{5}$ L$_{\\odot }$ ).", "Integrating the Planck curve gives a luminosity of 10$^{5}$ L$_{\\odot }$ .", "This luminosity is well above the nominal AGB limit at M$_{Bol} \\cong $ -7.0 mag.", "The object is more luminous and therefore more massive that the ILRTs which are close to the AGB limit, see the HR diagram (Fig.", "14) in [7].", "It is too red to be a foreground dwarf.", "Alternatively, it could be an H II region or the chance superposition of a background object such as an AGN, but we could not find any catalogued objects at its position.", "Assuming that the infrared source is in NGC 2748, with this luminosity, the candidate would of course be a massive star of $\\approx $ 25 M$_{\\odot }$ .", "It could be a very dusty red supergiant perhaps similar to the OH/IR stars in our galaxy.", "They are potential supergiants, but their distances are not known.", "[24] and [12] have identified several luminous, optically obscured stars in M33 and other nearby galaxies which may be similar to the supergiant OH/IR stars.", "The infrared source's mid-infrared color and dust temperature are similar to many of the sources listed by [12], although their objects are significantly more luminous with log L $\\sim $ 5.5 to 6.0 L$_{\\odot }$ .", "It is tempting to identify the infrared source with the erupting star, but more information was needed.", "Time was requested with Spitzer to confirm if the IRAC source had survived or changed after the eruption, but was denied.", "Based on their positions, stars A and B and the infrared source are all possible progenitors, but we favor a tentative identification with the infrared source.", "If the infrared source is the progenitor, the underlying star is most likely a cool or intermediate temperature evolved supergiant that may be transiting the HR Diagram to the blue, similar to more luminous examples such as VarA in M33.", "In this transition, the stars enter a period of enhanced instability that leads to high mass loss episodes.", "Followup imaging of PSN J09132750+7627410 with HST and Spitzer would be very worthwhile.", "Our first and highest quality spectra from 2015 February 16 were observed about six days after the reported discovery and the presumed maximum.", "The blue spectrum is shown in Figure 4.", "The blue and red spectra show strong narrow Balmer emission lines with prominent P Cygni features from H$\\alpha $ to H$\\epsilon $ .", "Three absorption minima are present in H$\\beta $ , H$\\gamma $ , H$\\delta $ and at least two are clearly identified in H$\\alpha $ .", "The H$\\alpha $ and H$\\beta $ profiles are shown in Figure 5 with the mulitple absorption minima identified.", "In addition to the narrow peaks, the hydrogen emission profiles all show the classic asymmetric Thomson scattering profile with prominent red wings extending to more than 2000 km sec$^{-1}$ at H$\\alpha $ and H$\\beta $ due to scattering off the electrons in the wind not Doppler motion.", "Because of the strong scattering wings, we measured the widths of the H$\\alpha $ and H$\\beta $ lines above where the profile begins to broaden.", "The “FWHM” of the H$\\alpha $ and H$\\beta $ narrow peaks are 379 km s$^{-1}$ and 368 km s$^{-1}$ , respectivelyThe FWHM was measured at 1.62 $\\times $ 10$^{-16}$ ergs s$^{-1}$ cm$^{-2}$ Å$^{-1}$ for H$\\alpha $ and at 1.25 $^{-16}$ ergs s$^{-1}$ cm$^{-2}$ Å$^{-1}$ for H$\\beta $.", "The narrow Balmer emission peaks have a mean velocity of 1685 km s$^{-1}$ , and the individual lines show no significant velocity shift in the two later spectra.", "Although, this is about 200 km s$^{-1}$ greater than the published Doppler velocity for NGC 2748 of +1476 km s$^{-1}$ , the object is in the outermost parts of the galaxy, and the velocity difference is consistent with rotation at its distance from the center [1].", "Figure: Blue spectrum from day 6Figure: The profiles of the Hα\\alpha (top) and Hβ\\beta (bottom) lines fromDay 6 showing the multiple P Cygni absorption features.The higher Hydrogen lines at $\\lambda $$\\lambda $ 3889,3835,3797, and 3770 Å are clearly visible in absorption and are blue-shifted relative to the Balmer emission lines by about 300 km s$^{-1}$ which we attribute to formation in the expanding wind or ejecta.", "The Fe II multiplet 42 lines are also prominent in emission with strong P Cygni absorption features, but with no more than one absorption minimum, blue-shifted by about 400 km s$^{-1}$ relative to the peak emission.", "So the Fe II absorption is not formed in the faster moving ejecta.", "Numerous other Fe II lines are present in emission but without P Cygni profiles.", "The Ca II K absorption line is present, and in the red, the strong, luminosity sensitive O I triplet at $\\lambda $ 7774 Å is in absorption.", "The first low-resolution spectrum described by [23] and observed within 24 hours of the discovery, does not show the multiple P Cyg absorptions.", "Our first spectrum was obtained only five days later, and although the difference could be real, we attribute it to the difference in spectral resolution, 14 Å vs. 3 Å in our spectra.", "The total width of the P Cyg absorption feature is the same in both spectra.", "All of the line identifications and measured velocities discussed in this section are given in Table 4.", "Although, it is common to quote the terminal velocity ($v_{\\infty }$ ) for the P Cygni profiles of hot stars, the terminal velocity, however, is derived from a stellar wind model fit to the profiles of resonance lines.", "The Hydrogen lines are not resonance lines.", "We also chose not to fit Gaussians to determine the blue edge velocity because of the reduced S/N in some of the spectra together with the complex profiles with multiple minima plus the strong scattering wings.", "Instead, we give the velocity at the absorption minimum measured relative to the emission peak which permits a well-controlled differential measurement.", "When our second spectrum was observed on Day 71, PSN J09132750+7627410 had already faded significantly.", "The blue spectrum has very poor S/N, and only H$\\beta $ is identified in emission, however, in the red, H$\\alpha $ is still strongly in emission (Figure 7) with multiple P Cygni absorption minima.", "The scattering wings have weakened considerably due to decreasing density in the expanding ejecta.", "The red wing is now measured to only 900 km s$^{-1}$ .", "In addition to the O I triplet, the Ca II near-infrared triplet has appeared plus Fe II and Fe I absorption lines and the K I doublet (Figure 6).", "The presence of these additional absorption lines suggests that the eruption has produced the optically thick cool wind observed in several post-maximum giant eruptions such as SN2011ht [8] and UGC 2773 OT2009-1 [22], [5], the red transients SN2008S and the NGC 300 OT, as well as the LBV/S Dor variables at maximum light.", "These absorption lines are offset by $\\approx $ 400 km s$^{-1}$ relative to the Hydrogen emission.", "Figure: The red spectra from Days 71 (red) and 99 (blue) illustrating the development of the absorption lines.Several additional absorption lines from neutral metals appeared in both the blue and the red spectra from Day 99.", "We note that the absorption lines present earlier show a redwards shift by about 50 - 100 km s$^{-1}$ between Days 71 and 99.", "The slow moving P Cyg absorption feature shows a similar redwards shift relative to the Hydrogen emission peak.", "This could be due either to a slowing of the dense wind or to possible infall back to the star.", "A slowing of the ejecta is often attributed to collision with previous circumstellar material but, in this case, the slow moving gas will have reached only 20 AU in 99 days while the two faster winds will have expanded to 63 and 91 AU, and further if ejected earlier.", "The dust formation radius will depend on the properties of the star, but is typically at least 100 – 200 AU and for the infrared source it is $\\approx $ 100 AU.", "So what is it colliding with?", "Despite a longer integration time the blue spectrum has relatively poor S/N, although H$\\beta $ still shows absorption minima.", "The red spectrum is shown in Figure 6 together with the Day 71 spectrum.", "For comparison, the three H$\\alpha $ profiles are shown together in Figure 7.", "Figure: The Hα\\alpha profiles from all three spectra illustrating the P Cygniabsorption shift to longer wavelengths with time.", "The top spectrum (blue)is from Day 6, green from Day 71 and red from Day 99.In general the spectra of PSN J09132750+7627410 are typical of non-terminal giant eruptions with strong narrow Hydrogen emission, Thomson scattering profiles due to the strong wind, and expansion velocities of a few 100 km s$^{-1}$ .", "The distinguishing characteristic of our spectra of this eruption however, are the multiple P Cygni absorption features present in several of the Balmer lines.", "The outflow velocities measured for the P Cygni absorption components from our three spectra are summarized in Table 5.", "Three distinct blue-shifted absorption features or minima are present at H$\\beta $ , H$\\gamma $ and H$\\delta $ in the spectra from Day 6 with blue-shifted velocities relative to their corresponding emission peaks at $\\sim $ -400, -1100 and -1600 km s$^{-1}$ , hereafter respectively called velocity components a, b, and c, see Figure 5.", "The two highest velocity minima, b and c, are clearly present at H$\\alpha $ ; component a, at the lowest velocity is very likely covered by the scattering wing.", "H$\\epsilon $ shows a classical P Cygni profile, although component c may be present as an unidentified absorption at 3968Å.", "Unfortunately, only the red spectrum from Day 71 has sufficient signal to noise for reliable line measurements.", "The wings on the H$\\alpha $ scattering profile have weakened considerably, and velocity component a is now clearly visible as a strong P Cygni feature.", "The two higher velocity minima, b and c, have weakened, and have merged into a single absorption feature although multiple components or structure are visible in the profile.", "In our third spectrum from Day 99, the highest velocity component c is no longer recognizable at H$\\alpha $ .", "There are now two strong absorption features and the blue edge of the high velocity component has shifted noticeably redwards.", "Two absorption minima corresponding to components a and b can still be clearly identified in the H$\\beta $ P Cygni profile and component c may still be present but much weaker.", "The multiple absorption features or P Cygni profiles could be due either to multiple outflows or ejections with different velocities and energies from the current eruption or be a remnant of the wind and prior mass loss episodes of the progenitor.", "If we assume that the lowest velocity feature at -400 km s$^{-1}$ is formed in the expanding ejecta in the current outburst, then there is clear evidence for two additional, separate winds or outflows.", "Their possible origin and implications for the eruption are discussed in the next section." ], [ "Discussion", "The absorption minima at $\\sim $ -400 km s$^{-1}$ plus the velocity difference between the absorption lines and emission peaks of 300-400 km s$^{-1}$ , and the FWHM of the narrow emission cores are all consistent with a relatively slow ejection velocity which we associate with the current brightening or eruption and the formation of the cool, dense wind.", "Expansion velocities on this order are higher than measured for LBV/S Dor variables (100 – 200 km s$^{-1}$ ) in their optically dense wind state or maximum light, but are observed in giant eruptions such as UGC 2773 OT1009-1 (350 km s$^{-1}$ ) and SN2009ip (550 km s$^{-1}$ ) [22], [5], SN2011ht 500 – 600 km s$^{-1}$ [8] and even $\\eta $ Car (600 km s$^{-1}$ ), although higher velocities have been reported in impostor eruptions such as SN2000ch (3000 km s$^{-1}$ ) [18].", "The outstanding question is the origin of the two higher velocity blue-shifted absorption minima, b and c, in PSN J09132750+7627410.", "We have identified four other objects in which multiple absorption minima have been observed; SN2005gj [27], SN2009ip [15] and two giant eruptions, SN2000ch [18] and SNHunt 248 [17].", "[27] measured velocities of 300 and 100 km s$^{-1}$ in two absorption troughs in the H$\\alpha $ and H$\\gamma $ lines in SN2005gj which they suggested were the remnants of the progenitor's previous mass loss state as an LBV.", "In contrast, much higher velocities of 3000 and 5300 km s$^{-1}$ were measured in two absorption minima in the H$\\beta $ line during SN2000ch's 2009-OT1 eruption [18], and similar minima appear to be present in the H$\\gamma $ line.", "Although, [17] did not specifically discuss them, multiple absorption minima can be seen in their first spectrum of SNHunt 248 for which they suggest an average outflow of $\\sim $ 1200 km s$^{-1}$ .", "In the peculiar SN2009ip multiple absorptions at high velocities are observed in the post-terminal eruption spectra, but were not reported in its prior eruptions when it was considered a non-terminal giant eruption.", "It is common in work on SN impostors and even terminal Type IIn SNe, to assume that evidence for additional outflows in their spectra or mass loss episodes in the ejecta are from separate preceding events.", "In several of these objects, SN2011ht, SN2009ip, SN1994W, SN2009kn, etc., enhanced mass loss precedes the main eruption by only a few months or years, as in SN2009ip.", "Instead of being viewed as entirely separate events occurring years or decades before, they could all be part of the same instability producing more than one eruption as part of the on-going episode or instability.", "In the case of slow moving ejecta, it is often assumed that it is due to a prior LBV/S Dor stage and that the progenitor was an LBV, but that might not be correct.", "In all of these cases we do not have a sufficient timeline to know.", "In PSN J09132750+7627410, the velocities at b and c are too high to be produced by the slow, dense wind of a presumed LBV progenitor or even LBVs in quiescence which have relatively slow winds for their corresponding hot supergiant spectral types.", "If either of these high velocity components are remnants of the progenitor's wind, then they imply a hot star.", "One of these components could be the remnant of the progenitor's wind and the second from a previous, and relatively recent, high mass loss episode.", "These high wind velocities are measured in late O-type and early B-type stars [13], but neither star A or B has an SED suggestive of hot star, and if the infrared source is the progenitor, the embedded star was most likely a cool or intermediate temperature supergiant.", "Alternatively, both may be from previous high mass loss episodes.", "Their much higher expansion velocities compared to the current event suggest much more energetic eruptions.", "Unfortunately there is no record of prior brightenings.", "However, the infrared source was observed for five years and was still present slightly more than a year prior to the eruption.", "If it was the progenitor, this may set a limit on when the prior ejections occurred, and an eruption, represented perhaps by one or both of these higher velocity outflows was responsible for the dust destruction.", "The dust condensation distance for the 780K infrared source is 100 AU.", "Velocity components c and b would have reached this distance in a fraction of a year.", "Here we suggest that the multiple P Cygni profiles were all formed during the current eruption which may have begun several weeks or even months before the brightening was observed, and are due to material ejected at different velocities or energies and possibly at different times.", "$\\eta $ Car provides an example of ejecta expanding at a range of velocities from the same giant eruption; from 600 km s$^{-1}$ in the Homunculus lobes to 1000 – 3000 km s$^{-1}$ in the outer ejecta ([30] and references therein).", "This possibility for PSN J09132750+7627410 is supported by the weakening and disappearance of the highest velocity component c as the eruption subsided and the ejecta expanded over the 99 days covered by our spectra.", "Formation during a single mass loss episode applies just as well to the two non-terminal objects described above.", "The multiple P Cyg minima in the first SNHunt 248 spectrum closely resemble the absorption troughs in our first spectrum of PSN J09132750+7627410, and were not apparent in the later spectra observed over the next month.", "Although most had lower resolution, one with higher resolution confirms that P Cygni minima were not present about one month after they were first observed.", "A series of spectra by [11] shows a lack of multiple minima at the same time in their highest resolution spectrum.", "Thus, the multiple P Cyg absorptions quickly disappeared and could have a common origin as multiple ejections during the high mass loss event.", "The velocities measured in SN2000ch during its 2009-OT1 eruption seem high compared to these other objects, but are comparable to the velocities reported in $\\eta $ Car's outer ejecta.", "Like SNHunt 248, the P Cyg absorptions are gone in spectra obtained a few weeks later.", "SN2005gj is considered to be a terminal explosion, but that doesn't rule out the ejection of material shortly before the eruption.", "SN2009ip is an example where lesser “giant eruptions” occurred several times during the three years prior to what is assumed to be the final event.", "The multiple absorptions observed in its post-maximum spectra with very high velocities of -5000 to -12000 km s$^{-1}$ , were produced during its final eruption and are considered evidence for separate asymmetric outflows [15].", "Similar processes may be occurring in the less energetic eruptions in the impostors.", "They did in $\\eta $ Car.", "In the two non-terminal giant eruptions, the multiple absorption minima weaken and disappear within a few weeks.", "In PSN J09132750+7627410 they were present for a couple of months.", "So it is possible that the evidence for multiple outflows during the high mass loss episodes is more common than has been reported due to their relatively brief or transient appearance, and to lower spectral resolution, especially in the early post-discovery spectra, as we found for PSN J09132750+7627410.", "We thank Kris Davidson for valuable discussion and suggestions.", "Research by R. Humphreys and M. Gordon on massive stars is supported by the National Science Foundation AST-1109394.", "J.", "C .", "Martin's collaborative work on luminous variables is supported by the National Science Foundation grant AST -1108890.", "Some 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.", "We also used data from the Pan-STARRS Survey for Transients.", "Operation of the Pan-STARRS1 telescope is supported by the National Aeronautics and Space Administration under Grant No.", "NNX12AR65G and Grant No.", "NNX14AM74G issued through the NEO Observation Program.", "Facilities: LBT/MODS1 lllll 0 pt 1 Journal of Spectroscopic Observations UT Date Instrument Exp.", "Time Wavelength Reference 11.98 Feb 2015 (Day 1) Asiago 1.8m 3400 – 8200$Å$ [23] 16.24 Feb 2015 (Day 6) LBT/MODS1 30m 3200$Å$ – 1$\\mu $ m 22.13 Apr 2015 (Day 71) \" 30m \" 20.16 May 2015 (Day 99) \" 60m \" llllllll 2 Multi-color Photometry U.T.", "Date B mag V mag r(P1) mag R mag i(P1) mag I mag Comment 10.92 Feb 2015 18.0 unfiltered; Cortini 11.15 Feb 2015 17.7 unfiltered; Yusa 11.23 Feb 2015 18.14 17.42 Kiyota 11.43 Feb 2015 17.8 unfiltered; Noguchi 11.98 Feb 2015 18.7 [23] 26.91 Feb 2015 18.34 $\\pm $ 0.03 Pan-STARRS 12.66 Mar 2015 18.83 $\\pm $ 0.08 18.58 $\\pm $ 0.12 17.54 $\\pm $ 0.05 Barber Obs.", "17.63 Mar 2015 18.69 $\\pm $ 0.11 18.22 $\\pm $ 0.12 17.58 $\\pm $ 0.05 17.40 $\\pm $ 0.09 Barber Obs.", "28.76 Mar 2015 18.80 $\\pm $ 0.02 Pan-STARRS 03.82 Apr 2015 18.94 $\\pm $ 0.03 Pan-STARRS 29.60 Apr 1015 $>$ 18.8 $>$ 18.9 $>$ 18.5 Barber Obs 19.67 May 2015 $>$ 19.0 $>$ 19.3 18.22 $\\pm $ 0.09 Barber Obs 13.73 Jan 2016 $>$ 19.2 $>$ 19.0 $>$ 18.6 Barber Obs 5.15 Feb 2016 $>$ 19.5 $>$ 19.2 $>$ 18.8 Barber Obs lllllll 3 HST/WFPC2 and Spitzer/IRAC Photometry Date Star F450W mag.", "F814W mag.", "3.6$\\mu $ m mag.", "4.5$\\mu $ m mag.", "ICRS J2000 Position 6 Jul 2001 A 24.43 $\\pm $ 0.20 23.79 $\\pm $ 0.15 ... ... 9:13:27.48 +76:27:41.1 \" B 26.08 $\\pm $ 1.28 24.00$\\pm $ 0.18 ... ... 9:13:27.38 +76:27:40.9 2 Dec 2009 ... ... ... 19.41 $\\pm $ 0.3 18.40 $\\pm $ 0.3 12 Apr 2010 ... ... ... 19.16 $\\pm $ 0.2 18.65 $\\pm $ 0.3 25 Jan 2014 ... ... ... 19.21 $\\pm $ 0.2 18.32 $\\pm $ 0.1 9:13:27.24 +76:27:41.2 lllll 4 Line Identifications and Measured Velocities in PSN J09132750+7627410 Line Velocity km s$^{-1}$ P Cyg Vel.", "km s$^{-1}$ Blue-Edge Vel.", "km s$^{-1}$ Red Wing km s$^{-1}$ Day 6 (16 Feb 2015) Emission Lines H$\\alpha $ 1607 , -1137, -1537 , -1337, -1817 2460 H$\\beta $ 1648 -466, -1154, -1559 -681, -1353, -1946 2200 H$\\gamma $ 1680 -412, -1027, -1622 -646, -1127, -1876 1650 H$\\delta $ 1745 -426, -1124, -1670 -717, -1255, -1953 1668 H$\\epsilon $ 1745 -428, , a , , Fe II(42) 4923.9Å 1682 -409 Fe II(42) 5018.4Å 1680 -348 Fe II(42) 5169.0Å 1706 -427 Absorption lines H 3889Å 1365 H 3835Å 1408 H 3797Å 1335 H 3770Å 1313 Ca II K 1391 O I 7774Å 1397 Day 71 (22 Apr 2015) Emission Lines H$\\alpha $ 1595 -396, -1171, -1453:: -727, -1764 914 Absorption lines O I 7774Å 1389 Ca II8498Å 1239 Ca II8542Å 1296 Ca II8662Å 1306 K I 7665Å 1211 K I 7699Å 1230 Fe II 6432Å 1240 Fe I 6494Å 1289 Day 99 (20 May 2015) Emission Lines H$\\alpha $ 1614 -364, -1161, -682, -1735 909 H$\\beta $ 1651 -316, -913, range -636, -1047, H$\\gamma $ 1701 Absorption lines O I 7774Å 1420 Ca II8498Å 1318 Ca II8542Å 1333 Ca II8662Å 1340 K I 7665Å 1295 K I 7699Å 1341 Fe II 6432Å 1288 Fe I 6494Å 1366 Fe I 8327Å 1329 Fe I 8387Å 1283 Na I 8194Å 1333 aAn unidentified line at 3968Å may be the blue-shifted component c with a corresponding velocity of -1773 km s$^{-1}$ .", "lll 5 Summary of the Outflow Velocities Date Velocity km s$^{-1}$ Lines Day 6 -433 H$\\beta $ , H$\\gamma $ , H$\\delta $ , H$\\epsilon $ -1110 H$\\alpha $ , H$\\beta $ , H$\\gamma $ , H$\\delta $ -1597 H$\\alpha $ , H$\\beta $ , H$\\gamma $ , H$\\delta $ -395 Fe II (42) Day 71 -396 H$\\alpha $ -1171 H$\\alpha $ -1453 H$\\alpha $ Day 99 -340 H$\\alpha $ , H$\\beta $ -1161, -913 H$\\alpha $ , H$\\beta $" ] ]
1606.04959
[ [ "Resonance interaction of two dipoles in optically active surroundings" ], [ "Abstract We study the resonance interaction between two quantum electric dipoles immersed in optically active surroundings.", "Quantum electrodynamics is employed to deal with dipole-vacuum interaction.", "Our results show that the optical activity of surroundings will not change the single atom behaviors while it can change the collective behaviors of the two dipoles, as well as greatly affect the dipole-dipole resonance interaction.", "Especially, if the orientations of two dipoles are orthogonal and respectively perpendicular to the interdipole axis, the interdipole resonance interaction can be established with the help of optically active surroundings while there is no resonance interaction in vacuum." ], [ "Introduction", "Resonance interaction (RI) [1], [2] is a mechanism describing two quantum emitters (one is excited and the other one is in ground state) with same transition frequencies interact with each other by exchanging virtual photons.", "RI represents the nature of the interaction between vacuum electromagnetic field and atomic system, and also plays important roles in broad application fields.", "For instance, RI can contribute to operation of solid state lasers and fluorescent tubes [3].", "In quantum physics, RI is very useful in creating entanglement [4], [5], and also plays important roles in laser cooling [6], creating cold molecules [7], [8] and quantum storage [9], [10], [11].", "The study of RI can contribute in developing precise measurement methods in chemistry and biology for metering distance between molecules  [12] or between domains in one protein which can provide information about protein conformation  [13], so it attracts the interests of chemists and biologists as well as physicists.", "Further more, RI between quantum emitters is the essence of Förster resonance energy transfer (FRET) [14], [15], while the latter one lies at the core of biophysics cause it represents the nature of intermolecule interaction, provides the foundation of several detection methods [16], [17] and plays important role in biological phenomena such as photosynthesis.", "The investigation into photosynthesis can help in increasing the efficiencies of solar cells (specially the dye-sensitized solar cells [18]), which is in favor of the usage of solar energy.", "In biological phenomena, RI generally takes place in living cells filled with solution of organic compounds, which are optically active in most cases.", "Similarly, the electrolyte solution used in dye-sensitized solar cells contains volatile organic compounds [19] and leads to an optically active surroundings.", "When the dipoles (which can be atoms, molecules, or just chromophores and fluorophores) are immersed in optically active solutions, the interaction between them must be affected by the optically active surroundings.", "The study of this effect can help in understanding the processes in natural photosynthesis, while the simulation of which can contribute in improving the efficiency of solar cells.", "We think this effect is important as well as interesting, but so far as we all know, there is an absence of detailed theoretical investigation of this effect.", "In this paper, we study the RI between two quantum dipoles which are immersed in optically active surroundings.", "Quantum electrodynamics is employed to deal with dipole-vacuum interaction.", "We give out the analytical expressions of interdipole RI in optically active surroundings as well as several numerical examples.", "We find that the collective behaviors of the dipole pair (containing both the collective radiation rates and the interaction induced level shifts) are affected by optically active surroundings while the single atom properties are not.", "When the orientations of two dipoles are orthogonal and respectively perpendicular to interdipole axis, the optical rotation of the surrounding medium can cause great interaction between the dipoles while there is no interaction in vacuum." ], [ "Dynamics of two dipoles interaction", "In this section, a brief review of the dynamical analysis of dipole-reservoir interaction will be given out.", "We calculate the dipole-light interaction problem in Schrödinger picture.", "The two dipoles are marked by 1 and 2.", "The quantum state of the system can be expressed by $\\left|\\psi \\right\\rangle =\\sum _{i=1,2}C_{i}\\left|i\\right\\rangle \\left|0\\right\\rangle +\\sum _{i=3,4}\\sum _{{k}\\lambda }D_{i,{k}\\lambda }\\left|i\\right\\rangle \\left|{k}\\lambda \\right\\rangle ,$ in which the basic states $\\left|1\\right\\rangle -\\left|4\\right\\rangle $ are $\\left|1\\right\\rangle =\\left|e_{1}g_{2}\\right\\rangle ,~\\left|2\\right\\rangle =\\left|g_{1}e_{2}\\right\\rangle ,~\\left|3\\right\\rangle =\\left|e_{1}e_{2}\\right\\rangle ,~\\left|4\\right\\rangle =\\left|g_{1}g_{2}\\right\\rangle .$ $\\left|e\\right\\rangle $ and $\\left|g\\right\\rangle $ are the quantum states of the dipoles.", "$\\left|g_{i}\\right\\rangle $ represents that the dipole $i$ is in ground state and $\\left|e_{j}\\right\\rangle $ means dipole $j$ is in excited state.", "For example, the ket $\\left|1\\right\\rangle =\\left|e_{1}g_{2}\\right\\rangle $ represents dipole 1 is excited while dipole 2 is in ground state.", "Electromagnet field state $\\left|{k}\\lambda \\right\\rangle $ means a photon with wave vector ${k}$ and polarization $\\lambda $ exists, while $\\left|0\\right\\rangle $ represents there is no photon (the electromagnetic vacuum state).", "$C_{i}$ and $D_{i,{k}\\lambda }$ are probability amplitudes.", "The total Hamiltonian $\\hat{H}$ can be departed into three parts: the electric dipole Hamiltonian $\\hat{H}_{D}$ , the electromagnetic Hamiltonian $\\hat{H}_{F}$ and the atom-light interaction Hamiltonian $\\hat{H}_{INT}$ .", "The dipole Hamiltonian can be written as $\\hat{H}_{D}=\\hbar \\sum _{i}\\omega _{i}\\hat{\\sigma }_{ii},$ in which $\\hat{\\sigma }_{ij}=\\left|i\\right\\rangle \\left\\langle j\\right|$ are dipolar transition operators.", "The electromagnetic Hamiltonian is $\\hat{H}_{F}\\left( {r}\\right) =\\sum _{{k}\\lambda }\\hbar \\omega _{k}\\left( \\hat{a}_{{k}\\lambda }^{\\dag }\\hat{a}_{{k}\\lambda }+\\frac{1}{2}\\right) ,$ where $\\hat{a}_{{k}\\lambda }^{\\dag }$ and $\\hat{a}_{{k}\\lambda }$ are creation and annihilation operators for photon of mode ${k}\\lambda $ , respectively.", "$\\hbar $ is modified Planck constant and $\\omega _{k}$ is the angular frequency of the photon.", "The interaction Hamiltonian writes $\\hat{H}_{INT}=-\\sum _{a=1,2}{\\hat{d}}_{a}\\cdot {\\hat{E}}\\left( {r}_{a}\\right) ,$ where ${\\hat{d}_{a}}=\\sum _{ij}{{d}}_{ij}^{a}\\hat{\\sigma }_{ij}$ are the electric dipole operators, and ${{d}}_{ij}^{a}$ are the dipole transition matrix elements.", "${\\hat{E}}$ is the electric field operator which is given by ${\\hat{E}}\\left( {r}\\right) =\\sum _{{k}\\lambda }A_{k}\\hat{a}_{{k}\\lambda }{W}\\left({k}\\lambda ,{r}\\right) +H.c,$ where $A_{k}$ is the normalization coefficients, and the electric wave mode function ${W}\\left( {k}\\lambda ,{r}\\right) $ is decided by the surrounding medium and the boundary conditions.", "With the initial condition that dipole 1 is excited and dipole 2 is unexcited, the probability amplitude can be worked out: $C_{1}\\left( t\\right) =e^{A_{L}t}\\frac{e^{A_{T}t}+e^{-A_{T}t}}{2},~C_{2}\\left(t\\right) =e^{A_{L}t}\\frac{e^{A_{T}t}-e^{-A_{T}t}}{2}.", "$ The expressions of coefficients $A_{L}$ and $A_{T}$ as well as calculation process are given in appendix A.", "Then we further define dipole-exchanging symmetric (DES) state $\\left|+\\right\\rangle $ and dipole-exchanging anti-symmetric (DEAS) state $\\left|-\\right\\rangle $ which is given by $\\left|\\pm \\right\\rangle =\\frac{1}{\\sqrt{2}}\\left( \\left|e_{1}g_{2}\\right\\rangle \\pm \\left|g_{1}e_{2}\\right\\rangle \\right) ,$ and the corresponding probability amplitudes are $C_{\\pm }=\\frac{1}{\\sqrt{2}}\\left( C_{1}\\pm C_{2}\\right) =\\frac{1}{\\sqrt{2}}e^{\\left( A_{L}\\pm A_{T}\\right) t}.$ It is obviously that $\\left|\\Re \\left( A_{L}\\pm A_{T}\\right)\\right|$ and $\\Im \\left( A_{L}\\pm A_{T}\\right)$ are the damping rates and the level shifts of DES and DEAS states, respectively.", "$A_{L}$ refers to single dipole mechanism and $A_{T}$ represents collective behavior.", "According to the solution we get above, the dipole-dipole interaction energy can be calculated as $E_{int}=-2\\hbar \\Im A_{T}\\left( \\left|C_{+}\\right|^{2}-\\left|C_{-}\\right|^{2}\\right) .", "$ The interdipole interaction energy is proportional to the collective level shift ($\\Im A_{T}$ ) times the population difference between DES and DEAS states." ], [ "Electromagnetic field in optically active medium", "In last section, we have given out a brief review of dipole-vacuum interaction.", "The general formulas are calculated, but to obtain the detail expressions, the mode function ${W}\\left({k}\\lambda ,{r}\\right)$ must be specified.", "In this section, we will give the function ${W}\\left( {k}\\lambda ,{r}\\right)$ in optically active medium.", "One beam of left (right) hand circularly polarized light with wave vector ${k}$ can be expressed by Jones vector as ${{W}}_{left/right}=\\frac{1}{\\sqrt{2}}\\left( {\\hat{e}}_{{k},1} \\pm i {\\hat{e}}_{{k},2}\\right) e^{i{k\\cdot r}},$ ${\\hat{e}}_{{k},1}$ and ${\\hat{e}}_{{k},2}$ are unit vectors which are orthogonal to each other and perpendicular to wave vector ${k}$ .", "$i$ is the imaginary unit.", "According to Fresnel's theorem, the phase velocities of left hand and right hand polarized light is different in optically active medium, then the general expression of light in optically active medium can be written as ${W}\\left( {k}\\lambda ,{r}\\right) =\\frac{1}{\\sqrt{2}} \\left( {\\hat{e}}_{{k},1}+s_{\\lambda }i{\\hat{e}}_{{k},2}\\right) e^{in_{\\lambda }{k\\cdot r}}, $ in which $\\lambda = L$ , $R$ represent the left and right hand circularly polarized components, respectively, and $s_{L}=1$ , $s_{R}=-1$ .", "$n_{\\lambda }$ are the refractive indexes for the two circularly polarized components and ${k}$ is wave vector in free space.", "Using Eq.", "(REF ), the expression of function $G_{aa\\left(b\\right)}$ in Eq.", "(REF ) can be derived $G_{11}\\left( {k}\\lambda \\right) =\\frac{1}{2}{d}_{1}\\cdot \\mathbf {M} \\left({k}\\lambda \\right)\\cdot {d}_{1}, ~~G_{12}\\left( {k}\\lambda \\right) =\\frac{1}{2}{d}_{1}\\cdot \\mathbf {M} \\left({k}\\lambda \\right)\\cdot {d}_{2} e^{-i n_{\\lambda } {k} \\cdot \\left({r}_{1}-{r}_{2}\\right)},\\\\G_{22}\\left( {k}\\lambda \\right) =\\frac{1}{2}{d}_{2}\\cdot \\mathbf {M} \\left({k}\\lambda \\right)\\cdot {d}_{2}, ~~G_{21}\\left( {k}\\lambda \\right) =\\frac{1}{2}{d}_{2}\\cdot \\mathbf {M} \\left({k}\\lambda \\right)\\cdot {d}_{1} e^{-i n_{\\lambda } {k} \\cdot \\left({r}_{2}-{r}_{1}\\right)}, $ in which $\\mathbf {M}\\left({k}\\lambda \\right)= \\left[{\\hat{e}}_{{k},1}{\\hat{e}}_{{k},1} +{\\hat{e}}_{{k},2}{\\hat{e}}_{{k},2} +s_{\\lambda }i \\left( {\\hat{e}}_{{k},1}{\\hat{e}}_{{k},2} -{\\hat{e}}_{{k},2}{\\hat{e}}_{{k},1} \\right)\\right].$" ], [ "Damping rates and level shifts", "After defining the functions $G_{ab}$ , Eqs.", "(REF ) can be calculated.", "We shall go to the continuum limit $\\sum _{{k}}\\rightarrow \\frac{V}{\\left( 2\\pi \\right) ^{3}}\\int d^{3}k,$ and make use of Eq.", "(REF ) to obtain the analytical expressions of $A_{L}$ and $A_{T}$ $&&A_{L}/\\Gamma _{0}=-\\sum _{\\lambda }\\frac{n_{\\lambda }}{4} +i \\sum _{\\lambda }\\frac{n_{\\lambda }}{2\\pi } \\int _{0}^{\\infty }d\\tilde{\\xi }\\frac{\\tilde{\\xi }^{4}}{\\tilde{\\xi }^{2}-1}, \\\\&&A_{T}/\\Gamma _{0}=-F_{1}\\left( R\\right) +i F_{2}\\left( R \\right),$ in which $F_{1}\\left( R\\right) &=&\\sum _{\\lambda }\\frac{3n_{\\lambda }}{8}{\\hat{d}}_{2}\\cdot {\\hat{d}}_{1}\\left[ \\frac{\\sin n_{\\lambda }k_{0}R}{n_{\\lambda }k_{0}R}+\\frac{\\cos n_{\\lambda }k_{0}R}{\\left(n_{\\lambda }k_{0}R\\right) ^{2}}-\\frac{\\sin n_{\\lambda }k_{0}R}{\\left(n_{\\lambda }k_{0}R\\right) ^{3}}\\right] \\nonumber \\\\&&-\\sum _{\\lambda }\\frac{3n_{\\lambda }}{8}{\\hat{d}}_{2}\\cdot {\\hat{R}\\hat{R}}\\cdot {\\hat{d}}_{1}\\left[ \\frac{\\sin n_{\\lambda }k_{0}R}{n_{\\lambda }k_{0}R}+3\\frac{\\cos n_{\\lambda }k_{0}R}{\\left( n_{\\lambda }k_{0}R\\right) ^{2}}-3\\frac{\\sin n_{\\lambda }k_{0}R}{\\left( n_{\\lambda }k_{0}R\\right) ^{3}}\\right] \\nonumber \\\\&&+\\sum _{\\lambda }\\frac{3n_{\\lambda }}{8}s_{\\lambda } \\left( {\\hat{d}}_{2}\\times {\\hat{d}}_{1}\\right) \\cdot {\\hat{R}}\\left[ \\frac{\\cos n_{\\lambda }k_{0}R}{n_{\\lambda }k_{0}R}-\\frac{\\sin n_{\\lambda }k_{0}R}{\\left( n_{\\lambda }k_{0}R\\right) ^{2}}\\right],$ and $F_{2}\\left( R\\right) &=&\\sum _{\\lambda }\\frac{3n_{\\lambda }}{8}{\\hat{d}}_{2}\\cdot {\\hat{d}}_{1} \\left[ \\frac{\\cos n_{\\lambda }k_{0}R}{n_{\\lambda }k_{0}R} -\\frac{\\sin n_{\\lambda }k_{0}R}{\\left(n_{\\lambda }k_{0}R\\right) ^{2}} -\\frac{\\cos n_{\\lambda }k_{0}R}{\\left(n_{\\lambda }k_{0}R\\right) ^{3}} \\right] \\nonumber \\\\&&-\\sum _{\\lambda }\\frac{3n_{\\lambda }}{8}{\\hat{d}}_{2}\\cdot {\\hat{R}\\hat{R}}\\cdot {\\hat{d}}_{1} \\left[ \\frac{\\cos n_{\\lambda }k_{0}R}{n_{\\lambda }k_{0}R} -3\\frac{\\sin n_{\\lambda }k_{0}R}{\\left( n_{\\lambda }k_{0}R\\right) ^{2}} -3\\frac{\\cos n_{\\lambda }k_{0}R}{\\left( n_{\\lambda }k_{0}R\\right) ^{3}} \\right] \\nonumber \\\\&&-\\sum _{\\lambda }\\frac{3n_{\\lambda }}{8}s_{\\lambda } \\left( {\\hat{d}}_{2}\\times {\\hat{d}}_{1}\\right) \\cdot {\\hat{R}}\\left[ \\frac{\\sin n_{\\lambda }k_{0}R}{n_{\\lambda }k_{0}R} +\\frac{\\cos n_{\\lambda }k_{0}R}{\\left( n_{\\lambda }k_{0}R\\right) ^{2}} \\right] \\nonumber \\\\&&-\\sum _{\\lambda }\\frac{3n_{\\lambda }}{8}s_{\\lambda } \\left( {\\hat{d}}_{2}\\times {\\hat{d}}_{1}\\right) \\cdot {\\hat{R}}\\frac{2}{\\pi } \\left[ \\frac{1}{n_{\\lambda }k_{0}R}I_{1} \\left(n_{\\lambda }k_{0}R\\right)+\\frac{1}{\\left( n_{\\lambda }k_{0}R\\right) ^{2}}I_{2}\\left(n_{\\lambda }k_{0}R\\right)\\right].", "$ We have defined $\\Gamma _{0} \\equiv \\left(k_{0}^{3}d^{2}\\right)/\\left(3\\hbar \\epsilon _{0}\\pi \\right) , $ which is the spontaneous radiation rate of a single dipole in free space.", "$k_{0}=\\omega _{0}/c$ where $\\omega _{0}$ is the resonance transition angular frequency of the dipoles and $c$ is the vacuum light speed.", "$d = \\left|{d}_{a} \\right|$ is the electric transition dipole momentum, and $\\epsilon _{0}$ is the vacuum permittivity.", "${\\hat{d}}_{a} ={d}_{a} / d$ are unit vectors that represent the polarization orientation of dipoles.", "The interdipole distance $R = \\left|{r}_{1}-{r}_{2}\\right|$ , and the unit vector ${\\hat{R}} = \\left({r}_{1}-{r}_{2}\\right)/R $ represents the interdipole axis.", "The two functions $I_{1}$ and $I_{2}$ in Eq.", "(REF ) are improper integrals defined by $I_{1} \\left(n_{\\lambda }k_{0}R\\right)=\\int _{0}^{\\infty }\\frac{\\tilde{\\xi }^{3}e^{-\\xi n_{\\lambda }k_{0}R}}{\\tilde{\\xi }^{2}+1}d\\tilde{\\xi },~~ I_{2}\\left(n_{\\lambda }k_{0}R\\right)=\\int _{0}^{\\infty }\\frac{\\tilde{\\xi }^{2}e^{-\\tilde{\\xi }n_{\\lambda }k_{0}R}}{\\tilde{\\xi }^{2}+1}d\\tilde{\\xi }.$ Up to here, the analytical expressions of $A_{L}$ and $A_{T}$ have been calculated.", "Associated with Eqs.", "(REF ) - (REF ), the behaviors of the dipole pair can be fully described.", "According to Eq.", "(REF ), we obtain the damping rates of DES and DEAS states $\\gamma _{\\pm } / \\Gamma _{0} = \\sum _{\\lambda } \\frac{ n_{\\lambda } }{4} \\pm F_{1}\\left( R \\right), $ as well as the level shifts due to the interdipole interaction $\\delta _{\\pm } / \\Gamma _{0} = \\sum _{\\lambda }\\frac{n_{\\lambda }}{2\\pi }\\int _{0}^{\\infty }d\\tilde{\\xi }\\frac{\\tilde{\\xi }^{4}}{\\tilde{\\xi }^{2}-1} \\pm F_{2}\\left( R \\right).", "$" ], [ "The behaviors of a single dipole", "Considering a situation that $R \\rightarrow \\infty $ , then $A_{T} \\rightarrow 0$ and $C_{1}\\left(t\\right)=\\exp \\left(A_{L} t\\right)$ while $C_{2}\\left(t\\right)=0$ .", "The physical picture of this result is that dipole 1 spontaneously emits photon while dipole 2 can never be excited.", "When the two dipole are very far away separated, they do not interact with each other and the population ($\\left|C_{1} \\right|^{2}$ ) on the excited state of dipole 1 (the initially excited dipole) decays in a rate of $\\sum _{\\lambda } \\Gamma _{0} n_{\\lambda }/ 2$ .", "That is why we say the single dipole mechanism is decided by $A_{L}$ .", "By defining the average refractive index $\\bar{n} = \\sum _{\\lambda } n_{\\lambda }/2, $ we find the spontaneous radiation rate of one single dipole immersed in optically active media is $\\gamma _{0} = \\bar{n} \\Gamma _{0}.", "$ Similarly, the imaginary part of $A_{L}$ is the vacuum Lamb shift of a single dipole, which is obviously divergent according to Eq.", "(REF ).", "This divergent integral can be renormalized by Bethe's method $\\int _{0}^{\\infty }d\\tilde{\\xi }\\frac{\\tilde{\\xi }^{4}}{\\tilde{\\xi }^{2}-1}\\rightarrow \\frac{1}{2} \\int _{0}^{m_{e}c/\\hbar k_{0}} d\\tilde{\\xi }\\frac{1}{\\tilde{\\xi }-1} =\\frac{1}{2} \\ln \\frac{m_{e}c}{\\hbar k_{0}}, $ in which $m_{e}$ is the observable mass of an electron.", "Using the expression of Eq.", "(REF ), we get the vacuum Lamb shift of a single dipole in optically active medium $\\delta _{Lamb}=\\bar{n} \\Gamma _{0} \\ln \\left(m_{e}c/\\hbar k_{0}\\right)/\\left(2 \\pi \\right).", "$ $\\delta _{Lamb}$ is position independent, which means it will not lead to any mechanical force.", "For this reason, we will not discuss the single dipole vacuum Lamb shift in the follows.", "$A_{L}$ represents the interaction between one single dipole and the vacuum electromagnetic field.", "According the analysis above, the radiation rate and level shift of a single dipole depend on the average rather then the difference of the refractive indexes for the two circularly polarized components, and the latter one decides the specific rotation.", "For this reason, we can say that the behaviors of a single dipole are affected by the average refractive indexes rather than the optical rotation of the surroundings." ], [ "The collective radiation rates and level shifts", "According to Eqs.", "(REF ) and (REF ), we have obtained the damping rates $\\left( \\gamma _{\\pm } / \\Gamma _{0}= \\frac{1}{2} \\bar{n} \\pm F_{1}\\right)$ as well as the level shifts $\\left( \\delta _{\\pm } / \\Gamma _{0}=\\delta _{Lamb}/\\Gamma _{0} \\pm F_{2} \\right)$ of DES and DEAS states.", "The total damping rates (and level shifts) are the single dipole part $A_{L}$ plus or minus the collective part $A_{T}$ .", "The value of $A_{T}$ is decided by the interaction between the dipoles via exchanging virtual photons, therefore it describes the collective behaviors of the dipole pair.", "By checking Eqs.", "(REF ) and (REF ), the most notable properties is that in optically active surroundings.", "The cross product (${\\hat{d}}_{1}\\times {\\hat{d}}_{2}$ ) of dipole 1 and dipole 2 contributes to the interdipole interaction, while this term does not appear in optically inactive case.", "This result is not difficult to understand.", "Imagining one situation that the orientations of the two dipoles are perpendicular to each other.", "If the dipoles are located in optically inactive surroundings, when one virtual photon are emitted by dipole 1, the polarization of the photon is also exactly perpendicular to dipole 2 and it is impossible to be absorbed.", "For the two dipoles can just interact with each other by exchanging virtual photons, they can not “feel\" each other in this perpendicular orientation situation.", "But when the dipoles are immersed in optically active surroundings, the polarization of the virtual photon emitted by the donor dipole will be rotated in propagation, and it may has a component parallel to the acceptor dipole then the photon can be absorbed.", "This effect make term (${\\hat{d}}_{1}\\times {\\hat{d}}_{2}$ ) appears in Eqs.", "(REF ) and (REF ), which respectively represent two-dipole collective radiation rate and level shift.", "Furthermore, according to Eqs.", "(REF ) and (REF ), the two circularly polarized components contribute differently to the interdipole interaction because of the different refractive indexes." ], [ "Numerical examples", "To visually show the results given above, we will give out three numerical examples in the follows.", "We plot the damping rates and the level shifts as functions of interdipole distance and compare the different behaviors of the dipole pair in optically active and inactive surroundings.", "To specify the parameters, we set the average refractive index of the surrounding medium to be 3 , and the specific rotation divided by wave vector $k$ to be $-1.5$ .", "We first consider a situation that the two dipoles are orthogonally polarized, and the orientations of them are perpendicular to the interdipole axis, as shown in Fig.", "REF .", "According to the physical analysis above, two orthogonally polarized dipoles in vacuum can not interact with each other while an optically active surroundings can help in establishing interdipole RI.", "To check this analysis, we plot the level shifts as well as the radiation rates in optically inactive and active surroundings.", "The results in the different situations are compared in Fig.", "REF .", "Figure: (Color online) The polarizations of the two atom are orthogonal andrespectively perpendicular to the interdipole axis, as shown in (d).", "Γ S =2γ + /Γ 0 \\Gamma _{S} = 2 \\protect \\gamma _{+} / \\Gamma _{0} represents the damping rateof the population of DES state defined in Eq.", "() whileΓ AS =2γ - /Γ 0 \\Gamma _{AS} = 2 \\protect \\gamma _{-} / \\Gamma _{0} refers to DEAS state.", "Inthe optically (a) inactive case, the level shift is a constant which meansthe two dipoles do not interaction with each other.", "When the dipoles areimmersed in (b) optically active medium, the RI can be built up by exchangesof virtual photons between the two dipoles.", "Furthermore, the dimensionlessdipole-dipole RI E int /ℏΓ 0 E_{int}/\\hbar \\Gamma _{0} at time Γ 0 t=1\\Gamma _{0}t=1 areplotted as functions of interdipole distance in (c).", "δ=δ + -δ - /Γ 0 \\protect \\delta = \\left(\\protect \\delta _{+} - \\protect \\delta _{-} \\right) / \\Gamma _{0} represents thedimensionless energy level difference between the DES and DEAS states, whilethese two states are degenerate without interdipole interaction.In the optically inactive case, according to Fig.", "REF (b), the level shift $\\delta $ is zero, which means the DES and DEAS states are degenerate and the two dipoles do not interact with each other.", "The damping rates of DES and DEAS state are $3\\Gamma _{0}$ , which indicate that dipole 1 can not feel the existence of dipole 2 and just decays alone.", "In the optically active case, the damping rates and level shifts vary with the interdipole distance in Fig.", "REF (c), which implies an RI between the dipoles.", "We plot the interdipole RI in the case of perpendicular dipole orientations as functions of interdipole distance in Fig.", "REF (c).", "For RI decays with time, we choose a time at $\\Gamma _{0}t=1$ , and use $\\hbar \\Gamma _{0}$ as the unit of interaction energy.", "The results of this numerical experiment agree with the physical analysis above, and typically shows that optically active surroundings can noticeably affect interdipole RI.", "Now we further consider a situation that the polarizations of the two dipoles are syntropic and perpendicular to the interdipole axis, as shown in Fig.", "REF .", "Figure: (Color online) The polarizations of the two atom are syntropic andrespectively perpendicular to the interdipole axis as shown in (d).", "Thevariations of collective radiation rates and level shift with respect tointerdipole distance RR are noticeably different in optically (a) inactiveand (b) active cases.", "The dimensionless dipole-dipole RI E int /ℏΓ 0 E_{int}/\\hbar \\Gamma _{0} at time Γ 0 t=1\\Gamma _{0}t=1 are plotted as functions of interdipoledistance in (c).", "RI behaves rather differently in optically active andinactive surroundings.The level shifts and the radiation rates in optically inactive and active cases are compared in Fig.", "REF (a) and (b).", "We see that for the optically inactive case, the first point of minimum of the level shift $\\delta $ appears in the neighbourhood of $k_{0}R = 3$ while it does near $k_{0}R = 2$ for the optically active case.", "Obviously that the curves in Fig.", "REF (a) and Fig.", "REF (b) are different, which means the resonance interaction of the dipoles are affected by the optical rotation of the medium.", "The dimensionless interaction energy $E_{int}/\\hbar \\Gamma _{0}$ in the case of syntropic dipole orientations at time $\\Gamma _{0}t=1$ are plotted in Fig.", "REF (c).", "Next we consider a situation that the dipoles are isotropically polarized, namely $d_{x}=d_{y}=d_{z}=d/\\sqrt{3}$ , as shown in Fig.", "REF .", "Figure: (Color online) The polarizations of the dipoles are isotropic (d x =d y =d z =d/3d_{x}=d_{y}=d_{z}=d/\\protect \\sqrt{3}) as shown in (d).", "The variations ofcollective radiation rates and level shift with respect to interdipoledistance RR are noticeably different in optically (a) inactive and (b) activecases.", "The dimensionless dipole-dipole RI E int /ℏΓ 0 E_{int}/\\hbar \\Gamma _{0} at timeΓ 0 t=1\\Gamma _{0}t=1 are plotted as functions of interdipole distance in (c).", "RIbehaves rather differently in optically active and inactive surroundings.The level shifts and the radiation rates in optically inactive and active cases are plotted in Fig.", "REF (a) and (b), which behave differently.", "For example, when the interdipole separation is about $k_{0}R =1.5 $ , the DEAS state decays faster in optically inactive surroundings while the DES state does in optically active case.", "The dimensionless interaction energy $E_{int}/\\hbar \\Gamma _{0}$ in the case of isotropic polarization at time $\\Gamma _{0}t=1$ are plotted in Fig.", "REF (c).", "These three numerical examples typically show the effect on RI of optically active surroundings.", "In studying RI or using it to develop measurement methods, the optical rotation of surroundings in which dipoles are immersed can not be ignored." ], [ "Conclusion", "In this paper, we have calculated the RI between two electric dipoles immersed in absorption free optically active surroundings.", "The analytical expressions of the collective radiation rates and level shifts are given out, which satisfy general dipole orientations and interdipole distances.", "Considering the optical rotation of surrounding medium, the cross product of two electric dipole moment appears while this term does not contribute to interdipole RI in optically inactive case.", "By comparing the RI in optically inactive and active cases numerically, we obtain that the interdipole RI are noticeably affected by the optical rotation of surroundings.", "Especially, in the case of that the orientations of two dipoles are orthogonal, the optical rotation of surrounding medium can help in building up interdipole RI while there is no interaction in situation of free space.", "Our result can help in studying the nature of RI between electric dipoles immersed in optically active medium, and may contribute in developing accurate methods of measurement in chemistry and biology." ], [ "Acknowledgements", "This work was supported by the National Natural Science Foundation of China under Grant Nos.", "11474095,  11274118,  11234003, and 11129402, and the Fundamental Research Funds for the Central Universities." ], [ "Appendix", "To get Eq.", "(REF ), we make use of Schrödinger equation $i\\hbar \\frac{\\partial \\left|\\psi \\right\\rangle }{\\partial t}=\\hat{H}\\left|\\psi \\right\\rangle , $ to get the equations of motion of the probability amplitudes $C_{i}$ and $D_{i, {k}\\lambda }$ $\\dot{C}_{1}=\\frac{i}{\\hbar }\\sum _{a=1,2}\\sum _{i^{\\prime }=3,4}\\sum _{{k}\\lambda } D_{i^{\\prime },{k}\\lambda }A_{k}{d}_{1i^{\\prime }}^{a} \\cdot {W}\\left( {k}\\lambda ,{r}_{a}\\right), \\\\\\dot{C}_{2}=\\frac{i}{\\hbar }\\sum _{a=1,2}\\sum _{i^{\\prime }=3,4}\\sum _{{k}\\lambda } D_{i^{\\prime },{k}\\lambda }A_{k}{d}_{2i^{\\prime }}^{a} \\cdot {W}\\left( {k}\\lambda ,{r}_{a}\\right), \\\\\\dot{D}_{3,{k}\\lambda }= -i\\left( \\omega _{k}+\\omega _{0}\\right)D_{3,{k}\\lambda } +\\frac{i}{\\hbar }\\sum _{a=1,2}\\sum _{i^{\\prime }=1,2} C_{i^{\\prime }}A_{k}^{\\ast }{d}_{3i^{\\prime }}^{a} \\cdot {W}^{\\ast }\\left( {k}\\lambda ,{r}_{a}\\right), \\\\\\dot{D}_{4,{k}\\lambda }= -i\\left( \\omega _{k}-\\omega _{0}\\right)D_{4,{k}\\lambda } +\\frac{i}{\\hbar }\\sum _{a=1,2}\\sum _{i^{\\prime }=1,2} C_{i^{\\prime }}A_{k}^{\\ast }{d}_{4i^{\\prime }}^{a} \\cdot {W}^{\\ast }\\left( {k}\\lambda ,{r}_{a}\\right).", "$ To solve these equations, we shall first calculate the formal solutions of Eqs.", "() and () $D_{3,{k}\\lambda }\\left( t\\right) =\\frac{i}{\\hbar }A_{k}^{\\ast }\\sum _{a=1,2}\\sum _{i^{\\prime }=1,2}{d}_{3i^{\\prime }}^{a}\\cdot {W}^{\\ast }\\left( {k}\\lambda ,{r}_{a}\\right) C_{i^{\\prime }}\\left( t\\right) \\left[ \\pi \\delta \\left( \\omega _{k}+\\omega _{0}\\right) -\\frac{i}{\\left( \\omega _{k}+\\omega _{0}\\right) }\\right], \\\\D_{4,{k}\\lambda }\\left( t\\right) =\\frac{i}{\\hbar }A_{k}^{\\ast }\\sum _{a=1,2}\\sum _{i^{\\prime }=1,2}{d}_{4i^{\\prime }}^{a}\\cdot {W}^{\\ast }\\left( {k}\\lambda ,{r}_{a}\\right) C_{i^{\\prime }}\\left( t\\right) \\left[ \\pi \\delta \\left( \\omega _{k}-\\omega _{0}\\right) -\\frac{i}{\\left( \\omega _{k}-\\omega _{0}\\right) }\\right], $ in which the Markov approximation has been used $\\int _{0}^{t}C_{i^{\\prime }}\\left( t^{\\prime }\\right) e^{-i\\left( \\omega _{k}-\\omega _{0}\\right) \\left( t-t^{\\prime }\\right) }dt^{\\prime } \\approx C_{i^{\\prime }}\\left( t\\right) \\left[ \\pi \\delta \\left( \\omega _{k}-\\omega _{0}\\right) -\\frac{i}{\\left( \\omega _{k}-\\omega _{0}\\right) }\\right].$ Substituting Eqs.", "(REF ) and () into Eqs.", "(REF ) and (), we can get the equations of $C_{1}$ and $C_{2}$ $\\dot{C}_{1}= A_{11}C_{1}\\left( t\\right) +A_{12}C_{2}\\left( t\\right), \\dot{C}_{2}= A_{22}C_{2}\\left( t\\right) +A_{21}C_{1}\\left( t\\right), $ and the coefficients $A_{ab}$ is defined by $A_{aa} =-\\frac{1}{\\hbar ^{2}} \\sum _{{k}\\lambda }\\left|A_{k}\\right|^{2} \\left\\lbrace G_{aa}\\left( {k}\\lambda \\right) \\pi \\delta \\left( \\omega _{k}-\\omega _{0}\\right) -G_{aa}\\left( {k}\\lambda \\right) \\frac{i}{\\left( \\omega _{k}-\\omega _{0}\\right) }-G_{bb}\\left({k}\\lambda \\right) \\frac{i}{\\left( \\omega _{k}+\\omega _{0}\\right) }\\right\\rbrace , \\nonumber \\\\A_{ab} =-\\frac{1}{\\hbar ^{2}} \\sum _{{k}\\lambda }\\left|A_{k}\\right|^{2} \\left\\lbrace G_{ba}\\left( {k}\\lambda \\right) \\pi \\delta \\left( \\omega _{k}-\\omega _{0}\\right) -G_{ba}\\left( {k}\\lambda \\right) \\frac{i}{\\left( \\omega _{k}-\\omega _{0}\\right) }-G_{ab}\\left({k}\\lambda \\right) \\frac{i}{\\left( \\omega _{k}+\\omega _{0}\\right) }\\right\\rbrace , $ in which $a \\ne b$ .", "The $G$ functions are defined as $G_{aa\\left(b\\right)}\\left( {k}\\lambda \\right) ={d}_{a}\\cdot {W}^{\\ast }\\left( {k}\\lambda ,{r}_{a}\\right) {W}\\left( {k}\\lambda ,{r}_{a\\left(b\\right)}\\right) \\cdot {d}_{a\\left(b\\right)} $ .", "The dipole transition matrix elements ${d}_{ij}^{a}$ have been denoted as ${d}_{a}$ for simplicity.", "The initial state is $C_{1}=1$ and $C_{2}=0$ , we further define $A_{L}=A_{11}=A_{22},A_{T}=A_{12}=A_{21},$ and it is not difficult to get solution (REF ) by solving Eq.", "(REF )." ] ]
1606.05149
[ [ "The mass discrepancy acceleration relation in early-type galaxies:\n extended mass profiles and the phantom menace to MOND" ], [ "Abstract The dark matter (DM) haloes around spiral galaxies appear to conspire with their baryonic content: empirically, significant amounts of DM are inferred only below a universal characteristic acceleration scale.", "Moreover, the discrepancy between the baryonic and dynamical mass, which is usually interpreted as the presence of DM, follows a very tight mass discrepancy acceleration (MDA) relation.", "Its universality, and its tightness in spiral galaxies, poses a challenge for the DM interpretation and was used to argue in favour of MOdified Newtonian Dynamics (MOND).", "Here, we test whether or not this applies to early-type galaxies.", "We use the dynamical models of fast-rotator early-type galaxies by Cappellari et al.", "based on ATLAS$^{3D}$ and SLUGGS data, which was the first homogenous study of this kind, reaching ~4 $R_e$, where DM begins to dominate the total mass budget.", "We find the early-type galaxies to follow an MDA relation similar to spiral galaxies, but systematically offset.", "Also, while the slopes of the mass density profiles inferred from galaxy dynamics show consistency with those expected from their stellar content assuming MOND, some profiles of individual galaxies show discrepancies." ], [ "Introduction", "The masses of galaxies, and in fact of any larger dynamically bound structure in the Universe, inferred from dynamics are found to exceed the masses of the observed baryons in these structures [68], [69], [46], [47], [7].", "This phenomenon is usually explained by postulating (non-baryonic) dark matter, which is also the backbone of structure formation in cosmological simulations [22], [8] and serves well in accounting for the characteristics of the cosmic microwave background radiation [6], [25].", "[50] demonstrated a surprising characteristic of this discrepancy between dynamical and baryonic mass: it occurs below a characteristic acceleration level.", "[60] reproduced such a characteristic acceleration scale in the $\\Lambda $ CDM context using semi-analytic models of galaxy formation, which were tuned to reproduce the Tully-Fisher relation.", "However, [50] also found that the amplitude of this discrepancy correlates with the acceleration, the so-called mass discrepancy acceleration (MDA) relation.", "Moreover, there is only small scatter about this relation, as confirmed by [37].", "This correlation, and even more its tightness, is surprising in the context of hierarchical structure formation and the large variety of possible merger trees for individual galaxies in $\\Lambda $ CDM cosmology (see also [37], [38], [64]).", "So far tests of the MDA relation have concentrated mainly on spiral galaxies.", "In these systems the H i gas makes it relatively easy to trace the dynamical mass far from the centre of the galaxy.", "However, spiral galaxies also possess large amounts of baryonic mass in the form of gas.", "The mass of this component is rather difficult to estimate accurately from observations and introduces uncertainties in the MDA relation.", "Early-type galaxies (ellipticals and lenticulars), due to their lower H i gas content, do not suffer from this problem, however they lack an easy to measure tracer.", "For this reason, studies addressing the MDA relation in early-type galaxies have had to largely rely on other tracers such as hot X-ray emitting gas or discrete tracers like planetary nebulae (PNe), globular clusters (GC), and satellite galaxies (e.g.", "[40]; see however [33]).", "While theoretical efforts trying to reproduce the MDA relation in the $\\Lambda $ CDM context are progressing (e.g.", "[24]), it is desirable to push the observational side further, and probe the MDA relation systematically for a sample of early-type galaxies down to low accelerations.", "Alternatively, the MDA relation, and its tightness, have been used to argue in favour of a modification of Newtonian dynamics (MOND; see, e.g.", "[32]; [67]).", "A number of previous studies have discussed whether early-type galaxies can be used to falsify the theory [41], [57], [66], [34], [44], [32], [48], [49], [20], [23].", "Cappellari:2015gtC+15 Recently, [19] carried out dynamical modelling of a sample of 14 early-type galaxies, empowered by the combination of the inner stellar kinematics from ATLAS$^{\\textrm {3D}}$ [15] and stellar kinematics reaching out to a median radius of about 4 half-light radii ($R_\\textrm {e}$ ; [4]) from the SLUGGS survey [9].", "This represents the first homogeneous, statistically meaningful sample of stellar kinematics of early-type galaxies (all of which are fast rotators as defined in [29]) reaching radii where dark matter is expected to dominate the mass budget (i.e.", "the dark matter fraction increases from $\\sim $ 1/3 to $\\sim $ 2/3 when measured within 2 and 4 $R_{\\rm e}$ , respectively).", "Here we use Cappellari:2015gt's mass modelling results to test whether or not early-type galaxies follow the MDA relation.", "Furthermore, we test whether or not MOND, which was originally designed to reproduce the rotation curves of spiral galaxies [39], fails in early-type galaxies.", "Finally, we also consider results from mass modelling based on GC kinematics for an extended sample, including slow rotators and reaching even larger radii, also from the SLUGGS survey [1]." ], [ "Sample, Data and Dynamical Modelling", "Our primary sample comprises 14 early-type galaxies in a stellar mass range of $1.5 \\times 10^{10} < M_* / \\textrm {M}_\\odot < 50 \\times 10^{10}$ [17].", "Mass models for these galaxies were built by Cappellari:2015gt using axisymmetric Jeans anisotropic modelling (JAM; [12]) with a stellar and a dark matter component.", "To ensure the method to be applicable, and to achieve a homogenous sample of nearly axisymmetric galaxies, Cappellari:2015gt restricted the sample to fast rotators.", "One input to the JAM modelling is the light distributions in these galaxies.", "Those were parametrized using Multi-Gaussian Expansions (MGE; [26], [11]) and were taken from various studies ([27], [14], [54], [55]; Cappellari:2015gt).", "Here, we use the same characterizations of the distributions of stars within the galaxies.", "Other galaxy parameters, such as half-light radius and distance, are taken from Cappellari:2015gt, and the velocity dispersion within $\\sigma _{\\rm e}$ from [17] and [35] for NGC 3115.", "The combination of data from ATLAS$^{\\textrm {3D}}$ [15] and SLUGGS [9] allowed Cappellari:2015gt to probe the stellar dynamics of the galaxies from their inner parts out to a median radius of $\\sim $ 4 $R_{\\textrm {e}}$ .", "The stellar kinematics were extracted in the optical (ATLAS$^{\\textrm {3D}}$ ; [28], [15]) and Ca triplet spectral region (SLUGGS; [4]), in both cases with pPXF [13].", "For the dynamical modelling the data were symmetrized, outliers were removed, and the two data sets combined.", "We refer the reader to Cappellari:2015gt for a description of the process.", "However, we do note the key characteristic of Cappellari:2015gt was the use of a very general parametrization for the dark halo in the modelling process, to be able to focus on the total density profile alone.", "Figure: Comparison of stellar mass-to-light ratios.", "The red squares show the mass-to-light ratios fromfits of the dynamical models to the observations assuming MOND ( (M/L) MOND (M/L)_{\\rm MOND}; see also Section ).Blue stars indicate the stellar (M/L) JAM ,* (M/L)_{{\\rm JAM},*} obtained directly from JAM of ATLAS 3 D^\\textrm {3D} data .", "The stellar population mass-to-light ratios from the same study, which assume a Salpeter IMF, are shown as green triangles, and the lines indicate a factor of 0.63approximating the corresponding value for a Kroupa IMF.The various methods for calculating stellar mass-to-light ratios reproduce the same qualitative trend of stellar mass-to-light ratio with velocity dispersion σ e \\sigma _e.The JAM models are very accurate and robust, which was confirmed by comparison with the results of [56] based on H i data, but despite the increase in radial extent when compared to similar earlier studies, they probe the dynamics to accelerations an order of magnitude higher than those typically probed in spiral galaxies.", "[1] recently applied the tracer mass estimator of [65] to the kinematics of the GC systems for a superset of early-type galaxies, mostly from the SLUGGS survey [9].", "This method estimates the enclosed mass from the line-of-sight velocities of the tracer population based on assumptions for the power-law slope of the gravitational potential, the power-law slope of the tracer density profile, and the orbital anisotropy.", "While being less reliable, these models allow us to get indications about trends beyond 5 $R_{\\textrm {e}}$ , and also for slow rotators.", "[1] analysed the dependence of their mass estimates on these assumptions, took into account corrections for non-sphericity and substructures in the tracer population, and compared their results to previous studies.", "Their total mass estimates with the assumption of isotropy agree with similar earlier studies within a factor of 1.6.", "We refer the reader for the details to [1] and adopt their estimates for isotropic orbits.", "The SLUGGS galaxies were selected to be representative early-type galaxies across various environments (see [9]) and span a wide mass range up to NGC 4486, i.e. M87.", "For 23 of the 25 SLUGGS galaxies [1] derived estimates for the total mass within $5\\,R_{\\textrm {e}}$ and beyond.", "Most of the galaxies have rich enough GC data sets to derive not only a single dynamical mass estimate, but also radial mass density profiles.", "This applies for 10 of the 14 galaxies of our primary sample.", "Beyond the galaxies common to both samples, the study of [1] comprises NGC 1400, NGC 3607, NGC 4564, NGC 5866 and also the slow rotators NGC 720, NGC 1407, NGC 3608, NGC 4365, NGC 4374, NGC 4486, and NGC 5846." ], [ "MDA relation in early-type galaxies", "Before analysing the MDA relation, we need to convert the stellar light to stellar mass, in order to estimate the expected accelerations.", "Several options are available.", "The stellar populations can be fitted with models to infer a stellar mass-to-light ratio.", "This requires knowledge of the stellar populations and needs to assume a stellar initial mass function (IMF), which is suspected to vary from galaxy to galaxy [62], [16], even in the framework of MOND [58].", "The stellar mass-to-light ratio $(M/L)_{{\\rm JAM},*}$ can also be obtained from the dynamical model.", "For ATLAS$^\\textrm {3D}$ , the ratio is then based on the assumption of Newtonian gravity [18].", "Here, we also compute the stellar mass-to-light ratio $(M/L)_{\\rm MOND}$ by fitting the dynamics expected from the light profile based on MOND to the observed dynamical profile, and with a spatially constant mass-to-light ratio as fitting parameter.", "This fit is dominated by the inner regions where density and flux are highest.", "We compare the various mass-to-light ratios in Fig.", "REF .", "Since the MGE used in the dynamical modelling came from observations with different photometric filters, we convert all mass-to-light ratios to the $r$ -band, using the photometric predictions of [63] and [43].", "The various ways of determining the mass-to-light ratio all show the same trend of an increase with increasing velocity dispersion $\\sigma _{\\rm e}$ within $R_{\\rm e}$ [14], [61].", "This trend is due to underlying changes of the stellar population characteristics, e.g.", "increasing age and metallicity (and increasing $M/L$ due to changes in the IMF).", "In the following we use the mass-to-light ratios from the MOND fitting $(M/L)_{\\rm MOND}$ for consistency with the comparisons in Section .", "In the next step the enclosed mass is calculated, both for the dynamical mass from JAM, which is calculated from the density profiles of Cappellari:2015gt, and for the stellar mass.", "The latter is obtained from the published MGEs of the galaxy light distribution and the spatially constant stellar mass-to-light ratio $(M/L)_{\\rm MOND}$ as follows.", "The mass of an axisymmetric MGE model, enclosed within a spherical shell of radius $r$ is given by $\\begin{split}& M(r) = \\sum _{i} M_{i} \\left\\lbrace \\textrm {erf}\\left[r/(\\sqrt{2}q_i\\sigma _i)\\right] \\,\\,\\, - \\vphantom{ { { \\exp \\left[-r^2/(2\\sigma _i^2)\\right] \\textrm {erf}\\left[r \\sqrt{1-q^2_i}/(\\sqrt{2}q_i\\sigma _i) \\right] } \\bigg /{ \\sqrt{1-q^2_i} }}} \\right.", "\\\\& \\left.", "{ { \\exp \\left[-r^2/(2\\sigma _i^2)\\right] \\textrm {erf}\\left[r \\sqrt{1-q^2_i}/(\\sqrt{2}q_i\\sigma _i) \\right] }\\bigg / { \\sqrt{1-q^2_i} }} \\right\\rbrace ,\\end{split}$ with the MGE parameters for width, flattening, and total mass ($\\sigma _i$ , $q_i$ , and $M_{i}$ ), for each Gaussian component respectively.", "This equation was obtained by integration of the density profile in terms of the MGE as given in footnote 11 of Cappellari:2015gt.", "In addition to stellar and dynamical mass, the Newtonian acceleration due to the baryons is needed for the MDA relation, and it is given in the spherical limit by $g_{N,*}(r) = G M_*(r) / r^2.$ We calculate uncertainties in a Monte Carlo fashion, by randomly perturbing the density profiles of Cappellari:2015gt within their errors (see Sec. ).", "However, rather than using only formal uncertainties, we try to be conservative and account for systematic uncertainties as follows.", "We adopt a 6% uncertainty in the overall mass normalization as inferred by [17] and an upper limit of 0.11 in the profile slope, derived from the observed scatter in Cappellari:2015gt.", "Therefore, we perturb the profiles by adding a random constant with $\\sigma = 0.025$ dex to log$(r)$ , and by adding a linear trend in $\\log \\rho (r)$ versus $\\log (r)$ with a random slope ($\\sigma =0.05$ ).", "We calculate and plot for each galaxy 100 realizations of this process.", "The resulting MDA relation is plotted for all 14 galaxies in the left-hand panel of Fig.", "REF .", "It shows that our fast rotators follow a relation similar to that found for spiral galaxies, with the dynamical to stellar mass discrepancy systematically increasing with decreasing acceleration at that radius.", "Figure: Mass discrepancy acceleration relation.", "Left panel: the dynamical-to-stellar mass ratio versus the acceleration caused by the stars (both quantities inferred through Newtonian dynamics).", "Right panel: MDA relation as in the other panel.", "Results based on GC system dynamics from are shown as blue plus signs and and green crosses for values at 5R e 5 R_\\textrm {e} and R max R_\\textrm {max}, respectively (see text), and larger symbols mark slow rotators.", "Typical error bars are indicated in the top right corner (these show random variations, while additional systematics of similar magnitude can be expected).", "For comparison, data for spiral galaxies from are included in both panels as grey points.", "The different colours of the lines denote the various galaxies, each with 100 realizations (see text).", "The black solid curve shows the relation as expected from MOND using the simple interpolating function (see Equation ), and the grey vertical dashed line indicates the corresponding value of a 0 =1.2×10 -10 a_0 = 1.2 \\times 10^{-10} m s -2 ^{-2}.", "The dash-dotted curve in the left panel shows the same MOND relation for a 0 =1.35×10 -10 a_0 = 1.35 \\times 10^{-10} m s -2 ^{-2}, and the dashed curve that for the standard interpolating function with a 0 =1.2×10 -10 a_0 = 1.2 \\times 10^{-10} m s -2 ^{-2}.", "While our sample follows an MDA relation with large scatter, it is offset from the comparison sample of spiral galaxies.", "The slow rotators in the right panel display still larger offsets." ], [ "Comparison to modified Newtonian Dynamics", "[39] introduced a modification to Newtonian dynamics as an alternative to dark matter in explaining flat rotation curves in spiral galaxies (for recent reviews see [32]; [10], and the entire Special Issue in which it was published).", "In Newtonian dynamics the acceleration caused by a spherical matter distribution within a radius $r$ is given by Equation (REF ).", "In MOND the acceleration $a$ felt by matter is modified.", "Under the assumption of spherical symmetry the two quantities are related by $g_N = \\mu (a/a_0)\\ a,$ with the interpolating function $\\mu $ and a characteristic acceleration scale $a_0 = 1.2 \\times 10^{-10}$ m s$^{-2}$ .", "For high accelerations ($a\\gg a_0$ ) $\\mu \\rightarrow 1$ , i.e.", "the modification vanishes and the Newtonian limit applies.", "Equation (REF ) can be easily inverted $a=\\nu \\left(g_N/a_0\\right)g_N,$ with $\\mu (x) = 1/\\nu [x\\mu (x)]$ .", "One common choice for the interpolating function $\\mu (x)$ , especially on galactic scales, is the simple interpolating function $\\mu (x)=x/(1+x)$ , so that $\\nu (y)=1/2 + \\sqrt{1/4 + 1/y}$ .", "The dynamically inferred mass $M_\\textrm {dyn}$ , when assuming Newtonian dynamics, is then given by [40] $M_\\textrm {dyn}(r) = a(r)\\ G^{-1} r^2 = \\nu \\left[ \\frac{G M_*(r)}{r^2 a_0} \\right] M_*(r).$ This offers a natural explanation for flat rotation curves in disc galaxies and for the baryonic Tully-Fisher relation (see, e.g., [32]), and it means a Newtonian observer concludes the presence of – from the MOND point of view `phantom' – dark matter given by $M_\\textrm {DM} = M_\\textrm {dyn} - M_*$ .", "In terms of the MDA this leads to a simple prediction, which is plotted as the black solid curve in Fig.", "REF (see also, e.g., [36] for a more theoretical account).", "Given the uncertainties the MOND curve looks like a rather good representation for our sample.", "Cappellari:2015gt found the galaxies to have nearly isothermal mass density profiles ($\\rho (r) \\sim r^{-2.2}$ ), with only small amounts of scatter, which was confirmed by [56] for a sample of early-type galaxies based on H i data.", "In Fig.", "REF the stellar density profile, the dynamical mass density profile from Cappellari:2015gt, and that predicted from the stellar density profile using MOND are shown.", "It can be seen that the MOND profiles reproduce the slopes of the dynamically determined profiles well.", "This is not a surprise, since galaxies have asymptotically isothermal profiles in MOND.", "However, when compared in more detail, for each galaxy individually, inconsistencies become noticeable.", "There are galaxies like NGC 3115 that start to diverge strongly from the MOND prediction at the outermost radii, but also galaxies like NGC 4473 and NGC 4494 for which features at smaller radii are not well reproduced.", "This becomes especially evident when considering the residual profiles in the right panel, which are calculated individually for each Monte Carlo representation." ], [ "Discussion", "The general reproduction of the MDA and the slopes of the density profile by MOND is intriguing at first.", "We note that this is achieved without fitting for the MOND acceleration scale, but using for all galaxies a value of $a_0=1.2\\times 10^{-10}$ m s$^{-2}$ , which was found by [5] for spiral galaxies, and which is still used in more recent studies [32].", "The sensitivity to the exact value is not very critical, as can be seen in the comparison curve in the MDA relation for another value $a_0=1.35\\times 10^{-10}$ m s$^{-2}$ , which was also used for spiral galaxies (e.g. [31]).", "However, as Cappellari:2015gt showed, the dynamical profiles can also be expected to be simple power-laws in the $\\Lambda $ CDM context out to about 10 $R_{\\textrm {e}}$ .", "Furthermore, the profile for individual cases are less consistent as mentioned above.", "Our analysis makes a few potentially limiting assumptions, such as a spatially constant mass-to-light ratio and anisotropy.", "Cappellari:2015gt accounted for two individual anisotropies in the inner and outer part.", "Also, for the calculation of the MONDian predictions we assumed spherical symmetry, which may seem to be a rather strong assumption given that the galaxies in our sample are – often flattened – fast rotators.", "[21] and [42] demonstrated that the expected deviations due to non-sphericity effects on the non-linear MOND equivalent of Poisson equation are however small even in flat spiral galaxies.", "The modelling uncertainties are dominated by systematics, which can be important in individual cases.", "For this reason significant progress can best be made in a statistical manner using large galaxy samples.", "With this in mind, the following could be considered as a more serious challenge for MOND than the mismatches for individual galaxies.", "We showed our comparisons with the MOND predictions when using the simple interpolating function, since it gives more consistent results for our sample than the standard one [30], [52], [66], [40], [20].", "While changing the MOND constant $a_0$ within the range of values used for spiral galaxies only leads to minor changes, switching to the standard interpolating function has more severe effects.", "The number of density profiles that are still consistent with MOND given our assumptions and estimates of uncertainties is roughly halved as compared to that when using the simple interpolating function.", "In the MDA relation this is even more evident.", "The calculation with the standard interpolating function largely underpredicts the mass discrepancy for our sample (Fig.", "REF ) and marks essentially the lower edge of the trend of observed MDA curves, while the MDA relation calculated with the simple interpolating function runs through the middle of this trend.", "However, the opposite is true for the comparison sample of spirals from [32], for which the standard interpolating function provides a superior representation.", "This is at odds with MOND, where there should be one universal interpolating function and MDA relation.", "The above is consistent with [33]'s ([33]) results from stellar dynamics.", "They concluded that, from their analysis of the MDA for a sample of early-type galaxies, the upturn of the MDA relation occurs at higher accelerations than for spiral galaxies, in disagreement with MOND (when using accelerations $a=v^2/r$ based on the dynamics as in their fig.", "19 our Fig.", "REF looks similar).", "Their sample included two of our galaxies and also slow rotators.", "At that time, [41] doubted that the change in the mass-to-light ratios, which is what [33] actually plotted, marked the onset of the mass discrepancy, and pointed out discrepancies between the results of [33] and the profiles of [45] for the galaxies common to both studies.", "Even with the combination of stellar kinematics from ATLAS$^\\textrm {3D}$ and SLUGGS surveys we only reach radii at which dark matter just starts to be dominant – and where the accelerations just decrease to values of the order of the MOND constant $a_0$ .", "Using GC kinematics from the SLUGGS survey, [1] studied dynamical models for a superset of galaxies reaching beyond $ 5\\, R_{\\textrm {e}}$ .", "The conclusions based on these should be considered more tentative, since they are less robust than those from the JAM modelling.", "Here, we are interested in the comparison to the MOND expectations.", "These were obtained by fitting the MOND profile to the JAM density profile via the (spatially constant) mass-to-light ratio as a fitting parameter.", "In Fig.", "REF , we vertically shift the density profiles of [1] so that they match the JAM profiles in the radial overlap region for comparison to the same MOND profiles.", "The [1] profiles are less smooth due to the discrete nature of the tracers, but give nonetheless some indication how the dynamically inferred density profiles in Fig.", "REF continue at larger radii.", "In some cases, e.g.", "NGC 821, the deviations from MOND increase, while in other cases, e.g.", "NGC 3377, the onset of the deviation appears less critical, since the continuation is close to consistent with MOND.", "Previous studies with dynamical tracers concluded that NGC 821 is MONDian to $\\sim $ 3.5 $R_\\textrm {e}$ , using PNe [41], while [49] could not reconcile NGC 3115 with MOND, similar to our analysis.", "The same applies to NGC 4278, which is in our analysis only marginally inconsistent.", "For the MDA relation, adding the [1] data is problematic, since their profiles are non-monotonic and `jumpy', again due to the discrete nature of the tracers.", "Instead, we use the total masses and dark matter fractions at 5 $R_\\textrm {e}$ and at $R_\\textrm {max}$ from their table A4.", "The resulting dynamical-to-stellar mass ratios and accelerations are shown in the right-hand panel of Fig.", "REF , including their 11 additional galaxies.", "The sample of fast rotators, which overlaps with our sample of galaxies with stellar dynamics, generally appears to follow the same MDA.", "The even larger scatter prevents conclusions as to which interpolating function performs better.", "However, the GC system data seem also to suggest that the slow rotators have systematically higher dynamical-to-stellar mass ratios, despite the large uncertainties.", "This is qualitatively consistent with [49], who generally found more severe inconsistencies with MOND for slow rotators based on binned kinematics of GC systems, also using SLUGGS data (see also [2] for an account of pressure supported dwarf spheroidals within MOND).", "While these tentative GC based conclusions do not share the robustness of those based on the JAM models (which were reinforced by [56]), they suggest an interesting trend of increasing offsets in the MDA relation from spiral galaxies to fast rotators to slow rotators.", "In the framework of MOND, these findings may be explained by unseen (normal) matter, which is known to be required in this context on the larger scales of galaxy groups and clusters [51], [3].", "In the above context it is noteworthy that [23] found for the ATLAS$^\\textrm {3D}$ galaxies with H i data that the baryonic Tully-Fisher relation has little scatter – which would be consistent with the MOND framework.", "However, [59] previously found an offset between the baryonic Tully-Fisher relation for spiral and early-type galaxies." ], [ "Summary", "We analysed the mass discrepancy acceleration (MDA) relation for early-type galaxies by using the dynamical models for the stellar dynamics of 14 fast rotators of [19].", "The range, robustness, and accuracy of these models allowed us to determine that, while the galaxies broadly follow such a relation, they are systematically offset from the comparison sample of spiral galaxies [32].", "This adds to the challenges found for MOND when comparing the dynamically determined profiles to the MOND predictions for individual galaxies.", "Meanwhile, the simulations of [24] demonstrated that the MDA for spiral galaxies could arise in $\\Lambda $ CDM from variations of the dark matter profile shape with galaxy mass, instead of a universal NFW profile.", "Our analysis predicts the corresponding models for fast-rotator early types to be offset from the MDA for spiral galaxies." ], [ "Acknowledgements", "We are grateful for the referee's suggestions, which improved the presentation of our results.", "JJ and DAF thank the ARC for financial support via DP130100388.", "MC acknowledges support from a Royal Society University Research Fellowship.", "LC was supported by the MIUR grant PRIN 2010-2011, project `The chemical and dynamical evolution of the Milky Way and Local Group Galaxies', prot.", "2010LY5N2T." ] ]
1606.05003
[ [ "Nonlinear optical conductivity of a generic two band systems, with\n application to doped and gapped graphene" ], [ "Abstract We present a general formulation to calculate the dynamic optical conductivity, beyond the linear response regime, of any electronic system whose quasiparticle dispersion is described by a two band model.", "Our phenomenological model is based on the optical Bloch equations.", "In the steady state regime it yields an analytic solution for the population inversion and the interband coherence, which are nonlinear in the optical field intensity, including finite doping and temperature effects.", "We explicitly show that the optical nonlinearities are controlled by a single dimensionless parameter which is directly proportional to the incident field strength and inversely proportional to the optical frequency.", "This identification leads to a unified way to study the dynamical conductivity and the differential transmission spectrum across a wide range of optical frequencies, and optical field strength.", "We use our formalism to analytically calculate the nonlinear optical conductivity of doped and gapped graphene, deriving the well known universal ac conductivity of $\\sigma_0={e^2}/4\\hbar$ in the linear response regime of low optical intensities (or equivalently high frequencies) and non-linear deviations from it which appear at high laser intensities (or low frequencies) including the impact of finite doping and band-gap opening." ], [ "Introduction", "Since it's discovery, graphene, a truly two-dimensional system, has been at the forefront of material research [Novoselov22102004,Nature438197(2005), 10.1038/nmat1849].", "On account of its linear dispersion and high carrier mobility, graphene has demonstrated remarkable electronic and optical properties [katsnelson2012graphene].", "In the linear response regime, with weak optical field induced momentum linearly coupling to the charge carriers in graphene, spectacular physical effects have been predicted and observed which are significantly enhanced or peculiar, when compared to their bulk counterparts.", "An early such surprise was the observation of a strong coupling of a single monolayer of carbon atoms to electro-magnetic radiation with almost a constant absorption coefficient of $2.3\\%$ over a broad range of optical frequencies.", "The corresponding optical conductivity is elegantly expressed in terms of universal constants, in the form $\\sigma (\\omega )= \\sigma _0 \\equiv {e^2}/{4\\hbar }$ [apl131905,Nair2008,PRL.100.117401,nphys989, Mak24082010].", "Several exotic predictions and observations followed, including ultra-high mobility in pristine graphene [Bolotin2008], Klein tunneling [PRL102026807(2009)], weak localization [PRL100056802(2008)] and quantum hall effect [Nature438197(2005), Science3151379(2007), Nature462196(2009)].", "Following the first prediction of universal optical conductivity for graphite honeycomb lattices [Ando2002], several theoretical works extended the formulation to graphene using the Dirac cone approximation for clean samples [PhysRevB.73.245411,Gusynin2006,PRB.75.165407], and with disorder [PRB.73.125411].", "Experimental observation of the universal optical conductivity [apl131905,Nair2008,PRL.100.117401,nphys989, Mak24082010] gave a strong impetus to the field and motivated further theoretical work.", "These include studying the bandstructure effects on optical conductivity beyond the Dirac cone approximation [StauberTB], analyzing effect of strain [EPL.92.67001], role of substrate [EPL.84.38002] and the effect of electron-electron interactions on the optical conductivity [PRL.98.216801, PRL.100.046403, PRB.78.085416, EPL.83.17005].", "In addition to the linear conductivity, several remarkable non-linear optical effects in graphene have also been explored [Zhang1, arXiv1106.4838, Mishchenko, arXiv150600534M, 0295-5075-79-2-27002, 0295-5075-84-3-37001, EPL.84.38002, PhysRevB.82.201402, JPCCMikhailov, Ruvinskii, arXiv150901209], motivating a plethora of applications[nphoton.2010.186, Nano.nn300989g, IEEEAvouris, nphoton.2012.262, nl102824h] based on broadband nonlinear optical properties of graphene.", "In particular, non-linear optical response along with linear dispersion in graphene imply higher harmonic generations, and the large velocities of carriers results in highly efficient electron-photon coupling.", "Furthermore, gapless excitation leads to effective resonant non-linear excitation in graphene.", "These interesting possibilities have motivated a plethora of studies based on non-linear response of these 2-d materials, ranging from microwave [Gusynin2006], terahertz [Mikhailov2009] to optical frequencies [nphoton.2010.186].", "Such response has led to second and third harmonics generation in the optical [Dean2009,Glazov2011] and THz domain [Crosse], frequency mixing ranging from microwave to optical excitations [PRL105097401(2010), NatPhot6554(2012)], self-phase modulation and optical Kerr effects [NanoLett115159(2011), Chu2012104], photon drag [PhysRevB.81.165441], THz driven chiral edge photo-currents [PhysRevLett.107.276601] or, dynamic Hall-effect driven by circularly polarized optical frequencies [PRL105227402(2010)].", "Such unusual effects triggered a range of applications including graphene based rf modulators [Nature47464(2011)], optically gated transistors [NatNano7363(2012), NanoLett141242(2014)], photo-detectors [NatNano9780€“793(2014)], graphene saturable absorbers for mode-locking [APL96111112(2010)] and even proposal for nonlinear interaction at the level of single photons [NJP17083031(2015)].", "Figure: Classification of different optical response regimes in the parameter space of optical frequency (in Hz) and the incident radiation field strenth (in units of V/m).", "The vertical black line marks the ω=γ 2 \\omega =\\gamma _2 boundary between the dirty or low frequency (left) and the clean or high frequency (right) limits.", "The black dashed line marks the ζ≡ev F E 0 /(ℏωγ 1 γ 2 )=1\\zeta \\equiv e v_{\\rm F} E_0/(\\hbar \\omega \\sqrt{\\gamma _1 \\gamma _2)} =1 line which is the boundary of the linear (bottom) and non-linear (top) response regimes.", "Note that for a given frequency and material (with fixed damping parameters) the linear to non-linear regime crossover occurs at smaller field strength.", "Here we have chosen v F =10 6 v_{\\rm F} =10^6 m/s, and {γ 1 ,γ 2 }={10 12 ,10 14 }\\lbrace \\gamma _1,\\gamma _2\\rbrace = \\lbrace 10^{12},10^{14}\\rbrace Hz, based on parameters for graphene (see Ref.", "[Zhang1] and the references therein).At a fundamental level, non-linear response functions serve as excellent tools for probing intrinsic material properties.", "This provides a rich class of information on material symmetry and selection rules, intricacies of band-structure, electron-spin relaxation and decoherence mechanisms that are otherwise hidden in the linear response regime.", "There is thus significant motivation to develop a unified theoretical framework to address non-linear response of low dimensional systems in general, that is applicable over a range of optical excitation frequencies, and a wide range of the incident optical field strength.", "Motivated by Mishchenko [Mishchenko], in this article we present a theoretical framework for calculating non-linear optical conductivity of a general two-band system, which is applicable over a large range of optical frequencies and field strengths.", "We recast the wave-function based approach of Ref.", "[Mishchenko], into a density matrix based approach and this allows us to incorporate the impact of finite temperature, finite doping (chemical potential) etc.", "into the non-linear conductivity calculations.", "In particular, we provide steady state solutions for the coupled Maxwell-Bloch equations for a two band system, in which we include electron-electron and electron-phonon scattering phenomenologically via the inter-band population inversion decay rate $\\gamma _1$ and the coherence decay rate $\\gamma _2$ .", "The steady state population inversion and the inter-band coherence is then used to calculate the optical conductivity for a generic two band systems, and the optical transmission spectrum for a two dimensional system in general.", "We apply the developed formulation to analyze the non-linear optical conductivity of doped and gapped graphene in detail.", "However, the formalism can also be applied to other systems such as bilayer graphene, Weyl semimetals, phosphorene etc.", "whose low energy electronic properties are captured by a two band model.", "A natural outcome of our formalism is that, with $\\gamma _1$ and $\\gamma _2$ as two phenomenological input parameters, it allows us to classify the parameter space in terms of the incident optical frequency ($\\omega $ ) and field strength ($E_0$ ) in four regimes: a) linear response in the clean regime, b) linear response in the dirty regime, c) non-linear response in the clean regime and finally d) the non-linear response in the dirty regime [see Fig.", "REF ].", "What we call the clean (dirty) regime can also be called the collision-less or high frequency limit (collisional or low frequency), and is quantified by the region $\\omega \\ge \\gamma _2$ ($\\omega \\ll \\gamma _2$ ).", "The linear ($\\zeta <1$ ) or nonlinear ($\\zeta >1$ ) response of the system is quantified by a single dimensionless parameter [Mishchenko], $\\zeta \\equiv \\frac{ e v_{\\rm F} E_0}{\\hbar \\omega \\sqrt{\\gamma _1 \\gamma _2}}~,$ where $v_{\\rm F}$ denotes a material dependent effective velocity.", "Non-linear optical effects start becoming dominant either on increasing the field strength keeping the frequency constant, or alternatively by decreasing the frequency while keeping the field strength constant.", "The manuscript is organized as follows: in Sec.", ", we describe a general two band systems and its optical response via the population inversion and coherence.", "This is followed by a discussion of the non-linear optical conductivity in Sec.", ", and the differential transmission of a freestanding two dimensional material in Sec. .", "In Sec.", "and Sec.", ", we discuss the dynamic conductivity of doped and gapped graphene respectively, in various limiting cases of linear response in clean limit (previously known), linear response in dirty limit, non-linear response in the clean limit and the most general case of non-linear response in the dirty limit.", "Finally we summarize our findings in Sec.", "." ], [ "Population inversion and coherence in a general two band system", "We start with a very generic two band electronic system, in presence of an electromagnetic radiation.", "The electromagnetic field is treated classically in the coulomb gauge, with the vector potential ${\\bf A}$ satisfying ${\\bf \\nabla }\\cdot {\\bf A} = 0$ and the scaler potential $\\Phi = 0$ , yielding $\\bf B = {\\bf \\nabla } \\times {\\bf A}$ and ${\\bf E} = - \\partial _t {\\bf A}$ .", "The Hamiltonian describing the dynamics of an electron in presence of an external electromagnetic field, is given by $H_{ \\rm em} = H_0 (\\hbar \\hat{\\bf k} \\rightarrow \\hbar \\hat{\\bf k}+ e{\\bf A} )$ , which is usually approximated as $H_{\\rm em} \\approx H_0 + e\\hbar ^{-1}{\\bf A} \\cdot \\nabla _{\\bf k} H_0 $ .", "Note that while $H_{\\rm em} $ is an approximation for general two band systems, it is exact for systems described by the two dimensional Dirac Hamiltonian, for which $H_0$ only depends linearly on the wave-vectors.", "Furthermore, from the perspective of calculating optical conductivity, this is akin to neglecting the diamagnetic part of the current (which anyway vanishes for Dirac systems since $\\partial ^2 H_0/\\partial k_i^2 = 0$ ), and only focussing on the paramagnetic part of the response function [StauberNJP].", "In the eigen-basis of $H_0$ , the effective Hamiltonian, can be rewritten as, $H_{\\rm em} = H_0^{\\prime } + \\bf A \\cdot {\\bf M}$ , where $H_0^{\\prime }$ is a diagonal matrix comprising of the dispersion of two bands.", "The elements of the matrix ${\\bf M}$ are defined by ${\\bf M}^{\\lambda \\lambda ^{\\prime }} ({\\bf k}) = e \\hbar ^{-1} \\langle \\psi ^\\lambda |{\\nabla _{\\bf k}} H_0 | \\psi ^{\\lambda ^{\\prime }} \\rangle $ .", "We consider light to be incident perpendicular to the sample, with negligible transverse momentum, such that it does not significantly alter the electron momentum.", "This sets a selection rule, allowing only vertical transitions in the momentum space.", "More explicitly, $H_{\\rm em} = H_0^{\\prime } + H_I$ , where we have $H_0^{\\prime } = \\sum _{\\bf k} \\varepsilon _{\\bf k}^c {a^c_{\\bf k}}^\\dagger a^c_{\\bf k} + \\varepsilon _{\\bf k}^v {a^v_{\\bf k}}^\\dagger a^v_{\\bf k}~,$ with ${a^c_{\\bf k}}({a^{c\\dagger }_{\\bf k}}) $ and ${a^v_{\\bf k}}({a^{v\\dagger }_{\\bf k}})$ being the annihilation (creation) operator for electron in the conduction and valance band respectively.", "The interaction part of the Hamiltonian is given by $\\frac{H_I}{\\hbar } = \\sum _{\\bf k} \\Omega ^{cc}_{\\bf k} {a^c_{\\bf k}}^\\dagger a^c_{\\bf k} + \\Omega ^{vv}_{\\bf k}{a^v_{\\bf k}}^\\dagger a^v_{\\bf k} + \\Omega ^{cv}_{\\bf k} {a^c_{\\bf k}}^\\dagger a^v_{\\bf k} + \\Omega ^{vc}_{\\bf k}{a^v_{\\bf k}}^\\dagger a^c_{\\bf k},$ where we have defined $\\hbar \\Omega ^{\\lambda \\lambda ^{\\prime }}_{\\bf k} = {\\bf M}^{\\lambda \\lambda ^{\\prime }}({\\bf k}) \\cdot {\\bf A}$ to be the Rabi frequencies.", "Let us now consider an arbitrary two band system, whose low-energy quasiparticle bands are described by the generic $2 \\times 2$ Hamiltonian, $H_0 = \\sum _{\\bf k}{\\bf h}_{\\bf k} \\cdot \\sigma ,$ where $ {\\bf h}_{\\bf k} = (h_{0{\\bf k}},h_{1{\\bf k}}, h_{2{\\bf k}}, h_{3{\\bf k}})$ is a vector composed of real scalar elements and $ \\sigma = (_2, \\sigma _x, \\sigma _y, \\sigma _z)$ is a vector composed of the identity and the Pauli matrices in two dimensions.", "The eigen energies of $H_0$ are given by $\\varepsilon _{\\bf k}^{\\lambda } = h_{0{\\bf k}} + \\lambda g_{\\bf k} $ where we have defined $g_{\\bf k} \\equiv \\sqrt{h_{1{\\bf k}}^2+h_{2{\\bf k}}^2+h_{3{\\bf k}}^2} $ , and $\\lambda =1$ (or $-1$ ) denotes the conduction (valance) band.", "The corresponding eigenvectors can be conveniently expressed as $\\psi ^{\\lambda } = \\lbrace \\cos \\theta _{\\lambda \\bf k}, \\sin \\theta _{\\lambda \\bf k} e^{i \\phi _{\\bf k}}\\rbrace $ , where $\\tan \\phi _{\\bf k} = h_{2 \\bf k}/h_{1 \\bf k}$ , and $\\tan \\theta _{\\lambda \\bf k} =(\\lambda g_{\\bf k}-h_{3 \\bf k}) / \\sqrt{h_{1\\rm k}^2 + h_{2 \\rm k}^2}$ .", "Note that for the special case of materials, such as graphene, for which $h_{0 \\bf k} = h_{3 \\bf k} = 0$ , we have $\\cos \\theta _{\\lambda \\bf k } = 1/ \\sqrt{2}$ , and $\\sin \\theta _{\\lambda \\bf k } = \\lambda /\\sqrt{2}$ .", "The optical matrix elements can accordingly be obtained in a very general form, as follows ${\\bf M}^{vv} &=& \\frac{e}{\\hbar g_{\\bf k}} \\Big ( g_{\\bf k}\\nabla _{\\bf k} h_{0 {\\bf k}} - \\sum _{i=1,2,3} h_{i\\bf k}\\nabla _{\\bf k} h_{i{\\bf k}}~\\Big ), \\\\{\\bf M}^{cc} &=& \\frac{e}{\\hbar g_{\\bf k}} \\Big ( g_{\\bf k}\\nabla _{\\bf k} h_{0 {\\bf k}} + \\sum _{i=1,2,3} h_{i\\bf k}\\nabla _{\\bf k} h_{i{\\bf k}}~\\Big ), \\\\{\\bf M}^{vc} &=& -\\frac{e}{\\hbar g_{\\bf k} h_{\\bf k}}\\Big (- h_{\\bf k}^2\\nabla _{\\bf k} h_{3 {\\bf k}} + (h_{1 {\\bf k}} h_{3 {\\bf k}} - i h_{2 {\\bf k}} g_{{\\bf k}}) \\nabla _{\\bf k} h_{1 {\\bf k}} \\nonumber \\\\& & + (h_{2 {\\bf k}} h_{3 {\\bf k}} + i h_{1 {\\bf k}} g_{{\\bf k}}) \\nabla _{\\bf k} h_{2 {\\bf k}} ~\\Big ), \\\\{\\bf M}^{cv} &=& \\left({\\bf M}^{vc}\\right)^*$ where we have defined $h_{\\bf k}^2 \\equiv h_{1{\\bf k}}^2 + h_{2{\\bf k}}^2$ .", "To describe the dynamics of the system, we consider the time evolution of the momentum resolved density matrix ($2 \\times 2$ ), whose diagonal elements are $\\rho _{11} = \\rho ^v_{\\bf k}$ , and $\\rho _{22} = \\rho ^c_{\\bf k}$ .", "Here $\\rho _{\\bf k}^\\lambda \\equiv \\langle {a_{\\bf k}^{\\lambda }}^\\dagger a_{\\bf k}^{\\lambda } \\rangle $ denotes the momentum resolved electron density in the valance and conduction bands.", "The off-diagonal elements of the density matrix are given by $\\rho _{12} = p_{\\bf k} \\equiv \\langle {a_{\\bf k}^{c}}^\\dagger a_{\\bf k}^v \\rangle $ , and $\\rho _{21} = p_{\\bf k}^*$ , with $p_{\\bf k}$ denoting the inter-band coherence or polarization.", "Using the equation of motion, $i \\hbar \\partial _t {\\hat{\\rho }}(t) = [H, \\hat{\\rho }]$ , we obtain the following equation for the population inversion, $n_{\\bf k} \\equiv \\rho ^{c}_{\\bf k} - \\rho ^{v}_{\\bf k}$ , as $\\partial _t{n_{\\bf k}} = 4 \\Im \\left[\\lbrace {\\Omega _{\\bf k}^{vc}}(t)\\rbrace ^* p_{\\bf k}(t)\\right]~.$ The corresponding inter-band coherence evolves as $\\partial _t{p_{\\bf k}} = i\\left[ \\omega _{\\bf k}+ \\Omega _{\\bf k}^{cc}(t) -\\Omega _{\\bf k}^{vv} (t)\\right] p_{\\bf k}(t)-i\\Omega _{\\bf k}^{vc}n_{\\bf k}(t)~,$ where $ \\hbar \\omega _{\\bf k} = \\varepsilon _{\\bf k}^c - \\varepsilon _{\\bf k}^v$ .", "To obtain the steady state (long time average response), we do a rotating wave approximation, where in we get rid of the fast oscillating terms (with frequencies $2 \\omega $ and higher), and retain the slow time dependence in Eqs.", "(REF ) and (REF ).", "To this end, one can substitute ${\\bf A}(t) = {\\bf e_0} \\omega ^{-1} E_0 \\cos {\\omega t} $ , with $\\bf {e_0}$ denoting the polarization direction.", "Keeping only the low frequency resonant terms (of the form of $e^{\\pm i(\\omega - \\omega _{\\bf k})}$ ), while neglecting all the high frequency ones ($ e^{\\pm i(\\omega +\\omega _{\\bf k})}$ ) leads to: $\\partial _t \\tilde{n} & =& 2 \\Im \\left[ \\tilde{\\Omega }^{vc*}_{\\bf k} \\tilde{p}_{\\bf k} \\right]~, \\\\\\partial _{t}\\tilde{p}_{\\bf k} & = & i (\\omega _{\\bf k}-\\omega ) \\tilde{p}_{\\bf k} - i \\tilde{\\Omega }^{vc}_{\\bf k}\\tilde{n}_{\\bf k}/2~,$ where $\\hbar \\tilde{\\Omega } = {\\bf M} \\cdot {\\bf e_0} ~E_0/\\omega $ , $\\tilde{p}_{\\bf k}(t) = {p}_{\\bf k}(t)e^{-i \\omega t}$ , and $ \\tilde{n}_{\\bf k} =n_{\\bf k}(t) $ .", "Here $\\tilde{p}, ~{\\rm and}~ \\tilde{n}_{\\bf k}$ , are almost frozen (vary very slowly) over the timescales of the order of $1/\\omega $ .", "Note that Eqs.", "(REF )-() do not include energy relaxation and decoherence mechanisms, arising due to electron-electron interactions, electron-phonon, electron-impurity scattering and other interactions.", "To include these effects, phenomenological damping terms are added in the above equations [meierbook] leading to $\\partial _t \\tilde{n}_{\\bf k} & =& 2 \\Im \\left[ \\tilde{\\Omega }^{vc*}_{\\bf k} \\tilde{p}_{\\bf k} \\right] - \\gamma _{1}(\\tilde{n}_{\\bf k}-n^{\\rm eq}_{\\bf k})~, \\\\\\partial _{t}\\tilde{p}_{\\bf k} & = & i (\\omega _{\\bf k}-\\omega ) \\tilde{p}_{\\bf k} - i \\tilde{\\Omega }^{vc}_{\\bf k}\\tilde{n}_{\\bf k}/2-\\gamma _{2}\\tilde{p}_{\\bf k}~.$ Here $\\gamma _{1}$ is the inverse of the relaxation time for the momentum resolved occupation number and $\\gamma _{2}$ is the inverse relaxation time of coherence.", "The equilibrium population inversion in absence of the optical field is $n^{\\rm eq}_{\\bf k} = f_{c\\bf k} - f_{v\\bf k}$ .", "The function $f_{a{\\bf k}} = [1+ \\exp {((\\epsilon _{\\bf k}^a-\\mu )/k_B T)}]^{-1}$ denotes the Fermi function ($k_B$ and $T$ are the Boltzmann constant and temperature, respectively).", "The relaxation rates $\\gamma _1$ and $\\gamma _2$ are usually dominated by electron-phonon and electron-electron interactions, respectively.", "Generally the damping rates are frequency dependent [Mishchenko], and can be modeled microscopically by self-consistently solving electron-electron (generally at a mean field level), electron-phonon and electron-photon coupling equations [Andreas1].", "For example in graphene, typical values of $\\gamma _{1} \\approx 10^{12}$ Hz, and $\\gamma _{2} \\approx 10^{14}$ Hz for optical frequencies [Zhang1].", "However, for simplicity of analysis, we choose these rates to be constants over the frequency range considered in this work.", "In the steady state regime, we can solve Eqs.", "(REF ) and () to obtain the following steady state values for the population inversion $\\frac{{\\tilde{n}}_{\\bf k}}{n^{\\rm eq}_{\\bf k}} = \\left(1 + \\frac{\\gamma _{2}| \\tilde{\\Omega }^{vc}|^2}{ \\gamma _{1} [(\\omega _{\\bf k}-\\omega )^2 + \\gamma _{2}^2] } \\right)^{-1}~,$ and the inter-band coherence $\\tilde{p}_{\\bf k} = \\frac{\\tilde{n}_{\\bf k}}{2} \\frac{\\tilde{\\Omega }^{vc}}{(\\omega _{\\bf k}-\\omega ) + i \\gamma _{2}}~.$ Before proceeding further it is instructive to express the steady state population inversion of Eq.", "(REF ), in the following form, $\\frac{{\\tilde{n}}_{\\bf k}}{n^{\\rm eq}_{\\bf k}} \\equiv G= \\left(1 + \\frac{|{\\bf M}^{vc} \\cdot {\\bf e_0}|^2}{e^2 v_{\\rm F}^2} \\zeta ^2 \\frac{ \\gamma _{2}^2}{ [(\\omega _{\\bf k}-\\omega )^2 + \\gamma _{2}^2] } \\right)^{-1}~,$ where $\\zeta $ is specified by Eq.", "(REF ).", "In Eq.", "(REF ), ${\\bf M}^{vc} \\cdot {\\bf e_0}/(e v_{\\rm F})$ is the dimensionless material dependent optical matrix element component which couples to the incident radiation.", "The term $\\gamma _2/{[(\\omega _{\\bf k}-\\omega )^2 + \\gamma _{2}^2]}$ in Eq.", "(REF ) is a Lorentzian centered around $\\omega = \\omega _{\\bf k}$ with half-width $\\gamma _{2}$ .", "It reduces to a Dirac-delta function $\\delta (\\omega -\\omega _{\\bf k})$ in the limiting case of $\\gamma _{2}/\\omega \\rightarrow 0$ .", "Furthermore, we note that $G$ can be expanded in a power series of $\\zeta ^2$ .", "Thus $\\zeta $ is a dimensionless effective field strength, which, for a fixed value of the optical frequency and the damping constants, distinguishes between the linear ($\\zeta \\ll 1$ ) and the nonlinear ($\\zeta \\ge 1$ ) response regime [Mishchenko].", "Additionally, for a fixed value of the optical frequency and effective field strength, the ratio $\\gamma _{2}/\\omega $ determines the clean/high frequency ($\\gamma _{2} \\ll \\omega $ ) or the dirty/low frequency limits ($\\gamma _{2} \\ge \\omega $ ).", "This classification of the system's response into linear and non-linear regimes, or alternatively into the clean and the dirty limit arises naturally in our formulation and will be used in the rest of the manuscript.", "Figure: The momentum resolved current J ˜ 𝐤x \\tilde{J}_{{\\bf k}x} density of Eq.", "(), in the k x -k y k_x-k_y plane for graphene in panels a), b) and c) and for massive graphene in panels d), e), and f).", "In panels b) and c) [panels e)-f) for massive case] we have a larger coherence decay rate γ 2 =ω/5\\gamma _2 = \\omega /5 as compared to a) for which γ 2 =ω/20\\gamma _2 = \\omega /20 and this show the impact of broadening of the current density around the circle k=ω/(2v F )k = \\omega /(2 v_{\\rm F}) at μ=0\\mu = 0.", "Further in panel c) for graphene [panel f) for massive graphene], we display the impact of a finite chemical potential μ=0.5ℏω\\mu = 0.5 \\hbar \\omega which manifests itself in the Pauli blocking of the momentum space for k<μ/ℏv F k < \\mu /\\hbar v_{\\rm F}.", "For more details see Eq.", "() for massless graphene, and Eq.", "() for the massive graphene case.Having obtained the steady state density matrix elements, we now proceed to calculate the induced current density and the nonlinear optical conductivity." ], [ "Nonlinear inter-band optical conductivity", "To obtain the inter-band contribution to the optical current and optical conductivity, let us consider the generic form of the current density for a $d$ -dimensional system: ${\\bf J}(t) = -g_s g_v (2 \\pi )^{-d}\\int d{\\bf k} ~{\\bf J}_{\\bf k}(t)$ , where $g_s$ ($g_v$ ) denotes the spin (valley) degeneracy factor, ${\\bf J}_{\\bf k}(t) =e {\\rm Tr}[\\rho _{\\bf k}(t) {\\bf v}_{\\bf k}(t) ]$ and the generalized velocity operator is given by $\\hat{\\bf v}_{\\bf k} = -i \\hbar ^{-1} [{\\bf r}, H_0] = \\hbar ^{-1} \\nabla _{\\bf k} H_0$ .", "The momentum dependent component of the particle current density is then ${\\bf J}_{\\bf k}(t) = 2 \\Re e [p_{\\bf k} {\\bf M}^{cv}_{\\bf k}] + \\sum _{\\lambda = c,v}\\rho _{\\bf k}^\\lambda {\\bf M}^{\\lambda \\lambda }~,$ In presence of particle-hole symmetry, as in graphene or gapped graphene, we have ${\\bf M}_{\\bf k}^{cc} = -{\\bf M}_{\\bf k}^{vv}$ .", "One can then rewrite Eq.", "(REF ) as ${\\bf J}_{\\bf k}(t) = 2 \\Re e [p_{\\bf k}(t) {\\bf M}^{cv}_{\\bf k}] + n_{\\bf k}(t) {\\bf M}^{cc}_{\\bf k}~.$ It is evident that the first term in Eq.", "(REF ) arises from the inter-band contribution, while the second term originates from the intra-band contributions.", "In Eq.", "(REF ), $p_{\\bf k} (t) = \\tilde{p}_{\\bf k} e^{i \\omega t}$ , and consequently the momentum resolved current density consists of three terms: $2\\Re e\\left[\\tilde{p}_{\\bf k}{\\bf M}_{\\bf k}^{cv}\\right]\\cos {\\omega t}$ , $-2\\Im m \\left[\\tilde{p}_{\\bf k}{\\bf M}_{\\bf k}^{cv}\\right]\\sin {\\omega t}$ and $\\tilde{n}_{\\bf k}{\\bf M}_{\\bf k}^{cc}$ .", "Of these, the term proportional to $\\cos {\\omega t}$ is the out of phase (with respect to the incident field) response of the system and it does not contribute to the dissipative part of the conductivity [Mishchenko].", "Hence it will be neglected in the rest of the article.", "Furthermore, for graphene and gapped graphene, it is easy to check that the intra-band term $\\tilde{n}_{\\bf k}{\\bf M}_{\\bf k}^{cc}$ , vanishes on performing the momentum sum.", "In fact, since we have considered only momentum conserving vertical transitions in calculating the electronic density matrix, it then follows that the intra-band part of Eq.", "(REF ) should vanish for all materials after the ${\\bf k}-$ integration.", "One therefore needs to focus only on the dissipative part of the current, which is captured by the in-phase (to the electric field) part of the response corresponding to the $\\sin (\\omega t)$ term.", "This part of the momentum resolved current, can be expressed as ${\\bf J}_{\\bf k} (t) = {\\bf {\\tilde{J}}}_{\\bf k} \\sin (\\omega t) $ , with ${\\bf {\\tilde{J}}}_{\\bf k} = - \\frac{E_0}{\\hbar \\omega }{\\tilde{n}}_{\\bf k} \\Im m \\left\\lbrace \\frac{({\\bf M}^{vc}_{\\bf k}\\cdot {\\bf e_0}){\\bf M}_{\\bf k}^{cv}}{\\omega _{\\bf k}-\\omega +i \\gamma _{2 }} \\right\\rbrace ~.$ The corresponding dynamical nonlinear optical conductivity can then be obtained by integrating the above expression and using $\\sigma _{ij} = |\\tilde{J}_i|/[E_0 ({\\bf e_0}\\cdot \\hat{j})]$ , where $E_0 ({\\bf e_0}\\cdot \\hat{j})$ is amplitude of the field along the $\\hat{j}$ direction.", "If one considers a linearly polarized light, say along the $x$ -direction, and restrict oneself to the longitudinal response only, then Eq.", "(REF ) reduces to $\\frac{{{\\tilde{J}}}_{{\\bf k}x}}{E_0} = - \\frac{{\\tilde{n}}_{\\bf k} |M^{vc}_x|^2}{\\hbar \\omega } \\Im m \\left\\lbrace \\frac{1}{\\omega _{\\bf k}-\\omega +i \\gamma _{2}} \\right\\rbrace ~.$ In Eq.", "(REF ), the total current, is in general `cut-off' or band-width dependent.", "For systems described by a continuum model, the ultraviolet momentum cutoff is inversely proportional to the lattice spacing.", "On the contrary, for systems described by a tight-binding model, the energy cutoff is typically the band-width of the system.", "Furthermore, there is static component of the current in the $\\omega \\rightarrow 0$ limit.", "This is unphysical, since in this limit, the inter-band current must vanish for a time-independent vector potential.", "Accordingly, this contribution needs to be subtracted from Eq.", "(REF ) as prescribed in Ref. [Falkovsky2007].", "The momentum resolved current density of Eq.", "(REF ) is shown in Fig.", "REF for graphene and massive graphene.", "Using Eq.", "(REF ), the longitudinal optical conductivity in the most general nonlinear-dirty case can be expressed as $\\sigma _{xx}(\\omega ) = \\frac{g_s g_v}{\\hbar \\omega (2 \\pi )^d} \\int d{\\bf k} |M^{vc}_x|^2 {\\tilde{n}}_{\\bf k} \\Im m \\left\\lbrace \\frac{1}{\\omega _{\\bf k}-\\omega +i \\gamma _{2}} \\right\\rbrace .$ We emphasize here that the non-linearity of the optical conductivity in Eq.", "(REF ), stems from the $n_{\\bf k}$ term, whose solution is obtained from the optical Bloch equation within RWA.", "Let us now consider the following limiting cases in which Eq.", "(REF ) simplifies.", "The limiting case for $\\gamma _2/\\omega \\ll 1$ ($\\gamma _2/\\omega \\ge 1$ ) corresponds to the clean (dirty) limit, and the limiting case for $\\zeta \\ll 1$ ($\\zeta \\ge 1$ ) is related to the linear (non-linear) response of the system." ], [ "Linear response in the clean limit: $\\zeta \\ll 1$ , and {{formula:d11db9f8-074a-4cc7-aff5-2cf76b52e97e}}", "In this limit, to zeroth order in $\\zeta $ , we have $n_{\\bf k} \\rightarrow {n_{\\bf k}^{\\rm eq}}$ , and converting the Lorentzian into a Dirac-delta function, we have $\\sigma _{ xx}^{\\rm lc}(\\omega ) = \\frac{-\\pi g_s g_v}{\\hbar \\omega (2 \\pi )^d} \\int d{\\bf k} |M^{vc}_x|^2 \\delta (\\omega _{\\bf k}-\\omega ) (f_{c\\bf k}-f_{v \\bf k})~.$ Note that the conductivity obtained above [Eq.", "(REF )], is identical to that obtained from the Kubo formalism.", "For systems with particle-hole symmetry and isotropic quasi-particle dispersion, such as graphene and massive graphene, Eq.", "(REF ) can be expressed as $\\sigma _{xx}^{\\rm lc}(\\omega ) = \\frac{\\pi g_s g_v g(\\omega , \\alpha , T)}{\\hbar \\omega (2 \\pi )^d} \\int d{\\bf k} |M^{vc}_x|^2 \\delta (\\omega _{\\bf k}-\\omega ) ~,$ where $\\alpha \\equiv {\\rm max}\\lbrace \\mu , \\Delta \\rbrace $ , with $\\Delta $ being half of the band-gap in a given semiconductor.", "The function $g(\\omega , \\alpha , T) \\equiv \\frac{1}{2}\\left[\\tanh \\left(\\frac{\\hbar \\omega + 2 \\alpha }{4 k_B T}\\right) + \\tanh \\left(\\frac{\\hbar \\omega - 2 \\alpha }{4 k_B T}\\right)\\right]~.$ In the zero temperature limit, $g(\\omega , \\alpha , T\\rightarrow 0) = \\Theta ( \\hbar \\omega /2 - |\\alpha |)$ , where $\\Theta (x)$ denotes the Heaviside step function." ], [ "Linear response in the dirty limit: $\\zeta \\ll 1$ , and {{formula:4ad0b9d3-58ca-46e0-8011-25f199393d9b}}", "As in the previous case, here again we can approximate $n_{\\bf k} \\rightarrow {n_{\\bf k}^{\\rm eq}}$ upto zeroth order in $\\zeta $ .", "However in this case the Lorentzian has to be retained in the integral.", "The corresponding conductivity is given as $\\sigma _{xx}^{\\rm ld}(\\omega ) = \\frac{- g_s g_v}{\\hbar \\omega (2 \\pi )^d} \\int d{\\bf k} |M^{vc}_x|^2 \\frac{\\gamma _{2} (f_{c\\bf k}-f_{v \\bf k})}{(\\omega _{\\bf k}-\\omega )^2+ \\gamma _{2}^2}~.$" ], [ "Non-linear response in the clean limit: $\\zeta \\ge 1$ , and {{formula:34d2a6a9-d47f-45d8-99e2-5efbda7dbad0}} ", "Here the Lorentzian can be approximated by a delta function, and from Eq.", "(REF ) one obtains $\\sigma _{xx}^{\\rm nc}(\\omega ) = \\frac{-\\pi g_s g_v}{\\hbar \\omega (2 \\pi )^d} \\int d{\\bf k} |M^{vc}_x|^2 \\frac{\\delta (\\omega _{\\bf k}-\\omega ) (f_{c \\bf k}-f_{v \\bf k})}{1+ |{M}^{vc}_{x}|^2 \\zeta ^2/(e^2 v_{\\rm F}^2)}~.$ For systems with particle-hole symmetry and isotropic quasi-particle dispersion, such as graphene and massive graphene, Eq.", "(REF ) can be approximated as $\\sigma _{xx}^{\\rm nc}= \\frac{\\pi g_s g_v g(\\omega , \\alpha , T)}{\\hbar \\omega (2 \\pi )^d} \\int d{\\bf k} |M^{vc}_x|^2 \\frac{\\delta (\\omega _{\\bf k}-\\omega )}{1+ \\zeta ^2 |{M}^{vc}_{x}|^2 /(e^2 v_{\\rm F}^2)}~,$ where $g(\\omega , \\alpha , T)$ is defined in Eq.", "(REF ).", "Below, we will explore the optical conductivity of doped and gapped graphene in all the four regimes.", "Note that the transverse (Hall like) conductivity, $\\sigma _{yx}(\\omega )$ will also have expressions similar to that of Eqs.", "(REF )-(REF ), with the substitution $|M^{vc}_x|^2 \\rightarrow M^{vc}_x M^{cv}_y $ in the numerator.", "Here we would like to emphasize that the formalism described above gives only the finite frequency part of the conductivity which cannot be extrapolated to the DC limit (limitation imposed due to RWA).", "The full paramagnetic conductivity, which includes the zero frequency Drude weight ($\\cal D$ ) is given by $ \\sigma _{\\rm total}(\\omega ) = \\pi {\\cal D} \\delta (\\omega ) + \\sigma (\\omega )$ .", "The Drude weight is explicitly given by ${\\cal D} = \\lim _{\\omega \\rightarrow 0} \\omega \\Im m[\\sigma (\\omega )]$ (see Ref.", "[StauberNJP] for details).", "The imaginary part of the optical conductivity can therefore be evaluated by making use of the Kramers-Kronig relations, as $\\Im m \\sigma (\\omega ) = \\frac{2}{\\pi \\omega } {\\cal P} \\int _0^\\infty d \\omega ^{\\prime } \\frac{\\omega ^{\\prime 2} \\Re e \\sigma (\\omega ^{\\prime })}{\\omega ^2-\\omega ^{\\prime 2}}~,$ where ${\\cal P}$ denotes the principal part of the integral.", "Experimentally the optical conductivity is probed via the differential optical transmission or reflection spectroscopy.", "Motivated by this, in the next Section, we explore the effect of non-linearity on the reflection and transmission spectrum." ], [ "Non-linear differential transmissivity and reflectivity for 2D materials", "The calculated non-linear conductivity has significant implications for optical experiments.", "In this section we determine the differential transmission (or equivalently the reflection coefficient) of a `free standing' two dimensional material, suspended in vacuum.", "Let us consider a linearly polarized incident field propagating normal to the plane of the 2D material.", "For definiteness, let us assume that the two-dimensional sample is freely suspended in the $x-y$ plane at $z = 0$ .", "An incoming field directed along the negative $\\hat{z}$ -direction can be decomposed into its components as, ${\\bf E} =\\left\\lbrace \\begin{array}{ll}{\\bf E}_I\\sin (\\omega t + qz) + {\\bf E}_R\\sin (\\omega t - qz) & , z > 0 \\\\{\\bf E}_T\\sin (\\omega t + qz) & , z < 0,\\end{array}\\right.$ where $q = \\omega /c$ is the photon momentum and $ {\\bf E}_i = (E_{0i}\\cos \\theta _0, E_{0i}\\sin \\theta _0)$ , with $ i = I,R~\\text{and}~T$ , denoting the incident, reflected and the transmitted components.", "$\\theta _0$ is the polarization angle with respect to the $\\hat{x}$ -axis.", "Interaction of the incident light with the carriers in the 2D sample, is modeled in the Maxwell's equations, via the total current density in the 2D ($\\hat{x}-\\hat{y}$ ) plane, having a vanishing thickness.", "This implies ${\\bf J}(t) \\equiv {\\bf J}(t)\\delta (z)$ .", "The spatiotemporal evolution of the $\\hat{x}$ component of the electromagnetic field is given by ${\\bf \\nabla }^2E_x -\\mu _0\\varepsilon _0\\frac{\\partial ^2E_x}{{\\partial t}^2} =\\mu _0\\partial _t J_x\\delta (z)~.$ Integrating the above equation across the two dimensional plane yields $\\partial _zE_x|_{z=0^{+}} - \\partial _zE_x|_{z=0^{-}} = \\mu _0\\partial _tJ_x~.$ Substituting Eq.", "(REF ) in Eq.", "(REF ), we obtain $(E_{I}-E_{R}-E_{T})\\cos \\theta _0 = \\frac{ c \\mu _0}{ \\omega \\cos (\\omega t)}\\partial _tJ_x~.$ Additionally, the continuity of the electromagnetic field across the 2D layer yields, $E_{I}+E_{R} = E_{T}.$ Furthermore, if we assume the material to be isotropic, with a conductivity tensor which has only diagonal elements, we have $J_x = \\sigma _{xx} \\sin (\\omega t) [\\omega E_T \\cos (\\theta _0)]$ .", "With this the non-linear transmissivity can be easily obtained to be, $T(\\omega ) \\equiv \\left| \\frac{E_{T}}{E_{I}} \\right|^2 = \\left[1 + \\frac{\\pi \\alpha _{\\rm fine}}{2} \\frac{\\sigma _{xx}(\\omega )}{\\sigma _0} \\right]^{-2} ~,$ where $\\alpha _{\\rm fine} \\equiv e^2/(4 \\pi \\hbar c \\epsilon _0)$ is the universal fine structure constant and $\\sigma _0 \\equiv e^2/(4 \\hbar )$ , is the so called universal ac conductivity of graphene.", "Here $\\sigma _{xx}(\\omega )$ is the non-linear longitudinal conductivity.", "The reflectivity $R(\\omega ) = |E_R/E_I|^2$ , can also be obtained in a similar fashion and it is given as $R(\\omega ) = \\left(\\frac{\\pi \\alpha _{\\rm fine}}{2} \\frac{\\sigma _{xx}(\\omega )}{\\sigma _0}\\right)^2\\left[1+ \\frac{\\pi \\alpha _{\\rm fine}}{2} \\frac{\\sigma _{xx}(\\omega )}{\\sigma _0} \\right]^{-2}~.$ The absorption coefficient is given by $\\alpha (\\omega ) \\equiv 1- T(\\omega ) - R(\\omega )$ and it denotes the fraction of light intensity which is either scattered by the surface atoms of the 2D material (Rayleigh scattering) or absorbed." ], [ "Nonlinear conductivity of doped graphene", "In this section we apply the general framework developed in Sec.", "and Sec.", "to calculate the non-linear optical conductivity of graphene.", "For simplicity, we use the effective low energy quasiparticle dispersion of graphene instead of the full tight-binding Hamiltonian.", "The corrections to the low energy dispersion on the universal conductivity has been shown to be $ \\sim \\hbar \\omega /(72 \\times 2.8 {\\rm eV})$ [StauberTB].", "Thus such effects can be safely neglected for all frequencies in the optical domain and below.", "For graphene, we have $\\varepsilon _{\\bf k}^c = -\\varepsilon _{\\bf k}^v = \\hbar v_{\\rm F} k$ , and ${\\bf M}^{vc} = {\\bf M}^{cv*} = iev_{\\rm F} \\tau \\lbrace \\sin {\\phi _{\\bf k}}, -\\cos {\\phi _{\\bf k}}\\rbrace $ where $k = ({k_x^2 + k_y^2})^{1/2}$ , $\\phi _{\\bf k} = \\tan ^{-1}(k_y/k_x)$ and $\\tau = +1$ ($-1$ ) for the $K$ ($K^{\\prime }$ ) valley.", "We now consider the non-linear optical conductivity of graphene arising from the inter-band transitions in various regimes.", "We start with the linear response in the clean limit of graphene, i.e., for $\\zeta \\ll 1$ and $\\gamma _2/\\omega \\ll 1$ limit.", "In this regime, using Eq.", "(REF ), one can obtain the finite temperature optical conductivity for graphene as $\\sigma _{xx}^{\\rm lc}(\\omega ) = \\frac{g_sg_ve^2}{16\\hbar } g(\\omega , \\mu , T)~.$ This has also been derived earlier from the Kubo formalism (see Eq.", "(25) of Ref. [StauberTB]).", "In Eq.", "(REF ), $g_s =2$ ($g_v =2$ ) denotes the spin (valley) degree of freedom in graphene and the function $g(\\omega , \\mu , T)$ is defined in Eq.", "(REF ).", "In the limiting case of $T \\rightarrow 0$ , including the spin and valley degeneracy factors, Eq.", "(REF ) reduces to, $\\sigma _{xx}^{\\rm lc}(\\omega ) = \\frac{e^2}{4\\hbar }\\Theta \\left(\\frac{\\hbar \\omega }{2} - |\\mu |\\right),$ which has a finite universal value as long as the optical excitation energy is greater than twice the chemical potential.", "Equation (REF ) yields the so called `universal' ac conductivity of graphene which was predicted in Refs.", "[Ando2002, Gusynin2006], and experimentally observed in Ref. [Nair2008].", "In the non-linear and clean limit, we have $\\zeta \\ge 1$ and $\\gamma _2/\\omega \\ll 1$ .", "Using Eq.", "(REF ), the optical conductivity for graphene takes the form $\\sigma _{xx}^{\\rm nc}(\\omega ) = \\frac{e^2}{4\\hbar }\\frac{2}{\\zeta ^2}\\left( 1 - \\frac{1}{(1 + \\zeta ^2)^{1/2}}\\right) g(\\omega , \\mu , T)~.$ As a consistency check we note that Eq.", "(REF ) reduces to Eq.", "(REF ) in the limiting case of $\\zeta \\rightarrow 0$ .", "Note that for the specific case of pristine graphene ($\\mu =0$ at zero temperature), Eq.", "(REF ) can also be derived by following the wave-function based approach and integrating Eq.", "(12) of Ref. [Mishchenko].", "Next we consider the linear response in the dirty limit for which $\\zeta \\ll 1$ and $\\gamma _2/\\omega \\ge 1$ .", "In this case the form of conductivity changes substantially and at zero temperature it is $\\sigma _{xx}^{\\rm ld}(\\omega ) = \\frac{e^2\\gamma _2}{4\\pi \\hbar \\omega }\\int _{\\frac{2|\\mu |}{\\hbar }}^{\\frac{2\\Lambda }{\\hbar }}d\\omega _{\\bf k}\\left[\\frac{\\omega _{\\bf k}}{(\\omega _{\\bf k}-\\omega )^2 + \\gamma _2^2}- (\\omega \\rightarrow 0) \\right]~,$ where $\\Lambda $ is the ultraviolet energy cutoff, which physically should correspond to half of the bandwidth.", "In a tight-binding model of graphene half of the bandwidth is $3\\times 2.8$ eV.", "Evaluating Eq.", "(REF ) we obtain $\\sigma _{xx}^{\\rm ld}(\\omega ) = \\frac{e^2}{4 \\pi \\hbar } \\left[f\\left(\\omega , 2 \\Lambda /\\hbar \\right) - f\\left(\\omega , 2|\\mu |/\\hbar \\right) \\right]~,$ where $f(\\omega , x) \\equiv \\tan ^{-1}\\left(\\frac{x-\\omega }{\\gamma _2}\\right)+\\frac{\\gamma _2}{2\\omega }\\ln \\left[\\frac{\\gamma _2^2 + (\\omega - x)^2 }{\\gamma _2^2 + x^2}\\right]~.$ The finite temperature generalization of Eq.", "(REF ) has to be calculated numerically.", "As a consistency check we note that Eq.", "(REF ) reduces to Eq.", "(REF ) in the limit $\\gamma _2 \\ll \\omega $ .", "For the most general case of the nonlinear-dirty regime, starting with Eq.", "(REF ), we express the dissipative part of the steady state current for each spin and valley of graphene ($\\hat{x}$ and $\\hat{y}$ components) as ${\\bf \\tilde{J}}_{\\bf k} = { \\tilde{J}}_{\\bf k}\\left(\\sin {\\phi _{\\bf k}},-\\cos {\\phi _{\\bf k}}\\right)$  .", "Here, ${ \\tilde{J}}_{\\bf k} = \\frac{e^2v_{\\rm F}^2E_0\\sin (\\phi _{\\bf k}-\\theta _0)\\gamma _{1}\\gamma _{2}n_{\\bf k}^{\\rm eq}}{\\hbar \\omega \\left\\lbrace \\gamma _{1}\\left[(\\omega _{\\bf k}-\\omega )^2 + \\gamma _{2}^2\\right]+ \\gamma _{2}|\\tilde{\\Omega }^{vc}|^2\\right\\rbrace }~,$ and $\\theta _0$ is the polarization angle of the linearly polarized field, with respect to the $\\hat{x}$ axis.", "We then have the total longitudinal conductivity for graphene as $\\sigma _{xx}(\\omega ) =(2 \\pi )^{-2} g_s g_v \\int d{\\bf k} \\ \\sigma ^{\\prime }_{xx}({\\bf k}) $ , where $\\omega \\sigma ^{\\prime }_{xx}({\\bf k}) = \\frac{e^2}{\\hbar } \\frac{v_{\\rm F}^2 \\gamma _{2} (f_{c \\bf k} -f_{v \\bf k}) \\sin ^2\\phi _{\\bf k} }{\\left[(\\omega _{\\bf k}-\\omega )^2 + \\gamma _{2}^2 (1+ \\zeta ^2 \\sin ^2\\phi _{\\bf k})\\right] } - (\\omega \\rightarrow 0).$ At $T \\rightarrow 0$ , the $k$ -integral of Eq.", "(REF ) has lower and upper limits of $2 |\\mu |/\\hbar v_{\\rm F}$ and $2 \\Lambda / \\hbar v_{\\rm F}$ respectively.", "It can therefore be evaluated exactly (as in Eq.", "(REF )) to obtain, $\\sigma _{xx}(\\omega ) = \\frac{e^2}{4 \\pi ^2 \\hbar } \\int _{0}^{2 \\pi } d \\phi _{\\bf k}\\sin ^2\\phi _{\\bf k} \\left[f_1(2 \\Lambda /\\hbar ) - f_1(2 |\\mu |/\\hbar )\\right]~,$ where, $f_1(\\omega , x) \\equiv \\frac{\\gamma _2}{\\gamma _{\\phi _{\\bf k}}}\\tan ^{-1}\\left(\\frac{x-\\omega }{\\gamma _{\\phi _{\\bf k}}}\\right)+\\frac{\\gamma _2}{2\\omega }\\ln \\left[\\frac{\\gamma _{\\phi _{\\bf k}}^2 + (\\omega - x)^2 }{\\gamma _{\\phi _{\\bf k}}^2 + x^2}\\right]~,$ and $\\gamma _{\\phi _{\\bf k}} = \\gamma _2 (1+ \\zeta ^2 \\sin ^2\\phi _{\\bf k} )^{1/2}$ .", "The $\\phi _{\\bf k}$ integral in Eq.", "(REF ), can also be done analytically, yielding a cumbersome expression without much insight.", "However, it is easy to check that in the linear response regime, $\\zeta \\rightarrow 0$ , $\\gamma _{\\phi _{\\bf k}} \\rightarrow \\gamma _2$ , and Eq.", "(REF ), reduces to Eq.", "(REF ), as expected.", "The optical conductivity of graphene, in all four regimes is shown in Fig.", "REF .", "The $\\zeta = 1$ line clearly marks the boundary of the non-linear response regime, with saturation effects dominating on the $\\zeta >1$ side.", "Interestingly enough, the `universal' optical conductivity of graphene $\\sigma _0 =e^2/(4 \\hbar )$ , seems to be valid only in the $\\omega > \\gamma _2$ and $\\zeta \\ll 1$ regime.", "Figure REF also suggests that non-linear optical saturation effects will become dominant with decreasing optical frequencies while keeping the laser intensity constant.", "The effect of non-linear optical conductivity on the transmission spectrum is highlighted in Fig.", "REF .", "An interesting observation from Eq.", "(REF ), is that the off-diagonal conductivity $\\sigma _{yx} (\\omega )$ (for a $\\hat{x}$ polarized field) vanishes for graphene.", "This is on account of the $\\phi _{\\bf k}$ integration vanishing for each valley.", "This is not the case for massive graphene where $\\sigma _{yx} (\\omega )$ is finite for each valley with opposite signs for the two valleys.", "Therefore the total $\\sigma _{yx} (\\omega )$ cancels out.", "This implies that one can possibly have a finite $\\sigma _{yx}$ in massive graphene if the two valleys can be made to have a different bandgap." ], [ "Nonlinear optical conductivity of gapped graphene", "We now proceed to discuss the case of gapped graphene.", "If a gap, $\\Delta $ is introduced in the band structure of graphene, say by growing it epitaxially on top of SiC [gap1gr, gap1gr2], then the effective Hamiltonian would take the form of Eq.", "(REF ), with $ h_{0{\\bf k}} = 0$ , $h_{1{\\bf k}} = \\hbar v_{\\rm F} k_x$ , $h_{2{\\bf k}} = \\hbar v_{\\rm F} k_y$ , and $ h_{3{\\bf k}} = \\Delta $ .", "In this case, we have $\\varepsilon _{\\bf k}^c = -\\varepsilon _{\\bf k}^v \\equiv g_{\\bf k} = (\\hbar ^2 v_{\\rm F}^2 k^2 + \\Delta ^2)^{1/2}$ .", "The $\\hat{x}$ and $\\hat{y}$ components of the inter-band optical matrix element are given by $\\frac{{\\bf M}^{vc}}{ev_{\\rm F}} = -\\left(\\frac{\\Delta \\cos \\phi _{\\bf k}}{g_{\\bf k}}- i \\tau \\sin \\phi _{\\bf k} , \\frac{\\Delta \\sin \\phi _{\\bf k}}{g_{\\bf k}} + i \\tau \\cos \\phi _{\\bf k} \\right),$ where $\\tau = +1$ ($-1$ ) for the $K$ -valley ($K^{\\prime }$ -valley).", "Figure: Color plot of the optical conductivity as a function of frequency and the electric field strength of the incident laser beam for (a) gapped graphene (Δ=0.065 eV =ℏ×10 14 \\Delta =0.065{\\rm eV} = \\hbar \\times 10^{14} Hz) with μ=0\\mu =0 and (b) gapped and doped graphene with μ=2Δ\\mu = 2 \\Delta .", "As in the case of graphene, the vertical solid white line at ω=γ 2 \\omega = \\gamma _2, and the dashed white line for ζ≡ev F E I /(ℏωγ 1 γ 2 )=1\\zeta \\equiv e v_{\\rm F} E_I/(\\hbar \\omega \\sqrt{\\gamma _1 \\gamma _2}) = 1] divide the parameter space into 4 regimes: linear clean (marked 'lc'), non-linear clean ('nc'), non-linear dirty ('nd') and linear dirty ('ld').", "Panel (c) show horizontal cuts from the upper two panels, i.e., the conductivity as a function of ω\\omega for different electric field strengths for both, the μ=0\\mu =0 case (solid lines), and the doped case of μ=0.13\\mu = 0.13 eV (dotted lines of the same color).", "The yellow circles show the excellent match of Eq.", "(), in the linear dirty limit, with the exact numerical results.Panel (d) displays vertical cuts from the upper two panels, i.e., the conductivity as a function of E I E_{I} for different frequencies with solid lines for the μ=0\\mu =0, and the dotted lines of the same color for μ=0.13\\mu = 0.13 eV.", "Here the yellow circles show the excellent match of Eq.", "(), in the non-linear clean limit, with the exact numerical results.", "In all the panels we have chosen v F =10 6 v_{\\rm F} =10^6 m/s, and {γ 1 ,γ 2 }={10 12 ,10 14 }\\lbrace \\gamma _1,\\gamma _2\\rbrace = \\lbrace 10^{12},10^{14}\\rbrace Hz.In the linear clean limit ($\\zeta \\ll 1,~\\gamma _2/\\omega \\ll 1$ ), using Eq.", "(REF ) in Eq.", "(REF ), we obtain the optical conductivity for massive graphene (with $g_s =g_v=2$ ) to be $\\sigma _{xx}^{\\rm lc,m}(\\omega ) = \\frac{g_sg_ve^2}{16\\hbar }\\left[1+ \\frac{4\\Delta ^2}{\\hbar ^2\\omega ^2}\\right]g(\\omega ,\\max (|\\mu |,\\Delta ),T),$ where function $g(x)$ is defined in Eq.", "(REF ).", "In the limiting case of $T \\rightarrow 0$ , $g(x) \\rightarrow \\Theta (x)$ and $\\sigma _{xx}^{\\rm lc,m}(\\omega ) = \\frac{e^2}{4\\hbar }\\left[1+ \\frac{4\\Delta ^2 }{\\hbar ^2\\omega ^2}\\right]\\Theta \\left(\\frac{\\hbar \\omega }{2} - \\max (|\\mu |,\\Delta )\\right).$ The above expression clearly suggests that the chemical potential is not important if it lies inside the gap ($\\mu < \\Delta $ ).", "This is a direct consequence of unavailability of any phase space below the bandgap for optical excitations.", "Evidently in the $\\Delta \\rightarrow 0$ limit, we recover the corresponding optical conductivity expression for graphene.", "In the nonlinear clean limit ($\\zeta \\ge 1$ and $\\gamma _2/\\omega \\ll 1$ ), we use Eq.", "(REF ) to calculate the optical conductivity at finite temperature.", "For massive Dirac systems in two dimensions, we obtain, $\\sigma _{xx}^{\\rm nc,m} = \\frac{e^2 g(\\omega ,\\alpha ,T)}{2\\hbar \\zeta ^2} \\left[ 1 - \\frac{1}{\\sqrt{1+\\zeta ^2}}\\left(1+ \\frac{4\\Delta ^2\\zeta ^2}{\\hbar ^2\\omega ^2} \\right)^{-1/2}\\right].$ In the limiting case of $\\Delta \\rightarrow 0$ , Eq.", "(REF ) reduces to Eq.", "(REF ) as expected.", "Next we consider the linear dirty limit ($\\zeta \\ll 1$ and $\\gamma _2/\\omega \\ge 1$ ) and with the help of Eq.", "(REF ), we arrive at the following expression in the $T \\rightarrow 0$ limit, $\\sigma _{xx}^{\\rm ld,m} = \\frac{e^2\\gamma _2}{4\\pi \\hbar \\omega }\\int _{\\frac{2 \\alpha }{\\hbar }}^{\\frac{2 \\Lambda }{\\hbar }}d\\omega _{\\bf k}\\left[\\frac{4 \\hbar ^{-2}\\Delta ^2 + \\omega _{\\bf k}^2}{\\omega _{\\bf k}\\left[(\\omega _{\\bf k}-\\omega )^2 + \\gamma _2^2\\right]} - (\\omega \\rightarrow 0) \\right],$ where $\\Lambda $ is the ultraviolet energy cutoff and $\\alpha = {\\rm max}\\lbrace |\\mu |, \\Delta \\rbrace $ .", "Evaluating Eq.", "(REF ) we obtain $\\sigma _{xx}^{\\rm ld,m}(\\omega ) = \\frac{e^2}{4 \\pi \\hbar } \\left[f_2\\left(\\omega , 2 \\Lambda /\\hbar \\right) - f_2\\left(\\omega , 2 \\alpha /\\hbar \\right) \\right]~,$ where we have defined, $f_2(\\omega , x) &\\equiv & \\left(1+y \\right) \\tan ^{-1}\\left(\\frac{x-\\omega }{\\gamma _2}\\right) - \\frac{\\gamma _2^2 - 4\\hbar ^{-2} \\Delta ^2}{2 \\gamma _2 \\omega } \\ln [x^2 + \\gamma _2^2 ] \\nonumber \\\\& + & \\frac{\\gamma _2(1-y)}{2\\omega } \\log [(x-\\omega )^2 + \\gamma _2^2] - \\frac{\\omega y}{\\gamma _2} \\log (x) ~, \\nonumber \\\\$ and $y \\equiv 4 \\hbar ^{-2}\\Delta ^2/(\\omega ^2 + \\gamma _2^2)$ .", "The finite temperature generalization of Eq.", "(REF ) has to be calculated numerically.", "As a simple check we note that as $\\Delta \\rightarrow 0$ , Eq.", "(REF ) reduces to Eq.", "(REF ).", "In the most general case, corresponding to the nonlinear dirty limit for massive graphene, the dissipative part of the current density which arises only from the inter-band contribution, is given by ${\\bf \\tilde{J}}_{\\bf k}^{\\rm m} = \\left({A}_{{\\bf k}}, {B}_{\\bf k}\\right)n^{\\rm eq}_{\\bf k}$ , where $A_{\\bf k} &=& \\frac{e^2v_{\\rm F}^2 E_0\\gamma _{1}\\left[ \\Delta ^2\\gamma _{2}\\cos \\theta _0+ \\hbar ^2 v_{\\rm F}^2 k^2 \\gamma _{2}\\sin (\\phi _{\\bf k}-\\theta _0)\\sin \\phi _{\\bf k} - \\tau \\Delta g_{\\bf k}(\\omega _{\\bf k}-\\omega )\\sin \\theta _0\\right]}{\\hbar \\omega g_{\\bf k}^2\\left\\lbrace \\gamma _{1}\\left[(\\omega _{\\bf k}-\\omega )^2+\\gamma _{2}^2\\right]+\\gamma _{2}|\\Omega _{\\bf k}^{cv}|^2\\right\\rbrace },\\\\ B_{\\bf k} &=& \\frac{e^2v_{\\rm F}^2 E_0\\gamma _{1}\\left[\\tau \\Delta g_{\\bf k}(\\omega _{\\bf k}-\\omega )\\cos \\theta _0- \\hbar ^2 v_{\\rm F}^2 k^2\\gamma _{2}\\sin (\\phi _{\\bf k}-\\theta _0)\\cos \\phi _{\\bf k}+\\Delta ^2\\gamma _{2}\\sin \\theta _0\\right]}{\\hbar \\omega g_{\\bf k}^2\\left\\lbrace \\gamma _{1} \\left[(\\omega _{\\bf k}-\\omega )^2+\\gamma _{2\\bf k}^2\\right]+\\gamma _{2}|\\Omega _{\\bf k}^{cv}|^2\\right\\rbrace } ~.$ However for this case, the $\\bf k$ integration of Eq.", "(REF ) has to be done numerically.", "Figure: Color plot of the nonlinear transmission as a function of frequency and the electric field strength of the incident laser beam for (a) pristine graphene with μ=0\\mu =0 and (b) doped graphene with μ=0.13\\mu = 0.13 eV (or equivalently μ/ℏ=2×10 14 \\mu /\\hbar = 2 \\times 10^{14} Hz).", "Panels (c) and (d) display horizontal and vertical cuts, respectively with the solid lines corresponding to the μ=0\\mu = 0 case, and the dotted lines representing the μ=0.13\\mu = 0.13 eV.", "Other parameters are identical to that of Fig.", ".The optical conductivity of massive graphene, in all four regimes is shown in Fig.", "REF .", "As in the case of graphene, the $\\zeta = 1$ line marks the boundary of the non-linear response regime, saturable absorption effects dominating beyond $\\zeta > 1$ .", "Note that the Kubo formula based result for the optical conductivity of massive graphene [Eq.", "(REF )], is valid only in the $\\omega > \\gamma _2$ and $\\zeta \\ll 1$ regime.", "The impact of non-linear optical conductivity on the transmission spectrum is highlighted in Fig.", "REF .", "Finally we note that in graphene $\\sigma _{yx}(\\omega ) $ was zero for a $\\hat{x}$ polarized light for each valley.", "This is not the case for massive graphene.", "We find that based on Eq.", "(), for a $\\hat{x}$ polarized light, each valley has a finite $\\sigma _{yx}(\\omega )$ , which in the clean linear response regime is given by $\\sigma _{yx}(\\omega ) = \\frac{2 \\sigma _0}{\\pi } \\sum _\\tau \\frac{\\tau \\Delta _{\\tau }}{\\hbar \\omega } \\log \\left(\\frac{2 ~{\\rm max}\\lbrace \\Delta _\\tau , \\mu \\rbrace }{2 ~{\\rm max}\\lbrace \\Delta _\\tau , \\mu \\rbrace - \\omega } \\right)~,$ where $\\tau = + 1$ ($-1$ ) corresponds to the $K$ ($K^{\\prime }$ ) valley and $\\Delta _\\tau $ is the corresponding gap.", "Note that in Eq.", "(REF ) the sign of $\\sigma _{yx}(\\omega )$ in the $K$ valley turns out to be opposite to that of the $K^{\\prime }$ valley and if they have the same gap, the total $\\sigma _{yx}(\\omega )$ vanishes.", "However, if a valley asymmetry can be induced in graphene or other Dirac material (by breaking time reversal symmetry) [Hunt1427], leading to different band gap at the $K$ and $K^{\\prime }$ valleys ($\\Delta _{K} \\ne \\Delta _{K^{\\prime }}$ ), then we can have a finite $\\sigma _{yx}(\\omega )$ ." ], [ "Conclusion", "In this paper, we have present a unified formulation to calculate the non-linear optical conductivity for a generic two band system.", "Our model is based on a steady state solution of the optical-Bloch equations which yields an analytic expression for the population inversion and the inter-band coherence.", "A natural outcome of our model is the appearance of the dimensionless parameter $\\zeta \\propto E_0/\\omega $ , which quantifies the degree of optical non-linearity in the system, which was first pointed out by Mishchenko in the context of graphene [Mishchenko].", "This implies that nonlinear saturation effects are stronger at lower frequencies for the same strength of the optical field strength.", "Furthermore, based on the parameter $\\zeta $ and the coherence decay rate $\\gamma _2$ , any optical two band system can be said to be in one of the four regimes: (1) linear clean where $\\zeta \\ll 1$ , and $\\gamma _2 \\ll \\omega $ , (2) linear dirty where $\\zeta \\ll 1$ , and $\\gamma _2 \\ge \\omega $ , (3) non-linear clean where $\\zeta \\ge 1$ , and $\\gamma _2 \\ll \\omega $ , and (4) non-linear dirty where $\\zeta \\ge 1$ , and $\\gamma _2 \\ge \\omega $ .", "These regimes present distinct signatures in the optical conductivity and the optical transmission and reflection spectrum.", "Having established a general formulation for any two band system, we explicitly study the non-linear optical conductivity of graphene and massive graphene using the effective low energy Hamiltonian, and find analytic expressions for the optical conductivity in various regimes, reproducing the results for the clean case in the linear response regime.", "We emphasize that the usually reported Kubo formula based results for the optical conductivity are generally valid only in the high frequency ($\\omega \\gg \\gamma _2$ ) and linear response ($\\zeta \\ll 1$ ) regimes.", "An obvious extension of this work is to include the electron-phonon and electron-electron interactions explicitly along with the optical Bloch equation.", "This will provide a natural microscopic model for population inversion and decoherence decay rates, $\\gamma _1$ and $\\gamma _2$ , which we have assumed to be constant in this paper.", "Along with this the effect of band bending, trigonal warping etc.", "can be included by considering a tight-binding model for the Hamiltonian as opposed to an effective low energy Hamiltonian, and that will also increase the validity of this formulation for a wide range of optical frequencies." ] ]
1606.05072
[ [ "A new tagger for hadronically decaying heavy particles at the LHC" ], [ "Abstract A new algorithm for the identification of boosted, hadronically decaying, heavy particles at the LHC is presented.", "The algorithm is based on the known procedure of jet clustering with variable distance parameter $R$ and adapts the jet size to its transverse momentum $p_T$.", "Subjets are found using a mass jump condition.", "The resulting algorithm - called Heavy Object Tagger with Variable $R$ (HOTVR) - features little algorithmic complexity and combines jet clustering, subjet finding and rejection of soft clusters in one sequence.", "While the HOTVR algorithm can be used for the identification of any heavy object decaying hadronically, e.g.", "W, Z, H, t, or possible new heavy resonances, this paper targets specifically the tagging of boosted top quarks.", "The studies presented here demonstrate a stable performance of the HOTVR algorithm in a wide range of top quark $p_T$, from low $p_T$, where the decay products can be resolved, to the region of boosted decays at high $p_T$." ], [ "Introduction", "The identification of hadronically decaying heavy Standard Model (SM) particles ($\\mathrm {W}$ , $\\mathrm {Z}$ , $\\mathrm {H}$ , $\\mathrm {t}$ ) is an important ingredient in an increasing number of SM analyses and searches for new physics at the LHC.", "For a particle with high energy, the large Lorentz factor leads to decay products which are collimated in the laboratory rest frame and result in a single jet.", "The task of separating these decays from the vast amount of background from QCD multijet production has been approached with a variety of jet substructure developments in recent years [1], [2], [3], [4], [5], [6], [7].", "The techniques face the challenge of a stable performance in significantly different kinematic regimes: from the region of low transverse momentum $p_\\mathrm {T}$ , where the decay products can be resolved, to the boosted regime of high $p_\\mathrm {T}$ .", "The existing algorithms can be classified into two approaches.", "The bottom-up approach extrapolates from the resolved into the boosted regime by successively combining small radius jets, similar to an exclusive jet clustering (e.g.", "the JADE algorithm [8], [9]).", "Modern algorithms have been devised for the task of heavy object identification; examples are the collection of jets in buckets [10], [11] or the recently proposed XCone algorithm [12], [13].", "These methods combine manageable complexity with promising performance, but have not been studied in experimental analyses so far.", "The larger number of algorithms follows the top-down approach which starts from large radius jets followed by subsequent declustering steps.", "These algorithms are based on jet clustering with a fixed distance parameter $R$ , where jet grooming methods like filtering [1], pruning [14], trimming [15] or soft drop [16] are used to remove soft radiation and contributions from the underlying event such that substructure observables like the jet mass reflect the hard underlying process.", "Alternatively the variable $R$ jet algorithm [17] can be used to dynamically reduce the jet distance parameter with increasing $p_\\mathrm {T}$ of the decaying particle.", "The algorithm was used in studies of new heavy resonances decaying to final states with two and four gluons [15], and also top quark, $\\mathrm {W}$ and Higgs decays at LHC energies [18], and a similar algorithm at energies of a future hadron collider [19].", "Additional substructure information like the $k_t $ splitting scales [20], N-subjettiness [21], [22], [23], energy correlation functions [24] or Qjets [25] are often used to further improve the performance of substructure algorithms.", "Combinations of these methods are used for the tagging of top quarks [26], [27], [28], [29], [30], where also more theoretically motivated taggers have been proposed [31], [32].", "The ATLAS and CMS collaborations have commissioned a number of the techniques mentioned above and studied their behaviour [33], [34], [35], [36], [37], [38], [39], [40].", "Several top-tagging algorithms have been employed successfully in various searches for new physics [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69] and in SM top quark measurements [70], [71] in LHC analyses with Run 1 data.", "At LHC's Run 2 the production rates of particles with high $p_\\mathrm {T}$ have increased and the importance of boosted analyses is further enhanced.", "Modifications of existing taggers have been proposed and studied in simulation [72], [73], [74], [75].", "In most cases, a modest performance improvement is contrasted with a significantly increased algorithmic complexity.", "A simple but robust algorithm [76] proposed by the ATLAS collaboration has a slightly reduced performance.", "In addition, recent developments of top-down taggers aim at closing the gap between the resolved and the boosted regime with rather complex algorithmic procedures using several clustering, declustering, mass drop and filtering steps.", "Examples are the scale invariant tagger [77] and the HEPTopTagger in OptimalR mode [30].", "In this work we introduce a new tagger useful in the resolved, the transition and the boosted regime, achieved with only little algorithmic complexity.", "The tagger is based on the variable $R$ jet algorithm [17], which adapts the jet distance parameter dynamically to the $p_\\mathrm {T}$ of the boosted object.", "A mass jump condition [78], [79] is included in the clustering process, which forms subjets reflecting the dynamics of the underlying hard decay, and enables efficient background suppression.", "The resulting Heavy Object Tagger with Variable $R$ (HOTVR) accommodates jet clustering, subjet finding and the rejection of soft radiation in one sequence, without the need of declustering and following grooming steps.", "In this paper we demonstrate the algorithm's properties and characteristics in hadronic top decays and leave studies of the decays of $\\mathrm {W}$ , $\\mathrm {Z}$ , $\\mathrm {H}$ and possible new resonances to future work.", "The paper is organised as follows.", "In section  the HOTVR algorithm is described.", "Its characteristics, free parameters and their influence on the jet and subjet clustering, the collinear and infrared safety and timing performance are discussed in section .", "The algorithm's performance for hadronic top quark quark decays and a comparison with other commonly used taggers is presented in section .", "A conclusion is given in section ." ], [ "The algorithm", "The HOTVR algorithm is based on the variable $R$ (VR) jet algorithm [17].", "Like all sequential recombination algorithms, it starts with an input list of pseudojetsWe use the notation pseudojet to denote entities entering the jet clustering.", "These can be partons, stable particles, reconstructed detector objects or combined objects from a previous clustering iteration.", "and continues the processing until the input list is empty.", "The algorithm uses the distance measures $d_{ij}$ and $d_{i\\mathrm {B}}$ , defined as $d_{ij} &= \\min \\left[ p_{\\mathrm {T},i}^{2n},p_{\\mathrm {T},j}^{2n} \\right] \\Delta R_{ij}^2 \\, ,\\\\d_{i\\mathrm {B}} &= p_{\\mathrm {T},i}^{2n} \\, R_\\mathrm {eff} ^2(p_{\\mathrm {T},i}) \\, , \\\\R_\\mathrm {eff} (p_\\mathrm {T}) &= \\frac{\\rho }{p_\\mathrm {T}} \\, .", "$ The value of $d_{ij}$ can be interpreted as distance between two pseudojets $i$ and $j$ , where $p_{\\mathrm {T},i}$ is the transverse momentum of pseudojet $i$ and $\\Delta R_{ij} = \\sqrt{(y_i - y_j)^2 +(\\phi _i - \\phi _j)^2}$ is the angular distance in rapidity $y$ and azimuth $\\phi $ between the pseudojets $i$ and $j$ .", "The value of $d_{i\\mathrm {B}}$ denotes the distance between pseudojet $i$ and the beam.", "For a fixed distance parameter of $R_\\mathrm {eff} = R$ in Eq.", "(), the anti-$k_t$  [80], Cambridge/Aachen (CA) [81], [82] and $k_t$  [83], [84] algorithms are obtained for the choices $n = -1, 0, 1$ , respectively.", "For the HOTVR algorithm $n=0$ is used, corresponding to CA clustering.", "However, in the VR algorithm $R_\\mathrm {eff}$ is an effective distance parameter, which scales with $1/p_\\mathrm {T} $ (cf.", "Eq.", "()) leading to broader jets at low $p_\\mathrm {T}$ and narrower jets at high $p_\\mathrm {T}$ .", "The scale $\\rho $ determines the slope of $R_\\mathrm {eff}$ .", "For robustness of the algorithm with respect to experimental effects a minimum and a maximum cut-off for $R_\\mathrm {eff}$ is introduced, $R_\\mathrm {eff} ={\\left\\lbrace \\begin{array}{ll}R_\\mathrm {min} & \\text{for } \\rho /p_\\mathrm {T} < R_\\mathrm {min} \\, , \\\\R_\\mathrm {max} & \\text{for } \\rho /p_\\mathrm {T} > R_\\mathrm {max} \\, , \\\\\\rho /p_\\mathrm {T} & \\text{else} \\, .\\end{array}\\right.", "}$ A known shortcoming of the VR algorithm is the clustering of additional radiation into jets in QCD multijet production, resulting in a higher jet $p_\\mathrm {T}$ on average and an increased rate once a $p_\\mathrm {T}$ selection is applied [17].", "The HOTVR algorithm approaches this issue by modifying the jet clustering procedure with a veto based on the invariant mass of the pseudojet pair, inspired by the recently proposed mass jump algorithm [78].", "The mass jump veto prevents the recombination of two pseudojets $i$ and $j$ if the combined invariant mass $m_{ij}$ is not large enough, $\\theta \\cdot m_{ij}>\\max \\left[m_i,m_j\\right] \\, .$ The parameter $\\theta $ determines the strength of the mass jump veto and can be chosen from the interval $[0,1]$ .", "The mass jump criterion (REF ) is only applied if the mass $m_{ij}$ is larger than a mass threshold $\\mu $ $m_{ij} > \\mu \\, .$ In case a mass jump is found and the $p_\\mathrm {T}$ of the pseudojets $i$ and $j$ fulfil $p_{\\mathrm {T}i,j} > p_\\mathrm {T,sub} $ the pseudojets are combined.", "The resulting pseudojet enters the next clustering step and the initial pseudojets are stored as separate subjets.", "In case the mass jump criterion is not fulfilled or the pseudojets are softer than $p_\\mathrm {T,sub}$ , the lighter pseudojet or the one too soft is removed from the list.", "This step reduces the effect of additional activity (soft radiation, underlying event, pile-up) and effectively stabilises the jet mass over a large range of $p_\\mathrm {T}$ .", "The full HOTVR algorithm can be summarised as follows.", "1) If the smallest distance parameter is $d_{i\\mathrm {B}}$ , store the pseudojet $i$ as jet and remove it from the input list of pseudojets.", "2) If the smallest distance parameter is $d_{ij}$ and $m_{ij} \\le \\mu $ , combine $i$ and $j$ .", "3) If the smallest distance parameter is $d_{ij}$ and $m_{ij} >\\mu $ , check the mass jump criterion $\\theta \\cdot m_{ij} >\\max [m_i,m_j]$ .", "a) If the mass jump criterion is not fulfilled, compare the masses of the two pseudojets and remove the one with the lower mass from the input list.", "b) If the mass jump criterion is fulfilled, check the transverse momenta of the subjets $i$ and $j$ .", "i) If $p_{\\mathrm {T},i} < p_\\mathrm {T,sub} $ or $p_{\\mathrm {T},j} < p_\\mathrm {T,sub} $ , remove the respective pseudojet from the input list.", "ii) Else, combine pseudojets $i$ and $j$ .", "Store the pseudojets $i$ and $j$ as subjets of the combined pseudojet.", "In case $i$ or $j$ have already subjets, associate their subjets with the combined pseudojet.", "4) Continue with 1) until the input list of pseudojets is empty.", "The algorithm results in jets with an effective size depending on $p_\\mathrm {T} $ and associated subjets.", "It incorporates jet finding, subjet finding and the rejection of soft radiation in one clustering sequence.", "The algorithm is available as plugin to FastJet  [85], [86] and can be obtained through the FastJet Contribs package [87].", "Its implementation is based on the implementations of the mass jump and VR algorithms in the FastJet Contribs packages ClusteringVetoPlugin 1.0.0 and VariableR 1.1.1, respectively.", "These implementations have been adapted and modified to make the HOTVR software an independent FastJet plugin." ], [ "Parameters, jet and subjet finding", "In total, the algorithm has six parameters, which are listed in Tab.", "REF .", "While the first three parameters steer the VR part of the algorithm, the last three define the mass jump condition.", "The default values given in the table have been optimised for top quark tagging in $\\mathrm {pp}$ collisions at $\\sqrt{s}$ =13$\\,\\mathrm {Te\\hspace{-1.00006pt}V}$ .", "The original VR algorithm is recovered for $\\mu \\rightarrow \\infty $ .", "In this case, for $\\rho \\rightarrow 0$ the algorithm is identical to the CA algorithm with a distance parameter of $R = R_\\mathrm {min} $ .", "Similarly, for $\\rho \\rightarrow \\infty $ the CA algorithm is obtained with $R = R_\\mathrm {max} $ .", "For values of $\\rho $ corresponding to the typical scale of an event ($m$ or $p_\\mathrm {T} $ in the range of $\\mathcal {O}(100\\,\\mathrm {Ge\\hspace{-1.00006pt}V})$ ) jets are clustered with an adaptive distance parameter between $R_\\mathrm {min} $ and $R_\\mathrm {max} $ .", "Higher values of $\\rho $ result in larger jet sizes.", "The number of subjets found is modified by the mass jump parameters $\\mu $ , $\\theta $ and $p_\\mathrm {T,sub}$ .", "Once the pseudojets become sufficiently heavy due to clustering, the mass jump threshold $\\mu $ results in a rejection of soft and light pseudojets.", "For a fixed value of $\\mu $ , the strength of this jet grooming depends on the parameters $\\theta $ and $p_\\mathrm {T,sub}$ .", "For $\\theta =1$ the condition (REF ) is always fulfilled and no pseudojets are rejected (equivalent to the case $\\mu \\rightarrow \\infty $ ).", "Conversely, the case of $\\theta =0$ results in a VR jet clustering which stops as soon as a jet mass of $\\mu $ is reached.", "The algorithm results in subjets with a maximum mass of $\\mu $ .", "Additional jet grooming is obtained by setting $p_\\mathrm {T,sub} >0$ .", "This results in subjets with a minimum $p_\\mathrm {T}$ of $p_\\mathrm {T,sub}$ , effectively removing soft radiation and improving the tagging performance at small $p_\\mathrm {T}$ of the heavy object.", "Figure: Two simulated tt ¯\\mathrm {t}\\overline{\\mathrm {t}} events clustered with the CA algorithm with distanceparameter R=0.8R=0.8 (left column) and with the HOTVR algorithm (right column).The top quarks have either low p T p_\\mathrm {T} (top row, Event 1) or high p T p_\\mathrm {T} (bottom row, Event 2).The two leading jets in the events are shown as coloured areas (orange/blue).The stable particles, input for the jet finders, are drawn as grey dots.The quarks from the top quark decay are depicted by red circles and areshown for illustration purposes only.", "In case of the HOTVR algorithmthe subjets are shaded from light to dark, corresponding to increasing p T p_\\mathrm {T}.", "Thegrey areas correspond to regions rejected by the mass jump criterion.The algorithm's behaviour is visualised in Fig.", "REF where two example $\\mathrm {t}\\overline{\\mathrm {t}}$ events, generated with Pythia 8 [88], [89], [90] at low $p_\\mathrm {T}$ (top row, Event 1) and at high $p_\\mathrm {T}$ (bottom row, Event 2), are clustered with the CA algorithm (left column) and with the HOTVR algorithm (right column).", "The active catchment areas of the hard jets are obtained using ghost particles [91] and are illustrated by the coloured (orange/blue) areasThe exact borders of the jet areas depend slightly on the specific configuration of the ghost particles..", "The impact of the VR part of the algorithm is nicely illustrated by the largely different jet sizes of the two events clustered with the HOTVR algorithm (right column).", "The grey regions in the right panels were rejected by the mass jump criterion and are not part of the HOTVR jets.", "This criterion has largest impact in events at low $p_\\mathrm {T}$ as exemplified in Event 1 (top, right).", "The HOTVR jets together with their subjets reproduce the kinematics of the top decay adequately, both at low and high $p_\\mathrm {T}$ , demonstrating a better adaptation to the decay topology than CA jets.", "A similar picture is obtained when comparing HOTVR jets to anti-$k_t$ jets.", "The HOTVR algorithm is infrared and collinear (IRC) safe, except for the unnatural parameter choice of $\\mu =0$ .", "For parameter choices corresponding to the original VR clustering, the HOTVR algorithm is trivially infrared and collinear (IRC) safe [17].", "Similarly, for choices of $\\mu >0$ the algorithm is IRC safe, as soft and collinear splittings do not generate mass.", "This has also been verified in a numerical test, where the stability of the jets as well as subjets found with the HOTVR algorithm was studied with respect to soft radiation and collinear splittings.", "The algorithm proved to be IRC safe with no events out of $10^{6}$ failing the test [92].", "For timing tests, and throughout this work, the FastJet 3.2.1 [85], [86] framework is used, together with FastJet Contribs version 1.024.", "Starting from FastJet version 3.2, advanced clustering strategies became available which led to substantial speed improvements, especially at high particle multiplicities.", "For this reason the run time of the algorithm has been studied for different particle multiplicity scenarios, low ${\\cal O} (50)$ , medium ${\\cal O} (300)$ and high ${\\cal O} (3000)$ .", "In Tab.", "REF the CPU time of the HOTVR algorithm with default parameters (cf.", "Tab.", "REF ) is compared to those of the CA jet algorithm [81], [82], the CMS top tagger [26], [27], the HEPTopTagger  [28], [29], the HEPTopTagger in OptimalR mode [30], the VR algorithm [17] as well as the mass jump algorithm [78].", "Table: CPU time comparison of various algorithms for low, medium and high particlemultiplicities.", "The values are normalized to the CPU time of the CA algorithm with R=0.8R = 0.8.For the various top taggers the CPU time listed includes the time for the underlying jet finding as well as for the top tagger specific processing steps.", "The developments in FastJet 3.2 result in a much faster runtime of the VR and HOTVR clustering, compared to previous versions (not shown).", "At low and medium multiplicities, the runtime of the HOTVR algorithm is comparable to that of the other top-tagging algorithms tested.", "At high multiplicities, it is about a factor four slower than the HEPTopTagger algorithms, but it is still fast enough for practical usesFor example, on a MacBook Pro with a 2.5 GHz Intel Core i5 processor and 16 GB 1600 MHz DDR3 Memory the runtime is about 25 ms per event for multiplicities of ${\\cal O} (3000)$ ..", "The original mass jump algorithm has not been updated to employ the new clustering strategies, which leads to a much worse performance at medium and high multiplicities." ], [ "Physics performance", "Studies of the physics performance are carried out using the event generator Pythia 8 [88], [89], [90].", "A $\\mathrm {p}\\mathrm {p} \\rightarrow \\mathrm {t}\\overline{\\mathrm {t}} $ sample is used as signal process, background events are obtained by simulating QCD dijet production in $\\mathrm {p}\\mathrm {p}$ collisions.", "For both samples a centre-of-mass energy of $\\sqrt{s}=13$$\\,\\mathrm {Te\\hspace{-1.00006pt}V}$ is used, the multiple parton interaction tune Monash 2013 [93] and the LO NNPDF2.3 QCD+QED [94] PDFs with $\\alpha _s(M_Z) = 0.130$ are employed.", "At this stage no additional $\\mathrm {p}\\mathrm {p}$ interactions during a single bunch crossing (pile-up) are simulatedWhile pile-up effects will worsen the overall performance of the algorithm, the change is not expected to be significant for moderate pile-up scenarios (up to 20–30 additional pile-up interactions)..", "Throughout this work, jets are clustered using all stable particles from the Pythia 8 output.", "In some studies, additional jets (labelled parton jets) are obtained using a list of all final state partonsFinal state partons are defined as partons which enter the hadronisation step.", "as input to the anti-$k_t$ algorithm with distance parameter $R=0.4$ with a minimum $p_\\mathrm {T}$ of 100 GeV.", "In case of $\\mathrm {t}\\overline{\\mathrm {t}}$ production, the top quark is effectively treated as stable for the purposes of defining the parton jet: after showering the top quarks are added to the parton list, and all partons from the top quark decay are removed.", "In case a matching between particle and parton jets is employed, the geometrical matching condition $\\Delta R < R_\\mathrm {eff} $ is used." ], [ "Reconstruction of masses and transverse momenta", "The key to the tagger's effectiveness is the accurate reconstruction of subjets originating from the top quark decay, achieved by the VR condition and the mass jump criterion.", "This leads to a stable peak position for the mass of top jets over a large range of jet $p_\\mathrm {T}$ , as shown in Fig.", "REF .", "The jet mass $m_\\mathrm {jet}$ distribution for jets with two different subjet multiplicity $N_\\mathrm {sub}$ selections is shown for two ranges in the $p_\\mathrm {T} $ of the parton jet matched to the particle jet.", "For $\\mathrm {t}\\overline{\\mathrm {t}} $ events with $N_\\mathrm {sub} \\ge 2$ the distributions feature a dominant peak, stable around the top quark massThe VR algorithm alone affects the jet mass distribution similarly to a trimming [15] procedure for anti-$k_t$ jets at high top quark $p_\\mathrm {T}$  [18].", "for fully merged decays, and two smaller peaks at lower masses corresponding to partially merged top quark decays.", "The requirement of $N_\\mathrm {sub} \\ge 3$ leads to a depletion of the two secondary peaks, while the peak around the mass of the top quark is hardly affected.", "At low $p_\\mathrm {T}$ (left) the top quark peak is wider with a larger tail and is situated on a larger plateau than at high $p_\\mathrm {T}$ (right) because of contributions from additional radiation which aggregate in the jet due to its large size.", "While this leads to a larger misidentification rate at low $p_\\mathrm {T}$ , it results in a non-vanishing efficiency already at top quark transverse momenta as low as $100\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ .", "For typical QCD jets a falling distribution is observed.", "The wide peak at mass values around 140$\\,\\mathrm {Ge\\hspace{-1.00006pt}V}$ observed at low $p_\\mathrm {T}$ (left) is a result of the subjet kinematics, where an angular separation of $\\Delta R = 1.0\\text{--}1.5$ leads to jet masses around this value.", "When changing the kinematics by relaxing the $p_\\mathrm {T,sub}$ requirement, the peak vanishes and a falling distribution is obtained.", "The width of this peak is reduced for intermediate ($400<p_\\mathrm {T} <600\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ ) transverse momenta (not shown) and a monotonically falling background distribution is obtained for values of $600<p_\\mathrm {T} <800\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ (right).", "Very similar distributions are obtained for values of $p_\\mathrm {T} >800\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ .", "Figure: Distribution of the jet mass (top),fractional leading subjet transverse momentum (middle)and minimum pairwise mass (bottom)for signal (black) and background (red) events as obtained with the HOTVR algorithmfor two different ranges in parton jet p T p_\\mathrm {T} .The distributions are shown for subjet multiplicities N sub ≥2N_\\mathrm {sub} \\ge 2 (dashed lines)and N sub ≥3N_\\mathrm {sub} \\ge 3 (solid lines).", "Note that the minimum pairwise mass is onlydefined for N sub ≥3N_\\mathrm {sub} \\ge 3.The distributions have been normalised to unit area for N sub ≥2N_\\mathrm {sub} \\ge 2.The distributions of the leading subjet's fractional transverse momentum $f_{p_\\mathrm {T}} = p_\\mathrm {T,1}/p_\\mathrm {T} $ is shown in Fig.", "REF (middle).", "Signal jets contain subjets with more evenly distributed transverse momenta, while for background jets the leading subjet carries a larger fractional $p_\\mathrm {T}$ on average.", "The variable $f_{p_\\mathrm {T}} $ shows good separation power between signal and background jets before a subjet multiplicity selection.", "After the requirement of $N_\\mathrm {sub} \\ge 3$ , the separation power is reduced, but the variable is still useful, especially at high $p_\\mathrm {T}$ .", "For jets with $N_\\mathrm {sub} \\ge 3$ , the distribution of the minimum pairwise mass $m_\\mathrm {min} $  [26], [27], defined as the minimum invariant mass of pairs of the three highest $p_\\mathrm {T}$ subjets $m_\\mathrm {min} = \\min [m_{12},m_{13},m_{23}]$ , is shown in Fig.", "REF (bottom) for two regions of $p_\\mathrm {T}$ of the parton jet.", "The distributions show a clear cut-off at the chosen value of the mass jump threshold ($\\mu =30$  GeV).", "Above this value the distribution is steeply falling for background jets, while $\\mathrm {t}\\overline{\\mathrm {t}}$ signal jets exhibit a pronounced peak around the value of the $\\mathrm {W}$ boson mass, as expected for top quark jets.", "The tail below the mass jump threshold is a result of light subjets combined with a heavier pseudojet, fulfilling the mass jump criterion in step 3) of the algorithm.", "Besides an adequate reconstruction of masses, algorithms should also be able to reconstruct the kinematics of the initial heavy particle.", "In particular the size of the catchment area, which is responsible for the amount of additional radiation clustered into the jets, and the intensity of the grooming procedure are critical components for the performance in this area.", "For an evaluation of the kinematic object reconstruction by the HOTVR algorithm, we calculate the $p_\\mathrm {T}$ ratio of the HOTVR jets and the matched parton jets containing a top quark.", "We find a mean value of the $p_\\mathrm {T}$ ratio of $1.0$ within small deviations of the order of 1%, independent of the parton jet $p_\\mathrm {T}$ .", "The widths of the $p_\\mathrm {T}$ ratio distributions are about 5%.", "This shows that the HOTVR algorithm is able to accurately reconstruct the kinematics of the heavy object with the parameter choice given above.", "For the discrimination of hadronically decaying top quarks from QCD multijets a selection based on simple cuts using commonly employed substructure variables has been implemented.", "The variables $m_\\mathrm {jet}$ and $m_\\mathrm {min}$ calculated from the HOTVR subjets are in principle sufficient for building a robust top tagger over a large region of $p_\\mathrm {T}$ .", "However, cuts on additional variables have been added to obtain a selection that allows a fair comparison with other top-tagging algorithms using similar selections.", "To ensure only a limited impact of not-included experimental effects (e.g.", "broadening of distributions) the cut values have not been optimised rigorously.", "Nevertheless, they result in an improved discrimination between signal and backgroundA more sophisticated selection based on multivariate analysis techniques or more complex observables might provide further performance improvement [73] over this simple approach.", "However, the aim of the studies presented here is a comparison of the performance of the HOTVR algorithm with existing algorithms..", "The following selection defines the standard working point of the HOTVR algorithm in top-tagging mode.", "The leading subjet is required to have a fractional transverse momentum with respect to the jet, $f_{p_\\mathrm {T}} = p_\\mathrm {T,1}/p_\\mathrm {T} <0.8$ , which ensures that the jet's momentum is distributed among its subjets and not carried by only the leading subjet.", "The number of subjets $N_\\mathrm {sub}$ is required to be $N_\\mathrm {sub} \\ge 3$ , which increases the probability of reconstructing fully merged top jets and rejects a fair amount of QCD jets.", "The jet mass is required to fulfil $140<m_\\mathrm {jet} <220\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ .", "The minimum pairwise mass has to fulfil $m_\\mathrm {min} > 50\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ .", "These selection criteria lead to similar subjet kinematics as obtained by the CMS and HEPTopTagger algorithms with default parameters.", "This provides the basis for the comparison made in the following.", "The signal efficiency and misidentification rate are studied using single variable receiver operating characteristic (ROC) curves.", "The signal efficiency $\\varepsilon _{S}$ is defined as the fraction of tagged jets matched to parton jets containing the top quark, with respect to all top quarks decaying hadronically.", "The background efficiency (or misidentification rate) $\\varepsilon _{B}$ is calculated as the fraction of tagged jets matched to parton jets in a QCD multijet sample, with respect to the total number of parton jets.", "Both, $\\varepsilon _{S}$ and $\\varepsilon _{B}$ therefore combine identification and matching efficiencies.", "These definitions allow for a comparison of different tagging algorithms, in particular using different choices of the jet distance parameter $R$ , since the reference $p_\\mathrm {T} $ is defined by the parton jet matched to the tagged jet and does not depend on the specifics of the tagging algorithm under study.", "Table: Settings of the top tagging algorithms used.", "The parameter RRis the distance parameter of the jet clustering.", "The definition of the parameters followsRef.", "for the CMS top tagger, Ref.", "for the HEPTopTagger and Ref.", "for the HEPTopTagger in OptimalR mode.In the following the performance of the HOTVR algorithm in top-tagging mode is compared with the performance of three top-tagging algorithms especially designed for dedicated regions of $p_\\mathrm {T}$ : the CMS top-tagger targets the region of high $p_\\mathrm {T}$ , the HEPTopTagger is designed for low $p_\\mathrm {T}$ and its improved version with OptimalR has been developed to extend its usability to higher $p_\\mathrm {T}$ .", "The free parameters of these taggers are listed in Tab.", "REF together with a choice of working points [95], [29], [73].", "The ROC curves are obtained by keeping the free parameters fixed at the values given and scanning only the N-subjettiness [21], [22], [23] ratio $\\tau _{3/2} = \\tau _{3}/\\tau _{2}$ with $\\beta =1$ .", "The choice of $\\tau _{3/2} $ as scanning variableThe usual procedure for obtaining the ROC curves by scanning the free parameters of each algorithm could provide misleading results in this case, as it cannot be ensured that the usage of additional or different scanning variables for a given tagger would not improve its performance considerably.", "ensures an unprejudiced comparison of the algorithms, which all rely on different reconstruction techniques and substructure variables, since this variable is not used in the definition of any of the taggers under study.", "Furthermore, $\\tau _{3/2} $ has been shown to improve the performance of existing taggers (see for example Refs.", "[73], [30]).", "Figure: Receiver operating characteristic (ROC) curves for different top tagging algorithms obtained froma scan of the variable τ 3/2 \\tau _{3/2} in four different p T p_\\mathrm {T} regions.In Fig.", "REF the ROC curves of the four top-tagging algorithms are shown for four different $p_\\mathrm {T}$ regions, where $p_\\mathrm {T}$ is defined by the parton jet matched to the tagged jet.", "The events were reweighted to obtain a flat $p_\\mathrm {T}$ spectrum such that all events in the interval have the same weight.", "At low $p_\\mathrm {T}$ ($200<p_\\mathrm {T} <400\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ , top left) the CMS top-tagging algorithm has very small efficiency due to the choice of $R = 0.8$ which results in jets not large enough to cluster all particles from the top quark decay chain.", "The HOTVR algorithm is able to provide a comparable performance as the two HEPTopTagger algorithms which were optimised for this $p_\\mathrm {T}$ region.", "For increasing values of $p_\\mathrm {T}$ the CMS tagger becomes more efficient, with a similar performance as the OptimalR HEPTopTagger starting from $p_\\mathrm {T} >600\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ .", "In the $p_\\mathrm {T}$ regions with $400<p_\\mathrm {T} <600\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ (top right) and $600<p_\\mathrm {T} <1000\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ (bottom left) the HOTVR algorithm shows a similar relation between $\\varepsilon _{S} $ and $\\varepsilon _{B} $ as the CMS and OptimalR HEPTopTagger, and is especially useful for high efficiencies.", "In the highest $p_\\mathrm {T}$ region considered ($1000<p_\\mathrm {T} <2000\\,\\mathrm {Ge\\hspace{-1.00006pt}V} $ , bottom right) the HOTVR algorithm features overall the best performance over all $\\varepsilon _{S} $ values, outperforming the CMS tagger, which was designed for the high $p_\\mathrm {T}$ region.", "In summary, the HOTVR algorithm shows a remarkably stable performance over a large range in $p_\\mathrm {T}$ with similar or even better performance than algorithms especially designed for certain $p_\\mathrm {T}$ regions.", "Detector reconstruction and resolution effects, which are not included in these studies, are expected to improve the performance of the HOTVR algorithm relative to the other algorithms studied [92]." ], [ "Conclusion", "A new algorithm for the reconstruction and identification of hadronically decaying heavy particles at the LHC has been introduced in this paper.", "The algorithm combines variable $R$ jet clustering with a veto based on a mass jump criterion.", "It performs jet and subjet finding, and the rejection of soft radiation in one sequence.", "This combination results in a stable determination of jet substructure variables like the jet mass over a large range in $p_\\mathrm {T}$ of the heavy object.", "In top-tagging mode the HOTVR algorithm provides an excellent ratio of signal to background efficiency at low top quark $p_\\mathrm {T}$ as well as at high $p_\\mathrm {T}$ , making the HOTVR algorithm useful in the regions of resolved and boosted decays at the same time.", "While we focussed on top tagging in this work, the algorithm is also applicable for the tagging of $\\mathrm {W}$ , $\\mathrm {Z}$ , $\\mathrm {H}$ or possible BSM resonances, where studies are ongoing.", "Because of its algorithmic simplicity combined with remarkable performance, this tagger could become a helpful ingredient for future boosted analyses at the LHC." ], [ "Acknowledgements", " We thank Michael Spannowsky for fruitful discussions during the development of the algorithm.", "We also thank Jesse Thaler for helpful suggestions on improvements of the document and for advise on speed improvements.", "This work is supported by the German Research Foundation (DFG) in the Collaborative Research Centre (SFB) 676 “Particles, Strings and the Early Universe” located in Hamburg.", "tocsectionReferences" ] ]
1606.04961
[ [ "Inscribed Matter Communication: Part I" ], [ "Abstract We provide a fundamental treatment of the molecular communication channel wherein \"inscribed matter\" is transmitted across a spatial gap to provide reliable signaling between a sender and receiver.", "Inscribed matter is defined as an ensemble of \"tokens\" (molecules, objects, and so on) and is inspired, at least partially, by biological systems where groups of individually constructed discrete particles ranging from molecules through membrane-bound structures containing molecules to viruses and organisms are released by a source and travel to a target -- for example, morphogens or semiochemicals diffuse from one cell, tissue or organism diffuse to another.", "For identical tokens that are neither lost nor modified, we consider messages encoded using three candidate communication schemes: a) token timing (timed release), b) token payload (composition), and c) token timing plus payload.", "We provide capacity bounds for each scheme and discuss their relative utility.", "We find that under not unreasonable assumptions, megabit per second rates could be supported at femtoWatt transmitter powers.", "Since quantities such as token concentration or bin-counting are derivatives of token arrival timing, individual token timing undergirds all molecular communication techniques.", "Thus, our modeling and results about the physics of efficient token-based information transfer can inform investigations of diverse theoretical and practical problems in engineering and biology.", "This work, Part I, focuses on the information theoretic bounds on capacity.", "Part II develops some of the mathematical and information-theoretic ideas that support the bounds presented here." ], [ "Introduction", "Scale-appropriate signaling methods become important as systems shrink to the nanoscale.", "For systems with feature sizes of microns and smaller, electromagnetic and acoustic communication become increasingly inefficient because energy coupling from the transmitter to the medium and from the medium to the receiver becomes difficult at usable frequencies.", "Biological systems, with the benefit of lengthy evolutionary experimentation, seem to have arrived at a ubiquitous solution to this signaling problem at small and not so small scales: use of “inscribed matter” (an ensemble of discrete particles) which travels through some material bearing a message from one entity to another.", "Broad classes of such particles include Molecules such as electronically activated species, ions, chemicals, biopolymers, and macromolecular complexes.", "Membrane-bound structures such as intra- and extracellular vesicles (for instance, exosomes, microvesicles, apoptotic bodies, ectosomes, endosomes, lysosomes, autophagosomes, and vacuoles) and intracellular organelles (for instance, nuclei, mitochondria, and chloroplasts).", "Cells such as stem cells, tumor cells, and hematocytes.", "Acellular, unicellular and multicellular life forms (organisms for brevity) such as viruses, viroids, phages, plasmids, bacteria, archaea, fungi, protists, plants, and animals.", "Objects such as matter in the natural world (for instance pollen grains, seeds, and proteinaceous aggregates such as prions); and human artifacts (for example, Voyager Golden Records).", "Studies of engineered nano-scale communication systems have focused on the encoding, transmission, and decoding of information using patterns of one category of discrete particles, namely molecules.", "A large portion of this work in “molecular communication” has considered time-varying concentration profiles of molecules as the fundamental signal measurement [1], [2], [3], [4], [5], [6], [7].", "However, concentration is a collective property of the process and masks the underlying physics of molecule release by the sender and capture by the receiver.", "This begs the questions of truly fundamental limits for communication using ensembles of molecules in particular, discrete particles more broadly, and what we term “tokens” in general.", "This paper is organized as follows: First, we discuss communication using inscribed matter from biological and engineering perspectives.", "We illustrate how scenarios spanning a wide range of spatial and temporal scales and from seemingly disparate disciplines can be understood within a unified framework: the token timing and/or token payload channel, a communication scheme wherein information is carried from sender to receiver by tokens via their timed release, their composition, or both.We will assume tokens always (eventually) arrive, and are removed from circulation upon first seizure by the receiver.", "This abstraction encompasses not only token timing but also the token concentration and token counting models prevalent in the molecular communication literature [1], [2], [3], [4], [5], [6], [7], [8], [9].", "Next, we describe the token timing channel wherein information is encoded only in the release time of identical tokens as opposed to inscribed onto tokens (tokens with payloads) or in the number of tokens released (token counting).", "Though seemingly limited, this pure timing model supplies the mathematical machinery to precisely consider both token payload and token counting communication schemes.", "To this end, we provide a mathematical formulation of token timing channel: identical tokens emitted with independent stochastic (but asymptotically assured, one-time) arrivals.", "We formalize the signaling model so that the typical energy-dependent asymptotic sequential channel use coding results based on mutual information between input and output can be applied [10].", "We then show how these results can be applied to token counting and tokens with payloads.", "We focus on molecular tokens – particularly DNA and protein sequences since their energy requirements (and information content) are well-understood – and show that information transfer using inscribed matter can be extremely efficient.", "We find that megabit per second rates could be supported theoretically with on the order of 100 femtoWatts of transmitter power.", "Finally, we explore how our studies and the attendant insights could aid biological understanding of and inform engineering approaches to inscribed matter communication.", "Networks of intercommunicating biological entities occur at whatever level one cares to consider: (macro)molecules, cells, tissues, organisms, populations, microbiomes, ecosystems, and so on.", "An ancient yet still widespread method for one entity to convey a message to another is via inscribed matter.", "The typical scenario is as follows: information-bearing discrete particles are released by a source, travel through a material, and are captured by a target where they are interpreted.", "The following examples illustrate the diversity and complexity of such inscribed matter communication (the particles are italicized).", "Electrons from an electron donor flow through an electron transport chain to an electron acceptor where the electrochemical gradient is used convert mechanical work into chemical energy as part of a cellular process such as photosynthesis or respiration.", "In microbial communities, electrons are transferred from one individual to another through bacterial nanowires (electrically conductive appendages), bacterial cables (thousands of individuals lined up end-to-end with electron donors located in the deeper regions of marine sediment and electron acceptors positioned in its upper layers where oxygen is more abundant), and biofilms (community members embedded in a self-produced three-dimensional matrix of extrapolymeric substances) [11].", "Free radicals produced from molecules in the nucleoplasm by the direct or indirect action of ionizing radiation diffuse to the genome where they alter/damage nucleotide bases and sugars.", "messenger RNA (mRNA) molecules transcribed from a eukaryotic genome in a nucleus migrate to ribosomes in the cytoplasm where they are translated into proteins.", "Acetylcholine (ACh) molecules released by a vertebrate motor neuron diffuse across the synapse to nicotinic ACh receptors on the plasma membrane of the muscle fiber where binding triggers muscular contraction.", "Homing endonuclease (HE) containing inteins self-excised from bacterial, archaeal, eukaryotic or viral host proteins home to a target site in the genome of the same or different organism where the genetic parasitic element reinserts itself into the intein-free allele of the host gene (horizontal dissemination); inteins without a functioning HE are mainly transferred vertically but may move horizontally along with the host gene.", "Ions, molecules, organelles, bacteria and viruses present in one cell travel through a thin membrane channel (tunneling nanotube) to the physically connected cell where they elicit a response.", "Semiochemicals (chemical substances or mixtures of volatile molecules) emitted by one individual travel to another of the same (pheromones) or different species (allelochemicals) where they elicit a response – allomones benefit only the sender, kairomone benefit only the receiver, and synomones benefit both.", "Extracellular vesicles secreted by all living cells – including bacteria, archaea and eukaryotes – and harboring specific cargo materials (for instance, proteins, nucleic acids, lipids, metabolites, antigens, and viruses) traverse the extracellular space or body fluids (for instance, blood and urine) to a local or distal recipient cell where they transfer their bioactive contents.", "Cargo-bearing molecular motors shuttle along a track system of cytoskeletal filaments to another point in the cell compartment where their freight such as vesicles containing molecules and tubes is unloaded.", "Single and clusters of metastatic cells that have escaped from a primary tumor circulate through the blood or lymph to a secondary organ site where, after extravasation, they can seed a new tumor.", "Organic particles such as microorganisms, fungal spores, small insects, and pollen grains associated with a macroorganism, geological site or geographic location relocate to another host or region where they influence the local biochemistry, geochemistry and climate [12]– long distance transport (including movement within and between continents and oceans) can occur via the same meteorological phenomena and processes, such as jetstreams and hurricanes, that translocate non-biological particles such as sea salt and dust.", "Crustal material ejected by a Solar System body travels to another body where if it carries microbial spores or building blocks such as amino acids, nucleobases and lipid-like molecules has the potential to seed life.", "Ejecta (potentially carrying microbial spores) travel from Ceres (the largest object in the asteroid belt which lies between the orbits of Mars and Jupiter) to terrestrial planets in the solar system (Earth, Mars or Venus) [13]; the presence of water on Ceres [14] suggests the dwarf planet's potential as a home for extraterrestrial life.", "Irrespective of the precise nature of the components of the inscribed matter communication system – the discrete particles (information carriers), source (sender), spatial gap (transmission medium), and target (receiver) – two fundamental questions are “How reliable is communication?,” and “How is useful information conveyed given constraints on resources?.” Here, we investigate token timing (discrete particle release and capture times) and token payload (energy required to manufacture discrete particles, to assemble symbolic strings from a set of building blocks – we do not consider the energy required for de novo synthesis of the building blocks).", "And although not explicitly stated, please note that our energy model could also include token sequestration, token ejection, and token transport, the active movement of discrete particles (the energetics of translocating vesicles by a molecular motor system which converts chemical or other form of energy into mechanical energy).", "The only requirement is that the energy cost per token is independent of the information carried.", "In the token timing channel model we will elaborate later on, tokens are neither lost nor modified: the number and makeup of the tokens emitted by the source is the same as the ones arriving at the target, all that differs are their times of emission and their times of arrival.", "While accommodating tokens that are delayed temporarily, our mathematical model does not directly consider tokens that are detained permanently, removed entirely, never arrive, or are changed en route.", "In the natural world, discrete particles often interact with the material through which they travel resulting in their immurement and ultimate removal or detention and eventual discharge.", "Examples include: Free radicals produced by radiolysis may react chemically with neighboring materials.", "mRNAs may be modified post-transcriptionally.", "ACh can be degraded by the enzyme acetylcholine esterase present in synapses.", "The random path of a semiochemical diffusing through air, soil or water may result in a trajectory that leads away from the destined individual.", "Circulating tumor cells may be destroyed by the immune system.", "Microscopic particles may be immobilized within mucus – the polymer-based hydrogel covering the inner linings of the body – depending on the density of the mucin network and environmental factors such as pH and ionic strength.", "Bacteria, particularly plant pathogens, present in the atmosphere can nucleate the formation of ice in clouds resulting in snow, rain and hail [12].", "Nonetheless, our model does provide an organizing principle for all forms of molecular communications since these sorts of impediments – token loss or corruption – can only decrease the capacity of the system we analyze.", "Furthermore, the analysis is “compartmental” in the sense that token corruption and loss can be treated separately without invalidating the fundamental “outer bound” results." ], [ "Communication using physical objects: an engineering perspective", "Inscribed matter can often be the most energy-efficient means of communication when delay can be tolerated.", "In fact, a once popular communication networks textbook [15] contains the passage: Never underestimate the bandwidth of a station wagon full of tapes hurtling down the highway.", "– A.S. Tanenbaum, Computer Networks, 4th ed., p. 91 This somewhat tongue-in-cheek “folklore” should come as no surprise.", "From early antiquity, private persons, governments, the military, press agencies, stockbrokers and others have used carrier pigeons to convey messages.", "Today, “sneakernets” [16] have been proposed as a low-latency high-fidelity network architecture for quantum computing across global distances: ships carry error-corrected quantum memories installed in cargo containers [17].", "Previous work on mobile wireless communication found that network capacity could be increased if delay-tolerant traffic was queued until the receiver and sender were close to one another – perhaps close enough to exchange physical storage media [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31].", "This recognition prompted a careful consideration of the energetics involved in delivery of physical messages, and a series of papers [32], [33], [34] revealed the surprising results that inscribed matter can be many orders of magnitude more efficient than radiative methods even with moderate delay constraints and over a variety of size scales.", "In fact, [34] showed that over interstellar distances (10k light years), inscribed matter could be on the order of $10^{15}$ times more energy-efficient than radiated messages, suggesting that evidence of extraterrestrial civilizations would more likely come from artifacts than from radio messages if energy requirements are a proxy for engineering difficulty [35].", "At the other end of the size scale there has been increasing interest in biologically-inspired inscribed matter communication at the nano/microscale [36] where the information carrier ranges from timed release of identical signaling agents to specially constructed information carriers [37], [38], [1], [2], [3], [39], [4], [5], [6], [7], [40].", "Although this field of molecular communication is in its infancy with seemingly futuristic application plans currently out of reach (for instance, in vivo biological signaling, surgical/medicinal/environmental microbot swarms, or process-on-a-chip), the theoretical potential rates and energy efficiencies, especially through media unfriendly to radiation, are sufficiently large [41] to warrant careful theoretical and practical consideration.", "From an engineering perspective, the basic idea of inscribed matter communication is very simple (FIGURE REF ).", "Information is coded in the structure of the signaling agent and/or its release time at the sender.", "These agents traverse some spatial gap to the receiver where they are captured and the information decoded.", "There are, of course, many details and variations on the theme.", "As explicitly mentioned for biological systems, the signaling agents (or tokens as we call them) could be identical, implying that timing (which includes time-varying concentration) is the only information carrier, or tokens could themselves carry data payloads (in addition to, or in lieu of timing).", "However, unlike the award-winning paper “Bits Through Queues” by Anantharam and Verdú [42] and later by Sundaresan and Verdú [43], [44], we do not know which arrival times correspond to which emission times.", "The “gap” (channel) could be a medium through which tokens diffuse stochastically, or some form of active transport might be employed.", "In addition, tokens could be deliberately “eaten” by gettering agents injected by the sender, channel or receiver.", "Similarly, tokens could be corrupted during passage through the channel or might simply get “lost” and never reach the receiver [39], [9].", "Furthermore, the reception process itself could be noisy.", "If we sought to mimic biological systems, a typical receptor structure is stereochemically matched to a particular signaling molecule (token) and the kinetics of the ligand binding/unbinding process must be considered as well as the number and density of receptors.", "Furthermore, a given receptor may preferentially bind to a ligand (token), but there may be other different or identical (but from another source) interfering ligands which bind to the same receptor.", "When one considers networks of molecular transceivers, this sort of “cross talk” or outright interference must be considered." ], [ "Inscribed matter communication: model distillation", "While the various engineering and biological scenarios require slightly different information theoretic formulations, they can all be understood within a unified framework: the identical token timing channel wherein Token release and capture timing is the only mechanism for information transfer.", "Tokens always (eventually) arrive at the receiver.", "Tokens are removed promptly from circulation (or deactivated) after first reception.", "Figure: An Abstraction of an Inscribed Matter Communication Channel.A sender transmits an ensemble of tokens (“inscribed matter”) to areceiver across a spatial gap (of length RR in the figure).", "The tokens are released at (unordered) times{T m }\\lbrace T_m\\rbrace , propagate through a transmission medium and are captured atcorresponding times {S m }\\lbrace S_m\\rbrace .", "For identical tokens, the receiver sees orderedarrivals {S → m }\\lbrace \\vec{S}_m \\rbrace which may differ in index from the unordered arrivals{S m }\\lbrace S_m\\rbrace .The identical token timing channel abstraction [19], [45], [46], [47], [41], [8] encompasses token concentration or token counting models since time-varying concentration (or token counts in “bit intervals”) at a receiver is a coarse-time approximation to the precise individual token timing model.", "The timing channel is also important for understanding information carriage via payload-charged tokens whose information packets may need resequencing at the receiver.", "That is, timing channel results provide tight bounds on resequencing overhead and are especially important if it is technologically difficult to construct tokens with large payloads.", "In addition, the operation of the timing channel also sheds light on channels wherein the number of tokens sent is the information carrier during signaling intervals [8].", "In addition, as mentioned at the end of section REF , the timing channel provides outer bounds since the uncertainty associated with various receptor and channel models can only decrease the information-carrying capacity of the channel via the data processing theorem [10].", "For instance, re-capture processes owing to receptor binding kinetics [39] and token loss (erasure) can only decrease the channel capacity.", "Likewise, token processing/corruption/loss can again only decrease channel capacity.", "Thus, the timing channel not only allows upper limits on capacity to be obtained, but also permits the overall channel to be treated as a cascade, each constituent of which can be analyzed separately and compared to identify potential information transfer bottlenecks.", "TABLE REF is a glossary of key quantities that will be discussed in what follows.", "For continuity and clarity, the identical table is included in the companion paper, Part-II [48].", "Table: Glossary of useful termsFollowing [49], [19], [46], [47], [41], assume emission of $M$ identical tokens at times $\\lbrace T_m\\rbrace $ , and their capture at times $\\lbrace S_m \\rbrace $ , $m = 1,2,\\cdots ,M$ .", "The duration of token $m$ 's first-passage between source and destination is $D_m$ .", "These $D_m$ are assumed i.i.d.", "with $f_{D_m}(d) = g(d) = G^{\\prime }(d)$ where $g(\\cdot )$ is some causal probability density with mean $\\frac{1}{{\\mu }}$ and cumulative distribution function (CDF) $G(\\cdot )$ .", "We also assume that $g(\\cdot )$ contains no singularities.", "Thus, the first portion of the channel is modeled as a sum of random $M$ -vectors cS= T+ Das shown in FIGURE REF prior to the sorting operation.", "We therefore have rCl fS(s) = 0 fT(t) fS|T(s|t) d t = 0s fT(t) m=1M g(sm - tm) d t = 0s fT(t) g(s- t) d twhere cg(s- t) = m=1M g(sm - tm) Figure: The token release with reordering Inscribed Matter communication channel.", "For token mmreleased at time T m T_m, the duration of its first-passage between the sender and receiver is D m D_mso it arrives at time S m S_m.", "The {S m }\\lbrace S_m \\rbrace are then sorted by order of arrival.", "Since the MMtokens are identical, the ordered arrival time S → m \\vec{S}_m may not correspond to S m S_m.We impose an emission deadline, $T_m \\le \\tau $ , $\\forall m \\in \\lbrace 1,2,\\cdots ,M \\rbrace $ .", "The associated emission time ensemble probability density $f_{{\\bf T}}({\\bf t})$ is assumed causal, but otherwise arbitrary.", "Had we imposed a mean constraint instead of a deadline, the channel between ${\\bf T}$ and ${\\bf S}$ would be the parallel version of Anantharam and Verdú's Bits Through Queues [42].", "However, since the tokens are identical we cannot necessarily determine which arrival corresponds to which emission time.", "That is, the final output of the channel is a reordering of the $\\lbrace s_m \\rbrace $ to obtain a set $\\lbrace \\vec{s}_m \\rbrace $ where $\\vec{s}_m \\le \\vec{s}_{m+1}$ , $m=1,2,\\cdots , M-1$ , as shown on the right hand side of FIGURE REF after the sorting operation.", "We write this relationship as c S= P(S) where $P_{{\\Omega }}(\\cdot )$ , ${{\\Omega }}=1,2,\\cdots ,M!$ , is a permutation operator and ${\\Omega }$ is that permutation index which produces ordered $\\vec{{\\bf S}}$ from the argument ${\\bf S}$ .", "We define $P_1(\\cdot )$ as the identity permutation operator, $P_1({\\bf s}) = {\\bf s}$ .", "We note that the event $S_i = S_j$ ($i \\ne j$ ) is of zero measure owing to the no-singularity assumption on $g(\\cdot )$ , Thus, for analytic convenience we will assume that $f_{{\\bf S}}({\\bf s}) = 0$ whenever two or more of the $s_m$ are equal and therefore that the $\\lbrace \\vec{s}_m \\rbrace $ are strictly ordered wherever $f_{\\vec{{\\bf S}}}(\\cdot ) \\ne 0$ (i.e., $\\vec{s}_m < \\vec{s}_{m+1}$ ).", "Thus, the density $f_{\\vec{{\\bf S}}}(\\vec{{\\bf s}})$ can be found by “folding” the density $f_{{\\bf S}}({\\bf s})$ about the hyperplanes described by one or more of the $s_m$ equal until the resulting probability density is nonzero only on the region where $s_m < s_{m+1}$ , $m=1,2,\\cdots ,M-1$ .", "Analytically we have c fS(s) = n=1M!", "fS(Pn(s)) Then, since $f_{S|T}(s|t) = g(s-t)$ , we can likewise describe $f_{\\vec{{\\bf S}}|{\\bf T}}({\\bf s}|{\\bf t})$ as c fS|T(s|t) = n=1M!", "g(Pn(s) - t) m=1M u([Pn(s)]m - tm) again for ${s_1 < s_2 < \\cdots < s_m}$ and zero otherwise.", "With exponential first-passage, $g(d) = {\\mu }e^{-{\\mu }d}u(d)$ , becomes c fS|T(s|t) = M e - i=1M (si - ti) ( n=1M!", "u(Pn(s) - t) ) again assuming $s_1 < s_2 < \\cdots < s_m$ .", "It is worth mentioning explicitly that equation () does not assume arguments $s_i \\ge t_i$ as might be implicit in equation ().", "Finally, the problem structure will allow us to make use of multi-dimensional function symmetry (hypersymmetry) arguments, $f({\\bf x}) = f(P_n({\\bf x}))$ $\\forall $ permutations $n$ .", "The following property of expectations of hypersymmetric functions over hypersymmetric random variables will later prove useful.", "Theorem 1 Hypersymmetric Expectation: Suppose $Q({\\bf x})$ is a hypersymmetric function, $Q({\\bf x}) = Q(P_k({\\bf x}))$ $\\forall k$ , and ${\\bf X}$ is a hypersymmetric random vector.", "Then, when $\\vec{{\\bf X}}$ is the ordered version of random vector ${\\bf X}$ we have c EX [Q(X) ] = EX [Q(X) ] Proof:  Theorem  REF  $\\tilde{{\\bf X}}$ is a deterministic function of ${\\bf X}$ ; i.e., $\\theta ({\\bf X}) = {\\tilde{{\\bf X}}}$ .", "Thus, cE [ Q(X) ] = E [ Q((X)) ] = E [ Q(X) ] where the last equality results from the hypersymmetry of $Q(\\cdot )$ .", "$\\bullet $ With these preliminaries done, we can now begin to examine the mutual information between the unordered emission times ${\\bf T}$ , the unordered arrival times ${\\bf S}$ , and the ordered (sorted) arrival times $\\vec{{\\bf S}}$ ." ], [ "Formalizing The Signaling Model", "To determine whether the mutual information between ${\\bf T}$ the input and $\\vec{{\\bf S}}$ the output is a measure of channel capacity, it suffices to have a signaling model which patently supports the usual asymptotically large block length and repeated independent sequential channel uses paradigm [10].", "In addition, we must also pay attention to the channel use energetics since lack of energy constraints can lead to unrealistic results.", "Thus, we have defined a channel use as the launch and capture of $M$ tokens under an emission deadline constraint, $\\tau $ , with the further constraint that c = M where ${\\lambda }$ , the token launch average intensity, has units of tokens per time.", "Equation (REF ) is implicitly a constraint on average power assuming a fixed per-token energy cost for construction/sequestration/release/delivery.", "We also note that the signaling interval $\\tau $ is now an explicit function of $M$ as in c = (M) = M So, consider FIGURE REF where sequential $M$ -token transmissions – channel uses – are depicted.", "We will assume a “guard interval” of some duration $\\gamma (M,\\epsilon )$ between successive transmissions so that all $M$ tokens are received before the beginning of the next channel use with probability ($1-\\epsilon $ ) for arbitrarily small $\\epsilon > 0$ .", "Figure: Successive MM-emission channel uses.For a given use of the token timing channel, the sender emits MM tokens overthe transmission interval τ(M)=M λ\\tau (M) = \\frac{M}{{\\lambda }}.γ(M,ϵ)\\gamma (M,\\epsilon ) is the waiting period (guard interval) before the next channel use.We further require that the average emission rate, ${M}/(\\tau (M) + \\gamma (M,\\epsilon ))$ satisfies c 0 M M(M) + (M,) = We then require that the last token arrival time $\\vec{S}_M$ occurs before the start of the next channel use with probability 1.", "That is, given arbitrarily small $\\epsilon $ we can always find a finite $M^*$ such that c Prob{SM (M) + (M,) } > 1-$\\forall M \\ge M^*$ .", "We now derive a sufficient condition on first-passage time densities for which equation (REF ) is true.", "Calculating a CDF for $\\vec{S}_M$ is in general difficult since emission times $T_m$ might not be independent.", "However, for a fixed emission interval $[0, \\tau (M)]$ we can readily calculate a worst case CDF for $\\vec{S}_M$ and thence an upper bound on the guard interval duration that satisfies the arrival condition of equation (REF ).", "That is, for a given emission schedule ${\\bf t}$ , the ${\\bf S}$ are conditionally independent and the CDF for the final arrival is cFSM|t(s|t) = m=1M G(s-tm)u(s-tm) so that cFSM(s) = 0(M) fT(t)m=1M G (s-tm)u(s-tm) d t However, it is easy to see that cFSM(s) GM(s-(M)) u(s-(M)) since $G(s-t)$ is monotone decreasing in $t$ .", "The end of the guard interval is $\\tau (M) + \\gamma (M,\\epsilon )$ , so the probability that the last arrival time $\\vec{{\\bf S}}_M$ occurs before the next signaling interval obeys c FSM((M) + (M,)) GM ( (M,) ) And to meet the requirement of equation (REF ) we must have c M GM ( (M,) ) = 1 which for convenience, we rewrite as c M M G ( (M,) ) = 0 If rewrite $\\log G \\left( \\gamma (M,\\epsilon ) \\right)$ in terms of the CCDF (complementary CDF) ${\\bar{G}}(\\cdot )$ (which must be vanishingly small in large $M$ if we are to meet the conditions of equation (REF )) and note that $\\log (1-x)\\approx -x$ for $x$ small, we have c- ( 1 - G((M,) ) ) G((M,) ) for sufficiently large $M$ .", "Thus, a first-passage distribution whose CCDF satisfies c M M G((M,) ) = 0 with some suitable $\\gamma (M,\\epsilon )$ will also allow satisfaction of equation (REF ).", "However, the satisfaction of equation (REF ) requires that $1/{\\bar{G}}\\left( \\gamma (M,\\epsilon ) \\right)$ be asymptotically supralinear in $M$ .", "We then note that since all first-passage times are non-negative random variables, the mean first-passage time is given by [50] c E[D] = 0 G(x) dx The integral of equation (REF ) exists iff $1/{\\bar{G}}(x)$ is asymptotically supralinear in $x$ .", "Thus, the existence of $E[D]$ in turn implies that choosing $\\gamma (M,\\epsilon ) = \\epsilon M$ allows satisfaction of equation (REF ).", "So, in the limit of vanishing $\\epsilon $ we then have c0 M M(M) + (M,) 0 1+ = and the energy requirement of equation (REF ) is met in the limit while assuring asymptotically independent sequential channel uses.", "The above development proves the following theorem: Theorem 2 Asymptotically Independent Sequential Channel Uses: Consider the channel use discipline depicted in FIGURE REF where tokens are emitted on an interval $[0,\\tau (M)]$ with $\\tau (M) = \\frac{M}{{\\lambda }}$ and guard intervals of duration $\\gamma (M,\\epsilon )$ are imposed between channel uses.", "If the mean first-passage time $E[D]$ is finite, then guard intervals can always be found such that the sequential channel uses approach asymptotic independence as $\\epsilon \\rightarrow 0$ , and the relative duration of the guard interval, $\\gamma (M,\\epsilon )$ vanishes compared to $\\tau (M)$ as $M \\rightarrow \\infty $ .", "Proof:  Theorem  REF  See the development leading to the statement of Theorem REF .", "$\\bullet $ Now, suppose the transport process from source to destination has infinite first passage time, implying that $1/{\\bar{G}}(x)$ is linear or sublinear in $x$ .", "Is asymptotically independent sequential channel use possible?", "The answer seems to be no.", "As a best case, the minimum probability of tokens arriving outside $\\tau (M) + \\gamma (M,\\epsilon )$ is obtained if all emissions occur at $t=0$ (see equation (REF )).", "Any other token emission distribution must have larger probability of interval overrun.", "For asymptotically independent sequential channel use we then must have, following equation (REF ) and equation (REF ), c M M G( M + (M,) ) = 0 We notice that the argument of ${\\bar{G}}(\\cdot )$ is at least linear in $M$ , and a linear-in-$M$ argument will not drive ${\\bar{G}}(\\cdot )$ to zero faster than $1/M$ because $1/{\\bar{G}}(\\cdot )$ is not supralinear.", "Thus, the argument of ${\\bar{G}}(\\cdot )$ must be supralinear in $M$ to drive ${\\bar{G}}\\left(\\frac{M}{\\lambda } + \\gamma (M,\\epsilon ) \\right)$ to zero faster than $1/M$ which in turn implies that $\\gamma (M,\\epsilon )$ must be supralinear in $M$ .", "However, if $\\gamma (M,\\epsilon )$ is supralinear in $M$ , then equation (REF ) cannot be satisfied and we have proved the following theorem: Theorem 3 Infinite Mean First Passage Does Not Allow Asymptotically Independent Sequential Channel Uses: Consider the channel use discipline depicted in FIGURE REF where tokens are emitted on an interval $[0,\\tau (M)]$ with $\\tau (M) = \\frac{M}{{\\lambda }}$ and guard intervals of duration $\\gamma (M,\\epsilon )$ are imposed between channel uses.", "If the mean first-passage time $E[D]$ is infinite, then guard intervals can never be found such that the sequential channel uses approach asymptotic independence as $\\epsilon \\rightarrow 0$ , and the relative duration of the guard interval, $\\gamma (M,\\epsilon )$ vanishes compared to $\\tau (M)$ as $M \\rightarrow \\infty $ .", "Proof:  Theorem  REF  See the development leading to the statement of Theorem REF .", "$\\bullet $ To summarize, if the mean first passage time exists, then asymptotically independent sequential channel uses are possible and the mutual information $I(\\vec{{\\bf S}};{\\bf T})$ is the proper measure of information transport through the channel.", "Conversely, if the mean first passage time is infinite, then asymptotically independent sequential channel uses are impossible and the associated channel capacity problem is ill-posed.", "It is worth noting that free-space diffusion (without drift) has infinite $E[D]$ .", "However, since all physical systems have finite extent, $E[D]$ is always finite for any realizable ergodic token transport process." ], [ "Channel Capacity Definitions", "The maximum $I(\\vec{{\\bf S}};{\\bf T})$ is the channel capacity in units of bits/nats per channel use.", "However, we will find it useful to define the maximum mutual information between ${\\bf T}$ and $\\vec{{\\bf S}}$ per token.", "That is, the channel capacity per token $C_q$ is c Cq(M, (M)) 1M fT() I(S;T) Since $\\tau (M)= M/{\\lambda }$ , it is easy to see that $C_q(M, \\tau (M))$ will be monotone increasing in $M$ since concatenation of two emission intervals with durations $\\tau (M/2)$ and $M/2$ tokens each is more constrained than a single interval of twice the duration $\\tau (M)$ with $M$ tokens.", "We can thus say that c Cq(M,(M)) 2 Cq(M/2, (M/2)) We can then define the limiting capacity in nats per token as c Cq M Cq(M,(M)) with no stipulation as yet to whether the limit exists or is bounded away from zero.", "Now consider the capacity per unit time.", "The duration of a channel use (or signaling epoch) is $\\tau (M) + \\gamma (M,\\epsilon )$ (see FIGURE REF ).", "Thus, for a given number $M$ of emissions per channel use and a probability $(1-\\epsilon )$ that all the tokens are received before the next channel use, we define the channel capacity in nats per unit time as rClCt(M,) fT() I(S;T)(M) + (M,) = Cq(M, (M)) (M(M) + (M,) ) which in the limits of $\\epsilon \\rightarrow 0$ and $M \\rightarrow \\infty $ becomes c0 M Ct(M,) = Cq via equation (REF ) and equation (REF ).", "The above development proves the following theorem: Theorem 4 Capacity of the identical-token timing channel: If the mean first-passage time $E[D]$ exists, then the channel capacity in nats per unit time obeys c Ct = Cq where $C_q$ is the capacity per token defined in equation (REF ) and ${\\lambda }$ is the average token emission rate.", "Proof:  Theorem  REF  See Theorem REF and the development leading to the statement of Theorem REF .", "$\\bullet $ It is worth noting that Theorem (REF ) is general and applies to any system with finite first-passage time.", "Now, we more carefully examine the mutual information $I(\\vec{{\\bf S}};{\\bf T})$ to determine whether the limits implied of equation (REF ) and equation (REF ) exist and are bounded away from zero." ], [ "Mutual Information Between Input ${\\bf T}$ and Output {{formula:1aa7fc4e-ce7f-4956-9be3-c8660fc877a7}}", "The mutual information between ${\\bf T}$ and ${\\bf S}$ is c I(S; T) = h(S) - h(S|T) = h(S) - M h(S|T) Since the $S_i$ given the $T_i$ are mutually independent each with density $g(s_i - t_i)$ , $h({\\bf S}|{\\bf T})$ does not depend on $f_{{\\bf T}}({\\bf t})$ .", "Thus, maximization of equation (REF ) is simply a maximization of $h({\\bf S})$ which is in turn maximized by maximizing the marginal $h(S)$ over the marginal $f_T(t)$ , a problem explicitly considered and solved in closed form for a mean $T_m$ constraint by Anantharam and Verdú in [42] and for a deadline constraint in [49], [48], both for exponential first-passage.", "The corresponding expression for the mutual information between ${\\bf T}$ and $\\vec{{\\bf S}}$ is c I(S; T) = h(S) - h(S|T) Unfortunately, $h(\\vec{{\\bf S}}|{\\bf T})$ now does depend on the input distribution and the maximization of $h(\\vec{{\\bf S}})$ is non-obvious.", "So, rather than attempting a brute force optimization of equation (REF ) by deriving order distributions $f_{\\vec{{\\bf S}}}(\\cdot )$ [39], we explore – with no loss of generality – simplifying symmetries.", "Consider that an emission vector ${\\bf t}$ and any of its permutations $P_n({\\bf t})$ produce statistically identical outputs $\\vec{{\\bf S}}$ owing to the reordering operation as depicted in FIGURES REF and  REF .", "Thus, any $f_{{\\bf T}}(\\cdot )$ which optimizes equation (REF ) can be “balanced” to form an optimizing input distribution which obeys c fT (t) = fT (Pn(t)) for $n=1,2,\\cdots ,M!$ and $P_n(\\cdot )$ the previously defined permutation operator (see equation ()).", "We can therefore restrict our search to hypersymmetric densities $f_{{\\bf T}}({\\bf t})$ as defined by equation (REF ).", "Now, hypersymmetric ${\\bf T}$ implies hypersymmetric ${\\bf S}$ which further implies that $f_{{\\bf S}}({\\bf s}) = f_{{\\bf S}}(P_k({\\bf s}))$ .", "The same non-zero corner and folding argument used in the derivation of equation () produces the following key theorem: Theorem 5 The entropy $h(\\vec{{\\bf S}})$ relative to the entropy $h({\\bf S})$ : If $f_{{\\bf T}}(\\cdot )$ is a hypersymmetric probability density function on emission times $\\lbrace T_m\\rbrace $ , $m=1,2,..,M$ , and the first-passage density $g(\\cdot )$ is non-singular, then the entropy of the time-ordered outputs $\\vec{{\\bf S}}$ is ch(S) = h(S) - M!", "Proof:  Theorem  REF  The hypersymmetry of $f_{{\\bf S}}({\\bf s})$ implies rClh(S) = -s M!", "fS(s) (M!", "fS(s) ) ds = -log M!", "-s M!", "fS(s) fS(s) ds = -log M!", "-s fS(s) fS(s) ds = -log M!", "+ h(S)  $\\bullet $ It is worth noting that hypersymmetric densities on ${\\bf T}$ are completely equivalent (from a mutual information maximization standpoint) to their “unbalanced” cousins.", "Remember that each and every $I(\\vec{{\\bf S}};{\\bf T})$ -maximizing $f_{{\\bf T}}(\\cdot )$ can be “balanced” and made into a hypersymmetric density without affecting the resulting value of $I(\\vec{{\\bf S}};{\\bf T})$ .", "Likewise, any hypersymmetric density has a corresponding ordered density that produces the same $I(\\vec{{\\bf S}};{\\bf T})$ .", "So, the assumption of hypersymmetric input densities is simply an analytic aid.", "Next we turn to $h(\\vec{{\\bf S}}|{\\bf T})$ .", "A zero-measure edge-folding argument on the conditional density is not easily applicable here, so we resort to some information-theoretic sleight of hand.", "As before we define $\\Omega $ as the permutation index number that produces an ordered output from ${\\bf S}$ so that $P_{\\Omega }({\\bf S}) = \\vec{{\\bf S}}$ .", "We first note the equivalence c { ,S} SThat is, specification of $\\lbrace \\Omega ,\\vec{{\\bf S}}\\rbrace $ specifies ${\\bf S}$ and vice versa because as in our derivation of $h(\\vec{{\\bf S}})$ , this equivalence requires that we exclude the zero-measure “edges” and “corners” of the density where two or more of the $s_i$ are equal.", "Thus, there is no ambiguity in the ${\\bf S}\\rightarrow \\vec{{\\bf S}}$ map.", "We then have, c h(S|T) = h(, S|T) = h(S|T) + H(|S, T) which also serves as an en passant definition for the entropy of a joint mixed distribution ($\\Omega $ is discrete while $\\vec{S}$ is continuous).", "We then rearrange equation (REF ) to prove a key theorem: Theorem 6 The Ordering Entropy, $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ : c h(S|T) = h(S|T) - H(|S, T) where $H(\\Omega |\\vec{{\\bf S}}, {\\bf T})$ , the ordering entropy, is the uncertainty about which $S_m$ corresponds to which $\\vec{S}_m$ given both ${\\bf T}$ and $\\vec{{\\bf S}}$ .", "Proof:  Theorem  REF  See equation (REF ).", "$\\bullet $ We note that c 0 H(|S, T) M!", "with equality on the right for any singular density, $f_{{\\bf T}}(\\cdot )$ , where all the $T_m$ are equal with probability 1.", "We can then, after assuming that $f_{{\\bf T}}(\\cdot )$ is hypersymmetric, write the ordered mutual information in an intuitively pleasing form: Theorem 7 The mutual information $I(\\vec{{\\bf S}};{\\bf T})$ relative to the mutual information $I({\\bf S};{\\bf T})$ : For a hypersymmetric density $f_{{\\bf T}}({\\bf t}) = f_{{\\bf T}}(P_k({\\bf t}))$ , $k=1,2, \\cdots , M!$ , the mutual information between launch times ${\\bf T}$ and ordered arrival times $\\vec{{\\bf S}}$ satisfies c I(S; T) = I(S;T) - ( M!", "- H(|S,T) ) Proof:  Theorem  REF  Combine Theorem REF and Theorem REF with equation (REF ).", "$\\bullet $ Put another way, an average information degradation of $\\log M!", "- H(\\Omega |\\vec{{\\bf S}},{\\bf T}) \\ge 0$ is introduced by the sorting operation, ${\\bf S}\\rightarrow \\vec{{\\bf S}}$ .", "Mutual information is convex in $f_{{\\bf T}}({\\bf t})$ and the space ${\\cal F}_{{\\bf T}}$ of feasible hypersymmetric $f_{{\\bf T}}({\\bf t})$ is convex.", "That is, for any two hypersymmetric probability functions $f_{{\\bf T}}^{(1)}$ and $f_{{\\bf T}}^{(2)}$ we have c fT(1)(t) + (1-) fT(2)(t) FT where $0 \\le \\kappa \\le 1$ .", "Thus, we can in principle apply variational [51] techniques to find that hypersymmetric $f_{{\\bf T}}(\\cdot )$ which attains the unique maximum of equation (REF ).", "However, in practice, direct application of this method leads to grossly infeasible $f_{{\\bf T}}(\\cdot )$ , implying that the optimizing $f_{{\\bf T}}(\\cdot )$ lies along some “edges” or in some “corners” of the convex search space." ], [ "An Analytic Bound for Ordering Entropy $H(\\Omega |\\vec{{\\bf S}}, {\\bf T})$", "The maximization of equation (REF ) hinges on specification of $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ , the ordering entropy given $\\vec{{\\bf S}}$ and ${\\bf T}$ .", "To determine analytic expressions for $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ , consider that given ${\\bf t}$ and $\\vec{{\\bf s}}$ , the probability that $\\vec{{\\bf s}}$ was produced by the $k^{\\mbox{th}}$ permutation of the underlying ${\\bf s}$ is c Prob(= k| s, t) = fS|T(Pk-1(s)| t) n=1M!", "fS|T(Pn(s)| t) where $\\vec{{\\bf s}}= P_{k}({\\bf s})$ .", "Some permutations will have zero probability (are inadmissible) since the specific $\\vec{{\\bf s}}$ and ${\\bf t}$ may render them impossible via the causality of $g(\\cdot )$ .", "Using equation (), the definition of entropy, and equation (REF ) we have rCl3lH(|s,t)     = - n=1M!", "[ g(Pn(s) - t)j=1M!g(Pj(s) - t) ] [ g(Pn(s) - t)j=1M!g(Pj(s) - t) ] and as might be imagined, equation (REF ) is difficult to work with in general.", "Nonetheless, let us define the number of nonzero terms in the sum of equation (REF ) as $|\\Omega |_{\\vec{{\\bf s}},{\\bf t}}$ .", "Now, consider that for exponential $g(\\cdot )$ , we can use equation () to write equation (REF ) as c Prob(= k| s, t) = u(Pk-1(s)- t) n=1M!", "u(Pn(s)- t) where ${\\bf u}(\\cdot )$ is a multidimensional unit step function.", "Equation (REF ) is a uniform probability mass function with $\\sum _{n=1}^{M!", "}{\\bf u}(P_n(\\vec{{\\bf s}})- {\\bf t}) = |\\Omega |_{\\vec{{\\bf s}},{\\bf t}}$ elements – the same as the number of non-zero terms in the sum of equation (REF ).", "Thus, c H(|s,t) n=1M!", "u(Pn(s)- t) for all possible causal first-passage time densities, $g(\\cdot )$ .", "In addition, it can be shown that exponential first-passage time is the only first-passage density which maximizes $H(\\Omega |\\vec{{\\bf s}},{\\bf t})$ , a result we state as a theorem: Theorem 8 A General Upper Bound for $H(\\Omega |\\vec{{\\bf s}},{\\bf t})$ : If we define the number of admissible combinations $\\lbrace P_n(\\vec{{\\bf s}}), {\\bf t}\\rbrace $ as c| |s,t n=1M!", "u(Pn(s)- t) where ${\\bf u}(\\cdot )$ is a multidimensional unit step function, then cH(|s,t) | |s,t with equality iff $g(\\cdot )$ is exponential.", "Proof:  Theorem  REF  We have already shown via equation (REF ) that exponential first passage renders $\\mbox{Prob}(\\Omega = k|\\vec{{\\bf s}}, {\\bf t})$ uniform.", "Now, consider that the probability mass function (PMF) of equation (REF ) can be written as cProb(= k| s, t) = g(Pk-1(s) - t)j=1M!g(Pj(s) - t) This PMF is uniform iff for all $n$ and $k$ where $P_n(\\vec{{\\bf s}})$ and $P_k(\\vec{{\\bf s}})$ are both causal with respect to ${\\bf t}$ we have c g(Pn(s) - t) = g(Pk(s) - t) That is, equation (REF ) must hold for all pairs $(P_n(\\vec{{\\bf s}}), {\\bf t})$ and $(P_k(\\vec{{\\bf s}}), {\\bf t})$ that are admissible.", "Since the maximum number of non-zero probability $\\Omega $ is exactly the cardinality of admissible $(P_n(\\vec{{\\bf s}}), {\\bf t})$ , any density which produces a uniform PMF over admissible $\\Omega $ thereby maximizes $H(\\Omega |\\vec{{\\bf s}},{\\bf t})$ , which proves the inequality.", "We then note that any given permutation of a list can be achieved by sequential pairwise swapping of elements.", "Thus, equation (REF ) is satisfied iff c g(x1 - t1)g(x2-t2) = g(x2 - t1)g(x1-t2) $\\forall $ admissible $\\lbrace (x_1,x_2)$ , $(t_1,t_2) \\rbrace $ .", "Rearranging equation (REF ) we have cg(x1-t1)g(x1-t2) = g(x2-t1)g(x2-t2) which implies that cg(x-t1)g(x-t2) = Constant w.r.t.", "$x$ Differentiation with respect to $x$ yields cg(x-t1)g(x-t2) - g(x-t1)g(x-t2)g2(x-t2) =0 which we rearrange to obtain cg(x-t1)g(x-t1) = g(x-t2)g(x-t2) which further implies that c g(x-t1)g(x-t1) = c since $t_1$ and $t_2$ are free variables.", "The only solution to equation (REF ) is cg(x) ecx Thus, exponential $g(\\cdot )$ is the only first-passage time density that can produce a maximum cardinality uniform distribution over $\\Omega $ given $\\vec{{\\bf s}}$ and ${\\bf t}$ – which completes the proof.", "$\\bullet $ Now consider that $\\left| \\Omega \\right|_{\\vec{{\\bf s}},{\\bf t}}$ , as defined in Theorem REF , is a hypersymmetric function of $\\vec{{\\bf s}}$ and ${\\bf t}$ and thus invariant under any permutation of its arguments $\\vec{{\\bf s}}$ or ${\\bf t}$ .", "That is, rCln=1M!", "u(Pn(s) - t) = n=1M!", "u(Pn(s)- t) = n=1M!", "u(Pn(s)- t) = n=1M!", "u(Pn(s)- t) because the summation is over all $M!$ permutations.", "Therefore, c | |s,t = | |s,t = | |s,t = | |s,t We must now enumerate this number of admissible permutations.", "Owing to equation (REF ) and Theorem REF we can assume time-ordered inputs $\\vec{{\\bf t}}$ with no loss of generality.", "So, let us define contiguous “bins” ${\\cal B}_k = \\lbrace t| t \\in [\\vec{t}_k, \\vec{t}_{k+1}) \\rbrace $ , $k=1,2,...,M$ ($\\vec{t}_{M+1} \\equiv \\infty $ ) and then define $\\sigma _m$ as bin occupancies.", "That is, $\\sigma _m = q$ if there are exactly $q$ arrivals in ${\\cal B}_m$ .", "The benefit of this approach is that the $\\sigma _m$ do not depend on whether $\\vec{{\\bf s}}$ or ${\\bf s}$ is used to count the arrivals.", "Thus, expectations can be taken over ${\\bf S}$ whose components are mutually independent given the ${\\bf t}$ and no order distributions for $\\vec{{\\bf S}}$ need be derived.", "To determine the random variable $\\left| \\Omega \\right|_{{\\bf S},\\vec{{\\bf t}}}$ we start by defining cm = j=1m j the total number of arrivals up to and including bin ${\\cal B}_m$ .", "Clearly $\\eta _m$ is monotonically increasing in $m$ with $\\eta _0 = 0$ and $\\eta _M = M$ .", "We then observe that the $\\sigma _m$ arrivals on $[\\vec{t}_m,\\vec{t}_{m+1})$ can be assigned to any of the $\\vec{t}_1, \\vec{t}_2, ..., \\vec{t}_m$ known emission times except for those $\\eta _{m-1}$ previously assigned.", "The number of possible new assignments is $(m-\\eta _{m-1})!/(m - \\eta _m)!$ which when applied iteratively leads to c | |S,t = m=1M (m - m-1)!", "(m - m)!", "= m=1M-1 (m + 1 - m) We then define the random variable cXi(m) = {ll 1 Si < tm+1 0 otherwise .", "for $i=1,2,...m$ .", "The PMF of $X_i^{(m)}$ is then cpXi(m)(x) = {ll G(tm+1 - ti) x = 1 G(tm+1 - ti) x = 0 .", "where we note that for a given $m$ , $X_i^{(m)}$ and $X_j^{(m)}$ are independent, $i \\ne j$ , and as previously defined, $G(\\cdot )$ is the CDF of the causal first-passage density $g(\\cdot )$ .", "$\\bar{G}(\\cdot ) = 1 - G(\\cdot )$ is the corresponding CCDF.", "We can then write cm = i=1m Xi(m) It is then convenient to define $\\bar{X}_i = 1 - X_i$ which allows us to define $\\bar{\\eta }_m = m -\\eta _m$ .", "We can then write c | |S,t = m=1M-1 (1 + m) Since we seek the expected value of equation (REF ), we can use equation (REF ) to calculate each individual $E_{{\\bf S}|\\vec{{\\bf t}}} [\\log (1 + \\bar{\\eta }_m) ]$ as c x (1 + i=1m xi) j=1m Gxj(tm+1 - tj) G1 -xj(tm+1 - tj) which allows us to define ${H^{\\uparrow }}({\\bf t})$ , an upper bound on $H(\\Omega |\\vec{{\\bf S}},{\\bf t})$ , as rClH(t) m=1M-1 x (1 + i=1m xi) j=1m Gxj(tm+1 - tj) G1 -xj(tm+1 - tj) where an ordering permutation on ${\\bf t}$ is part of the function ${H^{\\uparrow }}({\\bf t})$ .", "Equation (REF ) can be rearranged as rCl3lH(t) = =1M-1 (1 + )     m=M-1 |x| = j=1m Gxj(tm+1 - tj) G1 -xj(tm+1 - tj) We then note that crcl H(|S,T) ET [ ES|T [ H(|s, t) ] ] ET [ES|T [ | |s,t ] ] = ET [ES|T [ | |s,t ] ] follow from equation (REF ) in conjunction with Theorem REF and through hypersymmetric expectations (Theorem REF ) of hypersymmetric functions $\\left| \\Omega \\right|_{{\\bf s},\\vec{{\\bf t}}}$ (equation (REF )).", "Adding in the result of Theorem REF we have proven the following theorem: Theorem 9 A General and Computable Upper Bound for $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ : cE [ H(|s,t) ] H(|S,T) H(T) with equality iff the first-passage time density $g(\\cdot )$ is exponential.", "Proof:  Theorem  REF  See the development leading to the statement of Theorem REF .", "Theorem REF establishes equality iff the first passage density is exponential.", "$\\bullet $ Theorem REF gives us ${H^{\\uparrow }}({\\bf T})$ , a computable analytic upper bound for $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ , and an exact expression if the first-passage time is exponential." ], [ "Capacity Bounds For Timing Channels", "Despite significant effort, direct optimization of mutual information, $I(\\vec{{\\bf S}};{\\bf T})$ (see equation (REF )) remained elusive.", "The key issue is that $h({\\bf S})$ and $H(\\Omega |{\\bf T},\\vec{{\\bf S}})$ are “conflicting” quantities with respect to $f_{{\\bf T}}(\\cdot )$ .", "That is, independence of the $\\lbrace T_m \\rbrace $ favors larger $h({\\bf S})$ (i.e., $h({\\bf S}) \\le \\sum _m h(S_m)$ ) while tight correlation of the $\\lbrace T_m\\rbrace $ (as in $T_i=T_j$ , $i,j = 1,2,...,M$ ) produces the maximum $H(\\Omega |\\vec{{\\bf S}},{\\bf T}) = \\log M!$ .", "In light of these difficulties, we sought analytic expressions in the companion to this paper (Part II [48]) for $h({\\bf S})$ and $I({\\bf S};{\\bf T})$ which we restate here as Theorem REF and Theorem REF without proof.", "Theorem 10 Maximum $h(s)$ for exponential first passage under a deadline constraint: For first-passage time $D$ with density $f_D(d)= g(d) = {\\mu }e^{-{\\mu }d}$ , and launch time $T$ constrained to $[0,\\tau ]$ , the maximum entropy of $S=T+D$ is c fT() h(S) = ( e + ) The input density $f_T(\\cdot )$ which produces the maximum $h(S)$ is c rcl fT(t) = (t) 1e + + (t-) 1-ee+ + e+ (u(t) - u(t-)) Theorem 11 Maximum $I(S;T)$ For Exponential First Passage Under A Deadline Constraint: For first-passage time $D$ with density $f_D(d)= g(d) = {\\mu }^{-{\\mu }d}$ , and launch time $T$ constrained to $[0,\\tau ]$ , the maximum mutual information between $S=T+D$ and $T$ is c fT I(S;T) = (1 + e ) The definition of $C_q$ and $C_t$ in Theorem REF requires we consider the asymptotic value of $H(\\Omega |\\vec{{\\bf S}},{\\bf T})/M$ .", "A lower bound is provided in Part-II [48] assuming exponential first passage, a result we restate here as Theorem REF without proof.", "Theorem 12 Asymptotic $H(\\Omega |\\vec{{\\bf S}},{\\bf T})/M$ For Exponential First-Passage Under A Deadline Constraint ${\\bf T}\\in [{\\bf 0},{\\tau }]$ : For exponential first-passage with mean $1/{\\mu }$ , token launch intensity ${\\lambda }$ , and i.i.d.", "input distribution $f_{{\\bf T}}({\\bf t}) = \\prod _{m=1}^M f_T(t)$ where $f_T(\\cdot )$ maximizes $I(S;T)$ as in Theorem REF , the asymptotic ordering entropy per token is c M H(|S,T)M = k=2 e- kk!", "(k - 1) k!", "where ${\\rho }= {\\lambda }/{\\mu }$ is defined as a measure of system token “load” similar to a queueing system.", "We can rewrite the summation term in equation (REF ) more compactly noting that ck=1 kk!", "(k - 1) k!", "= =1 k= kk!", "(k - 1) Then ck= k 1k!", "= e - k=0- 1 k 1k!", "and ck= k k 1k!", "= k=-1 k 1k!", "can be used to obtain ck= kk!", "(k - 1) = 1(- 1)!", "-1 = 1!", "We then note that c p= e- !", "$\\ell = 0, 1, \\cdots , \\infty $ is a Poisson probability mass function and obtain the more compact c k=1 k (k/- 1) k!k!", "= 1 E[ ] Now turning toward capacity, equation (REF ) and Theorem REF are easily combined to show c1M I(S;T) - 1M M!", "(1 + e ) - 1MM!", "Then, since $\\tau = M/{\\lambda }$ we have c M 1M I(S;T) - 1M M!", "1 Noting that $I(\\vec{{\\bf S}};{\\bf T}) = I({\\bf S};{\\bf T}) - \\log M!", "+ H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ and $H(\\Omega |\\vec{{\\bf S}},{\\bf T})\\ \\ge 0$ proves the following theorem: Theorem 13 A Simple Lower Bound for $C_q$ under exponential first passage: Given a token launch intensity ${\\lambda }= M/\\tau $ and exponential first-passage time distribution with mean ${\\mu }^{-1}$ , the timing channel capacity $C_q({\\rho })$ in nats per token obeys c Cq() {-,0 } where ${\\rho }= \\frac{{\\lambda }}{{\\mu }}$ Proof:  Theorem  REF  See the development leading to the statement of Theorem REF .", "$\\bullet $ We can, however, combine equation (REF ), Theorem REF and equation (REF ) to obtain a better lower bound on capacity: Theorem 14 Lower Bound for $C_q$ and $C_t$ for Exponential First Passage: In the limit of large $M$ , with mean $1/{\\mu }$ exponential first-passage, the channel capacities, $C_q$ and $C_t$ must obey cCq() 1 + 1 E [ ] and cCt() ( 1 + 1 E [ ] ) where $\\ell $ is Poisson with PMF cp = e- !", "where ${\\rho }= {\\lambda }/{\\mu }$ .", "Proof:  Theorem  REF  Combine equation (REF ), Theorem REF and equation (REF ).", "$\\bullet $ Finally, from Part-II [48], [47] we have the following upper bound on $C_q$ (and the concomitant bound on $C_t = {\\lambda }C_q$ ) as: Theorem 15 Upper Bound for $C_q$ and $C_t$ for Exponential First Passage: If the first-passage density $f_D(\\cdot )$ is exponential with parameter ${\\mu }$ and the rate at which tokens are released is ${\\lambda }$ , then the capacity per token, $C_q$ is upper bounded by c Cq (1 + 4 ) and the capacity per unit time is upper bounded by c Ct (1 + 4 ) where ${\\rho }= \\frac{{\\lambda }}{{\\mu }}$ Proof:  Theorem  REF  Theorem 11 in Part-II [48] provides the bound for $C_q$ and application of Theorem REF provides the bound for $C_t$ .", "$\\bullet $ We have now concluded our treatment of the identical token timing channel.", "In the next sections we show how these results can be applied to molecular communication channels where tokens can carry information payloads." ], [ "Tokens With Payloads", "In sections REF through REF we developed all the machinery necessary to provide capacity bounds for channels with identical tokens where timing is the only means of information carriage.", "However, one can also imagine scenarios where the token itself carries information, much as a “packet” carries information over the Internet.", "That is, assume the token is a finite string of symbols over a finite alphabet.", "Having constructed tokens from these “building blocks,” a sender launches them into the channel and they are captured by a receiver.", "In this scenario a DNA sequence is a symbolic string drawn from a 4-character alphabet so that each nucleotide could carry 2 bits of information.", "Similarly, a protein sequence is a symbolic string drawn from a 20-character alphabet so that each amino acid could carry a little over 4 bits of information.", "Thus, a DNA token constructed from 100 nucleotides would carry 200 bits whereas a corresponding protein token would carry $>400$ bits.", "However, there are myriad other possibilities for coding information in structure.", "For example, a third major class of biological macromolecules, carbohydrates (polysaccharides), are linear and branched polymers constructed from a larger alphabet of monosaccharides.", "In addition to the composition information inherent in the makeup of a linear or non-linear concatenations of building blocks, one could imagine a layer of structural information as well [52] – as is the case with biological macromolecules where the spatiotemporal architecture of a polymer is as important as the order and frequencies of nucleotide, amino acid or monosaccharide residues in the sequence string.", "However, as the issue of “structural” information (the amount of information contained in an arbitrary 3-dimensional object) is as yet an open problem, we will not consider such constructions in detail.", "Nonetheless, the bounds we will derive are applicable to any method of information transfer wherein tokens carry information payload, either as string sequences or in some other structural way.", "So, for now consider only string tokens – as exemplified by DNA and protein sequences – where each token in the ensemble released by the sender carries a portion of the message.", "Thus, irrespective of their individual lengths, such “inscribed matter” tokens must be “strung together” to recover the original message, which implies that each token must be identifiable.", "Just as in human engineered systems like the Internet where information packets could arrive out of order, a sequence number could be appended to each packet to ensure proper reconstruction at the destination.", "Thus, given $M$ tokens per channel use, we could append $\\log M$ bits to each token.", "We will defer detailed discussion of this scenario until section  as this approach is asymptotically impractical with $\\log M$ tending toward $\\infty $ .", "Alternatively, one could employ gross differences to convey sequence information such as sending tokens of distinct lengths $1,2,\\cdots , K$ where $M= K(K+1)/2$ , and there may be other clever ways to embed structural side-information to establish token sequence.", "Nonetheless, the myriad possibilities notwithstanding, $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ provides the measure of essential token “overhead” or “side-information” (of any form) necessary to maintain proper sequence.", "Consider that operation of the timing channel involves construction of “blocks” $\\lbrace {\\bf t}_1, \\cdots ,{\\bf t}_N \\rbrace $ where each ${\\bf t}_n$ represents the launch schedule for $M$ tokens (a channel use).", "These blocks, “codewords” of blocklength $N$ , are launched into the channel.", "If capacity is not exceeded, the receiver can reliably recover the information embedded in the codewords and since we generally assume the receiver has access to the coding method, a correctly decoded message implies knowledge of the codewords $\\lbrace {\\bf t}_1, \\cdots , {\\bf t}_N \\rbrace $ .", "However, the channel imposes residual uncertainty about the mapping ${\\bf S}\\rightarrow \\vec{{\\bf S}}$ – the ambiguity about which $\\vec{s}_i$ is associated with which $s_j$ .", "For this reason, the payload-inscribed tokens cannot yet be correctly strung together to recover the message.", "However, given the observed arrivals $\\vec{{\\bf s}}$ and the correctly decoded ${\\bf t}$ , $H(\\Omega |\\vec{{\\bf s}}, {\\bf t})$ is the definition of the uncertainty about that ordering, $\\Omega $ .", "Likewise, the average uncertainty is $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ .", "Thus, the source coding theorem implies that at least $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ bits must be used, on average, to resolve the mapping ambiguity.", "So, consider a message ${\\bf P}$ to be carried as token payload that we break into equal size ordered submessages $p_m$ , $m=1,2,\\cdots , M$ .", "We can summarize the previous discussion as a theorem: Theorem 16 Sequencing Information for Tokens with Payload: If a message $P$ is broken into equal size “payload” submessages $\\lbrace p_m \\rbrace $ , $m=1,2,\\cdots , M$ and inscribed into otherwise identical tokens launched at times $\\lbrace t_m\\rbrace $ , we must provide, on average, additional “sequencing information” $\\frac{1}{M}H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ per token at the receiver to assure recovery of the full payload message ${\\bf P}= p_1p_2 \\cdots p_M$ .", "Proof:  Theorem  REF  Given arrivals $\\vec{{\\bf s}}$ and known departures ${\\bf t}$ , the uncertainty about the mapping between the $\\lbrace \\vec{s}_m \\rbrace $ and the $\\lbrace s_m\\rbrace $ (and thus the associated $\\lbrace t_m \\rbrace $ ) is exactly the ordering entropy $H(\\Omega |\\vec{{\\bf s}}, {\\bf t})$ .", "Considering $\\Omega $ as a letter from a random i.i.d.", "source, the source coding theorem [10], [53] requires at least $H(\\Omega |\\vec{{\\bf s}}, {\\bf t})$ bits on average to uniquely specify $\\Omega $ – or asymptotically over many channel uses, at least $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ bits on average.", "Therefore the information necessary at the receiver to recover the proper sequence and thence the message $P$ is greater than or equal to $\\frac{1}{M}H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ .", "$\\bullet $ It is important to note that we have not actually provided a method for message reconstruction, only a lower bound on the amount of “side information” necessary at the receiver to assure proper reconstruction.", "However, as a practical matter, the quantity $\\frac{1}{M}H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ does provide some guidance.", "In the worst case where the order of token arrival is completely random, $H(\\Omega |\\vec{{\\bf S}},{\\bf T}) = \\log M!$ which amounts to each packet carrying a header of size $\\frac{1}{M} \\log M!", "\\approx \\log M$ for large $M$ – essentially numbering the packets from 1 to $M$ .", "If $\\frac{1}{M}H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ is much smaller, then one could imagine cyclic packet numbers since smaller $\\frac{1}{M}H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ implies that packets launched far apart in time are unlikely to arrive out of order.", "The sequence header could then be commensurately smaller.", "In either case, the total amount information necessary to resolve the ordering $\\Omega $ is $\\frac{1}{M}H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ on average." ], [ "Energy Costs", "System energy is a critical resource which limits capacity in all communication systems.", "In the case of molecular communication, there are a variety of potential costs, most notably manufacture, launch and transport.", "So, assume the minimum cost of fabricating a token without a payload is $c_0$ Joules and with a payload $c_{1}$ Joules.", "Symbolic string tokens incur a “per character” cost which we define as $\\Delta c_{1}$ per character per token.", "For example, adding a nucleotide to double-stranded DNA requires 2 ATP ($1.6 \\times 10^{-19}$ J) while adding an amino acid to a protein requires 4 ATP ($3.2 \\times 10^{-19}$ J) [54].", "Apart from the per residue per token cost, there may be other energy involved in sequestration, release and/or token transport across a gap.", "However, the key assumption is constant energy use per token.", "Without considering the details as in [55] we will denote the combination of these and any other relevant energies as $c_e$ Joules per token.", "Thus, our power for the timing only channel is c PT = (c0 + ce) and for the timing plus payload channel, c PT+P ( c1 + ce + ( H(T)b+ K ) c1 ) where $K$ is the string length of information-laden tokens, and $b$ is the alphabet size used to construct the strings.", "For amino acids (alphabet) are used to construct proteins (strings), and in general, monomers are used to construct oligomers.", "The inequality in equation (REF ) results from the fact that ${H^{\\uparrow }}({\\bf T})$ is an upper bound on the ordering entropy $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ over all possible first-passage distributions (Theorem REF ).", "We also note that information could be carried only with payload (and not timing).", "The power budget would be identical to that of equation (REF ) except that ${H^{\\uparrow }}({\\bf T})$ would be replaced by $\\lim _{M\\rightarrow \\infty } \\min _{{\\bf t}} H(\\Omega |\\vec{{\\bf S}},{\\bf t})/M$ .", "However, since $H(\\Omega |\\vec{{\\bf S}},{\\bf T}) \\ge \\min _{{\\bf t}} H(\\Omega |\\vec{{\\bf S}},{\\bf t})$ , equation (REF ) provides an upper bound for the payload-only power as well." ], [ "Results", "We can now define the capacities for the token timing (only), token timing plus token payload and token payload (only) channels as follows: c CT = Cq() c CT+P = ( Cq() + K b ) and cCP = K b = CT+P - CT Figure: Lower bound (dashed line: Theorem ) and upper bound (solid line:Theorem ) for the token timing channel capacity C t C_t (in nats per passage time1/μ1/{\\mu }) as a function of channel load ρ{\\rho }, the ratio of the token launch rate λ{\\lambda }to the token uptake rate μ{\\mu }.In FIGURE REF we use Theorem REF and Theorem REF to plot lower and upper bounds for ${\\cal C}_T$ versus ${\\rho }$ , a proxy for power budget, ${\\cal P}$ assuming some unit cost per token ($c_0+c_e = 1$ ).", "It is important to note that first-passage time variance (jitter) produces disordered tokens.", "That is, the mean first-passage time is only a measure of channel latency – the “propagation delay” so to speak – and does not itself impact token order uncertainty.", "However, for exponential first-passage the standard deviation also happens to be the first passage time $1/{\\mu }$ .", "At small values of ${\\rho }$ the bounds are tight.", "At larger ${\\rho }$ the bounds diverge and the upper bound offers the tantalizing hint that timing channel capacity increases with increased token load ${\\rho }$ (see also [8], [9]).", "Unfortunately, we have as yet been unable to find an empirical density $f_{{\\bf T}}({\\bf t})$ which displays capacity growth similar to the upper bound and suspect that timing capacity flattens with increasing ${\\rho }$ owing to a more rapidly increasing probability of token confusion at the output.", "Figure: Lower bounds for the capacities of the token timing (ρ𝒞 T {\\rho }{\\cal C}_T)and token timing plus token payload (ρ𝒞 P+T {\\rho }{\\cal C}_{P+T}) channels as a function of powerbudget (𝒫 T {\\cal P}_T and 𝒫 P+T {\\cal P}_{P+T}) for DNA string tokens with exponential first-passagetimes (Theorem ).", "Capacity is in units bits per first-passage time 1/μ1/{\\mu }.", "Power isin units of 2-ATP (1.6×10 -19 1.6 \\times 10^{-19} J) per passage time 1/μ1/{\\mu } and a nucleotide residueis assumed to carry 2-bits of information.", "Solid lines: aggregate capacity of n=1,2,4n=1,2,4 separate(independent or parallel) token timing channels where DNA string tokens carry no informationpayload.", "Dashed/Dotted lines: aggregate capacity of token timing plus token payload channels forDNA string tokens of different lengths, K=1,2,4K=1,2,4-residue payloads.In FIGURE REF , we use Theorem REF to plot lower bounds in bits per first-passage time, $1/{\\mu }$ , as a function of power budget ${\\cal P}$ assuming DNA-based tokens.", "For token timing plus token payload signaling we show plots for $K=1,2,4$ DNA-residue tokens.", "For timing-only signaling we also include plots where different identifiable tokens (different molecule types or physically separate channels) are used (i.e., $n=1,2,4$ parallel timing channels as shown) for comparison with payload channels.", "We have assumed costs $c_0 =\\Delta c_1 = 2$ ATP.", "Furthermore, we assume $c_1 = c_e = \\Delta c_1$ since it seems likely that the absolute minimum energy for token release, $c_e$ , in a purely diffusive channel is probably comparable to the cost of creating (or breaking) the covalent bond used to append a nucleotide residue.", "If we assume $1/{\\mu }= 1$ ms, then the ordinate of FIGURE REF is in kbit/s and the abscissa is in units of $1.6 \\times 10^{-16}$ W. If $1/{\\mu }= {\\mu }s$ , (as might be the case for smaller gaps in a nano-system) the ordinate is in $Mbit/s$ and the abscissa is in units of $1.6\\times 10^{-13}$ W. These data rates are many many orders of magnitude larger than the fractional bit/second data rates previously reported for simple demonstrations of molecule communication [56], and the predicted power efficiencies are startling.", "Comparison of our results to [56] and others would be relatively straightforward if passage time jitter for the experimental setup were provided, although in [56] Avogradrian numbers of molecules were release with each alcohol “puff” so precise timing at the molecular level was not attempted.", "Finally, it is worth noting that increasing the rate at which tokens with payload are launched will increase the bit rate but not increase the required energy per bit.", "Of particular note, at low power, timing-only signaling provides the best rates while at higher power, inscribed matter tokens may be preferred.", "However, if it is difficult to synthesize long strings (heavily information-laden tokens), even a single bit of information (two distinguishable species used in parallel) markedly increases capacity." ], [ "Discussion & Conclusion", "We have provided a general and fundamental mathematical framework for molecular channels and derived some associated capacity bounds.", "We now discuss the results in the context of selected prior work and also touch upon ideas for further work suggested by the results.", "We separate these into two tranches: Engineering Implications where we consider how molecular communication can be extended to other known communication scenarios as well where we might look for inspiration from biology that has had eons to evolve solutions.", "Biological Implications where we consider how the results might impact/support known biology and suggest new avenues for investigation." ], [ "Engineering Implications", "Capacity Bounds and Coding Methods: Our upper bound on capacity $C_t$ , the timing capacity for identical tokens, is tight for low token load ${\\rho }$ but diverges for large ${\\rho }$ .", "However, no empirical distributions with rates higher than the lower bound have yet been found.", "So, does the capacity of the timing-only channel truly flatten with increasing ${\\rho }$ as in FIGURES REF and REF , or is there a benefit to increasing the intensity of timing-token release as suggested in [8], [9]?", "In addition, since exponential first-passage is not the worse case corruption, what is the minmax capacity of the molecular timing channel?", "Likewise, how much better than exponential might be other first-passage densities imposed by various physical channels, and what are good codes for reliable transmission of information over molecular channels?", "While we have focused on tokens in the form of linear symbolic strings, DNA and protein sequences in particular, what benefits might string tokens with a branched structure, exemplified by carbohydrates, confer for sequencing and/or payload?", "Should we vigorously pursue technology to produce large payload (many residue) tokens [57], or should a pool of smaller pre-fabricated payloads be used to deliver information?", "The bunching seen in FIGURE REF for payload tokens with increasing $K$ may suggest the latter when rapid token construction is difficult.", "That is, the capacity per power output does not scale linearly in $K$ owing to the increased power required by adding more bases to tokens.", "This implies a tradeoff between timing-only and increasingly larger payloads – completely aside from the fact that payload size, shape and composition can have an effect on transport properties.", "Precise Timing, Fuzzy Timing and Concentration: Of particular importance is establishing a careful quantitative relationship between our finest-grain timing model and other less temporally precise ones [37], [38], [1], [2], [3], [39], [4], [5], [6], [7], [40].", "To begin, consider that our model seems to imply infinitely precise control over the release times ${\\bf T}$ and infinite precision measurement of the arrival times $\\vec{{\\bf S}}$ .", "However, release time and measurement time imprecision are both easily incorporated into the transit time vector ${\\bf D}$ .", "Thus, application of our model to the “fuzzier” release and detection times associated with practical/real systems is straightforward.", "Put another way, first passage time jitter already imposes limits on timing precision.", "Thus, so long as timing precision is significantly better than passage time jitter, the bounds presented here will be moderately tight.", "In addition, we are hopeful that the upper bound of Theorem Theorem REF will be useful for evaluating molecular timing channel capacity for arbitrary first passage time distributions since it requires only knowledge of the timing channel capacity coupled to average properties of the corresponding input distribution.", "Concentration is derived from considering temporal windows and counting the arrivals within them.", "Therefore, via the data processing theorem, our precise timing model must undergird all concentration-based methods which, even with perfect concentration detection, cannot possibly exceed the capacity of the finest grain timing model presented here.", "Of particular note for the asymptotic nature of our analysis here, an individual emission schedule ${\\bf t}$ for large $M$ is exactly a temporal emission concentration sequence as time resolution coarsens.", "That said, we have not as yet tried to show a graceful degradation toward coarse timing concentration from precise timing.", "Regardless, the results here provide crisp bounds on the capacities derived from concentration-based models.", "As a specific example, there are channel models where information is carried by the number of molecules released and received (most recently, see [9], [58]).", "In this case, the capacity per channel use, $C_N$ , is upper bounded by $\\log (M+1)$ since between 0 and $M$ tokens can be released during a symbol interval of duration $\\tau (M)$ .", "Smaller $\\tau (M)$ increases the capacity in bits per second.", "Larger $M$ increases the capacity in bits per channel use.", "If we assume a fixed signaling interval $\\tau (M)$ during which $m=0,1, \\cdots M$ tokens are emitted, then we can also fix the average token rate ${\\lambda }$ as a proxy for power.", "Assuming a uniform distribution on the number of tokens sent we then have c(M) = M2 since the average number of tokens released is $M/2$ .", "Assuming exponential first passage, the probability that all tokens arrive by $\\tau (M)$ is minimized when all tokens are launched at $t=0$ .", "For exponential first passage and with arrival probability criterion $1-\\epsilon $ as in section REF we have c(M) = - 1 ( 1- (1-)1M ) which assures that even when $M$ tokens are emitted, they will all arrive before $\\tau (M)$ with probability $1-\\epsilon $ .", "However, equation (REF ) in combination with the power limit of equation (REF ) sets ${\\lambda }$ to c(M) = -M2 ( 1- (1-)1M ) Then, since $C_N \\le \\log (M+1)$ , after setting a successful channel use criterion of $1-\\epsilon $ we have cCN(M)(M) - (M+1)( 1- (1-)1M ) which we rewrite as cCN(M)(M) - (M+1)( 1- (1-)1M ) with normalized power constraint cP(M) = (M) = -M2 ( 1- (1-)1M ) However, except for small $\\epsilon $ , $\\frac{C_N(M)}{{\\mu }\\tau (M)}$ is not a reliable indicator of capacity since with increased $\\epsilon $ , $\\tau (M)$ decreases but the probability of intersymbol interference (ISI) increases.", "Since calculating the capacity of this channel with ISI is difficult, we roughly approximate by normalizing $\\frac{C_N(M)}{{\\mu }\\tau (M)}$ by the expected number of intervals over which a given emission burst of $M$ tokens will span (thereby corrupting potentially them).", "Noting that $(1-\\epsilon ^z)^M$ is the probability that all tokens arrive before the end of the $z+1^{\\mbox{st}}$ interval after emission we have rClz(M) = z=0(1 - (1 - z)M) = -n=1M M ()n (-1)n1-n and we obtain rClCN(M) CN(M)z(M) (M) ( M+1 ) - z(M) ( 1- (1-)1M ) as an approximation to capacity for the number/concentration channel.", "In FIGURE REF we plot the upper bound of $\\tilde{C}_N(M)$ in equation (REF ) versus ${\\cal P}(M)$ (i.e., parametrized in $M$ ) for a range of $\\epsilon $ as compared to the timing channel lower bound in FIGURE REF .", "The timing channel capacity lower bound is always significantly greater than $\\tilde{C}_N(M)$ .", "Nonetheless, the coding simplicity of the number/concentration channel could make it an attractive option.", "Figure: Lower bound for ρ𝒞 t \\rho {\\cal C}_t versus tokens per passage (dashed line) compared to C ˜ N \\tilde{C}_Nvs.", "tokens per passage for different values of ϵ\\epsilon as shown.", "Token construction and emission areassumed unit energy for both the timing channel and the number/concentration channel.Identifiable Tokens Without Payload: In section REF we mentioned the possibility of uniquely identifying each of $M$ emitted tokens with a sequence number of length $\\log M$ bits.", "We treat this scenario as distinct from ensemble timing channel coding which resolves residual orderering ambiguity (see section REF ) because if the tokens are individually identifiable, the potential emission schedules are not constrained to ensemble timing channel coding.", "Thus, the $M$ identifiable tokens constitute $M$ parallel single-token timing channels, which for exponential first passage have aggregate capacity $M \\log (1 + \\frac{M}{{\\rho }e})$ .", "However, ${\\rho }$ is limited by the power budget ${\\cal P}$ (in units of energy per passage, $1/{\\mu }$ ) cM P because each token requires $\\log M$ bits of sequencing information energy.", "Following REF we have ${\\lambda }\\tau (M)= M$ so the capacity in nats per passage time is cC = ( 1 + Me ) ( 1 + ePe ) the last inequality owed to equation (REF ).", "However, in the limit of $M \\rightarrow \\infty $ we have ${\\rho }\\rightarrow 0$ so we have c0 C = P in units of nats per passage time (and assuming unit per-bit cost of the token identfier string).", "Thus, the identifiable token timing channel capacity exceeds the identical token timing channel lower bound with increasing power budget and scales linearly in power as does the timing channel upper bound (see FIGURE REF ) – while still lying below it.", "Token Corruption and Receptor Noise: Certain channel properties we have so far ignored must also be studied, such as the potential for lost or corrupted tokens and potential binding noise at receptor sites.", "However, as previously stated, token erasure (tokens which do not arrive) or payload token corruption (tokens which are altered in passage) or receptor noise (tokens bind stochastically to the receptor) can only decrease capacity (via the data processing theorem).", "Thus, the results here provide upper bounds.", "Nonetheless it is worth considering how the analytic machinery developed might be modified to take into account such impediments.", "First, consider alteration of payload-carrying tokens en route.", "If the corruption is i.i.d.", "for each token, then the usual error correcting coding methods can be applied individually, or to the token ensemble.", "The resulting overall channel capacity will be degraded by the coding overhead necessary to preserve payload message integrity (including the sequencing headers).", "Then consider token erasure where a token never arrives (and is assume to not arrive in a later signaling interval).", "Since each signaling interval uses $M$ tokens, we will know whether tokens get “lost” in transit and can arbitrarily assign a faux arrival time to such tokens.", "However, the problem this poses for our analysis is two-fold.", "First, tokens released later in the signaling interval are more likely to be lost which implies that the first passage density is not identical for each token.", "Second, the first passage density for each token would then contain a singularity equal to the probability of loss, violating one of our key assumptions and making hypersymmetric probability density folding arguments invalid.", "That said, an erasure channel approach where tokens were deleted randomly from the output could be pursued and owing to its i.i.d.", "nature (with respect to which tokens were erased) would likely provide a worst case scenario, since the information associated with erasures being more likely to be derived from a later release would be absent.", "Finally, we have previously mentioned that a token may “arrive” multiple times owing to receptor binding kinetics.", "It was previously shown that given the first binding (first-passage) time the information content of subsequent bindings by the same token is nil [39].", "In addition, as shown in section REF , information-theoretically patent channel use requires that tokens from a given emission interval be eventually cleared at the receiver.", "Otherwise, lingering tokens can interfere with subsequent emission intervals.", "So, one could imagine that the rebinding process results in a characteristic finite-mean “burst” of arrivals associated with a given token which could perhaps be resolved into a single first-arrival time estimate – effectively adding more jitter to the first-passage time $D$ .", "If so, our model applies directly with appropriate modification.", "However, we have not attempted to analyze this scenario nor quantify the associated estimation noise.", "Thus, our results are most appropriately applied to systems where ligands bind tightly or where tokens are removed with high probability after first capture/detection.", "The addition of noisy ligand binding can only depress the capacity bounds presented here and we leave the question of exactly how much to future work.", "Interference and Multiple Users: Multi-user communication in a molecular setting is a critical question, and a better understanding of the single-user channel will certainly help with multi-user studies where transmissions interfere.", "There is some work to inspire an information-theoretic edifice [59] similar to how the current work builds on [42], but the multi-user molecular signaling problem has not yet been rigorously considered.", "Of particular interest would be a version of MIMO since FIGURE REF shows capacity benefits to parallel channels.", "One could imagine apposed arrays of emitters and receivers which could be engineered to collaborate to encode and decode information in a variety of ways, from parallel non-interfering channels to grossly interfering channels where joint/distributed coding might be employed.", "One could even imagine channels with chemically reactive species in which emitted tokens elicited spatially structured propagation of detectable reaction products [60], [61], [62].", "Other Applications: It is interesting to note that although our work is couched in terms of molecular communication, the notion of token inscription applies to any system where discrete emissions experience random transport delay between sender and receiver.", "The most obvious example is the Internet where packets experience variable delay and may arrive out of order.", "Our results provide crisp bounds on the amount of sequencing overhead necessary for proper message reconstruction and even suggest that (at least for low payload packets traveling over independent routes) timing information could be an interesting adjunct to payload, depending upon the amount of timing jitter between the source and destination.", "There is also the potential for cross pollination from biology to communication systems.", "As a simple example, there may be some selective benefit to hiding information from competitors.", "So, perhaps biological systems, where signaling chemicals are often detectable by other organisms, convey secrets over molecular communication channels in ways that can be mimicked in engineered systems.", "An obvious application, biosteganography [63], comes to mind in the context of tokens with payloads, although in such schemes timing plays no role as yet." ], [ "Biological Implications", "Clearly, the natural world clearly offers a dizzying array of processes and phenomena through which the same and different tasks, communication or otherwise, might be accomplished (see, for example, [64], [65], [66], [67], [68], [69]).", "It is no wonder therefore that communication theorists have plied their trade heavily in this scientific domain (for a relatively recent review, see [70]).", "Identifying the underlying mechanisms (signaling modality, signaling agent, signal transport, and so on) as well as the molecules and structures implementing the mechanisms is no small undertaking.", "Consequently, experimental biologists use a combination of prior knowledge and what can only be called instinct to choose those systems on which to expend effort.", "Guidance may be sought from evolutionary developmental biology – a field that compares the developmental processes of different organisms to determine their ancestral relationship and to discover how developmental processes evolved.", "Insights may be gained by using statistical machine learning techniques to analyze heterogeneous data such as the biomedical literature and the output of so-called “omics” technologies – genomics (genes, regulatory, and non-coding sequences), transcriptomics (RNA and gene expression), proteomics (protein expression), metabolomics (metabolites and metabolic networks), pharmacogenomics (how genetics affects hosts' responses to drugs), physiomics (physiological dynamics and functions of whole organisms), and so on.", "Frequently, the application of communication theory to biology starts by selecting a candidate system whose components and operations have been already elucidated to varying degrees using methods in the experimental and/or computational biology toolbox [71], [72] and then applying communication and/or information theoretic methods [70], [73], [74], [75], [69], [76], [77], [78].", "However, we believe that communication theory in general and information theory in particular are not mere system analysis tools for biology but new lenses on the natural world [79].", "Here, we have sought to demonstrate the potential of communication theory as an organizing principle for biology.", "That is, given energy constraints and some general physics of a problem, an information-theoretic treatment can be used to provide outer bounds on information transfer in a mechanism-blind manner.", "Thus, rather than simply elucidating and quantifying known biology, communication theory can winnow the plethora of possibilities (or even suggest new ones) amenable to experimental and computational pursuit.", "Likewise, general application of communication-theoretic principles to biology affords a new set of application areas for communication theorists.", "Examples of the implications of our main communication theoretic results are as follows: We have derived a model and methodology for determining the amount of information a system using chemical signaling can convey under given power constraints.", "Do (or How do) biological systems achieve this extremely – even outrageously – low value?", "Using tokens with large payloads can be very efficient.", "Is one example of this the transmission of hereditary material such as a genome over evolutionary time scales (periods spanning the history of groups and species)?", "If it is difficult to synthesize long strings (information-laden tokens), even a single bit of information (two distinguishable species) increases capacity.", "Is the transmission of hormones, semiochemicals and other small molecules over developmental time scales (periods spanning the life of individuals) indicative of some efficiency tradeoff?", "Although the production rate of tokens can be increased, the channel capacity might not increase if tokens are emitted at intervals smaller than the variability of arrival time.", "Does the material through which tokens travel hold the key to addressing this and the aforementioned questions and problems about engineered systems – particularly the issues of concentration versus timing, token corruption and interference and multiple users?", "In biological systems, discrete particles propagate through a cornucopia of substances en route from the source to the target: solids, liquids, gases or plasmas in the biosphere, lithosphere, atmosphere, hydrosphere, interstellar space or intergalactic space – for instance cytoplasm, nucleoplasm, mitochondrial matrix, extracellular matrix, extracellular polymeric substances, blood, lymph, phloem, xylem, bones, soil, rocks, air, water, steam, ice, and molecular clouds.", "Clearly, the “propagation delay” (jitter) experienced by a particular category and type of discrete particle is a function of the intrinsic physicochemical properties of the material.", "The standard deviation of arrival times will depend also on environmental factors such as temperature, pH, pressure and light.", "Our results could guide studies aimed at answering three key questions biologists ask of a living organism: How does it work?, How is it built?, and How did it get that way?", "[80].", "This is because our models of token timing, payload and timing+payload channels are inspired, at least in part, by fundamental “systems” problems about the dynamic and reciprocal relationship(s) between individuals in multicellular systems – whether microrganisms in communities or cells in metazoan tissues.", "Inscribed matter communication is a keystone of how individuals learn what to become or to be by a combination of internal and external cues and how, in turn, they teach others when to change or remain the same.", "Thus, the seemingly esoteric theoretical studies of channel capacities described here and discussed further in our companion paper might help pave the way to elucidating the origins (evolutionary developmental biology), generation (embryogenesis, and morphogenesis), maintenance (homeostasis, tolerance, and resilience), subversion (infectious and chronic diseases such as cancer and immune disorders), and decline (aging) of complex biological form and function [79].", "A key virtue of the token timing model is its implicit acknowledgment of the importance of the “transmission medium” in the spatial gap between sender and receiver and through which tokens move.", "Consider molecular inscribed matter communication from the microscopic to the macroscopic levels: within and between cells in tissues and organisms in (agro)ecosystems.", "Since the presence of obstacles influences the mobility of discrete particles through a material, a crowded environment will increase the mean arrival time relative to unhindered diffusion but should have less of an impact on the mean emission time.", "Decades of laboratory in vitro studies have promulgated the view of the cellular interior as a place proteins, nucleic acids, carbohydrates, lipids and other molecules exist as highly purified entities that act in isolation, diffusing more or less freely until they find their cognate binding partner.", "In its natural milieu however, a molecule lives and operates in an extremely structured, complex and confining environment: one where it is surrounded by other molecules of the same or different chemical nature, the bystanders in the crowd having positive or negative effects on its mobility, biochemistry and cell biology [81], [82].", "Widening the spatial and temporal horizon, semiochemicals diffusing through soil, water, and air mediate the complex ways crops, livestock, and microbes interact with one another.", "Whether molecule release and capture occurs among organisms in the above- and below-ground environments or between cells in the tissue microenvironment, the basic physics is similar.", "For this reason we feel that our fundamental treatment of inscribed matter communication presented here could help guide biological understanding and experimentation." ], [ "Acknowledgments", "Profound thanks are owed to A. Eckford, N. Farsad, S. Verdú and V. Poor for useful discussions and guidance.", "We are also extremely grateful to the editorial staff and the raft of especially careful and helpful anonymous reviewers.", "This work was supported in part by NSF Grant CDI-0835592." ] ]
1606.05023
[ [ "Crofton Formulae for Tensor-Valued Curvature Measures" ], [ "Abstract The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies.", "We prove a set of Crofton formulae for such tensorial curvature measures.", "These formulae express the integral mean of the tensorial curvature measures of the intersection of a given convex body with a uniform affine $k$-flat in terms of linear combinations of tensorial curvature measures of the given convex body.", "Here we first focus on the case where the tensorial curvature measures of the intersection of the given body with an affine flat is defined with respect to the affine flat as its ambient space.", "From these formulae we then deduce some new and also recover known special cases.", "In particular, we substantially simplify some of the constants that were obained in previous work on Minkowski tensors.", "In a second step, we explain how the results can be extended to the case where the tensorial curvature measure of the intersection of the given body with an affine flat is determined with respect to the ambient Euclidean space." ], [ "Introduction", "The classical Crofton formula is a major result in integral geometry.", "Its name originates from works of the Irish mathematician Morgan W. Crofton [4] on integral geometry in ${\\mathbb {R}}^{2}$ in the late 19th century.", "For a convex body $K$ (a non-empty, convex and compact set) in the $n$ -dimensional Euclidean space ${\\mathbb {R}}^{n}$ , $n \\in {\\mathbb {N}}$ , the classical Crofton formula (see [19]) states that $ \\int _{A(n, k)} V_{j}(K \\cap E) \\, \\mu _{k} (\\mathrm {d}E) = \\alpha _{n j k} V_{n - k + j}(K),$ for $k \\in \\lbrace 0, \\ldots , n \\rbrace $ and $j \\in \\lbrace 0, \\ldots , k \\rbrace $ , where $A(n, k)$ is the affine Grassmannian of $k$ -flats in ${\\mathbb {R}}^{n}$ , $\\mu _{k}$ denotes the motion invariant Haar measure on $A(n,k)$ , normalized as in [20], and $\\alpha _{n j k} >0$ is an explicitly known constant.", "Let $ {\\mathcal {K}}^{n} $ denote the set of convex bodies in ${\\mathbb {R}}^n$ .", "The functionals $V_{i} :{\\mathcal {K}}^n\\rightarrow {\\mathbb {R}}$ , for $i\\in \\lbrace 0,\\ldots ,n\\rbrace $ , appearing in (REF ), are the intrinsic volumes, which occur as the coefficients of the monomials in the Steiner formula $ V_{n} (K + \\epsilon B^{n})= \\sum _{j = 0}^{n} \\kappa _{n - j} V_{j} (K) \\epsilon ^{n - j},$ for a convex body $K \\in {\\mathcal {K}}^{n}$ and $\\epsilon \\ge 0$ (cf.", "[14]); here, as usual, $+$ denotes the Minkowski addition in ${\\mathbb {R}}^{n}$ and $\\kappa _{n}$ is the volume of the Euclidean unit ball $B^{n}$ in ${\\mathbb {R}}^{n}$ .", "Properties of the $V_i$ such as continuity, isometry invariance and additivity are derived from corresponding properties of the volume.", "A key result for the intrinsic volumes is Hadwiger's characterization theorem (see [7] and [14] ), which states that $V_{0}, \\ldots , V_{n}$ form a basis of the vector space of continuous and isometry invariant real-valued valuations on ${\\mathcal {K}}^{n}$ .", "A natural way to extend the Crofton formula is to apply the integration over the affine Grassmannian $A(n, k)$ to functionals which generalize the intrinsic volumes.", "One of these generalizations concerns the class of continuous and isometry covariant ${p}$ -valued valuations on ${\\mathcal {K}}^{n}$ , where ${p}$ denotes the vector space of symmetric tensors of rank $p \\in {\\mathbb {N}}_{0}$ over ${\\mathbb {R}}^n$ .", "The 0-valued valuations are simply the well-known and extensively studied intrinsic volumes.", "For the 1-valued (i.e.", "vector-valued) valuations, Hadwiger and Schneider [8] proved in 1971 a characterization theorem similar to the aforementioned real-valued case due to Hadwiger.", "In addition, they also established integral geometric formulae, including a Crofton formula [8].", "In 1997, McMullen [16] initiated a systematic investigation of this class of ${p}$ -valued valuations for general $p \\in {\\mathbb {N}}_0$ .", "Only two years later Alesker generalized Hadwiger's characterization theorem (see [2] and [14]) by showing that the vector space of continuous and isometry covariant ${p}$ -valued valuations on ${\\mathcal {K}}^{n}$ is spanned by the tensor-valued versions of the intrinsic volumes, the Minkowski tensors $\\Phi _{j}^{r,s}$ , where $j, r, s \\in {\\mathbb {N}}_{0}$ and $j < n$ , multiplied with suitable powers of the metric tensor in ${\\mathbb {R}}^{n}$ .", "In 2008, Hug, Schneider and Schuster proved a set of Crofton formulae for these Minkowski tensors (see [12]).", "Localizations of the intrinsic volumes yield other types of generalizations.", "The support measures are weakly continuous, locally defined and motion equivariant valuations on convex bodies with values in the space of finite measures on Borel subsets of ${\\mathbb {R}}^{n} \\times \\mathbb {S}^{n-1}$ , where $\\mathbb {S}^{n-1}$ denotes the Euclidean unit sphere in ${\\mathbb {R}}^{n}$ .", "These are determined by a local version of (REF ).", "Therefore, they are a crucial example of localizations of the intrinsic volumes.", "Furthermore, their marginal measures on Borel subsets of ${\\mathbb {R}}^{n}$ are called curvature measures and the ones on Borel subsets of $\\mathbb {S}^{n-1}$ are called surface area measures.", "In 1959, Federer [5] proved Crofton formulae for curvature measures, even in the more general setting of sets with positive reach.", "For further details and references, see also [14] and [14].", "Certain Crofton formulae for support measures were proved by Glasauer in 1997 [6].", "The combination of Minkowski tensors and localizations leads to another generalization of the intrinsic volumes.", "This topic has been explored by Schneider [18] and Hug and Schneider [9], [10] in recent years.", "They introduced particular tensorial support measures, the generalized local Minkowski tensors, and proved that they essentially span the vector space of isometry covariant and locally defined valuations on the space of convex polytopes ${\\mathcal {P}}^{n}$ with values in the ${p}$ -valued measures on ${\\mathcal {B}}({\\mathbb {R}}^{n} \\times \\mathbb {S}^{n-1})$ (see [14]).", "Under the additional assumption of weak continuity they extended this result to valuations on ${\\mathcal {K}}^{n}$ ; a summary of the required arguments is given in [14].", "The aim of the present article is to prove a set of Crofton formulae for similar functionals, which are localized in ${\\mathbb {R}}^{n}$ , the tensorial curvature measures or tensor-valued curvature measures.", "Here we first focus on the case where the tensorial curvature measures of the intersection of the given body with an affine flat are defined with respect to the affine flat as the ambient space (intrinsic viewpoint).", "In a second step, we demonstrate how the arguments can be extended to the case where the curvature measures are considered in ${\\mathbb {R}}^n$ (extrinsic viewpoint).", "The current approach combines main ideas of the previous works [12] and [9] and also links it to [3].", "A major advantage of the localization is that it naturally leads to a suitable choice of local tensor-valued measures for which the constants in the Crofton formulae are reasonably simple.", "From the general local results, we finally deduce various special consequences for the total measures, which are the Minkowski tensors that have been studied in [12].", "For the latter, we restrict ourselves to the translation invariant case, which simplifies the involved constants, but the general case can be treated similarly.", "In the case of the results for the extrinsic tensorial Crofton formulae, the connection to the approach in [3] via the methods of algebraic integral geometry is used and deepened, but this interplay will have to be explored further in future work.", "The structure of this contribution is as follows.", "In Section , we fix our notation and collect various auxiliary results which will be needed.", "Section contains the main results.", "We first state our findings for intrinsic tensorial curvature measures, then discuss some special cases and finally explain the extension to extrinsic tensorial curvature measures.", "The proofs of the results for the intrinsic case are given in Section .", "Section contains the arguments in the extrinsic setting.", "Some auxiliary results on sums of Gamma functions are provided in the final section." ], [ "Some Basic Tools", "In the following, we work in the $n$ -dimensional Euclidean space ${\\mathbb {R}}^{n}$ , equipped with its usual topology generated by the standard scalar product $\\cdot $ and the corresponding Euclidean norm ${\\,\\cdot \\,}$ .", "Recall that the unit ball centered at the origin is denoted by $B^n$ , its boundary (the unit sphere) is denoted by $\\mathbb {S}^{n-1}$ .", "For a topological space $X$ , we denote the Borel $\\sigma $ -algebra on $X$ by ${\\mathcal {B}}(X)$ .", "By $G(n, k)$ , for $k \\in \\lbrace 0, \\ldots , n\\rbrace $ , we denote the Grassmannian of $k$ -dimensional linear subspaces in ${\\mathbb {R}}^{n}$ , and we write $\\nu _k$ for the (rotation invariant) Haar probability measure on $G(n, k)$ .", "The directional space of an affine $k$ -flat $E \\in A(n, k)$ is denoted by $L(E) \\in G(n, k)$ , its orthogonal complement by $E^{\\perp } \\in G(n, n - k)$ , and the translate of $E$ by a vector $t \\in {\\mathbb {R}}^{n}$ is denoted by $E_{t} := E + t$ .", "For $k \\in \\lbrace 0,\\ldots , n\\rbrace $ , $l \\in \\lbrace 0, \\ldots k\\rbrace $ and $F \\in G(n, k)$ , we define $G(F, l) := \\lbrace L \\in G(n, l): L \\subset F \\rbrace $ .", "On $G(F, l)$ there exists a unique Haar probability measure $\\nu ^{F}_l$ invariant under rotations of ${\\mathbb {R}}^n$ mapping $F$ into itself and leaving $F^{\\perp }$ pointwise fixed.", "The orthogonal projection of a vector $x \\in {\\mathbb {R}}^{n}$ to a linear subspace $L$ of ${\\mathbb {R}}^{n}$ is denoted by $p_{L}(x)$ and its direction by $\\pi _{L}(x) \\in \\mathbb {S}^{n-1}$ , if $x \\notin L^{\\perp }$ .", "For two linear subspaces $L, L^{\\prime }$ of ${\\mathbb {R}}^{n}$ , the generalized sine function $[L, L^{\\prime }]$ is defined as follows.", "One extends an orthonormal basis of $L \\cap L^{\\prime }$ to an orthonormal basis of $L$ and to one of $L^{\\prime }$ .", "Then $[L, L^{\\prime }]$ is the volume of the parallelepiped spanned by all these vectors.", "The vector space of symmetric tensors of rank $p \\in {\\mathbb {N}}_{0}$ over ${\\mathbb {R}}^n$ is denoted by ${p}$ .", "The symmetric tensor product of two vectors $x, y \\in {\\mathbb {R}}^{n}$ is denoted by $xy$ and the $p$ -fold tensor product of a vector $x \\in {\\mathbb {R}}^{n}$ by $x^{p}$ .", "Identifying ${\\mathbb {R}}^{n}$ with its dual space via its scalar product, we interpret a symmetric tensor $a \\in {p}$ as a symmetric $p$ -linear map from $({\\mathbb {R}}^{n})^{p}$ to ${\\mathbb {R}}$ .", "One special tensor is the metric tensor $Q \\in {2}$ , defined by $Q(x, y) := x \\cdot y$ for $x,y \\in {\\mathbb {R}}^{n}$ .", "For an affine $k$ -flat $E \\in A(n, k)$ , $k \\in \\lbrace 0,\\ldots , n\\rbrace $ , the metric tensor $Q(E)$ in $E$ is defined by $Q(E)(x,y) := p_{L(E)} (x) \\cdot p_{L(E)} (y)$ for $x, y \\in {\\mathbb {R}}^{n}$ .", "Defining the tensorial curvature measures requires some preparation (see also [14]).", "For a convex body $K \\in {\\mathcal {K}}^{n}$ , we call the pair $(x, u) \\in {\\mathbb {R}}^{2n}$ a support element whenever $x$ is a boundary point of $K$ and $u$ is an outer unit normal vector of $K$ at $x$ .", "The set of all these support elements of $K$ is denoted by $\\operatorname{{\\rm Nor}}K \\subset \\Sigma ^{n} := {\\mathbb {R}}^{n} \\times \\mathbb {S}^{n-1}$ and called the normal bundle of $K$ .", "For $x \\in {\\mathbb {R}}^{n}$ , we denote the metric projection of $x$ onto $K$ by $p(K, x)$ , and define $u(K, x):= (x - p(K, x)) / {x - p(K, x)}$ for $x \\in {\\mathbb {R}}^{n} \\setminus K$ , the unit vector pointing from $p(K, x)$ to $x$ .", "For $\\epsilon >0$ and a Borel set $\\eta \\subset \\Sigma ^{n}$ , $M_{\\epsilon }(K, \\eta ) := \\left\\lbrace x \\in \\left( K + \\epsilon B^{n} \\right) \\setminus K \\colon \\left( p(K, x), u(K, x) \\right) \\in \\eta \\right\\rbrace $ is a local parallel set of $K$ which satisfies a local Steiner formula $ V_{n} (M_{\\epsilon }(K,\\eta )) = \\sum _{j = 0}^{n - 1} \\kappa _{n - j} \\Lambda _{j} (K, \\eta )\\epsilon ^{n - j}, \\qquad \\epsilon \\ge 0.$ This relation determines the support measures $\\Lambda _{0}(K, \\cdot ), \\ldots , \\Lambda _{n - 1} (K, \\cdot )$ of $K$ , which are finite Borel measures on ${\\mathcal {B}}(\\Sigma ^{n})$ .", "Obviously, a comparison of (REF ) and (REF ) yields $V_{j}(K) = \\Lambda _{j} (K, \\Sigma ^{n})$ .", "Now, for a convex body $K \\in {\\mathcal {K}}^{n}$ , a Borel set $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ and $j, r, s \\in {\\mathbb {N}}_{0}$ , the tensorial curvature measures are given by $\\phi _{j}^{r,s,0} (K, \\beta ):= \\omega _{n - j} \\int _{\\beta \\times \\mathbb {S}^{n-1}} x^r u^s \\, \\Lambda _j(K, \\mathrm {d}(x, u)),$ for $j\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ , where $\\omega _{n}$ denotes the $n-1$ -dimensional volume of $\\mathbb {S}^{n-1}$ , and by $\\phi _{n}^{r,0,0} (K, \\beta ): = \\int _{K \\cap \\beta } x^r \\, {\\mathcal {H}}^{n}(\\mathrm {d}x).$ If $K \\subset E \\in A(n, k)$ with $j < k \\le n$ , we denote the $j$ -th support measure of $K$ defined with respect to $E$ as the ambient space by $\\Lambda ^{(E)}_j(K, \\cdot )$ , which is a Borel measure on ${\\mathcal {B}}({\\mathbb {R}}^{n} \\times (L(E) \\cap \\mathbb {S}^{n-1}))$ , concentrated on $\\Sigma ^{(E)}:= E \\times (L(E) \\cap \\mathbb {S}^{n-1})$ with $L(E) \\in G(n, k)$ being the linear subspace parallel to $E$ .", "Then, we define the intrinsic tensorial curvature measures $\\phi _{j,E}^{r,s,0} (K, \\beta ): = \\omega _{k - j} \\int _{\\beta \\times (L(E) \\cap \\mathbb {S}^{n-1})} x^r u^s \\, \\Lambda ^{(E)}_j(K, \\mathrm {d}(x, u))$ and $\\phi _{k,E}^{r,0,0} (K, \\beta ) := \\int _{K \\cap \\beta } x^r \\, {\\mathcal {H}}^{k}(\\mathrm {d}x).$ For the sake of convenience, we extend the definition by $\\phi _{j}^{r,s,0} := 0$ (resp.", "$\\phi _{j,E}^{r,s,0} := 0$ ) for $j \\notin \\lbrace 0, \\ldots , n \\rbrace $ (resp.", "$j \\notin \\lbrace 0, \\ldots , k\\rbrace $ ) or $r \\notin {\\mathbb {N}}_{0}$ or $s \\notin {\\mathbb {N}}_{0}$ or $j = n$ (resp.", "$j= k$ ) and $s \\ne 0$ .", "We adopt the same convention for the Minkowski tensors and the generalized tensorial curvature measures introduced below.", "The tensorial curvature measures are natural local versions of the Minkowski tensors.", "For a convex body $K \\in {\\mathcal {K}}^{n}$ and $j,r, s \\in {\\mathbb {N}}_{0}$ , the latter are just the total measures $\\smash{\\Phi _{j}^{r,s}} (K) := \\smash{\\phi _{j}^{r,s,0}} (K,{\\mathbb {R}}^{n})$ and, if $K \\subset E \\in A(n, k)$ , an intrinsic version is given by $\\smash{\\Phi _{j,E}^{r,s}} (K) :=\\smash{\\phi _{j,E}^{r,s,0}} (K, {\\mathbb {R}}^{n})$ .", "These definitions of the Minkowski tensors differ slightly from the ones commonly used in the literature, as we slightly change the usual normalization (compare with the normalization used in [14]).", "The purpose of this change is to simplify the presentation of the main results of this article (and of future work).", "For a polytope $P \\in {\\mathcal {P}}^{n}$ and $j \\in \\lbrace 0, \\ldots , n \\rbrace $ , we denote the set of $j$ -dimensional faces of $P$ by ${\\mathcal {F}}_{j}(P)$ and the normal cone of $P$ at a face $F \\in {\\mathcal {F}}_{j}(P)$ by $N(P,F)$ .", "For a polytope $P \\in {\\mathcal {P}}^{n}$ and a Borel set $\\eta \\subset \\Sigma $ , the $j$ -th support measure is explicitly given by $\\Lambda _{j} (P, \\eta ) = \\frac{1}{\\omega _{n - j}}\\sum _{F \\in {\\mathcal {F}}_{j}(P)} \\int _{F} \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}}{\\mathbf {1}}_\\eta (x,u)\\, {\\mathcal {H}}^{n - j - 1} (\\mathrm {d}u) \\, {\\mathcal {H}}^{j} (\\mathrm {d}x)$ for $j\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ .", "For $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^n)$ , this yields $\\phi _{j}^{r,s,0} (P, \\beta ) = \\sum _{F \\in {\\mathcal {F}}_{j}(P)} \\int _{F \\cap \\beta } x^r \\, {\\mathcal {H}}^{j}(\\mathrm {d}x) \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} u^{s} \\, {\\mathcal {H}}^{n - j - 1} (\\mathrm {d}u)$ and, if $P \\subset E \\in A(n, k)$ and $j < k \\le n$ , $\\phi _{j,E}^{r,s,0} (P, \\beta ) = \\sum _{F \\in {\\mathcal {F}}_{j}(P)} \\int _{F \\cap \\beta } x^r \\, {\\mathcal {H}}^{j}(\\mathrm {d}x) \\int _{N_{E}(P,F) \\cap \\mathbb {S}^{n-1}} u^{s} \\, {\\mathcal {H}}^{k - j - 1} (\\mathrm {d}u),$ where $N_{E}(P, F) = N(P, F) \\cap L(E)$ is the normal cone of $P$ at the face $F$ , taken with respect to the subspace $L(E)$ .", "Of course, analogous representations are obtained for the (global) intrinsic and extrinsic Minkowski tensors.", "The Crofton formulae, which are stated in the next section, will naturally also involve the generalized tensorial curvature measures (see [14]) $\\phi _{j}^{r,s,1} (P, \\beta ) := \\sum _{F \\in {\\mathcal {F}}_{j}(P)} Q(F) \\int _{F \\cap \\beta } x^r \\, {\\mathcal {H}}^{j}(\\mathrm {d}x) \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} u^{s} \\, {\\mathcal {H}}^{n - j - 1} (\\mathrm {d}u),$ for $j \\in \\lbrace 1,\\ldots ,n-1\\rbrace $ , and, if $P \\subset E \\in A(n, k)$ and $0 < j < k \\le n$ , $\\phi _{j,E}^{r,s,1} (P, \\beta ) := \\sum _{F \\in {\\mathcal {F}}_{j}(P)} Q(F)\\int _{F \\cap \\beta } x^r \\, {\\mathcal {H}}^{j}(\\mathrm {d}x) \\int _{N_{E}(P,F)\\cap \\mathbb {S}^{n-1}} u^{s} \\, {\\mathcal {H}}^{k - j - 1} (\\mathrm {d}u).$ Due to Hug and Schneider [9] there exists a weakly continuous extension of the generalized tensorial curvature measures to ${\\mathcal {K}}^{n}$ .", "In fact, they proved such an extension for the generalized local Minkowski tensors, which are measures on ${\\mathcal {B}}({\\mathbb {R}}^{n} \\times \\mathbb {S}^{n-1})$ .", "Globalizing this in the $\\mathbb {S}^{n-1}$ -coordinate yields the result for the tensorial curvature measures.", "Apart from the easily verified relation $ \\phi _{n - 1}^{r,s,1} = Q\\phi _{n - 1}^{r,s,0} - \\phi _{n - 1}^{r,s + 2,0},$ the tensorial curvature measures and the generalized tensorial curvature measures are linearly independent.", "In contrast, McMullen [16] discovered basic linear relations for the (gobal) Minkowski tensors (see also [14]), and it was shown in [13] that these are essentially all linear dependences between the Minkowski tensors (see also [14]).", "Furthermore, McMullen [16] found relations for the global counterparts of the generalized tensorial curvature measures.", "In fact, the globalized form of (REF ) is a very special example of one of these relations.", "For the translation invariant Minkowski tensors $\\Phi _{j}^{{0},s} $ , these relations take a very simple form, nevertheless for our purpose they are essential in the proof of Theorem REF .", "To have a short notation for these translation invariant Minkowski tensors, we omit the first superscript and put $\\Phi _{j}^{s} := \\Phi _{j}^{{0},s}, \\qquad \\Phi _{j,E}^{s}:= \\Phi _{j,E}^{{0},s}.$ Then we can state the following very special case of McMullen's relations.", "Lemma 1 (McMullen) Let $P \\in {\\mathcal {P}}^n$ and $j, s \\in {\\mathbb {N}}_0$ with $j \\le n - 1$ .", "Then $\\tfrac{n - j + s}{s + 1} \\Phi _{j}^{s + 2}(P)= \\sum _{F \\in {\\mathcal {F}}_j (P)} Q (F^{\\perp }) {\\mathcal {H}}^{j} (F) \\int _{N(P,F)\\cap \\mathbb {S}^{n-1}} u^{s}\\, {\\mathcal {H}}^{n - j - 1} (\\mathrm {d}u).$ Note that this lemma is essentially a global result which is derived by applying a version of the divergence theorem." ], [ "Crofton Formulae", "In this article, for $0 \\le j \\le k < n$ and $i, s \\in {\\mathbb {N}}_0$ , we are first concerned with the Crofton integrals $ \\int _{A(n, k)} Q(E)^i \\,\\phi _{j,E}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E),$ which involve the intrinsic tensorial curvature measures, and the Crofton integrals $ \\int _{A(n, k)} Q(E)^i \\,\\Phi _{j,E}^{s} (K \\cap E) \\, \\mu _k(\\mathrm {d}E)$ for the global versions of the translation invariant intrinsic tensorial curvature measures, the translation invariant intrinsic Minkowski tensors obtained by setting $r=0$ .", "In the global case, we restrict our investigations mainly to these translation invariant intrinsic Minkowski tensors, general Crofton formulae have already been established in [12].", "Using the simplifications of the formulae obtained in the present work, the extrinsic formulae in [12], that is, Crofton formulae for the integrals $ \\int _{A(n, k)}\\phi _{j}^{r,s,0} (K \\cap E,\\beta \\cap E) \\, \\mu _k(\\mathrm {d}E)$ can be simplified accordingly.", "We explain this in detail in the case where $j=k-1$ .", "The connection to [3] turns out to be crucial for simplifying the constants if $s$ is odd.", "However, for even $s$ the current approach works completely independently." ], [ "Crofton Formulae for Intrinsic Tensorial Curvature\nMeasures", "In this section we state the formulae for the integrals given in (REF ) and (REF ).", "We start with the local versions, where we distinguish the cases $j = k$ and $j < k$ .", "Theorem 1 Let $K \\in {\\mathcal {K}}^n$ , $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^n)$ and $i, k, r, s \\in {\\mathbb {N}}_0$ with $k < n$ .", "Then $& \\int _{A(n, k)} Q(E)^i \\, \\phi _{k,E}^{r,s,0} (K \\cap E,\\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) = \\frac{\\Gamma (\\frac{n}{2})\\Gamma (\\frac{k}{2} + i)}{\\Gamma (\\frac{n}{2} +i)\\Gamma (\\frac{k}{2})} \\, Q^i \\phi _{n}^{r,0,0} (K, \\beta )$ if $s=0$ ; for $s\\ne 0$ the integral is zero.", "If $s = 0$ in Theorem REF , then we interpret the coefficient of the tensor on the right-hand side as 0, if $k = 0$ and $i \\ne 0$ , and as 1, if $k = i = 0$ .", "A global version of Theorem REF is obtained by simply setting $\\beta ={\\mathbb {R}}^n$ .", "Next we turn to case $j<k$ .", "Theorem 2 Let $K \\in {\\mathcal {K}}^n$ , $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^n)$ and $i, j, k, r, s \\in {\\mathbb {N}}_0$ with $j < k <n$ and $k > 1$ .", "Then $[3] \\int _{A(n, k)} Q(E)^i \\, \\phi _{j,E}^{r,s,0}(K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\={}& \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor +i} Q^z \\bigl ( \\lambda _{n, k, j, s, i, z}^{(0)} \\, \\phi _{n - k+ j}^{r,s + 2i - 2z,0} (K, \\beta ) + \\lambda _{n, k, j, s, i,z}^{(1)} \\, \\phi _{n - k + j}^{r,s + 2i - 2z - 2,1} (K,\\beta ) \\bigr ),$ where for $\\varepsilon \\in \\lbrace 0, 1 \\rbrace $ we set $\\gamma _{n, k, j} :={} & \\binom{n - k + j - 1}{j}\\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi },\\\\\\lambda _{n, k, j, s, i, z}^{(\\varepsilon )} := {} & \\sum _{p =0}^{i} \\sum _{q = (z - p + \\varepsilon )^+}^{\\lfloor \\frac{s}{2}\\rfloor + i - p} (-1)^{p + q - z} \\binom{i}{p} \\binom{s + 2i -2p}{2q} \\binom{p + q - \\varepsilon }{z} \\Gamma (q + \\tfrac{1}{2})\\\\& \\times \\frac{\\Gamma (\\frac{j + s}{2} + i - p - q + 1)}{\\Gamma (\\frac{n - k + j + s}{2} + i - p + 1)} \\frac{\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n + 1}{2} + p + q)} \\vartheta _{n, k, j, p,q}^{(\\varepsilon )},\\\\\\vartheta _{n, k, j, p, q}^{(0)} := {} & (n - k + j) \\big (\\tfrac{k - 1}{2} + p\\big ), \\qquad \\vartheta _{n, k, j, p, q}^{(1)} :=p(n - k) - q(k - 1).$ If $j=k-1$ , then the tensorial curvature measures and the generalized tensorial curvature measures are linearly dependent.", "In this case, the right-hand side can be expressed as a linear combination of the tensor-valued curvature measures $Q^z\\phi _{n -1}^{r,s + 2i - 2z,0} (K, \\,\\cdot \\,)$ , whereas the measures $Q^z\\phi _{n - 1}^{r,s + 2i - 2z,1} (K, \\,\\cdot \\,)$ are not needed.", "An explicit description of this case is given in Corollary REF for $i=0$ and in (REF ) for $i \\in {\\mathbb {N}}_{0}$ .", "If the additional metric tensor is omitted as a weight function, that is in the case $i = 0=p$ , then the coefficients $\\lambda _{n, k,j, s, 0, z}^{\\smash{(\\varepsilon )}}$ in Theorem REF simplify to a single sum.", "Apparently, the coefficients in Theorem REF are not well defined in the (excluded) case $k = 1$ and $j=0$ , as $\\Gamma (0)$ is involved in the numerator of $\\lambda _{n, 1, 0, s, i,z}^{\\smash{(\\varepsilon )}}$ .", "Although this issue can be resolved by a proper interpretation of the (otherwise ambiguous) expression $\\Gamma (p)\\cdot p=\\Gamma (p+1)$ as 1 for $p=0$ , we prefer to state and derive this case separately.", "In fact, our analysis leads to substantial simplifications of the constants, as our next result shows.", "Theorem 3 Let $K \\in {\\mathcal {K}}^n$ , $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^n)$ and $i, r, s \\in {\\mathbb {N}}_0$ .", "Then $\\int _{A(n, 1)} Q(E)^i \\, \\phi _{0,E}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& = \\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{s + 1}{2} + i)}{\\pi \\Gamma ( \\frac{n + s + 1}{2} + i) } \\sum _{z = 0}^{\\frac{s}{2} + i} (-1)^{z} \\binom{\\frac{s}{2} + i}{z} \\frac{1}{1 -2z} \\, Q^{\\frac{s}{2} + i - z} \\phi _{n - 1}^{r,2z,0} (K,\\beta )$ for even $s$ .", "If $s$ is odd, then $\\int _{A(n, 1)} Q(E)^i \\, \\phi _{0,E}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) = \\frac{\\Gamma (\\frac{n}{2}) \\Gamma (\\frac{s}{2} + i + 1)}{\\sqrt{\\pi }\\Gamma (\\frac{n + s + 1}{2} + i)} \\, Q^{\\frac{s - 1}{2} + i} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ Note that in Theorem REF the Crofton integral is expressed only by tensorial curvature measures $\\phi _{n -1}^{r,z,0}$ (multiplied with suitable powers of the metric tensor), whereas generalized tensorial curvature measures are not needed.", "A global version of Theorem REF is obtained by simply setting $\\beta = {\\mathbb {R}}^n$ .", "A translation invariant, global version of Theorem REF allows us to combine several of the summands on the right-hand side of the formula.", "Theorem 4 Let $K \\in {\\mathcal {K}}^n$ and $i, j,k, s \\in {\\mathbb {N}}_0$ with $j < k < n$ and $k > 1$ .", "Then $\\int _{A(n, k)} Q(E)^i \\, \\Phi _{j,E}^{s} (K \\cap E) \\,\\mu _k(\\mathrm {d}E) = \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\lambda _{n, k, j, s, i, z}^{(0)} \\, Q^z\\Phi _{n - k + j}^{s + 2i - 2z} (K),$ where $\\gamma _{n, k, j}$ and $\\lambda _{n, k, j, s, i, z}^{(0)}$ are defined as in Theorem REF , but $[1] \\vartheta _{n, k, j, s, i ,z , p, q}^{(0)} := (n - k + j)\\bigl (\\tfrac{k - 1}{2} + p\\bigr ) - \\bigl ( p(n -k) - q(k - 1) \\bigr ) \\bigl ( 1 + \\tfrac{k - j - 1}{s + 2i - 2z- 1} (1 - \\tfrac{z}{p + q})\\bigr )$ replaces $\\vartheta _{n, k, j, p, q}^{(0)}$ , except if $s$ is odd and $z=\\lfloor \\frac{s}{2}\\rfloor + i$ , where $\\lambda _{n, k, j,s, i, \\lfloor \\frac{s}{2}\\rfloor + i}^{(0)}:=0$ .", "In Theorem REF , if $p = q = 0$ , then the definition of $\\lambda _{n, k, j, s, i, z}^{\\smash{(0)}}$ implies that also $z=0$ and thus, $\\vartheta _{n, k, j, s, i ,0 , 0,0}^{\\smash{(0)}}$ is well-defined with $\\tfrac{z}{p + q} = 1$ ." ], [ "Some Special Cases", "In the following, we restrict to the case $i=0$ of Crofton formulae for unweighted intrinsic Minkowski tensors or tensorial curvature measures.", "Corollary 1 Let $K \\in {\\mathcal {K}}^n$ and $k,j, s \\in {\\mathbb {N}}_0$ with $0 \\le j < k < n$ .", "Then $\\int _{A(n, k)} \\Phi _{j,E}^{s} (K \\cap E) \\, \\mu _k(\\mathrm {d}E) = \\delta _{n, k, j, s} \\sum _{z=0}^{\\lfloor \\frac{s}{2} \\rfloor }\\eta _{n, k, j, s, z} \\, Q^z \\Phi _{n - k + j}^{s - 2z} (K),$ where $\\delta _{n, k, j, s} : ={} & \\binom{n - k + j - 1}{j}\\frac{\\Gamma (\\frac{n - k + 1}{2}) \\Gamma (\\frac{k + 1}{2})}{\\pi \\Gamma (\\frac{n - k + j + s}{2} + 1)},\\\\\\eta _{n, k, j, s, z} := {} & \\sum _{q = z}^{\\lfloor \\frac{s}{2}\\rfloor } (-1)^{q - z} \\binom{s}{2q} \\binom{q}{z} \\Gamma (q +\\tfrac{1}{2}) \\frac{\\Gamma (\\frac{j + s}{2} - q + 1) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n + 1}{2} + q)}\\\\& \\times \\left( \\tfrac{n - k + j}{2} + q + \\tfrac{(k - j - 1)(q - z)}{s - 2z - 1} \\right),$ but $\\eta _{n, k, j, s, \\lfloor \\frac{s}{2} \\rfloor } := 0$ if $s$ is odd.", "Specific choices of $s$ Next we collect some special cases of Corollary REF , which are obtained for specific choices of $s \\in {\\mathbb {N}}_{0}$ by applications of Legendre's duplication formula and elementary calculations.", "Corollary 2 Let $K \\in {\\mathcal {K}}^n$ and $k, j \\in {\\mathbb {N}}_0$ with $0 \\le j < k <n$ .", "Then $\\int _{A(n, k)} \\Phi _{j,E}^{2} (K \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& = \\frac{\\Gamma (\\frac{k + 1}{2})\\Gamma (\\frac{n - k + j +1}{2})}{\\Gamma (\\frac{n + 3}{2})\\Gamma (\\frac{j + 1}{2})}\\bigl ( \\tfrac{n - k}{4 (n - k + j)} \\, Q \\Phi _{n - k +j}^{0} (K) + \\tfrac{n - k + nj + j}{2(n - k + j)} \\,\\Phi _{n - k + j}^{2} (K) \\bigr ).$ Corollary 3 Let $K \\in {\\mathcal {K}}^n$ and $k,j \\in {\\mathbb {N}}_0$ with $0 \\le j < k < n$ .", "Then $\\int _{A(n, k)} \\Phi _{j,E}^{3} (K \\cap E) \\,\\mu _k(\\mathrm {d}E) = \\frac{j + 1}{n - k + j + 1} \\frac{\\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{n - k + j}{2})}{\\Gamma (\\frac{n + 1}{2}) \\Gamma (\\frac{j}{2})} \\Phi _{n- k + j}^{3} (K).$ As $\\Gamma (\\frac{j}{2})^{-1} = 0$ , for $j = 0$ , the integral in Corollary REF equals 0 in this case.", "However, as the integrand on the left-hand side is already 0, this is not surprising.", "The same is true for any odd number $s \\in {\\mathbb {N}}$ and $j = 0$ .", "Corollary REF immediately leads to a result which was obtained and applied by Bernig and Hug in [3].", "Corollary 4 Let $K \\in {\\mathcal {K}}^n$ .", "Then $\\int _{A(n, 2)} \\Phi _{1,E}^{3} (K \\cap E) \\,\\mu _k(\\mathrm {d}E) = \\binom{n}{2}^{-1} \\Phi _{n - 1}^{3}(K).$ $ $ The choice $j = k - 1$ Furthermore, we obtain simple Crofton formulae for the specific choice $j = k - 1$ in the local and in the global case.", "Corollary 5 Let $K \\in {\\mathcal {K}}^n$ , $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^n)$ and $k, r, s \\in {\\mathbb {N}}_0$ with $1 < k <n$ .", "Then $\\int _{A(n, k)} \\phi _{k-1,E}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) = \\delta _{n, k, k - 1, s}\\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\xi _{n, k, s, z}Q^z \\phi _{n - 1}^{r,s-2z,0} (K, \\beta ),$ where $\\xi _{n, k, s, z} := \\sum _{q = z}^{\\lfloor \\frac{s}{2}\\rfloor } (-1)^{q - z} \\binom{s}{2q} \\binom{q}{z} \\Gamma \\left(q + \\tfrac{1}{2} \\right) \\frac{\\Gamma (\\frac{k + s +1}{2} - q) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n- 1 }{2} + q)}.$ Corollary REF will be derived from Theorem REF in the same way as Theorem REF is proved.", "More specifically, relation (REF ) is applied, which can be considered as a local version of Lemma REF in the particular case $j = n - 1$ .", "Although $k=1$ is excluded in Corollary REF , the result is formally consistent with Theorem REF (for $i=0$ ), which can be checked by simplifying the coefficients $\\xi _{n, 1, s, z}$ with the help of Zeilberger's algorithm.", "A global version of Corollary REF is obtained by setting $\\beta = {\\mathbb {R}}^n$ .", "Finally, Theorem REF can be globalized to give a result, which was obtained in [15] by a completely different approach.", "Corollary 6 Let $K \\in {\\mathcal {K}}^n$ and $s\\in {\\mathbb {N}}_0$ .", "Then $\\int _{A(n, 1)} \\Phi _{0,E}^{s} (K \\cap E) \\,\\mu _k(\\mathrm {d}E) = \\frac{2 \\omega _{n + s + 1}}{\\pi \\omega _{s + 1} \\omega _n} \\sum _{z = 0}^{\\frac{s}{2} } \\frac{\\left( -1 \\right)^{z}}{1 - 2z} \\binom{\\frac{s}{2}}{z}Q^{\\frac{s}{2} - z} \\Phi _{n - 1}^{2z} (K)$ for even $s$ .", "For odd $s$ the integral on the left-hand side equals 0.", "Note that if $s \\in {\\mathbb {N}}$ is odd, then the Crofton integral in Theorem REF is a non-zero measure, as the tensorial curvature measures $\\phi _{n -1}^{r,1,0}(K,\\,\\cdot \\,)$ are non-zero (if the underlying set $K$ is at least $(n-1)$ -dimensional), whereas $\\Phi _{n -1}^{1} \\equiv 0$ in the global case considered in Corollary REF ." ], [ "Crofton Formulae for Extrinsic Tensorial Curvature\nMeasures", "In the following, we state Crofton formulae for tensorial curvature measures for $j = k - 1$ .", "The method also applies to the cases where $j\\le k-2$ , but it remains to be explored to which extent the constants can be simplified then.", "As for the intrinsic versions, we have to distinguish between the cases $k > 1$ and $k = 1$ .", "We start with the former.", "Theorem 5 Let $K \\in {\\mathcal {K}}^{n}$ , $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ and $k, r, s \\in {\\mathbb {N}}_{0}$ with $1 < k < n$ .", "Then $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) =\\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\kappa _{n, k, s, z} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ),$ where $\\kappa _{n, k, s, z}: = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s + 1}{2}) \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\frac{\\Gamma (\\tfrac{n - k}{2} + z) \\Gamma (\\tfrac{k + s - 1}{2} - z)}{\\Gamma (\\frac{s}{2} - z + 1) z!", "}$ if $z \\ne \\frac{s - 1}{2}$ , and $\\kappa _{n, k, s, \\frac{s - 1}{2}} : = \\pi ^{\\frac{n - k - 1}{2}} \\frac{2k (n + s - 2)}{(n - 1) (n - k + s - 1)} \\frac{\\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s}{2} + 1)}{ \\Gamma (\\frac{n + s + 1}{2}) }.$ In Theorem REF , if $s$ is odd the coefficient $\\kappa _{n, k, s, ({s - 1})/ 2}$ has to be defined separately, as the proof shows.", "(In fact, the difference amounts to a factor $ {k (n + s - 2)} [{(k - 1) (n + s - 1)}]^{-1}$ .)", "For even $s$ , the constants involved in the proof of Theorem REF can be simplified by a direct calculation to arrive at the asserted result.", "However, if $s$ is odd, we need the connection to the work [3] to simplify the constants.", "Since this connection breaks down for $z=(s-1)/2$ , $s$ odd, a separate direct calculation is required for this case, and this finally yields the correct constant in (REF ).", "The result is also consistent with the special case $k=1$ which is considered next.", "For $k = 1$ the Crofton integrals can be represented with a single functional, as the following theorem shows.", "Theorem 6 Let $K \\in {\\mathcal {K}}^{n}$ , $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ and $r, s \\in {\\mathbb {N}}_{0}$ .", "Then $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) = \\pi ^{\\frac{n - 2}{2}} \\frac{\\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\lfloor \\frac{s + 1}{2} \\rfloor + \\frac{1}{2})}{\\Gamma (\\frac{n}{2} + \\lfloor \\frac{s + 1}{2} \\rfloor )} Q^{\\lfloor \\frac{s}{2} \\rfloor } \\phi _{n - 1}^{r,s - 2\\lfloor \\frac{s}{2} \\rfloor ,0} (K, \\beta ).$ It can be easily checked that the result for $k=1$ can be obtained from the one for $k>1$ by a formal specialization and proper interpretation of expressions which a priori are not well defined.", "For this to work, it is indeed crucial that for odd values of $s$ and $z=(s-1)/2$ the definition in (REF ) applies.", "In [3], an alternative basis of the vector space of continuous, translation invariant and rotation covariant ${p}$ -valued valuations on ${\\mathcal {K}}^{n}$ was introduced, based on the trace free part of the Minkowski tensors, which was called the $\\Psi $ -basis.", "In the same spirit (but locally and with the current normalization), we now define $\\psi _{k}^{r,s,0} := \\phi _{k}^{r,s,0} + \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 1}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} Q^{j} \\phi _{k}^{r,s - 2j,0}$ for $r,s\\in {\\mathbb {N}}_0$ and $k\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ .", "Interpreting this definition in the right way if $n=2$ and $s=0$ (where $\\psi _{k}^{r,0,0}=\\phi _{k}^{r,0,0}$ ), we can also write $\\psi _{k}^{r,s,0} = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} Q^{j} \\phi _{k}^{r,s - 2j,0}.", "$ In particular, $\\psi _{k}^{r,s,0}=\\phi _{k}^{r,s,0}$ for $s\\in \\lbrace 0,1\\rbrace $ .", "Conversely, we have $\\phi _{k}^{r,s,0} = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - 2j)}{\\Gamma (\\frac{n}{2} + s - j)} Q^{j} \\psi _{k}^{r,s - 2j,0}.", "$ Although this will not be needed explicit, it shows how we can switch between a $\\phi $ -representation and a $\\psi $ -representation of tensorial curvature measures.", "The main advantage of the new local tensor valuations given in (REF ) is that the Crofton formula takes a particularly simple form.", "Corollary 7 Let $K \\in {\\mathcal {K}}^{n}$ , $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ , and let $k, r, s \\in {\\mathbb {N}}_{0}$ with $0 < k < n$ .", "Then $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\,\\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\pi ^{\\frac{n - k}{2}} \\frac{k - 1}{n - 1} \\frac{\\Gamma (\\frac{n}{2}) \\Gamma (\\frac{k + s - 1}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\frac{ \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n - k + s + 1}{2})} \\,\\psi _{n - 1}^{r,s,0} (K, \\beta ).$ For $r = 0$ and $\\beta = {\\mathbb {R}}^{n}$ , Corollary REF coincides with [3] (in the case corresponding to $j=k-1$ ).", "If $s \\in \\lbrace 0, 1 \\rbrace $ , then $\\psi _{k}^{r,s,0} = \\phi _{k}^{r,s,0}$ and Corollary REF coincides with Theorem REF (resp.", "Theorem REF , for $k = 1$ ).", "If $k = 1$ , then the integral in Corollary REF vanishes, except for $s \\in \\lbrace 0, 1 \\rbrace $ ." ], [ "Proofs of the Main Results", "In this section, we first recall some results from [12].", "Then we prove an integral formula which is required in the following.", "Finally, all ingredients are combined for the proofs of our main theorems.", "A basic tool is the following transformation formula (see [12]).", "It can be used to carry out an integration over linear Grassmann spaces recursively.", "The result is also true for $k=1$ , but in this case the outer integration on the right-hand side is trivial.", "Lemma 2 Let $u \\in \\mathbb {S}^{n-1}$ and let $h: G(n, k) \\rightarrow {p}$ be an integrable function for $k, p \\in {\\mathbb {N}}_{0}$ , $0 < k < n$ .", "Then $\\int _{G(n, k)} h(L) \\, \\nu _k (\\mathrm {d}L) =\\frac{\\omega _{k}}{2 \\omega _{n}} \\int _{G(u^\\perp , k -1)} \\int _{-1}^{1} \\int _{U^{\\perp } \\cap u^{\\perp } \\cap \\mathbb {S}^{n-1}} {t}^{k - 1} ( 1 - t^2 )^{\\frac{n - k - 2}{2}}\\\\& \\times h \\bigl ( \\operatorname{{\\rm span}}\\bigl \\lbrace U, t u + \\sqrt{1 - t^2} w\\bigr \\rbrace \\bigr ) \\, {\\mathcal {H}}^{n - k - 1} (\\mathrm {d}w) \\, \\mathrm {d}t \\,\\nu ^{u^\\perp }_{k - 1} (\\mathrm {d}U).$ The next results are derived from the previous one (see [12]).", "Lemma 3 Let $i, k \\in {\\mathbb {N}}_0$ with $k \\le n$ .", "Then $\\int _{G(n, k)} Q(L)^i \\, \\nu _{k} (\\mathrm {d}L) = \\frac{\\Gamma (\\frac{n}{2}) \\Gamma (\\frac{k}{2} + i)}{\\Gamma (\\frac{n}{2} + i)\\Gamma (\\frac{k}{2})} Q^{i}.$ In Lemma REF , we interpret the coefficient of the tensor on the right-hand side as 0, if $k = 0$ and $i\\ne 0$ , and as 1, if $k = i = 0$ , as $\\Gamma (0)^{-1} := 0$ and $\\frac{\\Gamma (a)}{\\Gamma (a)}=1$ for all $a \\in {\\mathbb {R}}$ .", "Lemma 4 Let $i \\in {\\mathbb {N}}_0$ , $k,r\\in \\lbrace 0, \\ldots , n \\rbrace $ with $k + r \\ge n$ , and let $F \\in G(n, r)$ .", "Then $\\int _{G(n, k)} [F, L]^{2} Q(L)^{i} \\, \\nu _{k} (\\mathrm {d}L) ={} & \\frac{r!", "k!}{n!", "(k + r - n)!}", "\\frac{\\Gamma ( \\frac{n}{2} + 1) \\Gamma ( \\frac{k}{2} + i)}{\\Gamma ( \\frac{n}{2} +i + 1)\\Gamma ( \\frac{k}{2} + 1)}\\\\& \\times \\bigl ( ( \\tfrac{k}{2} + i ) Q^i + i \\tfrac{k - n}{r} Q^{i - 1} Q(F) \\bigr ).$ We interpret the second summand on the right-hand side of Lemma REF as 0, if $i = 0$ , which is consistent with [12].", "If $r =0$ , we also interpret the second summand as 0 and the integral on the left equals $Q^i$ .", "Finally, we state the following integral formula (see [12]), which is a special case of [17].", "Lemma 5 Let $P \\in {\\mathcal {P}}^n$ be a polytope, $L \\in G(n, k)$ for $0 \\le j < k < n$ and let $g: {\\mathbb {R}}^n \\times (\\mathbb {S}^{n-1}\\cap L) \\rightarrow be a measurablebounded function.", "Then{\\begin{@align*}{1}{-1}[3] \\int _{L^\\perp } \\int _{L_t \\times (L \\cap \\mathbb {S}^{n-1})} g(x, u) \\, \\Lambda _j^{(L_t)}(P \\cap L_t, \\mathrm {d}(x,u)) \\, {\\mathcal {H}}^{n - k}(\\mathrm {d}t)\\\\= {} & \\frac{1}{\\omega _{k - j}} \\sum _{F \\in {\\mathcal {F}}_{n - k +j}(P)} \\, \\int _{F \\times (N(P,F) \\cap \\mathbb {S}^{n-1})} g(x,\\pi _L(u)) {p_L(u)}^{j - k} [ F, L ]^2 \\, {\\mathcal {H}}^{n -1}(\\mathrm {d}(x, u)).\\end{@align*}}$" ], [ "Auxiliary Integral Formulae", "With the preliminary results from [12] we are able to establish the following integral formula, which is a slightly modified version of [12].", "Proposition 1 Let $i, j, k, s\\in {\\mathbb {N}}_0$ with $j < k < n$ and $k > 1$ , $F \\in G(n, n - k +j)$ and $u \\in F^{\\perp } \\cap \\mathbb {S}^{n-1}$ .", "Then $\\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\& = \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2}\\rfloor + i} \\bigl ( \\lambda _{n, k, j, s, i, z}^{(0)} u^2+ \\lambda _{n, k, j, s, i, z}^{(1)} Q(F) \\bigr ) Q^{z} u^{s+ 2i - 2z - 2},$ where the coefficients are defined as in Theorem REF .", "Lemma REF yields $[3] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\frac{\\omega _{k}}{2 \\omega _{n}} \\int _{G(u^\\perp , k -1)} \\int _{-1}^{1} \\int _{U^{\\perp } \\cap u^{\\perp } \\cap \\mathbb {S}^{n-1}}{t}^{k - 1} ( 1 - t^2 )^{\\frac{n - k - 2}{2}} \\pi _{\\operatorname{{\\rm span}}\\lbrace U, t u + \\sqrt{1 - t^2} w \\rbrace }(u)^s\\\\& \\times Q \\bigl ( \\operatorname{{\\rm span}}\\bigl \\lbrace U, t u + \\sqrt{1 - t^2} w\\bigr \\rbrace \\bigr )^i {p_{\\operatorname{{\\rm span}}\\lbrace U, t u + \\sqrt{1 - t^2} w\\rbrace }(u)}^{j - k}\\\\& \\times \\bigl [ F, \\operatorname{{\\rm span}}\\bigl \\lbrace U, t u + \\sqrt{1 - t^2} w\\bigr \\rbrace \\bigr ]^2 \\, {\\mathcal {H}}^{n - k - 1} (\\mathrm {d}w) \\, \\mathrm {d}t \\,\\nu ^{u^\\perp }_{k - 1} (\\mathrm {d}U).$ As $Q \\bigl ( \\operatorname{{\\rm span}}\\bigl \\lbrace U, t u + \\sqrt{1 - t^2} w \\bigr \\rbrace \\bigr ) & = Q ( U ) + \\bigl ( {t} u + \\sqrt{1 - t^2} \\operatorname{{\\rm sign}}(t) w \\bigr )^2,\\\\\\pi _{\\operatorname{{\\rm span}}\\lbrace U, t u + \\sqrt{1 - t^2} w \\rbrace }(u) & = {t} u +\\sqrt{1 - t^2} \\operatorname{{\\rm sign}}(t) w,\\\\{p_{\\operatorname{{\\rm span}}\\lbrace U, t u + \\sqrt{1 - t^2} w \\rbrace }(u)} & ={t},\\\\\\bigl [ F, \\operatorname{{\\rm span}}\\bigl \\lbrace U, t u + \\sqrt{1 - t^2} w \\bigr \\rbrace \\bigr ] & = [ F, U ]^{(u^\\perp )} {t}$ hold for all $t \\in [-1, 1] \\setminus \\lbrace 0 \\rbrace $ , we obtain $[3] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\frac{\\omega _{k}}{2 \\omega _{n}} \\int _{G(u^\\perp , k -1)} \\int _{-1}^{1} \\int _{U^{\\perp } \\cap u^{\\perp } \\cap \\mathbb {S}^{n-1}}{t}^{j + 1} ( 1 - t^2 )^{\\frac{n - k - 2}{2}} \\bigl ( [ F,U ]^{(u^\\perp )} \\bigr )^2 \\bigl ( {t} u + \\sqrt{1 - t^2} w \\bigr )^s\\\\& \\times \\bigl ( Q( U ) + ( {t} u + \\sqrt{1 - t^2} w )^2 \\bigr )^i \\, {\\mathcal {H}}^{n - k - 1} (\\mathrm {d}w) \\, \\mathrm {d}t \\,\\nu ^{u^\\perp }_{k - 1} (\\mathrm {d}U),$ where we used the fact that the integration with respect to $w$ is invariant under reflections in the origin.", "Then we apply the binomial theorem to the terms $( Q ( U ) + ( |t| u + \\sqrt{1 -t^2} w )^2 )^i$ and $( |t| u + \\sqrt{1 - t^2} w )^{s + 2p}$ and get $[3] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\frac{\\omega _{k}}{2 \\omega _{n}} \\sum _{p = 0}^{i}\\sum _{q = 0}^{s + 2p} \\binom{i}{p} \\binom{s + 2p}{q} \\int _{G(u^\\perp , k - 1)} \\int _{-1}^{1} |t|^{j + s + 2p - q + 1} (1 - t^2 )^{\\frac{n - k + q - 2}{2}} \\, \\mathrm {d}t\\\\& \\times \\int _{U^{\\perp } \\cap u^{\\perp } \\cap \\mathbb {S}^{n-1}} w^{q} \\,{\\mathcal {H}}^{n - k - 1} (\\mathrm {d}w) \\bigl ( [ F, U ]^{(u^\\perp )} \\bigr )^2u^{s + 2p - q} Q ( U )^{i - p} \\, \\nu ^{u^\\perp }_{k - 1} (\\mathrm {d}U).$ Since $\\int _{U^{\\perp } \\cap u^{\\perp } \\cap \\mathbb {S}^{n-1}} w^{q} \\, {\\mathcal {H}}^{n - k- 1} (\\mathrm {d}w) = {\\mathbf {1}} \\lbrace q \\text{ even}\\rbrace 2 \\frac{\\omega _{n -k + q}}{\\omega _{q + 1}} Q(U^\\perp \\cap u^\\perp )^{\\frac{q}{2}},$ we deduce from the definition of the Beta function and its relation to the Gamma function that $[3] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\frac{\\omega _{k}}{\\omega _{n}} \\sum _{p = 0}^{i}\\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + p} \\binom{i}{p}\\binom{s + 2p}{2q} \\frac{\\Gamma (\\frac{j + s}{2} + p - q + 1)\\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - k + j + s}{2}+ p + 1)} \\frac{\\omega _{n - k + 2q}}{\\omega _{2q + 1}}\\\\& \\times u^{s + 2p - 2q} \\int _{G(u^\\perp , k - 1)} Q (U^{\\perp } \\cap u^{\\perp } )^q \\bigl ( [ F, U ]^{(u^\\perp )}\\bigr )^2 Q ( U )^{i - p} \\, \\nu ^{u^\\perp }_{k - 1} (\\mathrm {d}U).$ Applying the binomial theorem to $Q ( U^{\\perp } \\cap u^{\\perp })^q = (Q(u^{\\perp }) - Q (U))^q$ yields $[3] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L) \\nonumber \\\\= {} & \\frac{\\Gamma (\\frac{n}{2})}{\\sqrt{\\pi }\\Gamma (\\frac{k}{2})} \\sum _{p = 0}^{i} \\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + p} \\sum _{y = 0}^{q} (-1)^{y} \\binom{i}{p} \\binom{s +2p}{2q} \\binom{q}{y} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{j + s}{2} + p - q + 1)}{\\Gamma (\\frac{n - k + j + s}{2} + p + 1)} \\nonumber \\\\& \\times u^{s + 2p - 2q} Q\\bigl ( u^{\\perp } \\bigr )^{q - y} \\int _{G(u^\\perp , k - 1)} \\bigl ( [ F, U ]^{(u^\\perp )}\\bigr )^2 Q ( U )^{i - p + y} \\, \\nu ^{u^\\perp }_{k - 1} (\\mathrm {d}U).", "$ We conclude from Lemma REF , which is applied in $u^\\perp $ to the remaining integral on the right-hand side of (REF ), $[3] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\frac{(n - k + j)!", "(k - 1)!", "}{\\sqrt{\\pi }(n - 1)!", "j!", "}\\frac{\\Gamma (\\frac{n}{2}) \\Gamma ( \\frac{n + 1}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma ( \\frac{k + 1}{2})} \\sum _{p =0}^{i} \\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + p}\\binom{i}{p} \\binom{s + 2p}{2q} \\Gamma (q + \\tfrac{1}{2})\\\\& \\times \\frac{\\Gamma (\\frac{j + s}{2} + p - q + 1)}{\\Gamma (\\frac{n - k + j + s}{2} + p + 1)} u^{s + 2p - 2q}\\sum _{y = 0}^{q} (-1)^{y} \\binom{q}{y} \\frac{\\Gamma ( \\frac{k- 1}{2} + i - p + y)}{\\Gamma ( \\frac{n + 1}{2} + i - p +y)}\\\\& \\times \\!\\begin{aligned}[t]\\Bigl ( &\\big ( \\tfrac{k - 1}{2} + i - p + y \\big ) Q \\bigl (u^\\perp \\bigr )^{i - p + q} + \\tfrac{k - n}{n - k + j} (i - p + y) Q \\bigl ( u^\\perp \\bigr )^{i - p + q - 1} Q(F) \\Bigr ).\\end{aligned}$ Lemma REF from Section  applied twice to the summations with respect to $y$ and Legendre's duplication formula applied three times to the Gamma functions involving $n$ , $k$ and $n - k$ yield together with the definitions of $\\gamma _{n, k, j}$ and $\\vartheta _{n, k, j, p,q}^{\\smash{(\\varepsilon )}}$ , $\\varepsilon \\in \\lbrace 0, 1\\rbrace $ , $[3] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\gamma _{n, k, j} \\sum _{p = 0}^{i} \\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i - p} \\binom{i}{p} \\binom{s + 2i -2p}{2q} \\Gamma (q + \\tfrac{1}{2})\\\\& \\times \\frac{\\Gamma (\\frac{j + s}{2} + i - p - q + 1)}{\\Gamma (\\frac{n - k + j + s}{2} + i - p + 1)} \\frac{\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n + 1}{2} + p + q)}\\\\& \\times u^{s + 2i - 2p - 2q} \\Bigl ( \\vartheta _{n, k, j, p,q}^{(0)} Q \\bigl ( u^\\perp \\bigr )^{p + q} - \\vartheta _{n, k,j, p, q}^{(1)} Q \\bigl ( u^\\perp \\bigr )^{p + q - 1} Q(F)\\Bigr ),$ where we changed the order of summation with respect to $p$ .", "From the binomial theorem applied to $Q(u^\\perp )^{p + q} = (Q -u^2)^{p + q}$ we obtain $[2] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\gamma _{n, k, j} \\sum _{p = 0}^{i} \\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i - p} \\binom{i}{p} \\binom{s + 2i -2p}{2q} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{j + s}{2} + i - p - q + 1)}{\\Gamma (\\frac{n - k + j + s}{2} + i - p+ 1)}\\\\& \\times \\frac{\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n -k}{2} + q)}{\\Gamma (\\frac{n + 1}{2} + p + q)} \\biggl (\\sum _{z = 0}^{p + q} (-1)^{p + q - z} \\binom{p + q}{z}\\vartheta _{n, k, j, p, q}^{(0)} Q^{z} u^{s + 2i - 2z}\\\\& + \\sum _{z = 0}^{p + q - 1} (-1)^{p + q - z } \\binom{p + q -1}{z} \\vartheta _{n, k, j, p, q}^{(1)} Q^z u^{s + 2i - 2z -2} Q(F) \\biggr ).$ A change of the order of summation, such that we sum with respect to $z$ first, gives $\\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j- k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\& = \\gamma _{n, k , j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\bigl ( \\lambda _{n, k, j, s, i, z}^{(0)} u^2 +\\lambda _{n, k, j, s, i, z}^{(1)} Q(F) \\bigr ) Q^{z} u^{s + 2i -2z - 2},$ which concludes the proof.", "Next we state the special case of Proposition REF where $k = 1$ .", "Proposition 2 Let $i, s \\in {\\mathbb {N}}_0$ , $F \\in G(n, n - 1)$ and $u \\in F^{\\perp } \\cap \\mathbb {S}^{n-1}$ .", "Then $\\int _{G(n, 1)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{-1} [ F, L ]^2 \\, \\nu _1(\\mathrm {d}L)\\\\& = \\frac{\\Gamma (\\frac{n}{2}) \\Gamma (\\frac{s + 1}{2} + i)}{\\pi \\Gamma (\\frac{n + s + 1}{2} + i)} \\sum _{z = 0}^{\\frac{s}{2}+ i} (-1)^{z} \\binom{\\frac{s}{2} + i}{z} \\frac{1}{1 - 2z}u^{2z} Q^{\\frac{s}{2} + i - z}$ for even $s$ .", "If $s$ is odd, then $\\int _{G(n, 1)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{-1} [ F, L]^2 \\, \\nu _k(\\mathrm {d}L) = \\frac{\\Gamma (\\frac{n}{2})\\Gamma (\\frac{s}{2} + i + 1)}{\\sqrt{\\pi }\\Gamma (\\frac{n + s +1}{2} + i)} u Q^{\\frac{s - 1}{2} + i}.$ The proof basically works as the proof of Proposition REF .", "But we do not need to apply Lemma REF as (REF ) simplifies to $[3] \\int _{G(n, 1)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{-1} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\= {} & \\frac{\\Gamma (\\frac{n}{2})}{\\pi } \\sum _{p = 0}^{i}\\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + p} \\sum _{y = 0}^{q}(-1)^{y} \\binom{i}{p} \\binom{s + 2p}{2q} \\binom{q}{y} \\Gamma (q+ \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{s}{2} + p - q + 1)}{\\Gamma (\\frac{n + s + 1}{2} + p)}\\\\& \\times u^{s + 2p - 2q} Q \\bigl ( u^{\\perp } \\bigr )^{q - y}\\int _{G(u^\\perp , 0)} \\bigl ( [ F, U ]^{(u^\\perp )} \\bigr )^2 Q (U )^{i - p + y} \\, \\nu ^{u^\\perp }_{k - 1} (\\mathrm {d}U).$ Since the remaining integral on the right-hand side equals 1, if $p = i$ and $y = 0$ , and in all the other cases it equals 0, we obtain $\\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j- k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\& = \\frac{\\Gamma (\\frac{n}{2})}{\\pi } \\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\binom{s + 2i}{2q} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{s}{2} + i - q + 1)}{\\Gamma (\\frac{n + s + 1}{2} + i)} u^{s + 2i - 2q} Q \\bigl ( u^{\\perp }\\bigr )^{q}.$ Applying the binomial theorem to $Q ( u^{\\perp } )^{q} = (Q -u^2)^{q}$ yields $[1] \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s{p_L(u)}^{j - k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\& = \\frac{\\Gamma (\\frac{n}{2})}{\\pi } \\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\sum _{z = 0}^{q} (-1)^{q - z}\\binom{s + 2i}{2q} \\binom{q}{z} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{s}{2} + i - q + 1)}{\\Gamma (\\frac{n + s + 1}{2}+ i)} u^{s + 2i - 2z} Q^{z}.$ A change of the order of summation and Legendre's duplication formula applied to the Gamma functions involving $q$ give $\\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j- k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\& = \\frac{(s + 2i)!", "\\Gamma (\\frac{n}{2})}{2^{s + 2i}\\Gamma (\\frac{n + s + 1}{2} + i)} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\frac{1}{z!}", "\\sum _{q = z}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\frac{(-1)^{q - z} }{\\Gamma (\\frac{s + 1}{2} +i - q)(q - z)!}", "u^{s + 2i - 2z} Q^{z}.$ If $s$ is even, we conclude from Lemma REF applied to the summation with respect to  $q$ and from another application of Legendre's duplication formula that $\\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j- k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L)\\\\& = \\frac{\\Gamma (\\frac{n}{2}) \\Gamma (\\frac{s + 1}{2} + i)}{\\pi \\Gamma (\\frac{n + s + 1}{2} + i)} \\sum _{z = 0}^{\\frac{s}{2}+ i} (-1)^{\\frac{s}{2} + i - z + 1} \\binom{\\frac{s}{2} + i}{z}\\frac{1}{s + 2i - 2z - 1} u^{s + 2i - 2z} Q^{z}.$ A change of the order of summation with respect to $z$ then yields the assertion.", "On the other hand, if $s$ is odd, the binomial theorem gives, for $\\lfloor \\frac{s}{2} \\rfloor + i \\ne z$ , $\\sum _{q = z}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\frac{(-1)^{q -z} }{\\Gamma (\\frac{s + 1}{2} + i - q)(q - z)!}", "& = \\frac{1}{(\\lfloor \\frac{s}{2} \\rfloor + i - z)!}", "\\sum _{q = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i - z} (-1)^{q} \\binom{\\lfloor \\frac{s}{2}\\rfloor + i - z}{q} \\nonumber \\\\& = \\frac{1}{(\\lfloor \\frac{s}{2} \\rfloor + i - z)!}", "(1 -1)^{\\lfloor \\frac{s}{2} \\rfloor + i - z} \\nonumber \\\\& = 0.", "$ For $\\lfloor \\frac{s}{2} \\rfloor + i = z$ , the sum on the left-hand side of (REF ) equals 1.", "Hence, we finally obtain $\\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u)}^{j - k} [ F, L]^2 \\, \\nu _k(\\mathrm {d}L) = \\frac{\\Gamma (\\frac{n}{2})\\Gamma (\\frac{s}{2} + i + 1)}{\\sqrt{\\pi }\\Gamma (\\frac{n + s +1}{2} + i)} u Q^{\\lfloor \\frac{s}{2} \\rfloor + i},$ if $s$ is odd." ], [ "The Proofs for the Intrinsic Case", "Now all tools are available which are needed to prove the main theorems.", "We start with the proof of Theorem REF .", "Let $L \\in G(n, k)$ and $t \\in L^\\perp $ .", "Then we have $\\phi _{k,L_t}^{r,s,0}(K \\cap L_t, \\beta \\cap L_t) = {\\mathbf {1}} \\lbrace s = 0 \\rbrace \\int _{K \\cap \\beta \\cap L_t} x^r \\, {\\mathcal {H}}^{k}(\\mathrm {d}x)$ and thus, for $s \\ne 0$ , $\\int _{A(n, k)} Q(E)^i \\phi _{k,E}^{r,s,0} (K\\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E)\\\\& = \\int _{G(n, k)} \\int _{L^\\perp } Q(L_t)^i\\phi _{k,L_t}^{r,s,0} (K \\cap L_t, \\beta \\cap L_t) \\,{\\mathcal {H}}^{n - k} (\\mathrm {d}t) \\, \\nu _k(\\mathrm {d}L) = 0.$ Furthermore, for $s = 0$ Fubini's theorem yields $\\int _{A(n, k)} Q(E)^i \\phi _{k,E}^{r,0,0} (K\\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E)\\\\& = \\int _{G(n, k)} Q(L)^i \\int _{L^\\perp } \\int _{K \\cap \\beta \\cap L_t} x^r \\, {\\mathcal {H}}^{k}(\\mathrm {d}x) \\, {\\mathcal {H}}^{n - k} (\\mathrm {d}t)\\, \\nu _k(\\mathrm {d}L)\\\\& = \\int _{G(n, k)} Q(L)^i \\, \\nu _k(\\mathrm {d}L) \\int _{K \\cap \\beta } x^r \\, {\\mathcal {H}}^{n}(\\mathrm {d}x).$ Then we conclude the proof with Lemma REF and the definition of $\\phi _{n}^{r,0,0}$ .", "We turn to the proof of Theorem REF .", "First, we prove the formula for a polytope $P \\in {\\mathcal {P}}^n$ .", "The general result then follows by an approximation argument.", "As a matter of convenience, we name the integral of interest $I$ .", "Then Lemma REF yields $I & = \\omega _{k - j} \\int _{G(n, k)} Q(L)^i \\int _{L^\\perp }\\int _{L_t \\times (L \\cap \\mathbb {S}^{n-1})} {\\mathbf {1}}_{\\beta } (x) x^r u^s \\Lambda _j^{(L_t)}(P \\cap L_t, \\mathrm {d}(x, u)) \\, {\\mathcal {H}}^{n - k} (\\mathrm {d}t) \\, \\nu _k(\\mathrm {d}L)\\\\& = \\sum _{F \\in {\\mathcal {F}}_{n - k + j}(P)} \\int _{F \\cap \\beta } x^r{\\mathcal {H}}^{n - k + j} (\\mathrm {d}x) \\int _{G(n, k)} Q(L)^i\\\\& \\qquad \\qquad \\qquad \\times \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} \\pi _L(u)^s\\Vert p_L(u) \\Vert ^{j - k} [ F, L ]^2 \\, {\\mathcal {H}}^{k - j - 1}(\\mathrm {d}u) \\,\\nu _k(\\mathrm {d}L).$ With Fubini's theorem we conclude $I = {} & \\sum _{F \\in {\\mathcal {F}}_{n - k + j}(P)} \\int _{F \\cap \\beta }x^r {\\mathcal {H}}^{n - k + j} (\\mathrm {d}x) \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} \\nonumber \\\\& \\times \\int _{G(n, k)} Q(L)^i \\pi _L(u)^s {p_L(u) }^{j -k} [ F, L ]^2 \\, \\nu _k(\\mathrm {d}L) \\, {\\mathcal {H}}^{k - j - 1}(\\mathrm {d}u).", "$ Then we obtain from Proposition  REF $[3] I = \\gamma _{n, k, j} \\sum _{F \\in {\\mathcal {F}}_{n - k +j}(P)} \\int _{F \\cap \\beta } x^r {\\mathcal {H}}^{n - k + j} (\\mathrm {d}x)\\\\& \\times \\!\\begin{aligned}[t]\\biggl (& \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i}\\lambda _{n, k, j, s, i, z}^{(0)} Q^z \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}}u^{s + 2i - 2z} \\, {\\mathcal {H}}^{k - j - 1}(\\mathrm {d}u)\\\\& + \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i - 1}\\lambda _{n, k, j, s, i, z}^{(1)} Q^z Q(F) \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} u^{s + 2i - 2z - 2} \\, {\\mathcal {H}}^{k - j - 1}(\\mathrm {d}u)\\biggr ).\\end{aligned}$ With the definition of the tensorial curvature measures we get $I = {} & \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2}\\rfloor + i} \\lambda _{n, k, j, s, i, z}^{(0)} Q^z \\phi _{n -k + j}^{r,s + 2i - 2z,0} (P, \\beta ) + \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor +i - 1} \\lambda _{n, k, j, s, i, z}^{(1)} Q^z \\phi _{n - k +j}^{r,s + 2i - 2z - 2,1} (P, \\beta ).$ Combining the two sums yields the assertion in the polytopal case.", "As pointed out before, there exists a weakly continuous extension of the generalized tensorial curvature measures $\\smash{\\phi _{n- k + j}^{r,s + 2i - 2z - 2,1}}$ from the set of all polytopes to ${\\mathcal {K}}^{n}$ .", "The same is true for the tensorial curvature measures $\\phi _{n- k + j}^{r,s + 2i - 2z,0}$ .", "Hence, approximating a convex body $K \\in {\\mathcal {K}}^{n}$ by polytopes yields the assertion in the general case.", "Now we prove Theorem REF , which deals with the case $k=1$ excluded in the statement of Theorem REF .", "The proof basically works as the one of Theorem REF .", "Again, we prove the formula for a polytope $P \\in {\\mathcal {P}}^n$ .", "We call the integral of interest $I$ and proceed as in the previous proof in order to obtain (REF ).", "Now we apply Proposition REF and obtain $I = {} & \\frac{\\Gamma (\\frac{n}{2}) \\Gamma (\\frac{s + 1}{2} +i)}{\\pi \\Gamma (\\frac{n + s + 1}{2} + i)} \\sum _{z =0}^{\\frac{s}{2} + i} (-1)^{z} \\binom{\\frac{s}{2} + i}{z}\\frac{1}{1 - 2z}Q^{\\frac{s}{2} + i - z} \\\\& \\times \\sum _{F \\in {\\mathcal {F}}_{n - 1}(P)} \\int _{F \\cap \\beta } x^r{\\mathcal {H}}^{n - k + j} (\\mathrm {d}x) \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} u^{2z} \\,{\\mathcal {H}}^{0}(\\mathrm {d}u),$ if $s$ is even.", "Hence, we conclude the assertion with the definition of $\\phi _{n - 1}^{r,2z,0}$ .", "If $s$ is odd, Proposition REF yields $I = \\frac{\\Gamma (\\frac{n}{2}) \\Gamma (\\frac{s}{2} + i + 1)}{\\sqrt{\\pi }\\Gamma (\\frac{n + s + 1}{2} + i)} Q^{\\frac{s - 1}{2} + i}\\phi _{n - 1}^{r,1,0} (P, \\beta ).$ As sketched in the proof of Theorem REF , the general result follows by an approximation argument.", "For the proof of Theorem REF , we first globalize Theorem REF and then apply Lemma REF to treat the appearing tensors $\\smash{\\phi _{n - k + j}^{0,s + 2i - 2z - 2,1}}$ .", "We only prove the formula for a polytope $P \\in {\\mathcal {P}}^n$ .", "As before, the general result follows by an approximation argument.", "We briefly write $I$ for the Crofton integral under investigation.", "Starting from the special case of Theorem REF where $r = 0$ and $\\beta ={\\mathbb {R}}^n$ , we obtain $I = {} & \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2}\\rfloor + i} \\lambda _{n, k, j, s, i, z}^{(0)} Q^z \\Phi _{n- k + j}^{s + 2i - 2z} (P) + \\gamma _{n, k, j} \\sum _{z =0}^{\\lfloor \\frac{s}{2} \\rfloor + i - 1} \\lambda _{n, k, j, s,i, z}^{(1)} Q^z\\\\& \\qquad \\times \\sum _{F \\in {\\mathcal {F}}_{n - k + j}(P)} Q(F) {\\mathcal {H}}^{n -k + j} (F) \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} u^{s + 2i - 2z - 2} \\,{\\mathcal {H}}^{k - j - 1} (\\mathrm {d}u).$ With $Q(F) = Q - Q(N(P, F))$ and Lemma REF we get $\\sum _{F \\in {\\mathcal {F}}_{n - k + j}(P)} Q(F) {\\mathcal {H}}^{n - k +j}(F) \\int _{N(P,F) \\cap \\mathbb {S}^{n-1}} u^{s + 2i - 2z - 2} \\, {\\mathcal {H}}^{k- j - 1} (\\mathrm {d}u)\\\\& = Q \\Phi _{n - k + j}^{s + 2i - 2z - 2} (P) - \\tfrac{k -j + s + 2i - 2z - 2}{s + 2i - 2z - 1} \\Phi _{n - k +j}^{s + 2i - 2z} (P)$ and thus $I = {} & \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2}\\rfloor + i} \\lambda _{n, k, j, s, i, z}^{(0)} Q^z \\Phi _{n- k + j}^{s + 2i - 2z} (P)\\\\& + \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor +i - 1} \\lambda _{n, k, j, s, i, z}^{(1)} Q^z \\bigl ( Q\\Phi _{n - k + j}^{s + 2i - 2z - 2} (P) - \\tfrac{k - j + s+ 2i - 2z - 2}{s + 2i - 2z - 1} \\Phi _{n - k + j}^{s +2i - 2z} (P) \\bigr ).$ Combining these sums yields $I = \\gamma _{n, k, j} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\bigl ( \\lambda _{n, k, j, s, i, z}^{(0)} + \\lambda _{n,k, j, s, i, z - 1}^{(1)} - \\tfrac{k - j + s + 2i - 2z - 2}{s + 2i - 2z - 1} \\lambda _{n, k, j, s, i, z}^{(1)} \\bigr ) Q^z\\Phi _{n - k + j}^{s + 2i - 2z} (P).$ In fact, we have $\\lambda _{n, k, j, s, i, -1}^{\\smash{(1)}} = 0$ and, furthermore for even $s$ , as the sum with respect to $q$ is empty, $\\lambda _{n, k, j, s, i, \\lfloor \\frac{s}{2} \\rfloor +i}^{\\smash{(1)}}$ also vanishes.", "On the other hand, for odd $s$ , as $\\Phi _{n - k + j}^{1} \\equiv 0$ , the last summand of the sum with respect to $z$ actually vanishes and thus its coefficient does not have to be determined and is defined as zero.", "Hence, we obtained a representation of the integral with the desired Minkowski tensors.", "It remains to determine the coefficients explicitly.", "First, we consider the case where ($k >1$ and) $z \\in \\lbrace 1, \\ldots , \\lfloor \\frac{s}{2} \\rfloor + i - 1\\rbrace $ .", "We get $[3] \\lambda _{n, k, j, s, i, z}^{(0)} + \\lambda _{n,k, j, s, i, z - 1}^{(1)}\\\\= {} & \\sum _{p = 0}^{i} \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor + i - p} (-1)^{p + q - z} \\binom{i}{p} \\binom{s +2i - 2p}{2q} \\binom{p + q}{z} \\Gamma (q + \\tfrac{1}{2})\\\\& \\times \\frac{\\Gamma (\\frac{j + s}{2} + i - p - q + 1)\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - k + j + s}{2} + i - p + 1)\\Gamma (\\frac{n +1}{2} + p + q) }\\\\& \\times \\Bigl ( (n- k + j)(\\tfrac{k - 1}{2} + p) - \\tfrac{z}{p + q} \\bigl (p(n - k) - q(k - 1)\\bigr ) \\Bigr )$ and $\\lambda _{n, k, j, s, i, z}^{(1)} = {} & \\sum _{p = 0}^{i}\\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor + i - p}(-1)^{p + q - z} \\binom{i}{p} \\binom{s + 2i - 2p}{2q} \\binom{p+ q}{z} \\Gamma (q + \\tfrac{1}{2}) \\nonumber \\\\& \\times \\frac{\\Gamma (\\frac{j + s}{2} + i - p - q + 1)\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - k + j + s}{2} + i - p + 1)\\Gamma (\\frac{n +1}{2} + p + q)} \\nonumber \\\\& \\times \\tfrac{p + q - z}{p + q} \\bigl (p(n - k) - q(k -1)\\bigr ).", "$ Hence we conclude $[3] \\lambda _{n, k, j, s, i, z}^{(0)} + \\lambda _{n,k, j, s, i, z - 1}^{(1)} - \\tfrac{k - j + s + 2i - 2z - 2}{s + 2i - 2z - 1} \\lambda _{n, k, j, s, i, z}^{(1)}\\\\= {} & \\sum _{p = 0}^{i} \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor + i - p} (-1)^{p + q - z} \\binom{i}{p} \\binom{s +2i - 2p}{2q} \\binom{p + q}{z} \\Gamma (q + \\tfrac{1}{2})\\\\& \\times \\frac{\\Gamma (\\frac{j + s}{2} + i - p - q + 1)\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - k + j + s}{2} + i - p + 1)\\Gamma (\\frac{n +1}{2} + p + q) }\\\\& \\times \\left( (n - k + j) (\\tfrac{k - 1}{2} + p) - \\tfrac{p(n - k) - q(k - 1)}{p + q} \\left( p + q + \\tfrac{(k - j- 1)(p + q - z)}{s + 2i - 2z - 1} \\right) \\right).$ The case $z = \\lfloor \\frac{s}{2} \\rfloor + i$ , for even $s$ , follows similarly.", "For $z = 0$ , we have $\\lambda _{n, k, j, s,i, -1}^{(1)} = 0$ and (REF ) still holds, if one cancels the remaining $\\frac{p + q - z}{p + q} =1$ .", "Finally, we provide the argument for Corollary REF , which is the special case of Theorem REF obtained for $i=0$ and $j+1=k\\ge 2$ .", "With the specific choices of the indices, we obtain $\\lambda ^{(\\varepsilon )}_{n,k,k-1,s,0,z}= {} & \\sum _{q = z +\\varepsilon }^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{q - z}\\binom{s }{2q} \\binom{ q - \\varepsilon }{z} \\Gamma (q + \\tfrac{1}{2})\\\\& \\times \\frac{\\Gamma (\\frac{k + s + 1}{2} - q )}{\\Gamma (\\frac{n + s + 1}{2} )} \\frac{\\Gamma (\\frac{k - 1}{2} )\\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n + 1}{2} + q)}\\vartheta _{n, k, k-1, 0, q}^{(\\varepsilon )},$ with $\\vartheta _{n, k, k-1, 0, q}^{(0)} = \\tfrac{1}{2}(n -1) (k-1), \\qquad \\vartheta _{n, k, k-1, 0, q}^{(1)} := - q(k - 1),$ and $\\gamma _{n, k, k-1} = \\binom{n - 2}{k-1} \\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi }.$ Let us denote the Crofton integral by $I$ .", "Then Theorem REF implies that $I= {} &\\gamma _{n,k,k-1}\\sum _{z=0}^{\\lfloor \\frac{s}{2}\\rfloor }Q^z\\bigl (\\lambda ^{(0)}_{n,k,k-1,s,0,z}-\\lambda ^{(1)}_{n,k,k-1,s,0,z}\\bigr )\\phi _{n -1}^{r,s - 2z ,0}(K,\\beta )\\\\& + \\gamma _{n,k,k-1}\\sum _{z=1}^{\\lfloor \\frac{s}{2}\\rfloor +1}Q^z \\lambda ^{(1)}_{n,k,k-1,s,0,z-1} \\phi _{n-1}^{r,s - 2z ,0}(K,\\beta )\\\\= {} &\\gamma _{n,k,k-1}\\sum _{z=0}^{\\lfloor \\frac{s}{2}\\rfloor }Q^z\\bigl (\\underbrace{\\lambda ^{(0)}_{n,k,k-1,s,0,z}+\\lambda ^{(1)}_{n,k,k-1,s,0,z-1}-\\lambda ^{(1)}_{n,k,k-1,s,0,z}}_{=:\\lambda }\\bigr ) \\phi _{n-1}^{r,s - 2z ,0}(K,\\beta ),$ where $\\lambda = {} & \\frac{\\Gamma (\\frac{k - 1}{2} )}{\\Gamma (\\frac{n + s+ 1}{2} )}\\sum _{q=z}^{\\lfloor \\frac{s}{2} \\rfloor }(-1)^{q -z} \\binom{s }{2q} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + s + 1}{2} - q )\\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n +1}{2} + q)}\\\\& \\times \\!\\begin{aligned}[t]\\biggl [&\\binom{q}{z}\\frac{1}{2}(n-1)(k-1)-\\binom{q-1}{z-1}(-1)q(k-1) -\\binom{q-1}{z}(-1)q(k-1)\\biggr ]\\end{aligned}\\\\= {} &\\frac{\\Gamma (\\frac{k - 1}{2} )}{\\Gamma (\\frac{n + s + 1}{2})}\\sum _{q=z}^{\\lfloor \\frac{s}{2} \\rfloor }(-1)^{q - z}\\binom{s }{2q} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k +s + 1}{2} - q )\\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n +1}{2} + q)}\\binom{q}{z}(k-1)\\left(\\tfrac{n-1}{2} +q\\right)\\\\= {} &2\\frac{\\Gamma (\\frac{k +1}{2} )}{\\Gamma (\\frac{n + s + 1}{2})}\\sum _{q=z}^{\\lfloor \\frac{s}{2} \\rfloor }(-1)^{q - z}\\binom{s }{2q}\\binom{q}{z} \\Gamma (q + \\tfrac{1}{2})\\frac{\\Gamma (\\frac{k + s + 1}{2} - q )\\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n -1}{2} + q)},$ from which the assertion follows." ], [ "The Proofs for the Extrinsic Case", "Our starting point is a relation, due to McMullen, which relates the intrinsic and the extrinsic Minkowski tensors (see [16]).", "Its proof can easily be localized (see [21]).", "Combining this localization with the relation $Q=Q(E)+Q(E^\\perp )$ , where $E\\subset {\\mathbb {R}}^n$ is any $k$ -flat, we obtain the following lemma.", "Lemma 6 Let $j, k, r, s \\in {\\mathbb {N}}_{0}$ with $j < k < n$ , let $K \\in {\\mathcal {K}}^{n}$ with $K \\subset E \\in A(n , k)$ and $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ .", "Then $\\phi _{j}^{r,s,0} (K, \\beta ) & = \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - j + s}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{k - j + s}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "Q^{l} Q(E)^{m - l} \\phi _{j,E}^{r,s - 2m,0} (K, \\beta ).$ We start with the proof of Theorem REF , for which we use Theorem REF after an application of Lemma REF .", "Lemma REF for $j=k-1$ gives $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} Q^{l} \\\\& \\qquad \\qquad \\times \\int _{A(n, k)} Q(E)^{m - l} \\phi _{k - 1,E}^{r,s - 2m,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E).$ For $j=k-1$ we can argue as in the proof of Corollary REF to see that Theorem REF implies that $& \\int _{A(n, k)} Q(E)^{i} \\phi _{k - 1,E}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\nonumber \\\\& \\qquad = \\gamma _{n, k, k - 1} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\lambda _{n, k, k - 1, s, i, z} \\, Q^z \\phi _{n - 1}^{r,s + 2i - 2z,0} (K \\cap E, \\beta \\cap E), $ where $\\lambda _{n, k, k - 1, s, i, z} & = (k - 1) \\sum _{p = 0}^{i} \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor + i - p} (-1)^{p + q - z} \\binom{i}{p} \\binom{s + 2i - 2p}{2q} \\binom{p + q}{z} \\\\& \\qquad \\qquad \\times \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + s + 1}{2} + i - p - q)}{\\Gamma (\\frac{n + s + 1}{2} + i - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q)}.$ (Of course, for $i=0$ we recover Corollary REF .)", "Hence, we obtain $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor - l} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\\\& \\qquad \\qquad \\times \\lambda _{n, k, k - 1, s - 2m, m - l, z} \\, Q^{l + z} \\phi _{n - 1}^{r,s - 2l - 2z,0} (K, \\beta ) .$ An index shift of the summation with respect to $z$ yields $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} \\sum _{z = l}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\\\& \\qquad \\qquad \\times \\lambda _{n, k, k - 1, s - 2m, m - l, z - l} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ Changing the order of summation gives $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\nonumber \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{z} \\sum _{m = l}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\nonumber \\\\& \\qquad \\qquad \\times \\lambda _{n, k, k - 1, s - 2m, m - l, z - l} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).", "$ The coefficients of the tensorial curvature measures on the right-hand side of (REF ) do not depend on the choice of $r \\in {\\mathbb {N}}_{0}$ or $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ .", "Thus, we can set $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) = \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor }\\kappa _{n, k, s, z} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ),$ where the coefficient $\\kappa _{n, k, s, z}$ is uniquely defined in the obvious way.", "By choosing $r = 0$ and $\\beta = {\\mathbb {R}}^{n}$ , we can compare this to the Crofton formula for translation invariant Minkowski tensors in [3].", "In fact, since the functionals $Q^{z} \\phi _{n - 1}^{0,s - 2z,0} (K, {\\mathbb {R}}^n)$ , $z\\in \\lbrace 0,\\ldots ,\\lfloor s/2\\rfloor \\rbrace \\setminus \\lbrace ({s-1})/{2}\\rbrace $ , are linearly independent, we can conclude from the Crofton formula for the translation invariant Minkowski tensors in [3] that $\\kappa _{n, k, s, z} = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s + 1}{2}) \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\frac{\\Gamma (\\tfrac{n - k}{2} + z) \\Gamma (\\tfrac{k + s - 1}{2} - z)}{\\Gamma (\\frac{s}{2} - z + 1) z!", "}$ for $z \\ne ({s - 1})/ 2$ .", "If $z = ( {s - 1} )/2$ , then $\\phi _{n - 1}^{0,s - 2z,0} (K, {\\mathbb {R}}^{n}) = \\Phi _{n - 1}^{1}(K) = 0$ , and hence we do not get any information about the corresponding coefficient from the global theorem.", "Consequently, we have to calculate $\\kappa _{n, k, s, ( {s - 1} )/2}$ directly, which is what we do later in the proof.", "But first we demonstrate that the coefficients of the tensorial curvature measures in (REF ) can be determined also by a direct calculation if $s$ is even.", "In fact, we obtain $S & := \\sum _{m = l}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\lambda _{n, k, k - 1, s - 2m, m - l, z - l} \\\\& \\phantom{:}= (k - 1) \\sum _{m = l}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{p = l}^{m} \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{m + l + p + q - z} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& \\qquad \\times \\binom{m}{l} \\binom{m - l}{p - l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\\\& \\qquad \\times \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ Changing the order of summation gives $S & = (k - 1) \\sum _{p = l}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{l + q - z} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\\\& \\qquad \\times \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)} \\\\& \\qquad \\times \\sum _{m = p}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m + p} \\binom{m}{l} \\binom{m - l}{p - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!", "}.$ We denote the sum with respect to $m$ by $T$ and conclude $T & = \\sum _{m = p}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m + p} \\binom{m}{l} \\binom{m - l}{p - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& = \\frac{1}{l!", "(p - l)!}", "\\sum _{m = p}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m + p} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, (m - p)!", "(s - 2m)!", "}.$ An index shift yields $T & = \\frac{1}{2^{s} l!", "(p - l)!}", "\\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{m} \\frac{2^{s - 2p - 2m} \\Gamma (\\frac{s + 1}{2} - p - m)}{m!", "(s - 2p - 2m)!", "}.$ Legendre's duplication formula gives $T & = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - p - m + 1)}.$ If $s$ is even, the binomial theorem yields $T & = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!", "(\\frac{s}{2} - p)!}", "\\sum _{m = 0}^{\\frac{s}{2} - p} (-1)^{m} \\binom{\\frac{s}{2} - p}{m} \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!", "(\\frac{s}{2} - p)!}", "(1 - 1)^{\\frac{s}{2} - p} \\\\& = {\\mathbf {1}}\\lbrace p = \\frac{s}{2} \\rbrace \\frac{\\sqrt{\\pi }}{2^{s} l!", "(\\frac{s}{2} - l)!", "}.$ Hence, we obtain $S & = \\frac{(k - 1) \\sqrt{\\pi }}{2^{s} l!", "(\\frac{s}{2} - l)!}", "\\sum _{q = (z - \\frac{s}{2})^+}^{0} (-1)^{l + q - z} \\binom{\\frac{s}{2} + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + 1}{2} - q)}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n + s - 1}{2} + q - l)} \\\\& = (-1)^{l - z} \\frac{(k - 1) \\sqrt{\\pi }\\Gamma (\\tfrac{1}{2})}{2^{s} l!", "(\\frac{s}{2} - l)!}", "\\binom{\\frac{s}{2} - l}{z - l} \\frac{\\Gamma (\\frac{k + 1}{2})}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l) \\Gamma (\\frac{n - k}{2})}{\\Gamma (\\frac{n + s - 1}{2} - l)} \\\\& = (-1)^{l - z} \\frac{\\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{n - k}{2})}{\\Gamma (\\frac{n + 1}{2})} \\frac{(k - 1) \\pi }{2^{s} l!", "(\\frac{s}{2} - l)!}", "\\binom{\\frac{s}{2} - l}{z - l} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l)}{\\Gamma (\\frac{n + s - 1}{2} - l)}.$ Furthermore, Legendre's duplication formula yields $s!", "S & = (-1)^{l - z} \\frac{(k - 1) \\sqrt{\\pi }\\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{n - k}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + 1}{2})}\\underbrace{\\binom{\\frac{s}{2}}{l} \\binom{\\frac{s}{2} - l}{z - l}}_{ = \\binom{\\frac{s}{2}}{z} \\binom{z}{l}} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l)}{\\Gamma (\\frac{n + s - 1}{2} - l)}.$ Thus, we obtain $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}}}{\\Gamma (\\frac{n - k + s + 1}{2})} \\frac{(k - 1) \\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{n - k}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + 1}{2})} \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\\\& \\qquad \\qquad \\times \\sum _{l = 0}^{z} (-1)^{l - z} \\binom{z}{l} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l)}{\\Gamma (\\frac{n + s - 1}{2} - l)} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ From Lemma REF we conclude $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}}}{\\Gamma (\\frac{n - k + s + 1}{2})} \\frac{(k - 1) \\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\Gamma (\\tfrac{k + s - 1}{2} - z) \\Gamma (\\tfrac{n - k}{2} + z) Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ With $\\gamma _{n, k, k - 1} = \\binom{n - 2}{k - 1} \\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi } = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 1)!}", "\\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi }$ we get $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\frac{(n - 2)!", "}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\tfrac{n - k + 1}{2}) }{(n - k - 1)!}", "\\frac{\\Gamma (\\frac{k + 1}{2})}{(k - 2)!}", "\\frac{\\pi ^{\\frac{n - k - 1}{2}} \\Gamma (\\frac{s + 1}{2})}{2 \\Gamma (\\frac{n + s - 1}{2}) \\Gamma (\\frac{n - k + s + 1}{2})} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\Gamma (\\tfrac{k + s - 1}{2} - z) \\Gamma (\\tfrac{n - k}{2} + z) Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ Legendre's formula applied three times gives $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})}\\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + s - 1}{2}) \\Gamma (\\frac{n - k + s + 1}{2})} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\Gamma (\\tfrac{k + s - 1}{2} - z) \\Gamma (\\tfrac{n - k}{2} + z) Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ),$ which confirms the coefficients for even $s$ .", "On the other hand, if $s$ is odd, then Lemma REF yields $T & = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\sum _{m = 0}^{\\frac{s - 1}{2} - p} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - p - m + 1)} \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\bigg ( \\sum _{m = 0}^{\\frac{s + 1}{2} - p} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - p - m + 1)} - (-1)^{\\frac{s + 1}{2} - p} \\frac{1}{(\\frac{s + 1}{2} - p)!", "\\Gamma (\\frac{1}{2})} \\bigg ) \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\bigg ( (-1)^{\\frac{s + 1}{2} - p} \\frac{1}{\\sqrt{\\pi }(- s + 2p) (\\frac{s + 1}{2} - p)!", "}- (-1)^{\\frac{s + 1}{2} - p} \\frac{1}{\\sqrt{\\pi }(\\frac{s + 1}{2} - p)!}", "\\bigg ) \\\\& = (-1)^{\\frac{s - 1}{2} - p} \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\frac{1}{\\sqrt{\\pi }(\\frac{s + 1}{2} - p)!}", "(\\tfrac{1}{s - 2p} + 1) \\\\& = (-1)^{\\frac{s - 1}{2} - p} \\frac{1}{2^{s - 1} (s - 2p) (\\frac{s - 1}{2} - p)!", "l!", "(p - l)!}", "\\\\& = (-1)^{\\frac{s - 1}{2} - p} \\frac{2 \\Gamma (\\frac{s}{2} + 1)}{\\sqrt{\\pi }(s - 2p) s!}", "\\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l}.$ Hence, we obtain $s!", "\\sum _{l = 0}^{z} S & = \\frac{2 (k - 1) \\Gamma (\\frac{s}{2} + 1)}{\\sqrt{\\pi }} \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} \\sum _{q = (z - p)^+}^{\\frac{s - 1}{2} - p} (-1)^{\\frac{s - 1}{2} + l + p + q - z} \\frac{1}{(s - 2p)} \\\\& \\qquad \\times \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\\\& \\qquad \\times \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ This yields $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\quad = 2 (k - 1) \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k - 1}{2}} \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{z = 0}^{\\frac{s - 1}{2}} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ) \\\\& \\qquad \\times \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} \\sum _{q = (z - p)^+}^{\\frac{s - 1}{2} - p} (-1)^{\\frac{s - 1}{2} + l + p + q - z} \\frac{1}{(s - 2p)} \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\\\& \\qquad \\times \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ With $\\gamma _{n, k, k - 1} = \\binom{n - 2}{k - 1} \\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi } = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 1)!}", "\\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi }$ we get $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 2)!}", "\\frac{\\pi ^{\\frac{n - k - 3}{2}} \\Gamma (\\frac{n - k + 1}{2}) \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{z = 0}^{\\frac{s - 1}{2}} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ) \\\\& \\quad \\times \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} \\sum _{q = (z - p)^+}^{\\frac{s - 1}{2} - p} (-1)^{\\frac{s - 1}{2} + l + p + q - z} \\frac{1}{(s - 2p)} \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\\\& \\quad \\times \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ We denote the threefold sum with respect to $l$ , $p$ and $q$ by $R$ .", "Hence, $R$ multiplied with the factor in front of the sum with respect to $z$ equals $\\kappa _{n, k, s, z}$ .", "A direct calculation for $R$ still remains an open task.", "However, for the proof this is not required.", "Finally, if $s$ is odd we calculate the only so far unknown coefficient $\\kappa _{n, k, s, ( {s - 1})/ 2}$ .", "For $z = ( {s - 1})/ 2$ we see that the sum over $q$ only contains one summand, namely $q = ( {s - 1})/ 2 - p$ .", "Hence, we obtain $R & = \\Gamma (\\tfrac{k}{2} + 1) \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2} + l} \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\Gamma (\\tfrac{s}{2} - p) \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k + s - 1}{2} - p)}{\\Gamma (\\frac{n + s + 1}{2} - p) \\Gamma (\\frac{n + s}{2} - l - 1)} \\\\& = \\Gamma (\\tfrac{k}{2} + 1) \\sum _{p = 0}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2}} \\binom{\\frac{s - 1}{2}}{p} \\Gamma (\\tfrac{s}{2} - p) \\frac{\\Gamma (\\frac{n - k + s - 1}{2} - p)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\sum _{l = 0}^{p} (-1)^{l} \\binom{p}{l} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l)}{\\Gamma (\\frac{n + s}{2} - l - 1)}.$ Then Lemma REF yields $R & = \\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2}) \\Gamma (\\frac{n - k + s - 1}{2})}{\\Gamma (\\frac{n + s}{2} - 1)} \\sum _{p = 0}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2} + p} \\binom{\\frac{s - 1}{2}}{p} \\frac{\\Gamma (\\frac{s}{2} - p)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\\\& = \\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2}) \\Gamma (\\frac{n - k + s - 1}{2})}{\\Gamma (\\frac{n + s}{2} - 1)} \\sum _{p = 0}^{\\frac{s - 1}{2}} (-1)^{p} \\binom{\\frac{s - 1}{2}}{p} \\frac{\\Gamma (\\frac{1}{2} + p)}{\\Gamma (\\frac{n}{2} + 1 + p)}.$ Again, we apply Lemma REF and obtain $R & = \\sqrt{\\pi }\\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2})}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\frac{n + s}{2}) \\Gamma (\\frac{n - k + s - 1}{2})}{\\Gamma (\\frac{n + s}{2} - 1) \\Gamma (\\frac{n + s + 1}{2}) }.$ Thus, we conclude $\\kappa _{n, k, s, \\frac{s - 1}{2}} & = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 2)!}", "\\frac{\\pi ^{\\frac{n - k - 3}{2}} \\Gamma (\\frac{n - k + 1}{2}) \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2})} R \\\\& = \\pi ^{\\frac{n - k - 2}{2}} \\frac{(n - 2)!", "}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2})}{(k - 2)!}", "\\frac{\\Gamma (\\frac{n - k + 1}{2})}{(n - k - 1)!}", "\\frac{(n + s - 2) \\Gamma (\\frac{s}{2} + 1)}{(n - k + s - 1) \\Gamma (\\frac{n + s + 1}{2}) }.$ Applying three times Legendre's formula gives $\\kappa _{n, k, s, \\frac{s - 1}{2}} = \\pi ^{\\frac{n - k - 1}{2}}\\frac{2k (n + s - 2)}{(n - 1) (n - k + s - 1)}\\frac{\\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s}{2} + 1)}{ \\Gamma (\\frac{n + s + 1}{2}) },$ which completes the argument.", "Next we prove Theorem REF .", "As in the previous proof, one can compare the Crofton integral to the global one obtained in [3].", "However, we deduce it directly from Theorem REF .", "Lemma REF yields $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 1}{2}} s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "Q^{l} \\\\& \\qquad \\qquad \\times \\int _{A(n, 1)} Q(E)^{m - l} \\phi _{0,E}^{r,s - 2m,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E).$ If $s \\in {\\mathbb {N}}_0$ is even, we conclude from Theorem REF $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 1}{2}} s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{m = 0}^{\\frac{s}{2}} \\sum _{l = 0}^{m} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& \\qquad \\qquad \\times \\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{s + 1}{2} - l)}{ \\pi \\Gamma ( \\frac{n + s + 1}{2} - l) } \\sum _{z = 0}^{\\frac{s}{2} - l} (-1)^{z} \\binom{\\frac{s}{2} - l}{z} \\frac{1}{1 - 2z} \\, Q^{\\frac{s}{2} - z} \\phi _{n - 1}^{r,2z,0} (K, \\beta ).$ A change of the order of summation yields $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 1}{2}} s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{\\frac{s}{2}} \\sum _{m = l}^{\\frac{s}{2}} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& \\qquad \\qquad \\times \\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{s + 1}{2} - l)}{ \\pi \\Gamma ( \\frac{n + s + 1}{2} - l) } \\sum _{z = 0}^{\\frac{s}{2} - l} (-1)^{z} \\binom{\\frac{s}{2} - l}{z} \\frac{1}{1 - 2z} \\, Q^{\\frac{s}{2} - z} \\phi _{n - 1}^{r,2z,0} (K, \\beta ).$ Legendre's duplication formula gives for the sum with respect to $m$ , which we denote by $S$ , $S & = \\frac{\\sqrt{\\pi }}{2^{s}} \\sum _{m = l}^{\\frac{s}{2}} (-1)^{m - l} \\binom{m}{l} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - m + 1)} \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\sum _{m = 0}^{\\frac{s}{2} - l} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - l - m + 1)}.$ As seen before, we conclude from the binomial theorem $S & = \\frac{\\sqrt{\\pi }}{2^{s} (\\frac{s}{2} - l)!", "l!}", "\\sum _{m = 0}^{\\frac{s}{2} - l} (-1)^{m} \\binom{\\frac{s}{2} - l}{m} \\\\& = {\\mathbf {1}}\\lbrace l = \\frac{s}{2} \\rbrace \\frac{\\Gamma (\\frac{s + 1}{2})}{s!", "}.$ Hence, we obtain $\\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E)& = \\frac{\\pi ^{\\frac{n - 3}{2}} \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + s}{2})} \\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{1}{2})}{\\Gamma ( \\frac{n + 1}{2}) } Q^{\\frac{s}{2}} \\phi _{n - 1}^{r,0,0} (K, \\beta ) \\\\& = \\pi ^{\\frac{n - 2}{2}}\\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + s}{2}) \\Gamma ( \\frac{n + 1}{2}) } Q^{\\frac{s}{2}} \\phi _{n - 1}^{r,0,0} (K, \\beta ).$ On the other hand, if $s \\in {\\mathbb {N}}$ is odd, we conclude from Theorem REF $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{m = 0}^{ \\frac{s-1}{2} } \\sum _{l = 0}^{m} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\frac{\\Gamma (\\frac{s}{2} - l + 1)}{\\Gamma (\\frac{n + s + 1}{2} - l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ A change of the order of summation yields $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{ \\frac{s-1}{2} } \\sum _{m = l}^{\\frac{s-1}{2}} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\frac{\\Gamma (\\frac{s}{2} - l + 1)}{\\Gamma (\\frac{n + s + 1}{2} - l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ Legendre's duplication formula gives for the sum with respect to $m$ , which we denote by $S$ , $S & = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\sum _{m = 0}^{\\frac{s-1}{2} - l} (-1)^{m} \\frac{1}{ m!", "\\Gamma (\\frac{s}{2} - l - m + 1)}.$ Then Lemma REF yields $S & = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\bigg ( \\sum _{m = 0}^{\\frac{s + 1}{2} - l} (-1)^{m} \\frac{1}{ m!", "\\Gamma (\\frac{s}{2} - l - m + 1)} - (-1)^{\\frac{s + 1}{2} - l} \\frac{1}{ (\\frac{s + 1}{2} - l)!", "\\Gamma (\\frac{1}{2})} \\bigg ) \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\bigg ( (-1)^{\\frac{s - 1}{2} - l} \\frac{1}{ \\sqrt{\\pi }(s - 2l) (\\frac{s + 1}{2} - l)!}", "- (-1)^{\\frac{s + 1}{2} - l} \\frac{1}{ \\sqrt{\\pi }(\\frac{s + 1}{2} - l)!}", "\\bigg ) \\\\& = (-1)^{\\frac{s - 1}{2} - l} \\frac{1}{2^{s - 1} l!", "(s - 2l) (\\frac{s - 1}{2} - l)!", "}.$ Hence, we obtain $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{2^{s} \\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2} - l} \\frac{1}{l!", "(\\frac{s - 1}{2} - l)!}", "\\frac{\\Gamma (\\frac{s}{2} - l)}{\\Gamma (\\frac{n + s + 1}{2} - l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{2^{s} \\Gamma (\\frac{s + 1}{2}) \\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{\\frac{s - 1}{2}} (-1)^{l} \\binom{\\frac{s - 1}{2}}{l} \\frac{\\Gamma (l + \\frac{1}{2})}{\\Gamma (\\frac{n + 2}{2} + l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ Then Lemma REF gives $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{s!", "}{2^{s} \\Gamma (\\frac{s + 1}{2})} \\frac{\\pi ^{\\frac{n - 1}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n + s + 1}{2}) \\Gamma (\\frac{n + 1}{2})} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ Finally, the assertion follows from Legendre's duplication formula.", "Finally, we show that the Crofton formula has a very simple form in the $\\psi $ -representation of tensorial curvature measures.", "The cases $s\\in \\lbrace 0,1\\rbrace $ are checked directly, hence we can assume $s\\ge 2$ in the following.", "Using (REF ) we get $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\,\\mu _{k}(\\mathrm {d}E) \\nonumber \\\\& \\qquad = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor }(-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2})\\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} Q^{j} \\nonumber \\\\& \\qquad \\qquad \\qquad \\times \\int _{A(n, k)} \\phi _{k - 1}^{r,s - 2j,0}(K \\cap E, \\beta \\cap E) \\,\\mu _{k}(\\mathrm {d}E).", "$ Then, for $k \\ne 1$ , Theorem REF yields $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\, \\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor - j} \\kappa _{n, k, s - 2j, z} \\, Q^{z + j} \\phi _{n - 1}^{r,s - 2j - 2z,0} (K, \\beta ) \\\\& \\qquad = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{z = j}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} \\kappa _{n, k, s - 2j, z - j} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ),$ where $\\kappa _{n, k, s - 2j, z - j} = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s + 1}{2} - j) \\Gamma (\\frac{s}{2} - j + 1)}{\\Gamma (\\frac{n - k + s + 1}{2} - j) \\Gamma (\\frac{n + s - 1}{2} - j)} \\frac{\\Gamma (\\tfrac{n - k}{2} + z - j) \\Gamma (\\tfrac{k + s - 1}{2} - z)}{\\Gamma (\\frac{s}{2} - z + 1) (z - j)!", "},$ if $z \\ne (s - 1) /2$ .", "On the other hand, if $z = (s - 1) /2$ , then the coeffcient needs to be multiplied by the factor $\\frac{k(n + s - 2j - 2)}{(k - 1)(n + s - 2j - 1)}$ (see the comment after the proof of Theorem REF ).", "Applying Legendre's duplication formula twice, we thus obtain $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\,\\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})} \\frac{s!", "}{2^{s}} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\frac{\\Gamma (\\tfrac{k + s - 1}{2} - z)}{z!", "\\Gamma (\\frac{n}{2} + s - 1) \\Gamma (\\frac{s}{2} - z + 1)} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ) \\\\& \\qquad \\qquad \\times \\sum _{j = 0}^{z} (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (\\frac{n}{2} + s - j - 1) \\Gamma (\\tfrac{n - k}{2} + z - j)}{\\Gamma (\\frac{n - k + s + 1}{2} - j) \\Gamma (\\frac{n + s - 1}{2} - j)} \\\\& \\qquad \\qquad \\times \\left( 1 - {\\mathbf {1}}\\lbrace z = \\tfrac{s - 1}{2} \\rbrace \\left(1 - \\tfrac{k(n + s - 2j - 2)}{(k - 1)(n + s - 2j - 1)} \\right) \\right),$ Denoting the sum with respect to $j$ by $S_{z}$ , an application of Lemma REF shows that $S_{z} & = \\sum _{j = 0}^{z} (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (\\frac{n}{2} + s - j - 1) \\Gamma (\\tfrac{n - k}{2} + z - j)}{\\Gamma (\\frac{n - k + s + 1}{2} - j) \\Gamma (\\frac{n + s - 1}{2} - j)} \\nonumber \\\\& = (-1)^{z} \\frac{\\Gamma (\\frac{n - k}{2}) \\Gamma (\\frac{s + 1}{2}) \\Gamma (\\frac{k + s - 1}{2}) \\Gamma (\\frac{n}{2} + s - z - 1)}{\\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2}) \\Gamma (\\frac{s + 1}{2} - z) \\Gamma (\\frac{k + s - 1}{2} - z)}, $ for $z \\ne (s-1)/ 2$ and $k>1$ .", "On the other hand, for $z = ( {s - 1} )/2 =: t$ , we obtain from Lemma REF and Lemma REF (since $s > 1$ and thus $t > 0$ ) that $S_{t} & = \\tfrac{k}{k - 1} \\sum _{j = 0}^{t} (-1)^{j} \\binom{t}{j} \\left( 1 - \\tfrac{1}{n + 2t - 2j} \\right) \\frac{\\Gamma (\\frac{n}{2} + 2t - j) \\Gamma (\\tfrac{n - k}{2} + t - j)}{\\Gamma (\\frac{n - k}{2} + t - j + 1) \\Gamma (\\frac{n}{2} + t - j)} \\\\& = \\tfrac{k}{k - 1} \\bigg ( \\sum _{j = 0}^{t} (-1)^{j} \\binom{t}{j} \\frac{\\Gamma (\\frac{n}{2} + 2t - j) \\Gamma (\\tfrac{n - k}{2} + t - j)}{\\Gamma (\\frac{n - k}{2} + t - j + 1) \\Gamma (\\frac{n}{2} + t - j)} \\\\& \\qquad \\qquad - \\sum _{j = 0}^{t} (-1)^{j} \\binom{t}{j} \\tfrac{1}{\\frac{n - k}{2} + t - j} \\frac{\\Gamma (\\frac{n}{2} + 2t - j)}{\\Gamma (\\frac{n}{2} + t - j + 1)} \\bigg ) \\\\& = (-1)^{t} \\frac{\\Gamma (\\frac{n - k}{2}) \\Gamma (t + 1) \\Gamma (\\frac{k}{2} + t)}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2} + t + 1)},$ which coincides with (REF ) for $z = ({s - 1})/ 2$ .", "Thus, we have $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\,\\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}} \\Gamma (\\frac{n}{2}) \\Gamma (\\frac{k + s - 1}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\frac{s!", "\\Gamma (\\frac{s + 1}{2})}{2^{s}} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{z} \\frac{\\Gamma (\\frac{n}{2} + s - z - 1)}{z!", "\\Gamma (\\frac{n}{2} + s - 1) \\Gamma (\\frac{s}{2} - z + 1) \\Gamma (\\frac{s + 1}{2} - z)} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ Applying Legendre's duplication formula twice, we get $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\, \\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{n}{2}) \\Gamma (\\frac{k + s - 1}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\\\& \\qquad \\qquad \\times \\frac{1}{\\sqrt{\\pi }} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{z} \\binom{s}{2z} \\frac{\\Gamma (z + \\frac{1}{2})\\Gamma (\\frac{n}{2} + s - z - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ With (REF ) we obtain the assertion for $k \\ne 1$ .", "On the other hand, if $k = 1$ , then Theorem REF yields for (REF ) that $& \\int _{A(n, 1)} \\psi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\,\\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 3}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n + 1}{2}) \\Gamma (\\frac{n}{2} + s - 1)} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1) \\Gamma (\\lfloor \\frac{s + 1}{2} \\rfloor - j + \\frac{1}{2})}{\\Gamma (\\frac{n}{2} + \\lfloor \\frac{s + 1}{2} \\rfloor - j)} \\\\& \\qquad \\qquad \\qquad \\times Q^{\\lfloor \\frac{s}{2} \\rfloor } \\phi _{n - 1}^{r, s - 2\\lfloor \\frac{s}{2} \\rfloor ,0} (K, \\beta ).$ Denoting the sum with respect to $j$ by $S$ and applying Legendre's duplication formula three times, we conclude that $S = \\sqrt{\\pi }\\Gamma (\\lfloor \\tfrac{s + 1}{2} \\rfloor + \\tfrac{1}{2}) \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{\\lfloor \\frac{s}{2} \\rfloor }{j} \\frac{\\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + \\lfloor \\frac{s + 1}{2} \\rfloor - j)}.$ Since $s\\ge 2$ , Lemma REF yields $S = 0$ due to (REF ), and hence the assertion." ], [ "Sums of Gamma Functions", "In this section, we state four basic identities involving sums of Gamma functions.", "Lemma 7 Let $q \\in {\\mathbb {N}}_0$ and $a, b >0$ .", "Then $\\sum _{y=0}^q (-1)^{y} \\binom{q}{y} \\frac{\\Gamma (a + y)}{\\Gamma (b + y)} = \\frac{\\Gamma (a) \\Gamma (b - a + q)}{\\Gamma (b + q) \\Gamma (b - a)}.$ Under the additional assumption $a<b$ , this lemma can be found as Lemma 15.6.4 in [1], which is also proved there.", "Since this case is not sufficient for our purposes, we deduce the current more general version via Zeilberger's algorithm.", "The factor $\\Gamma (b - a + q)$ in Lemma REF does not cause any problems in case $a - b - q \\in {\\mathbb {N}}_{0}$ , as the also appearing $\\Gamma (b - a)$ cancels out the singularity, see (REF ).", "We set $F(q, y) := (-1)^{y} \\binom{q}{y} \\frac{\\Gamma (a + y)}{\\Gamma (b + y)},$ for which we see that $F(q, y) = 0$ if $y \\notin \\lbrace 0, \\ldots , q \\rbrace $ , and $f(q) := \\sum _{y = 0}^{q} F(q, y).$ Furthermore, we define the function $G(q, y) :={\\left\\lbrace \\begin{array}{ll}\\frac{y (b + y - 1)}{q - y + 1} F(q, y), \\qquad & \\text{ for } y \\in \\lbrace 0, \\ldots , q\\rbrace , \\\\G(q, q) -(b + q) F(q + 1, q) & \\\\\\qquad + (b - a + q) F(q, q), \\qquad & \\text{ for } y = q + 1, \\\\0, & \\text{ else}.\\end{array}\\right.", "}$ A direct calculation yields $-(b + q - 1) F(q, y) + (b - a + q - 1) F(q - 1, y) = G(q - 1, y + 1) - G(q - 1, y)$ for $y \\in {\\mathbb {N}}_{0}$ .", "Summing this relation over $y \\in \\lbrace 0, \\ldots , q \\rbrace $ gives $- (b + q - 1) f(q) + (b - a + q - 1) f(q - 1) = 0$ and thus $f(q) & = \\frac{b - a + q - 1}{b + q - 1} f(q - 1) \\\\& = \\frac{(b - a + q - 2)(b - a + q - 1)}{(b + q - 2) (b + q - 1)} f(q - 2) \\\\& \\ \\, \\vdots \\\\& = \\frac{(b - a) \\cdots (b - a + q - 1)}{b \\cdots (b + q - 1)} f(0) \\\\& = \\frac{\\Gamma (b - a + q) \\Gamma (b)}{\\Gamma (b + q) \\Gamma (b - a)} f(0),$ where $\\frac{\\Gamma (b - a + q)}{\\Gamma (b - a)} = (b - a) \\cdots (b - a + q - 1)$ is well-defined, even for $a - b \\in {\\mathbb {N}}$ .", "With $f(0) = \\frac{\\Gamma (a)}{\\Gamma (b)}$ we obtain the assertion.", "Lemma 8 Let $a \\in {\\mathbb {N}}_0$ .", "Then $ \\sum _{q = 0}^{a} \\frac{(-1)^{q}}{\\Gamma (a - q + \\frac{1}{2}) q!}", "= \\frac{(-1)^{a}}{\\sqrt{\\pi }(1 - 2a)a !", "}.$ For the sum $S$ on the left-hand side of the asserted equation, we obtain $S = \\sum _{q = 0}^{a} \\left( \\frac{2q}{2a - 1} \\frac{(-1)^{q} }{\\Gamma (a - q + \\frac{1}{2}) q!}", "+ \\frac{2q + 2}{2a - 1} \\frac{(-1)^{q} }{\\Gamma (a - q - \\frac{1}{2}) (q + 1)!}", "\\right),$ where we use that $(-\\frac{1}{2})\\Gamma (-\\frac{1}{2})=\\sqrt{\\pi }$ .", "Due to cancellation in this telescoping sum, the assertion follows immediately.", "Finally, we establish the following lemmas.", "Lemma 9 Let $a, b, c \\in {\\mathbb {R}}$ and $z \\in {\\mathbb {N}}_{0}$ with $a > z\\ge 0$ and $b > 0$ .", "Then $& \\sum _{j = 0}^{z} (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (a - j) \\Gamma (b + z - j)}{\\Gamma (c - j) \\Gamma (a + b - c - j + 1)} \\\\& \\qquad = (-1)^{z} \\frac{\\Gamma (a - z) \\Gamma (b)}{\\Gamma (a + b - c + 1) \\Gamma (c)} \\frac{\\Gamma (a - c + 1)}{\\Gamma (a - c +1 - z )} \\frac{\\Gamma (c - b)}{\\Gamma (c - b - z)}.$ The factor $\\Gamma (a - c + 1)$ (resp.", "$\\Gamma (c - b)$ ) in Lemma REF does not cause any problems for $c - a \\in {\\mathbb {N}}$ (resp.", "$b - c \\in {\\mathbb {N}}_{0}$ ), as the also appearing $\\Gamma (a - c + 1 - z )$ (resp.", "$\\Gamma (c - b - z)$ ) cancels out the singularity.", "On the other hand, in our applications of the lemma, we only need the cases where $a-c+1>z$ and $c-b>z$ .", "We set $F(z, j) := (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (a - j) \\Gamma (b + z - j)}{\\Gamma (c - j) \\Gamma (a + b - c - j + 1)},$ for $j \\in \\lbrace 0, \\ldots , z \\rbrace $ , and $F(z, j) = 0$ in all other cases, and $f(z) := \\sum _{j = 0}^{z} F(z,j).$ Furthermore, we define the function $G(z, j) :={\\left\\lbrace \\begin{array}{ll}- \\frac{j (a - j) (b + z - j)}{z - j + 1} F(z, j), \\qquad & \\text{ for } j \\in \\lbrace 0, \\ldots , z \\rbrace , \\\\G(z, z) + (a - z - 1) F(z + 1, z) & \\\\\\qquad + (c - b - z - 1) (a - c - z) F(z, z), \\qquad & \\text{ for } j = z + 1, \\\\0, & \\text{ otherwise}.\\end{array}\\right.", "}$ A direct calculation yields $(a - z) F(z, j) + (c - b - z) (a - c - z + 1) F(z - 1, j) = G(z - 1, j + 1) - G(z - 1, j)$ for $j \\in {\\mathbb {N}}_{0}$ .", "Summing this relation over $j \\in \\lbrace 0, \\ldots , z \\rbrace $ gives $(a - z) f(z) + (c - b - z) (a - c - z + 1) f(z - 1) = 0$ and thus $f(z) & = - \\frac{(c - b - z) (a - c - z + 1)}{a - z} f(z - 1) \\\\& = \\frac{(c - b - z) (c - b - z + 1) (a - c - z + 1) (a - c - z + 2)}{(a - z)(a - z + 1)} f(z - 2) \\\\& \\ \\, \\vdots \\\\& = (-1)^{z} \\frac{(c - b - z) \\cdots (c - b - 1) (a - c - z + 1) \\cdots (a - c)}{(a - z) \\cdots (a - 1)} f(0) \\\\& = (-1)^{z} \\frac{\\Gamma (c - b) \\Gamma (a - c + 1) \\Gamma (a - z)}{\\Gamma (c - b - z) \\Gamma (a - c+1 - z ) \\Gamma (a)} f(0),$ where $\\frac{\\Gamma (c - b)}{\\Gamma (c - b - z)} = (c - b - z) \\cdots (c - b - 1)$ is well-defined, even for $b - c \\in {\\mathbb {N}}_{0}$ , and a similar statement holds for $ {\\Gamma (a - c + 1)} /\\Gamma (a - c +1- z )$ .", "With $f(0) = \\frac{\\Gamma (a) \\Gamma (b)}{\\Gamma (c) \\Gamma (a + b - c + 1)}$ we obtain the assertion.", "Lemma 10 Let $a, b \\in {\\mathbb {R}}$ with $a, b > 0$ and $t \\in {\\mathbb {N}}$ .", "Then $& \\sum _{j = 0}^{t} (-1)^{j} \\frac{1}{b + j} \\binom{t}{j} \\frac{\\Gamma (a + t + j)}{\\Gamma (a + 1 + j)} = \\frac{\\Gamma (a - b + t) \\Gamma (b) \\Gamma (t + 1)}{\\Gamma (a - b + 1) \\Gamma (b + t + 1)}.$ The factor $\\Gamma (a - b + t)$ in Lemma REF does not cause any problems for $b - a - t \\in {\\mathbb {N}}_{0}$ , as the also appearing $\\Gamma (a - b + 1)$ cancels out the singularity.", "In our application of the lemma, we will additionally know that $a>b$ .", "We set $F(t, j) := (-1)^{j} \\frac{1}{b + j} \\binom{t}{j} \\frac{\\Gamma (a + t + j)}{\\Gamma (a + 1 + j)},$ for which we see that $F(t, j) = 0$ if $j \\notin \\lbrace 0, \\ldots , t \\rbrace $ , and $f(t) := \\sum _{j = 0}^{t} F(t, j).$ Furthermore, we define the function $G(t, j) :={\\left\\lbrace \\begin{array}{ll}\\frac{j (a + j) (a + 2t + 1) (t^2 + t(a + 1) - j + 1) (b + j)}{t (t - j + 1) (a + t) (a + t + 1)} F(t, j), \\qquad & \\text{ for } j \\in \\lbrace 0, \\ldots , t \\rbrace , \\\\G(t, t) - (b + t + 1) F(t + 1, t) & \\\\\\qquad + (t + 1) (a - b + t) F(t, t), \\qquad & \\text{ for } j = t + 1, \\\\0, & \\text{ otherwise}.\\end{array}\\right.", "}$ A direct calculation yields $- (b + t) F(t, j) + t (a - b + t - 1) F(t - 1, j) = G(t - 1, j + 1) - G(t - 1, j)$ for $j \\in {\\mathbb {N}}_{0}$ .", "Summing this relation over $j \\in \\lbrace 0, \\ldots , t \\rbrace $ gives $- (b + t) f(t) + t (a - b + t - 1) f(t - 1) = 0$ and thus $f(t) & = \\frac{t (a - b + t - 1)}{b + t} f(t - 1) \\\\& = \\frac{(t - 1) t (a - b + t - 2) (a - b + t - 1)}{(b + t - 1) (b + t)} f(t - 2) \\\\& \\ \\, \\vdots \\\\& = \\frac{2 \\cdots t (a - b + 1) \\cdots (a - b + t - 1)}{(b + 2) \\cdots (b + t)} f(1) \\\\& = \\frac{\\Gamma (t + 1) \\Gamma (a - b + t) \\Gamma (b + 2)}{\\Gamma (a - b + 1) \\Gamma (b + t + 1)} f(1).$ With $f(1) = \\frac{1}{b} - \\frac{1}{b + 1} = \\frac{1}{b (b + 1)}$ we obtain the assertion." ], [ "The Proofs for the Extrinsic Case", "Our starting point is a relation, due to McMullen, which relates the intrinsic and the extrinsic Minkowski tensors (see [16]).", "Its proof can easily be localized (see [21]).", "Combining this localization with the relation $Q=Q(E)+Q(E^\\perp )$ , where $E\\subset {\\mathbb {R}}^n$ is any $k$ -flat, we obtain the following lemma.", "Lemma 6 Let $j, k, r, s \\in {\\mathbb {N}}_{0}$ with $j < k < n$ , let $K \\in {\\mathcal {K}}^{n}$ with $K \\subset E \\in A(n , k)$ and $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ .", "Then $\\phi _{j}^{r,s,0} (K, \\beta ) & = \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - j + s}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{k - j + s}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "Q^{l} Q(E)^{m - l} \\phi _{j,E}^{r,s - 2m,0} (K, \\beta ).$ We start with the proof of Theorem REF , for which we use Theorem REF after an application of Lemma REF .", "Lemma REF for $j=k-1$ gives $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} Q^{l} \\\\& \\qquad \\qquad \\times \\int _{A(n, k)} Q(E)^{m - l} \\phi _{k - 1,E}^{r,s - 2m,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E).$ For $j=k-1$ we can argue as in the proof of Corollary REF to see that Theorem REF implies that $& \\int _{A(n, k)} Q(E)^{i} \\phi _{k - 1,E}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\nonumber \\\\& \\qquad = \\gamma _{n, k, k - 1} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor + i} \\lambda _{n, k, k - 1, s, i, z} \\, Q^z \\phi _{n - 1}^{r,s + 2i - 2z,0} (K \\cap E, \\beta \\cap E), $ where $\\lambda _{n, k, k - 1, s, i, z} & = (k - 1) \\sum _{p = 0}^{i} \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor + i - p} (-1)^{p + q - z} \\binom{i}{p} \\binom{s + 2i - 2p}{2q} \\binom{p + q}{z} \\\\& \\qquad \\qquad \\times \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + s + 1}{2} + i - p - q)}{\\Gamma (\\frac{n + s + 1}{2} + i - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q)}.$ (Of course, for $i=0$ we recover Corollary REF .)", "Hence, we obtain $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor - l} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\\\& \\qquad \\qquad \\times \\lambda _{n, k, k - 1, s - 2m, m - l, z} \\, Q^{l + z} \\phi _{n - 1}^{r,s - 2l - 2z,0} (K, \\beta ) .$ An index shift of the summation with respect to $z$ yields $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} \\sum _{z = l}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\\\& \\qquad \\qquad \\times \\lambda _{n, k, k - 1, s - 2m, m - l, z - l} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ Changing the order of summation gives $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\nonumber \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k}{2}} s!", "}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{z} \\sum _{m = l}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\nonumber \\\\& \\qquad \\qquad \\times \\lambda _{n, k, k - 1, s - 2m, m - l, z - l} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).", "$ The coefficients of the tensorial curvature measures on the right-hand side of (REF ) do not depend on the choice of $r \\in {\\mathbb {N}}_{0}$ or $\\beta \\in {\\mathcal {B}}({\\mathbb {R}}^{n})$ .", "Thus, we can set $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) = \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor }\\kappa _{n, k, s, z} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ),$ where the coefficient $\\kappa _{n, k, s, z}$ is uniquely defined in the obvious way.", "By choosing $r = 0$ and $\\beta = {\\mathbb {R}}^{n}$ , we can compare this to the Crofton formula for translation invariant Minkowski tensors in [3].", "In fact, since the functionals $Q^{z} \\phi _{n - 1}^{0,s - 2z,0} (K, {\\mathbb {R}}^n)$ , $z\\in \\lbrace 0,\\ldots ,\\lfloor s/2\\rfloor \\rbrace \\setminus \\lbrace ({s-1})/{2}\\rbrace $ , are linearly independent, we can conclude from the Crofton formula for the translation invariant Minkowski tensors in [3] that $\\kappa _{n, k, s, z} = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s + 1}{2}) \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\frac{\\Gamma (\\tfrac{n - k}{2} + z) \\Gamma (\\tfrac{k + s - 1}{2} - z)}{\\Gamma (\\frac{s}{2} - z + 1) z!", "}$ for $z \\ne ({s - 1})/ 2$ .", "If $z = ( {s - 1} )/2$ , then $\\phi _{n - 1}^{0,s - 2z,0} (K, {\\mathbb {R}}^{n}) = \\Phi _{n - 1}^{1}(K) = 0$ , and hence we do not get any information about the corresponding coefficient from the global theorem.", "Consequently, we have to calculate $\\kappa _{n, k, s, ( {s - 1} )/2}$ directly, which is what we do later in the proof.", "But first we demonstrate that the coefficients of the tensorial curvature measures in (REF ) can be determined also by a direct calculation if $s$ is even.", "In fact, we obtain $S & := \\sum _{m = l}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\lambda _{n, k, k - 1, s - 2m, m - l, z - l} \\\\& \\phantom{:}= (k - 1) \\sum _{m = l}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{p = l}^{m} \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{m + l + p + q - z} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& \\qquad \\times \\binom{m}{l} \\binom{m - l}{p - l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\\\& \\qquad \\times \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ Changing the order of summation gives $S & = (k - 1) \\sum _{p = l}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{q = (z - p)^+}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{l + q - z} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\\\& \\qquad \\times \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)} \\\\& \\qquad \\times \\sum _{m = p}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m + p} \\binom{m}{l} \\binom{m - l}{p - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!", "}.$ We denote the sum with respect to $m$ by $T$ and conclude $T & = \\sum _{m = p}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m + p} \\binom{m}{l} \\binom{m - l}{p - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& = \\frac{1}{l!", "(p - l)!}", "\\sum _{m = p}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{m + p} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, (m - p)!", "(s - 2m)!", "}.$ An index shift yields $T & = \\frac{1}{2^{s} l!", "(p - l)!}", "\\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{m} \\frac{2^{s - 2p - 2m} \\Gamma (\\frac{s + 1}{2} - p - m)}{m!", "(s - 2p - 2m)!", "}.$ Legendre's duplication formula gives $T & = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor - p} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - p - m + 1)}.$ If $s$ is even, the binomial theorem yields $T & = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!", "(\\frac{s}{2} - p)!}", "\\sum _{m = 0}^{\\frac{s}{2} - p} (-1)^{m} \\binom{\\frac{s}{2} - p}{m} \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!", "(\\frac{s}{2} - p)!}", "(1 - 1)^{\\frac{s}{2} - p} \\\\& = {\\mathbf {1}}\\lbrace p = \\frac{s}{2} \\rbrace \\frac{\\sqrt{\\pi }}{2^{s} l!", "(\\frac{s}{2} - l)!", "}.$ Hence, we obtain $S & = \\frac{(k - 1) \\sqrt{\\pi }}{2^{s} l!", "(\\frac{s}{2} - l)!}", "\\sum _{q = (z - \\frac{s}{2})^+}^{0} (-1)^{l + q - z} \\binom{\\frac{s}{2} + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + 1}{2} - q)}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n + s - 1}{2} + q - l)} \\\\& = (-1)^{l - z} \\frac{(k - 1) \\sqrt{\\pi }\\Gamma (\\tfrac{1}{2})}{2^{s} l!", "(\\frac{s}{2} - l)!}", "\\binom{\\frac{s}{2} - l}{z - l} \\frac{\\Gamma (\\frac{k + 1}{2})}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l) \\Gamma (\\frac{n - k}{2})}{\\Gamma (\\frac{n + s - 1}{2} - l)} \\\\& = (-1)^{l - z} \\frac{\\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{n - k}{2})}{\\Gamma (\\frac{n + 1}{2})} \\frac{(k - 1) \\pi }{2^{s} l!", "(\\frac{s}{2} - l)!}", "\\binom{\\frac{s}{2} - l}{z - l} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l)}{\\Gamma (\\frac{n + s - 1}{2} - l)}.$ Furthermore, Legendre's duplication formula yields $s!", "S & = (-1)^{l - z} \\frac{(k - 1) \\sqrt{\\pi }\\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{n - k}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + 1}{2})}\\underbrace{\\binom{\\frac{s}{2}}{l} \\binom{\\frac{s}{2} - l}{z - l}}_{ = \\binom{\\frac{s}{2}}{z} \\binom{z}{l}} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l)}{\\Gamma (\\frac{n + s - 1}{2} - l)}.$ Thus, we obtain $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}}}{\\Gamma (\\frac{n - k + s + 1}{2})} \\frac{(k - 1) \\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{n - k}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + 1}{2})} \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\\\& \\qquad \\qquad \\times \\sum _{l = 0}^{z} (-1)^{l - z} \\binom{z}{l} \\frac{\\Gamma (\\frac{k + s - 1}{2} - l)}{\\Gamma (\\frac{n + s - 1}{2} - l)} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ From Lemma REF we conclude $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}}}{\\Gamma (\\frac{n - k + s + 1}{2})} \\frac{(k - 1) \\Gamma (\\frac{k + 1}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\Gamma (\\tfrac{k + s - 1}{2} - z) \\Gamma (\\tfrac{n - k}{2} + z) Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ With $\\gamma _{n, k, k - 1} = \\binom{n - 2}{k - 1} \\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi } = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 1)!}", "\\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi }$ we get $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\frac{(n - 2)!", "}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\tfrac{n - k + 1}{2}) }{(n - k - 1)!}", "\\frac{\\Gamma (\\frac{k + 1}{2})}{(k - 2)!}", "\\frac{\\pi ^{\\frac{n - k - 1}{2}} \\Gamma (\\frac{s + 1}{2})}{2 \\Gamma (\\frac{n + s - 1}{2}) \\Gamma (\\frac{n - k + s + 1}{2})} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\Gamma (\\tfrac{k + s - 1}{2} - z) \\Gamma (\\tfrac{n - k}{2} + z) Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ Legendre's formula applied three times gives $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})}\\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + s - 1}{2}) \\Gamma (\\frac{n - k + s + 1}{2})} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{ \\frac{s}{2} } \\binom{\\frac{s}{2}}{z} \\Gamma (\\tfrac{k + s - 1}{2} - z) \\Gamma (\\tfrac{n - k}{2} + z) Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ),$ which confirms the coefficients for even $s$ .", "On the other hand, if $s$ is odd, then Lemma REF yields $T & = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\sum _{m = 0}^{\\frac{s - 1}{2} - p} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - p - m + 1)} \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\bigg ( \\sum _{m = 0}^{\\frac{s + 1}{2} - p} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - p - m + 1)} - (-1)^{\\frac{s + 1}{2} - p} \\frac{1}{(\\frac{s + 1}{2} - p)!", "\\Gamma (\\frac{1}{2})} \\bigg ) \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\bigg ( (-1)^{\\frac{s + 1}{2} - p} \\frac{1}{\\sqrt{\\pi }(- s + 2p) (\\frac{s + 1}{2} - p)!", "}- (-1)^{\\frac{s + 1}{2} - p} \\frac{1}{\\sqrt{\\pi }(\\frac{s + 1}{2} - p)!}", "\\bigg ) \\\\& = (-1)^{\\frac{s - 1}{2} - p} \\frac{\\sqrt{\\pi }}{2^{s} l!", "(p - l)!}", "\\frac{1}{\\sqrt{\\pi }(\\frac{s + 1}{2} - p)!}", "(\\tfrac{1}{s - 2p} + 1) \\\\& = (-1)^{\\frac{s - 1}{2} - p} \\frac{1}{2^{s - 1} (s - 2p) (\\frac{s - 1}{2} - p)!", "l!", "(p - l)!}", "\\\\& = (-1)^{\\frac{s - 1}{2} - p} \\frac{2 \\Gamma (\\frac{s}{2} + 1)}{\\sqrt{\\pi }(s - 2p) s!}", "\\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l}.$ Hence, we obtain $s!", "\\sum _{l = 0}^{z} S & = \\frac{2 (k - 1) \\Gamma (\\frac{s}{2} + 1)}{\\sqrt{\\pi }} \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} \\sum _{q = (z - p)^+}^{\\frac{s - 1}{2} - p} (-1)^{\\frac{s - 1}{2} + l + p + q - z} \\frac{1}{(s - 2p)} \\\\& \\qquad \\times \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\Gamma (q + \\tfrac{1}{2}) \\\\& \\qquad \\times \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ This yields $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& \\quad = 2 (k - 1) \\gamma _{n, k, k - 1} \\frac{\\pi ^{\\frac{n - k - 1}{2}} \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{z = 0}^{\\frac{s - 1}{2}} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ) \\\\& \\qquad \\times \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} \\sum _{q = (z - p)^+}^{\\frac{s - 1}{2} - p} (-1)^{\\frac{s - 1}{2} + l + p + q - z} \\frac{1}{(s - 2p)} \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\\\& \\qquad \\times \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ With $\\gamma _{n, k, k - 1} = \\binom{n - 2}{k - 1} \\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi } = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 1)!}", "\\frac{\\Gamma (\\frac{n - k + 1}{2})}{2 \\pi }$ we get $& \\int _{A(n, k)} \\phi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _k(\\mathrm {d}E) \\\\& = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 2)!}", "\\frac{\\pi ^{\\frac{n - k - 3}{2}} \\Gamma (\\frac{n - k + 1}{2}) \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2})} \\sum _{z = 0}^{\\frac{s - 1}{2}} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ) \\\\& \\quad \\times \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} \\sum _{q = (z - p)^+}^{\\frac{s - 1}{2} - p} (-1)^{\\frac{s - 1}{2} + l + p + q - z} \\frac{1}{(s - 2p)} \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\binom{s - 2p}{2q} \\binom{p + q - l}{z - l} \\\\& \\quad \\times \\Gamma (q + \\tfrac{1}{2}) \\frac{\\Gamma (\\frac{k + s + 1}{2} - p - q)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k}{2} + q)}{\\Gamma (\\frac{n - 1}{2} + p + q - l)}.$ We denote the threefold sum with respect to $l$ , $p$ and $q$ by $R$ .", "Hence, $R$ multiplied with the factor in front of the sum with respect to $z$ equals $\\kappa _{n, k, s, z}$ .", "A direct calculation for $R$ still remains an open task.", "However, for the proof this is not required.", "Finally, if $s$ is odd we calculate the only so far unknown coefficient $\\kappa _{n, k, s, ( {s - 1})/ 2}$ .", "For $z = ( {s - 1})/ 2$ we see that the sum over $q$ only contains one summand, namely $q = ( {s - 1})/ 2 - p$ .", "Hence, we obtain $R & = \\Gamma (\\tfrac{k}{2} + 1) \\sum _{l = 0}^{z} \\sum _{p = l}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2} + l} \\binom{\\frac{s - 1}{2}}{p} \\binom{p}{l} \\Gamma (\\tfrac{s}{2} - p) \\frac{\\Gamma (\\frac{k - 1}{2} + p - l) \\Gamma (\\frac{n - k + s - 1}{2} - p)}{\\Gamma (\\frac{n + s + 1}{2} - p) \\Gamma (\\frac{n + s}{2} - l - 1)} \\\\& = \\Gamma (\\tfrac{k}{2} + 1) \\sum _{p = 0}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2}} \\binom{\\frac{s - 1}{2}}{p} \\Gamma (\\tfrac{s}{2} - p) \\frac{\\Gamma (\\frac{n - k + s - 1}{2} - p)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\sum _{l = 0}^{p} (-1)^{l} \\binom{p}{l} \\frac{\\Gamma (\\frac{k - 1}{2} + p - l)}{\\Gamma (\\frac{n + s}{2} - l - 1)}.$ Then Lemma REF yields $R & = \\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2}) \\Gamma (\\frac{n - k + s - 1}{2})}{\\Gamma (\\frac{n + s}{2} - 1)} \\sum _{p = 0}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2} + p} \\binom{\\frac{s - 1}{2}}{p} \\frac{\\Gamma (\\frac{s}{2} - p)}{\\Gamma (\\frac{n + s + 1}{2} - p)} \\\\& = \\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2}) \\Gamma (\\frac{n - k + s - 1}{2})}{\\Gamma (\\frac{n + s}{2} - 1)} \\sum _{p = 0}^{\\frac{s - 1}{2}} (-1)^{p} \\binom{\\frac{s - 1}{2}}{p} \\frac{\\Gamma (\\frac{1}{2} + p)}{\\Gamma (\\frac{n}{2} + 1 + p)}.$ Again, we apply Lemma REF and obtain $R & = \\sqrt{\\pi }\\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2})}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\frac{n + s}{2}) \\Gamma (\\frac{n - k + s - 1}{2})}{\\Gamma (\\frac{n + s}{2} - 1) \\Gamma (\\frac{n + s + 1}{2}) }.$ Thus, we conclude $\\kappa _{n, k, s, \\frac{s - 1}{2}} & = \\frac{(n - 2)!", "}{(n - k - 1)!", "(k - 2)!}", "\\frac{\\pi ^{\\frac{n - k - 3}{2}} \\Gamma (\\frac{n - k + 1}{2}) \\Gamma (\\frac{s}{2} + 1)}{\\Gamma (\\frac{n - k + s + 1}{2})} R \\\\& = \\pi ^{\\frac{n - k - 2}{2}} \\frac{(n - 2)!", "}{\\Gamma (\\frac{n + 1}{2})} \\frac{\\Gamma (\\tfrac{k}{2} + 1) \\Gamma (\\frac{k - 1}{2})}{(k - 2)!}", "\\frac{\\Gamma (\\frac{n - k + 1}{2})}{(n - k - 1)!}", "\\frac{(n + s - 2) \\Gamma (\\frac{s}{2} + 1)}{(n - k + s - 1) \\Gamma (\\frac{n + s + 1}{2}) }.$ Applying three times Legendre's formula gives $\\kappa _{n, k, s, \\frac{s - 1}{2}} = \\pi ^{\\frac{n - k - 1}{2}}\\frac{2k (n + s - 2)}{(n - 1) (n - k + s - 1)}\\frac{\\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s}{2} + 1)}{ \\Gamma (\\frac{n + s + 1}{2}) },$ which completes the argument.", "Next we prove Theorem REF .", "As in the previous proof, one can compare the Crofton integral to the global one obtained in [3].", "However, we deduce it directly from Theorem REF .", "Lemma REF yields $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 1}{2}} s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{m = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{l = 0}^{m} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "Q^{l} \\\\& \\qquad \\qquad \\times \\int _{A(n, 1)} Q(E)^{m - l} \\phi _{0,E}^{r,s - 2m,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E).$ If $s \\in {\\mathbb {N}}_0$ is even, we conclude from Theorem REF $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 1}{2}} s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{m = 0}^{\\frac{s}{2}} \\sum _{l = 0}^{m} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& \\qquad \\qquad \\times \\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{s + 1}{2} - l)}{ \\pi \\Gamma ( \\frac{n + s + 1}{2} - l) } \\sum _{z = 0}^{\\frac{s}{2} - l} (-1)^{z} \\binom{\\frac{s}{2} - l}{z} \\frac{1}{1 - 2z} \\, Q^{\\frac{s}{2} - z} \\phi _{n - 1}^{r,2z,0} (K, \\beta ).$ A change of the order of summation yields $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 1}{2}} s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{\\frac{s}{2}} \\sum _{m = l}^{\\frac{s}{2}} (-1)^{m - l} \\binom{m}{l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\\\& \\qquad \\qquad \\times \\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{s + 1}{2} - l)}{ \\pi \\Gamma ( \\frac{n + s + 1}{2} - l) } \\sum _{z = 0}^{\\frac{s}{2} - l} (-1)^{z} \\binom{\\frac{s}{2} - l}{z} \\frac{1}{1 - 2z} \\, Q^{\\frac{s}{2} - z} \\phi _{n - 1}^{r,2z,0} (K, \\beta ).$ Legendre's duplication formula gives for the sum with respect to $m$ , which we denote by $S$ , $S & = \\frac{\\sqrt{\\pi }}{2^{s}} \\sum _{m = l}^{\\frac{s}{2}} (-1)^{m - l} \\binom{m}{l} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - m + 1)} \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\sum _{m = 0}^{\\frac{s}{2} - l} (-1)^{m} \\frac{1}{m!", "\\Gamma (\\frac{s}{2} - l - m + 1)}.$ As seen before, we conclude from the binomial theorem $S & = \\frac{\\sqrt{\\pi }}{2^{s} (\\frac{s}{2} - l)!", "l!}", "\\sum _{m = 0}^{\\frac{s}{2} - l} (-1)^{m} \\binom{\\frac{s}{2} - l}{m} \\\\& = {\\mathbf {1}}\\lbrace l = \\frac{s}{2} \\rbrace \\frac{\\Gamma (\\frac{s + 1}{2})}{s!", "}.$ Hence, we obtain $\\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E)& = \\frac{\\pi ^{\\frac{n - 3}{2}} \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + s}{2})} \\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{1}{2})}{\\Gamma ( \\frac{n + 1}{2}) } Q^{\\frac{s}{2}} \\phi _{n - 1}^{r,0,0} (K, \\beta ) \\\\& = \\pi ^{\\frac{n - 2}{2}}\\frac{\\Gamma ( \\frac{n}{2} ) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{n + s}{2}) \\Gamma ( \\frac{n + 1}{2}) } Q^{\\frac{s}{2}} \\phi _{n - 1}^{r,0,0} (K, \\beta ).$ On the other hand, if $s \\in {\\mathbb {N}}$ is odd, we conclude from Theorem REF $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{m = 0}^{ \\frac{s-1}{2} } \\sum _{l = 0}^{m} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\frac{\\Gamma (\\frac{s}{2} - l + 1)}{\\Gamma (\\frac{n + s + 1}{2} - l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ A change of the order of summation yields $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{\\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{ \\frac{s-1}{2} } \\sum _{m = l}^{\\frac{s-1}{2}} (-1)^{m - l} \\frac{\\Gamma (\\frac{s + 1}{2} - m)}{4^{m} \\, m!", "(s - 2m)!}", "\\binom{m}{l} \\frac{\\Gamma (\\frac{s}{2} - l + 1)}{\\Gamma (\\frac{n + s + 1}{2} - l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ Legendre's duplication formula gives for the sum with respect to $m$ , which we denote by $S$ , $S & = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\sum _{m = 0}^{\\frac{s-1}{2} - l} (-1)^{m} \\frac{1}{ m!", "\\Gamma (\\frac{s}{2} - l - m + 1)}.$ Then Lemma REF yields $S & = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\bigg ( \\sum _{m = 0}^{\\frac{s + 1}{2} - l} (-1)^{m} \\frac{1}{ m!", "\\Gamma (\\frac{s}{2} - l - m + 1)} - (-1)^{\\frac{s + 1}{2} - l} \\frac{1}{ (\\frac{s + 1}{2} - l)!", "\\Gamma (\\frac{1}{2})} \\bigg ) \\\\& = \\frac{\\sqrt{\\pi }}{2^{s} l!}", "\\bigg ( (-1)^{\\frac{s - 1}{2} - l} \\frac{1}{ \\sqrt{\\pi }(s - 2l) (\\frac{s + 1}{2} - l)!}", "- (-1)^{\\frac{s + 1}{2} - l} \\frac{1}{ \\sqrt{\\pi }(\\frac{s + 1}{2} - l)!}", "\\bigg ) \\\\& = (-1)^{\\frac{s - 1}{2} - l} \\frac{1}{2^{s - 1} l!", "(s - 2l) (\\frac{s - 1}{2} - l)!", "}.$ Hence, we obtain $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{2^{s} \\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{\\frac{s - 1}{2}} (-1)^{\\frac{s - 1}{2} - l} \\frac{1}{l!", "(\\frac{s - 1}{2} - l)!}", "\\frac{\\Gamma (\\frac{s}{2} - l)}{\\Gamma (\\frac{n + s + 1}{2} - l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 2}{2}} \\Gamma (\\frac{n}{2}) s!", "}{2^{s} \\Gamma (\\frac{s + 1}{2}) \\Gamma (\\frac{n + s}{2})} \\sum _{l = 0}^{\\frac{s - 1}{2}} (-1)^{l} \\binom{\\frac{s - 1}{2}}{l} \\frac{\\Gamma (l + \\frac{1}{2})}{\\Gamma (\\frac{n + 2}{2} + l)} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ Then Lemma REF gives $& \\int _{A(n, 1)} \\phi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\, \\mu _1(\\mathrm {d}E) \\\\& \\qquad = \\frac{s!", "}{2^{s} \\Gamma (\\frac{s + 1}{2})} \\frac{\\pi ^{\\frac{n - 1}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n + s + 1}{2}) \\Gamma (\\frac{n + 1}{2})} \\, Q^{\\frac{s - 1}{2}} \\phi _{n - 1}^{r,1,0} (K, \\beta ).$ Finally, the assertion follows from Legendre's duplication formula.", "Finally, we show that the Crofton formula has a very simple form in the $\\psi $ -representation of tensorial curvature measures.", "The cases $s\\in \\lbrace 0,1\\rbrace $ are checked directly, hence we can assume $s\\ge 2$ in the following.", "Using (REF ) we get $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\,\\mu _{k}(\\mathrm {d}E) \\nonumber \\\\& \\qquad = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor }(-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2})\\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} Q^{j} \\nonumber \\\\& \\qquad \\qquad \\qquad \\times \\int _{A(n, k)} \\phi _{k - 1}^{r,s - 2j,0}(K \\cap E, \\beta \\cap E) \\,\\mu _{k}(\\mathrm {d}E).", "$ Then, for $k \\ne 1$ , Theorem REF yields $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\, \\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor - j} \\kappa _{n, k, s - 2j, z} \\, Q^{z + j} \\phi _{n - 1}^{r,s - 2j - 2z,0} (K, \\beta ) \\\\& \\qquad = \\frac{1}{\\sqrt{\\pi }} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\sum _{z = j}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} \\kappa _{n, k, s - 2j, z - j} \\, Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ),$ where $\\kappa _{n, k, s - 2j, z - j} = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})} \\frac{\\Gamma (\\frac{s + 1}{2} - j) \\Gamma (\\frac{s}{2} - j + 1)}{\\Gamma (\\frac{n - k + s + 1}{2} - j) \\Gamma (\\frac{n + s - 1}{2} - j)} \\frac{\\Gamma (\\tfrac{n - k}{2} + z - j) \\Gamma (\\tfrac{k + s - 1}{2} - z)}{\\Gamma (\\frac{s}{2} - z + 1) (z - j)!", "},$ if $z \\ne (s - 1) /2$ .", "On the other hand, if $z = (s - 1) /2$ , then the coeffcient needs to be multiplied by the factor $\\frac{k(n + s - 2j - 2)}{(k - 1)(n + s - 2j - 1)}$ (see the comment after the proof of Theorem REF ).", "Applying Legendre's duplication formula twice, we thus obtain $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\,\\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2})} \\frac{s!", "}{2^{s}} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } \\frac{\\Gamma (\\tfrac{k + s - 1}{2} - z)}{z!", "\\Gamma (\\frac{n}{2} + s - 1) \\Gamma (\\frac{s}{2} - z + 1)} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ) \\\\& \\qquad \\qquad \\times \\sum _{j = 0}^{z} (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (\\frac{n}{2} + s - j - 1) \\Gamma (\\tfrac{n - k}{2} + z - j)}{\\Gamma (\\frac{n - k + s + 1}{2} - j) \\Gamma (\\frac{n + s - 1}{2} - j)} \\\\& \\qquad \\qquad \\times \\left( 1 - {\\mathbf {1}}\\lbrace z = \\tfrac{s - 1}{2} \\rbrace \\left(1 - \\tfrac{k(n + s - 2j - 2)}{(k - 1)(n + s - 2j - 1)} \\right) \\right),$ Denoting the sum with respect to $j$ by $S_{z}$ , an application of Lemma REF shows that $S_{z} & = \\sum _{j = 0}^{z} (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (\\frac{n}{2} + s - j - 1) \\Gamma (\\tfrac{n - k}{2} + z - j)}{\\Gamma (\\frac{n - k + s + 1}{2} - j) \\Gamma (\\frac{n + s - 1}{2} - j)} \\nonumber \\\\& = (-1)^{z} \\frac{\\Gamma (\\frac{n - k}{2}) \\Gamma (\\frac{s + 1}{2}) \\Gamma (\\frac{k + s - 1}{2}) \\Gamma (\\frac{n}{2} + s - z - 1)}{\\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2}) \\Gamma (\\frac{s + 1}{2} - z) \\Gamma (\\frac{k + s - 1}{2} - z)}, $ for $z \\ne (s-1)/ 2$ and $k>1$ .", "On the other hand, for $z = ( {s - 1} )/2 =: t$ , we obtain from Lemma REF and Lemma REF (since $s > 1$ and thus $t > 0$ ) that $S_{t} & = \\tfrac{k}{k - 1} \\sum _{j = 0}^{t} (-1)^{j} \\binom{t}{j} \\left( 1 - \\tfrac{1}{n + 2t - 2j} \\right) \\frac{\\Gamma (\\frac{n}{2} + 2t - j) \\Gamma (\\tfrac{n - k}{2} + t - j)}{\\Gamma (\\frac{n - k}{2} + t - j + 1) \\Gamma (\\frac{n}{2} + t - j)} \\\\& = \\tfrac{k}{k - 1} \\bigg ( \\sum _{j = 0}^{t} (-1)^{j} \\binom{t}{j} \\frac{\\Gamma (\\frac{n}{2} + 2t - j) \\Gamma (\\tfrac{n - k}{2} + t - j)}{\\Gamma (\\frac{n - k}{2} + t - j + 1) \\Gamma (\\frac{n}{2} + t - j)} \\\\& \\qquad \\qquad - \\sum _{j = 0}^{t} (-1)^{j} \\binom{t}{j} \\tfrac{1}{\\frac{n - k}{2} + t - j} \\frac{\\Gamma (\\frac{n}{2} + 2t - j)}{\\Gamma (\\frac{n}{2} + t - j + 1)} \\bigg ) \\\\& = (-1)^{t} \\frac{\\Gamma (\\frac{n - k}{2}) \\Gamma (t + 1) \\Gamma (\\frac{k}{2} + t)}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k}{2} + t + 1)},$ which coincides with (REF ) for $z = ({s - 1})/ 2$ .", "Thus, we have $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E) \\,\\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k + 1}{2}} \\Gamma (\\frac{n}{2}) \\Gamma (\\frac{k + s - 1}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\frac{s!", "\\Gamma (\\frac{s + 1}{2})}{2^{s}} \\\\& \\qquad \\qquad \\times \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{z} \\frac{\\Gamma (\\frac{n}{2} + s - z - 1)}{z!", "\\Gamma (\\frac{n}{2} + s - 1) \\Gamma (\\frac{s}{2} - z + 1) \\Gamma (\\frac{s + 1}{2} - z)} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ Applying Legendre's duplication formula twice, we get $& \\int _{A(n, k)} \\psi _{k - 1}^{r,s,0} (K \\cap E, \\beta \\cap E)\\, \\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{k - 1}{n - 1} \\frac{\\pi ^{\\frac{n - k}{2}} \\Gamma (\\frac{n}{2}) \\Gamma (\\frac{k + s - 1}{2}) \\Gamma (\\frac{s + 1}{2})}{\\Gamma (\\frac{k}{2}) \\Gamma (\\frac{n - k + s + 1}{2}) \\Gamma (\\frac{n + s - 1}{2})} \\\\& \\qquad \\qquad \\times \\frac{1}{\\sqrt{\\pi }} \\sum _{z = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{z} \\binom{s}{2z} \\frac{\\Gamma (z + \\frac{1}{2})\\Gamma (\\frac{n}{2} + s - z - 1)}{\\Gamma (\\frac{n}{2} + s - 1)} Q^{z} \\phi _{n - 1}^{r,s - 2z,0} (K, \\beta ).$ With (REF ) we obtain the assertion for $k \\ne 1$ .", "On the other hand, if $k = 1$ , then Theorem REF yields for (REF ) that $& \\int _{A(n, 1)} \\psi _{0}^{r,s,0} (K \\cap E, \\beta \\cap E) \\,\\mu _{k}(\\mathrm {d}E) \\\\& \\qquad = \\frac{\\pi ^{\\frac{n - 3}{2}} \\Gamma (\\frac{n}{2})}{\\Gamma (\\frac{n + 1}{2}) \\Gamma (\\frac{n}{2} + s - 1)} \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{s}{2j} \\frac{\\Gamma (j + \\frac{1}{2}) \\Gamma (\\frac{n}{2} + s - j - 1) \\Gamma (\\lfloor \\frac{s + 1}{2} \\rfloor - j + \\frac{1}{2})}{\\Gamma (\\frac{n}{2} + \\lfloor \\frac{s + 1}{2} \\rfloor - j)} \\\\& \\qquad \\qquad \\qquad \\times Q^{\\lfloor \\frac{s}{2} \\rfloor } \\phi _{n - 1}^{r, s - 2\\lfloor \\frac{s}{2} \\rfloor ,0} (K, \\beta ).$ Denoting the sum with respect to $j$ by $S$ and applying Legendre's duplication formula three times, we conclude that $S = \\sqrt{\\pi }\\Gamma (\\lfloor \\tfrac{s + 1}{2} \\rfloor + \\tfrac{1}{2}) \\sum _{j = 0}^{\\lfloor \\frac{s}{2} \\rfloor } (-1)^{j} \\binom{\\lfloor \\frac{s}{2} \\rfloor }{j} \\frac{\\Gamma (\\frac{n}{2} + s - j - 1)}{\\Gamma (\\frac{n}{2} + \\lfloor \\frac{s + 1}{2} \\rfloor - j)}.$ Since $s\\ge 2$ , Lemma REF yields $S = 0$ due to (REF ), and hence the assertion." ], [ "Sums of Gamma Functions", "In this section, we state four basic identities involving sums of Gamma functions.", "Lemma 7 Let $q \\in {\\mathbb {N}}_0$ and $a, b >0$ .", "Then $\\sum _{y=0}^q (-1)^{y} \\binom{q}{y} \\frac{\\Gamma (a + y)}{\\Gamma (b + y)} = \\frac{\\Gamma (a) \\Gamma (b - a + q)}{\\Gamma (b + q) \\Gamma (b - a)}.$ Under the additional assumption $a<b$ , this lemma can be found as Lemma 15.6.4 in [1], which is also proved there.", "Since this case is not sufficient for our purposes, we deduce the current more general version via Zeilberger's algorithm.", "The factor $\\Gamma (b - a + q)$ in Lemma REF does not cause any problems in case $a - b - q \\in {\\mathbb {N}}_{0}$ , as the also appearing $\\Gamma (b - a)$ cancels out the singularity, see (REF ).", "We set $F(q, y) := (-1)^{y} \\binom{q}{y} \\frac{\\Gamma (a + y)}{\\Gamma (b + y)},$ for which we see that $F(q, y) = 0$ if $y \\notin \\lbrace 0, \\ldots , q \\rbrace $ , and $f(q) := \\sum _{y = 0}^{q} F(q, y).$ Furthermore, we define the function $G(q, y) :={\\left\\lbrace \\begin{array}{ll}\\frac{y (b + y - 1)}{q - y + 1} F(q, y), \\qquad & \\text{ for } y \\in \\lbrace 0, \\ldots , q\\rbrace , \\\\G(q, q) -(b + q) F(q + 1, q) & \\\\\\qquad + (b - a + q) F(q, q), \\qquad & \\text{ for } y = q + 1, \\\\0, & \\text{ else}.\\end{array}\\right.", "}$ A direct calculation yields $-(b + q - 1) F(q, y) + (b - a + q - 1) F(q - 1, y) = G(q - 1, y + 1) - G(q - 1, y)$ for $y \\in {\\mathbb {N}}_{0}$ .", "Summing this relation over $y \\in \\lbrace 0, \\ldots , q \\rbrace $ gives $- (b + q - 1) f(q) + (b - a + q - 1) f(q - 1) = 0$ and thus $f(q) & = \\frac{b - a + q - 1}{b + q - 1} f(q - 1) \\\\& = \\frac{(b - a + q - 2)(b - a + q - 1)}{(b + q - 2) (b + q - 1)} f(q - 2) \\\\& \\ \\, \\vdots \\\\& = \\frac{(b - a) \\cdots (b - a + q - 1)}{b \\cdots (b + q - 1)} f(0) \\\\& = \\frac{\\Gamma (b - a + q) \\Gamma (b)}{\\Gamma (b + q) \\Gamma (b - a)} f(0),$ where $\\frac{\\Gamma (b - a + q)}{\\Gamma (b - a)} = (b - a) \\cdots (b - a + q - 1)$ is well-defined, even for $a - b \\in {\\mathbb {N}}$ .", "With $f(0) = \\frac{\\Gamma (a)}{\\Gamma (b)}$ we obtain the assertion.", "Lemma 8 Let $a \\in {\\mathbb {N}}_0$ .", "Then $ \\sum _{q = 0}^{a} \\frac{(-1)^{q}}{\\Gamma (a - q + \\frac{1}{2}) q!}", "= \\frac{(-1)^{a}}{\\sqrt{\\pi }(1 - 2a)a !", "}.$ For the sum $S$ on the left-hand side of the asserted equation, we obtain $S = \\sum _{q = 0}^{a} \\left( \\frac{2q}{2a - 1} \\frac{(-1)^{q} }{\\Gamma (a - q + \\frac{1}{2}) q!}", "+ \\frac{2q + 2}{2a - 1} \\frac{(-1)^{q} }{\\Gamma (a - q - \\frac{1}{2}) (q + 1)!}", "\\right),$ where we use that $(-\\frac{1}{2})\\Gamma (-\\frac{1}{2})=\\sqrt{\\pi }$ .", "Due to cancellation in this telescoping sum, the assertion follows immediately.", "Finally, we establish the following lemmas.", "Lemma 9 Let $a, b, c \\in {\\mathbb {R}}$ and $z \\in {\\mathbb {N}}_{0}$ with $a > z\\ge 0$ and $b > 0$ .", "Then $& \\sum _{j = 0}^{z} (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (a - j) \\Gamma (b + z - j)}{\\Gamma (c - j) \\Gamma (a + b - c - j + 1)} \\\\& \\qquad = (-1)^{z} \\frac{\\Gamma (a - z) \\Gamma (b)}{\\Gamma (a + b - c + 1) \\Gamma (c)} \\frac{\\Gamma (a - c + 1)}{\\Gamma (a - c +1 - z )} \\frac{\\Gamma (c - b)}{\\Gamma (c - b - z)}.$ The factor $\\Gamma (a - c + 1)$ (resp.", "$\\Gamma (c - b)$ ) in Lemma REF does not cause any problems for $c - a \\in {\\mathbb {N}}$ (resp.", "$b - c \\in {\\mathbb {N}}_{0}$ ), as the also appearing $\\Gamma (a - c + 1 - z )$ (resp.", "$\\Gamma (c - b - z)$ ) cancels out the singularity.", "On the other hand, in our applications of the lemma, we only need the cases where $a-c+1>z$ and $c-b>z$ .", "We set $F(z, j) := (-1)^{j} \\binom{z}{j} \\frac{\\Gamma (a - j) \\Gamma (b + z - j)}{\\Gamma (c - j) \\Gamma (a + b - c - j + 1)},$ for $j \\in \\lbrace 0, \\ldots , z \\rbrace $ , and $F(z, j) = 0$ in all other cases, and $f(z) := \\sum _{j = 0}^{z} F(z,j).$ Furthermore, we define the function $G(z, j) :={\\left\\lbrace \\begin{array}{ll}- \\frac{j (a - j) (b + z - j)}{z - j + 1} F(z, j), \\qquad & \\text{ for } j \\in \\lbrace 0, \\ldots , z \\rbrace , \\\\G(z, z) + (a - z - 1) F(z + 1, z) & \\\\\\qquad + (c - b - z - 1) (a - c - z) F(z, z), \\qquad & \\text{ for } j = z + 1, \\\\0, & \\text{ otherwise}.\\end{array}\\right.", "}$ A direct calculation yields $(a - z) F(z, j) + (c - b - z) (a - c - z + 1) F(z - 1, j) = G(z - 1, j + 1) - G(z - 1, j)$ for $j \\in {\\mathbb {N}}_{0}$ .", "Summing this relation over $j \\in \\lbrace 0, \\ldots , z \\rbrace $ gives $(a - z) f(z) + (c - b - z) (a - c - z + 1) f(z - 1) = 0$ and thus $f(z) & = - \\frac{(c - b - z) (a - c - z + 1)}{a - z} f(z - 1) \\\\& = \\frac{(c - b - z) (c - b - z + 1) (a - c - z + 1) (a - c - z + 2)}{(a - z)(a - z + 1)} f(z - 2) \\\\& \\ \\, \\vdots \\\\& = (-1)^{z} \\frac{(c - b - z) \\cdots (c - b - 1) (a - c - z + 1) \\cdots (a - c)}{(a - z) \\cdots (a - 1)} f(0) \\\\& = (-1)^{z} \\frac{\\Gamma (c - b) \\Gamma (a - c + 1) \\Gamma (a - z)}{\\Gamma (c - b - z) \\Gamma (a - c+1 - z ) \\Gamma (a)} f(0),$ where $\\frac{\\Gamma (c - b)}{\\Gamma (c - b - z)} = (c - b - z) \\cdots (c - b - 1)$ is well-defined, even for $b - c \\in {\\mathbb {N}}_{0}$ , and a similar statement holds for $ {\\Gamma (a - c + 1)} /\\Gamma (a - c +1- z )$ .", "With $f(0) = \\frac{\\Gamma (a) \\Gamma (b)}{\\Gamma (c) \\Gamma (a + b - c + 1)}$ we obtain the assertion.", "Lemma 10 Let $a, b \\in {\\mathbb {R}}$ with $a, b > 0$ and $t \\in {\\mathbb {N}}$ .", "Then $& \\sum _{j = 0}^{t} (-1)^{j} \\frac{1}{b + j} \\binom{t}{j} \\frac{\\Gamma (a + t + j)}{\\Gamma (a + 1 + j)} = \\frac{\\Gamma (a - b + t) \\Gamma (b) \\Gamma (t + 1)}{\\Gamma (a - b + 1) \\Gamma (b + t + 1)}.$ The factor $\\Gamma (a - b + t)$ in Lemma REF does not cause any problems for $b - a - t \\in {\\mathbb {N}}_{0}$ , as the also appearing $\\Gamma (a - b + 1)$ cancels out the singularity.", "In our application of the lemma, we will additionally know that $a>b$ .", "We set $F(t, j) := (-1)^{j} \\frac{1}{b + j} \\binom{t}{j} \\frac{\\Gamma (a + t + j)}{\\Gamma (a + 1 + j)},$ for which we see that $F(t, j) = 0$ if $j \\notin \\lbrace 0, \\ldots , t \\rbrace $ , and $f(t) := \\sum _{j = 0}^{t} F(t, j).$ Furthermore, we define the function $G(t, j) :={\\left\\lbrace \\begin{array}{ll}\\frac{j (a + j) (a + 2t + 1) (t^2 + t(a + 1) - j + 1) (b + j)}{t (t - j + 1) (a + t) (a + t + 1)} F(t, j), \\qquad & \\text{ for } j \\in \\lbrace 0, \\ldots , t \\rbrace , \\\\G(t, t) - (b + t + 1) F(t + 1, t) & \\\\\\qquad + (t + 1) (a - b + t) F(t, t), \\qquad & \\text{ for } j = t + 1, \\\\0, & \\text{ otherwise}.\\end{array}\\right.", "}$ A direct calculation yields $- (b + t) F(t, j) + t (a - b + t - 1) F(t - 1, j) = G(t - 1, j + 1) - G(t - 1, j)$ for $j \\in {\\mathbb {N}}_{0}$ .", "Summing this relation over $j \\in \\lbrace 0, \\ldots , t \\rbrace $ gives $- (b + t) f(t) + t (a - b + t - 1) f(t - 1) = 0$ and thus $f(t) & = \\frac{t (a - b + t - 1)}{b + t} f(t - 1) \\\\& = \\frac{(t - 1) t (a - b + t - 2) (a - b + t - 1)}{(b + t - 1) (b + t)} f(t - 2) \\\\& \\ \\, \\vdots \\\\& = \\frac{2 \\cdots t (a - b + 1) \\cdots (a - b + t - 1)}{(b + 2) \\cdots (b + t)} f(1) \\\\& = \\frac{\\Gamma (t + 1) \\Gamma (a - b + t) \\Gamma (b + 2)}{\\Gamma (a - b + 1) \\Gamma (b + t + 1)} f(1).$ With $f(1) = \\frac{1}{b} - \\frac{1}{b + 1} = \\frac{1}{b (b + 1)}$ we obtain the assertion." ] ]
1606.05131
[ [ "Mortality and life expectancy forecasting for a group of populations in\n developed countries: A multilevel functional data method" ], [ "Abstract A multilevel functional data method is adapted for forecasting age-specific mortality for two or more populations in developed countries with high-quality vital registration systems.", "It uses multilevel functional principal component analysis of aggregate and population-specific data to extract the common trend and population-specific residual trend among populations.", "If the forecasts of population-specific residual trends do not show a long-term trend, then convergence in forecasts may be achieved.", "This method is first applied to age- and sex-specific data for the United Kingdom, and its forecast accuracy is then further compared with several existing methods, including independent functional data and product-ratio methods, through a multi-country comparison.", "The proposed method is also demonstrated by age-, sex- and state-specific data in Australia, where the convergence in forecasts can possibly be achieved by sex and state.", "For forecasting age-specific mortality, the multilevel functional data method is more accurate than the other coherent methods considered.", "For forecasting female life expectancy at birth, the multilevel functional data method is outperformed by the Bayesian method of \\cite{RLG14}.", "For forecasting male life expectancy at birth, the multilevel functional data method performs better than the Bayesian methods in terms of point forecasts, but less well in terms of interval forecasts.", "Supplementary materials for this article are available online." ], [ "Introduction", "Many statistical methods have been proposed for forecasting age-specific mortality rates [16], [5], [7], [20], [65], [67].", "Of these, a significant milestone in demographic forecasting was the work by [44].", "They applied a principal component method to age-specific mortality rates and extracted a single time-varying index of the level of mortality rates, from which the forecasts are obtained by a random-walk with drift.", "The method has since been extended and modified.", "For example, [60] proposed the age-period-cohort Lee-Carter method; [35] proposed a functional data model that utilizes nonparametric smoothing and high-order principal components; [20] and [68] considered Bayesian techniques for Lee-Carter model estimation and forecasting; and [49] extended the Lee-Carter method to model the rotation of age patterns for long-term projections.", "These works mainly focused on forecasting mortality for a single population, or several populations individually.", "However, individual forecasts, even when based on similar extrapolative procedures, may imply increasing divergence in mortality rates in the long run, counter to the expected and observed trend toward a global convergence [48], [54], [46].", "Thus, joint modeling mortality for two or more populations simultaneously is paramount, as it allows one to model the correlations among two or more populations, distinguish between long-term and short-term effects in the mortality evolution, and explore the additional information contained in the experience of other populations to further improve forecast accuracy.", "These populations can be grouped by sex, state, ethnic group, socioeconomic status and other attributes.", "In these cases, it is often desirable to produce coherent forecasts that do not diverge over time (e.g., in demography, [48], [48], [4], [4], [2], [2], [59], [59], [57], [57], [46], [46], [58], [58], [62], [62]; in actuarial science, [38], [38], [47], [47], [9], [10], [19], [19]).", "The definition of coherent in demography varies, but here it means joint modeling of populations, and further that the mortality forecasts do not overlap.", "In the case of two-sex populations, there may be common features in the groups of data that can first be captured with the common principal components.", "Further, we can prevent the forecasts of the groups from diverging by requiring the difference in each sex-specific principal component scores to be stationary for different populations $i$ and $j$ , so that $\\limsup _{t\\rightarrow \\infty } \\text{E}||f_{t,i} - f_{t,j}||<\\infty , \\quad \\text{for all} \\ i \\ \\text{and} \\ j,$ where $\\text{E}||f_{t,i} - f_{t,j}|| = \\int _{\\mathcal {I}}[f_{t,j}(x) - f_{t,i}(x)]^2dx$ is the $L_2$ norm, $f_t(x)$ represents age-specific mortality for year $t$ , and $\\mathcal {I}$ denotes a function support range.", "The problem of jointly forecasting mortality rates for a group of populations has been considered by [42], [48], [43], [17] and [62] in the context of the Lee-Carter model.", "These authors proposed the augmented common factor model that extracts a common trend for a group of populations, while acknowledging their individual differences in level, age pattern and short-term trend [48].", "On the other hand, [32] proposed a functional data model to jointly model the gap between female and male age-specific mortality rates, and [58] proposed a Bayesian method to jointly model the gap between female and male life expectancies at birth.", "Based on the work of [48], a general framework is presented by [43] for forecasting life expectancy at birth as the sum of a common trend and the population-specific trend.", "Coherent forecasting in the framework of [44]'s [44] model has recently been extended to the coherent functional data model by [32].", "These authors proposed the product-ratio method, which models the product and ratio functions of the age-specific mortality rates of different populations through a functional principal component decomposition, and forecasts age- and sex-specific mortality coherently by constraining the forecast ratio function via stationary time-series model.", "The forecasts of product and ratio functions are obtained using the independent functional data method given in [35]; the forecast product and ratio functions are then transformed back into the male and female age-specific mortality rates.", "Illustrated by empirical studies, they found that the product-ratio method generally gives slightly less accurate female mortality forecasts and produces much more accurate male mortality forecasts than the independent functional data method, in which the latter one does not impose a coherent structure.", "As an extension of [48] and [32], we consider a multilevel functional data model motivated by the work of [18], [14], [13] and [23], among many others.", "The objective of the multilevel functional data method is to model multiple sets of functions that may be correlated among groups.", "In this paper, we apply this technique to forecast age-specific mortality and life expectancy at birth for a group of populations.", "We found the multilevel functional data model captures the correlation among populations, models the forecast uncertainty through Bayesian paradigm, and is adequate for use within a probabilistic population modeling framework [59].", "Similar to the work of [48], [43], [17] and [46], the multilevel functional data model captures the common trend and the population-specific trend.", "It produces forecasts that are comparable with the ones from the product-ratio method, which themselves are also more accurate than the independent functional data method for male age-specific mortality and life expectancy forecasts.", "The multilevel functional data model is described in Section .", "In Section , we outline the differences among the multilevel functional data, augmented common factor and independent functional data methods.", "In Section , we illustrate the multilevel functional data method by applying it to the age- and sex-specific mortality rates for the United Kingdom (UK).", "In Section , we compare the point and interval forecast accuracy among five methods for 32 populations.", "In Section , we investigate the performance of the multilevel functional data method with the age-, and sex- and state-specific mortality rates in Australia.", "In Section , we provide some concluding remarks, along with some reflections on how the method presented here can be further extended.", "More information on some theoretical properties of multilevel functional principal component decomposition are deferred to the Supplementary Material A [64]." ], [ "A multilevel functional data model", "We first present the problem in the context of forecasting male and female age-specific mortality rates, although the method can easily be generalized to any number of populations.", "Let $y_t^{j}(x_i)$ be the log central mortality rates observed at the beginning of each year for year $t=1,2,\\dots ,n$ at observed ages $x_1,x_2,\\dots ,x_p$ where $x$ is a continuous variable, $p$ is the number of ages, and superscript $j$ represents either male or female in the case of two populations.", "Following the functional data framework, we assume there is an underlying continuous and smooth function $f_t^{j}(x)$ that is observed at discrete data points with error.", "That is $y_t^{j}(x_i) = f_t^{j}(x_i) + \\delta _t^{j}(x_i)\\varepsilon ^{j}_{t,i},$ where $x_i$ represents the center of each age or age group for $i=1,\\dots ,p$ , $\\varepsilon _{t,i}^{j}$ is an independent and identically distributed (iid) standard normal random variable for each age in year $t$ , and $\\delta _{t}^{j}(x_i)$ measures the variability in mortality at each age in year $t$ for the $j^{\\text{th}}$ population.", "Together, $\\delta _t^{j}(x_i)\\varepsilon ^{j}_{t,i}$ represents the smoothing error.", "Let $m_t^j(x_i) = \\exp \\left\\lbrace y_t^j(x_i)\\right\\rbrace $ be the observed central mortality rates for age $x_i$ in year $t$ and define $N_t^j(x_i)$ to be the total $j^{\\text{th}}$ population of age $x_i$ at 1st January of year $t$ .", "The observed mortality rate approximately follows a binomial distribution with estimated variance $\\text{Var}\\left[m_t^j(x_i)\\right] \\approx \\frac{m_t^j(x_i)\\times \\left[1-m_t^j(x_i)\\right]}{N_t^j(x_i)}.$ Via Taylor's series expansion, the estimated variance associated with the log mortality rate is given by $\\left(\\widehat{\\delta }_t^j\\right)^2(x_i)\\approx \\text{Var}\\left\\lbrace \\ln \\left[m_t^j(x_i)\\right]\\right\\rbrace = \\frac{1-m_t^j(x_i)}{m_t^j(x_i) \\times N_t^j(x_i)}.$ Since $m_t^j(x_i)$ is often quite small, $(\\delta _t^j)^2(x_i)$ can be approximated by a Poisson distribution with estimated variance $\\left(\\widehat{\\delta }_t^j\\right)^2(x_i) \\approx \\frac{1}{m_t^j(x_i)\\times N_t^j(x_i)}.$ As suggested by [35], we smooth mortality rates using weighted penalized regression splines with a partial monotonic constraint for ages above 65, where the weights are equal to the inverse variances given in (REF ).", "The weights are used to model heterogeneity (different variances) in mortality across different ages.", "Let the weights be the inverse variances $w_t^j(x_i) = 1/\\big [(\\delta _t^j)^2(x_i)\\big ]$ , the penalized regression spline can be written as: $\\widehat{f}_t^j(x_i) = \\operatornamewithlimits{\\arg \\!\\min }_{\\theta _t(x_i)}\\sum ^M_{i=1}w_t^j(x_i)\\Big |y_t^j(x_i) - \\theta _t(x_i)\\Big |+\\alpha \\sum ^{M-1}_{i=1}\\Big |\\theta ^{^{\\prime }}_t(x_{i+1})-\\theta ^{^{\\prime }}_t(x_i)\\Big |,$ where $i$ represents different ages (grid points) in a total of $M$ grid points, $\\alpha $ is a smoothing parameter, and $^{^{\\prime }}$ symbolizes the first derivative of a function.", "While the $L_1$ loss function and the $L_1$ roughness penalty are employed to obtain robust estimates, the monotonic increasing constraint helps to reduce the noise from estimation of older ages [26].", "In the multilevel functional data model, we first apply (REF ) to smooth multiple sets of curves from different populations that may be correlated.", "The multilevel functional data model can be related to a two-way functional analysis of variance model studied by [52], [15] and [71], it is a special case of the general `functional mixed model' proposed in [51].", "In the case of two populations, the basic idea is to decompose curves among different populations into an average of total mortality $\\mu (x)$ , a sex-specific deviation from the averaged total mortality $\\eta ^{j}(x)$ , a common trend across populations $R_t(x)$ , a sex-specific residual trend $U_t^{j}(x)$ , and measurement error $e_t^{j}(x)$ with finite variance $(\\sigma ^2)^{j}$ .", "The common and sex-specific residual trends are modeled by projecting them onto the eigenvectors of covariance operators of the aggregate and population-specific centered stochastic processes, respectively.", "To express our idea, the smoothed mortality rate at year $t$ can be written as: $f_t^{j}(x) = \\mu (x) + \\eta ^{j}(x) + R_t(x) + U_t^{j}(x),\\qquad x\\in \\mathcal {I}.$ To ensure identifiability, we assume two stochastic processes $R(x)$ and $U^j(x)$ are uncorrelated but we allow correlations among their realizations.", "Because the centered stochastic processes $R(x)$ and $U^j(x)$ are unknown in practice, the population eigenvalues and eigenfunctions can only be approximated through a set of realizations $\\mathbf {R}(x) = \\left\\lbrace R_1(x),\\dots ,R_n(x)\\right\\rbrace $ and $\\mathbf {U}^j(x) = \\left\\lbrace U_1^j(x),\\dots ,U_n^j(x)\\right\\rbrace $ .", "From the covariance function of $\\mathbf {R}(x)$ , we can extract a set of functional principal components and their corresponding scores, along with a set of residual functions.", "Based on the covariance function of residual functions, we can then extract a second set of functional principal components and their associated scores.", "While the first functional principal component decomposition captures the common trend from total mortality rates, the second functional principal component decomposition captures the sex-specific residual trend.", "The sample versions of the aggregate mean function, sex-specific mean function deviation, common trend, and sex-specific residual trend, for a set of dense and regularly spaced functional data, can be estimated by: $\\widehat{\\mu }(x)&=\\frac{1}{n}\\sum ^n_{t=1}f_t^{\\text{T}}(x), \\\\\\widehat{\\eta }^{j}(x) &= \\widehat{\\mu }^{j}(x)-\\widehat{\\mu }(x),\\\\\\widehat{R}_t(x) &= \\sum ^{\\infty }_{k=1}\\widehat{\\beta }_{t,k}\\widehat{\\phi }_k(x) \\approx \\sum ^K_{k=1}\\widehat{\\beta }_{t,k}\\widehat{\\phi }_k(x),\\\\\\widehat{U}^{j}_t(x) &= \\sum ^{\\infty }_{l=1}\\widehat{\\gamma }^{j}_{t,l}\\widehat{\\psi }_l^{j}(x) \\approx \\sum ^L_{l=1}\\widehat{\\gamma }^{j}_{t,l}\\widehat{\\psi }_l^{j}(x),$ where $\\lbrace f_1^{\\text{T}}(x),\\dots ,f_n^{\\text{T}}(x)\\rbrace $ represents a set of smoothed functions for the age-specific total mortality; $\\widehat{\\mu }(x)$ represents the simple average of the total mortality, whereas $\\widehat{\\mu }^j(x)$ represents the simple average of females or males; $\\lbrace \\widehat{\\mathbf {\\beta }}_k = (\\widehat{\\beta }_{1,k},\\dots ,\\widehat{\\beta }_{n,k}); k=1,\\dots ,K\\rbrace $ represents the $k^{\\text{th}}$ sample principal component scores of $\\mathbf {R}(x)$ , $\\mathbf {\\Phi }=\\left[\\widehat{\\phi }_1(x),\\dots ,\\widehat{\\phi }_{K}(x)\\right]$ are the corresponding orthogonal sample eigenfunctions in a square integrable function space.", "Similarly, $\\lbrace \\widehat{\\mathbf {\\gamma }}_l^j = (\\widehat{\\gamma }^{j}_{1,l},\\dots ,\\widehat{\\gamma }^{j}_{n,l}); l=1,\\dots ,L\\rbrace $ represents the $l^{\\text{th}}$ sample principal component scores of $\\mathbf {U}^{j}(x)$ , $\\mathbf {\\Psi }=\\left[\\widehat{\\psi }_1^{j}(x),\\dots ,\\widehat{\\psi }_L^{j}(x)\\right]$ are the corresponding orthogonal sample eigenfunctions, $K$ , $L$ are truncation lags.", "As two stochastic processes $R(x)$ and $U^j(x)$ are uncorrelated, $\\widehat{\\mathbf {\\beta }}_k$ are uncorrelated with $\\widehat{\\mathbf {\\gamma }}_l^j$ .", "Substituting Equations (REF )– () into Equations (REF )– (REF ), we obtain $y_t^{j}(x)=\\widehat{\\mu }(x)+\\widehat{\\eta }^{j}(x)+\\sum ^K_{k=1}\\widehat{\\beta }_{t,k}\\widehat{\\phi }_k(x)+\\sum ^L_{l=1}\\widehat{\\gamma }_{t,l}^{j}\\widehat{\\psi }_l^{j}(x)+e_t^{j}(x)+\\delta ^j_t(x)\\varepsilon _{t}^{j},$ where $\\widehat{\\beta }_{t,k}\\sim \\text{N}\\Big (0,\\widehat{\\lambda }_k\\Big )$ , and $\\widehat{\\lambda }_k$ represents the $k^{\\text{th}}$ eigenvalue of empirical covariance operator associated with the common trend; $\\widehat{\\gamma }_{t,l}^{j}\\sim \\text{N}\\left(0,\\widehat{\\lambda }_l^{j}\\right)$ , and $\\widehat{\\lambda }_l^{j}$ represents the $l^{\\text{th}}$ eigenvalue of empirical covariance operator associated with the sex-specific residual trend; and $e_t^{j}(x)\\sim N\\left(0, (\\widehat{\\sigma }^2)^j\\right)$ represents model errors due to finite truncation.", "Selecting the number of principal components, $K$ and $L$ , is an important practical issue.", "Four common approaches are cross validation [61], Akaike's information criterion [70], bootstrap method [25], and explained variance [13], [12].", "We use a cumulative percentage of total variation to determine $K$ and $L$ .", "The optimal numbers of $K$ and $L$ are determined by: $K &= \\operatornamewithlimits{\\arg \\!\\min }_{K: K\\ge 1}\\left\\lbrace \\sum _{k=1}^K\\widehat{\\lambda }_k\\Big /\\sum _{k=1}^{\\infty }\\widehat{\\lambda }_k{1}{\\big \\lbrace \\widehat{\\lambda }_k>0\\big \\rbrace }\\ge P_1\\right\\rbrace ,\\\\L &= \\operatornamewithlimits{\\arg \\!\\min }_{L: L\\ge 1}\\left\\lbrace \\sum _{l=1}^L\\widehat{\\lambda }_l^j\\Big /\\sum _{l=1}^{\\infty }\\widehat{\\lambda }_l^j{1}{\\big \\lbrace \\widehat{\\lambda }_l^j>0\\big \\rbrace }\\ge P_2\\right\\rbrace ,$ where ${1}\\lbrace \\cdot \\rbrace $ denotes a binary indicator function.", "Following [12], we chose $P_1=P_2=0.9$ .", "An important parameter is the proportion of variability explained by aggregate data, which is the variance explained by the within-cluster variability [18].", "A possible measure of within-cluster variability is given by: $\\frac{\\sum ^{\\infty }_{k=1}\\lambda _k}{\\sum ^{\\infty }_{k=1}\\lambda _k+\\sum ^{\\infty }_{l=1}\\lambda _l}=\\frac{\\int _{\\mathcal {I}} \\text{Var}\\left[\\mathbf {R}(x)\\right]dx}{\\int _{\\mathcal {I}}\\text{Var}\\left[\\mathbf {R}(x)\\right]dx + \\int _{\\mathcal {I}}\\text{Var}\\left[\\mathbf {U}^{j}(x)\\right]dx}.$ When the common factor can explain the main mode of total variability, the value of within-cluster variability is close to 1.", "For multiple populations, the other important parameter is the total variability for a population, given by $\\frac{1}{n}\\sum ^n_{t=1}[f_t(x) - \\bar{f}(x)][f_t(w) - \\bar{f}(w)], \\qquad x, w\\in \\mathcal {I}.$ This allows us to identify the population with larger variability.", "Conditioning on the estimated principal components $\\mathbf {\\Phi }$ , $\\mathbf {\\Psi }$ and continuous functions $\\mathbf {y}^{j}=\\Big [y_1^{j}(x),\\dots ,y_n^{j}(x)\\Big ]$ , the $h$ -step-ahead point forecasts of $y_{n+h}^{j}(x)$ are given by: $\\widehat{y}_{n+h|n}^{j}(x)&=\\text{E}\\left[y_{n+h}(x)|\\mu (x), \\eta (x), \\mathbf {\\Phi },\\mathbf {\\Psi },\\mathbf {y}^{j}\\right]\\\\&=\\widehat{\\mu }(x)+\\widehat{\\eta }^{j}(x)+\\sum ^K_{k=1}\\widehat{\\beta }_{n+h|n,k}\\widehat{\\phi }_k(x)+\\sum ^L_{l=1}\\widehat{\\gamma }^{j}_{n+h|n,l}\\widehat{\\psi }_l^{j}(x),$ where $\\widehat{\\beta }_{n+h|n,k}$ and $\\widehat{\\gamma }_{n+h|n,l}^{j}$ are the forecast principal component scores, obtained from a univariate time-series forecasting method, such as the random walk with drift (rwf) or autoregressive integrated moving average (ARIMA)$(p,d,q)$ model.", "The automatic algorithm of [33] is able to choose the optimal orders $p, q$ and $d$ automatically.", "$d$ is selected based on successive Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit-root test [41].", "KPSS tests are used for testing the null hypothesis that an observable time series is stationary around a deterministic trend.", "We first test the original time series for a unit root; if the test result is significant, then we test the differenced time series for a unit root.", "The procedure continues until we obtain our first insignificant result.", "Having identified $d$ , the orders of $p$ and $q$ are selected based on the Akaike information criterion [1] with a correction for finite sample sizes.", "The maximum likelihood method can then be used to estimate these parameters.", "It is noteworthy that a multivariate time-series method, such as vector autoregressive model, can also be used to model and forecast stationary principal component scores [3].", "[32] used the autoregressive fractionally integrated moving average (ARFIMA) in the product-ratio method (see Section REF ), which allows non-integer values for the difference parameter, to forecast the principal component scores.", "For any two populations, convergent forecasts are obtained when $\\left\\lbrace \\widehat{\\gamma }_{n+h|n,l}^{\\text{F}}-\\widehat{\\gamma }_{n+h|n,l}^{\\text{M}}\\right\\rbrace $ is stationary for each $l$ .", "As pointed out by [48], if $\\left\\lbrace \\widehat{\\gamma }_{n+h|n,l}^{\\text{F}}-\\widehat{\\gamma }_{n+h|n,l}^{\\text{M}}; l=1,\\dots ,L\\right\\rbrace $ has a trending long-term mean, the Li and Lee method fails to achieve convergence.", "As an extension of the Li and Lee method, the proposed method may also fail to achieve convergence.", "However, if the common mean function and common trend capture the long-term effect, the Li-Lee and multilevel functional data methods produce convergent forecasts, as the forecasts of residual trends would be flat.", "To quantify forecast uncertainty, the interval forecasts of $y_{n+h}^{j}(x)$ can be obtained through a Bayesian paradigm equipped with Markov chain Monte Carlo (MCMC) for estimating all variance parameters and drawing samples from the posterior of principal component scores.", "Given errors are assumed to be normally distributed, a hierarchical regression model is able to capture fixed and random effects [57], [27].", "With a set of MCMC outputs, the forecasts of future sample path are given by: $\\widehat{y}_{n+h|n}^{b,j}(x) = \\ & \\text{E}\\left[y_{n+h}(x)|\\mu (x), \\eta (x), \\mathbf {\\Phi },\\mathbf {\\Psi },\\mathbf {y}^{j}\\right]\\\\= \\ & \\widehat{f}_{n+h}^{b,j}(x) + \\widehat{\\delta }_{n+h}^{b,j}(x)\\varepsilon _{n+h}^{b,j}, \\\\= \\ & \\widehat{\\mu }(x)+\\widehat{\\eta }^{j}(x)+\\sum ^K_{k=1}\\widehat{\\beta }_{n+h|n,k}^b\\widehat{\\phi }_k(x)+\\sum ^L_{l=1}\\widehat{\\gamma }_{n+h|n,l}^{b,j}\\widehat{\\psi }_l^{j}(x)+ \\\\&\\widehat{e}_{n+h}^{b,j}(x)+\\widehat{\\delta }_{n+h}^{b,j}(x)\\varepsilon _{n+h}^{b,j},$ for $b=1,\\dots ,B$ .", "We first simulate $\\left\\lbrace \\widehat{\\beta }^b_{1,k},\\dots ,\\widehat{\\beta }^b_{n,k}\\right\\rbrace $ drawn from its full conditional density, and then obtain $\\widehat{\\beta }_{n+h|n,k}^b$ using a univariate time-series forecasting method for each simulated sample; similarly, we first simulate $\\left\\lbrace \\widehat{\\gamma }^{b,j}_{1,l},\\dots ,\\widehat{\\gamma }^{b,j}_{n,l}\\right\\rbrace $ drawn from its full conditional density, and then obtain $\\widehat{\\gamma }_{n+h|n,l}^{b,j}$ for each simulated sample; $\\left(\\widehat{\\sigma }^2\\right)^{b,j}$ is drawn from its full conditional density.", "The derivation of full conditional densities is given in the Supplement B [64], while some WinBUGS computation code is presented in the Supplement C [64].", "As we pre-smooth the functional data, we must add the smoothing error $\\widehat{\\delta }_{n+h}^{b,j}(x)\\varepsilon _{n+h}^{b,j}$ , where $\\widehat{\\delta }_{n+h}^{b,j}(x)$ is simulated from its posterior and $\\varepsilon _{n+h}^{b,j}$ is drawn from $N(0,1)$ .", "The total number of MCMC draws is 20,000 iterations, the first 10,000 iterations are used for the burn-in, whereas the remaining 10,000 iterations are recorded.", "Among these recorded draws, we keep every 10th draw in order to reduce autocorrelation.", "The prediction interval is constructed from the percentiles of the bootstrapped mortality forecasts.", "The point and interval forecasts of life expectancy are obtained from the forecast age-specific mortality rates using the life table method [55].", "In this paper, we focus on forecasting life expectancy at birth, described simply as life expectancy hereafter." ], [ "Relationship to the augmented common factor method", "The multilevel functional data method can be viewed as a generalization of the augmented common factor method of [48].", "They proposed the following model for the two-sex case, which can be expressed using a functional data model notation: $y_t^j(x_i) = \\widehat{\\mu }^j(x_i)+\\widehat{\\beta }_t\\widehat{\\phi }(x_i)+\\widehat{\\gamma }_{t}^j\\widehat{\\psi }^j(x_i)+e_t^j(x_i),$ where $x_i$ represents a discrete age or age group, $\\widehat{\\mu }^j(x_i)$ is the age- and sex-specific mean, $(\\widehat{\\beta }_1,\\dots ,\\widehat{\\beta }_n)$ is the mortality index of the common factor, which can be forecast by random walk with drift; $\\widehat{\\phi }(x_i)$ is the first estimated principal component of the common factor of [44]'s [44] model (based on log mortality), and it measures the sensitivity of the log total mortality to changes in $\\lbrace \\beta _1,\\dots ,\\beta _n\\rbrace $ over time; $\\widehat{\\gamma }_t^j$ is the time component of the additional factor, and it can be forecast by an autoregressive (AR) process of order 1; $\\widehat{\\psi }^j(x_i)$ is the first estimated principal component of the residual matrix that is specific to males or females; and $e_t^j(x_i)$ is the error term.", "$\\widehat{\\beta }_t\\widehat{\\phi }(x_i)$ specifies the long-term trend in mortality change and random fluctuations that are common for all populations, whereas $\\widehat{\\gamma }_t^j\\widehat{\\psi }^j(x_i)$ describes the short-term changes that are specific only for $j^{\\text{th}}$ population.", "The augmented common factor model takes into account the mortality trends in all populations by applying the Lee-Carter method twice, subject to identifiability constraints $\\sum _{i=1}^p\\widehat{\\phi }(x_i)=1$ and $\\sum ^n_{t=1}\\widehat{\\beta }_t=0$ .", "The eventual constant ratio between the age-specific mortality rates will thus be adjusted to the short term according to the population-specific deviations from the common pattern and trend [37].", "If the $|\\widehat{\\gamma }_{n+h|n}^{\\text{F}}-\\widehat{\\gamma }_{n+h|n}^{\\text{M}}|$ values become constant, this model leads to non-divergent forecasts in the long run but not necessarily in the short term in the case of two populations [48].", "There are two main differences between the proposed multilevel functional data method and [48]'s [48] method.", "First, [48]'s [48] method uses a single principal component to capture the largest amount of variation.", "In contrast, the multilevel functional data method includes the option of incorporating more than just one component by selecting the number of components based on the cumulative percentage of total variation in the data [13], [12].", "An examination of the residual contour plots can help to reveal the existence of any systematic patterns not being accounted for.", "In such cases, the additional principal components capture patterns in the data that may not necessarily be explained by the first principal component.", "As noted by [32], the use of multiple principal components does not introduce additional model complexity because the scores are uncorrelated and components are orthogonal by construction.", "In a similar vein, [6] considered up to three components in total when analyzing data of both sexes combined, and found that clustering in the residuals was diminished after the addition of extra components.", "[17] modeled five countries' data simultaneously with a number of components, and [46] modeled Australian female and male mortality and life expectancy jointly using more than one component.", "The second main difference between the proposed multilevel functional data method and that of [48] is that the latter restricted the univariate time-series forecasting method to be random-walk with drift for $\\widehat{\\beta }_t$ and AR(1) for $\\widehat{\\gamma }_{t}^j$ .", "These choices for the univariate time-series forecasting method may not necessarily be optimal for a given time series.", "In contrast, we implemented the auto.arima algorithm of [33], which selects the optimal order of ARIMA process based on the corrected Akaike information criterion." ], [ "Relationship to the product-ratio method", "Let us again consider modeling mortality in the two-sex case.", "The product-ratio method begins by obtaining the product and ratio functions of all series.", "The product function can be seen as the sum of all series in the log scale, whereas the ratio function can be seen as the differences among series in the log scale.", "It first applies an independent functional data method to forecast the future realizations of product and ratio functions, then transforms the forecasts of product and ratio functions back to the original male and female age-specific mortality rates.", "The convergent forecasts are achieved through the ARFIMA modeling of the ratio function, which implicitly prevents it from diverging in a long-run.", "This constraint ultimately results in a better forecast accuracy than the independent functional data method for males, but worse forecast accuracy for females.", "A possible explanation is that the product-ratio method improves the goodness of fit for males at the cost of reduced goodness of fit for females.", "The prediction intervals of mortality are constructed based on the normality assumption in [32], although it is possible to use a bootstrap method [34].", "In contrast, in the multilevel functional data method, the prediction intervals of mortality were constructed based on Bayesian paradigm.", "The validity of Bayesian paradigm for principal component scores has been given in [18].", "For a small sample size, a Bayesian sampling technique is known to produce more accurate interval forecast accuracy than the one based on the normality assumption [11]." ], [ "Application to UK age- and sex-specific mortality", "Age- and sex-specific raw mortality data for the UK between 1922 and 2009 are available from the [29].", "For each sex in a given calendar year, the mortality rates obtained by the ratio between “number of deaths\" and “exposure to risk\", are arranged in a matrix for age and calendar year.", "By analyzing the changes in mortality as a function of both age $x$ and year $t$ , it can be seen that mortality rates have shown a gradual decline over time.", "To provide an idea of this evolution, we present the functional time-series plot for male and female log mortality rates in Figure REF .", "Mortality rates dip from their early childhood high, climb in the teen years, stabilize in the early 20s, and then steadily increase with age.", "We further notice that for both males and females, mortality rates are declining over time, especially in the younger and older ages.", "Despite the higher male mortality rates in comparison to females, the difference becomes smaller and smaller over years at the older ages.", "Figure: Observed and smoothed age-specific male and female log mortality rates in the UK.", "Data from the distant past are shown in light gray, and the most recent data are shown in dark gray.In the top panel of Figure REF , we display the estimated common mean function $\\widehat{\\mu }(x)$ , first estimated common principal component $\\widehat{\\phi }_1(x)$ and corresponding principal component scores $\\left\\lbrace \\widehat{\\beta }_{1,1},\\dots ,\\widehat{\\beta }_{n,1}\\right\\rbrace $ along with 30-years-ahead forecasts.", "The first common functional principal component captures more than 98% of the total variation in the age-specific total mortality.", "In the middle panel of Figure REF , we display the estimated mean function deviance of females from the overall mean function $\\widehat{\\eta }^{\\text{F}}(x)$ , first functional principal component for females $\\widehat{\\psi }_1^{\\text{F}}(x)$ and corresponding principal component scores $\\left\\lbrace \\widehat{\\gamma }_{1,1}^{\\text{F}},\\dots ,\\widehat{\\gamma }_{n,1}^{\\text{F}}\\right\\rbrace $ with 30-years-ahead forecasts.", "In the bottom panel of Figure REF , we display the estimated mean function deviance of males from the overall mean function $\\widehat{\\eta }^{\\text{M}}(x)$ , first functional principal component for males $\\widehat{\\psi }_1^{\\text{M}}(x)$ and corresponding principal component scores $\\left\\lbrace \\widehat{\\gamma }_{1,1}^{\\text{M}},\\dots ,\\widehat{\\gamma }_{n,1}^{\\text{M}}\\right\\rbrace $ with 30-years-ahead forecasts.", "In this data set, the first three functional principal components explain at least 90% of the remaining 10% total variations for both females and males.", "Due to limited space, we present only the first functional principal component, which captures more than 64% and 50% of the remaining 10% total variations for females and males, respectively.", "Based on (REF ), the proportion of variability explained by the total mortality is 94% for females and 95% for males, respectively.", "Figure: Estimated common mean function, first common functional principal component, and associated scores for UK total mortality (top); estimated mean function deviation for females, first functional principal component, and associated scores for UK female mortality (middle); estimated mean function deviation for males, first functional principal component, and associated scores for UK male mortality (bottom).", "The dark and light gray regions show the 80% and 95% prediction intervals, respectively.From Figure REF , it is apparent that the basis functions are modeling different movements in mortality rates: $\\widehat{\\phi }_1(x)$ primarily models mortality changes in children and adults, $\\widehat{\\psi }_1^{\\text{F}}(x)$ models mortality changes between late-teens and 40, and $\\widehat{\\psi }_1^{\\text{M}}(x)$ models the differences between young adults and those over 60.", "From the forecast common principal component scores, the mortality changes in children and adults are likely to continue in the future with increasing forecast uncertainty.", "From the forecasts of sex-specific principal component scores, there are no clear trends associated with each sub-population, as the forecasts would be flat.", "Thus, it is likely to achieve convergent forecasts between female and male sub-populations.", "In the first column of Figure REF , we plot the historical mortality sex ratios (Male/Female) from 1922 to 1979, alongside the 30-years-ahead forecasts of mortality sex ratios from 1980 to 2009 by the non-coherent forecasting methods, namely Lee and Carter's method and the independent functional data method.", "In the second column, we show the 30-years-ahead forecasts of mortality sex ratios from 1980 to 2009, using coherent forecasting methods, including Li and Lee's method, and the product-ratio and multilevel functional data methods.", "We found that all the coherent forecasting methods exhibit a quite similar pattern, with much smaller sex ratios than the non-coherent forecasting methods.", "Our results confirm the expected trend toward convergence, where the gap in mortality forecasts between males and females gradually converges to a constant for each age.", "The convergent forecasts demonstrate biological characteristics, for example, the mortality of females has been lower than that of males, it would be counter-intuitive if forecasts of the recent convergence of mortality which has been observed in many developed countries leads to the opposite situation.", "Our results further reflect the importance of joint modeling, which has already been adopted for the official mortality projection in New Zealand [69].", "Figure: 30-years-ahead forecasts of mortality sex ratios from 1980 to 2009 in the UK data using Lee and Carter's method, Li and Lee's method, the independent functional data method, the product-ratio method, and the multilevel functional data method (rwf).", "The forecast curves are plotted using a rainbow color palette; the most recent forecast curves are shown in red, whereas the long-term forecast curves are shown in purple." ], [ "Multi-country comparison", "While joint modeling mortality for multiple populations offers the advantage of avoiding possible undesirable divergence in the forecasts, little is known about whether these methods can improve forecast accuracy at various lengths of forecast horizon.", "In order to investigate the forecast accuracy of the multilevel functional data method, we consider 15 other developed countries for which data are also available in the [29].", "These raw mortality rates are shown in Table REF , along with their respective data periods, within-cluster variability in (REF ) and total variance in (REF ).", "The selected countries are all developed countries with relatively long data series commencing at or before 1950.", "It was desirable to have a long available data period, in order to obtain consistent sample estimators [8].", "Including the UK data, 32 sex-specific populations were obtained for all analyses.", "Note that the age groups are single years of age from 0 to 94 and then a single age group for 95 and above, in order to avoid the excessive fluctuations at older ages.", "Table: Data period and within-cluster variability for each country." ], [ "Evaluation of point forecast accuracy", "We split our age- and sex-specific data into a training sample (including data from years 1 to $(n-30)$ ) and a testing sample (including data from years $(n-29)$ to $n$ ), where $n$ represents the total number of years in the data.", "The length of the fitting period differs by country (see Table REF ).", "We implement a rolling origin approach, following [32] and [65].", "A rolling origin analysis of a time-series model is commonly used to assess model and parameter stabilities over time.", "A common technique to assess the constancy of a model's parameter is to compute parameter estimates and their forecasts over a rolling origin of a fixed size through the sample [72].", "The advantage of the rolling origin approach is that it allows us to assess the point and interval forecast accuracy among methods for different forecast horizons.", "With the initial training sample, we produce one- to 30-year-ahead forecasts, and determine the forecast errors by comparing the forecasts with actual out-of-sample data.", "As the training sample increases by one year, we produce one- to 29-year-ahead forecasts and calculate the forecast errors.", "This process continues until the training sample covers all available data.", "We compare these forecasts with the holdout samples to determine the out-of-sample point forecast accuracy.", "To measure overall point forecast accuracy and bias, we use the root mean squared forecast error (RMSFE), mean absolute forecast error (MAFE), and mean forecast error (MFE), averaged across ages and forecasting years.", "Averaged over 16 countries, they are defined as: $\\text{RMSFE}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\sqrt{\\frac{1}{(31-h)\\times p}\\sum ^{n}_{k=n-30+h}\\sum ^p_{i=1}\\left[m_k^{c}(x_i)-\\widehat{m}_k^c(x_i)\\right]^2},\\\\\\text{MAFE}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\frac{1}{(31-h)\\times p}\\sum ^{n}_{k=n-30+h}\\sum ^p_{i=1}\\left|m_k^c(x_i)-\\widehat{m}_k^c(x_i)\\right|,\\\\\\text{MFE}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\frac{1}{(31-h)\\times p}\\sum ^{n}_{k=n-30+h}\\sum ^p_{i=1}\\left[m_k^c(x_i)-\\widehat{m}_k^c(x_i)\\right],$ where $m_k^c(x_i)$ denotes mortality rate at year $k$ in the forecasting period for age $x_i$ in country $c$ , and $\\widehat{m}_k^c(x_i)$ denotes the point forecast.", "The ordering of the 16 countries are given in Table REF .", "The RMSFE and MAFE are the average of squared and absolute errors and they measure forecast precision regardless of sign.", "The MFE is the average of errors and it measures bias." ], [ "Evaluation of interval forecast accuracy", "To assess interval forecast accuracy, we use the interval score of [22] [21].", "For each year in the forecasting period, one-year-ahead to 30-year-ahead prediction intervals were calculated at the $(1-\\alpha )\\times 100\\%$ nominal coverage probability.", "We consider the common case of symmetric $(1-\\alpha )\\times 100\\%$ prediction interval, with lower and upper bounds that are predictive quantiles at $\\alpha /2$ and $1-\\alpha /2$ , denoted by $m_k(x_l)$ and $m_k(x_u)$ for a given year $k$ .", "As defined by [22], a scoring rule for the interval forecast of mortality at age $x_i$ is: $S_{\\alpha }\\left[m_k(x_l), m_k(x_u); m_k(x_i)\\right] = \\left[m_k(x_u)-m_k(x_l)\\right]+&\\frac{2}{\\alpha }[m_k(x_l)-m_k(x_i)]{1}\\lbrace m_k(x_i)<m_k(x_l)\\rbrace +\\\\&\\frac{2}{\\alpha }\\left[m_k(x_i)-m_k(x_u)\\right]{1}\\lbrace m_k(x_i)>m_k(x_u)\\rbrace ,$ where $\\alpha $ denotes the level of significance, customarily $\\alpha =0.2$ .", "The interval score rewards for a narrow prediction interval, if and only if the true observation lies within the prediction interval.", "The optimal score is achieved when $m_k(x_i)$ lies between $m_k(x_l)$ and $m_k(x_u)$ , and the distance between $m_k(x_l)$ and $m_k(x_u)$ is minimal.", "From different ages, countries and years in the forecasting period, the mean interval score averaged across 16 countries is defined by: $\\bar{S}_{\\alpha }(h) &= \\frac{1}{16\\times (31-h)\\times p}\\sum ^{16}_{c=1}\\sum _{k=n-30+h}^{n}\\sum _{i=1}^{p}S_{\\alpha ,k}^{c}[m_k(x_l),m_k(x_u);m_k(x_i)].$" ], [ "Multi-country comparison of point forecast accuracy", "Based on the averaged MAFE and RMSFE across 30 horizons shown in Table REF , the Lee-Carter method performs overall the worst among the methods considered.", "[45] and [49] stated that mortality at older ages has been declining more quickly (on a log scale) than at younger ages, which contradicts the stationarity assumption of mortality improvement in the Lee-Carter method.", "Thus, it has been systematically under-predicting improvements in life expectancy over time.", "This confirms the fact that progress in life expectancy has been and continues to rise [53].", "Table: Point forecast accuracy of mortality and life expectancy for females and males by method, as measured by the averaged MAFE, RMSFE, and MFE.", "For mortality, the forecast errors were multiplied by 100 in order to keep two decimal places.", "The minimal forecast errors are underlined for females and males, whereas the minimal overall forecast error is highlighted in bold.", "FDM represents functional data model.The functional data methods use the automatic ARIMA algorithm for selecting the optimal difference operator $d$ , for which the mortality improvement will then be stationary.", "Generally, the functional data methods give more accurate forecasts than the Lee-Carter and Li-Lee methods.", "The independent functional data method performs consistently the best for forecasting female mortality, followed by the multilevel functional data (arima) and product-ratio methods.", "The superiority of the independent functional data method over the coherent forecasting methods is manifested by a population with small variabilities over age and time, such as in female mortality.", "In terms of male and overall forecast errors, the product-ratio and multilevel functional data methods perform similarly: they both produce more accurate forecasts than those from the independent functional data method.", "From the averaged MFE across 30 horizons, the coherent forecasting methods produce less-biased forecasts than the non-coherent forecasting methods for males.", "The independent functional data method gives the least-biased forecasts of female mortality.", "For male mortality, the product-ratio method and multilevel functional data method (arima) perform about the same in terms of bias, and they both produce less-biased forecasts than the ones from the independent functional data method.", "With the forecast age-specific mortality, we can also forecast life expectancy [55].", "Based on the averaged MAFE, RMSFE, and MFE across 30 horizons, we again found that the functional data methods generally give smaller overall forecast errors and bias across two sexes, in comparison to the Lee-Carter and Li-Lee methods.", "The independent functional data method performs the best for forecasting female life expectancy, followed by the multilevel functional data (arima) and product ratio methods.", "For male data, the multilevel functional data method (rwf) gives the most accurate point forecasts.", "The product-ratio and multilevel functional data methods both produce more accurate point forecasts than the ones from the independent functional data method.", "Of the two approaches, the multilevel functional data method (arima) performs the best based on simple averaging of the forecast errors over two sub-populations.", "To achieve optimal point forecast accuracy and bias, the independent functional data method should be used for forecasting female mortality and life expectancy, whereas the product-ratio or multilevel functional data method (rwf) should be implemented for forecasting male mortality and male life expectancy, respectively.", "Based on the simple average of two sub-populations, the multilevel functional data method (arima) generally performs the overall best in all.", "With respect to the automatic ARIMA and random-walk with drift (rwf), the automatic ARIMA method is recommended to forecast principal component scores in the multilevel functional data method for age-specific female mortality and life expectancy.", "In contrast, the rwf method is suitable to forecast principal component scores for age-specific male mortality and life expectancy." ], [ "Multi-country comparison of interval forecast accuracy", "The prediction intervals for age-specific mortality are obtained from (REF ), and the prediction intervals for life expectancy are obtained from the percentiles of simulated life expectancies.", "The simulation method takes the nonlinear relationship between age-specific mortality and life expectancy into account, thus giving an asymmetric prediction interval [32].", "Based on the averaged mean interval scores shown in Table REF , the independent functional data method produces the most accurate forecasts for female mortality, followed by the multilevel functional data (arima) method.", "For male mortality, the multilevel functional data model (rwf) performs the best, followed by the Li-Lee method.", "Averaged across both sexes, the multilevel functional data method (arima) performs the best.", "For forecasting female life expectancy, the multilevel functional data method (arima) produces the most accurate interval forecasts, followed by the independent functional data method.", "For forecasting male life expectancy, the multilevel functional data method (rwf) gives the best interval forecast accuracy.", "Averaged across both sexes, the multilevel functional data method (arima) performs the best.", "Table: Interval forecast accuracy of mortality and life expectancy for females and males by method, as measured by the averaged mean interval score.", "For mortality, the mean interval scores were multiplied by 100 in order to keep two decimal places.Apart from the mean forecast errors and mean interval scores, we also consider the maximum absolute forecast error, maximum root squared forecast error, and maximum interval score, for measuring the extreme point and interval errors across different ages and years in the forecasting period.", "Their results in the multi-country comparison are included in the supplement D [64]." ], [ "Comparison between the functional data models and a Bayesian method", "[58] proposed a Bayesian hierarchical model for joint probabilistic projection of male and female life expectancies that ensures coherence between them by projecting the gap between female life expectancy and male life expectancy.", "This method starts with probabilistic projection of life expectancy for females obtained from a Bayesian hierarchical model, then models the gap in life expectancy between females and males.", "The probabilistic projection of life expectancy for males can be obtained by combining the former two quantities.", "Computationally, this method is implemented in the bayesLife package [63] in R [56].", "In Tables REF and REF , we compare the forecast accuracy between the multilevel functional data and Bayesian methods for forecasting life expectancy.", "Table: Point and interval forecast accuracy between the multilevel functional data method and Bayesian method for forecasting female life expectancy at birth (e(0)).", "Using the data until 1979, we forecast the e(0) for years 1984, 1989, 1994, 1999, 2004 and 2009.Table: Point and interval forecast accuracy between the multilevel functional data method and Bayesian method for forecasting male life expectancy at birth (e(0)).", "Using the data until 1979, we forecast the e(0) for years 1984, 1989, 1994, 1999, 2004 and 2009.For females, the Bayesian method is recommended.", "For males, the multilevel functional data method is preferable, in terms of point forecast accuracy.", "In terms of interval forecast accuracy, the Bayesian method is slightly advantageous for long-term forecasts.", "We found that the Bayesian (a simpler and direct) method outperforms the multilevel functional data method for long-term projection of life expectancy.", "The Bayesian method shows a superior interval forecast accuracy for two reasons: 1) the Bayesian method uses the historical life expectancy data to produce forecasts, whereas the multilevel functional data method uses the historical age-specific mortality to produce these age-specific mortality rate forecasts, which are then combined non-linearly to give life expectancy forecasts.", "Oftentimes, the direct forecasting method outperforms the indirect forecasting method.", "2) the Bayesian method uses the prior information to assist its forecasts, in particular at longer forecast horizon.", "By contrast, the multilevel functional data method is a time-series extrapolation, which works reasonably well in the short time.", "However, it does not work well for long term.", "Given that different changes are at play at different phases of a mortality transition, the age components of change in the past are not necessarily informative of longer-term future.", "Application to Australian age- and sex- and state-specific mortality First, we consider the age- and state-wise total mortality rates from 1950 to 2003 in Australia, available in the addb package of [30] in R [56].", "This data set contains mortality rates for six states of Australia: Victoria (VIC), New South Wales (NSW), Queensland (QLD), South Australia (SA), Western Australia (WA), and Tasmania (TAS).", "The Australian Capital Territory and the Northern Territory are excluded from the analysis due to many missing values in the available data.", "In Figure REF , we show the estimated overall mean function $\\widehat{\\mu }(x)$ , first common functional principal component $\\widehat{\\phi }_1(x)$ and corresponding scores $\\left\\lbrace \\widehat{\\beta }_{1,1},\\dots ,\\widehat{\\beta }_{n,1}\\right\\rbrace $ with 30-years-ahead forecasts.", "The first common functional principal component accounts for at least 90% of total variation in the total mortality.", "The retained number of functional principal components for each state is the one that explains at least 90% of the remaining 10% total variations in the data.", "Due to limited space, we present only the first principal components for the six states, which explain 27%, 68%, 26%, 22%, 22%, and 28% of the remaining 10% total variations for VIC, NSW, TAS, QLD, SA, WA, respectively.", "Based on (REF ), the proportion of variability explained by the aggregate data (the simple average of total mortality across states) is $71\\%, 71\\%, 33\\%, 63\\%, 50\\%$ , and $50\\%$ for VIC, NSW, TAS, QLD, SA, WA, respectively.", "Figure: The first common functional principal component and its associated scores for the aggregate mortality data (top), followed by the first functional principal component and associated scores for the state-wise total age-specific mortality rates in VIC, NSW, TAS, QLD, SA and WA, respectively.", "The dark and light gray regions show the 80% and 95% prediction intervals.In Figure REF , we also show the estimated mean function deviation, first state-specific functional principal component $\\widehat{\\psi }_1^s(x)$ and principal component scores $\\lbrace \\widehat{\\gamma }^s_{1,1},\\dots ,\\widehat{\\gamma }^s_{n,1}\\rbrace $ with 30-years-ahead forecasts, where $s$ denotes a state.", "The convergence in forecasts is likely to be achieved by the multilevel functional data method, because the forecasts of principal component scores for each state do not show a long-term trend, with the exception of NSW.", "From a statistical perspective, this may be because the NSW has the largest proportion of variability that can not be explained by the aggregate data.", "From a social perspective, NSW is the state that attracts the most migrants in Australia (http://www.abs.gov.au/ausstats/[email protected]/mf/3412.0).", "Figure REF shows 30-years-ahead forecasts of median log mortality rates and life expectancy from 2004 to 2033 for all states, for the independent functional data, product-ratio and multilevel functional data methods.", "We focus on these three methods in this application, because they generally outperform the Lee-Carter and Li-Lee methods as demonstrated in Section .", "For the independent functional data method, the gap in mortality and life expectancy forecasts among states diverges.", "In contrast, the product-ratio and multilevel functional data methods are quite similar, and the gaps between female and male age-specific mortality and life expectancy converge, respectively.", "Figure: Based on historical mortality rates (1950–2003), we forecast future mortality rates and life expectancy from 2004 to 2033, for the independent functional data, product-ratio, and multilevel functional data methods.Comparisons of point and interval forecast accuracy Table REF displays the point and interval forecast accuracy for both age- and state-specific total mortality rates and life expectancy at each forecast horizon.", "As measured by the averaged MAFE, RMSFE, MFE and averaged mean interval score across 30 horizons, the independent functional data method performs the worst, whereas the multilevel functional data method (rwf) performs the best, for forecasting age- and state-specific total mortality and life expectancy.", "As the product-ratio and multilevel functional data methods perform similarly, it is paramount to incorporate correlation among sub-populations in forecasting, as this allows us to search for characteristics within and among series.", "Table: Point and interval forecast accuracy of mortality and life expectancy (e(0)) across different states by method and forecast horizon, as measured by the averaged MAFE, RMSFE, MFE, and averaged mean interval score.", "The minimal forecast errors are underlined for each state, whereas the minimal overall forecast error is highlighted in bold.", "Application to Australian age-, sex- and state-specific mortality We extend the multilevel functional data method to two or more sub-populations in a hierarchy.", "This is related to hierarchical/grouped time series [31].", "A grouped structure is depicted in the two-level hierarchical diagram, presented in Figure REF .", "Figure: A two-level hierarchical tree diagram.Following a bottom-up hierarchical structure, we first extract a common trend from the total mortality within each state.", "For the $j^{\\text{th}}$ population in state $s$ , the multilevel functional data model can be written as: $f_t^{j,s}(x) = \\mu ^{j,s}(x) + R_t^{s}(x) + U_t^{j, s}(x), $ where $f_t^{j,s}(x)$ represents the female or male mortality in state $s$ at year $t$ ; $\\mu ^{j,s}(x)$ is the mean function of female or male mortality in state $s$ ; $R_t^{s}(x)$ captures the common trend across two populations for a state; and $U_t^{j, s}(x)$ captures the sex-specific residual trend for a state.", "Based on (REF ), the proportion of variability explained by the total mortality in each state is 65%, 69%, 25%, 53%, 43%, and 37% for females, and 59%, 59%, 22%, 54%, 41%, and 38% for males.", "We can also extract the common trend from the averaged mortality across all states for females and males.", "For the $j^{\\text{th}}$ population in state $s$ , the multilevel functional data model can be written as: $f_t^{j,s}(x) = \\mu ^{j,s}(x) + S_t^{j}(x)+W_t^{j,s}(x),$ where $S_t^{j}(x)$ captures the common trend across six populations; and $W_t^{j,s}(x)$ captures the state-specific residual trend.", "By combining (REF ) and (REF ), we obtain $f_t^{j,s}(x) &= \\mu ^{j,s}(x) + \\frac{R_t^{s}(x) + U_t^{j, s}(x) + S_t^{j}(x)+W_t^{j,s}(x)}{2}.$ Table: Point forecast errors (×100\\times 100) of mortality across states and sexes by method, as measured by the averaged MAFE, RMSFE, and MFE.", "The minimal forecast errors are underlined for each state and each sex, whereas the minimal overall forecast error is highlighted in bold.Table: Point forecast accuracy of life expectancy across states and sexes by method, as measured by the averaged MAFE, RMSFE, and MFE.", "The minimal forecast errors are underlined for each state and each sex, whereas the minimal overall forecast error is highlighted in bold.Table: Interval forecast accuracy of mortality and life expectancy across states and sexes by method, as measured by the averaged mean interval score.", "The minimal forecast errors are underlined for each state and each sex, whereas the minimal overall forecast error is highlighted in bold.Tables REF , REF and REF show the point and interval forecast accuracy among different functional data methods.", "As measured by the averaged MAFE, RMSFE, MFE and averaged mean interval score across 30 horizons, the multilevel functional data method (rwf) gives the smallest errors for forecasting female mortality rate and life expectancy, as well as the smallest overall errors, whereas the product-ratio method produces the most accurate forecasts for male mortality rate and life expectancy.", "Apart from the expected error loss function, we also consider the maximum point and interval forecast error criteria.", "Their results are also included in the supplement D [64].", "Conclusion In this paper, we adapt the multilevel functional data model to forecast age-specific mortality and life expectancy for a group of populations.", "We highlight the relationships among the adapted multilevel functional data, augmented common factor method and product-ratio method.", "As demonstrated by the empirical studies consisting of two populations, we found that the independent functional data method gives the best forecast accuracy for females, whereas the multilevel functional data and product-ratio methods produce more accurate forecasts for males.", "Based on their averaged forecast errors, the multilevel functional data method (arima) should be used in the case of two sub-populations, in particular for females.", "In the case of more than two populations, it is evident that the multilevel functional data and product-ratio methods consistently outperform the independent functional data method.", "The multilevel functional data method (rwf) gives the most accurate mortality and life expectancy forecasts for age- and state-specific total mortality.", "When we further disaggregated the age- and state-specific total mortality by sex, we found that the multilevel functional data method (rwf) should be used for forecasting female mortality and life expectancy, whereas the product-ratio method should be applied for forecasting male mortality and life expectancy.", "The superiority of the product-ratio and multilevel functional data methods over the independent functional data method is manifested by a population with large variability over age and year.", "For example, the male data generally show greater variability over age and year than do the female data; as a result the product-ratio and multilevel functional data methods perform better in terms of forecast accuracy than the independent functional data method.", "Because the product-ratio and multilevel functional data methods produce better forecast accuracy than the independent functional data method overall, this may lead to their use by government agencies and statistical bureaus involved in short-term demographic forecasting.", "For long-term forecast horizons, any time-series extrapolation methods, including the proposed one, may not be accurate as the underlying model may no longer be optimal.", "Given that different changes are at play in different phases of a mortality transition, the age components of change in the past are not necessarily informative of the longer-term future.", "By incorporating prior knowledge, the Bayesian method of [58] demonstrated the superior forecast accuracy of the long-term projection of life expectancy.", "A limitation of the current study is that the comparative analysis among the five methods focuses on errors that aggregate over all age groups for one- to 30-step-ahead mortality forecasts.", "In future research, it is possible that the analysis of the forecast errors for certain key age groups, such as those above 65, might shed light on the results of more detailed analysis.", "For a relatively long time series, geometrically decaying weights can be imposed on the computation of functional principal components [34] for achieving potentially improved forecast accuracy.", "In addition, the product-ratio and multilevel functional data methods could be applied to model and forecast other demographic components, such as age-specific immigration, migration, and population size by sex or other attributes for national and sub-national populations.", "Reconciling these forecasts across different levels of a hierarchy is worthwhile to investigate in the future [66].", "Supplement to: “Mortality and life expectancy forecasting for a group of populations in developed countries: A multilevel functional data method.\"", "by H. L. Shang This supplement contains a PDF divided into four sections.", "Supplement A: Some theoretical properties of multilevel functional principal component decomposition; Supplement B: Derivation of posterior density of principal component scores and other variance parameters; Supplement C: WinBUGS computational code used for sampling principal component scores and estimating variance parameters from full conditional densities; Supplement D: Additional results for point and interval forecast accuracy of mortality and life expectancy, based on maximum forecast error measures.", "Supplement to “Mortality and life expectancy forecasting for a group of populations in developed countries: A multilevel functional data method by H. L. Shang Supplement A: Some theoretical properties of multilevel functional principal component decomposition Let $R$ and $U^{j}$ be two stochastic processes defined on a compact set $\\mathcal {I}$ , with finite variance.", "The covariance functions of $R$ and $U^{j}$ are defined to be the function $\\mathcal {K}: \\mathcal {I}\\times \\mathcal {I}\\rightarrow R$ , such that $\\mathcal {K}^R(w,v) &= \\text{cov}\\lbrace R(w), R(v)\\rbrace = \\text{E}\\left\\lbrace [R(w)-\\mu (w)]\\otimes [R(v)-\\mu (v)]\\right\\rbrace ,\\\\\\mathcal {K}^{U^{j}}(w,v) &= \\text{cov}\\left\\lbrace U^{j}(w), U^{j}(v)\\right\\rbrace = \\text{E}\\left\\lbrace [U^{j}(w)-\\mu (w)]\\otimes [U^{j}(v)-\\mu (v)]\\right\\rbrace ,$ where $\\otimes $ represents the tensor product and $j$ represents the index of sub-populations.", "In a finite dimension, the tensor product reduces to matrix multiplication.", "Mercer's theorem [36] provides the following consistent spectrum decomposition, $\\mathcal {K}^R(w,v) &= \\text{cov}\\left\\lbrace R(w), R(v)\\right\\rbrace = \\sum ^{\\infty }_{k=1}\\lambda _k \\phi _k(w)\\phi _k(v),\\\\\\mathcal {K}^{U^{j}}(w,v) &= \\text{cov}\\left\\lbrace U^{j}(w), U^{j}(v)\\right\\rbrace = \\sum ^{\\infty }_{l=1}\\lambda ^{j}_l\\psi _l^{j}(w)\\psi _l^{j}(v),$ where $\\lambda _1\\ge \\lambda _2\\ge \\dots $ are the ordered population eigenvalues and $\\phi _k(\\cdot )$ is the $k^{\\text{th}}$ orthonormal eigenfunction of $\\mathcal {K}^{R}(\\cdot ,\\cdot )$ in the $L^2$ norm.", "Similarly, $\\lambda ^{j}_1\\ge \\lambda ^{j}_2\\ge \\dots $ are the ordered population eigenvalues and $\\psi _l^{j}(\\cdot )$ is the $l^{\\text{th}}$ orthonormal eigenfunction of $\\mathcal {K}^{U^{j}}(\\cdot ,\\cdot )$ in the $L^2$ norm.", "With Mercer's lemma, stochastic processes $R$ and $U^{j}$ can be expressed by the Karhunen-Loève expansion [39], [50].", "In practice, we reduce the dimensionality of functional data by truncating the infinite series to finite dimension, such as the first $K$ number of principal components [70], [24], [28].", "These can be expressed as: $R_t(x) &= \\sum ^{\\infty }_{k=1}\\beta _{t,k}\\phi _k(x)\\approx \\sum ^{K}_{k=1}\\beta _{t,k}\\phi _k(x),\\\\U_t^{j}(x) &= \\sum ^{\\infty }_{l=1}\\gamma _{t,l}^{j}\\psi _{l}^{j}(x)\\approx \\sum ^{L}_{l=1}\\gamma _{t,l}^{j}\\psi _{l}^{j}(x),$ where $\\beta _{t,k}=\\int _{\\mathcal {I}} R_t(x)\\phi _k(x)dx$ , $\\gamma _{t,l}^{j}=\\int _{\\mathcal {I}}U_t^{j}(x)\\psi _l^{j}(x)dx$ are the uncorrelated principal component scores with $\\text{E}(\\beta _{t,k}) = \\text{E}\\left(\\gamma _{t,l}^{j}\\right)=0$ , $\\text{Var}(\\beta _{t,k})=\\lambda _k<\\infty $ , $\\text{Var}(\\gamma _{t,l}^{j})=\\lambda _l^{j}<\\infty $ , $K$ and $L$ represent the retained numbers of principal components, and $\\mathcal {I}$ represents the domain of $x$ variable, such as $x\\in [0,95+]$ in our context.", "Appendix B: Derivation of posterior density of principal component scores We present derivations for the multilevel functional data model, including its specification and full conditional densities.", "The full conditionals are also given in [18], which provides a foundation for this work.", "Here, we extend it by adding an additional stochastic variance for the pre-smoothing step.", "This stochastic variance takes into account the varying uncertainty across observations.", "$ \\left\\lbrace \\begin{array}{ll}y_t^j(x_i) = f_t^j(x_i)+\\delta _t^j(x_i)\\epsilon _{t,i}^j \\\\f_t^j(x_i) = \\mu (x_i) + \\eta ^j(x_i) + \\sum ^K_{k=1}\\beta _{t,k}\\phi _k(x_i) + \\sum ^L_{l=1}\\gamma _{t,l}^j\\psi _l^j(x_i)+\\varepsilon _t^j(x_i) \\\\\\beta _{t,k}\\sim N\\left(0,\\lambda _k\\right); \\gamma _{t,l}^j\\sim N\\left(0,\\lambda _l^{j}\\right); \\varepsilon _t^j(x_i)\\sim N(0,(\\sigma ^2)^j); \\delta _t^j(x_i) \\sim N(0, (\\kappa _i^2)^j) \\\\\\frac{1}{(\\sigma ^2)^j} \\sim \\text{Gamma}(\\alpha _1,\\alpha _2)\\end{array} \\right.", "$ The full conditional density of inverse error variance given other parameters is $1/\\left(\\sigma ^2\\right)^j\\big |\\text{others} \\sim \\text{Gamma}\\left(\\alpha _1^{\\text{post}},\\alpha _2^{\\text{post}}\\right),$ where $\\alpha _1^{\\text{post}} &= \\alpha _1 + \\frac{1}{2}Jnp \\\\\\alpha _2^{\\text{post}} &= \\alpha _2 + \\frac{1}{2}\\sum ^J_{j=1}\\sum ^n_{t=1}\\sum ^p_{i=1}\\left[\\varepsilon _t^j(x_i)\\right]^2$ and $\\varepsilon _t^j(x_i) = f_t^j(x_i) - \\mu (x_i) - \\eta ^j(x_i) - \\sum ^K_{k=1}\\beta _{t,k}\\phi _k(x_i) - \\sum ^L_{l=1}\\gamma _{t,l}^j\\psi _l^j(x_i),$ where $J$ denotes the number of populations, $n$ denotes the sample size, and $p$ denotes the total number of age groups.", "The full conditional density of principal component scores for the common trend given other parameters is $\\beta _{t,k}\\big |\\text{others}\\sim N\\left(\\mu _{\\beta _{t,k}}^{\\text{post}}, v_{\\beta _{t,k}}^{\\text{post}}\\right)$ where $\\mu _{\\beta _{t,k}}^{\\text{post}} &= \\frac{\\lambda _k J \\sum ^p_{i=1}\\phi _k(x_i)^2}{\\lambda _k J \\sum ^p_{i=1}\\phi _k(x_i)^2 + (\\sigma ^2)^j}\\cdot \\frac{\\sum ^J_{j=1}\\sum ^p_{i=1}\\phi _k(x_i)\\left[\\varepsilon _t^j(x_i)+\\beta _{t,k}\\phi _k(x_i)\\right]}{J\\sum ^p_{i=1}\\phi _k(x_i)^2}, \\\\v_{\\beta _{t,k}}^{\\text{post}} &= \\frac{\\lambda _k (\\sigma ^2)^j}{\\lambda _k J\\cdot \\sum _{i=1}^p\\phi _k(x_i)^2+(\\sigma ^2)^j},$ where $\\lambda _k$ denotes the $k$ th eigenvalue of the common covariance function.", "The full conditional density of principal component scores for the population-specific residual trend given other parameters is $\\gamma _{t,l}^j|\\text{others} \\sim N\\left(\\mu _{\\gamma _{t,l}^j}^{\\text{post}}, v_{\\gamma _{t,l}^j}^{\\text{post}}\\right),$ where $\\mu _{\\gamma _{t,l}^j}^{\\text{post}} &= \\frac{\\lambda _l^j\\cdot \\sum ^{p}_{i=1}\\psi _l^j(t_i)^2}{\\lambda _l^j\\cdot \\sum _{i=1}^p\\psi _l^j(x_i)^2+(\\sigma ^2)^j}\\cdot \\frac{\\sum _{i=1}^p\\phi _k(x_i)\\left[\\varepsilon _{t}^j(x_i)+\\gamma _{t,l}^j\\psi _l^j(x_i)\\right]}{\\sum _{i=1}^p\\psi _l^{j}(x_i)^2}, \\\\v_{\\gamma _{t,l}^j}^{\\text{post}}&=\\frac{\\lambda _l^j(\\sigma ^2)^j}{\\lambda _l^j\\cdot \\sum _{i=1}^p\\psi _l^j(x_i)^2+(\\sigma ^2)^j},$ where $\\lambda _l^j$ denotes the $l$ th eigenvalue of the population-specific covariance function.", "Since the first step involves a nonparametric smoothing with heteroscedastic of unknown form.", "We can incorporate this nonparametric smoothing step in our Markov chain Monte Carlo (MCMC) iterations.", "For different ages or age groups, variances are unequal as shown in equation (2.4) of the main manuscript.", "Following the early work by [40], we consider a linear regression with heterscedastic errors and its Bayesian computation algorithm is documented in [40] Let $(\\omega _1,\\omega _2,\\dots ,\\omega _p)=\\left[1/\\delta ^2(x_1),1/\\delta ^2(x_2),\\dots ,1/\\delta ^2(x_p)\\right]$ be the precision parameters for different ages.", "Consider the following Gamma prior for $\\omega _i$ : $\\pi (\\omega _i) = f_{\\text{G}}(1,v_{\\omega }),\\qquad i=1,2,\\dots ,p,$ where the prior for $\\omega _i$ depends upon a hyperparameter $v_{\\omega }$ and assume that each precision $\\omega _i$ comes from the same distribution, but can differ from each other in values.", "Each of the conditional posteriors for $\\omega _i$ has the form of a Gamma density, given by $\\pi (\\omega _i|v_{\\omega }, \\text{others}) &= f_{\\text{G}}\\left(\\frac{v_{\\omega }+1}{\\sum ^n_{t=1}\\left[y_t(x_i) - f_t(x_i)\\right]^2+v_{\\omega }}, v_{\\omega }+1 \\right), \\\\\\pi (v_{\\omega }|\\omega _i,\\text{other}) &\\propto \\left(\\frac{v_{\\omega }}{2}\\right)^{p\\cdot \\frac{v_{\\omega }}{2}}\\Gamma \\left(\\frac{v_{\\omega }}{2}\\right)^{-p}e^{-\\eta \\cdot v_{\\omega }},$ where $\\eta = \\frac{1}{v_{\\omega }}+\\frac{1}{2}\\sum ^p_{i=1}\\left[\\ln \\left(\\frac{1}{\\omega _i}\\right)+\\omega _i\\right]$ , and $\\Gamma (\\cdot )$ denotes a Gamma function.", "Supplement C: WinBUGS code used for estimating variance parameters Statistical software WinBUGS is used to estimate variances in the principal component scores and error function.", "From the estimated variances, the principal component scores and error function are simulated from normal distributions with zero mean.", "Below is a modified version of WinBUGS given by [13], for modeling age- and sex-specific mortality rates.", "R model {   for (i in 1:N_subj)   { \tfor (t in 1:N_obs) \t{ \t\tW_1[i,t] ~ dnorm(m_1[i,t], taueps_1) \t\tW_2[i,t] ~ dnorm(m_2[i,t], taueps_2) \t\tm_1[i,t] <- X[i,t] + U_1[i,t] \t\tm_2[i,t] <- X[i,t] + U_2[i,t]   \t\tX[i,t] <- inprod(xi[i,], psi_1[t,]) \t\tU_1[i,t] <- inprod(zi[i,], psi_2[t,]) \t\tU_2[i,t] <- inprod(fi[i,], psi_3[t,]) \t} \tfor(k in 1:dim_space_b) \t{ \t\txi[i,k] ~ dnorm(0.0, ll_b[k]) \t} \tfor(l in 1:dim_space_w) \t{ \t\tzi[i,l]  ~ dnorm(0.0, ll_w[l]) \t} \tfor(j in 1:dim_space_f) \t{\t \t\tfi[i,j] ~ dnorm(0.0, ll_f[j]) \t}   }\t   for(k in 1:dim_space_b)   { \tll_b[k] ~ dgamma(1.0E-3, 1.0E-3) \tlambda_b[k] <- 1/ll_b[k]   }   for(l in 1:dim_space_w)   { \tll_w[l] ~ dgamma(1.0E-3, 1.0E-3) \tlambda_w[l] <- 1/ll_w[l]   }   for(j in 1:dim_space_f)   {  \tll_f[j] ~ dgamma(1.0E-3, 1.0E-3) \tlambda_f[j] <- 1/ll_f[j]   }   taueps_1 ~ dgamma(1.0E-3, 1.0E-3)   taueps_2 ~ dgamma(1.0E-3, 1.0E-3) } The definition of all variables is given below: $\\text{N}\\_\\text{subj}$ is the number of subjects (sample size) $\\text{N}\\_\\text{obs}$ is the number of observations within subjects $\\text{W}\\_1$ [i,t] and $\\text{W}\\_2$ [i,t] are the functional observations at the aggregated level and sex-specific level, for subject i at time t. Both matrices $\\text{W}\\_1$ [,] and $\\text{W}\\_2$ [,] are $\\text{N}\\_\\text{subj} \\times \\text{N}\\_\\text{obs}$ , are loaded as data and may contain missing observations $\\text{m}\\_1$ [i,t] and $\\text{m}\\_2$ [i,t] are the smoothed means of $\\text{W}\\_1$ [i,t] and $\\text{W}\\_2$ [i,t], respectively, are unknown and their joint distribution is simulated $\\text{X}$ [i,t] is the mean process at the aggregated level.", "$\\text{X}$ [,] is a $\\text{N}\\_\\text{subj}\\times \\text{N}\\_\\text{obs}$ dimensional matrix of parameters that are estimated from the model $\\text{U}\\_1$ [i,t] and $\\text{U}\\_2$ [i,t] are the sex-specific mean process at the individual level.", "$\\text{U}\\_1$ [i,t] and $\\text{U}\\_2$ [i,t] are the $\\text{N}\\_\\text{subj} \\times \\text{N}\\_\\text{obs}$ dimensional matrices of parameters that are estimated from the model $\\text{psi}\\_1$ [t,], $\\text{psi}\\_2$ [t,], $\\text{psi}\\_3$ [t,] are eigenfunctions at both the aggregated level and sex-specific level, evaluated at the time t. The matrices $\\text{psi}\\_1$ , $\\text{psi}\\_2$ , $\\text{psi}\\_3$ are $\\text{N}\\_\\text{obs}\\times \\text{K}\\_1$ , $\\text{N}\\_\\text{obs} \\times \\text{K}\\_2$ , $\\text{N}\\_\\text{obs} \\times \\text{K}\\_3$ , where $\\text{K}\\_1$ is the number of retained components that explains at least 90% of total variations in total mortality data, $\\text{K}\\_2$ and $\\text{K}\\_3$ are the number of retained components that explains at least 90% of the remaining 10% total variations in sex-specific data.", "The matrices of $\\text{psi}\\_1$ , $\\text{psi}\\_2$ , $\\text{psi}\\_3$ do not contain any missing value, and are loaded as data $\\text{xi}$ [i,k] are the scores for the subject i on the kth eigenfunction $\\text{psi}\\_1$ [t,k] $\\text{zi}$ [i,l] are the scores for the subject i on the lth eigenfunction $\\text{psi}\\_2$ [t,l] $\\text{fi}$ [i,j] are the scores for the subject i on the jth eigenfunction $\\text{psi}\\_3$ [t,j] $\\text{ll}\\_{\\text{b}}$ [k] are the precisions for the distribution of scores $\\text{xi}$ [i,k] $\\text{ll}\\_{\\text{w}}$ [l] are the precisions for the distribution of scores $\\text{zi}$ [i,l] $\\text{ll}\\_{\\text{f}}$ [j] are the precisions for the distribution of scores $\\text{fi}$ [i,j] $\\text{taueps}\\_1$ is the precision of the error process due to imperfect observations of $\\text{W}\\_1$ [i,t] around its smooth mean $\\text{m}\\_1$ [i,t].", "This is a parameter of the model that is estimated $\\text{taueps}\\_2$ is the precision of the error process due to imperfect observations of $\\text{W}\\_2$ [i,t] around its smooth mean $\\text{m}\\_2$ [i,t].", "This is a parameter of the model that is estimated All precision priors are Gamma priors with mean 1 and variance 1000 Supplement D: Additional results for point and interval forecast accuracy of mortality and life expectancy Apart from the averaged forecast error criteria, we also consider the maximum absolute forecast error (Max AFE), maximum root squared forecast error (Max RSFE), and maximum interval score for measuring the extreme errors across different ages ($x_i$ ) and years in the forecasting period (year $k$ ).", "Averaging across 16 countries, they are defined as $\\text{Max AFE}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\max _{k,i}\\left|m_k^c(x_i)-\\widehat{m}_k^c(x_i)\\right|, \\\\\\text{Max RSFE}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\sqrt{\\max _{k,i}\\left[m_k^c(x_i)-\\widehat{m}_k^c(x_i)\\right]^2}, \\\\\\text{Max interval score}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\max _{k,i} S_{\\alpha ,k}^{c}(x_l,x_u;x_i).$ Tables REF to REF present the Max AFE, Max RSFE, and Max interval score for comparing point and interval forecast accuracies of the age-specific mortality and life expectancy by method, in the case of two populations.", "Table: Point and interval forecast accuracy of mortality and life expectancy for females and males by method, as measured by the Max AFE, Max RSFE and Max interval score.", "For mortality, the forecast errors were multiplied by 100, in order to keep two decimal places.", "The minimal forecast errors are underlined for female and male data given in Section 5, whereas the minimal overall forecast error is highlighted in bold.Table: Point and interval forecast accuracy of mortality and life expectancy across different states (described in Section 6.1) by method, as measured by the Max AFE, Max RSFE, and maximum interval score.", "The minimal forecast errors are underlined for each state in Section 6, whereas the minimal overall forecast error is highlighted in bold.Table: Point and interval forecast accuracy of mortality (×100\\times 100) across states and sexes (described in Section 6.2) by method, as measured by the Max AFE, Max RSFE, and maximum interval score.", "The minimal forecast errors are underlined for female and male data and their average, whereas the minimal overall forecast error is highlighted in bold.Table: Point and interval forecast accuracy of life expectancy across states and sexes (described in Section 6.2) by method, as measured by the Max AFE, Max RSFE, and maximum interval score.", "The minimal forecast errors are underlined for female and male data and their average, whereas the minimal overall forecast error is highlighted in bold." ], [ "Application to Australian age- and sex- and state-specific mortality", "First, we consider the age- and state-wise total mortality rates from 1950 to 2003 in Australia, available in the addb package of [30] in R [56].", "This data set contains mortality rates for six states of Australia: Victoria (VIC), New South Wales (NSW), Queensland (QLD), South Australia (SA), Western Australia (WA), and Tasmania (TAS).", "The Australian Capital Territory and the Northern Territory are excluded from the analysis due to many missing values in the available data.", "In Figure REF , we show the estimated overall mean function $\\widehat{\\mu }(x)$ , first common functional principal component $\\widehat{\\phi }_1(x)$ and corresponding scores $\\left\\lbrace \\widehat{\\beta }_{1,1},\\dots ,\\widehat{\\beta }_{n,1}\\right\\rbrace $ with 30-years-ahead forecasts.", "The first common functional principal component accounts for at least 90% of total variation in the total mortality.", "The retained number of functional principal components for each state is the one that explains at least 90% of the remaining 10% total variations in the data.", "Due to limited space, we present only the first principal components for the six states, which explain 27%, 68%, 26%, 22%, 22%, and 28% of the remaining 10% total variations for VIC, NSW, TAS, QLD, SA, WA, respectively.", "Based on (REF ), the proportion of variability explained by the aggregate data (the simple average of total mortality across states) is $71\\%, 71\\%, 33\\%, 63\\%, 50\\%$ , and $50\\%$ for VIC, NSW, TAS, QLD, SA, WA, respectively.", "Figure: The first common functional principal component and its associated scores for the aggregate mortality data (top), followed by the first functional principal component and associated scores for the state-wise total age-specific mortality rates in VIC, NSW, TAS, QLD, SA and WA, respectively.", "The dark and light gray regions show the 80% and 95% prediction intervals.In Figure REF , we also show the estimated mean function deviation, first state-specific functional principal component $\\widehat{\\psi }_1^s(x)$ and principal component scores $\\lbrace \\widehat{\\gamma }^s_{1,1},\\dots ,\\widehat{\\gamma }^s_{n,1}\\rbrace $ with 30-years-ahead forecasts, where $s$ denotes a state.", "The convergence in forecasts is likely to be achieved by the multilevel functional data method, because the forecasts of principal component scores for each state do not show a long-term trend, with the exception of NSW.", "From a statistical perspective, this may be because the NSW has the largest proportion of variability that can not be explained by the aggregate data.", "From a social perspective, NSW is the state that attracts the most migrants in Australia (http://www.abs.gov.au/ausstats/[email protected]/mf/3412.0).", "Figure REF shows 30-years-ahead forecasts of median log mortality rates and life expectancy from 2004 to 2033 for all states, for the independent functional data, product-ratio and multilevel functional data methods.", "We focus on these three methods in this application, because they generally outperform the Lee-Carter and Li-Lee methods as demonstrated in Section .", "For the independent functional data method, the gap in mortality and life expectancy forecasts among states diverges.", "In contrast, the product-ratio and multilevel functional data methods are quite similar, and the gaps between female and male age-specific mortality and life expectancy converge, respectively.", "Figure: Based on historical mortality rates (1950–2003), we forecast future mortality rates and life expectancy from 2004 to 2033, for the independent functional data, product-ratio, and multilevel functional data methods." ], [ "Comparisons of point and interval forecast accuracy", "Table REF displays the point and interval forecast accuracy for both age- and state-specific total mortality rates and life expectancy at each forecast horizon.", "As measured by the averaged MAFE, RMSFE, MFE and averaged mean interval score across 30 horizons, the independent functional data method performs the worst, whereas the multilevel functional data method (rwf) performs the best, for forecasting age- and state-specific total mortality and life expectancy.", "As the product-ratio and multilevel functional data methods perform similarly, it is paramount to incorporate correlation among sub-populations in forecasting, as this allows us to search for characteristics within and among series.", "Table: Point and interval forecast accuracy of mortality and life expectancy (e(0)) across different states by method and forecast horizon, as measured by the averaged MAFE, RMSFE, MFE, and averaged mean interval score.", "The minimal forecast errors are underlined for each state, whereas the minimal overall forecast error is highlighted in bold." ], [ "Application to Australian age-, sex- and state-specific mortality", "We extend the multilevel functional data method to two or more sub-populations in a hierarchy.", "This is related to hierarchical/grouped time series [31].", "A grouped structure is depicted in the two-level hierarchical diagram, presented in Figure REF .", "Figure: A two-level hierarchical tree diagram.Following a bottom-up hierarchical structure, we first extract a common trend from the total mortality within each state.", "For the $j^{\\text{th}}$ population in state $s$ , the multilevel functional data model can be written as: $f_t^{j,s}(x) = \\mu ^{j,s}(x) + R_t^{s}(x) + U_t^{j, s}(x), $ where $f_t^{j,s}(x)$ represents the female or male mortality in state $s$ at year $t$ ; $\\mu ^{j,s}(x)$ is the mean function of female or male mortality in state $s$ ; $R_t^{s}(x)$ captures the common trend across two populations for a state; and $U_t^{j, s}(x)$ captures the sex-specific residual trend for a state.", "Based on (REF ), the proportion of variability explained by the total mortality in each state is 65%, 69%, 25%, 53%, 43%, and 37% for females, and 59%, 59%, 22%, 54%, 41%, and 38% for males.", "We can also extract the common trend from the averaged mortality across all states for females and males.", "For the $j^{\\text{th}}$ population in state $s$ , the multilevel functional data model can be written as: $f_t^{j,s}(x) = \\mu ^{j,s}(x) + S_t^{j}(x)+W_t^{j,s}(x),$ where $S_t^{j}(x)$ captures the common trend across six populations; and $W_t^{j,s}(x)$ captures the state-specific residual trend.", "By combining (REF ) and (REF ), we obtain $f_t^{j,s}(x) &= \\mu ^{j,s}(x) + \\frac{R_t^{s}(x) + U_t^{j, s}(x) + S_t^{j}(x)+W_t^{j,s}(x)}{2}.$ Table: Point forecast errors (×100\\times 100) of mortality across states and sexes by method, as measured by the averaged MAFE, RMSFE, and MFE.", "The minimal forecast errors are underlined for each state and each sex, whereas the minimal overall forecast error is highlighted in bold.Table: Point forecast accuracy of life expectancy across states and sexes by method, as measured by the averaged MAFE, RMSFE, and MFE.", "The minimal forecast errors are underlined for each state and each sex, whereas the minimal overall forecast error is highlighted in bold.Table: Interval forecast accuracy of mortality and life expectancy across states and sexes by method, as measured by the averaged mean interval score.", "The minimal forecast errors are underlined for each state and each sex, whereas the minimal overall forecast error is highlighted in bold.Tables REF , REF and REF show the point and interval forecast accuracy among different functional data methods.", "As measured by the averaged MAFE, RMSFE, MFE and averaged mean interval score across 30 horizons, the multilevel functional data method (rwf) gives the smallest errors for forecasting female mortality rate and life expectancy, as well as the smallest overall errors, whereas the product-ratio method produces the most accurate forecasts for male mortality rate and life expectancy.", "Apart from the expected error loss function, we also consider the maximum point and interval forecast error criteria.", "Their results are also included in the supplement D [64]." ], [ "Conclusion", "In this paper, we adapt the multilevel functional data model to forecast age-specific mortality and life expectancy for a group of populations.", "We highlight the relationships among the adapted multilevel functional data, augmented common factor method and product-ratio method.", "As demonstrated by the empirical studies consisting of two populations, we found that the independent functional data method gives the best forecast accuracy for females, whereas the multilevel functional data and product-ratio methods produce more accurate forecasts for males.", "Based on their averaged forecast errors, the multilevel functional data method (arima) should be used in the case of two sub-populations, in particular for females.", "In the case of more than two populations, it is evident that the multilevel functional data and product-ratio methods consistently outperform the independent functional data method.", "The multilevel functional data method (rwf) gives the most accurate mortality and life expectancy forecasts for age- and state-specific total mortality.", "When we further disaggregated the age- and state-specific total mortality by sex, we found that the multilevel functional data method (rwf) should be used for forecasting female mortality and life expectancy, whereas the product-ratio method should be applied for forecasting male mortality and life expectancy.", "The superiority of the product-ratio and multilevel functional data methods over the independent functional data method is manifested by a population with large variability over age and year.", "For example, the male data generally show greater variability over age and year than do the female data; as a result the product-ratio and multilevel functional data methods perform better in terms of forecast accuracy than the independent functional data method.", "Because the product-ratio and multilevel functional data methods produce better forecast accuracy than the independent functional data method overall, this may lead to their use by government agencies and statistical bureaus involved in short-term demographic forecasting.", "For long-term forecast horizons, any time-series extrapolation methods, including the proposed one, may not be accurate as the underlying model may no longer be optimal.", "Given that different changes are at play in different phases of a mortality transition, the age components of change in the past are not necessarily informative of the longer-term future.", "By incorporating prior knowledge, the Bayesian method of [58] demonstrated the superior forecast accuracy of the long-term projection of life expectancy.", "A limitation of the current study is that the comparative analysis among the five methods focuses on errors that aggregate over all age groups for one- to 30-step-ahead mortality forecasts.", "In future research, it is possible that the analysis of the forecast errors for certain key age groups, such as those above 65, might shed light on the results of more detailed analysis.", "For a relatively long time series, geometrically decaying weights can be imposed on the computation of functional principal components [34] for achieving potentially improved forecast accuracy.", "In addition, the product-ratio and multilevel functional data methods could be applied to model and forecast other demographic components, such as age-specific immigration, migration, and population size by sex or other attributes for national and sub-national populations.", "Reconciling these forecasts across different levels of a hierarchy is worthwhile to investigate in the future [66]." ], [ "Supplement to: “Mortality and life expectancy forecasting for a group of populations in developed countries: A multilevel functional data method.\" by H. L. Shang", "This supplement contains a PDF divided into four sections.", "Supplement A: Some theoretical properties of multilevel functional principal component decomposition; Supplement B: Derivation of posterior density of principal component scores and other variance parameters; Supplement C: WinBUGS computational code used for sampling principal component scores and estimating variance parameters from full conditional densities; Supplement D: Additional results for point and interval forecast accuracy of mortality and life expectancy, based on maximum forecast error measures.", "Supplement to “Mortality and life expectancy forecasting for a group of populations in developed countries: A multilevel functional data method by H. L. Shang Supplement A: Some theoretical properties of multilevel functional principal component decomposition Let $R$ and $U^{j}$ be two stochastic processes defined on a compact set $\\mathcal {I}$ , with finite variance.", "The covariance functions of $R$ and $U^{j}$ are defined to be the function $\\mathcal {K}: \\mathcal {I}\\times \\mathcal {I}\\rightarrow R$ , such that $\\mathcal {K}^R(w,v) &= \\text{cov}\\lbrace R(w), R(v)\\rbrace = \\text{E}\\left\\lbrace [R(w)-\\mu (w)]\\otimes [R(v)-\\mu (v)]\\right\\rbrace ,\\\\\\mathcal {K}^{U^{j}}(w,v) &= \\text{cov}\\left\\lbrace U^{j}(w), U^{j}(v)\\right\\rbrace = \\text{E}\\left\\lbrace [U^{j}(w)-\\mu (w)]\\otimes [U^{j}(v)-\\mu (v)]\\right\\rbrace ,$ where $\\otimes $ represents the tensor product and $j$ represents the index of sub-populations.", "In a finite dimension, the tensor product reduces to matrix multiplication.", "Mercer's theorem [36] provides the following consistent spectrum decomposition, $\\mathcal {K}^R(w,v) &= \\text{cov}\\left\\lbrace R(w), R(v)\\right\\rbrace = \\sum ^{\\infty }_{k=1}\\lambda _k \\phi _k(w)\\phi _k(v),\\\\\\mathcal {K}^{U^{j}}(w,v) &= \\text{cov}\\left\\lbrace U^{j}(w), U^{j}(v)\\right\\rbrace = \\sum ^{\\infty }_{l=1}\\lambda ^{j}_l\\psi _l^{j}(w)\\psi _l^{j}(v),$ where $\\lambda _1\\ge \\lambda _2\\ge \\dots $ are the ordered population eigenvalues and $\\phi _k(\\cdot )$ is the $k^{\\text{th}}$ orthonormal eigenfunction of $\\mathcal {K}^{R}(\\cdot ,\\cdot )$ in the $L^2$ norm.", "Similarly, $\\lambda ^{j}_1\\ge \\lambda ^{j}_2\\ge \\dots $ are the ordered population eigenvalues and $\\psi _l^{j}(\\cdot )$ is the $l^{\\text{th}}$ orthonormal eigenfunction of $\\mathcal {K}^{U^{j}}(\\cdot ,\\cdot )$ in the $L^2$ norm.", "With Mercer's lemma, stochastic processes $R$ and $U^{j}$ can be expressed by the Karhunen-Loève expansion [39], [50].", "In practice, we reduce the dimensionality of functional data by truncating the infinite series to finite dimension, such as the first $K$ number of principal components [70], [24], [28].", "These can be expressed as: $R_t(x) &= \\sum ^{\\infty }_{k=1}\\beta _{t,k}\\phi _k(x)\\approx \\sum ^{K}_{k=1}\\beta _{t,k}\\phi _k(x),\\\\U_t^{j}(x) &= \\sum ^{\\infty }_{l=1}\\gamma _{t,l}^{j}\\psi _{l}^{j}(x)\\approx \\sum ^{L}_{l=1}\\gamma _{t,l}^{j}\\psi _{l}^{j}(x),$ where $\\beta _{t,k}=\\int _{\\mathcal {I}} R_t(x)\\phi _k(x)dx$ , $\\gamma _{t,l}^{j}=\\int _{\\mathcal {I}}U_t^{j}(x)\\psi _l^{j}(x)dx$ are the uncorrelated principal component scores with $\\text{E}(\\beta _{t,k}) = \\text{E}\\left(\\gamma _{t,l}^{j}\\right)=0$ , $\\text{Var}(\\beta _{t,k})=\\lambda _k<\\infty $ , $\\text{Var}(\\gamma _{t,l}^{j})=\\lambda _l^{j}<\\infty $ , $K$ and $L$ represent the retained numbers of principal components, and $\\mathcal {I}$ represents the domain of $x$ variable, such as $x\\in [0,95+]$ in our context.", "Appendix B: Derivation of posterior density of principal component scores We present derivations for the multilevel functional data model, including its specification and full conditional densities.", "The full conditionals are also given in [18], which provides a foundation for this work.", "Here, we extend it by adding an additional stochastic variance for the pre-smoothing step.", "This stochastic variance takes into account the varying uncertainty across observations.", "$ \\left\\lbrace \\begin{array}{ll}y_t^j(x_i) = f_t^j(x_i)+\\delta _t^j(x_i)\\epsilon _{t,i}^j \\\\f_t^j(x_i) = \\mu (x_i) + \\eta ^j(x_i) + \\sum ^K_{k=1}\\beta _{t,k}\\phi _k(x_i) + \\sum ^L_{l=1}\\gamma _{t,l}^j\\psi _l^j(x_i)+\\varepsilon _t^j(x_i) \\\\\\beta _{t,k}\\sim N\\left(0,\\lambda _k\\right); \\gamma _{t,l}^j\\sim N\\left(0,\\lambda _l^{j}\\right); \\varepsilon _t^j(x_i)\\sim N(0,(\\sigma ^2)^j); \\delta _t^j(x_i) \\sim N(0, (\\kappa _i^2)^j) \\\\\\frac{1}{(\\sigma ^2)^j} \\sim \\text{Gamma}(\\alpha _1,\\alpha _2)\\end{array} \\right.", "$ The full conditional density of inverse error variance given other parameters is $1/\\left(\\sigma ^2\\right)^j\\big |\\text{others} \\sim \\text{Gamma}\\left(\\alpha _1^{\\text{post}},\\alpha _2^{\\text{post}}\\right),$ where $\\alpha _1^{\\text{post}} &= \\alpha _1 + \\frac{1}{2}Jnp \\\\\\alpha _2^{\\text{post}} &= \\alpha _2 + \\frac{1}{2}\\sum ^J_{j=1}\\sum ^n_{t=1}\\sum ^p_{i=1}\\left[\\varepsilon _t^j(x_i)\\right]^2$ and $\\varepsilon _t^j(x_i) = f_t^j(x_i) - \\mu (x_i) - \\eta ^j(x_i) - \\sum ^K_{k=1}\\beta _{t,k}\\phi _k(x_i) - \\sum ^L_{l=1}\\gamma _{t,l}^j\\psi _l^j(x_i),$ where $J$ denotes the number of populations, $n$ denotes the sample size, and $p$ denotes the total number of age groups.", "The full conditional density of principal component scores for the common trend given other parameters is $\\beta _{t,k}\\big |\\text{others}\\sim N\\left(\\mu _{\\beta _{t,k}}^{\\text{post}}, v_{\\beta _{t,k}}^{\\text{post}}\\right)$ where $\\mu _{\\beta _{t,k}}^{\\text{post}} &= \\frac{\\lambda _k J \\sum ^p_{i=1}\\phi _k(x_i)^2}{\\lambda _k J \\sum ^p_{i=1}\\phi _k(x_i)^2 + (\\sigma ^2)^j}\\cdot \\frac{\\sum ^J_{j=1}\\sum ^p_{i=1}\\phi _k(x_i)\\left[\\varepsilon _t^j(x_i)+\\beta _{t,k}\\phi _k(x_i)\\right]}{J\\sum ^p_{i=1}\\phi _k(x_i)^2}, \\\\v_{\\beta _{t,k}}^{\\text{post}} &= \\frac{\\lambda _k (\\sigma ^2)^j}{\\lambda _k J\\cdot \\sum _{i=1}^p\\phi _k(x_i)^2+(\\sigma ^2)^j},$ where $\\lambda _k$ denotes the $k$ th eigenvalue of the common covariance function.", "The full conditional density of principal component scores for the population-specific residual trend given other parameters is $\\gamma _{t,l}^j|\\text{others} \\sim N\\left(\\mu _{\\gamma _{t,l}^j}^{\\text{post}}, v_{\\gamma _{t,l}^j}^{\\text{post}}\\right),$ where $\\mu _{\\gamma _{t,l}^j}^{\\text{post}} &= \\frac{\\lambda _l^j\\cdot \\sum ^{p}_{i=1}\\psi _l^j(t_i)^2}{\\lambda _l^j\\cdot \\sum _{i=1}^p\\psi _l^j(x_i)^2+(\\sigma ^2)^j}\\cdot \\frac{\\sum _{i=1}^p\\phi _k(x_i)\\left[\\varepsilon _{t}^j(x_i)+\\gamma _{t,l}^j\\psi _l^j(x_i)\\right]}{\\sum _{i=1}^p\\psi _l^{j}(x_i)^2}, \\\\v_{\\gamma _{t,l}^j}^{\\text{post}}&=\\frac{\\lambda _l^j(\\sigma ^2)^j}{\\lambda _l^j\\cdot \\sum _{i=1}^p\\psi _l^j(x_i)^2+(\\sigma ^2)^j},$ where $\\lambda _l^j$ denotes the $l$ th eigenvalue of the population-specific covariance function.", "Since the first step involves a nonparametric smoothing with heteroscedastic of unknown form.", "We can incorporate this nonparametric smoothing step in our Markov chain Monte Carlo (MCMC) iterations.", "For different ages or age groups, variances are unequal as shown in equation (2.4) of the main manuscript.", "Following the early work by [40], we consider a linear regression with heterscedastic errors and its Bayesian computation algorithm is documented in [40] Let $(\\omega _1,\\omega _2,\\dots ,\\omega _p)=\\left[1/\\delta ^2(x_1),1/\\delta ^2(x_2),\\dots ,1/\\delta ^2(x_p)\\right]$ be the precision parameters for different ages.", "Consider the following Gamma prior for $\\omega _i$ : $\\pi (\\omega _i) = f_{\\text{G}}(1,v_{\\omega }),\\qquad i=1,2,\\dots ,p,$ where the prior for $\\omega _i$ depends upon a hyperparameter $v_{\\omega }$ and assume that each precision $\\omega _i$ comes from the same distribution, but can differ from each other in values.", "Each of the conditional posteriors for $\\omega _i$ has the form of a Gamma density, given by $\\pi (\\omega _i|v_{\\omega }, \\text{others}) &= f_{\\text{G}}\\left(\\frac{v_{\\omega }+1}{\\sum ^n_{t=1}\\left[y_t(x_i) - f_t(x_i)\\right]^2+v_{\\omega }}, v_{\\omega }+1 \\right), \\\\\\pi (v_{\\omega }|\\omega _i,\\text{other}) &\\propto \\left(\\frac{v_{\\omega }}{2}\\right)^{p\\cdot \\frac{v_{\\omega }}{2}}\\Gamma \\left(\\frac{v_{\\omega }}{2}\\right)^{-p}e^{-\\eta \\cdot v_{\\omega }},$ where $\\eta = \\frac{1}{v_{\\omega }}+\\frac{1}{2}\\sum ^p_{i=1}\\left[\\ln \\left(\\frac{1}{\\omega _i}\\right)+\\omega _i\\right]$ , and $\\Gamma (\\cdot )$ denotes a Gamma function.", "Supplement C: WinBUGS code used for estimating variance parameters Statistical software WinBUGS is used to estimate variances in the principal component scores and error function.", "From the estimated variances, the principal component scores and error function are simulated from normal distributions with zero mean.", "Below is a modified version of WinBUGS given by [13], for modeling age- and sex-specific mortality rates.", "R model {   for (i in 1:N_subj)   { \tfor (t in 1:N_obs) \t{ \t\tW_1[i,t] ~ dnorm(m_1[i,t], taueps_1) \t\tW_2[i,t] ~ dnorm(m_2[i,t], taueps_2) \t\tm_1[i,t] <- X[i,t] + U_1[i,t] \t\tm_2[i,t] <- X[i,t] + U_2[i,t]   \t\tX[i,t] <- inprod(xi[i,], psi_1[t,]) \t\tU_1[i,t] <- inprod(zi[i,], psi_2[t,]) \t\tU_2[i,t] <- inprod(fi[i,], psi_3[t,]) \t} \tfor(k in 1:dim_space_b) \t{ \t\txi[i,k] ~ dnorm(0.0, ll_b[k]) \t} \tfor(l in 1:dim_space_w) \t{ \t\tzi[i,l]  ~ dnorm(0.0, ll_w[l]) \t} \tfor(j in 1:dim_space_f) \t{\t \t\tfi[i,j] ~ dnorm(0.0, ll_f[j]) \t}   }\t   for(k in 1:dim_space_b)   { \tll_b[k] ~ dgamma(1.0E-3, 1.0E-3) \tlambda_b[k] <- 1/ll_b[k]   }   for(l in 1:dim_space_w)   { \tll_w[l] ~ dgamma(1.0E-3, 1.0E-3) \tlambda_w[l] <- 1/ll_w[l]   }   for(j in 1:dim_space_f)   {  \tll_f[j] ~ dgamma(1.0E-3, 1.0E-3) \tlambda_f[j] <- 1/ll_f[j]   }   taueps_1 ~ dgamma(1.0E-3, 1.0E-3)   taueps_2 ~ dgamma(1.0E-3, 1.0E-3) } The definition of all variables is given below: $\\text{N}\\_\\text{subj}$ is the number of subjects (sample size) $\\text{N}\\_\\text{obs}$ is the number of observations within subjects $\\text{W}\\_1$ [i,t] and $\\text{W}\\_2$ [i,t] are the functional observations at the aggregated level and sex-specific level, for subject i at time t. Both matrices $\\text{W}\\_1$ [,] and $\\text{W}\\_2$ [,] are $\\text{N}\\_\\text{subj} \\times \\text{N}\\_\\text{obs}$ , are loaded as data and may contain missing observations $\\text{m}\\_1$ [i,t] and $\\text{m}\\_2$ [i,t] are the smoothed means of $\\text{W}\\_1$ [i,t] and $\\text{W}\\_2$ [i,t], respectively, are unknown and their joint distribution is simulated $\\text{X}$ [i,t] is the mean process at the aggregated level.", "$\\text{X}$ [,] is a $\\text{N}\\_\\text{subj}\\times \\text{N}\\_\\text{obs}$ dimensional matrix of parameters that are estimated from the model $\\text{U}\\_1$ [i,t] and $\\text{U}\\_2$ [i,t] are the sex-specific mean process at the individual level.", "$\\text{U}\\_1$ [i,t] and $\\text{U}\\_2$ [i,t] are the $\\text{N}\\_\\text{subj} \\times \\text{N}\\_\\text{obs}$ dimensional matrices of parameters that are estimated from the model $\\text{psi}\\_1$ [t,], $\\text{psi}\\_2$ [t,], $\\text{psi}\\_3$ [t,] are eigenfunctions at both the aggregated level and sex-specific level, evaluated at the time t. The matrices $\\text{psi}\\_1$ , $\\text{psi}\\_2$ , $\\text{psi}\\_3$ are $\\text{N}\\_\\text{obs}\\times \\text{K}\\_1$ , $\\text{N}\\_\\text{obs} \\times \\text{K}\\_2$ , $\\text{N}\\_\\text{obs} \\times \\text{K}\\_3$ , where $\\text{K}\\_1$ is the number of retained components that explains at least 90% of total variations in total mortality data, $\\text{K}\\_2$ and $\\text{K}\\_3$ are the number of retained components that explains at least 90% of the remaining 10% total variations in sex-specific data.", "The matrices of $\\text{psi}\\_1$ , $\\text{psi}\\_2$ , $\\text{psi}\\_3$ do not contain any missing value, and are loaded as data $\\text{xi}$ [i,k] are the scores for the subject i on the kth eigenfunction $\\text{psi}\\_1$ [t,k] $\\text{zi}$ [i,l] are the scores for the subject i on the lth eigenfunction $\\text{psi}\\_2$ [t,l] $\\text{fi}$ [i,j] are the scores for the subject i on the jth eigenfunction $\\text{psi}\\_3$ [t,j] $\\text{ll}\\_{\\text{b}}$ [k] are the precisions for the distribution of scores $\\text{xi}$ [i,k] $\\text{ll}\\_{\\text{w}}$ [l] are the precisions for the distribution of scores $\\text{zi}$ [i,l] $\\text{ll}\\_{\\text{f}}$ [j] are the precisions for the distribution of scores $\\text{fi}$ [i,j] $\\text{taueps}\\_1$ is the precision of the error process due to imperfect observations of $\\text{W}\\_1$ [i,t] around its smooth mean $\\text{m}\\_1$ [i,t].", "This is a parameter of the model that is estimated $\\text{taueps}\\_2$ is the precision of the error process due to imperfect observations of $\\text{W}\\_2$ [i,t] around its smooth mean $\\text{m}\\_2$ [i,t].", "This is a parameter of the model that is estimated All precision priors are Gamma priors with mean 1 and variance 1000 Supplement D: Additional results for point and interval forecast accuracy of mortality and life expectancy Apart from the averaged forecast error criteria, we also consider the maximum absolute forecast error (Max AFE), maximum root squared forecast error (Max RSFE), and maximum interval score for measuring the extreme errors across different ages ($x_i$ ) and years in the forecasting period (year $k$ ).", "Averaging across 16 countries, they are defined as $\\text{Max AFE}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\max _{k,i}\\left|m_k^c(x_i)-\\widehat{m}_k^c(x_i)\\right|, \\\\\\text{Max RSFE}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\sqrt{\\max _{k,i}\\left[m_k^c(x_i)-\\widehat{m}_k^c(x_i)\\right]^2}, \\\\\\text{Max interval score}(h) &= \\frac{1}{16}\\sum ^{16}_{c=1}\\max _{k,i} S_{\\alpha ,k}^{c}(x_l,x_u;x_i).$ Tables REF to REF present the Max AFE, Max RSFE, and Max interval score for comparing point and interval forecast accuracies of the age-specific mortality and life expectancy by method, in the case of two populations.", "Table: Point and interval forecast accuracy of mortality and life expectancy for females and males by method, as measured by the Max AFE, Max RSFE and Max interval score.", "For mortality, the forecast errors were multiplied by 100, in order to keep two decimal places.", "The minimal forecast errors are underlined for female and male data given in Section 5, whereas the minimal overall forecast error is highlighted in bold.Table: Point and interval forecast accuracy of mortality and life expectancy across different states (described in Section 6.1) by method, as measured by the Max AFE, Max RSFE, and maximum interval score.", "The minimal forecast errors are underlined for each state in Section 6, whereas the minimal overall forecast error is highlighted in bold.Table: Point and interval forecast accuracy of mortality (×100\\times 100) across states and sexes (described in Section 6.2) by method, as measured by the Max AFE, Max RSFE, and maximum interval score.", "The minimal forecast errors are underlined for female and male data and their average, whereas the minimal overall forecast error is highlighted in bold.Table: Point and interval forecast accuracy of life expectancy across states and sexes (described in Section 6.2) by method, as measured by the Max AFE, Max RSFE, and maximum interval score.", "The minimal forecast errors are underlined for female and male data and their average, whereas the minimal overall forecast error is highlighted in bold." ] ]
1606.05067
[ [ "Local H\\\"older continuity of the isoperimetric profile in complete\n noncompact Riemannian manifolds with bounded geometry" ], [ "Abstract For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\\left(1-\\frac{1}{n}\\right)$-H\\\"older continuous function and so in particular it is continuous.", "Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers.", "We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own interest about the approximation of finite perimeter sets with finite volume by open bounded with smooth boundary ones of the same volume.", "Finally the upper semicontinuity of the isoperimetric profile for every metric (not necessarily complete) is shown." ], [ "Introduction", "In this paper we always assume that all the Riemannian manifolds $(M, g)$ considered are smooths with smooth Riemannian metric $g$ .", "We denote by $V_g$ the canonical Riemannian measure induced on $M$ by $g$ , and by $A_g$ the $(n-1)$ -Hausdorff measure associated to the canonical Riemannian length space metric $d$ of $M$ , by $¶_g(\\Omega , U)$ the perimeter in $U\\subseteq M$ with respect to the metric $g$ of a finite perimeter set $\\Omega \\subseteq M$ , here $U$ is an open set, by $|Du|_g$ we denote the positive Radon measure represented by the total variation of the distributional gradient of a $BV$ -function $u$ having domain $M$ .", "For each $k\\in $ we denote by $\\mathbb {M}_k^n$ the $n$ -dimensional space form of constant sectional curvature equal to $k$ .", "When it is already clear from the context, explicit mention of the metric $g$ will be suppressed.", "When dealing with finite perimeter sets or locally finite perimeter sets we will denote the reduced boundary by $\\partial ^*\\Omega $ , whenever the topological boundary $\\partial \\Omega $ is smooth the reduced boundary coincides with the topological boundary $\\partial \\Omega $ .", "For this reason we will denote $¶(\\Omega ):=¶(\\Omega , M)=A(\\partial ^*\\Omega )=A(\\partial \\Omega )$ when no confusion may rise, and for every finite perimeter set $\\Omega ^{\\prime }$ we always choose a representative $\\Omega $ (i.e., that differs from $\\Omega ^{\\prime }$ by a set of Riemannian measure 0), such that $\\partial \\Omega =\\overline{\\partial ^*\\Omega }$ , where $\\partial \\Omega $ is the topological boundary of $\\Omega $ .", "At this point we give the definition of the isoperimetric profile function which is our main object of study in this paper." ], [ "The isoperimetric profile", "Typically in the literature, the isoperimetric profile function of $M$ (or briefly, the isoperimetric profile) $I_M:[0,V(M)[\\rightarrow [0,+\\infty [$ , is defined by $I_M(v):= \\inf \\lbrace A(\\partial \\Omega ): \\Omega \\in \\tau _M, V(\\Omega )=v \\rbrace ,$ where $\\tau _M$ denotes the set of relatively compact open subsets of $M$ with smooth boundary.", "However there is a more general context in which to consider this notion that will be better suited to our purposes.", "Namely, we can give a weak formulation of the preceding variational problem replacing the set $\\tau _M$ with the family $\\tilde{\\tau }_M$ of subsets of finite perimeter of $M$ .", "Let $M$ be a Riemannian manifold of dimension $n$ (possibly with infinite volume).", "We denote by $\\tilde{\\tau }_M$ the set of finite perimeter subsets of $M$ .", "The function $\\tilde{I}_M:[0,V(M)[\\rightarrow [0,+\\infty [$ defined by $\\tilde{I}_M(v):= \\inf \\lbrace \\mathcal {P}(\\Omega ): \\Omega \\in \\tilde{\\tau }_M, V(\\Omega )=v\\rbrace ,$ is called the weak isoperimetric profile function (or shortly the isoperimetric profile) of the manifold $M$ .", "If there exists a finite perimeter set $\\Omega \\in \\tilde{\\tau }_M$ satisfying $V(\\Omega )=v$ , $\\tilde{I}_M(V(\\Omega ))=A(\\partial ^*\\Omega )= \\mathcal {P}(\\Omega )$ such an $\\Omega $ will be called an isoperimetric region, and we say that $\\tilde{I}_M(v)$ is achieved.", "There are many others possible definitions of isoperimetric profile corresponding to the minimization over various differents sets of admissible domains, as stated in the following definition.", "For every $v\\in [0, +\\infty [$ , let us define $I^{*}_{M}(v):=inf\\lbrace A(\\partial _{top}\\Omega ): \\Omega \\subset M, \\partial _{top}\\Omega \\: \\text{is}\\: C^{\\infty }, V(\\Omega )=v\\rbrace ,\\\\\\tilde{I}_{M}^{*}(v):=inf\\lbrace \\mathcal {P}_{M}(\\Omega ):\\Omega \\subset M, \\Omega \\in \\tilde{\\tau }_{M} , V(\\Omega )=v, diam(\\Omega )<+\\infty \\rbrace ,$ where $diam(\\Omega ):=\\sup \\lbrace d(x,y):x,y\\in \\Omega \\rbrace $ denotes the diameter of $\\Omega $ .", "Trivially one have $I_M\\ge I^{*}_{M}\\ge \\tilde{I}_M$ and $I_M\\ge \\tilde{I}^*_{M}\\ge \\tilde{I}_M$ .", "However as we will see in Theorem $\\ref {Thm:Equivalence}$ , all of these definitions are actually equivalents, in the sense that the infimum remains unchanged, i.e., $I_M=\\tilde{I}_M$ ." ], [ "Main Results", "If $M^n$ is an arbitrary complete Riemannian manifold, then $I_M(v)=\\tilde{I}_{M}^{*}(v)=\\tilde{I}_M(v)=I^{*}_{M}(v)$ .", "The proof of this fact involves actually very natural ideas.", "In spite of this it is technical and we have found no written traces in the literature, unless Lemma 2 of [11] that deal with the case of a compact domain of $^n$ as an ambient space.", "Hence we provided ourselves a proof based on Lemma which have an independent interest, because it gives an approximation theorem of a finite perimeter set by open relatively compact sets with smooth boundary of the same volume and for this constitutes a refinement of a more classical approximation theorem of finite perimeter sets by members of $\\tau _M$ that one can find in the literature (see for example the books of [8], [1], or in the paper [7]).", "The equivalence stated in Theorem REF allows us to consider elements of $\\tau _M$ or $\\tilde{\\tau }_M$ according to what is more convenient in subsequent arguments.", "This observation is used in a crucial way when we prove Theorem $\\ref {Main}$ and Corollary REF .", "This latter could be considered as a corollary of Lemma .", "Let $M^n$ be an $n$ -dimensional Riemannian manifold (possibly incomplete, or possibly complete not necessarily with bounded geometry).", "Then $I_M$ is upper semicontinuous.", "A complete Riemannian manifold $(M, g)$ , is said to have bounded geometry if there exists a constant $k\\in \\mathbb {R}$ , such that $Ric_M\\ge k(n-1)$ (i.e., $Ric_M\\ge (n-1)kg$ in the sense of quadratic forms) and $V(B_{(M,g)}(p,1))\\ge v_0$ for some positive constant $v_0$ , where $B_{(M,g)}(p,r)$ is the geodesic ball (or equivalently the metric ball) of $M$ centered at $p$ and of radius $r> 0$ .", "[Local $\\left(1-\\frac{1}{n}\\right)$ -Hölder continuity of the isoperimetric profile] Let $M^n$ be a complete smooth Riemannian manifold with bounded geometry.", "Then there exists a positive constant $C=C(n,k)$ such that for every $v,v^{\\prime }\\in ]0, V(M)[$ satisfying $|v-v^{\\prime }|\\le \\frac{1}{C(n,k)}\\min \\left(v_0,\\left(\\frac{v}{I_M(v)+C(n,k)}\\right)^n\\right)$ , we have $\\left|I_M(v)-I_M(v^{\\prime })\\right|\\le C(n,k)\\left(\\frac{|v-v^{\\prime }|}{v_0}\\right)^{\\frac{n-1}{n}}.$ In particular $I_M$ is continuous on $[0,V(M)[$ .", "Let $f:(X,d)\\rightarrow $ and $\\alpha \\in [0,1]$ , we say that $f$ is locally $\\alpha $ -Hölder continuous on $X$, for every $z\\in X$ there exist $\\delta _z, C_z>0$ such that for every $x,y\\in X$ satisfying $|x-z|, |y-z|\\le \\delta _z$ we have $|f(x)-f(y)|\\le C_z|x-y|^{\\alpha }$ .We say that $f$ is uniformly locally $\\alpha $ -Hölder continuous on $X$, if there exist two constants $\\delta , C>0$ such that for every $x,y\\in X$ satisfying $d(x,y)\\le \\delta $ we have $|f(x)-f(y)|\\le C|x-y|^{\\alpha }$ .", "We say that $f$ is (globally) $\\alpha $ -Hölder continuous on $X$, if there exists $C>0$ such that $|f(x)-f(y)|\\le C|x-y|^{\\alpha }$ for every $x,y\\in X$ .", "We call the various constants $C_z, C$ appearing in this definition the Hölder constants of $f$.", "[Local $\\frac{n-1}{n}$ -Hölder continuity of the isoperimetric profile] Let $M^n$ be a complete smooth Riemannian manifold with bounded geometry and $v\\in ]0, V(M)[$ .", "Then there exists positive constants $\\delta =\\delta (n,k,v_0,v)>0$ , if $k\\le 0$ , $\\delta =\\delta (n,k,v_0,, v, V(M))$ , if $k>0$ , and $C=C(n,k)>0$ , such that for every $v_1,v_2\\in [v-\\delta , v+\\delta ]$ we have $\\left|I_M(v_1)-I_M(v_2)\\right|\\le C(n,k)\\left(\\frac{|v_1-v_2|}{v_0}\\right)^{\\frac{n-1}{n}}.$ Moreover, if $V(M)=+\\infty $ then $I_M$ is uniformly locally $\\frac{n-1}{n}$ -Hölder continuous on $[\\bar{v}, +\\infty [$ , for every $\\bar{v}>0$ .", "If $V(M)=+\\infty $ then $I_M$ is globally $\\frac{n-1}{n}$ -Hölder continuous on every interval $[a,b]\\subset ]0, +\\infty [$ with Hölder constant $\\bar{C}$ depending on $n,k,v_0, a,b$ .", "If $V(M)<+\\infty $ , then $I_M$ is globally $\\frac{n-1}{n}$ -Hölder continuous on $[\\bar{v},V(M)-\\bar{v}]$ , for every $\\bar{v}\\in ]0,\\frac{V(M)}{2}[$ .", "Unfortunately $\\lim _{a\\rightarrow 0^+}\\bar{C}(n,k,v, a,b)=+\\infty $ and $\\lim _{b\\rightarrow 0^+}\\bar{C}(n,k,v, a,b)=+\\infty $ .", "Observe that in the statement of the preceding Corollary the Hölder constant $C$ does not depend on $v_0$ and $v$ , but just $\\delta $ depends on them.", "At our actual knowledge, it is still an open question wether or not we can prove global $\\frac{n-1}{n}$ -Hölder continuity of $I_M$ on an arbitrary proper interval $[0,b]\\subset [0, V(M)[$ or on the entire interval $[0,V(M)[$ , or at least unifom local $\\frac{n-1}{n}$ -Hölder continuity on $[0, V(M)[$ , when we assume the manifold $M$ to be with bounded geometry and with $V(M)=+\\infty $ .", "The next fact to be observed is that it is worth to have a proof of the continuity or Hölder continuity of the isoperimetric profile, because in general the isoperimetric profile function of a complete Riemannian manifold is not continuous.", "In case of manifolds with density, in Proposition 2 of [2] is exhibited an example of a manifold with density having discontinuous isoperimetric profile.", "To exhibit a complete Riemannian manifold with a discontinuous isoperimetric profile is a more subtle and difficult task that was performed by the second author and Pierre Pansu in [14], for manifolds of dimension $n\\ge 3$ .", "In spite of these quite sophisticated counterexamples the class of manifolds admitting a continuous isoperimetric profile is vast, for an account of the existing literature on the continuity results obtained for $I_M$ , one could consult the introduction of [16] and the references therein.", "If $M$ is compact, classical compactness arguments of geometric measure theory combined with the direct method of the calculus of variations provide a short proof of the continuity of $I_{M}$ in any dimension $n$ , [2] Proposition 1.", "Finally, if $M$ is complete, non-compact, and $V(M)<+\\infty $ , an easy consequence of Theorem $2.1$ in [17] yields the possibility of extending the same compactness argument valid in the compact case and to prove the continuity of the isoperimetric profile, see for instance Corollary 2.4 of [15].", "A careful analysis of Theorem 1 of [13] about the existence of generalized isoperimetric regions, leads to the continuity of the isoperimetric profile $I_M$ in manifolds with bounded geometry satisfying some other assumptions on the geometry of the manifold at infinity, of the kind considered by the second author and A. Mondino in [10], i.e., for every sequence of points diverging to infinity, there exists a pointed smooth manifold $(M_{\\infty }, g_{\\infty }, p_{\\infty })$ such that $(M,g,p_j)\\rightarrow (M_{\\infty }, g_{\\infty }, p_{\\infty })$ in $C^0$ -topology.", "This proof is independent from that of Theorem REF .", "This is not the case for general complete infinite-volume manifolds $M$ .", "Recently Manuel Ritoré (see for instance [16]) showed that a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function has continuous and nondecreasing isoperimetric profile $\\tilde{I}_M$ .", "Particular cases of these manifolds are Cartan-Hadamard manifolds and complete noncompact manifolds with strictly positive sectional curvatures.", "In [16] as in our Theorem REF the major difficulty consists in finding a suitable way of subtracting a volume to an almost minimizing region.", "The aim of this paper is to prove Theorem REF in which we give a very short and quite elementary proof of the continuity of $I_M$ when $M$ is a complete noncompact Riemannian manifold of bounded geometry and even better we show that $I_M$ is actually a locally $C^{1-\\frac{1}{n}}(]0,V(M)[)$ function.", "The reason which allow us to achieve this goal, is that in bounded geometry it is always possible to add or subtract to a finite perimeter set a small ball that captures a fixed fraction of volume (depending only from the bounds of the geometry) centered at points close to it.", "Corollary REF ensures upper semicontinuity, so the problems appears when we try to prove lower semicontinuity.", "To prove lower semicontinuity we need some kind of compactness that is expressed here by a bounded geometry condition.", "Geometrically speaking our assumptions of bounded geometry ensures that the manifold at infinity is thick enough to permit to place a small geodesic ball $B$ close to an arbitrary domain $D$ in such a way $V(B\\cap D)$ recovers a controlled fraction of $V(D)$ and this fraction depends only on $V(D)$ and the bounds on the geometry $n, v_0, k$ , see Definition REF for the exact meaning of $n$ , $v_0$ , $k$ .", "The proof that we present here uses only metric properties of the manifolds with bounded geometry and for this reason it is still valid when suitably reformulated in the context of metric measure spaces.", "One can find similar ideas already in the metric proof of continuity of the isoperimetric profile contained in [5].", "For the full generality of the results we need that the spaces have to be doubling, satisfying a 1-Poincaré inequality and a curvature dimension condition.", "This class of metric spaces includes for example manifolds with density as well as subRiemannian manifolds.", "Following the arguments contained in [4] we can obtain another proof of the continuity of the isoperimetric profile under our assumptions of bounded geometry but with the extra assumption of the existence of isoperimetric regions of every volume, which is less general of our own proof of Theorem $\\ref {Main}$ , because in Theorem $\\ref {Main}$ we do not need to assume any kind of existence of isoperimetric regions.", "In spite of this the Heintze-Karcher type arguments used in [4] have an advantage because they permits to give a uniform bound on the length of the mean curvature vector of the generalized isoperimetric regions (i.e., left and right derivatives of $I_M$ ) with volumes inside an interval $[a, b]\\subset ]0, V(M)[$ , depending only on $a$ and $b$ .", "Finally, we mention that just with Ricci bounded below and existence of isoperimetric regions the arguments of [4] fails and we cannot prove the continuity of the isoperimetric profile, for this we need a noncollapsing condition on the volume of geodesic balls as in our definition of bounded geometry.", "It remains still an open question whether $Ricci$ bounded below and existence of isoperimetric regions for every volume implies continuity of the isoperimetric profile in presence of collapsing.", "We are not able to extend to this setting the arguments of $\\cite {BP}$ , neither to provide a counterexample, because the manifolds with discontinuous isoperimetric profile constructed in $\\cite {NardulliPansuDiscontinuous}$ have $Ricci$ curvature tending to $-\\infty $ ." ], [ "Plan of the article", " Section $\\ref {1}$ constitutes the introduction of the paper.", "We state the main results of the paper.", "In Section $\\ref {Sec:Weak}$ we prove that $\\tilde{I}_M=I_M$ .", "In section $\\ref {Sec:Continuity}$ we prove the local $C^{1-\\frac{1}{n}}$ -Hölder continuity of the isoperimetric profile in bounded geometry, i.e., Theorem $\\ref {MainLemma}$ and Corollary $\\ref {Main}$ without assuming existence of isoperimetric regions." ], [ "Acknowledgements", "The second author is indebted to Pierre Pansu for inspiring this paper and then to Pierre Pansu, Frank Morgan, Andrea Mondino, and Luigi Ambrosio for useful discussions on the topics of this article.", "The first author wish to thank the CAPES for financial support for the period in which he was a Ph.D. student at IM-UFRJ.", "Finally we want to thank a lot the anonymous referee whose comments contributed to improve both the results and the presentation of the proofs contained in this paper." ], [ "Equivalence of the weak and strong formulation", "As the example $3.53$ of [1] shows, in general we can have finite perimeter sets with positive perimeter and void interior that are not equivalent to any other set of finite perimeter with non void interior.", "So the question of putting a ball inside or outside a set of finite perimeter is a genuine technical problem.", "On the other hand, following [6] Theorem 1, it is always possible to put a small ball inside and outside an isoperimetric region.", "As a general remark a result of Federer (the reader could consult [1] Theorem $3.61$ ) states that for a given set of finite perimeter $E$ the density is either 0 or $\\frac{1}{2}$ or 1, $\\mathcal {H}^{n-1}$ -a.e.", "$x\\in M$ , moreover points of density 1 always exist $V$ -a.e.", "inside $D$ , because of the Lebesgue's points Theorem applied to the characteristic function of any $V$ -measurable set of $M$ .", "About this topic the reader could consult the book [8] Example $5.17$ .", "Thus $V(D)>0$ ensures the existence of at least one point $p$ belonging to $D$ of density 1, which is enough for the aims of our proofs.", "In view of these facts to prove Theorem $\\ref {Thm:Equivalence}$ we need to make a construction which replace a finite perimeter set by one of the same volume with a small ball inside and one outside, by adding a small geodesic ball (with smooth boundary) to a point of density 0 and subtracting a small geodesic ball to a point of density 1 taking care of not altering the volume.", "This enables us to obtain again a finite perimeter set of the same volume with a perimeter that is a small perturbation of the original one and that in addition have the property that we can put inside and outside a small ball.", "This construction legitimate us to apply mutatis mutandis the arguments of the proof of Lemma 1 of [11] to get the isovolumic approximation Lemma and then to conclude the proof of Theorem $\\ref {Thm:Equivalence}$ .", "Our adapted version of Lemma 1 of [11] is the following lemma.", "Let $\\Omega _1\\in \\tilde{\\tau }_M$ with $V(\\Omega _1)<+\\infty $ , such that there exists two geodesic balls satisfying $B(x_1, r_1)\\subset \\Omega _1$ and $B(x_2,r_2)\\cap \\Omega _1=\\emptyset $ , with $0<r_1<inj_M(x_1)$ and $0<r_2<inj_M(x_2)$ .", "We set $v^*:=\\min \\left\\lbrace V(B(x_1, \\frac{r_1}{2})), V(B(x_2, \\frac{r_2}{2}))\\right\\rbrace $ .", "For any $v\\in [0,v^*]$ we denote by $R_{i,v}$ a radius such that $V(B(x_i,R_{i,v}))=v$ and by $S(x,r)$ the sphere of radius $r$ and center $x$ .", "Let us define $f_{\\Omega _1}(v):=\\max \\left\\lbrace \\sup _{0\\le t\\le R_{1,v}}A(S(x_1,t)), \\sup _{0\\le t\\le R_{2,v}}A(S(x_2,t))\\right\\rbrace .$ Then for any $\\varepsilon >0$ and any $v\\in ]V(\\Omega _1)-v^*, V(\\Omega _1)+v^*[$ , there exists $\\Omega _2\\in \\tau _M$ such that $V(\\Omega _2)=v$ and $¶(\\Omega _2)\\le ¶(\\Omega _1)+f_{\\Omega _1}(|v-V(\\Omega _1)|)+\\frac{\\varepsilon }{4}.$ We observe that if $M$ is noncompact and $\\Omega $ bounded, then we always have $Interior(\\Omega ^c)\\ne \\emptyset $ .", "[of Lemma ] By the proof of the claim p. 105 of [7], there exists a sequence of $BV$ -functions $(u_l)$ on $M$ such that $\\lim _l||u_l-\\chi _{\\Omega _1}||_1=0$ , $|Du_l|(M)=¶(\\Omega _1)$ and each $u_l$ has compact support $K_l$ .", "Note that we can assume that $B(x_1, r_1)\\subset K_l$ .", "Moreover, construction the $u_l$ satisfy $0\\le u_l\\le \\chi _{\\Omega _1}$ , which gives $K_l\\subset \\Omega _1$ .", "Considering a smooth positive kernel $\\rho $ with compact support the mollified functions $u_{j,l}=u_l*\\rho _{\\frac{1}{j}}$ satisfy $0\\le u_{j,l}\\le 1$ , $\\lim _{j\\rightarrow +\\infty }||u_{j,l}-u_l||_1=0$ , $\\lim _{l}|Du_{j,l}|(M)=|Du_l|(M)$ and for $j$ large enough the support $K_{j,l}$ of $u_{j,l}$ satisfies $B(x_1,\\frac{r}{2})\\cap K_{j,l}=\\emptyset $ .", "As explained in [7] to perform a convolution on a manifold one have just to use a partition of unity associated to finite sets of local charts covering the compact support of $u_l$ and then mollify in each local chart.", "By a diagonal argument we extract a subsequence $v_l=u_{j,l}$ , satisfying $0\\le v_l\\le 1$ , $\\lim _{l}||v_l-\\chi _{\\Omega _1}||_1=0$ , $\\lim _l|Dv_l|(M)=¶(\\Omega _1)$ , and for $l$ large enough the support $C_l$ of $v_l$ satisfies $B(x_1,\\frac{r_1}{2})\\subset C_l$ and $B(x_2,\\frac{r_2}{2})\\cap C_l=\\emptyset $ .", "Putting $F_t^l:=\\lbrace x\\in M:v_l(x)>t\\rbrace $ and using the Fleming-Rishel Theorem (compare Theorem $4.3$ of [3]) we have $¶(\\Omega _1)=\\lim _l|Dv_l|=\\lim _l\\int _0^1¶(F_t^l)dt\\ge \\int _0^1\\liminf _l¶(F_t^l)dt.$ An application of Sard's Theorem ensures that the sets $F_t^l$ are smooth for almost every $t\\in ]0,1[$ .", "Thus for every $l$ we can choose a $t\\in ]0,1[$ (depending on $l$ ), such that $\\liminf _l¶(F_t^l)\\le ¶(\\Omega _1)$ .", "Moreover, we have $|V(F_t^l)-V(\\Omega _1)|\\le V(F_t^l\\setminus \\Omega _1)+V(\\Omega _1\\setminus F_t^l)$ and $V(F_t^l\\setminus \\Omega _1)\\le \\frac{1}{t}||v_l-\\chi _{\\Omega _1}||_1,$ $V(\\Omega _1\\setminus F_t^l)\\le \\frac{1}{1-t}||v_l-\\chi _{\\Omega _1}||_1.$ Since we have $|v-V(\\Omega _1)|<v_0$ , we can choose $l$ large enough to get $|v-V(\\Omega _1)|+\\frac{||v_l-\\chi _{\\Omega _1}||_1}{t(1-t)}<v^*,$ which yields for $l$ large enough $|V(F_t^l)-v|<v^*$ .", "Hence by subtracting $B(x_1, R_{1,V(F_t^l)-v})$ or adding $B(x_2, R_{2,v-V(F_t^l)})$ to $F_t^l$ , we obtain a bounded open set with smooth topological boundary and volume $v$ and perimeter equal to $¶(F_t^l)+A(S(x_{i,l}, R_{i,l})\\le ¶(F_t^l)+f_{\\Omega _1}(|v-V(F_t^l)|),$ where $R_{i,l}:=R_{2,v-V(F_t^l)}$ if $V(F_t^l)<v$ and $R_{i,l}:=R_{1,V(F_t^l)-v}$ , if $V(F_t^l)<v$ and $R_{i,l}=0$ if $V(F_t^l)=v$ otherwise.", "We finally get $\\Omega _2$ for any $l$ large enough and we conclude the proof.", "We can state now the next lemma which permits to approximate an arbitrary finite perimeter set with another one having the same volume and two holes (balls), one inside and the other outside it.", "Before stating the next lemma just let us mention that for a set $X$ inside a topological space we denote by $Interior(X)=\\mathring{X}$ the set of its interior points.", "Let $M$ be a Riemannian manifold and $\\Omega \\in \\tilde{\\tau }_M$ be a set of finite perimeter with finite volume $V(\\Omega )\\in ]0, V(M)[$ .", "For any $\\varepsilon >0$ , there exists a set of finite perimeter $\\tilde{\\Omega }\\subseteq M$ and two geodesic balls $B(x_1, r_1)$ , and $B(x_2, r_2)$ such that $V(\\Omega )=V(\\tilde{\\Omega })$ , $B(x_1, r_1)\\subset \\Omega _1$ , $B(x_2, r_2)\\cap \\tilde{\\Omega }=\\emptyset $ , and $¶(\\tilde{\\Omega })\\le ¶(\\Omega )+\\frac{\\varepsilon }{4}.$ Consider an arbitrary set $\\Omega \\in \\tilde{\\tau }_M$ and take two distinct points $x_1\\in \\Omega $ and $x_2\\in \\Omega ^c$ of density $\\Theta (x_1, V\\Omega )=1$ and $\\Theta (x_2, V\\Omega )=0$ , where $\\Theta (p, V\\Omega ):=\\lim _{r\\rightarrow 0^+}\\frac{V(\\Omega \\cap B(p, r))}{\\omega _nr^n}$ , for every $p\\in M$ .", "By $\\omega _n$ we denote the volume of the ball of radius 1 in $^n$ .", "Consider the two continuous functions $f_1, f_2:I\\rightarrow $ , where $I:=[0, r_0[$ such that $f_1(r):=V(\\Omega \\cap B_M(x_1, r))$ , $f_2(r):=V(\\Omega ^c\\cap B_M(x_2, r))$ .", "The radius $r_0$ could be chosen small enough to have $B_M(x_1, r_1)\\cap B_M(x_2, r_2)=\\emptyset $ for every $r_1, r_2\\in I$ and such that there exist $r_1, r_2\\in I$ satisfying the property $f_1(r_1)=f_2(r_2)$ and $\\partial B_M(x_1, r_1),\\partial B_M(x_2, r_2)$ smooths (for this last property it is enough to take $r_0$ less than the injectivity radius at $x_1$ and $x_2$ ).", "Then we set $\\tilde{\\Omega }:=[\\Omega \\setminus B_M(x_1,r_1)]\\mathring{\\cup }[\\Omega ^c\\cap B_M(x_2, r_2)]=[\\Omega \\setminus B_M(x_1,r_1)]\\cup B_M(x_2, r_2).$ As it is easy to see $V(\\tilde{\\Omega })=V(\\Omega )$ , $|¶(\\tilde{\\Omega })-¶(\\Omega )|\\le \\sum _{i=1}^2[A(\\partial B_M(x_i, r_i))+¶(\\Omega , B_M(x_i, r_i))],$ $V(\\Omega \\Delta \\tilde{\\Omega })=f_1(r_1)+f_2(r_2),$ $\\mathring{\\tilde{\\Omega }}\\ne \\emptyset $ , and $Interior(\\tilde{\\Omega }^c)\\ne \\emptyset $ .", "It is straightforward to verify that the right hand sides of (REF ) and (REF ) converge to zero when the radii $r_1$ and $r_2$ go to zero and the theorem easily follows.", "As an easy consequence of Lemmas and we have the following isovolumic approximation lemma.", "Let $\\Omega \\in \\tilde{\\tau }_M$ be a finite perimeter set with $V(\\Omega )<+\\infty $ , $V(\\Omega ), V(\\Omega ^c)>0$ , where $\\Omega ^c:=M\\setminus \\Omega $ .", "Then there exists a sequence $\\Omega _{k}\\in \\tau _{M}$ such that $V(\\Omega _k)=V(\\Omega )$ and $\\Omega _k$ converges to $\\Omega $ in the sense of finite perimeter sets.", "[of Lemma ] Let us assume that $\\Omega \\in \\tilde{\\tau }_M$ is bounded, then for any arbitrary $\\varepsilon >0$ , the Lemma $\\ref {Lemma:1ModicaReferee}$ applied to the finite perimeter set $\\tilde{\\Omega }$ given by Lemma applied to $\\Omega $ , permits to find $\\tilde{\\Omega }_{\\varepsilon }\\in \\tau _M$ such that $V(\\tilde{\\Omega }_{\\varepsilon })=V(\\tilde{\\Omega })=V(\\Omega )$ and $V(\\tilde{\\Omega }_{\\varepsilon }\\Delta \\tilde{\\Omega })\\le \\frac{\\varepsilon }{2},$ $|¶(\\tilde{\\Omega }_{\\varepsilon })-¶(\\tilde{\\Omega })|\\le \\frac{\\varepsilon }{2}.$ These last two inequalities combined with (REF ) and (REF ) imply that $V(\\tilde{\\Omega }_{\\varepsilon }\\Delta \\Omega )\\le \\varepsilon ,$ $|¶(\\tilde{\\Omega }_{\\varepsilon })-¶(\\Omega )|\\le \\varepsilon .$ Now we are ready to prove Theorem $\\ref {Thm:Equivalence}$ .", "[of Theorem $\\ref {Thm:Equivalence}$ ] Taking into account Remark $\\ref {Rem:Trivialinequality}$ , it is easy to check that to prove the theorem, it is enough to show the nontrivial inequality $I_M(v)\\le \\tilde{I}_M(v)$ for every $v\\in [0, V(M)[$ .", "To this aim, let us consider $\\varepsilon >0$ and $\\Omega \\in \\tilde{\\tau }_M$ , with $V(\\Omega )=v$ .", "By Lemma $\\ref {Lemma:Smoothisovolumic}$ there is a sequence $\\Omega _k\\in \\tau _M$ such that $V(\\Omega _k)=v$ , and $(\\Omega _k)$ converges to $\\Omega $ in the sense of finite perimeter sets.", "In particular we have that $\\lim _{k\\rightarrow +\\infty }¶(\\Omega _k)=¶(\\Omega )$ .", "On the other hand by definition we have that $I_M(v)\\le ¶(\\Omega _k)$ for every $k\\in \\mathbb {N}$ .", "Passing to limits leads to have $I_M(v)\\le ¶(\\Omega ),$ for every $\\Omega \\in \\tilde{\\tau }_M$ with $V(\\Omega )=v$ .", "Taking the infimum in (REF ) when $\\Omega $ runs over $\\tilde{\\tau }_M$ keeping $V(\\Omega )$ fixed and equal to $v$ , we infer that $I_M(v)\\le \\tilde{I}_M(v)$ .", "This completes the proof.", "In the remaining part of this section we prove Corollary $\\ref {CorRes:Uppersemicontinuity}$ .", "[of Corollary $\\ref {CorRes:Uppersemicontinuity}$ ] In view of Theorem $\\ref {Thm:Equivalence}$ we actually prove that $\\tilde{I}_M$ is upper semicontinuous.", "For any $v\\in ]0, V(M)[$ and any $\\varepsilon >0$ , consider a finite perimeter set $\\Omega $ such that $V(\\Omega )=v$ and $¶(\\Omega )\\le \\frac{\\varepsilon }{4}$ .", "We then apply Lemma to it, which gives us $\\Omega _1$ such that $V(\\Omega _1)=v$ , $¶(\\Omega _1)\\le \\tilde{I}_M(v)+\\frac{\\varepsilon }{2}$ , and a $\\bar{v}=\\bar{v}_{\\Omega _{1,\\varepsilon }}$ such that for any $w\\in ]v-\\bar{v}, v+\\bar{v}[$ there exists $\\Omega _2\\in \\tilde{\\tau }_M$ satisfying $V(\\Omega _2)=w$ and $¶(\\Omega _2)\\le I_M(v)+f(|w-v|)+\\frac{3\\varepsilon }{4}$ , where $f$ is given by (REF ).", "By the very definition of isoperimetric profile we have immediately that $I_M(w)\\le I_M(v)+f(|w-v|)+\\frac{3\\varepsilon }{4}.$ Now, the function $f$ depends only on $\\Omega _1$ , satisfies $f(0)=0$ and is continuous at 0.", "So there exists $v_1\\in ]0, \\bar{v}[$ such that $f(|w-v|)\\le \\frac{\\varepsilon }{4}$ for every $w\\in ]v-v_1, v+v_1[$ , which gives the upper semicontinuity in $v$ .", "By the arbitrariness of $v$ the corollary readily follows." ], [ "Local Hölder continuity of $I_M$ in bounded geometry", "For the needs of the proof of Theorem $\\ref {MainLemma}$ we restate here a version of Lemma $2.5$ of [13] that we will use in the sequel.", "[Lemma $2.5$ of [13]] There is a constant $c=c(n,k)$ , with $0<c<1$ such that for any Riemannian manifold $M^n$ with bounded geometry, any radius $0<r\\le 1$ , any set $D\\in \\tilde{\\tau }_M$ with $V(D)<+\\infty $ , there is a point $p\\in M$ such that $V(B(p,r)\\cap D)\\ge c\\min \\lbrace v_0r^n, \\left(\\frac{V(D)}{¶(D)}\\right)^n\\rbrace .$ The proof of the preceding Lemma is essentially the same as in Lemma $2.5$ of [13].", "Now we can start the proof of Theorem $\\ref {MainLemma}$ .", "[of Theorem $\\ref {MainLemma}$ ] As a preliminary remark we observe that it is enough to prove the theorem thinking to the definition of $\\tilde{I}_M$ when it is more useful for our reasoning.", "Let $\\varepsilon \\in ]0,1]$ .", "By Theorem $\\ref {Thm:Equivalence}$ we can get $\\Omega \\in \\tau _M$ with $V(\\Omega )=w$ and $¶(\\Omega )\\le I_M(w)+\\varepsilon $ .", "When $M$ is not compact, there exists a ball $B(x_2, 1)$ not intersecting $\\Omega $ (that could be chosen compact).", "Then for every $v^{\\prime }\\in ]w,w+v_0[$ there exists $r_{v^{\\prime }}\\le 1$ such that $\\Omega _1=\\Omega \\mathring{\\cup } B_M(x_2, r_{v^{\\prime }})$ satisfies $V(\\Omega _1)=v^{\\prime }$ and $I_M(v^{\\prime })\\le ¶(\\Omega _1)\\le ¶(\\Omega )+¶(B_M(x_2, r_{v^{\\prime }}))\\le I_M(w)+\\varepsilon +C(n,k)r_{v^{\\prime }},$ where the last inequality comes from the spherical Bishop-Gromov's theorem (which asserts that when $Ric_g\\ge (n-1)kg$ the area of spheres are less than the area of corresponding spheres in space form of constant curvature $k$ ) and from the value of the area of the spheres in constant curvature.", "Since by Bishop-Gromov's Theorem we have $\\frac{v_0r_{v^{\\prime }}^n}{C_1(n,k)}\\le V(B(x_2, r_{v^{\\prime }})=v^{\\prime }-v$ , Inequality (REF ) gives us $I_M(v^{\\prime })\\le I_M(v) +\\varepsilon +C_2(n,k)\\left(\\frac{v^{\\prime }-v}{v_0}\\right)^{\\frac{n-1}{n}}.$ The case $v^{\\prime }\\le v$ needs more work.", "Let us apply Lemma to $\\Omega $ , we get for any $v^{\\prime }\\in ]v-v_1, v[$ , where $v_1=c\\min \\left\\lbrace v_0,\\left(\\frac{v}{I_M(v)+\\varepsilon }\\right)^n\\right\\rbrace $ , then we have $V\\left(\\Omega \\cap B\\left(p, \\left(\\frac{v-v^{\\prime }}{cv_0}\\right)^{\\frac{1}{n}}\\right)\\right)\\ge \\min \\left\\lbrace v-v^{\\prime }, c\\left(\\frac{v}{I_M(v)+\\varepsilon }\\right)^n\\right\\rbrace =v-v^{\\prime },$ and so there exists a $r_{v^{\\prime }}\\le \\left(\\frac{v-v^{\\prime }}{cv_0}\\right)^{\\frac{1}{n}}$ such that $\\Omega _2:=\\Omega \\setminus B(p,r_{v^{\\prime }})$ has volume $v^{\\prime }$ and so, by the spherical Bishop-Gromov's Theorem, we get $I_M(v^{\\prime })\\le ¶(\\Omega _2)\\le ¶(\\Omega )+¶(B_M(p,r_{v^{\\prime }}))\\le I_M(v)+\\varepsilon +C_2(n,k)\\left(\\frac{v^{\\prime }-v}{v_0}\\right)^{\\frac{n-1}{n}}.$ Now, we can let $\\varepsilon $ tends to 0 in $(\\ref {Eq:ProofMain0})$ and $(\\ref {Eq:ProofMain1})$ .", "If we have $v^{\\prime }\\le v$ , then we get the result combining $(\\ref {Eq:ProofMain1})$ and $(\\ref {Eq:ProofMain0})$ where we exchange $v$ and $v^{\\prime }$ .", "If $v\\le v$ , we first control $I_M(v^{\\prime })$ by $I_M(v)$ using $(\\ref {Eq:ProofMain0})$ and then apply $(\\ref {Eq:ProofMain1})$ with $v$ and $v^{\\prime }$ exchanged.", "Combined with $(\\ref {Eq:ProofMain0})$ we conclude the proof in the case $V(M)=+\\infty $ .", "If $V(M)<+\\infty $ we can just take as $\\Omega $ an isoperimetric region of volume $v$ (which exists always), then apply the arguments leading to (REF ) to $M\\setminus \\Omega $ and consider as a competitor the finite perimeter set $\\Omega ^{\\prime }:=\\Omega \\cup B_M(p,r_{v^{\\prime }})$ , then it is straightforward to adapt the preceding arguments to conclude the proof.", "At this point, we are ready to prove Corollary $\\ref {Main}$ .", "Lemma $3.5$ [9] states that whenever $(M,g)$ have $Ric_g\\ge (n-1)k$ then the perimeter of a geodesic ball in $M$ enclosing volume $v$ , have no more perimeter than a geodesic ball in $\\mathbb {M}_k^n$ enclosing the same volume, this is used to prove Proposition $3.2$ of [10] which states that if $(M^n,g)$ is a complete Riemannian manifold with $Ric_g\\ge (n-1)k$ , then $I_M\\le I_{\\mathbb {M}_k^n}$ .", "But we know a lot about $I_{\\mathbb {M}_k^n}$ , for example that it is a continuous strictly increasing function and that for every $w>0$ , $I_{\\mathbb {M}_k^n}(w)$ is achieved by a geodesic ball enclosing volume $w$ and we will use these informations several times in the sequel.", "For each $v\\in ]0,V(M)[$ it is a trivial matter to determine $\\eta _v>0$ such that $[v-\\eta _v, v+\\eta _v]\\subset ]0, V(M)[$ (for example to put $\\eta _v=\\min (\\frac{v}{2},\\frac{V(M)-v}{2})$ it is sufficient for our purposes).", "Put $\\delta :=\\frac{1}{2}\\min \\left\\lbrace \\eta _v, \\frac{1}{C(n,k)}\\min \\left(v_0,\\left(\\frac{v-\\eta _v}{I_{\\mathbb {M}^n_k}(v+\\eta _v)+C(n,k)}\\right)^n\\right)\\right\\rbrace .$ It is easy to check that $\\delta =\\delta (n,k,v_0, V(M),v)$ .", "Using Theorem $\\ref {MainLemma}$ we obtain the validity of $(\\ref {Eq:ResMainStatement})$ for every $v_1,v_2\\in ]v-\\delta ,v+\\delta [$ .", "To show the local uniform $\\frac{n-1}{n}$ -Hölder continuity away from zero we set $\\delta ^{\\prime }:=\\inf _{v\\in [\\bar{v},V(M)[}\\lbrace \\delta (n,k,v_0,v)\\rbrace =\\delta ^{\\prime }(n,k,v_0)$ .", "It is easy to see that $\\delta ^{\\prime }>0$ because $v\\mapsto \\delta (n,k,v_0,v)$ is a continuous function of $v$ .", "Readily follows that $(\\ref {Eq:ResMainStatement})$ holds for every $v_1,v_2\\in [\\bar{v}, V(M)[$ satisfying $|v_1-v_2|\\le \\delta ^{\\prime }$ .", "Furthermore, if we assume that $V(M)<+\\infty $ we can divide the interval $[\\bar{v}, V(M)-\\bar{v}]$ in a finite number of interval whose length is less that $\\delta ^{\\prime }$ .", "Then it is straightforward to prove that for all $v_1,v_2\\in [\\bar{v}, V(M)-\\bar{v}]$ we have $|I_M(v_1)-I_M(v_2)|\\le \\left(\\left[\\frac{V(M)-2\\bar{v}}{\\delta ^{\\prime }}\\right]+1\\right)C(n,k)\\left(\\frac{|v_1-v_2|}{v_0}\\right)^{\\frac{n-1}{n}}.$ To finish the proof it is enough to remark that for every $v\\in [a,b]$ it holds $\\delta _v>\\delta (n,k,v_0, a, b)=\\frac{1}{C(n,k)}\\min \\left(v_0, \\left(\\frac{a}{I_{\\mathbb {M}^n_k}(b)+C(n,k)}\\right)^{n}\\right)>0,$ which ensures that $I_M$ is uniformly locally continuous on $[a,b]$ .", "With this in mind it is a standard task to conclude the global $\\frac{n-1}{n}$ -Hölder continuity of $I_M$ and to complete the proof.", "tocsectionReferences Abraham Muñoz Flores Departamento de Matemática Instituto de Matemática UFRJ-Universidade Federal do Rio de Janeiro, Brasil email: [email protected] and Departamento de Geometria e Representação Gráfica Instituto de Matemática e Estatística UERJ-Universidade Estadual do Rio de Janeiro email: [email protected] Stefano Nardulli Departamento de Matemática Instituto de Matemática UFRJ-Universidade Federal do Rio de Janeiro, Brasil email: [email protected]" ] ]
1606.05020
[ [ "Ginzburg - Landau Expansion in BCS - BEC Crossover Region of Disordered\n Attractive Hubbard Model" ], [ "Abstract We have studied disorder effects on the coefficients of Ginzburg - Landau (GL) expansion for attractive Hubbard model within the generalized DMFT+Sigma approximation for the wide region of the values of attractive potential U - from the weak-coupling limit, where superconductivity is described by BCS model, towards the strong coupling, where superconducting transition is related to Bose - Einstein condensation (BEC) of compact Cooper pairs.", "For the case of semi-elliptic initial density of states disorder influence on the coefficients A and B before the square and the fourth power of the order parameter is universal for at all values of electronic correlations and is related only to the widening of the initial conduction band (density of states) by disorder.", "Similar universal behavior is valid for superconducting critical temperature T_c (the generalized Anderson theorem) and specific heat discontinuity at the transition.", "This universality is absent for the coefficient C before the gradient term, which in accordance with the standard theory of \"dirty\" superconductors is strongly suppressed by disorder in the weak-coupling region, but can slightly grow in BCS - BEC crossover region, becoming almost independent of disorder in the strong coupling region.", "This leads to rather weak disorder dependence of the penetration depth and coherence length, as well as the slope of the upper critical magnetic field at T_c, in BCS - BEC crossover and strong coupling regions." ], [ "Introduction", "Ilya Mikhailovich Lifshitz was one of the creators of the modern theory of disordered systems [1].", "Among his numerous contributions in this field we only mention the general formulation of the concept of self – averaging [2] and the method of optimal fluctuation for the description of “Lifshitz tails” in the electron density of states [3].", "These ideas and approaches are widely used now in many fields of the theory of disordered systems, even those which initially were outside the scope of his personal scientific interests.", "The studies of disorder effects in superconductors have a rather long history.", "The pioneer works by Abrikosov and Gor'kov [5], [4], [6] and Anderson [7] had been devoted to the limit of weakly disordered metal ($p_Fl\\gg 1$ , where $p_F$ is Fermi momentum and $l$ is the mean free path) and weakly coupled superconductors, well described by BCS theory [8].", "The notorious Anderson theorem [7], [8] on $T_c$ of superconductors with “normal” (spin independent) disorder was proved in this limit under the assumption of self – averaging superconducting order – parameter [8], [9], [10].", "The generalizations for the case of strong enough disorder ($p_Fl\\sim 1$ ) were also mainly done under the same assumption, though it can be explicitly shown, that self – averaging of the order parameter is violated close to Anderson metal – insulator transition [9], [10].", "Here, the ideas originating from Ref.", "[3] are of primary importance [11].", "The problem of superconductivity in disordered systems in the limit of strongly coupled Cooper pairs, including the region of BCS – BEC crossover, was not well studied until recently.", "In fact, the problem of superconductivity in the case of strong enough pairing interactions was considered for a long enough time [12].", "Significant progress here was achieved by Nozieres and Schmitt-Rink [13], who proposed an effective method to study the crossover from BCS behavior in the weak coupling region towards Bose – Einstein condensation of Cooper pairs in the strong coupling region.", "One of the simplest models, where we can study the BCS – BEC crossover, is Hubbard model with attractive interaction.", "The most successful theoretical approach to describe strong electronic correlations in Hubbard model (both repulsive and attractive) is the dynamical mean field theory (DMFT) [14], [15], [16].", "The attractive Hubbard model was already studied within this approach in a number of papers [17], [18], [19], [20], [21].", "However, there are only few papers, where disorder effects in BCS – BEC crossover region were taken into account.", "In recent years we have developed the generalized DMFT+$\\Sigma $ approach to Hubbard model [22], [23], [24], [25], which is very convenient for the studies of different “external” (with respect to DMFT) interactions, such as pseudogap fluctuations [22], [23], [24], disorder scattering [26], [27] and electron – phonon interaction [28]).", "This approach is also well suited to the analysis of two – particle properties, such as dynamic (optical) conductivity [26], [29].", "In Ref.", "[21] we have used this approach to analyze the single – particle properties of the normal (non – superconducting) phase and optical conductivity of the attractive Hubbard model.", "Further on, DMFT+$\\Sigma $ approach was used to study disorder influence on superconducting transition temperature, which was calculated within Nozieres – Schmitt-Rink approach [30], [31].", "The general review of DMFT+$\\Sigma $ approach was given in Ref.", "[25], and the review of this approach to disordered Hubbard model (both repulsive and attractive) was recently presented in Ref.", "[32].", "In this paper we investigate Ginzburg – Landau (GL) expansion for disordered attractive Hubbard model including the BCS – BEC crossover region and the limit of strong coupling.", "Coefficients of GL – expansion in BCS – BEC crossover region were studied in a number of papers [33], [34], [35], but there were no previous studies of disorder effects, except our recent paper [36], where we have considered only the case of homogeneous GL – expansion and demonstrated certain universal behavior of GL – coefficients on disorder (reflecting the generalized Anderson theorem).", "Below we mainly concentrate on the study of the GL – coefficient before the gradient term, where such universal behavior is just absent.", "Here we limit ourselves to the case of weak enough disorder ($p_Fl\\gg 1$ ), neglecting the effects of Anderson localization, which can significantly change the behavior of this coefficient in the limit of strong disorder [9], [10]." ], [ "Hubbard model within DMFT+$\\Sigma $ approach", "We shall consider the disordered paramagnetic Hubbard model with attractive interaction.", "The Hamiltonian is written as: $H=-t\\sum _{\\langle ij\\rangle \\sigma }a_{i\\sigma }^{\\dagger }a_{j\\sigma }+\\sum _{i\\sigma }\\epsilon _{i}n_{i\\sigma }-U\\sum _{i}n_{i\\uparrow }n_{i\\downarrow },$ where $t>0$ is the transfer integral between the nearest neighbors on the lattice, $U$ is the Hubbard – like on site attraction, $n_{i\\sigma }=a_{i\\sigma }^{\\dagger }a_{i\\sigma }^{{\\phantom{\\dagger }}}$ is electron number operator at site $i$ , $a_{i\\sigma }$ ($a_{i\\sigma }^{\\dagger }$ ) is electron annihilation (creation) operator at $i$ -th site and spin $\\sigma $ .", "Local energy levels $\\epsilon _{i}$ are assumed to be independent and random at different sites.", "To use the standard “impurity” diagram technique we assume the Gaussian statistics for energy levels $\\epsilon _{i}$ : $\\mathcal {P}(\\epsilon _{i})=\\frac{1}{\\sqrt{2\\pi }\\Delta }\\exp \\left(-\\frac{\\epsilon _{i}^2}{2\\Delta ^2}\\right)$ Parameter $\\Delta $ here is the measure of disorder strength, while the Gaussian random field of energy levels introduces the “impurity” scattering, which is considered using the standard approach, using the averaged Green's functions [37].", "The generalized DMFT+$\\Sigma $ approach [22], [23], [24], [25] adds to the standard DMFT [14], [15], [16] an additional “external” electron self – energy $\\Sigma _{\\bf p}(\\varepsilon )$ (in general case momentum dependent), which is produced by additional interactions outside the DMFT, which gives an effective procedure to calculate both single – particle and two – particle properties [26], [29], [32].", "The success of this approach is related to choice of the single – particle Green's function in the following form: $G(\\varepsilon ,{\\bf p})=\\frac{1}{\\varepsilon +\\mu -\\varepsilon ({\\bf p})-\\Sigma (\\varepsilon )-\\Sigma _{\\bf p}(\\varepsilon )},$ where $\\varepsilon ({\\bf p})$ – is the “bare” electron dispersion, while the full self – energy is the additive sum the local self – energy $\\Sigma (\\varepsilon )$ , determined from DMFT, and “external” $\\Sigma _{\\bf p}(\\varepsilon )$ .", "Thus we neglect all the interference processes between of Hubbard and “external” interactions.", "This allows us to conserve the general structure of self – consistent equations of the standard DMFT [14], [15], [16].", "At the same time, at each step of DMFT iterations the “external” self – energy $\\Sigma _{\\bf p}(\\varepsilon )$ is recalculated using some approximate calculation scheme, corresponding to the form of additional interaction, while the local Green's function is dressed by $\\Sigma _{\\bf p}(\\varepsilon )$ at each step of DMFT procedure.", "Here, in the impure Hubbard model, the “external” self – energy entering DMFT+$\\Sigma $ is taken in the simplest form (self – consistent Born approximation), which neglects all diagrams with intersecting lines of impurity scattering, so that: $\\Sigma _{\\bf p}(\\varepsilon )\\rightarrow \\tilde{\\Sigma }(\\varepsilon )=\\Delta ^2\\sum _{\\bf p}G(\\varepsilon ,{\\bf p}),$ where $G(\\varepsilon ,{\\bf p})$ is the single – electron Green's function (REF ) and $\\Delta $ is the amplitude of site disorder.", "To solve the effective Anderson impurity model of DMFT throughout this paper we used the numerical renormalization group (NRG) algorithm [38].", "All calculations below were done for the case of the quarter – filled band (n=0.5 electrons per lattice site).", "Further on we shall consider the model of the “bare” conduction band with semi – elliptic density of states (per unit cell and single spin projection): $N_0(\\varepsilon )=\\frac{2}{\\pi D^2}\\sqrt{D^2-\\varepsilon ^2}$ where $D$ defines the band half – width.", "This is a rather good approximation for three – dimensional case.", "In Ref.", "[31] we have given an analytic proof that in DMFT+$\\Sigma $ approximation for disordered Hubbard model with semi – elliptic density of states all disorder effects in single – particle properties, calculated in DMFT+$\\Sigma $ (with the use of self – consistent Born approximation (REF )) are reduced to conduction band – widening by disorder, i.e.", "to the replacement (in the density of states) $D\\rightarrow D_{eff}$ , where $D_{eff}$ is the effective half – width of the band in the presence of disorder scattering: $D_{eff}=D\\sqrt{1+4\\frac{\\Delta ^2}{D^2}}.$ so that the “bare” density of states (in the absence of correlations,$U=0$ ) becomes: $\\tilde{N}_{0}(\\xi )=\\frac{2}{\\pi D_{eff}^2}\\sqrt{D_{eff}^2-\\varepsilon ^2}$ conserving its semi – elliptic form.", "It should be noted, that for different models of the “bare” conduction band disorder can also change the form of the density of states, so that such universal disorder effects in single – properties is absent.", "However, in the limit of strong enough disorder almost any initial density of states actually acquires semi – elliptic form, restoring this universal dependence on disorder [31].", "The temperature of superconducting transition in attractive Hubbard model within DMFT was calculated in a number of papers [17], [18], [20], analyzing both from the Cooper instability of the normal phase [17] (divergence of Cooper susceptibility) and from the disappearance of superconducting order parameter [18], [20].", "In Ref.", "[21] we determined the critical temperature from instability of the normal phase (instability of DMFT iteration procedure).", "The results obtained were in good agreement with the results of Refs.", "[17], [18], [20].", "Besides that, in Ref.", "[21] to calculate $T_c$ we have used the Nozieres – Schmitt-Rink approach [13], showing that this approach allows qualitatively, though approximately, describes the BCS – BEC crossover region.", "In Refs.", "[30], [31] we used the combination of Nozieres – Schmitt-Rink approach and DMFT+$\\Sigma $ for detailed studies of disorder influence on the temperature of superconducting transition and the number of local pairs.", "In this approach we determine $T_c$ from the following equation [31]: $1=\\frac{U}{2}\\int _{-\\infty }^{\\infty }d\\varepsilon \\tilde{N}_0(\\varepsilon )\\frac{th\\frac{\\varepsilon -\\mu }{2T_c}}{\\varepsilon -\\mu }.$ with chemical potential $\\mu $ for different $U$ and $\\Delta $ being determined from DMFT+$\\Sigma $ calculations, i.e.", "from the standard equation for the number of electrons (band filling), defined by the Green's function (REF ).", "This allows us to find $T_c$ for the wide range of the model parameters, including the BCS – BEC crossover region and the limit of strong coupling, as well as for the different disorder levels.", "This reflects the physical meaning of Nozieres – Schmitt-Rink approximation: in the weak coupling region transition temperature is controlled by the equation for Cooper instability (REF ), while in the strong coupling limit it is determined by the temperature of Bose condensation of compact Cooper pairs, which is controlled by chemical potential.", "Figure: Universal dependence of superconducting critical temperature on Hubbardattraction for different levels of disorder.In Fig.", "REF we show the universal dependence of superconducting critical temperature $T_c$ on Hubbard attraction for different levels of disorder obtained in Ref.", "[31].", "This is a manifestation of the generalized Anderson theorem.", "In the weak coupling region $T_c$ is well described by BCS model (dashed line in Fig.", "REF shows $T_c$ determined by Eq.", "(REF ) with chemical potential independent of $U$ and obtained for the quarter – filled “bare” band), while in the strong coupling region $T_c$ is determined by the condition of Bose condensation of Cooper pairs giving $\\sim t^2/U$ dependence (corresponding to inverse mass dependence of compact Bosons), passing through a characteristic maximum at $U/2D_{eff}\\sim 1$ in BCS – BEC crossover region." ], [ "Ginzburg – Landau expansion", "Ginzburg – Landau expansion for the difference of free energies of superconducting and normal phases can be written in the standard form: $F_{s}-F_{n}=A|\\Delta _{\\bf q}|^2+q^2 C|\\Delta _{\\bf q}|^2+\\frac{B}{2}|\\Delta _{\\bf q}|^4,$ where $\\Delta _{\\bf q}$ is the Fourier component of the order parameter.", "Figure: Diagrammatic representation of Ginzburg – Landau expansion.Microscopically GL – expansion (REF ) is determined by diagrams of loop – expansion for the free energy of electrons in an “external” field of random fluctuations of order parameter with small wave – vector ${\\bf q}$ [10], [37] shown in Fig.REF (where fluctuations are represented by dashed lines).", "In disordered system, the use here of the standard impurity diagram technique implicitly assumes the self – averaging nature of the order parameter [8], [9], [10].", "Within the framework of Nozieres – Schmitt-Rink approach [13] the loops with two and four Cooper vertices, shown in Fig.REF , do not contain contributions from attractive Hubbard interaction (as in weak coupling theory) and are “dressed” only by disorder (impurity) scattering In the absence of disorder this approach gives the same results for GL – coefficients as in Refs.", "[33], [34], [35], where the functional integral for free energy was analyzed via Hubbard – Stratonovich transformation, reducing it to the functional integral over arbitrary fluctuations of superconducting order parameter.", "However, the chemical potential here, which has an important dependence on the strength of interaction $U$ and determines the condition of Bose condensation of Cooper pairs, should be calculated in the framework of DMFT+$\\Sigma $ approximation, as it was done in Refs.", "[30], [31] in calculations of $T_c$ .", "In Ref.", "[36] we have shown that in this approach GL – coefficients $A$ and $B$ are determined by the following expressions: $A(T)=\\frac{1}{U}-\\int _{-\\infty }^{\\infty }d\\varepsilon \\tilde{N}_0(\\varepsilon )\\frac{th\\frac{\\varepsilon -\\mu }{2T}}{2(\\varepsilon -\\mu )},$ $B=\\int _{-\\infty }^{\\infty }\\frac{d\\varepsilon }{2(\\varepsilon -\\mu )^3}\\left(th\\frac{\\varepsilon -\\mu }{2T}-\\frac{(\\varepsilon -\\mu )/2T}{ch^2\\frac{\\varepsilon -\\mu }{2T}}\\right)\\tilde{N}_0(\\varepsilon ),$ For $T\\rightarrow T_c$ coefficient $A(T)$ takes the usual form: $A(T)\\equiv \\alpha (T-T_c).$ In BCS weak coupling limit we obtain the standard expressions for $\\alpha $ and $B$ [37]: $\\alpha _{BCS}=\\frac{\\tilde{N}_0(\\mu )}{T_c},\\qquad B_{BCS}=\\frac{7\\zeta (3)}{8\\pi ^2 T_c^2}\\tilde{N}_0(\\mu ).$ so that coefficients $A$ and $B$ are determined only by disorder widened density of states $\\tilde{N}_0(\\varepsilon )$ and chemical potential $\\mu $ .", "Then, in the case of semi – elliptic density of states their dependence on disorder is described by the simple replacement $D\\rightarrow D_{eff}$ and we have universal dependencies of $\\alpha $ and $B$ (properly normalized by powers of $2D_{eff}$ ) on $U/2D_{eff}$ , as shown in Fig.", "REF .", "Both $\\alpha $ and $B$ drop fast with the growth of interaction $U/2D_{eff}$ .", "Figure: Universal dependence of Ginzburg – Landau coefficients α\\alpha (a)and BB (b) on the strength of Hubbard attraction for different levels ofdisorder.It should be noted that Eqs.", "(REF ) and (REF ) for coefficients $A$ and $B$ were obtained in Ref.", "[36] using the exact Ward identities and remain valid also in the limit of strong disorder (up to Anderson localization).", "Correspondingly, in the limit of strong disorder the coefficients $A$ and $B$ depend on disorder only via appropriate dependence of the density of states.", "Dependence on disorder, related only to the band widening by $D\\rightarrow D_{eff}$ , is also observed for specific heat discontinuity at the critical temperature [36], determined by coefficients $\\alpha $ and $B$ : $C_s(T_c)-C_n(T_c)=T_c\\frac{\\alpha ^2}{B}.$ In Fig.", "REF we show the universal dependence of specific heat discontinuity on $U/2D_{eff}$ .", "In BCS limit specific heat discontinuity grows with coupling, while in BEC limit it drops with $U/2D_{eff}$ , passing through maximum at $U/2D_{eff}\\approx 0.55$ in BCS – BEC crossover region.", "This behavior of specific heat discontinuity is mainly related to the similar dependence of $T_c$ (cf.", "Fig.", "REF ), as $\\frac{\\alpha ^2}{B}$ in Eq.", "(REF ) only smoothly depends on the coupling strength.", "Figure: Universal dependence of specific heat discontinuity on U/2D eff U/2D_{eff}for different disorder levels.From diagrammatic representation of GL – expansion shown in Fig.", "REF it is clear, that coefficient $C$ is determined by the $q^2$ term in the expansion of the two – particle loop (first term in Fig.", "REF ) in powers of $q$ .", "Then we obtain: $C=-T\\lim _{q \\rightarrow 0}\\sum _{n, \\bf p, \\bf {p^{\\prime }}}\\frac{\\Phi _{\\bf p\\bf {p^{\\prime }}}( \\varepsilon _n,{\\bf q})-\\Phi _{\\bf p\\bf {p^{\\prime }}}( \\varepsilon _n,0)}{q^2},$ where $\\Phi _{\\bf p,\\bf {p^{\\prime }}}( \\varepsilon _n,{\\bf q})$ is two – particle Green's function in Cooper channel “dressed” (in Nozieres – Schmitt-Rink approximation) only by impurity scattering.", "To determine the coefficient $C$ we again use the exact Ward identity, derived by us in Ref.", "[29]: $G(\\varepsilon _n,{\\bf p_{+}})-G(-\\varepsilon _n,-{\\bf p_{-}})=-\\sum _{\\bf p^{\\prime }}\\Phi _{\\bf pp^{\\prime }}(\\varepsilon _n,{\\bf q})\\left[(G_0^{-1}(\\varepsilon _n,{\\bf p^{\\prime }_{+}})-G_0^{-1}(-\\varepsilon _n,-{\\bf p^{\\prime }_{-}})\\right],$ where ${\\bf p_{\\pm }}={\\bf p} \\pm \\frac{\\bf q}{2}$ , $G_0(\\varepsilon _n,{\\bf p})=\\frac{1}{\\varepsilon _n+\\mu -\\varepsilon _{\\bf p}}$ is the “bare” single – particle Green's function at Fermion Matsubara frequencies $\\varepsilon _n$ , while $G(\\varepsilon _n,{\\bf p})$ is the single – particle Green's function “dressed” only by impurity scattering.", "Introducing the notation $\\Delta G(\\varepsilon _n,{\\bf p})=G(\\varepsilon _n,{\\bf p_+})-G(-\\varepsilon _n,-{\\bf p_-})$ and using the symmetry $\\varepsilon ({\\bf p})=\\varepsilon (-{\\bf p})$ and $G(\\varepsilon _n,-{\\bf p})=G(\\varepsilon _n,{\\bf p})$ we rewrite the Ward identity as: $\\Delta G(\\varepsilon _n,{\\bf p})=-\\sum _{\\bf p^{\\prime }}\\Phi _{\\bf pp^{\\prime }}(\\varepsilon _n,{\\bf q})(2i\\varepsilon _n-\\Delta \\varepsilon _{\\bf p^{\\prime }}),$ where $\\Delta \\varepsilon _{\\bf p}=\\varepsilon _{\\bf p_+}-\\varepsilon _{\\bf p_-}$ .", "Then we can perform here summation over ${\\bf p}$ (also with additional multiplication by $\\Delta \\varepsilon _{\\bf p}$ ) to obtain the following system of equations: $\\sum _{\\bf {p}}\\Delta G(\\varepsilon _n,{\\bf p})&=&-2i\\varepsilon _n\\Phi _{0}(\\varepsilon _n,{\\bf q})+\\Phi _{1}(\\varepsilon _n,{\\bf q})\\nonumber \\\\\\sum _{\\bf {p}}\\Delta \\varepsilon _{\\bf p}\\Delta G(\\varepsilon _n,{\\bf p})&=&-2i\\varepsilon _n\\Phi _{1}(\\varepsilon _n,{\\bf q})+\\sum _{\\bf p,p^{\\prime }}\\Delta \\varepsilon _{\\bf p}\\Phi _{\\bf pp^{\\prime }}(\\varepsilon _n,{\\bf q})\\Delta \\varepsilon _{\\bf p^{\\prime }},$ where $\\Phi _{0}(\\varepsilon _n,{\\bf q})=\\sum _{\\bf pp^{\\prime }}\\Phi _{\\bf pp^{\\prime }}(\\varepsilon _n,{\\bf q})$ , $\\Phi _{1}(\\varepsilon _n,{\\bf q})=\\sum _{\\bf pp^{\\prime }}\\Phi _{\\bf pp^{\\prime }}(\\varepsilon _n,{\\bf q})\\Delta \\varepsilon _{\\bf p^{\\prime }}=\\sum _{\\bf pp^{\\prime }}\\Delta \\varepsilon _{\\bf p}\\Phi _{\\bf pp^{\\prime }}(\\varepsilon _n,{\\bf q})$ .", "Then, excluding $\\Phi _{1}(\\varepsilon _n,{\\bf q})$ from this system of equations, we obtain: $\\sum _{\\bf {p}}\\Delta \\varepsilon _{\\bf p}\\Delta G(\\varepsilon _n,{\\bf p})=-2i\\varepsilon _n\\sum _{\\bf {p}}\\Delta G(\\varepsilon _n,{\\bf p})-(2i\\varepsilon _n)^2\\Phi _{0}(\\varepsilon _n,{\\bf q})+\\sum _{\\bf pp^{\\prime }}\\Delta \\varepsilon _{\\bf p}\\Phi _{\\bf pp^{\\prime }}(\\varepsilon _n,{\\bf q})\\Delta \\varepsilon _{\\bf p^{\\prime }}.$ All terms in Eq.", "(REF ) are functions of ${\\bf q}^2$ .", "Let us write down two lowest – order terms of $q^2$ – expansion of Eq.", "(REF ).", "The $\\sim q^0$ term is: $\\Phi _{0}(\\varepsilon _n,{\\bf q}=0)=-\\frac{\\sum _{\\bf {p}}\\Delta G(\\varepsilon _n,{\\bf p})}{2i\\varepsilon _n}.$ As there is no dependence on the direction of ${\\bf q}$ we choose ${\\bf q}=(q,0,0)$ .", "Then $\\sim q^2$ terms are written as: $\\varphi (\\varepsilon _n,{\\bf q}=0){(2i\\varepsilon _n)}^2=\\sum _{\\bf {p}\\bf {p^{\\prime }}}v_x\\Phi _{\\bf {pp^{\\prime }}}(\\varepsilon _n,{\\bf q}){v_x}^{\\prime }-\\lim _{q \\rightarrow 0} \\frac{\\sum _{\\bf p}\\Delta \\varepsilon _{\\bf p}\\Delta G(\\varepsilon _n,{\\bf p})}{q^2},$ where $v_x=\\frac{\\partial \\varepsilon _{\\bf p}}{\\partial p_x}$ and $\\varphi (\\varepsilon _n,{\\bf q}=0)=\\lim _{q \\rightarrow 0} \\frac{\\Phi _{0}(\\varepsilon _n,{\\bf q})-\\Phi _{0}(\\varepsilon _n,0)}{q^2}$ .", "For weak enough disorder we can neglect localization corrections and consider the two – particle loop in “ladder” approximation for disorder scattering.", "Then, due to vector nature of vertices, all vertex corrections vanish due to angular integration and we obtain: $\\sum _{\\bf {p}\\bf {p^{\\prime }}}v_x\\Phi _{\\bf {pp^{\\prime }}}(\\varepsilon _n,{\\bf q}){v_x}^{\\prime }=\\sum _{\\bf p}{v_x}^2G(\\varepsilon _n,{\\bf p})G(-\\varepsilon _n,{\\bf p})$ For the case of isotropic spectrum we have: $\\lim _{q \\rightarrow 0} \\frac{\\sum _{\\bf p}\\Delta \\varepsilon _{\\bf p}\\Delta G(\\varepsilon _n,{\\bf p})}{q^2}=-\\frac{1}{2}\\sum _{\\bf p}\\frac{{\\partial }^2\\varepsilon _{\\bf p}}{\\partial {p_x}^2}(G(\\varepsilon _n,{\\bf p})+G(-\\varepsilon _n,{\\bf p})).$ As a result, we can write $C$ coefficient (REF ) as: $C=-T\\sum _{n}\\frac{\\sum _{\\bf p}{v_x}^2G(\\varepsilon _n,{\\bf p})G(-\\varepsilon _n,{\\bf p})+\\frac{1}{2}\\sum _{\\bf p}\\frac{{\\partial }^2\\varepsilon _{\\bf p}}{\\partial {p_x}^2}(G(\\varepsilon _n,{\\bf p})+G(-\\varepsilon _n,{\\bf p}))}{{(2i\\varepsilon _{n})}^2}.$ After the standard summation over Matsubara frequencies we obtain: $C=-\\frac{1}{8\\pi }\\int _{-\\infty }^{\\infty }d\\varepsilon \\frac{th\\frac{\\varepsilon }{2T}}{\\varepsilon }\\sum _{\\bf p}\\left({v_x}^2Im\\frac{G^{R}(\\varepsilon ,{\\bf p})G^{A}(-\\varepsilon ,{\\bf p})}{\\varepsilon +i\\delta }+\\frac{{\\partial }^2\\varepsilon _{\\bf p}}{\\partial {p_x}^2}Im\\frac{G^{R}(\\varepsilon ,{\\bf p})}{\\varepsilon +i\\delta }\\right).$ Finally $C$ coefficient is expressed as: $C=-\\frac{1}{8\\pi }\\int _{-\\infty }^{\\infty }d\\varepsilon \\frac{th\\frac{\\varepsilon }{2T}}{\\varepsilon ^2}\\sum _{\\bf p}\\left({v_x}^2Im(G^{R}(\\varepsilon ,{\\bf p})G^{A}(-\\varepsilon ,{\\bf p}))+\\frac{{\\partial }^2\\varepsilon _{\\bf p}}{\\partial {p_x}^2}ImG^{R}(\\varepsilon ,{\\bf p})\\right)+\\nonumber \\\\\\frac{1}{16T}\\sum _{\\bf p}\\left({v_x}^2Re(G^{R}(0,{\\bf p})G^{A}(0,{\\bf p}))+\\frac{{\\partial }^2\\varepsilon _{\\bf p}}{\\partial {p_x}^2}ReG^{R}(0,{\\bf p})\\right).$ The procedure to calculate velocity $v_x$ and its derivative $\\frac{{\\partial }^2\\varepsilon _{\\bf p}}{\\partial {p_x}^2}$ in the model with semi – elliptic density of states was discussed in detail in Ref.", "[26].", "In the absence of disorder ($\\Delta =0$ ) we replace $G\\rightarrow G_0$ and the expression for $C$ takes the following form: $C=-\\frac{1}{8\\pi }\\sum _{\\bf p}\\left(\\frac{{v_x}^2}{{(\\varepsilon _{\\bf p}-\\mu )}^2}-\\frac{{\\partial }^2\\varepsilon _{\\bf p}}{\\partial {p_x}^2}\\frac{1}{\\varepsilon _{\\bf p}-\\mu }\\right)\\left(\\frac{th\\frac{\\varepsilon _{\\bf p}-\\mu }{2T}}{\\varepsilon _{\\bf p}-\\mu }-\\frac{1}{2T}\\right)$ In the weak coupling BCS limit in the absence of disorder the coefficient $C$ reduces to the standard expression [37]: $C_{BCS}=\\frac{7\\zeta (3)}{16\\pi ^2 T_c^2}N_0(\\mu )\\frac{v_F^2}{d},$ where $v_F$ is Fermi velocity, $d$ – dimensionality of space.", "Semi – elliptic density of states is a good approximation for $d=3$ .", "As noted above disorder influence on $C$ is not reduced to a simple replacement $N_0\\rightarrow \\tilde{N}_0$ , so that even in the BCS weak coupling limit (in contrast to coefficients $\\alpha $ and $B$ (cf.", "(REF )) we can not derive for $C$ a compact expression, similar to (REF )." ], [ "Main results", "Let us discuss now the main results of our calculations for the gradient term coefficient $C$ of GL – expansion and the related physical characteristics, such as the coherence length, penetration depth and the slope of the upper critical magnetic field at $T_c$ .", "The coherence length at given temperature $\\xi (T)$ determines the characteristic scale of order – parameter $\\Delta $ inhomogeneities: $\\xi ^2(T)=-\\frac{C}{A}.$ Coefficient $A$ changes its sign at the critical temperature $A=\\alpha (T-T_c)$ , so that $\\xi (T)=\\frac{\\xi }{\\sqrt{1-T/T_c}},$ where we have introduced the coherence length as: $\\xi =\\sqrt{\\frac{C}{\\alpha T_c}}.$ In the weak coupling limit and in the absence of disorder it is written in the standard form [37]: $\\xi _{BCS}=\\sqrt{\\frac{C_{BCS}}{\\alpha _{BCS} T_c}}=\\sqrt{\\frac{7\\zeta (3)}{16\\pi ^2 d}}\\frac{v_F}{T_c}.$ Penetration depth of magnetic field into superconductor is defined as: $\\lambda ^2(T)=-\\frac{c^2}{32 \\pi e^2}\\frac{B}{A C}.$ Thus: $\\lambda (T)=\\frac{\\lambda }{\\sqrt{1-T/T_c}},$ where we have introduced: $\\lambda ^2=\\frac{c^2}{32 \\pi e^2}\\frac{B}{\\alpha C T_c},$ which in the absence of disorder has the form: $\\lambda ^2_{BCS}=\\frac{c^2}{32 \\pi e^2}\\frac{B_{BCS}}{\\alpha _{BCS} C_{BCS} T_c}=\\frac{c^2}{16 \\pi e^2}\\frac{d}{N_0(\\mu )v_F^2}.$ Note that $\\lambda _{BCS}$ does not depend on $T_c$ , and correspondingly on the coupling strength, so that it is convenient for normalization of penetration depth $\\lambda $ (REF ) for arbitrary $U$ and $\\Delta $ .", "Close to $T_c$ the upper critical field $H_{c2}$ is defined via GL – coefficients as: $H_{c2}=\\frac{\\Phi _0}{2 \\pi \\xi ^2(T)}=-\\frac{\\Phi _0}{2 \\pi }\\frac{A}{C},$ where $\\Phi _0=c \\pi /e$ is magnetic flux quantum.", "Then the slope of the upper critical field at $T_c$ is given by: $\\frac{dH_{c2}}{dT}= \\frac{\\Phi _0}{2 \\pi }\\frac{\\alpha }{C}.$ Figure: Dependence of the coefficient CC on the strength of Hubbard attractionfor different levels of disorder (aa – lattice parameter).", "(a) — all energy units are normalized by the width of the “bare”conduction band 2D2D.", "(b) — all energy units normalized by effective band width 2D eff 2D_{eff}.", "(c) — coefficient CC normalized by its value C BCS C_{BCS} in the weak couplinglimit and in the absence of disorder.In Fig.", "REF we show the dependencies of coefficient $C$ on the strength of Hubbard attraction for different disorder levels.", "It is seen that $C$ drops fast with the growth of the coupling constant.", "Especially fast this drop is in the weak coupling region (see insert in Fig.", "REF (a)).", "Being essentially a two – particle characteristic coefficient $C$ does not demonstrate universal dependencies on disorder, similar to $\\alpha $ and $B$ coefficients, as is clearly seen from Fig.", "REF (b).", "Fig.", "REF (c) shows the coupling strength dependence of $C$ normalized by its BCS value (REF ) in the absence of disorder.", "Figure: Dependence of coefficient CC, normalized by its value in the absenceof disorder, on disorder levels for different values of Hubbard attraction UU.In Fig.", "REF we show the dependencies of $C$ on disorder for different values of coupling strength $U/2D$ .", "In the weak coupling limit ($U/2D=0.1$ ) we observe fast enough drop of $C$ with the growth of disorder in the region of weak enough disorder scattering.", "However, in the region of strong enough disorder we can observe even the growth of $C$ with disorder, related mainly to noticeable band widening at high disorder levels and respective drop in the effective coupling $U/2D_{eff}$ .", "For intermediate couplings ($U/2D=0.4 - 0.6$ ) coefficient $C$ only demonstrates some weak growth with disorder.", "In BEC limit ($U/2D>1$ ) coefficient $C$ is practically independent of disorder.", "Figure: Dependence of coherence length, normalized by lattice parameter aa,on Hubbard attraction UU for different disorder levels.", "Insert: fast growth ofcoherence length in weak coupling BCS limit.Let us now discuss the physical characteristics.", "Dependence of coherence length on the strength of Hubbard attraction is shown in Fig.", "REF .", "We can see that in the weak coupling region (cf.", "insert in Fig.REF ) the coherence length drops fast with the growth of $U$ at any disorder level, reaching the values of the order of lattice spacing $a$ at the intermediate couplings $U/2D \\sim 0.4-0.6$ .", "The further growth of the coupling strength leads only to small changes of coherence length.", "Figure: Dependence of penetration depth, normalized by its BCS value in theabsence of disorder, on Hubbard attraction UU for different disorder levels.In Fig.", "REF we show the dependence of penetration depth, normalized by its BCS value in the absence of disorder (REF ), on Hubbard attraction $U$ for different levels of disorder.", "In the absence of disorder scattering penetration depth grows with coupling.", "Disorder in BCS weak coupling limit leads to fast growth of penetration depth (for “dirty” BCS superconductors $\\lambda \\sim l^{-1/2}$ , where $l$ is the mean free path).", "In BEC strong coupling region disorder only slightly diminishes the penetration depth (cf.", "Fig.", "REF (a)).", "Figure: Dependence of the upper critical magnetic field slope on Hubbardattraction UU for different disorder levels.Dependence of the slope of the upper critical filed $dH_{c2}\\equiv (dH_{c2}/dT)_{T=T_c}$ on Hubbard attraction for different disorder levels is shown in Fig.", "REF .", "For any value of disorder scattering the slope of the upper critical field grows with coupling.", "However, in the limit of weak disorder we observe the fast growth of the slope with $U$ in the limit of weak enough attraction, while in the strong coupling limit the slope is weakly dependent on $U/2D$ .", "Figure: Dependence of coherence length on disorder for different values of Hubbardattraction.", "(a) — coherence length normalized by lattice parameter aa.Insert: coherence length dependence on disorder in the limit of weak coupling.", "(b) — coherence length normalized by its value in the absence of disorder.In Fig.", "REF we show the dependence of coherence length $\\xi $ on disorder for different values of coupling.", "In BCS weak coupling limit and for weak enough disorder we observe the standard “dirty” superconductors dependence $\\xi \\sim l^{1/2}$ , i.e.", "the coherence length drops with the growth of disorder (cf.", "insert in Fig.", "REF (a)).", "However, for strong enough disorder the coherence length starts to grow with disorder (cf.", "insert in Fig.", "REF (a) and Fig.", "REF (b)), which is mainly related to the noticeable widening of the initial band by disorder and appropriate drop of $U/2D_{eff}$ .", "With further growth of the coupling strength $U/2D \\ge 0.4-0.6$ the coherence length $\\xi $ becomes of the order of the lattice parameter and is almost independent of disorder.", "In particular, in strong coupling BEC limit for $U/2D=1.4, 1.6$ the growth of disorder to very large values ($\\Delta /2D=0.5$ ) leads to the drop of coherence length by the factor of two (cf.", "Fig.", "REF (b)).", "Figure: Dependence of penetration depth (a) and Ginzburg – Landau parameter (b)on disorder for different values of Hubbard attraction.", "Insert: the growth ofGL – parameter with disorder in the weak coupling limit.Dependence of penetration depth on disorder for different values of Hubbard attraction is shown in Fig.", "REF (a).", "In the limit of weak coupling in accordance with the theory of “dirty” superconductors disorder leads to the growth of penetration depth $\\lambda \\sim l^{-1/2}$ .", "With the increase of the coupling strength this growth of penetration depth with disorder slows down and in the limit of very strong coupling $U/2D=1.4, 1.6$ penetration depth even slightly diminishes with the growth of disorder.", "In Fig.", "REF (b) we show the disorder dependence of dimensionless Ginzburg – Landau parameter $\\kappa = \\lambda / \\xi $ .", "We can see that in the weak coupling limit GL – parameter grows fast with disorder (cf.", "insert in Fig.", "REF (b)) in accordance with the theory of “dirty” superconductors, where $\\kappa \\sim l^{-1}$ .", "With the increase of the coupling the growth of GL – parameter with disorder slows down and in the limit of strong coupling $U/2D>1$ parameter $\\kappa $ is practically independent of disorder.", "Figure: Dependence of the slope of the upper critical field (a) and theslope normalized by its value in the absence of disorder (b) on the level ofdisorder for different values of Hubbard attraction.", "Insert:the growth of the slope with disorder in the weak coupling limit.In Fig.", "REF we show the dependence of the slope of the upper critical magnetic field on disorder.", "In the weak coupling limit we again observe the typical “dirty” superconductor behavior — the slope grows with disorder (cf.", "Fig.", "REF (a) and the insert in Fig.", "REF (b)).", "For the intermediate coupling region ($U/2D=0.4 - 0.8$ ) the slope of the upper critical field is practically independent of disorder.", "In the limit of very strong coupling at small disorder the slope of the upper critical field can even slightly diminish, but in the limit of strong disorder the slope grows with the growth of disorder scattering." ], [ "Conclusion", "In the framework of DMFT+$\\Sigma $ generalization of dynamic mean field theory we have studied the effects of disorder on the coefficients of Ginzburg – Landau expansion and the related physical characteristics of superconductors close to $T_c$ in attractive Hubbard model.", "To study the GL – coefficients we have used the combination of DMFT+$\\Sigma $ approach and Nozieres – Schmitt-Rink approximation.", "Calculations were performed for the wide range of the values of attractive potential $U$ , from the weak coupling region ($U/2D_{eff}\\ll 1$ ), where instability of the normal phase and superconductivity are well described by BCS model, up to the strong coupling limit ($U/2D_{eff}>1$ ), where the superconducting transition is related to the Bose – Einstein condensation of compact Cooper pairs.", "The growth of the coupling strength $U$ leads to fast drop of all GL – coefficients.", "Coherence length $\\xi $ drops fast with the growth of the coupling strength and for $U/2D \\sim 0.4$ becomes of the order of the lattice parameter and only slightly changes with the further growth of the coupling.", "Penetration depth in “clean” superconductors grows with $U$ , while in “dirty” case it drops in the weak coupling region and grows in BEC limit, passing through the minimum in the intermediate (crossover) region of $U/2D\\sim 0.4-0.8$ .", "The slope of the upper critical magnetic field grows with $U$ .", "Specific heat discontinuity grows with Hubbard attraction $U$ in the weak coupling region and diminishes in the strong coupling region, passing through the maximum at $U/2D_{eff}\\approx 0.55$ .", "Disorder influence on the critical temperature $T_c$ , GL – coefficients $A$ and $B$ and specific heat discontinuity is universal — their change is related only to conduction band widening by disorder scattering, i.e.", "to the replacement $D \\rightarrow D_{eff}$ .", "Thus, both in BCS – BEC crossover region and in the strong coupling limit both critical temperature and GL – coefficients $A$ and $B$ obey the generalized Anderson theorem — all the influence of disorder reduces to disorder change of the density of states.", "GL – coefficient $C$ was studied here in the “ladder” approximation for disorder scattering.", "Disorder influence upon $C$ is not universal and is not related purely to the conduction band widening by disorder.", "In the limit of weak coupling $U/2D_{eff}\\ll 1$ the behavior of $C$ and the related physical characteristics are well described by the usual theory of “dirty” superconductors.", "Both $C$ and coherence length drops fast with the growth of disorder, while the penetration depth and the slope of the upper critical magnetic field grow with disorder.", "In the region of BCS – BEC crossover and in the BEC limit the coefficient $C$ and all physical characteristics are only weakly dependent on disorder.", "In particular, in BEC limit both the coherence length and penetration depth are only slightly suppressed with the growth of disorder, so that the GL – parameter $\\kappa $ is practically independent of disorder.", "This work was supported by RSF grant 14-12-00502." ] ]
1606.05125
[ [ "Cost Effective Campaigning in Social Networks" ], [ "Abstract Campaigners are increasingly using online social networking platforms for promoting products, ideas and information.", "A popular method of promoting a product or even an idea is incentivizing individuals to evangelize the idea vigorously by providing them with referral rewards in the form of discounts, cash backs, or social recognition.", "Due to budget constraints on scarce resources such as money and manpower, it may not be possible to provide incentives for the entire population, and hence incentives need to be allocated judiciously to appropriate individuals for ensuring the highest possible outreach size.", "We aim to do the same by formulating and solving an optimization problem using percolation theory.", "In particular, we compute the set of individuals that are provided incentives for minimizing the expected cost while ensuring a given outreach size.", "We also solve the problem of computing the set of individuals to be incentivized for maximizing the outreach size for given cost budget.", "The optimization problem turns out to be non trivial; it involves quantities that need to be computed by numerically solving a fixed point equation.", "Our primary contribution is, that for a fairly general cost structure, we show that the optimization problems can be solved by solving a simple linear program.", "We believe that our approach of using percolation theory to formulate an optimization problem is the first of its kind." ], [ "Introduction ", "Online social networking platforms are being increasingly used by campaigners, activists and marketing managers for promoting ideas, brands and products.", "In particular, the ability to recommend news articles [1], videos, and even products [2] by friends and acquaintances through online social networking platforms is being increasingly recognized by marketing gurus as well as political campaigners and activists.", "Influencing the spread of content through social media enables campaigners to mold the opinions of a large group of individuals.", "In most cases, campaigners and advertisers aim to spread their message to as many individuals as possible while respecting budget constraints.", "This calls for a judicious allocation of limited resources, like money and manpower, for ensuring highest possible outreach, i.e., the proportion of individuals who receive the message.", "Individuals share information with other individuals in their social network using Twitter tweets, Facebook posts or simply face to face meetings.", "These individuals may in turn pass the same to their friends and so on, leading to an information epidemic.", "However, individuals may also become bored or disillusioned with the message over time and decide to stop spreading it.", "Past research suggests that such social effects may lead to opinion polarization in social systems [3].", "This can be exploited by a campaigner who desires to influence such spreading or opinion formation by incentivizing individuals to evangelize more vigorously by providing them with referral rewards in the form of discounts, cash back or other attractive offers.", "Due to budget constraints, it may not be feasible to incentivize all, or even a majority of the population.", "Individuals have varying amount of influence over others, e.g., ordinary individuals may have social connections extending to only close family and friends, while others may have a large number of social connections which can enable them to influence large groups [4].", "Thus, it would seem that incentivizing highly influential individuals would be the obvious strategy.", "However, recruiting influential people can be very costly, which may result in the campaigner running out of funds after recruiting just a handful of celebrities, which in turn may result in suboptimal outreach size.", "A resource constrained campaigner, for a given cost budget, may want to maximize the proportion of informed individuals, while other campaigners who care more about campaign outreach than resource costs, may desire to minimize costs for achieving a given number of informed individuals.", "We address both the resource allocation challenges by formulating and solving two optimization problems with the help of bond percolation theory.", "A similar problem of preventing epidemics through vaccinations has received a lot of attention [5], [6], [7], [8], [9].", "However, in these problems the cost of vaccination is uniform for all individuals, and hence it is sufficient to calculate the minimum number of vaccinations.", "Information diffusion can also be maximized by selecting an optimal set of seeds, i.e., individuals best suited to start an epidemic [10], [11], [12].", "This is different from our strategy which involves incentivizing individuals to spread the message.", "It is possible to address the problem posed here using optimal control theory, which involves computing the optimal resource allocation in real time for ensuring maximum possible outreach size by a give deadline [13], [14], [15], [16], [17].", "However, the optimal control solution is not only difficult to compute, but also very hard to implement as it requires a centralized real time controller.", "Furthermore, recent work, [13], [14], [15], [16], [17], on optimal campaigning in social networks does not address the problem of minimizing the cost while gurantering an outreach size.", "Our formulation allows us to solve both the problems.", "Our model assumes two types of individuals viz.", "the `ordinary' and the `selected', and they are connected to one another through a social network.", "Before the campaign starts, the selected individuals are incentivized to spread the message more vigorously than the ordinary.", "We use the Susceptible Infected Recovered (SIR) model for modeling the information epidemic.", "For a given set of selected individuals, we first calculate the size of the information outbreak using network percolation theory, and then find the set of selected nodes which, 1. minimizes the cost for achieving a given proportion of informed individuals, and 2. maximize the fraction of informed individual for a given cost budget.", "We believe that our approach of using percolation theory to formulate an optimization problem is the first of its kind.", "The detailed model description can be found in Sec.", ", percolation analysis in Sec.", ", the problem formulation in Sec.", ", numerical results in Sec.", ", and finally conclusions are discussed in Sec.", "." ], [ "Model ", "We divide the total population of $N$ individuals in two types: the ordinary (type 1) and the selected (type 2).", "Before the campaign starts selected individuals are provided incentives to spread the information more vigorously.", "These individuals are connected with one another through a social network, which is represented by an undirected graph (network).", "Nodes represent individuals while a link embodies the communication pathways between individuals.", "Let $P(k)$ be the degree distribution of the social network.", "For analytical tractability, we assume that the network is uncorrelated [18].", "We generate an uncorrelated network using the configuration model [19].", "A sequence of $N$ integers, called the degree sequence, is obtained by sampling the degree distribution.", "Thus each node is associated with an integer which is assumed to be the number of half edges or stubs associated with the node.", "Assuming that the total number of stubs is even, each stub is chosen at random and joined with another randomly selected stub.", "The process continues until all stubs are exhausted.", "Self loops and multiple edges are possible, but the number of such self loops and multiple edges goes to zero as $N \\rightarrow \\infty $ with high probability.", "We assume that $N$ is large but finite.", "Let $\\phi (k)$ be the proportion of individuals with $k$ degrees that are provided incentives for vigorously spreading the message, i.e., proportion of nodes with degree $k$ that are type 2 nodes.", "The goal is to find the optimum $\\phi (k)$ for maximizing the epidemic size (or minimizing the cost).", "The actual individuals can be identified by sampling from a population of individuals with degree $k$ with probability $\\phi (k)$ .", "We assume that the information campaign starts with a randomly chosen individual, who may pass the information to her neighbors, who in turn may pass the same to their neighbors and so on.", "However, as the initial enthusiasm wanes, individuals may start loosing interest in spreading the information message.", "This is similar to the diffusion of infectious diseases in a population of susceptible individuals.", "Since, we account for individuals loosing interest in spreading the message, we use a continuous time SIR process to model the information diffusion.", "The entire population can be divided into three classes, those who haven't heard the message (susceptible class), those who have heard it and are actively spreading it (infected class) and those who have heard the message but have stopped spreading it (recovered class).", "Let $\\beta _1$ be the rate of information spread for an ordinary node (Type 1), while $\\beta _2$ for a selected node (Type 2).", "In other words, the probability that a type $i$ individual `infects' her susceptible neighbors in small time $dt$ is $\\beta _idt +o(dt)$ .", "Note that this is independent of the type of the susceptible node.", "Let $\\mu _i$ be the rate at which type $i$ infected individuals move to the recovered state.", "The larger the $\\mu _i$ the lesser the time an individual spends in spreading the message.", "Since type 2 individuals are incentivized to spread information more vigorously, $\\beta _2 > \\beta _1$ and $\\mu _2 < \\mu _1$ .", "Let $T_i$ be the probability that a type $i$ infected node infects its susceptible neighbors (any type) before it recovers ($i \\in \\lbrace 0,1\\rbrace $ ).", "It can be easily shown that $T_i = \\frac{\\beta _i}{\\beta _i + \\mu _i}$ , see [20].", "Therefore, $T_2 > T_1$ .", "$T_i$ can be interpreted as the probability that a link connecting type $i$ infected node to any susceptible node is occupied.", "We refer to such links as type $i$ links and $T_i$ the occupation probability for link of type $i$ .", "This mapping allows us to apply bond percolation theory for obtaining the size of the information epidemic [21]." ], [ "Analysis ", "We first aim to calculate the proportion of individuals who have received the message, or in other words, the proportion of recovered individuals at $t \\rightarrow \\infty $ .", "Let $P(k^{\\prime } \\mid k)$ be the probability of encountering a node of degree $k^{\\prime }$ by traversing a randomly chosen link from a node of degree $k$ .", "In other words, $P(k^{\\prime } \\mid k)$ is the probability that a node with degree $k$ has a neighbor with degree $k^{\\prime }$ .", "For a network generated by configuration model, $P(k^{\\prime } \\mid k)= \\frac{k^{\\prime }P(k^{\\prime })}{\\langle k \\rangle }$ [21], where $\\langle k^i \\rangle $ is the $i^{th}$ moment of $P(k)$ .", "Let $q$ be the probability of encountering a type 2 node by traversing a randomly chosen link from a node of degree $k$ .", "Therefore, $q = \\sum \\limits _{k^{\\prime }=1}^{\\infty }Pr($ Neighboring node is type 2 $\\mid $ neighboring node has degree $k^{\\prime })\\cdot Pr($ Neighboring node has degree $k^{\\prime }\\mid $ original node has degree $k)$ .", "$q= \\frac{1}{\\langle k \\rangle }\\sum _{k=1}^{\\infty }k\\phi (k)P(k)$ The probability that a randomly chosen node has $k_1$ type 1 and $k_2$ type 2 neighbors $= P(k_1,k_2) = \\sum \\limits _{k:k=k_1+k_2}^{\\infty } Pr(k_1,k_2\\mid $ node has degree $k)P(k) $ .", "For a large $N$ , the event that a given node has degree $k$ , can be approximated to be independent of the event that another node, having a common neighbor with the given node, has degree $k^{\\prime }$ .", "This is true since the degree sequence is generated by independent samples from the distribution, and for a large $N$ the effect of sampling without replacement is negligible.", "The probability that a node is selected (type 2), is a function of its degree, hence the event that a node is type 1 (or 2) is independent of the event that any other node is type 1 (or 2).", "This allows us to write: $P(k_1,k_2)= {k_1+k_2 \\atopwithdelims ()k_2}q^{k_2}(1-q)^{k_1}P(k_1+k_2)$ Let $Q(k)$ be the excess degree distribution, i.e., the degree distribution of a node arrived at by following a randomly chosen link without counting that link.", "For the configuration model $Q(k) = (k+1)P(k+1)/ <k>$ .", "Let $Q(k_1,k_2)$ be the excess degree distribution for connections to type 1 and type 2 nodes.", "$Q(k_1,k_2) = {k_1+k_2 \\atopwithdelims ()k_2}q^{k_2}(1-q)^{k_1}Q(k_1+k_2)$ Let $\\tilde{P}(\\tilde{k}_1,\\tilde{k}_2)$ and $\\tilde{Q}(\\tilde{k}_1,\\tilde{k}_2)$ be the distribution and the excess distribution of the number of type 1 and type 2 neighbors that have received the information message.", "In other words the distribution and the excess distribution of type $i$ occupied links.", "$\\tilde{P}(\\tilde{k}_1,\\tilde{k}_2) &= \\sum _{k_1=\\tilde{k}_1}^{\\infty }\\sum _{k_2=\\tilde{k}_2}^{\\infty }P(k_1,k_2)\\prod _{i=1}^{2}{k_i \\atopwithdelims ()\\tilde{k}_i}T_i^{\\tilde{k}_i}(1-T_i)^{k_i-\\tilde{k}_i} \\\\\\tilde{Q}(\\tilde{k}_1,\\tilde{k}_2) &= \\sum _{k_1=\\tilde{k}_1}^{\\infty }\\sum _{k_2=\\tilde{k}_2}^{\\infty }Q(k_1,k_2)\\prod _{i=1}^{2}{k_i \\atopwithdelims ()\\tilde{k}_i}T_i^{\\tilde{k}_i}(1-T_i)^{k_i-\\tilde{k}_i}$ Table: List of probability generating functions.The probability generating functions for the distributions used in the analysis above are listed in Table REF .", "For example $G(u_1,u_2)$ is given by : $G(u_1,u_2) = \\sum \\limits _{k_1,k_2=0}^{\\infty } u_1^{k_1}u_2^{k_2}P(k_1,k_2)$ Now, $\\tilde{G}(u_1,u_2)$ is given by $& \\sum _{\\tilde{k}_1,\\tilde{k}_2}^{\\infty }u_1^{\\tilde{k}_1}u_2^{\\tilde{k}_2}\\sum _{k_1 = \\tilde{k}_1}\\sum _{k_2=\\tilde{k}_2}P(k_1,k_2)\\prod _{i=1}^{2}{k_i \\atopwithdelims ()\\tilde{k}_i}T_i^{\\tilde{k}_i}(1-T_i)^{k_i-\\tilde{k}_i} \\\\&= \\sum _{k_1,k_2}^{\\infty }(1+(u_1-1)T_1)^{k_1}(1+(u_2-1)T_2)^{k_2}P(k_1,k_2) \\\\&= G\\left(1+(u_1-1)T_1,1+(u_2-1)T_2\\right)$ Similarly, $\\tilde{F}(u_1,u_2) = F(1+(u_1-1)T_1,1+(u_2-1)T_2)$ A component is a small cluster of nodes that have received the information message.", "By small we mean that the cluster is finite and does not scale with the network size.", "However, at the phase transition, the average size of the cluster diverges (as $N \\rightarrow \\infty $ ).", "An information epidemic outbreak is possible only when the average size of the cluster diverges.", "In this regime the component is termed as a giant connected component (GCC) and it grows with the network size.", "Let $\\tilde{H}_i(u_1,u_2)$ be the generating function of the distribution of the number of type 1 and type 2 nodes in a component arrived at from a type $i$ link.", "Let $\\tilde{J}_i(u_1,u_2)$ and $\\tilde{J}(u_1,u_2)$ be the generating functions of the distribution of the number of type 1 and type 2 nodes in a component arrived at from node $i$ and a randomly chosen node, respectively.", "Figure: (Color Online) Illustration of components.", "The red boxes represent the components reached by type 1 link while green boxes represent components reached by a type 2 link.", "A type 2 node is represented by a green circle while a red circle represents a type 1 node.Let random variable $Y_i$ be the number of type 1 and 2 nodes, that have received the message, in a component arrived at from a type $i$ link.", "The probability of encountering closed loops in finite cluster is $O(N^{-1})$ [20] which can be neglected for large $N$ .", "The tree like structure of the cluster allows us to write the size of the component encountered by traversing the link, as the sum of the size of components encountered after traversing the links emanating from the node at the end of the initial link.", "This is illustrated in Fig.", "REF .", "Hence, $Y_i$ can be written as: $Y_i = 1 + \\tilde{K}_1Y_1 + \\tilde{K}_2Y_2$ where random variable $\\tilde{K}_i$ is the number of type $i$ neighbors of the end node of type $i$ link that have received the message; the arrival link is not counted (excess degree).", "Since, the size of the components along different links are mutually independent (absence of loops) we can write the above equation in terms of probability generating functions.", "$\\tilde{H}_i(u_1,u_2) &= u_i\\tilde{H}_1(u_1,u_2)^{\\tilde{K}_1}\\tilde{H}_2(u_1,u_2)^{\\tilde{K}_2} \\\\&= u_i \\sum _{\\tilde{k}_1,\\tilde{k}_2}^{\\infty } \\tilde{H}_1^{k_1}(u_1,u_2)\\tilde{H}_2^{k_2}(u_1,u_2)\\tilde{Q}(\\tilde{k}_1,\\tilde{k}_2) \\\\&= u_i\\tilde{F}(\\tilde{H}_1(u_1,u_2) ,\\tilde{H}_2(u_1,u_2) ) \\\\$ Which can also be written as $\\tilde{H}_i(u_1,u_2) = u_iF\\left(1+(\\tilde{H}_1(u_1,u_2)-1)T_1,1+(\\tilde{H}_2(u_1,u_2)-1)T_2 \\right) $ Similarly, $\\tilde{J}(u_1,u_2)$ can be expressed as : $\\tilde{J}_i(u_1,u_2) &= u_i\\sum _{\\tilde{k}_1,\\tilde{k}_2}^{\\infty } \\tilde{H}_1^{k_1}(u_1,u_2)\\tilde{H}_2^{k_2}(u_1,u_2)\\tilde{P}(\\tilde{k}_1,\\tilde{k}_2) \\\\&= u_i\\tilde{G}(\\tilde{H}_1(u_1,u_2) ,\\tilde{H}_2(u_1,u_2) ) \\\\\\tilde{J}(u_1,u_2) &= (1-p)\\tilde{J}_1(u_1,u_2) +p \\tilde{J}_2(u_1,u_2)$ where $p$ is the probability of choosing a type 2 node, $p = \\sum \\limits _{k=1}^{\\infty }P(k)\\phi (k)$ .", "The following theorem describes the phase transition conditions required for an outbreak and the size of the such an outbreak.", "The proof can be found in .", "Theorem 1 The condition required for a small cluster to become a giant connected component is given by: $\\tilde{\\nu }\\ge 1 $ , where $\\tilde{\\nu } = T_1\\sum _{k_1,k_2}^{\\infty }k_1Q(k_1,k_2) + T_2\\sum \\limits _{k_1,k_2}^{\\infty }k_2Q(k_1,k_2)$ and the proportion of nodes in the giant connected component (size of GCC) is given by $1-\\psi $ , $\\psi = \\sum _{k_1,k_2}^{\\infty }(1+(u^*-1)T_1)^{k_1}(1+(u^*-1)T_2)^{k_2}P(k_1,k_2)$ where $u^*$ is the solution of the fixed point equation $u = \\sum _{k_1,k_2}^{\\infty }(1+(u-1)T_1)^{k_1}(1+(u-1)T_2)^{k_2}Q(k_1,k_2)$ The size of the information epidemic outbreak can now be used for formulating the optimization problem." ], [ "Problem Formulation ", "Providing incentives in the form of referral rewards for low degree nodes, or sponsorship offers for celebrities (high degree nodes) is costly.", "Since, the cost is a function of the degree let $c(k)$ be the cost of providing incentivizing a node with degree $k$ .", "The average cost, $\\bar{c}(\\phi )$ , is given by $\\sum \\limits _{k=1}^{\\infty }c(k)Pr($ node is selected $\\mid $ node has degree $k)P(k) = \\sum \\limits _{k=1}^{\\infty } c(k)\\phi (k)P(k)$ .", "The proportion of type 2 individuals is given by $\\sum \\limits _{k=1}^{\\infty }\\phi (k)P(k)$ .", "We formulate two optimization problems, viz., one which minimizes cost while enforcing a lower bound on the epidemic size, and the other which maximizes the epidemic size for a given cost budget.", "For both the problems, the evaluation of the size of the epidemic requires one to numerically solve a fixed point equation.", "Thus, there is no straightforward method to solve the optimization problem such as the Karush Kuhn Tucker (KKT) conditions, because evaluating the objective function requires one to solve a fixed point equation.", "We show that this problem can be reduced to a linear program, which can then be solved easily using any off the shelf LP solver." ], [ "Cost minimization problem", "Providing guarantees on the minimum number of individuals who will be informed about the campaign is appropriate for campaigns with large funding, such as election campaigns where message penetration is more important than the cost.", "The guarantee on epidemic size is written as a constraint to the optimization problem.", "The cost $\\bar{c}(\\phi )$ is minimized subject to $1 - \\psi \\ge \\gamma $ where $\\gamma \\ \\in \\ [0,1]$ and $\\phi $ is the control variable.", "If $\\gamma = 0$ , the constraint becomes $\\tilde{\\nu } \\le 1$ , as $\\gamma = 0$ implies $\\psi = 1$ which is the same as $\\tilde{\\nu } \\le 1$ .", "A finite amount of money, may put a constraint on the number of type 2 individuals.", "The proportion of type 2 individuals is given by $\\sum _{k=1}^{\\infty } \\phi (k)P(k)$ .", "This translates in to the constraint : $\\sum _{k=1}^{\\infty } \\phi (k)P(k) \\le B$ , where budget $B \\ \\in \\ [0,1]$ .", "The following theorem which is our principle contribution allows us to solve a possible non convex problem by solving a linear program.", "The key insight is that the probability of outbreak is monotonically decreasing in $q$ , which then allows one to write the optimization problem as a linear program.", "The intuition behind this claim is that since $q$ is the probability of finding a type 2 node on a randomly chosen link, increase in $q$ is equivalent to the increase in number of type 2 individuals resulting in a higher epidemic size.", "Theorem 2 If $T_2 > T_1$ , then $\\psi \\ \\in \\ (0,1)$ , is strictly decreasing with respect to $q$ , i.e, $\\frac{d\\psi }{dq} <0$ for all $q \\ \\in \\ [0,1] $ .", "For the $\\psi =0$ case $(\\tilde{\\nu }\\ge 1)$ , $\\tilde{\\nu }$ is strictly increasing with respect to $q$ , i.e, $\\frac{d\\tilde{\\nu }}{dq} > 0, \\ \\forall \\ \\ q \\ \\in \\ [0,1]$ , where $q = \\frac{1}{\\langle k \\rangle }\\sum \\limits _{k=1}^{\\infty }k\\phi (k)P(k)$ .", "The proof follows from Lemmas REF and REF detailed in Appendix .", "Since, $\\frac{d\\psi }{dq} < 0$ , the epidemic size constraint can be written as $\\frac{1}{\\langle k \\rangle }\\sum \\limits _{k=1}^{\\infty }k\\phi (k)P(k) \\ge q^*$ , where $ \\psi (q) \\mid _{q=q^*} \\ = 1-\\gamma $ .", "The optimization problem can now be written as follows: $\\begin{aligned}& \\underset{\\phi }{\\text{minimize}}\\ \\ \\ \\ \\sum _{k=1}^{\\infty } c(k)\\phi (k)P(k) \\\\& \\text{subject to:} \\ \\ \\\\& \\frac{1}{\\langle k\\rangle }\\sum _{k=1}^{\\infty }k\\phi (k)P(k) \\ge q^* \\\\& \\sum _{k=1}^{\\infty } \\phi (k)P(k) \\le B \\\\& 0 \\le \\phi \\le 1 \\end{aligned}$ The above problem is a linear program which can be solved by any off-the-shelf LP solver.", "The optimization problem described above may not be feasible for all values of $T_1$ and for all possible degree distributions.", "Assume, $B=1$ , the problem becomes infeasible if $1 - \\psi \\le \\gamma $ when $T_2$ is at the maximum possible value, i.e., all individuals are incentivized and yet $1 - \\psi \\le \\gamma $ ." ], [ "Epidemic Size Maximization Problem", "We now look at the problem of maximizing the information epidemic size (outreach) in a resource constrained scenario.", "More, specifically we study a scenario where the cost budget is finite.", "Thus the outbreak size $1-\\psi $ must be maximized subject to a cost constraint.", "Since $\\frac{d\\psi }{dq} < 0$ , maximizing $q$ is equivalent to maximizing $1-\\psi $ .", "Thus the problem is equivalent to the following linear program.", "$\\begin{aligned}& \\underset{\\phi }{\\text{maximize}}\\ \\ \\ \\ \\sum _{k=1}^{\\infty }k\\phi (k)P(k) \\\\& \\text{subject to:} \\ \\ \\\\& \\sum _{k=1}^{\\infty } c(k)\\phi (k)P(k) \\le C \\\\& \\sum _{k=1}^{\\infty } \\phi (k)P(k) \\le B \\\\& 0 \\le \\phi \\le 1 \\end{aligned}$ The linear program can now be solved using any standard linear programing solver.", "Note that constants $T_1, T_2$ do not play any role in problem (REF ), while they do play a role in problem (REF ) because $q^*$ is a function of $T_1$ and $T_2$ ." ], [ "Numerical Results ", "As an illustration, we study the solution of the optimization problem for a linear cost, i.e., $c(k)=k$ .", "The higher the degree, the higher the cost.", "Note that even if cost is non linear in $k$ , the optimization problem remains a linear program.", "In the real world, the cost may be different, but whatever the cost function, the solution can be obtained by simply solving a linear program.", "We used an uncorrelated random graph generated using the configuration model technique with power law degree distribution ($P(k)\\propto k^{- \\alpha }$ ), $\\alpha = 2.5$ ." ], [ "Cost Minimization Problem", "We solved the cost minimization linear program using the `linprog' MATLAB solver; $q^*$ was computed numerically using the bisection method.", "In Fig.", "REF , we plot the solution $\\phi $ for different values of $T_1$ .", "The solution shows that only about $50 \\%$ of high degree nodes need to be incetivized for $T_1$ values ranging from $0.3$ to $0.43$ .", "As $T_1$ decreases from $0.47$ to $0.3$ , the proportion of high degree nodes that are incentivized remain fairly constant (50%), while the proportion of incentivized low degree nodes increase.", "In Fig.", "REF , we plot the optimal proportion of individuals that need to be incentivized for achieving the given outreach size." ], [ "Epidemic Size Maximization Problem", "The solution, $\\phi $ , is very similar to the one in problem (REF ), and hence we do not show it here.", "In Fig.", "REF , we plot the size of the epidemic for varying cost budget $C$ .", "As expected, the epidemic size increases with $C$ because higher the budget, the higher the proportion of incentivized individuals.", "However, at some point epidemic size saturates, this is because all nodes have been incentivized and therefore nothing more can be done to increase the outreach size.", "This is verified by Fig.", "REF , the fraction of type 2 nodes hit 1, when $C=3$ ." ], [ "Conclusion and Future Work", "To summarize, we studied the problem of maximizing information spreading in a social networks.", "More specifically, we considered a scenario where individuals are incentivized to vigorously spread the campaign message to their neighbors, and we proposed a mechanism to identify the individuals who should be incetivized.", "Using bond percolation theory we calculated the size of the information epidemic outbreak and the conditions for the occurrence of such outbreaks.", "We then formulated an optimization problem for minimizing the expected cost of incentivizing individuals while providing guarantees on the information epidemic size.", "Although the optimization problem could not be addressed using standard analytical tools, Theorem REF enabled us to compute the global optimum by solving a linear program.", "We believe that our approach of using percolation theory to formulate an optimization problem is the first of its kind.", "For the sake of analytical tractability we assumed an uncorrelated network, however in reality real world social networks have positive degree-degree correlations [22].", "Such networks with positive degree associativity percolate more easily compared to uncorrelated networks [23], [24].", "Therefore, for the problem of minimizing cost the given campaign size could be achieved with a slightly lesser cost, while in the second problem, the theoretical optimal size would be a lower bound and the actual campaign size would be slightly larger than the theoretical.", "Apart from positive degree associativity, social networks are also found to contain community structures [22].", "The presence of communities may slow down information spreading leading to a reduction in the campaign size.", "This may happen as most links point inside the community rather than outside it, thus localizing the information spread [25].", "However, if the network contains high degree nodes that bridge different communities, then incetivizing such nodes may substantially increase the campaign size.", "A similar finding was reported in [26], where authors investigated usefulness of targeted vaccinations on nodes that bridge communities.", "Although SIR models are widely used to model epidemics, they have some limitations.", "They fail to capture the fact that individuals may stop spreading when they perceive that most of their neighbors already known the information.", "This is captured by the Maki-Thompson model [27] which forces the recovery rate to be an increasing function of the number of informed individuals she contacts.", "Thus the recovery rate for an infected node is a function of her degree.", "An SIR process has a fixed recovery rate and hence the current results would approximately hold for an Maki Thompson process on Erdos-Renyi networks, where every node on average has the same degree.", "However, our results for SIR may not generalize for the Maki Thompson spread model on scale free networks.", "High degree nodes may have a higher chance of being connected to informed individuals which may lead them to stop spreading to other uninformed nodes.", "An interesting extension to this problem, which was suggested by the anonymous referee, is to compute a targeted incentivization strategy for two interacting campaigns.", "For example, the campaigner may want to maximize campaign $A$ given that campaign $B$ , which has either run its course or is simultaneously running along with $A$ , either reinforces or hinders campaign $A$ .", "This is an important problem since such interacting campaigns are often observed during parliamentary or presidential elections.", "Although the current results may not shed much light on such questions, we believe that they lay the foundation for investigating such problems which we hope to address in the future." ], [ "Proof of Theorem 3.1 ", "Let $\\langle s_1 \\rangle $ and $\\langle s_2 \\rangle $ be the average number of type 1 and type 2 nodes in the component.", "The expected number of nodes in the component, $<s>$ , is given by: $\\langle s \\rangle &= \\langle s_1 \\rangle + \\langle s_2 \\rangle \\\\&= \\frac{\\partial }{\\partial u_1}\\tilde{J}(u_1,u_2) \\biggr |_{u=1} + \\frac{\\partial }{\\partial u_2}\\tilde{J}(u_1,u_2) \\biggr |_{u=1}$ After differentiating and simplifying, $\\langle s_1 \\rangle $ can be written as: $\\langle s_1 \\rangle = (1-p) + \\langle \\tilde{k}_1 \\rangle \\tilde{H}_1^{^{\\prime }}(1,1) + \\langle \\tilde{k}_2 \\rangle \\tilde{H}_2^{^{\\prime }}(1,1)$ where $\\langle \\tilde{k}_i \\rangle = \\sum \\limits _{\\tilde{k}_1,\\tilde{k}_2}^{\\infty }\\tilde{k}_i \\tilde{P}(\\tilde{k}_1,\\tilde{k}_2)$ and $\\tilde{H}_i^{^{\\prime }}(1,1) = \\frac{\\partial }{\\partial u_1} \\tilde{H}_i(u_1,u_2) \\biggr |_{u = 1}$ $\\tilde{H}_i^{^{\\prime }}(1,1)$ can be obtained by differentiating equation (REF ).", "$\\tilde{H}_1^{^{\\prime }}(1,1)&= 1 + T_1 \\bar{k}_1\\tilde{H}_1^{^{\\prime }}(1,1) + T_2 \\bar{k}_2\\tilde{H}_2^{^{\\prime }}(1,1) \\\\\\tilde{H}_2^{^{\\prime }}(1,1)&= T_1 \\bar{k}_1\\tilde{H}_1^{^{\\prime }}(1,1) + T_2 \\bar{k}_2\\tilde{H}_2^{^{\\prime }}(1,1)$ where $\\bar{k}_i = \\sum \\limits _{k_1,k_2}^{\\infty }k_iQ(k_1,k_2)$ .", "Solving the two simultaneous equations we obtain $\\tilde{H}_1^{^{\\prime }}(1,1)= \\frac{1-T_2\\bar{k}_2}{1 - T_1\\bar{k}_1 - T_2\\bar{k}_2} $ and $\\tilde{H}_2^{^{\\prime }}(1,1)= \\frac{T_1\\bar{k}_1}{1 - T_1\\bar{k}_1 - T_2\\bar{k}_2}$ .", "Substituting in the expression for $\\langle s_1 \\rangle $ we get.", "$\\langle s_1 \\rangle = (1-p) + \\frac{\\langle \\tilde{k}_1\\rangle (1-T_2\\bar{k}_2) + \\langle \\tilde{k}_2\\rangle T_1\\bar{k}_1 }{1 - T_1\\bar{k}_1 - T_2\\bar{k}_2}$ One can similarly show that: $\\langle s_2 \\rangle = p + \\frac{\\langle \\tilde{k}_1 \\rangle T_2\\bar{k}_2 + \\langle \\tilde{k}_2 \\rangle (1-T_1\\bar{k}_1) }{1 - T_1\\bar{k}_1 - T_2\\bar{k}_2}$ Therefore, $\\langle s \\rangle = 1 + \\frac{\\langle \\tilde{k}_1\\rangle + \\langle \\tilde{k}_2\\rangle }{1 - T_1\\bar{k}_1 -T_2 \\bar{k}_2}$ Thus, when $T_1\\bar{k}_1 + T_2\\bar{k}_2 \\ge 1$ , $\\langle s \\rangle $ is no longer finite, it morphs into a giant connected component, or in other words there is an information epidemic outbreak.", "Assume that a giant connected component of exists ($\\tilde{\\nu } \\ge 1$ ).", "For any given node let $z_i$ be the probability that one of its type $i$ links does not lead to the giant connected component.", "The probability that a randomly chosen node is not a part of the GCC is given by $\\psi &= \\sum _{\\tilde{k}_1,\\tilde{k}_2}^{\\infty }z_1^{k_1}z_2^{k_2}P(k_1,k_2) \\\\$ Now, $z_i$ can be written as $Pr($ link is not occupied $)$ + $Pr($ link is occupied and the neighbor is not connected to the GCC$)$ .", "By occupied we mean that the node at the end of the link is a believer.", "Mathematically this can be written as: $z_1 &= 1 - T_1 + T_1\\sum _{k_1,k_2}^{\\infty }z_1^{k_1}z_2^{k_2}Q(k_1,k_2) \\\\z_2 &= 1 - T_2 + T_2\\sum _{k_1,k_2}^{\\infty }z_1^{k_1}z_2^{k_2}Q(k_1,k_2)$ Simplifying we obtain, $(z_1 -1)/T_1 = (z_2 -1)/T_2 $ .", "Let $u := (z_1 -1)/T_1 + 1$ .", "Hence, $z_i = 1 + (u-1)T_i$ .", "Note that $z_i$ is bounded from below by $1-T_i$ and bounded from above by 1, and hence $0 \\le u \\le 1$ .", "Substituting this in above equations we obtain the desired result: $\\psi = \\sum _{k_1,k_2}^{\\infty }(1+(u-1)T_1)^{k_1}(1+(u-1)T_2)^{k_2}P(k_1,k_2)$ where $u$ must satisfy $u = \\sum _{k_1,k_2}^{\\infty }(1+(u-1)T_1)^{k_1}(1+(u-1)T_2)^{k_2}Q(k_1,k_2)$" ], [ "Lemmas required for Theorem 4.1 ", "Lemma REF is used in the proof of Lemma REF and REF .", "Lemma 3 For all $a ,\\ b \\ \\in \\ [0,1] $ and $ k_1 + k_2 \\le n, \\ n \\ \\in \\ Z^+$ and any arbitrary $f:Z\\rightarrow R$ the following is true: $&\\sum _{k_1=0}^{n}\\sum _{k_2=0}^{n-k_1}f(k_1+k_2)k_2 {k_1+ k_2 \\atopwithdelims ()k_2}a^{k_2-1}b^{k_1} \\\\ -& \\sum _{k_1=0}^{n}\\sum _{k_2=0}^{n-k_1}f(k_1+k_2)k_1 {k_1 + k_2 \\atopwithdelims ()k_2}a^{k_2}b^{k_1-1} = 0$ We can switch the indices in the second term, i.e., $&\\sum _{k_1=0}^{n}\\sum _{k_2=0}^{n-k_1}f(k_1+k_2)k_1 {k_1 + k_2 \\atopwithdelims ()k_2}a^{k_2}b^{k_1-1} \\\\&=\\sum _{k_1=0}^{n}\\sum _{k_2=0}^{n-k_1}f(k_1+k_2)k_2 {k_1 + k_2 \\atopwithdelims ()k_2}a^{k_1}b^{k_2-1}$ Hence, $LHS = &\\sum _{k_1=0}^{n}\\sum _{k_2=0}^{n-k_1} f(k_1+k_2)k_2 {k_1+ k_2 \\atopwithdelims ()k_2}\\bigg {(}a^{k_2-1}b^{k_1} - a^{k_1}b^{k_2-1} \\bigg {)} \\nonumber \\\\& = \\sum _{k_1=0}^{n}\\sum _{k_2=0}^{n-k_1} g(k_1,k_2) $ We now count the number of terms in the above equation and show that they are even.", "An expression indexed by a specific $k_1$ and $k_2$ denotes a term, e.g, $g(1,1)$ is a term.", "The total number of terms in the summation $= \\sum \\limits _{i=1}^{n+1}i = \\frac{(n+1)(n+2)}{2}$ .", "Out of those, $n+1$ terms are 0 due to the $k_2$ multiplier ($k_2 = 0$ for $k_1 = 0 \\ \\text{to} \\ n$ ).", "Additionally, when $k_2 = k_1+1$ equation (REF ) is zero.", "The total number of terms when $k_2 = k_1 +1 $ is given by $*{\\frac{n+1}{2}}$ .", "Since, these terms are zero, subtracting out these terms from the total number of terms results in $&\\frac{(n+1)(n+2)}{2} - (n+1) - *{\\frac{n+1}{2}} \\\\&= \\frac{n^2}{2} \\ \\text{for }n \\text{ even}\\\\& = \\frac{(n-1)(n+1)}{2} \\ \\text{for }n \\text{ odd}$ Thus, the remaining terms are even for both $n$ odd and even.", "This allows us to pair the terms.", "Consider one such pairing: the term with indices $k_1, \\ k_2$ are paired with a term with indices $\\hat{k}_1, \\ \\hat{k}_2$ where $\\hat{k}_2 = k_1 + 1$ and $\\hat{k}_1 = k_2 -1$ .", "If we sum these two terms we obtain $& g(k_1,k_2) + g(\\hat{k}_1,\\hat{k}_2) \\\\& = f(k_1+k_2)a^{k_2-1}b^{k_1}\\left( \\frac{k_2(k_1+k_2)!}{k_1!k_2!}", "- \\frac{k_2(k_1+k_2)!}{k_1!k_2!}", "\\right)\\\\&+ f(k_1+k_2)a^{k_1}b^{k_2-1}\\left( \\frac{(k_1+1)(k_1+k_2)!}{(k_2-1)!(k_1+1)!}", "- \\frac{(k_1+1)(k_1+k_2)!}{(k_2-1)!(k_1+1)!}", "\\right) \\\\&=0$ Thus, the summation of the remaining terms is zero, which completes the proof.", "Lemma 4 If $T_2 >T_1$ then $\\tilde{\\nu }$ is strictly increasing with respect to $q$ , i.e, $\\frac{d\\tilde{\\nu }}{dq} > 0, \\ \\forall \\ q \\ \\in \\ [0,1]$ .", "$\\frac{d\\tilde{\\nu }}{dq} = & T_1\\sum _{k_1,k_2}^{\\infty }k_1Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}\\left(k_2q^{k_2-1}r^{k_1} - k_1q^{k_2}r^{k_1-1}\\right) \\\\&+ T_2\\sum _{k_1,k_2}^{\\infty }k_2Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}\\left(k_2q^{k_2-1}r^{k_1} - k_1q^{k_2}r^{k_1-1}\\right)$ where $r = 1-q$ .", "Let, $a_1 &= \\sum _{k_1,k_2}^{\\infty }k_1Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}\\left(k_2q^{k_2-1}r^{k_1} - k_1q^{k_2}r^{k_1-1}\\right) \\\\a_2 &= \\sum _{k_1,k_2}^{\\infty }k_2Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}\\left(k_2q^{k_2-1}r^{k_1} - k_1q^{k_2}r^{k_1-1}\\right)$ Adding $a_1$ and $a_2$ we get $a_1 + a_2 = \\sum _{k_1,k_2}^{\\infty }(k_1+k_2)Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}\\left(k_2q^{k_2-1}r^{k_1} - k_1q^{k_2}r^{k_1-1}\\right)$ Since $N$ is large but finite, $Q(k_1+k_2)=0$ for $k_1+k_2 > k_{max}$ , where $k_{max}$ is the maximum degree.", "From Lemma REF , $a_1 + a_2 = 0$ .", "Now we prove that $a_2 > 0$ .", "Let $k_1 + k_2 = m$ .", "$a_2 = \\sum _{m=1}^{k_{max}}Q(m)\\bigg [ \\frac{1}{q}\\sum _{k_2=0}^{m}k_2^2{m \\atopwithdelims ()k_2}q^{k_2}r^{m-k_2} -\\frac{1}{r}\\sum _{k_1=0}^{m}k_1(m-k_1){m \\atopwithdelims ()k_1}q^{m-k_1}r^{k_1}\\bigg ]$ The summations are the second moments of a binomial random variable.", "$E \\left[X^2 \\right] = Var\\left[X\\right] + E \\left[X \\right]^2$ , $E[X] = mq, \\ Var[X] = mqr$ .", "$a_2 &= \\sum _{m=1}^{k_{max}}Q(m)\\bigg [\\frac{1}{q}(mqr + m^2q^2) -\\frac{1}{r}(m^2r - mqr-m^2r^2)\\bigg ] \\\\&= \\sum _{m=1}^{k_{max}}Q(m)m \\\\&>0$ Since $T_2>T_1$ , $T_1a_1 + T_2a_2 > 0$ , which completes the proof.", "Lemma 5 For $\\psi \\ \\in \\ (0,1)$ , if $T_2 >T_1$ then $\\psi $ is strictly decreasing with respect to $q$ , i.e, $\\frac{d\\psi }{dq} <0, \\ \\forall \\ q \\ \\in \\ [0,1]$ .", "Let, $\\psi = g(u^*,q)$ where $u^*$ is the solution of the fixed point equation $u = f(u,q)$ .", "$g(u^*,q) &= \\sum _{k_1,k_2}^{\\infty }\\alpha ^{k_1}\\beta ^{k_2}P(k_1+k_2) {k_1+k_2 \\atopwithdelims ()k_2}q^{k2}(1-q)^{k_1} \\\\f(u,q)&=\\sum _{k_1,k_2}^{\\infty }\\alpha ^{k_1}\\beta ^{k_2}Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}q^{k2}(1-q)^{k_1}$ where $\\alpha = 1+(u^*-1)T_1$ and $\\beta = 1+(u^*-1)T_2$ .", "We first show that the solution to the fixed point equation is strictly decreasing with $q$ .", "Let us consider the behavior of the R.H.S of the fixed point equation, $f(u,q)$ , w.r.t.", "$q$ .", "Now $\\frac{\\partial f(u,q)}{\\partial q} =&\\sum _{k_1,k_2}^{\\infty }\\alpha ^{k_1}\\beta ^{k_2}Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}\\left(k_2q^{k2-1}r^{k_1}-k_1q^{k2}r^{k_1-1}\\right) \\\\&=\\beta \\sum _{k_1,k_2}^{\\infty }Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}k_2(\\beta q)^{k2-1}(\\alpha r)^{k_1} \\\\&-\\alpha \\sum _{k_1,k_2}^{\\infty }Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}k_1(\\beta q)^{k_2}(\\alpha r)^{k_1-1}$ From Lemma REF , $&\\sum _{k_1,k_2}^{\\infty }Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}k_2(\\beta q)^{k2-1}(\\alpha r)^{k_1} \\\\&-\\sum _{k_1,k_2}^{\\infty }Q(k_1+k_2){k_1+k_2 \\atopwithdelims ()k_2}k_1(\\beta q)^{k_2}(\\alpha r)^{k_1-1} \\\\& = 0$ Now $\\alpha > \\beta $ because $T_2 > T_1$ , which implies $ \\frac{\\partial f(u,q)}{\\partial q} <0$ .", "We use the Implicit Function Theorem for computing the sign of $\\frac{\\partial u^*}{\\partial q}$ .", "Let $h(u,q) = f(u,q) - u$ .", "According to the Implicit Function Theorem when $h(u,q) = 0$ $\\frac{du}{dq} = -\\frac{\\frac{\\partial }{\\partial q}h(u,q)}{\\frac{\\partial }{\\partial u}h(u,q)}$ We now show that the derivative exists and is greater than zero.", "The numerator $\\frac{\\partial h(u,q)}{\\partial q} = \\frac{\\partial f(u,q)}{\\partial q} < 0 $ .", "The denominator is given by $ \\frac{\\partial h(u,q)}{\\partial u} = \\frac{\\partial f(u,q)}{\\partial u} - 1 $ Claim: $\\frac{\\partial f(u,q)}{\\partial u} < 1$ when $h(u,q)=0$ .", "We prove this by contradiction.", "Assume the contrary, i.e., $\\frac{\\partial f(u,q)}{\\partial u} \\ge 1$ .", "It can be easily shown that $\\frac{\\partial f(u,q)}{\\partial u} > 0 $ and $\\frac{\\partial ^2 f(u,q)}{\\partial u^2} > 0$ for all $u, \\ q \\ \\in [0,1]$ .", "Thus $f$ is a convex function in $u$ for any fixed $q$ .", "Also $f(0,q) > 0$ for all $q \\ \\in [0,1]$ .", "Now $h(u,q) = 0$ implies that $u=f(u,q)$ , or in other words the curve $f(u,q)$ intersects the line passing through the origin with slope 1.", "Since we assumed $\\frac{\\partial f(u,q)}{\\partial u} \\ge 1$ , i.e., the derivative of $f(u,q)$ is greater than equal to 1 at the intersection, and since $f(u,q)$ is monotone increasing in $u$ , the curve $f(u,q)$ will never again intersect the line passing through the origin with slope 1.", "Therefore the equation $u=f(u,q)$ has a unique fixed point.", "However, this is a contradiction since $u=1$ is always a fixed point and since $\\psi > 0$ there is another fixed point less than 1.", "Hence, $\\frac{\\partial f(u,q)}{\\partial u} \\ge 1$ is impossible, and therefore $\\frac{\\partial f(u,q)}{\\partial u} < 1$ .", "Thus, the derivative exists and is less than 0 as $\\frac{\\partial h(u,q)}{\\partial q} <0$ and $ \\frac{\\partial }{\\partial u}h(u,q)<0$ .", "Since we assumed $h(u,q)=0$ , the derivative $\\frac{du}{dq}$ is the derivative of the fixed point w.r.t $q$ , i.e., it can be represented as $\\frac{du^*}{dq}$ , where $u^*$ is the fixed point The function $g(u^*,q)$ has the same structure as the function $f(u,q)$ , and hence using the same procedure it can be shown that $\\frac{\\partial g(u,q)}{\\partial q} < 0$ .", "The total derivative $\\frac{d\\psi }{dq}$ is given by: $\\frac{d\\psi }{dq} = \\frac{\\partial g}{\\partial q} + \\frac{\\partial g}{\\partial u}\\frac{du^*}{dq}$ Since all the terms on the right hand side of the above equation are negative ($g$ is a non decreasing function of u), $\\frac{d\\psi }{dq} < 0$ ." ] ]
1606.05209
[ [ "Power Series with Coefficients from a Finite Set" ], [ "Abstract We prove in this paper that a multivariate D-finite power series with coefficients from a finite set is rational.", "This generalizes a rationality theorem of van der Poorten and Shparlinski in 1996." ], [ "Introduction ", "In his thesis [14], Hadamard began the study of the relationship between the coefficients of a power series and the properties of the function it represents, especially its singularities and natural boundaries.", "Two special cases of the problem have been extensively studied: one is on power series with integer coefficients and the other is on power series with finitely many distinct coefficients.", "In the first case, Fatou [12] in 1906 proved a lemma on rational power series with integer coefficients, which is now known as Fatou's lemma [31].", "The next celebrated result is the Pólya-Carlson theorem, which asserts that a power series with integer coefficients and of radius of convergence 1 is either rational or has the unit circle as its natural boundary.", "This theorem was first conjectured in 1915 by Pólya [23] and later proved in 1921 by Carlson [6].", "Several extensions of the Pólya-Carlson theorem have been presented in [24], [22], [13], [29], [20], [33], [1].", "In the second case, Fatou [12] was also the first to investigate power series with coefficients from a finite set.", "The study was continued by Pólya [23] in 1916, Jentzsch [15] in 1917, Carlson [5] in 1918 and finally Szegő [34], [35] in 1922 settled the question by proving the following beautiful theorem (see [25] and [9] for its proof and related results).", "Theorem 1.1 (Szegő, 1922) Let $F = \\sum f(n) x^n$ be a power series with coefficients from a finite values of $ {\\mathbb {C}}$ .", "If $F$ is continuable beyond the unit circle then it is a rational function of the form $F = P(x)/(1-x^m)$ , where $P$ is a polynomial and $m$ a positive integer.", "Szegő's theorem was generalized in 1945 by Duffin and Schaeffer [10] by assuming a weaker condition that $f$ is bounded in a sector of the unit circle.", "In 2008, P. Borwein et al.", "in [4] gave a shorter proof of Duffin and Schaeffer's theorem.", "By using Szegő's theorem, van der Poorten and Shparlinski proved the following result [36].", "Theorem 1.2 (van der Poorten and Shparlinski, 1996) Let $F = \\sum f(n) x^n$ be a power series with coefficients from a finite values of $ {\\mathbb {Q}}$ .", "If $f(n)$ satisfies a linear recurrence equation with polynomial coefficients, then $F$ is rational.", "A univariate sequence $f: {\\mathbb {N}}\\rightarrow K$ is P-recursive if it satisfies a linear recurrence equation with polynomial coefficients in $K[n]$ .", "A power series $F = \\sum f(n)x^n$ is D-finite if it satisfies a linear differential equation with polynomial coefficients in $K[x]$ .", "By [30], a sequence $a(n)$ is P-recursive if and only if the power series $F := \\sum f(n)x^n$ is D-finite.", "The notion of D-finite power series can be generalized to the multivariate case (see Definition REF ).", "Our main result is the following multivariate generalization of Theorem REF .", "Theorem 1.3 Let $K$ be a field of characteristic zero, and let $\\Delta $ be a finite subset of $K$ .", "Suppose that $f:\\mathbb {N}^d\\rightarrow \\Delta $ with $d\\ge 1$ is such that $F(x_1,\\ldots ,x_d):=\\sum _{(n_1,\\ldots ,n_d)\\in \\mathbb {N}^d} f(n_1,\\ldots ,n_d)x_1^{n_1}\\cdots x_d^{n_d} \\in K[[x_1,\\ldots ,x_d]]$ is $D$ -finite.", "Then $F$ is rational.", "We note that a multivariate rational power series $F(x_1, \\cdots ,x_d) = \\sum f(n_1,\\ldots ,n_d)x_1^{n_1}\\cdots x_d^{n_d}$ with all coefficients in $\\lbrace 0,1\\rbrace $ has a very restricted form.", "In particular, the set $E$ of $(n_1,\\ldots ,n_d)\\in \\mathbb {N}^d$ for which $f(n_1,\\ldots ,n_d)\\ne 0$ is semilinear; that is there exist $n\\in {\\mathbb {N}}$ and finite subsets $V_0, \\ldots V_n$ of $ {\\mathbb {N}}^d$ , and $b_1, \\ldots , b_n\\in {\\mathbb {N}}^d$ such that $E = V_0 \\bigcup \\left\\lbrace \\bigcup _{i=1}^n \\left(b_i + \\sum _{v\\in V_i} v\\cdot \\mathbb {N}\\right)\\right\\rbrace .$ Although this result is known, we are unaware of a reference and give a proof of this fact in Proposition REF .", "The remainder of this paper is organized as follows.", "The basic properties of D-finite power series are recalled in Section .", "The proof of Theorem REF is given in Section .", "In Section , we present several applications of our main theorem on generating functions over nonnegative points on algebraic varieties." ], [ "D-finite power series", "Throughout this paper, we let $ {\\mathbb {N}}$ denote the set of all nonnegative integers.", "Let $K$ be a field of characteristic zero and let $K( { \\mathbf {x}})$ be the field of rational functions in several variables $ { \\mathbf {x}}= x_1, \\ldots , x_d$ over $K$ .", "By $K[[ { \\mathbf {x}}]]$ we denote the ring of formal power series in $ { \\mathbf {x}}$ over $K$ and by $K(( { \\mathbf {x}}))$ we denote the field of fractions of $K[[ { \\mathbf {x}}]]$ .", "For two power series $F = \\sum f(i_1, \\ldots , i_d)x_1^{i_1} \\cdots x_d^{i_d}$ and $G = \\sum g(i_1, \\ldots , i_d)x_1^{i_1} \\cdots x_d^{i_d}$ , the Hadamard product of $F$ and $G$ is defined by $F {\\odot }G = \\sum f(i_1, \\ldots , i_d)g(i_1, \\ldots , i_d) x_1^{i_1} \\cdots x_d^{i_d}.$ Let $D_{x_1}, \\ldots , D_{x_d}$ denote the derivations on $K(( { \\mathbf {x}}))$ with respect to $x_1, \\ldots , x_d$ , respectively.", "Definition 2.1 ([17]) A formal power series $F(x_1, \\ldots , x_d) \\in K[[ { \\mathbf {x}}]]$ is said to be D-finite over $K( { \\mathbf {x}})$ if the set of all derivatives $D_{x_1}^{i_1}\\cdots D_{x_d}^{i_d}(F)$ with $i_j \\in {\\mathbb {N}}$ span a finite-dimensional $K( { \\mathbf {x}})$ -vector subspace of $K(( { \\mathbf {x}}))$ .", "Equivalently, for each $i\\in \\lbrace 1, \\ldots , d\\rbrace $ , $F$ satisfies a nontrivial linear partial differential equation of the form $\\left\\lbrace p_{i, m_i} D_{x_i}^{m_i} + p_{i, m_1-1}D_{x_i}^{m_i-1} + \\cdots + p_{i, 0}\\right\\rbrace F = 0 \\quad \\text{with~$p_{i, j}\\in K[ { \\mathbf {x}}]$}.$ The notion of D-finite power series was first introduced in 1980 by Stanley [30], and has since become ubiquitous in algebraic combinatorics as an important part of the study of generating functions (see [32]).", "We recall some closure properties of this class of power series.", "Proposition 2.2 ([18]) Let $ {\\mathcal {D}}$ denote the set of all D-finite power series in $K[[ { \\mathbf {x}}]]$ .", "Then (i) $ {\\mathcal {D}}$ forms a subalgebra of $K[[ { \\mathbf {x}}]]$ , i.e., if $F, G \\in {\\mathcal {D}}$ and $\\alpha , \\beta \\in K$ , then $\\alpha F + \\beta G \\in {\\mathcal {D}}$ and $FG\\in {\\mathcal {D}}$ .", "(ii) $ {\\mathcal {D}}$ is closed under the Hadamard product, i.e., if $F, G \\in {\\mathcal {D}}$ , then $F {\\odot }G \\in {\\mathcal {D}}$ .", "(iii) If $F(x_1, \\ldots , x_d)$ is D-finite, and $\\alpha _1(y_1, \\ldots , y_d),\\ldots ,\\alpha _d(y_1,\\ldots ,y_d)\\in K[[y_1,\\ldots ,y_d]]$ are algebraic over $K(y_1, \\ldots , y_d)$ and the substitution makes sense, then $F(\\alpha _1, \\ldots , \\alpha _d)$ is also D-finite over $K(y_1, \\ldots , y_d)$ .", "In particular, if $F(x_1, \\ldots , x_d)$ is D-finite and the evaluation of $F$ at $x_d=1$ makes sense, then $F(x_1, \\ldots , x_{d-1}, 1)$ is D-finite.", "The coefficients of a D-finite power series are highly structured.", "In the univariate case, a power series $f = \\sum a(n)x^n$ is D-finite if and only if the sequence $a(n)$ is P-recursive, i.e., it satisfies a linear recurrence equation with polynomial coefficients in $n$  [30].", "The structure in the multivariate case is much more profound, which was explored by Lipshitz in [18].", "We continue this exploration to study the position of nonzero coefficients.", "To this end, we recall a notion of size in the semigroup $( {\\mathbb {N}}, +)$ , introduced by Bergelson et al.", "[2] for general semigroups.", "A subset $S\\subseteq {\\mathbb {N}}$ is syndetic if there is some positive integer $C$ such that if $n\\in S$ then $n+i\\in S$ for some $i\\in \\lbrace 1,\\ldots ,C\\rbrace $ .", "Lemma 2.3 Let $K$ be a field of characteristic zero and let $ G(x_1,\\ldots ,x_d)=\\sum _{(n_1,\\ldots ,n_d)\\in \\mathbb {N}^d} g(n_1,\\ldots ,n_d)x_1^{n_1}\\cdots x_d^{n_d}\\in K[[ { \\mathbf {x}}]]$ be a $D$ -finite power series over $K( { \\mathbf {x}})$ .", "Then the set $\\lbrace n\\in \\mathbb {N} \\mid \\exists (n_1,\\ldots ,n_{d-1})\\in \\mathbb {N}^{d-1}~{\\rm such~that}~g(n_1,\\ldots ,n_{d-1},n)\\ne 0\\rbrace $ is either finite or syndetic.", "We let $L$ denote the field of fractions of $K[[x_1,\\ldots ,x_{d-1}]]$ .", "Then we may regard $G$ as a power series in $L[[x_d]]$ and it is $D$ -finite in $x_d$ over $L(x_d)$ and it is straightforward to see that the lemma reduces to the univariate case.", "Thus we now assume that $G(x)=\\sum g(n)x^n \\in L[[x]]$ is $D$ -finite.", "Then there exist $m\\ge 1$ , distinct nonnegative integers $a_1=0, \\ldots ,a_m$ , and nonzero polynomials $P_1,\\ldots ,P_m\\in L[z]$ such that $\\sum _{j=1}^m P_j(n) g(n+a_j)=0$ for all sufficiently large $n$ .", "Then there is some $M$ such that $P_1(n)\\cdots P_m(n)\\ne 0$ for $n>M$ .", "If $m=1$ then we then see that $g(n)=0$ for $n>M$ .", "Thus we assume that $m>1$ .", "Then if $n>M$ and $g(n)$ is nonzero then $g(n+a_j)$ is nonzero for some $1<j\\le m$ and so we see that the set of $n$ for which $g(n)$ is nonzero is syndetic." ], [ "Proof of the main theorem", "The proof of Theorem REF by van der Poorten and Shparlinski is based on the fact that any univariate D-finite power series represents an analytic function with only finitely many poles [30], so it is impossible to have the unit circle as its natural boundary.", "Then their result follows from Szegő's theorem.", "The singularities of analytic functions represented by multivariate D-finite power series are much more involved.", "It is not known how to extend Szegő's theorem to the multivariate case.", "Thus new ideas are needed in order to generalize Theorem REF to the multivariate case.", "Before the proof of our main theorem, we first prove a lemma about finitely generated $ {\\mathbb {Z}}$ -algebras.", "Lemma 3.1 Let $R$ be a finitely generated $\\mathbb {Z}$ -algebra that is an integral domain of characteristic zero and suppose that $x$ is a nonzero element of $R$ .", "Then there exists a finite set of prime numbers $\\lbrace p_1,\\ldots ,p_m\\rbrace $ such that if $n$ is a positive integer such that $x\\in nR$ then $n$ is an element of the multiplicative semigroup of $\\mathbb {N}$ generated by $p_1,\\ldots ,p_m$ .", "Let $U$ denote the group of units of $R$ .", "By a result of Roquette [26] (or see [16]) we have that $U$ is a finitely generated abelian group and so $U_0$ , the subgroup of $U$ generated by the rational numbers in $U$ is a finitely generated subgroup of $\\mathbb {Q}^*$ .", "In particular, there exist prime numbers $q_1,\\ldots , q_t$ such that every positive rational number in $U$ is in the multiplicative subgroup of $\\mathbb {Q}^*$ generated by $q_1,\\ldots ,q_t$ .", "Thus if $x$ is a unit and $x\\in nR$ then $n$ is an integer unit of $R$ and hence in the semigroup generated by $q_1,\\ldots ,q_t$ .", "Hence we may assume that $x$ is not a unit.", "Let $Q_1\\cap \\cdots \\cap Q_d$ be the primary decomposition of the ideal $xR$ and let $P_j$ denote the radical of $Q_j$ for $j=1,\\ldots ,d$ .", "Since each $P_j$ is (proper) prime ideal, it follows that there is at most one prime number in $P_j$ for $j=1,\\ldots ,d$ .", "We now let $\\lbrace p_1,\\ldots ,p_m\\rbrace $ denote the union of the prime numbers in $P_1,\\ldots ,P_d$ and $\\lbrace q_1,\\ldots ,q_t\\rbrace $ .", "Now suppose that $x\\in pR$ with $p$ a prime number.", "We claim that $p$ must be in the finite set $\\lbrace p_1,\\ldots ,p_m\\rbrace $ .", "To see this, we have $x=pr$ for some $r\\in R$ and so for each $i\\in \\lbrace 1,\\ldots ,d\\rbrace $ we have $pr\\in Q_i$ .", "Since $Q_i$ is primary that gives that either $r\\in Q_i$ or there is some $k\\ge 1$ such that $p^k\\in Q_i$ .", "Now if $r\\in Q_i$ for every $i\\in \\lbrace 1,\\ldots ,d\\rbrace $ then we see that $r$ is in the intersection of the $Q_i$ and hence in $xR$ .", "This then gives that $r=xa$ for some $a\\in R$ and so we have $x=pr=pax.$ Since $x$ is nonzero and $R$ is an integral domain, this then gives that $p\\in U$ and so by the above remarks we must have $p\\in \\lbrace q_1,\\ldots ,q_t\\rbrace \\subseteq \\lbrace p_1,\\ldots ,p_m\\rbrace $ .", "Alternatively, we have that there is some $i$ such that $p^k\\in Q_i$ for some $k\\ge 1$ .", "Then $p\\in P_i$ and so $p\\in \\lbrace p_1,\\ldots ,p_m\\rbrace $ .", "To finish the proof, observe that if $n$ is not in the multiplicative semigroup generated by $p_1,\\ldots ,p_m$ then there is some prime $q\\notin \\lbrace p_1,\\ldots ,p_m\\rbrace $ such that $q$ divides $n$ and so $x\\notin nR$ since $nR \\subseteq qR$ and $x\\notin qR$ .", "The result follows.", "We prove this by induction on $d$ .", "When $d=0$ , $F$ is constant and there is nothing to prove.", "We now suppose that the result holds whenever $d<k$ and we consider the case when $d=k$ .", "Since $F$ is $D$ -finite, we have that $F(x_1,\\ldots ,x_k)$ satisfies a nontrivial linear differential equation of the form $\\sum _{j=0}^{\\ell } P_j(x_1,\\ldots ,x_k) \\partial _{x_k}^j F = 0,$ where $P_0,\\ldots ,P_{\\ell }$ are polynomials in $K[x_1, \\ldots , x_k]$ .", "Translating this into a relation for the coefficients of $F$ , we see that there exists some positive integer $N$ and polynomials $Q_{a_1,\\ldots ,a_k}(t)\\in K[t]$ for $(a_1,\\ldots ,a_k)\\in \\lbrace -N,\\ldots ,N\\rbrace ^k$ , not all zero, such that $\\sum _{-N\\le a_1,\\ldots ,a_k\\le N} Q_{a_1,\\ldots ,a_k}(n_k) f(n_1-a_1,\\ldots ,n_k-a_k)=0$ for all $(n_1,\\ldots ,n_k)\\in \\mathbb {N}^k$ , where we take $f(i_1,\\ldots ,i_k)=0$ if some $i_j$ is negative.", "By dividing our polynomials $Q_{a_1,\\ldots ,a_k}(t)$ by $t^a$ for some nonnegative integer $a$ if necessary, we may assume that $q(a_1,\\ldots ,a_k):=Q_{a_1,\\ldots ,a_k}(0)$ is nonzero for some $(a_1,\\ldots ,a_k)\\in \\lbrace -N,\\ldots ,N\\rbrace ^k$ .", "We now let $R$ denote the $\\mathbb {Z}$ -subalgebra of $K$ generated by $\\Delta $ and by the coefficients of $Q_{a_1,\\ldots ,a_k}(t)\\in K[t]$ with $(a_1,\\ldots ,a_k)\\in \\lbrace -N,\\ldots ,N\\rbrace ^k$ .", "Then $R$ is finitely generated.", "By construction, we have $\\sum _{-N\\le a_1,\\ldots ,a_k\\le N} q(a_1,\\ldots ,a_k) f(n_1-a_1,\\ldots ,n_k-a_k)\\in n_kR$ for all $(n_1,\\ldots ,n_k)\\in \\mathbb {N}^k$ .", "Now let $\\Gamma $ denote the set of all numbers of the form $\\sum _{-N\\le a_1,\\ldots ,a_k\\le N} q(a_1,\\ldots ,a_k) s(a_1,\\ldots ,a_k)$ with $s(a_1,\\ldots ,a_k)\\in \\Delta \\cup \\lbrace 0\\rbrace $ .", "Then $\\Gamma $ is a finite set.", "By Lemma REF , there is a finite set of prime numbers $p_1,\\ldots ,p_m$ such that for each nonzero $x\\in \\Gamma $ we have that if $n$ is a positive integer with $x\\in nR$ then $n$ is in the semigroup generated by $p_1,\\ldots ,p_m$ .", "In particular, $\\sum _{-N\\le a_1,\\ldots ,a_k\\le N} q(a_1,\\ldots ,a_k) f(n_1-a_1,\\ldots ,n_k-a_k)=0$ whenever $n_k$ is not in the multiplicative semigroup generated by $p_1,\\ldots ,p_m$ .", "Equivalently, $G(x_1,\\ldots ,x_k):=F(x_1,\\ldots , x_k) \\left( \\sum _{0\\le a_1,\\ldots ,a_k\\le N} q(a_1,\\ldots ,a_k) x_1^{a_1}\\ldots x_k^{a_k}\\right)x_1^N\\cdots x_k^N$ has the property that $g(n_1,\\ldots ,n_k)=0$ whenever $n_k\\ge N$ and $n_k-N$ is not in the semigroup generated by $p_1,\\ldots ,p_m$ , where $g(n_1,\\ldots ,n_k)$ denotes the coefficient of $x_1^{n_1}\\cdots x_k^{n_k}$ in $G(x_1,\\ldots ,x_k)$ .", "Since $G$ is just $F$ multiplied by a polynomial, $G(x_1,\\ldots ,x_k)$ is $D$ -finite by Proposition REF  (i); moreover, all coefficients of $G$ lie in the finite set $\\Gamma $ .", "But now Lemma REF gives that there is some positive integer $M$ such that $g(n_1,\\ldots ,n_k)=0$ whenever $n_k>M$ since a translate of the multiplicative semigroup generated by $p_1,\\ldots ,p_m$ cannot be syndetic.", "Thus we have $G=\\sum _{i=0}^M G_i(x_1,\\ldots ,x_{k-1})x_k^i$ for some power series $G_0,\\ldots ,G_M\\in K[[x_1,\\ldots ,x_{k-1}]]$ .", "Then for $i\\in \\lbrace 0,\\ldots ,M\\rbrace $ , we have that $G_i x_k^i$ is the Hadamard product of $G$ with $x_k^i \\prod _{j=1}^{k-1} (1-x_j)^{-1}$ and so each $G_i x_k^i$ is $D$ -finite by Proposition REF  (ii).", "Then specializing $x_k=1$ gives each $G_i$ is $D$ -finite by Proposition REF  (iii).", "Since each $G_i$ has coefficients in a finite set, we see by the induction hypothesis that each $G_i$ is rational and so $G$ is rational.", "But this now gives that $F$ is rational by our definition of $G$ , completing the proof." ], [ "Generating functions over nonnegative integer points on algebraic varieties", "Let $V\\subseteq {\\mathbb {A}}_{K}^d$ be an affine algebraic variety over an algebraically closed field $K$ of characteristic zero.", "We define the generating function over nonnegative integer points on $V$ by $ F_V(x_1, \\ldots , x_d) := \\sum _{(n_1, \\ldots , n_d)\\in V \\cap {\\mathbb {N}}^d}x_1^{n_1} \\cdots x_d^{n_d}.$ Then one can ask the following questions about the properties of $F_V$ that often reflect the global geometric structure of $V$ : When $F_V$ is zero?", "This is Hilbert Tenth Problem when $K$ is the field of rational numbers.", "In 1970, Matiyasevich [21], [8] proved that this problem is undecidable.", "When $F_V$ is a polynomial?", "If so, $V$ has only finitely many nonnegative integer points.", "Siegel's theorem on integral points answers this question for a smooth algebraic curve C of genus $g\\ge 1$ defined over a number field $K$  [3].", "When $F_V$ is a rational function?", "This is always true when the variety $V$ is defined by linear polynomials with integer coefficients [31].", "When $F_V$ is $D$ -finite?", "By our main theorem, we see that this question is the same as question (3), by taking $f(n_1, \\ldots , n_d) =1$ if $(n_1, \\ldots , n_d) \\in V\\cap {\\mathbb {N}}^d$ and $f(n_1, \\ldots , n_d)=0$ otherwise (see Corollary REF ).", "When $F_V$ satisfies an algebraic differential equation?More precisely, we say that a power series $F(x_1, \\ldots , x_d)\\in K[[x_1, \\ldots , x_d]]$ is differentially algebraic if the transcendence degree of the field generated by all of the derivatives $D_{x_1}^{i_1}\\cdots D_{x_d}^{i_d}(F)$ with $i_j \\in {\\mathbb {N}}$ over $K(x_1, \\ldots , x_d)$ is finite.", "If a power series is not differentially algebraic, then it is called transcendentally transcendental.", "For a nice survey on transcendentally transcendental functions, see Rubel [27].", "Corollary 4.1 Let $V\\subseteq \\mathbb {A}^d_K$ be an affine variety over an algebraically closed field $K$ of characteristic zero.", "Then the power series $ F_V(x_1, \\ldots , x_d) := \\sum _{(n_1,\\ldots ,n_d)\\in V\\cap \\mathbb {N}^d} x_1^{n_1}\\cdots x_d^{n_d}$ is $D$ -finite if and only if it is rational.", "To show an application of this corollary, let us consider the linear system $A { \\mathbf {x}}= 0$ , where $A$ is a $d\\times m$ matrix with integer entries.", "Let $E$ be the set of all vectors $(n_1, \\ldots , n_d) \\in {\\mathbb {N}}^d$ such that $A { \\mathbf {x}}=0$ .", "We now give a proof of the following classical theorem in enumerative combinatorics.", "Theorem 4.2 (Theorem 4.6.11 in [31]) The generating function $F_E(x_1, \\ldots , x_d) := \\sum _{{(n_1, \\ldots , n_d)\\in E}} x_1^{n_1}\\cdots x_d^{n_d}$ represents a rational function of $x_1, \\ldots , x_d$ .", "By Corollary REF , it suffices to show that $F_E$ is D-finite.", "We first recall a fact proved by Lipshitz in [17] that if the power series $G( { \\mathbf {x}}) = \\sum g(n_1, \\ldots , n_d)x_1^{n_1} \\cdots x_d^{n_d}$ is D-finite and $C\\subseteq {\\mathbb {N}}^d$ is the set of elements of $\\mathbb {N}^d$ satisfying a finite set of inequalities of the form $\\sum a_i n_i + b\\ge 0$ , where the $a_i, b\\in {\\mathbb {Z}}$ , then the power series $H( { \\mathbf {x}}) := \\sum _{(n_1, \\ldots n_k)\\in C} g(n_1, \\dots , n_d)x_1^{n_1}\\cdots x_d^{n_d}$ is D-finite.", "Note that $R(x_1, \\ldots , x_d) := \\sum x_1^{n_1}\\cdots x_d^{n_d} = 1/\\prod _{i=1}^d(1-x_i)$ is D-finite and any equality $\\sum a_i n_i=0$ is equivalent to two inequalities $\\sum a_i n_i \\ge 0$ and $\\sum (-a_i) n_i \\ge 0$ .", "Then the D-finiteness of $F_E$ follows from the fact.", "We now derive some properties of an algebraic variety $E$ from the generating function $F_E$ when $d=2$ .", "We first prove a basic result that is probably well-known, but for which we are unaware of a reference.", "Proposition 4.3 Let  $F(x_1, \\cdots ,x_d) = \\sum f(n_1,\\ldots ,n_d)x_1^{n_1}\\cdots x_d^{n_d}\\in {\\mathbb {Q}}[[x_1,\\ldots ,x_d]]$ with $f(n_1,\\ldots ,n_d)\\in \\lbrace 0, 1\\rbrace $ for all $(n_1,\\ldots ,n_d)\\in \\mathbb {N}^d$ .", "Then $F$ is rational if and only if the support set $E := \\lbrace (n_1,\\ldots ,n_d)\\in {\\mathbb {N}}^d\\mid f(n_1, \\ldots ,n_d) \\ne 0\\rbrace $ of $F$ is semilinear (see Equation (REF ) for the definition of semilinearity).", "The sufficiency follows from Theorem REF .", "For the other direction assume that $F(x_1, \\ldots ,x_d)$ is rational.", "Then the sequence $f: {\\mathbb {N}}^d \\rightarrow \\lbrace 0, 1\\rbrace $ has a rational generating function over any finite field $ {\\mathbb {F}}_p$ , where $p$ is a prime number.", "(This follows from the fact that we can express the generating series for $F$ of the form $P/Q$ with $P$ and $Q$ polynomials in which the gcd of the collection of coefficients of $P$ and $Q$ is 1.)", "By Salon's theorem [28], which is an multi-dimensional extension of the theorem by Christol, Kamae, Mendès France, and Rauzy [7], the sequence $f: {\\mathbb {N}}^d \\rightarrow \\lbrace 0, 1\\rbrace $ is $p$ -automatic for every prime number $p$ .", "Then the Cobham-Semenov theorem [11] implies that the support set $E$ of $f$ is semilinear.", "We now use this result in the special case when $d=2$ .", "Theorem 4.4 Let $p(x, y)\\in K[x, y]$ be a nonzero polynomial satisfying that the generating function $F_p(x, y) := \\sum _{\\begin{array}{c}(n, m)\\in {\\mathbb {N}}^2 \\\\ p(n, m)=0\\end{array}} x^ny^m$ is rational.", "Then $p = c\\cdot f\\cdot g$ , where $c\\in K$ is a constant, $f$ is a product of linear polynomials in $x$ and $y$ with integer coefficients and $g$ has only finite roots in $ {\\mathbb {N}}^2$ .", "Let $p = p_1 \\cdots p_r$ with $p_i$ irreducible over $K$ .", "Assume that $p_1, \\ldots , p_m$ have only finitely many zeros in $ {\\mathbb {N}}^2$ and that $p_i$ with $i>m$ has infinitely many roots in $ {\\mathbb {N}}^2$ .", "Then let $g = p_1\\cdots p_m$ .", "We show that $p_{m+1}, \\ldots , p_{r}$ are, up to scalar multiplication, polynomials of the form $ax+by+c$ with $a, b, c\\in {\\mathbb {Z}}$ .", "By Proposition REF , the set $E$ of all nonnegative points $(n, m)$ on the curve $p(x, y)=0$ is semilinear.", "Now suppose that $E$ is infinite.", "Then if the subset $V_i$ in (REF ) is not contained in a line in $ {\\mathbb {Z}}^2$ through the origin, then the set $b_i + \\sum _{v\\in V_i} v\\cdot \\mathbb {N}$ is Zariski dense in the plane, which is impossible since $E$ is contained in the zero set of a nonzero polynomial.", "Thus we see that after refining our decomposition of $E$ if necessary, we may assume that each $|V_i| = 1$ for $i>0$ .", "Let $q$ be any irreducible factor of $p$ having infinitely many zeros in $ {\\mathbb {N}}^2$ .", "Then there is some $V_i = \\lbrace v\\rbrace \\subseteq {\\mathbb {N}}^2$ with $i>0$ , such that $q(b_i + vn)=0$ for infinitely many $n\\in {\\mathbb {N}}$ .", "Write $b_i = (c, d)$ and $v = (a, b)$ .", "Then $q(c+an, d+bn)=0$ for infinitely many $n\\in {\\mathbb {N}}$ and so $q(c+at, d+bt)=0$ for all $t\\in K$ .", "Hence the linear polynomial $ay-bx-(da-cb)$ divides $q$ .", "Since $q$ is irreducible over $K$ , then $q= \\lambda (ay-bx-(da-cb))$ for some constant $\\lambda \\in K$ .", "This completes the proof.", "The theorem as above cannot be extended to the case when $d>2$ as shown in the following example.", "Example 4.5 Let $p = x-y + 2z^2 + zy^2$ .", "We claim that $E := \\lbrace (n, n, 0)\\mid n\\in {\\mathbb {N}}\\rbrace $ is the set of all zeros of $p$ in $ {\\mathbb {N}}^3$ .", "Suppose that $(a, b, c)$ is another $ {\\mathbb {N}}^3$ -point with $c$ nonzero.", "Then $a+2c^2 + cb^2=b$ and so $c(2c+b) = 2c^2+cb^2\\le b$ since $a$ is nonnegative.", "But if $c$ is strictly positive then we must have $2c+b\\le c(2c+b)\\le b$ , which gives $c\\le 0$ , a contradiction.", "Now the corresponding generating function is equal to $1/((1-x)(1-y))$ which is rational, but the polynomial $p$ is not of the integer-linear form up to scalar multiplication.", "As in the first question, we can show that it is undecidable to test whether the generating function $F_V$ for an arbitrary algebraic variety $V$ is D-finite or not.", "Let $P\\in {\\mathbb {Q}}[x_1, \\ldots , x_d]$ be any polynomials over $ {\\mathbb {Q}}$ in $x_1, \\ldots , x_d$ and let $V$ be the algebraic variety defined by $V := \\lbrace (a_1, \\ldots , a_d, b, c)\\in \\overline{ {\\mathbb {Q}}} \\mid P(a_1, \\ldots , a_d)^2 + (b-c^2)^2=0\\rbrace .$ The undecidability follows from the equivalence that the generating function $F_V$ is D-finite if and only if $P$ has no root in $ {\\mathbb {N}}^d$ .", "Clearly, $F_V = 0$ if $P$ has no root in $ {\\mathbb {N}}^d$ and then it is D-finite.", "Now suppose that $P$ has at least one root in $ {\\mathbb {N}}^d$ .", "Then the generating function $F_V$ is of the form $F_V = \\sum _{\\begin{array}{c}(n_1, \\ldots , n_d, m)\\in {\\mathbb {N}}^{d+1}\\\\ P(n_1, \\ldots , n_d)=0\\end{array}} x_1^{n_1}\\cdots x_d^{n_d} y^{m^2}z^m.$ It is sufficient to show that $G_V(x_1, \\ldots , x_d, y):=F_V(x_1, \\ldots , x_d, y, 1)$ is not D-finite.", "Clearly, the set $\\lbrace m\\mid \\text{$\\exists (n_1, \\ldots , n_d) \\in {\\mathbb {N}}^d$ such that $g(n_1, \\ldots , n_d, m)\\ne 0$} \\rbrace $ is the set of square numbers, which is neither finite nor syndetic.", "Thus $G_V$ is not D-finite by Lemma REF .", "Example 4.6 Let $p= x^2-y \\in K[x, y]$ .", "Then the associated generating function is $F(x, y) = \\sum _{m\\ge 0} x^{m}y^{m^2}$ .", "Since $p$ is not of the integer-linear form, $F(x, y)$ is not D-finite by Theorem REF .", "Actually, we can show that $F(x, y)$ is transcendentally transcendental.", "Suppose that $F(x, y)$ is differentially algebraic.", "Then it satisfies a nontrivial algebraic differential equation $Q(x, y, F, D_x(F), \\ldots , D_x^r(F)) = 0$ , where $r\\in {\\mathbb {N}}$ and $Q\\in K[z_1, z_2, \\ldots , z_{r+3}]$ .", "Note that the evaluation of a power series at $y=2$ gives a ring homomorphism $e_{2}:K[[x, y]]\\rightarrow K[[x]]$ and we have a commuting square $ \\begin{array}{ccc} K[[x,y]] & \\stackrel{e_{2}}{\\longrightarrow } & K[[x]] \\\\ \\big \\downarrow & ~ & \\big \\downarrow \\\\K[[x,y]] & \\stackrel{e_{2}}{\\longrightarrow } & K[[x]],\\end{array}\\\\$ where both vertical maps are differentiation with respect to $x$ .", "It follows that $F(x, 2) = \\sum _{m\\ge 0} 2^{m^2}x^{m}$ is also differentially algebraic.", "This leads to a contradiction with the fact proved by Mahler in [19] on the rate of coefficient growth of a differentially algebraic power series, since $2^{m^2}\\gg (m!", ")^c$ for any positive constant $c$ .", "This example motivates us to formulate the following conjecture, which can be viewed as an analogue of the Pólya-Carlson theorem in the context of algebraic geometry and differential algebra.", "Conjecture 4.7 Let $V\\subseteq \\mathbb {A}^d_K$ be an affine variety over an algebraically closed field $K$ of characteristic zero.", "Then the power series $ F_V(x_1, \\ldots , x_d) := \\sum _{(n_1,\\ldots ,n_d)\\in V\\cap \\mathbb {N}^d} x_1^{n_1}\\cdots x_d^{n_d}$ is either rational or transcendentally transcendental." ] ]
1606.04986
[ [ "High-power ultrafast Yb:fiber laser frequency combs using commercially\n available components and basic fiber tools" ], [ "Abstract We present a detailed description of the design, construction, and performance of high-power ultrafast Yb:fiber laser frequency combs in operation in our laboratory.", "We discuss two such laser systems: an 87 MHz, 9 W, 85 fs laser operating at 1060 nm and an 87 MHz, 80 W, 155 fs laser operating at 1035 nm.", "Both are constructed using low-cost, commercially available components, and can be assembled using only basic tools for cleaving and splicing single-mode fibers.", "We describe practical methods for achieving and characterizing low-noise single-pulse operation and long-term stability from Yb:fiber oscillators based on nonlinear polarization evolution.", "Stabilization of the combs using a variety of transducers, including a new method for tuning the carrier-envelope offset frequency, is discussed.", "High average power is achieved through chirped-pulse amplification in simple fiber amplifiers based on double-clad photonic crystal fibers.", "We describe the use of these combs in several applications, including ultrasensitive femtosecond time-resolved spectroscopy and cavity-enhanced high-order harmonic generation." ], [ "Introduction", "Originally intended for the precise measurement of optical frequencies, femtosecond optical frequency combs have since found many other applications outside of their original purpose.", "[1], [2] They are now used for the calibration of astronomical spectrographs,[3] laser ranging,[4] high-order harmonic generation,[5] attosecond physics,[6], [7] and direct frequency comb spectroscopy,[8], [9] among other things.", "In our lab, for example, we have recently demonstrated a large improvement in the sensitivity of ultrafast optical spectroscopy using frequency comb methods.", "[10] Most frequency comb lasers operate in the near infrared, based on Ti:sapphire, Er:fiber, or Yb-based gain media, but most spectroscopic applications of frequency combs lie in other regions of the electromagnetic spectrum.", "It is then desirable to shift the comb to other spectral regions using nonlinear optical techniques,[11], [12], [13], [14] but doing this with high efficiency requires high peak powers.", "Thus for a frequency comb with a useful repetition rate and comb spacing, high average power is needed.", "Ytterbium-based systems then stand out as providing an excellent platform for average power scaling due to the very small quantum defect of Yb and the capability of high doping in both glasses and crystals.", "[15], [16], [17] Since the first Yb-based femtosecond lasers,[18], [19] progress in this field has moved at a rapid pace using fibers, thin disks, and slabs.", "Yb:fibers are particularly attractive for average power scaling, due to the large surface area to volume ratio of fibers, large gain bandwidth,[15] and the availability of double-clad fibers for use with low-brightness (and thus low-cost) pump diodes.", "Indeed, kW scale femtosecond lasers have been reported using high-power Yb:fiber amplifier systems,[20], [21] and amplified Yb-combs have demonstrated high phase coherence.", "[22] Yb:fiber has sufficient gain bandwidth to support sub-200 fs pulses through linear chirped pulse amplification (CPA), in which the total nonlinear phase shift accumulated in the amplifier chain is kept less than 1 radian.", "[23] Recently, shorter pulses have been generated at high average power through nonlinear amplification,[24], [25] in which nonlinear propagation in the gain fibers is harnessed for generating additional spectral components.", "With narrow linewidth and controllable combs,[22], [26], [27], [28] the simultaneous combination of high peak and average power can be obtained through enhancement in passive optical cavities.", "[29], [30], [31], [26] The literature regarding this development has been confined to conference proceedings and specialty optics journals, and it requires quite a bit of know-how to go from this literature to a working femtosecond fiber laser.", "Indeed, most femtosecond lasers used in laboratory research are still based on Ti:sapphire, and there is a much larger community that is familiar with the operation of Ti:sapphire lasers than fiber-based systems.", "One can build a fiber laser for a fraction of the cost of an equivalent Ti:sapphire system,[32] and it is much simpler, but there is a learning curve to climb: What pump diodes do I use?", "Is that cheap current controller quiet enough?", "How do you mount the fibers?", "How fast does the pump diode need to turn off to save the fiber amplifier if the seed is lost?", "These are the types of questions we encountered when starting to build fiber lasers in 2013 - and they can be fiendishly difficult to find in the literature.", "In this article we present a detailed account of the design, construction, and operation of two high-power Yb:fiber laser frequency combs that we built in our laboratory over the past three years.", "The literature regarding Yb:fiber lasers is vast, and we are not attempting here to provide a comprehensive review.", "Rather, we hope to provide a practical guide to those not already intimately familiar with the details of fiber lasers.", "A block-diagram outlining both lasers is shown in figure REF .", "The lasers we describe in this article are made from all commercial components, most of which are stock items at major distributors, and can be assembled with only basic fiber tools.", "For example, a fiber splicer capable of splicing standard single-mode fibers is sufficient and you do not need more expensive models capable of handling polarization maintaining fibers or photonic crystal fibers (PCF), which cost many times more.", "Figure: Overview of the laser systems.", "Chirped pulse amplification using a fiber-based stretcher and simple PCF amplifiers provides minimal complexity.", "More detailed schematics are shown in figures , , and and component lists and mechanical drawings are provided in the supplemental material.In section , we describe the construction of Yb:fiber oscillators passively mode locked using nonlinear polarization evolution (NPE) and run near zero net cavity group delay dispersion (GDD) for the lowest noise and narrowest comb-tooth linewidth.", "[28] The physics and many implementations of these oscillators has been discussed extensively in the literature (for a review see reference ChongRepProgPhys2015), but NPE mode locking has a reputation for being finicky, and this literature is not necessarily helpful when standing over the laser and trying to make it work.", "We instead focus on practical procedures for achieving and characterizing mode-locked single-pulse operation, low noise, and long-term stability.", "We also discuss stabilization of the comb, including a new method for actuating on the comb's carrier-envelope offset frequency using the intracavity grating separation.", "In section REF , we describe a simple scheme for amplifying the comb to 9 W of average power and 85fs pulse duration at 1060 nm using CPA in a one-stage PCF amplifier.", "In section REF , we describe CPA to 80 W using a PCF rod amplifier.", "Both amplifiers use fiber-based pulse stretching with anomalous third order dispersion fibers,[34], [27] which require careful initial design, but then dramatically simplify the mechanical design of the laser system.", "In section , we describe several applications of these lasers.", "Detailed lists of all components appear in the supplemental material." ], [ "Yb:fiber oscillators", "The large nonlinear phase shifts accumulated when an ultrashort pulse propagates for a distance in the confined space of an optical fiber can give rise to many phenomena.", "[35], [36], [37], [38] Inside a laser cavity, this high nonlinearity can allow for mode-locked operation over a very wide range of parameters.", "Unlike Ti:sapphire oscillators, in which a few standard designs that emerged in the 1990's [39], [40] are found in most ultrafast laser labs, there are many fiber oscillator designs working at repetition rates from 100 kHZ [41] to 10 GHz,[42] and the literature presents a large and daunting landscape to navigate.", "We attempt a brief summary here with the goal of putting our lasers in context.", "Mode-locked fiber lasers can be broadly classified by their net cavity GDD and the saturable loss mechanism by which they are mode locked.", "[16], [43] With large anomalous GDD, soliton-like pulse shaping produces nearly chirp-free pulses, but with limited power.", "[44], [16] Lasers working with large normal GDD, even with all normal dispersion elements,[43] can support wave-breaking free pulses of very large energy.", "For example, Baumgartl et al.", "[45] have even demonstrated 66 W of average power and $\\mu $ J pulses directly from an oscillator without subsequent amplification.", "However, for the quietest operation, with both the lowest phase and amplitude noise most suitable for comb applications, it is desirable to operate the laser near net zero cavity GDD.", "[28], [46], [47], [48], [22] Unlike Er-doped fiber lasers operating at 1.5 $\\mu $ m, where it is easy to make fibers with normal or anomalous dispersion, silica fibers predominantly have normal dispersion in the 1.0-1.1 $\\mu $ m range amplified by Yb, so that dispersion compensation is usually done with a free-space dispersive delay line [28], [49], [50] or fiber Bragg gratings.", "[22], [48] Fiber Bragg gratings can allow for all-fiber designs, but require very careful design before assembly, as the dispersion is not adjustable.", "Oscillators with a free-space dispersive delay line allow tuning to find zero dispersion.", "As we show in section REF , with transmission gratings, fine-tuning of the grating separation in such a delay line using piezo-electric actuators can also be used to control the comb's carrier-envelope offset frequency.", "For mode locking, NPE in fiber[38], [37], [51] provides a fast artificial saturable absorber[52] that does not require any special components, but is sensitive to temperature or humidity changes.", "On the other hand, lasers based on real saturable absorbers, such as semiconductor saturable absorber mirrors (SESAM), can be made very environmentally stable, but typically have larger phase and amplitude noise.", "[53], [54]The quiet Yb-oscillators described in referencesSchibliNatPhot2008 and RuehlOptLett2010 actually use a combination of both NPE and the SESAM In our lasers, we use NPE mode locking, and have observed free-running comb-tooth linewidths less than 30 kHz and residual intensity noise less than -130 dBc/Hz for frequencies above 10 kHz (see figure REF ).", "We have also observed reasonable long-term stability in a laboratory setting (more details below).", "Figure: NPE ring oscillator design.", "I = Faraday isolator, WDM = wavelength division multiplexer, QWP = quarter-wave plate, HWP = half-wave plate, PBS = polarizing beam splitter, SM = single-mode, EOM = electro-optic modulator, G = grating, RP = roof reflecting prism, M = mirror.", "Inset: Typical mode-locked spectra for oscillator A (red) centered at 1060 nm and oscillator B (blue) centered at 1035 nm, both operating near zero net GDD." ], [ "Oscillator Construction", "The basic layout for both Yb:fiber oscillators is shown in figure REF .", "The fiber section provides gain and nonlinearity while the components in the free space section compensate the dispersion of the fiber, manipulate the polarization, and actuate on the pulse's round-trip group delay and carrier-envelope offset phase.", "The main differences between the two lasers are that (1) the design wavelengths of the components are different to accommodate the different center wavelengths.", "Oscillator A is designed to operate at 1060 nm and oscillator B is designed to operate at 1035 nm.", "(2) Oscillator B operates with much lower residual third-order dispersion (TOD) due to a shorter electro-optic modulator (EOM) and larger pitch diffraction gratings (B = 600 groove/mm vs. A = 1000 groove/mm).", "Complete lists of all the components can be found in the supplemental material, but here we briefly highlight some important aspects.", "The pump laser is a fiber Bragg grating stabilized diode laser operating at 976 nm (Oclaro LC96L76P-20R).", "Basic current and temperature controllers from Thorlabs (LDC210C and TED200C) are used to drive the pump laser.", "The noise specifications of this diode current controller are sufficient to obtain low-noise operation because the effect of high-frequency pump power fluctuations is suppressed by the low-frequency roll-off of the pump-modulation transfer function observed for these Yb:fiber oscillators, shown in figure REF a).", "Although the pump laser's fiber is polarization maintaining (PM), we simply splice this onto non-PM fiber for injection into the oscillator cavity with a fused wavelength division multiplexer (WDM).", "In addition to the WDM, a polarization insensitive isolator (I1) is used to isolate the pump laser from the oscillator light.", "The fiber assembly is terminated on each end with anti-reflection (AR) coated angled FC/APC connectors which are then plugged into a fiber coupler lens assembly (Thorlabs PAF-X-5-C).", "These AR coated fiber tips are obtained simply by splicing the ends of AR coated patch cables (Thorlabs P4-980AR-2) to the ends of the gain fiber and WDM fiber pigtail.", "The use of connectorized fiber tips enables replacing the fiber assembly with minimal realignment and also allows for the rough alignment of the cavity using another fiber coupled laser, if desired.", "Single-mode Yb-doped gain fiber can be purchased from a variety of companies (Nufern, Thorlabs, Cor-active, ...) with a variety of Yb doping concentrations.", "We have used either YB1200-4/125 or YB1200-6/125DC from Thorlabs with similar results.", "The fiber assembly is spliced together with a basic optical fiber fusion splicer.", "In our lab, we use a refurbished Ericsson FSU 995FA.", "The specifics of the fiber lengths are important for a few reasons, and can be found in the supplemental material.", "The first is that mode locking depends upon the amount of nonlinearity in the fiber[56] and the more fiber there is, particularly following the gain fiber, the easier it is to mode lock.", "Second, if the AR coated fiber tips become damaged, this is usually due to the gain fiber being too long, and is not remedied simply by reducing the pump power.", "However, the overall length of the assembly and the relative lengths of the different sections does not have be controlled with high precision.", "By changing the fiber assembly, we have operated oscillators with repetition rates from 70 MHz to 97 MHz with the same free-space section and obtained similar performance.", "In the free-space section, zeroth-order waveplates are used for polarization control and tuning of the laser, a Faraday isolator ensures uni-directional operation, an EOM enables fast actuation on the effective cavity length, and a pair of transmission gratings is used for dispersion compensation.", "A polarization beam splitter cube (PBS) is used as an output coupler, reflecting vertically polarized light out of the cavity.", "The first diffraction grating (G1) is mounted on a manual translation stage for finding zero dispersion, and a piezo-electric transducer (PZT) for fine-tuning the comb's carrier-envelope offset frequency.", "A right angle prism with an AR coated hypotenuse (Thorlabs PS908H-C) serves as the retroreflector in the dispersive delay line, changing the beam height by 5 mm and allowing the beam which initially crossed above to be reflected by D-shaped mirror (M1).", "In section REF , we describe the PZT and EOM actuators more carefully using fixed-point analysis.", "[53], [57] For suppression of acoustic noise and mechanical vibration, both oscillators are enclosed in aluminum sheet metal boxes sided with “egg-crate\" style sound damping foam and built on honeycomb optical breadboards that are supported on the optical table with a 5/8\" thick piece of sorbothane rubber.", "Light is coupled from the oscillator to the amplifier chain via the single-mode fiber pigtail of the stretcher fiber module." ], [ "Alignment, mode locking, and long-term stability", "Since the stress-induced birefringence of the coiled fiber assembly is unknown, finding the correct positions of the waveplates for NPE mode locking is a somewhat random process.", "If one simply randomly rotates the three waveplates, this amounts to searching a three-dimensional space.", "We have instead developed a reliable procedure for finding mode locking that simplifies the search considerably.", "First, the oscillator is aligned with the goal of minimizing the pump power necessary for lasing, minimizing the CW lasing threshold, which includes optimizing beam alignment and iterative rotation of polarizers.", "The pump power is then increased to around 200 mW, well above the minimum pump power that can maintain mode locking, which is about 100 mW.", "The quarter-wave plate just before the input fiber coupler (QWP2) is rapidly rotated a few degrees back and forth.", "If mode locking is not achieved, the half-wave plate (HWP) should be stepped a few degrees and then the QWP2 rotation repeated.", "This process should be repeated until mode locking is achieved, usually within a few iterations.", "Once the first quarter-wave plate (QWP1) is set for lowest CW lasing threshold, it is generally not necessary to rotate it to find mode locking.", "A major problem with NPE Yb:fiber oscillators is that they are prone to multi-pulsing,[58] or the production of more than one pulse circulating in the cavity.", "The separation between pulses can occur on many different time scales, and thus one needs a range of instruments to detect it.", "Unlike a Ti:sapphire oscillator, we have frequently observed multi-pulsing to occur with the particularly troublesome separation of 1-500 ps: too short to measure with a typical oscilloscope, but too long to observe easily as interference fringes in the optical spectrum.", "To be able to detect multi-pulsing at all separations, we employ a combination of three instruments: (1) a low resolution USB optical spectrometer for small pulse separations $<$ 3 ps, (2) a simple scanning autocorrelator using a GaAsP two-photon photodiode [59] for the 1-50 ps range, and (3) a fast photodiode (Electro Optics Technology Inc. ET-3010) and sampling oscilloscope (Tektronix 11801C with an SD-26 sampling head, 20 GHz equivalent bandwidth) for longer timescales.", "A collinear, interferometric autocorrelator is preferred so that one can align it well enough to have confidence in the alignment for longer stage travels, and a two-photon photodiode simplifies the nonlinear signal detection.", "When multi-pulsing occurs, the first step is to make sure there is nothing terribly wrong with the oscillator.", "Specifically, check the CW lasing threshold and make sure it is low (typically less than 50 mW).", "Turn the waveplates, starting with either the HWP or QWP2, until it stops multi-pulsing.", "Once stable mode locking is found, one can search for the lowest noise.", "The grating spacing for net zero GDD can in principle be calculated using the material parameters for the fibers and other optical elements in the cavity, and this is a good place to start, but it is generally necessary to fine tune this spacing once mode locked.", "The GDD can be measured using the technique of Knox,[60] changing the center wavelength either by inserting a knife edge into the dispersed beam or rotating QWP2 slightly (or both).", "While tuning the grating separation, we monitor oscillator performance using two metrics that can be evaluated quickly: (1) The oscillator relative intensity noise (RIN), measured on a low noise, high bandwidth photodiode, and (2) the free-running heterodyne beat between the oscillator and a narrow linewidth ($<$ 1 kHz) CW Nd:YAG laser (Innolight Mephisto).", "Cingöz et al.", "[61] showed that the phase noise and the RIN are correlated, and as reported by Nugent-Glandorf et al.", "[28] the laser comb-tooth linewidth depends strongly on the net cavity GDD.", "Indeed, we have observed optical linewidths ranging between 2 MHz and less than 30 kHz this way, depending on the grating separation.", "An important question is: with what precision do I have to find zero GDD?", "Previously, some authors have emphasized the importance of being slightly normal.", "[22], [16] In our lab, we have observed very similar performance on either side of zero dispersion, within approximately $\\pm 2000$ fs$^2$ in both oscillators, in agreement with references Nugent-GlandorfOptLett2011 and SongOE2011.", "We have also found that being near zero dispersion is a necessary, but not sufficient, condition to obtaining low-noise performance, and the noise can also depend on the details of NPE and the waveplate angles.", "Once satisfactory mode-locked performance is found, we leave the oscillator on indefinitely, and have enjoyed stable hands-free operation for many months at a time in a laboratory setting with reasonable temperature and humidity control ($\\pm $ 1 $^{\\circ }$ C, 20-60% relative humidity).", "The parts of the laser that in principle have finite lifetimes, the pump diode and the gain fiber, are inexpensive.", "We have not observed significant degradation of the pump diode performance over three years of nearly continuous operation.", "However, we have observed that the gain fibers can fail after about one year of continuous operation.", "The main symptom of this is that the laser just won't mode lock.", "Lasing thresholds and output powers are similar, but stable mode-locked operation is not re-attained until the gain fiber is replaced, or a new fiber assembly with a fresh gain fiber is installed in the oscillator." ], [ "Comb Stabilization", "The key element defining an optical frequency comb is that its comb teeth are evenly spaced to an extraordinary precision.", "[63], [64] This occurs naturally in mode-locked lasers and can also occur in other comb-generating systems such as microresonantors [65] and broad-band electro-opticially modulated light fields.", "[66] Once even spacing is established, the comb has two degrees of freedom that determine the frequencies of the comb teeth.", "Usually this is expressed in terms of the electronically countable repetition rate $f_{\\text{rep}}$ and carrier-envelope offset frequency $f_0$ via the familiar comb formula $\\nu _n = n f_{\\scriptsize \\mbox{rep}} + f_{0}$ where $n$ is an integer and the $\\nu _n$ are the optical frequencies.", "Indeed, for self-referenced combs this may be the most sensible parameterization, as it is $f_{\\scriptsize \\mbox{rep}}$ and $f_0$ that are actively controlled.", "However, for optically referenced combs, combs coupled to cavities, or when discussing the effects of actuators or noise sources, the discussion is often simplified by using a fixed point analysis,[67], [57], [53] writing the comb's optical frequencies as $\\nu _n = (n-n^{*})f_{\\scriptsize \\mbox{rep}} +\\nu _{n^{*}}$ where $n^*$ is an integer representing a fixed point of the frequency comb that does not change due to a particular perturbation such as noise or intentional actuation on the laser.", "In the fixed point picture, one considers the comb teeth simply expanding and contracting around the fixed point via changes in $f_{\\scriptsize \\mbox{rep}}$ .", "The larger the frequency difference is between the fixed point and a particular comb tooth, the more the frequency of that comb tooth changes due to the perturbation.", "Since the frequency comb has two degrees of freedom, one needs two feedback loops and two actuators to stabilize the comb.", "Ideally, these two feedback loops would have zero cross-talk.", "For example, if one directly stabilizes $f_{\\scriptsize \\mbox{rep}}$ and $f_0$ , ideally one actuator would actuate only on $f_{\\scriptsize \\mbox{rep}}$ and the other only on $f_0$ .", "In practice, this goal is almost never reached exactly, which is acceptable as long as one feedback loop can be significantly slower than the other, such that the faster loop can adiabatically track and correct for the cross-talk from the competing loop.", "For coupling a frequency comb to an optical cavity, or locking the frequency comb to another optical reference, it is desirable to have one fast actuator with its fixed point far from the optical frequency and another actuator with fixed point near the optical frequency.", "This allows the fast actuator to have large travel at optical frequencies that are being stabilized, and the second actuator to simply cause the comb to breath around this locked point.", "Figure: Transducer transfer functions.", "a) Pump amplitude to laser amplitude modulation transfer function.", "b) Voltage to phase modulation transfer functions for the intracavity EOMs in oscillators A and B.A commonly used combination of actuators to accomplish this is a fast intracavity EOM [68], [69], [70], [71], [72], [73], [74] for cavity length changes, with fixed point near DC ($n^{*}\\sim 0$ ), and the pump laser power for changing the intracavity pulse's round trip phase shift, with fixed point near the optical carrier frequency.", "[53], [57], [75] While fast bandwidth can be obtained via actuating on the pump current in some laser designs,[61] the bandwidth attainable with this actuator depends on the details of the laser and population inversion dynamics.", "[76] In the current ring cavity design with lower loss and smaller gain than the laser in reference CingozOptLett2011, and thus lower relaxation oscillation frequency, we have observed the bandwidth of pump power modulation to be quite limited, as shown in figure REF a).", "To record this data, we modulate the pump diode current sinusoidally and record the amplitude modulation on the comb light with a photodiode and a spectrum analyzer.", "The pump current to pump power transfer function (not shown) was independently verified to be flat out past 1 MHz modulation frequencies with this setup, confirming that the roll-off is due to transfer function of the Yb laser.", "In our lasers we use bulk EOMs for fast (multi-MHz bandwidth) actuation with a fixed point near DC, and instead of the pump power we use $\\mu $ m-scale piezo-electric adjustments of the grating spacing, which we show below has a fixed point near the optical frequency.", "For the EOM, we use simple bulk-crystal EOMs.", "Previous authors [73] have used short (few mm) EOM crystals due to concern over higher order dispersion.", "However, short crystals require multi-kV voltages to produce substantial phase shifts, and this is challenging to provide with high bandwidth.", "We have found that the remarkable tolerance of Yb:fiber oscillators to uncompensated higher-order dispersion[77] enables the use of longer crystals with correspondingly lower voltage requirements.", "Oscillator A uses a commercial phase modulator (Thorlabs EO-PM-NR-C2) with a 40 mm MgO doped Lithium Niobate crystal and a $V_{\\pi }$ of 250 V, and oscillator B uses a home-built EOM with a 4 mm LiTaO$_3$ crystal (United Crystals) compressed between brass and acrylic fixtures[78] and a $V_{\\pi }$ of approximately 3 kV.", "As shown in figure REF b), both EOMs allow phase modulation with multi-MHz bandwidth without piezo-elastic resonances that have limited previous efforts.", "[73] To record this data, we drive the EOM with a sinusoidal voltage and record the amplitude of phase-modulation side bands on the heterodyne beat with the CW Nd:YAG laser, taking care of the fact that the intracavity EOM modulates the laser's frequency but the sidebands on the beat report on the phase modulation depth.", "In both lasers we supply a 2 MHz sinusoidal voltage to the EOM to put frequency modulation (FM) sidebands on the comb that enable Pound-Drever-Hall (PDH) locking of the combs to passive optical cavities.", "[30], [79], [80] We measured the residual amplitude modulation (RAM) on the output light of the oscillator to be less than -90 dBc when driving the EOMs with a 20 V (peak to peak) sine wave (more than what is typically required for PDH locking).", "The EOM alignment can be fine-tuned in situ by minimizing this RAM.", "For actuation on the grating spacing, the first grating is glued to a ring PZT (Noliac NAC2125) using Loctite Hysol 1C-LV epoxy (also sold under the trade name Torr-seal), and this allows $ > 10$ kHz of bandwidth before encountering mechanical resonances.", "Shifting the carrier-envelope offset phase of a pulse by actuation on the grating separation of a pulse compressor has been employed for carrier-envelope offset phase stabilization in amplified Ti:sapphire lasers (after the amplifier chain),[81] but to our knowledge this is first report of doing this inside a laser cavity.", "Here we derive the resulting frequency shifts for a transmission grating geometry and show that for transmission gratings operated in Littrow condition, the fixed point is at the optical carrier frequency $\\nu _{\\scriptsize \\mbox{optical}}$ , such that the change in $f_0$ is approximately $\\nu _{\\scriptsize \\mbox{optical}}/f_{\\scriptsize \\mbox{rep}}$ larger than the change in $f_{\\scriptsize \\mbox{rep}}$ .", "For the parallel grating pulse compressor illustrated in figure REF , the total phase shift for one pass through the grating pair is given by:[82] $\\phi _{g}(\\omega ) = \\frac{\\omega }{c}p(\\omega ) - \\frac{2\\pi }{d}G\\tan (\\beta )$ where $\\omega =2\\pi \\nu $ is the angular frequency, $\\phi _{g}(\\omega )$ is the spectral phase, $p(\\omega )$ is the frequency dependent optical path length through the compressor, $\\beta $ is the angle of diffraction determined from the grating equation, $\\sin (\\alpha ) + \\sin (\\beta ) = \\lambda /d$ , with $\\alpha $ the angle of incidence measured from normal, $d$ is the grating pitch, $\\lambda $ is the wavelength, and $G$ is the distance between the gratings measured perpendicular to the grating surfaces.", "The second term in equation (REF ) accounts for the $2 \\pi $ phase shift encountered by the light for each grating groove traversed and must be included to obtain correct results.", "[82] Careful inspection of the angle-dependent path length shows that $\\frac{d \\phi _{g}}{dG} = \\frac{\\omega }{c} \\left( \\frac{1}{\\cos (\\beta )} - \\frac{\\cos (\\alpha +\\beta )}{\\cos (\\beta )} \\right) - \\frac{2\\pi }{d}\\tan (\\beta )$ Now one is tempted to locate the fixed point, $\\omega ^*$ , by setting equation (REF ) equal to zero and solving for $\\omega $ , but this is not generally correct because mode locking demands that the comb teeth remain evenly spaced, and thus the differential phase shift between comb tooth $n+1$ and $n$ must be the same as the differential phase shift between comb tooth $n+2$ and comb tooth $n+1$ .", "Enforcing this fact that the comb has only two degrees of freedom amounts to linearizing the spectral phase using the phase shifts obtained near the optical carrier frequency, viz.", "$\\frac{d\\phi _{\\scriptsize \\mbox{comb}}}{dG} = \\left.", "\\frac{d \\phi _{g}}{dG} \\right|_{\\omega _0} + \\left.", "\\frac{d \\tau }{dG} \\right|_{\\omega _0}(\\omega - \\omega _0)$ where $\\tau = d\\phi _{g}/d\\omega $ is the frequency dependent group delay, which is evaluated at the optical carrier frequency $\\omega _0$ in equation (REF ).", "The fixed point is then given by $\\omega ^* = \\omega _0 - \\left( \\left.", "\\frac{d \\phi _{g}}{dG} \\right|_{\\omega _0} \\right) \\left({\\left.", "\\frac{d \\tau }{dG} \\right|_{\\omega _0}} \\right)^{-1}$ At the Littrow condition, $\\alpha = \\beta (\\omega _0) = \\sin ^{-1}(\\pi c/ \\omega _0 d)$ , one can show that the phase shift due to changing the grating separation, $d \\phi _{g}/dG$ , is identically zero and the fixed point is thus at the optical carrier frequency.", "For the more realistic scenario that the gratings end up slightly off-Littrow, one can use equations (REF ), (REF ), and (REF ) along with the relation $d f_{\\scriptsize \\mbox{rep}}/dG = -f_{\\scriptsize \\mbox{rep}}^2 \\left.", "d\\tau /dG \\right|_{\\omega _0}$ in order to determine the changes in comb tooth frequencies.", "One can also derive relations for the changes in $f_{\\scriptsize \\mbox{rep}}$ and $f_0$ .", "For two passes through the grating pair, under Littrow conditions, the result is: Figure: Transmission grating geometry.", "Illustration of the notation for the intracavity grating compressor.", "The grating spacing is adjusted slightly (ΔG\\Delta G) using a PZT to control the comb's carrier-envelope offset frequency.", "The arrow indicates the direction of positive ΔG\\Delta G.$\\frac{df_{\\scriptsize \\mbox{rep}}}{dG}=~-\\frac{4\\pi f^2_{\\scriptsize \\mbox{rep}}}{ \\omega _o d}\\tan (\\alpha ).$ $\\frac{d f_0}{dG}~= -n^* \\frac{df_{\\scriptsize \\mbox{rep}}}{dG} = \\frac{2 f_{\\scriptsize \\mbox{rep}}}{d}\\tan (\\alpha ).$ Figure: Grating acutation.", "a) 1064 nm beat frequency change with grating separation.", "b) Repetition rate change with grating separation.", "The data indicate that the fixed point is near the optical frequency.We have experimentally verified this analysis by recording changes in the comb repetition frequency and an optical comb tooth near 1064 nm with oscillator A when voltage is applied to the grating PZT.", "The data are shown in figure REF .", "The repetition rate changes are measured using a photodiode and a frequency counter.", "The changes in the optical frequency are measured by recording the beat frequency of an unstabilized heterodyne beat between the comb and the CW Nd:YAG laser.", "Linear fits to the data give slopes of $\\frac{d \\nu _{\\scriptsize \\mbox{beat}}}{dV} = 0.14 $ MHz/V and $\\frac{d f_{\\scriptsize \\mbox{rep}}}{d V} = 0.39$ Hz/V.", "The number of comb teeth between 1064 nm and the fixed point can then be simply calculated from the ratio $\\Delta \\nu _{\\scriptsize \\mbox{beat}} / \\Delta f_{\\scriptsize \\mbox{rep}} = 3.3 \\times 10^5$ .", "So the fixed point lies only approximately 30 THz away from the optical carrier frequency of 283 THz.", "This is consistent with the above equations and $\\alpha $ deviating from the Littrow angle by approximately 4 degrees, which is realistic given our ability to initially set the grating angle in the laser and the $\\alpha $ dependence of the grating's diffraction efficiency.", "The grating can also be used to make a shift purely in $f_0$ , if the grating is moved parallel to its surface, similar to the motion of a sound wave in an acousto-optic frequency shifter.", "Here the phase shift is simply $2 \\pi $ per grating pitch moved,[83] and the fixed point is at $\\nu = \\infty $ .", "Using both parallel and perpendicular motions, in principle one could completely control the comb with only $\\mu $ m-scale motions of the grating alone, as the two motions have different fixed points.", "Amplification of continuous wave lasers to high average power in Yb:fiber is straightforward, but amplification of femtosecond pulses presents additional complications.", "The long length of fiber presents a large amount of dispersion even for large mode area (LMA) fibers, and it is much more difficult to avoid accumulated nonlinear phase shifts than in bulk solid-state lasers.", "Designers of ultrafast fiber lasers usually take one of two approaches: embrace nonlinearity [36], [35], [24], [84], [25] or use stretchers and compressors with very large GDD to avoid it.", "[27], [85], [86], [20], [87] For comb applications, linear amplification, in which the $B$ -integral, or accumulated nonlinear phase shift throughout the amplifier chain, is less than one, is generally preferred because then the amplified comb's coherence properties are determined mainly by the oscillator.", "In nonlinear amplification, amplitude noise from the high-power pump diodes in the amplifier chain could write phase noise on the amplified comb,[88], [16] although we are aware of some recent efforts using high-power nonlinear fiber amplification for comb applications.", "[89], [90] In our lasers, we have used linear CPA, but have strived to maintain modest stretcher/compressor dispersion by (1) seeding the amplifiers with very broad spectra and (2) maximizing the mode area of the seed light throughout the amplifier chain, even if this means seeding amplifiers below saturation.", "For simplicity and low-cost, we make use of fiber stretchers based on anomalous third-order dispersion depressed cladding fibers (OFS) and grating compressors based on inexpensive polymer transmission gratings (Wasatch Photonics).", "While we do not quite reach transform limited pulses on either laser system, the benefits of the simplicity of this scheme have outweighed the slightly reduced performance." ], [ "9 W amplifier at 1060 nm", "A schematic of the 9 W CPA scheme is shown in figure REF .", "The light from oscillator A is coupled into a stretcher fiber module custom made by OFS Specialty Photonics Division with FC/APC connectorized SMF-980 fiber pigtails.", "Between the stretcher and the following Faraday isolator (I1), about half the seed power is lost.", "After the dichroic mirror (D1), which is used to isolate the pump light and seed (or signal) light, 15-20 mW is launched into the 5 m amplifier fiber.", "The amplifier fiber is a doped, LMA (760 $\\mu $ m$^2$ ) PCF terminated with sealed ends and copper SMA 905 connectors, purchased from NKT Photonics (aerogain Flex 5.0).", "This fiber is end pumped with a 30 W, 915 nm pump diode (nLight Element).", "After the second dichroic mirror (D2), the beam is expanded to 4.2 mm ($1/e^2$ diameter), sent through another Faraday isolator, and compressed using a pair of polymer transmission gratings (Wasatch Photonics) and a roof reflector (M3).", "Below, we discuss these features in more detail and the design decision processes behind them.", "The coiled amplifier fiber is supported on a circular aluminum plate.", "The pump end, where the optical power is the highest, is mounted in a water-cooled copper clamshell assembly.", "Detailed drawings of the copper clamshell can be found in the supplemental material (figure S3).", "The seed end of the amplifier fiber is screwed into an SMA connector (Thorlabs HFB001) mounted on a flexure stage (Thorlabs MicroBlock MBT616D).", "The heavy copper mode-stripper assembly of the fiber is further supported by shims placed on the flexure stage.", "The output of the stretcher fiber and pump diode fiber are mounted on flexure stages in similar fashion.", "For delivering the pump light, the pump diode fiber pigtail is spliced onto the end of an AR coated multimode patch cable (Thorlabs M105L02S-B).", "The pump light is launched counter-propagating to the amplified seed light to reduce the accumulated nonlinear phase shift ($B$ -integral) in the amplifier fiber.", "Despite not being “all-fiber\", we have observed consistent performance from this mechanical setup without alignment for more than 2 years of operation.", "The saturation power for the LMA PCF amplifier is more than 200 mW, and many previously reported amplifier systems using these PCF amplifiers have employed a fiber pre-amplifier with smaller mode area between the oscillator and the PCF amplifier [27], [87], [86], [89], [90] in order to seed the power amplifier at saturation.", "Since we are seeding the amplifier well below the saturation power, a threshold-like behavior is observed in the amplified power vs. pump power curves shown in figure REF a), reducing the efficiency of the amplifier.", "While this one-stage amplification scheme is less efficient, it is much simpler due to (1) the lack of all the pre-amplifier components and (2) the pulses do not have to be stretched as much to avoid nonlinearity, since all the high power propagation is done in LMA fiber.", "A smaller stretching/compression ratio allows for looser tolerances on matching the higher-order dispersion of the stretcher and compressor.", "The 5 Watts of pump power wasted before the amplifier reaches saturation are not really of consequence, due to the low cost of high-power pump diodes.", "Table: Dispersion budget for the 1060 nm laser.", "FS = fused silica.", "TGG = Terbium Gallium Garnet.Two concerns with underseeding the amplifier are (1) noise due to amplified spontaneous emission (ASE),[38] and (2) catastrophic damage to the amplifier fiber due to self-lasing and Q-switching.", "Regarding (1), despite the expectation of increased ASE, the measured RIN spectrum of the amplified light (see pre-amplifier curve in figure REF ) indicates that the main source of noise on the amplified light is due to the pump diode RIN, not ASE.", "Regarding (2), while we do not know what the lowest necessary seed power is to avoid catastrophic damage, we can say that we have run these PCF amplifiers (both at 1035 nm and 1060 nm) for many hours with seed powers as low as 10 mW without observing damage.", "We continuously monitor the seed light with a fast (100 MHz bandwidth) photodiode (PD) and a simple interlock circuit, shown in figure REF , which immediately shuts off the pump diode in the event that the RF power from the photodiode drops below a set threshold, indicating reduced power or loss of mode locking.", "Pump diode drivers from VueMetrix Inc. shut off in less than 50 microseconds upon receiving an electronic signal, much shorter than the energy storage time in Yb of approximately 1 ms. [15] A similar interlock system is used for the amplifiers of the 80 W laser discussed in section REF .", "Another feature of this amplifier system to note is the operating wavelengths.", "The amplified light is at 1060 nm and the pump light is at 915 nm, whereas most ultrafast Yb amplifiers are pumped at 975 nm and amplify light at 1030-1040 nm, where the absorption and emission cross sections are largest.", "[15] We use 915 nm for the pump wavelength because the absorption feature at 975 nm is narrow, requiring tight control over the pump diode wavelength for efficient pumping.", "While the absorption cross section at 915 nm is three times lower, the absorption maximum there is also much broader, which loosens the requirements for controlling the pump diode wavelength, and thus temperature, considerably.", "With the long 5 m PCF, more than 90% of the pump light is still absorbed.", "For the amplified wavelength, we operate this 9 W laser at 1060 nm, far to the red of the Yb emission maximum, because one can amplify with considerably less gain narrowing, and this has also been employed in a few other linear CPA designs.", "[27], [22] Figure REF shows the output spectrum of the amplified laser with more than 30 nm of bandwidth.", "Figure: 80 W amplifier layout.", "Chirped pulses are amplified first in a 2.5 m flexible photonic crystal fiber and then in a 0.8 m rod fiber.", "Component lists can be found in the supplemental material.A dispersion budget for the CPA system following oscillator A is shown in table REF .", "The oscillator pulses are stretched to approximately 100 ps duration in the fiber stretcher module and compressed to $<$ 100 fs after amplification using a Treacy-style compressor with 1250 groove/mm transmission gratings (Wasatch Photonics).", "We measure an overall compressor efficiency of 77%, corresponding to a diffraction efficiency of $(77\\%)^{1/4}$ = 94%.", "Figure REF c) shows a second harmonic generation (SHG) frequency resolved optical gating (FROG) trace taken with the laser at full power using a commercial FROG system (Mesa Photonics FROGscan Ultra).", "While a free-space Offner-type stretcher[91], [92] would allow for more tunability than the fiber stretcher module, and perhaps better compensation of higher order dispersion,[93] it would also add substantial mechanical complexity and cost.", "The fiber stretcher module is alignment free, and we have observed nearly transform-limited performance in both CPA systems using these for these stretcher modules." ], [ "80 W Amplifier at 1035 nm", "Figure REF shows a higher power laser capable of 80 W average power via two-stage amplification.", "The PCF pre-amplifier is similar to the system described in section REF , except that a shorter 2.5 m fiber is used due to the larger gain and absorption of Yb at 1035 nm.", "For the second stage, a 0.8 m long Yb-doped PCF rod (NKT Photonics, aeroGAIN-ROD-PM85-POWER) with a 3,400 $\\mu $ m$^2$ mode field area, end pumped by a 200 W pump diode module (LIMO, LIMO200-F200-DL980-S1886) is used for the rod amplifier.", "When seeding the rod with 6.4 W, 96 W emerges with excellent beam quality when pumping the rod with 200 W of pump power.", "The large mode field diameter of the rod allows the laser to maintain linear amplification to $\\mu $ J pulse energies using only a modest stretcher dispersion of 5.3 ps$^2$ , or stretched pulse durations of only approximately 300 ps.", "The available rod fibers are shorter than flexible PCF, necessitating pumping at 975 nm and amplification in the more conventional 1030-1040 nm region to achieve efficient gain.", "With the amplified center wavelength at 1035 nm, we observe significant gain narrowing[94], [95] in the amplifier chain, as shown in figure REF b).", "However, we are still able to compress the rod amplifier output to very clean 155 fs pulses, as shown in figure REF c).", "Pumping at 975 nm requires tighter control over the pump diode wavelength, increasing the cost of the pump diode.", "In our system we do not actively control the temperature, and this is the reason for the nonlinear amplified power vs. pump power curves of figure REF - the pump laser wavelength is changing as the power is increased (see figure S2 in the supplemental material).", "Figure: Rod amplifier performance.", "a) Output power of the Yb:fiber rod amplifier, pumped by a counterpropagating 975 nm pump diode.", "The beam mode at lower power and full power is shown for both the output of the rod fiber and the compressor.", "b) Laser spectra measured from the oscillator and amplifier output.", "c) Raw FROG trace at 80 W (inset) and retrieved pulse shapes of the compressed pulses compared to the transform limit calculated from the spectrum in b) for both low and high power operation.The rod fiber is supported along its length by a water-cooled aluminum V-groove, held loosely by only two pieces of kapton tape.", "The end caps, the parts that are critical for alignment, are rigidly located in V-shaped jaws mounted on the optical table separate from the V-groove.", "The high pump power of the rod amplifier can bring complications.", "When pumped with 200 W, more than 40 W of pump light comes out from the seed end of rod fiber with NA=0.5, which could heat the mount of the lens L5 or the mirror mount for D2, causing a dangerous drift in the seed light alignment.", "We dump the pump light safely using two water-cooled black-anodized aluminum beam dumps, the annular BD2 to protect the lens mount and BD1 to collect the pump light transmitted through the dichroic mirror (D2).", "Mechanical drawings for BD2 can be found in the supplemental material.", "Table: Dispersion budget for the 1035 nm laser.", "FS stands for fused silica.", "TGG = Terbium Gallium Garnet.Another problem with high average power is thermal lensing and distortions of the beam quality.", "As shown in figure REF , an excellent, nearly Gaussian, spatial profile is observed in the amplified light from the rod amplifier at both low and high power.", "However, even at low power, we observe distortions of the beam after four diffractions from the inexpensive polymer gratings.", "At higher power, despite the large expanded beam size of 5.8 mm $1/e^2$ diameter after the telescope (L8 and L9), thermal lensing is observed, and the output mode measured 1.8 m after the compressor, is significantly smaller.", "Inspection of the beam at various points indicates that the thermal lensing occurs either in this telescope or the Terbium Gallium Garnet crystal of the optical isolator.", "However, despite the obvious thermal lensing, we can still obtain nearly constant coupling efficiency to the TEM$_{00}$ mode of an external femtosecond enhancement cavity (fsEC) for the full power range of the laser by simply changing the lens spacing in a mode-matching telescope between the compressor and the fsEC.", "One of the authors (T. K. Allison) has observed similar behavior for the laser described in reference RuehlOptLett2010.", "The dispersion budget for the 80 W comb is shown in table REF and the raw FROG trace and retrieved pulse shape are shown in figure REF c).", "The compressor has an overall efficiency of 86%, corresponding to a grating diffraction efficiency of $(0.86)^{1/4} = 96\\%$ .", "Clean 155 fs pulses are observed at both high and low power.", "The absence of pulse distortion at high power indicates that linear amplification has been achieved.", "Figure REF shows the RIN measured in various parts of the laser system.", "The pump diode RIN spectra are nearly flat out to frequencies of 1 MHz at -100 dBc/Hz for the 30 W pump diode and -85 dBc/Hz for the 200 W pump diode.", "The 200 W pump diode is driven by a 1500 W power supply from TDK Lambda (GEN 20-76).", "For the both the pre-amplifier and the rod amplifier, the storage time of the Yb:fiber gain medium provides a low-pass filter to the pump diode RIN.", "Figure REF also shows the RIN spectrum for the intracavity light of a fsEC operating with 11 kW of intracavity power.", "The cavity is locked to the comb using the PDH method[79], [80] and a PZT on one of the fsEC mirrors.", "A servo bump is observed at $\\sim $ 30 kHz due to the finite bandwidth of the PZT lock, but for lower frequencies an intracavity RIN level is obtained nearly equal to the RIN of the comb.", "Lower RIN for a high-power Yb:fiber laser has been obtained recently by Wunram et al.,[87] however the dominant noise source was not identified in this work.", "Figure: Intensity noise.", "RIN spectra at various places throughout the 80 W laser system.", "The RIN spectrum of the 9 W laser is similar to the blue curve." ], [ "Applications", "In our laboratory, the primary application of these frequency combs involves coupling them to passive optical cavities[30], [96] for either performing spectroscopy[10] or generating high-order harmonics.", "[97] Here we give examples using these lasers in both applications." ], [ "Ultrasensitive femtosecond optical spectroscopy", "Ultrafast optical spectroscopy methods, such as transient absorption spectroscopy and 2D spectroscopy, are widely used across many disciplines.", "However, these techniques are typically restricted to optically thick samples, such as solids and liquid solutions, since the sensitivity of femtosecond-time resolved optical spectroscopy has lagged far behind that of linear spectroscopy methods.", "[98], [99], [100] Recently we have demonstrated the cavity-enhancement of femtosecond time-resolved measurements using the 9 W comb described here, enabling ultrafast optical spectroscopy in dilute molecular beams.", "[10] Complete details of the optical setup are given in reference ReberOptica2016 but here we give a brief summary.", "Light from the 9 W comb is frequency doubled and sent to two 4-mirror ring cavities whose foci intersect in a supersonic expansion of I$_2$ seeded in a noble carrier gas.", "A two point locking scheme is used to stabilize the laser to the probe cavity.", "The center of the comb is tightly locked to the probe fsEC using the PDH method[30], [79], [80] and the long EOM in oscillator A.", "A bias tee is used to provide both PDH FM sidebands and a high voltage feedback signal to the EOM.", "Using a wideband high voltage amplifier (Thorlabs inc. HVA200), we achieve approximately 100 kHz bandwidth in this feedback loop.", "A second, slower, PDH feedback loop monitors a separate part of the optical spectrum and moves the intracavity grating separation to maintain resonance for the rest of the comb.", "In reference ReberOptica2016 we reported measurements taken in a helium-seeded supersonic expansion using a 700 $\\mu $ m diameter round nozzle, where there is no clustering between the He and the I$_2$ molecules.", "If the expansion is instead seeded with Argon under the right conditions, the effects of clustering can be observed as the backing pressure is increased.", "Figure REF shows transient absorption data taken for I$_2$ in an Argon-seeded expansion, 6 mm away from a 5 mm long $\\times $ 200 $\\mu $ m wide slit nozzle at three different stagnation pressures.", "The nozzle is 3D printed in PEEK plastic (Arevo Labs Inc.) for compatibility with I$_2$ .", "As the backing pressure is increased from 260 Torr to 460 Torr (absolute), the effects of clustering are clearly seen as the coherent transient peak at time zero increases and subsequent oscillations are suppressed as the I$_2$ molecules collide with the caging Argon atoms.", "[101] Figure: Ultrasensitive transient absorption spectroscopy.", "The comb from the 1060 nm laser is frequency doubled and coupled to two femtosecond enhancement cavities for performing ultrafast spectroscopy, as in ref.", "ReberOptica2016.", "Shown are pump-probe traces in a supersonic expansion of molecular iodine and Argon gas, 6 mm above a 200 μ\\mu m wide slit nozzle.", "As the Argon pressure is increased, Argon atoms cluster around the I 2 _2 molecules, quenching the excited state vibrational wavepacket.", "Curves are offset for clarity." ], [ "Cavity-enhanced high-order harmonic generation and extreme-UV frequency combs.", "High-power ultrafast lasers are now routinely used to drive high harmonic generation (HHG), providing extreme ultraviolet (XUV) ultrashort pulses with a table-top setup, and also attosecond pulses.", "[102], [103] This is a highly nonlinear process, during which electrons are ionized by a strong laser field and then recombine with their parent ions, generating coherent light at high photon energies.", "Conventionally, HHG is realized by focusing high energy ($>$ 100 $\\mu $ J) laser pulses to intensities of more than 10$^{13}$ W/cm$^2$ intensity in a noble gas.", "The repetition rate is then limited by the average laser power available, usually to less than 100 kHz, but many applications such as XUV frequency metrology,[12] surface photoemission,[104] and photoionization coincidence methods[105] demand higher repetition rates.", "Figure: Cavity-enhanced high-harmonic generation.", "Spectrum of harmonics generated from Krypton gas via cavity-enhanced HHG at 87 MHz repetition rate.", "Photon fluxes correspond to values measured at the exit slit of a time-preserving monochromator, corresponding to the flux directly deliverable to experiments.An elegant way to achieve this is via resonant enhancement of the fundamental pulse train in a fsEC.", "[29], [30] Although generating high harmonics in these cavities was originally demonstrated as early as 2005,[106], [107] since then through the use of higher power driving lasers and understanding of the intracavity extreme-nonlinear optics[108], [109], [110], [111] the power from cavity-enhanced HHG systems has increased by more than 6 orders of magnitude, to more the 100 $\\mu $ W/harmonic (at the gas jet) for  20 eV harmonics generated in Xenon.", "[12], [108], [110], [112], [113] In our lab, we have built a cavity-enhanced HHG system for the purpose of performing time-resolved surface photoemission experiments at high repetition rate.", "The 1035 nm comb described in this paper is coupled to a 6 mirror femtosecond enhancement cavity with a 1% input coupler and roughly 15 $\\mu $ m focus size.", "Krypton gas is injected at the focus using a small glass capillary backed with 760 Torr pressure and harmonics are coupled out of the cavity using a 250 $\\mu $ m thick sapphire plate at Brewster's angle for the fundamental.", "The harmonics are then passed through a home-built time-preserving monochromator similar to that described by Frasetto et al.", "[114] Figure REF shows the spectrum of harmonics after the monochromator exit slit measured using an Al-coated silicon photodiode (Opto Diode AXUV100Al) as the grating angle is rocked.", "For calculating the y-axis, the manufacturer's quantum efficiency specifications were used without additional calibration.", "More than $10^{11}$ photons per second emerge from the monochromator in the 19th harmonic at 23 eV, or approximately 1400 photons/pulse.", "The low pulse energy and high repetition rate make this source an ideal system for time-resolved photoelectron spectroscopy or time-resolved photoelectron microscopy experiments which are limited by space charge effects.", "[115], [116]" ], [ "Acknowledgements", "This material is based upon work supported by the Air Force Office of Scientific Research under award numbers FAFA9550-16-1-0164 and FA9550-13-1-0109, and the National Science Foundation under award number 1404296.", "When we embarked on this laser building project three years ago, we were helped immensely by S. A. Diddams, G. Ycas, and L. Nugent-Glandorf at the National Institute of Standards and Technology in Boulder, CO and D. J. Jones and A. K. Mills at the University of British Columbia.", "We also thank R. Jason Jones at the University of Arizona for useful discussions.", "Supplementary information for high-power ultrafast Yb:fiber laser frequency combs using commercially available components and basic fiber tools X. L. Li, M. A. R. Reber, C. Corder, Y. Chen, P. Zhao, and T. K. Allison Stony Brook University, Stony Brook, NY 11794-3400 USA The tables in this section list the detailed information for the parts used in our laser system (oscillators and amplifiers).", "Table: List of parts in the oscillators (figure 2 in the paper).", "Part numbers are from Thorlabs unless otherwise indicated.Table: List of parts in the 9 W optical amplifier (figure 6 in the paper).", "Part numbers are from Thorlabs unless otherwise indicated.", "GDD = group delay dispersion.", "TOD = third-order dispersion.", "f = focal length.", "MFD = mode-field diameter.Table: List of parts in the 80 W optical amplifier (figure 8 in the paper.)", "Part numbers are from Thorlabs unless otherwise indicated.Figure S1 shows how the beam dump (BD2 in the 80 W laser system) works to protect the lens mount from exposure to the high power pump light.", "Due to the large numerical aperture (NA=0.5) of the inner cladding of the 0.8 m photonic-crystal fiber (PCF) rod, the powerful pump laser coming out of the fiber has a very big divergence angle and will heat the mount of the lens if the water-cooled beam dump is absent.", "The beam dump blocks most of the pump light but allows the seed light to go through.", "In the 80 W amplifier, as shown in figure S2, the output wavelength of the 975 nm pump diode changes with the operation temperature and power in the range of 967 nm to 977 nm, causing an increasing amplification efficiency while turning up the pump power (figure 9a) in the paper).", "The temperature shown is just for the cooling water flowing into the pump diode.", "The pump diode temperature is not controlled.", "Figure S3 shows the pump end of the coiled large mode area (LMA) double-clad PCF (5 m in the 9 W laser and 2.5 m in the 80 W laser).", "The pump power and the seed power are both low at the seed end, so we only cool the pump end of the fiber (25 W pump laser and 12 W seed laser is guided in the pump end).", "The pump end, together with copper SMA905 termination (NKT Photonics) is placed in a water-cooled copper clamshell.", "The termination is wrapped in a piece of indium foil to increase the thermal conduction with the copper clamshell.", "Figure: NO_CAPTIONFigure S4 shows the cooling solution for the NKT 0.8 m PCF rod.", "The rod fiber, whose diameter is 1 mm, is placed in the V-groove of 1 inch aluminum pipe, held loosely by only two pieces of kapton tape, and water flows between the water ports.", "The ends of the pipe (0.5 inch inner diameter) are plugged with NPT fittings.", "As seen in figure S5, the end cap of the rod fiber has a dimension of 6 mm (OD) $\\times $ 5 mm (length).", "So it can not be placed in the V-groove of the aluminum pipe.", "A separation V-jaw supports the end cap, as shown in figure 5S.", "Figure: NO_CAPTION" ] ]
1606.05234
[ [ "Dark matter substructure modelling and sensitivity of the Cherenkov\n Telescope Array to Galactic dark halos" ], [ "Abstract Hierarchical structure formation leads to a clumpy distribution of dark matter in the Milky Way.", "These clumps are possible targets to search for dark matter annihilation with present and future $\\gamma$-ray instruments.", "Many uncertainties exist on the clump distribution, leading to disputed conclusions about the expected number of detectable clumps and the ensuing limits that can be obtained from non-detection.", "In this paper, we use the CLUMPY code to simulate thousands of skymaps for several clump distributions.", "This allows us to statistically assess the typical properties (mass, distance, angular size, luminosity) of the detectable clumps.", "Varying parameters of the clump distributions allows us to identify the key quantities to which the number of detectable clumps is the most sensitive.", "Focusing our analysis on two extreme clump configurations, yet consistent with results from numerical simulations, we revisit and compare various calculations made for the Fermi-LAT instrument, in terms of number of dark clumps expected and the angular power spectrum for the Galactic signal.", "We then focus on the prospects of detecting dark clumps with the future CTA instrument, for which we make a detailed sensitivity analysis using open-source CTA software.", "Based on a realistic scenario for the foreseen CTA extragalactic survey, and accounting for a post-trial sensitivity in the survey, we show that we obtain competitive and complementary limits to those based on long observation of a single bright dwarf spheroidal galaxy." ], [ "Introduction", "Dark matter (DM) indirect detection in $\\gamma $ -rays was first discussed almost 40 years ago [1], [2] in the context of the diffuse $\\gamma $ -ray emission.", "The most promising DM candidate is a weakly interacting massive particle (WIMP), which produces secondary $\\gamma $ -rays originating from the particles' self-annihilation [3].", "As the signal from self-annihilating DM roughly goes as the DM density squared divided by the distance squared, the most promising targets result from an interplay between being close and/or massive, highly concentrated DM reservoirs.", "This made the Galactic centre (GC) a prime target, as first considered in [4].", "However, it was soon recognised that setting constraints on DM would be limited by astrophysical background at GeV energies [5].", "Higher energies face similar challenges, as illustrated by the first H.E.S.S.", "observations of the GC [6], and following DM interpretations [7].", "Owing to their potentially high DM densities and small astrophysical backgrounds, dwarf spheroidal (dSph) galaxies orbiting the Milky Way [8] and dark clumps [9], [10] were proposed next.", "In particular, the question of the DM clump population boost of the Galactic signal [11], [12] and/or the detectability of individual clumps with future $\\gamma $ -ray satellites and ground-based instruments [13], [14], [12] was raised.", "Since these pioneering studies, a steady progress has been made in estimating the prospects of DM indirect detection (or in setting constraints from non-detection) with the $\\gamma $ -ray sky.", "The launch of the Fermi $\\gamma $ -ray Space Telescope in 2008 with its main instrument, the Large Area Telescope (LAT) [15], was a huge step forward in exploring with unprecedented sensitivity the DM parameter space (see [16] for a review).", "Thanks to the all-sky survey observing strategy, the Fermi-LAT collaboration has set constraints on $\\gamma $ -ray lines [17], [18], [19], [20] or excesses in the $\\gamma $ -ray continuum spectrum of the diffuse Galactic and extragalactic emission [21], [22], [23], galaxy clusters [24], [25], dSph galaxies [26], [27], [28], [29], [30], [31], and dark clumps [28].", "For dSph galaxies, whose known number is still growing from recent optical surveys, a stacking strategy allowed to exclude a thermal relic annihilation cross-section $\\langle \\sigma v\\rangle $ of a few times $10^{-26}~{cm^3~s^{-1}}$ for WIMP masses $m_{\\chi }< 100$  GeV [30], [31].", "Constraints from ground-based Cherenkov telescopes are competitive with Fermi-LAT limits to constrain DM particle masses above a few hundreds of GeV.", "These pointed instruments use a different observation strategy: the best constraints are obtained from DM searches in regions around the GC, leading to limits of $\\langle \\sigma v\\rangle \\lesssim 3 \\times 10^{-24}~{cm^3~s^{-1}}$ for $m_{\\chi }{}\\sim $  TeV [32], [33]; for dSph galaxies, the constraints either come from a stacking analysis (VERITAS [34], [35] and H.E.S.S.", "[36]) or from a deep observation of a single highly ranked object (MAGIC [37], [38]), leading to limits of $\\langle \\sigma v\\rangle \\lesssim 1-3 \\times 10^{-24}~{cm^3~s^{-1}}$ .", "To take full advantage of this wealth of data and to set the most reliable and effective limits on DM, the modelling of the DM signal received as much attention.", "Beside the complex GC region (see, e.g., the Fermi-LAT analysis [39]), improvements were made in the DM modelling of dSph galaxies [40], [41], galaxy clusters [42], [43], [44], [45], [46], and diffuse emissions (see [47] for a recent review).", "To increase the sensitivity to DM indirect detection, multiwavelength analyses [48], [49] and observations of dSph galaxies [50], [51], [52] and galaxy clusters [53], [54] have been carried out.", "Enhancements of the signal-to-noise ratio arise from a better characterisation of $\\gamma $ -ray anisotropies and cross-correlation with other tracers (e.g., [55]).", "Note that most of these calculations at both the Galactic and extragalactic level depend on the hierarchical structure formation and on the survival and distribution of subhalos in their host halos.", "In particular, the detectability of dark clumps observed as $\\gamma $ -ray point sources with no counterparts at other wavelengths or from their imprint in the angular power spectrum (APS) over the diffuse DM background (that can be boosted by these same micro-halos), depends crucially on the modelling of these clumps.", "Their properties are investigated by heavy numerical simulations [56].", "If state-of-the-art Milky Way like simulated halos, such as Via Lactea [57], Aquarius [58], and more recently COCO [59] reach a mass resolution of a few $10^5$ solar masses, this must be compared with the very uncertain minimal mass of the subhalos (set by the DM candidate couplings) that could be ten orders of magnitude below (e.g., [60]).", "Numerical simulations of the early universe have confirmed that such subhalos might survive until today [61], a result also deduced from theoretical calculations (see [62] for a review).", "This paper revisits the question of the detectability of dark clumps for present and future $\\gamma $ -ray instruments.", "The prospects for the Fermi-LAT instrument were discussed by many authors in the light of high resolution numerical simulations [63], [64], [65], [66], [67], or based on semi-analytical approaches which extrapolate the clump population down to the mass of the smallest micro-halos [68], [69], [70], [71], [72], [73].", "Whereas recent efforts are turned towards a better discrimination between astrophysical or DM signals in sources with no counterparts in the Fermi-LAT catalogue [74], [75], [76], [77], [78], [79], the constraints that can be set depend ultimately on the number of expected DM clumps, which is still disputed [80], [81].", "The sensitivity to dark clumps for the future Cherenkov Telescope Array (CTA) [82], [83] is discussed in [84], but a more up-to-date estimate is made here based on the foreseen CTA extragalactic survey [85].", "We use the CLUMPY code [86], [87] to evaluate the impact of the clump distribution uncertainties on the ensuing $\\gamma $ -ray signal.", "Hundreds of Monte Carlo (MC) realisations are run per configuration to estimate the resulting uncertainties on the number of clumps, and to characterise the typical mass and distance of these detectable clumps.", "We also use realistic instrument responses in a plausible large-sky survey scenario to assess the sensitivity of CTA to these clumps.", "This provides a complementary view of CTA capabilities against pointed targets that will be part of the CTA DM programme [88], [89], [90], [91], [92].", "The paper is organised as follows: in sec:basics, we present our modelling approach for the DM distribution in the Galaxy, focusing in particular on seven physical parameters important to the Galactic substructure distribution; sec:globalprop presents the resulting $\\gamma $ -ray flux from the different models and describes several cross checks to demonstrate the consistency of our modelling; in sec:fermi, we use our findings to revisit the possibility that DM subhalos might be present in the 3FGL catalogue of Fermi-LAT; sec:cta presents the sensitivity of the future CTA to detect DM subhalos within its planned extragalactic sky survey and we finally discuss and summarise our findings in sec:discussion.The article is followed by a detailed appendix.", "In app:definitions, we shortly review different definitions of cosmological matter overdensities used throughout this paper.", "In app:meanmedian, we present useful formulae to analytically describe power-law source count distributions, and we use these results for a convergence study of the angular power spectra in app:APSconvergence.", "In app:mvirJ, we present the derivation and the mass and annihilation factor for the spectroscopically confirmed satellite galaxies in the Milky Way, including recently discovered objects.", "In app:CTAanalysisdetails, we provide additional details about the CTA analysis performed in sec:cta.", "Throughout the paper, we denote the mean of quantities $Q$ with a bar, $\\overline{Q}$ , and median values with a tilde, $\\widetilde{Q}$ .", "Global properties of the Galactic host halo are denoted with capital letters (mass $M$ , positions $R$ within the halo, distance $D$ to observer), and properties of individual subhalos with lowercase letters (mass $m$ at distance $d$ , positions $r$ within the subhalo).", "When referring to the brightest subhalo, variables are indicated with an asterisk, $Q^{\\star }$ ." ], [ "Modelling the $\\gamma $ -ray-emission from Galactic DM subhalos", "To assess detection prospects of Galactic DM subhalos, we explore various parameter sets for the substructure density.", "The average total Galactic halo density is left unchanged." ], [ "Modelling approach", "We use the CLUMPY codehttp://lpsc.in2p3.fr/clumpy to model the $\\gamma $ -ray emission from Galactic DM subhalos.", "We refer the reader to [86], [87] for an extensive description of the CLUMPY code features and validation.", "CLUMPY has been used previously to study DM annihilation and/or decay in dSph galaxies [93], [94], [95], [96], [97], [98], [99] and galaxy clusters [100], [46], [101].", "For the purpose of this work, we mainly use CLUMPY in the so-called `skymap mode' which allows the fast computation of full-sky maps of DM annihilation or decay signals.", "We focus on DM annihilation only, for which the expected DM differential $\\gamma $ -ray flux at energy $E$ , in the direction $\\vec{k}=(\\psi , \\vartheta )$ and per solid angle $\\mathrm {d}\\Omega $ reads $\\frac{\\mathrm {d}\\Phi }{\\mathrm {d}E\\,\\mathrm {d}\\Omega }(E,\\,\\vec{k})=\\frac{\\mathrm {d}\\Phi ^{\\rm PP}}{\\mathrm {d}E}(E)\\times \\frac{\\mathrm {d}J}{\\mathrm {d}\\Omega }(\\vec{k}) \\,,$ where $\\frac{\\mathrm {d}\\Phi ^{\\rm PP}}{\\mathrm {d}E}(E)= \\frac{1}{4\\pi }\\frac{\\langle \\sigma v\\rangle }{m_{\\chi }^{2}\\delta }\\,\\sum _{f}\\frac{\\mathrm {d}N^{f}_{\\gamma }}{\\mathrm {d}E}\\, B_{f}\\,,$ and where the $J$ -factor is generically written as $J(\\vec{k},\\,\\Delta \\Omega ) = \\int \\limits _{\\Delta \\Omega }\\; \\int \\limits _{\\rm {l.o.s}} \\!\\!", "\\rho ^2 \\mathrm {d}l\\,\\mathrm {d}\\Omega = \\int \\limits _{0}^{2\\pi }\\int \\limits _0^{\\theta _{\\text{int}}}\\int \\limits _{\\rm {l.o.s}} \\!\\!", "\\rho ^2(\\vec{k};\\;l,\\,\\theta ,\\,\\phi )\\, \\mathrm {d}l\\,\\sin \\theta \\,\\mathrm {d}\\theta \\,\\mathrm {d}\\phi \\,.$ In these equations, $m_{\\chi }$ is the mass of the DM particle $\\chi $ , $\\langle \\sigma v\\rangle $ is the velocity-averaged annihilation cross-section, and $\\mathrm {d}N_{\\gamma }^f/\\mathrm {d}E$ and $B_f$ correspond to the spectrum and branching ratio of annihilation channel $f$ .", "The parameter $\\delta $ is $\\delta =2$ for a Majorana and $\\delta =4$ for a Dirac particle, and we choose $\\delta \\equiv 2$ in the remainder of this paper.", "In CLUMPY, $m_{\\chi }$ , $\\langle \\sigma v\\rangle $ , and $B_f$ are user-defined parameters, from which the $\\gamma $ -ray annihilation spectrum is calculated (based on the parametrisations of [102]).", "The DM density $\\rho $ is integrated along the line of sight (l.o.s.", "), and up to a maximum angular distance $\\theta _{\\text{int}}$ .", "The overall DM density can be written $\\rho _{\\rm tot}=\\rho _{\\rm sm}+\\rho _{\\rm subs}$ , where $\\rho _{\\rm sm}$ corresponds to the smooth component, and $\\rho _{\\rm subs}$ corresponds to the substructures of the Galactic DM halo.", "Generating skymaps with CLUMPY starts from setting DM properties: smooth DM profile, spatial and mass distribution of Galactic substructures, halo mass-concentration relation, DM particle mass, and annihilation/decay channels.", "The computation has been optimised as to draw only subhalos that outshine the mean DM signal (set by a user-defined precision), leading to a decomposition of the substructure signal $J_{\\rm subs}^{\\rm tot} = {J_{\\rm drawn}}+ \\langle J_{\\rm subs}\\rangle $ into two components: ${J_{\\rm drawn}}$ is the signal from the substructures drawn in a realisation of the skymaps, and $\\langle J_{\\rm subs}\\rangle $ is the average signal from all `unresolved' halosFor legibility purpose, we define $\\langle J_{\\rm subs}\\rangle $ to be the sum of $\\langle J_{\\rm subs}\\rangle $ and $J_{\\rm cross-prod}$ as defined in [87]., i.e., faint subhalos whose intrinsic $J$ -factors do not pass the threshold defined from the precision level required by the user.", "Additional levels of clustering within subhalos are also considered using this average description.", "We refer the reader to [86], [87] for details on the computation of these quantities.", "For the purpose of this work, suffice to say that the higher the precision requirement, the more halos are drawn and the smaller is the contribution of $\\langle J_{\\rm subs}\\rangle $ .", "A convergence analysis with respect to the precision requirement is presented in app:APSconvergence.", "The flexibility of CLUMPY allows the user to easily explore various models and configurations, calibrated (but not limited) to the results of various $\\Lambda $ CDM numerical simulations such as the Aquarius [58], Phoenix [103] or Via Lactea II (VL II) [104] simulations, as well as their hydrodynamical updates [105], [106], [107]." ], [ "Milky Way dark matter halo", "The total DM density profile $\\overline{\\rho }_{\\rm tot}$ of the Milky Way (MW) is modelled with an Einasto profile with a slope $\\alpha _{\\rm E}=0.17$ and scale radius $R_{\\rm s} = 15.14\\,\\rm {kpc}$ , as suggested by [108].", "The normalisation of the profile is computed to satisfy $\\rho _\\odot \\equiv \\rho (R_\\odot =8\\,\\text{kpc})= 0.4\\,\\mathrm {GeV\\,cm^{-3}}$ , as estimated by [109] (see also [110], [111] for higher estimates).", "Although $R_\\odot $ and $\\rho _\\odot $ suffer from large uncertainties [112], [113], [114], their exact values are subdominant for the purpose of our study.", "All our calculations use a maximum radius $R_{\\rm MW}=260\\,\\mathrm {kpc}$ of the MW DM halo, yielding a total MW mass $M_{\\rm MW}=1.1\\times 10^{12}\\,\\rm M_\\odot $ , in agreement with [115].", "As discussed in [86], we define the average substructure density, $\\overline{\\rho }_{\\rm subs}{} = f_{\\rm subs}\\times M_{\\rm MW}\\times \\mathrm {d}P/\\mathrm {d}V$ , where $\\mathrm {d}P/\\mathrm {d}V$ is the spatial distribution of Galactic substructures, and $f_{\\rm subs}$ is the global fraction of the MW mass contained in subhalos.", "Given $\\overline{\\rho }_{\\rm tot}$ and $\\overline{\\rho }_{\\rm subs}$ , the smooth MW profile then is defined as $\\rho _{\\rm sm} = \\overline{\\rho }_{\\rm tot}- \\overline{\\rho }_{\\rm subs}$ .", "At sufficient angular distance from the GC ($\\theta \\gtrsim 10{^{\\circ }}$ ), this smooth component results in a diffuse background flux, $J_{\\rm sm}$ , that is found negligible by at least 3 orders of magnitude when compared to the flux of resolved subhalos or to the residual background of CTA (see sec:CTAbackground)." ], [ "Substructure properties and set of models", "Substructures in their host halo are characterised by their mass and spatial distribution, as well as the description of the DM distribution within each subhalo.", "In the $\\Lambda $ CDM hierarchical structure formation scenario, small structures collapse first and then fall and merge into larger structures.", "Subject to tidal forces, the `unevolved' initial substructure distribution turns into an `evolved' population whose properties differ from those of field halos [116].", "Semi-analytical models as well as numerical simulations have been used to characterise the properties of these substructures down (or extrapolated down) to the smallest mass scale.", "The spread in the various results yield significant uncertainties, e.g., on the cosmological annihilation signal [117].", "To assess the detectability of dark clumps, we build sets of models by varying seven important properties of Galactic substructures, as described in tab:model-parameters.", "For all these models, the threshold mass of the smallest and most massive subhalos are fixed to $m_{\\rm min}=[10^{-6}]{\\rm M_\\odot }$ [118] and $m_{\\rm max} = 0.01\\,M_{\\rm tot}$ [119] respectively: increasing $m_{\\rm max}$ would slightly increase the median $J$ -factor of the brightest subhalo, leaving unchanged our conclusions; decreasing $m_{\\rm min}$ amounts to adding an extra-population of very low-mass halos that would contribute to the average $\\langle J_{\\rm subs}\\rangle $ component, the effect being dependent on the slope $\\alpha _m$ of the mass distribution (see below).", "For the angular resolutions considered in our calculations, the brightest resolved objects always outshine the diffuse DM background emission from $J_{\\rm sm}+ \\langle J_{\\rm subs}\\rangle $ , moreover, for CTA, the diffuse background is dominated by the instrumental residual background and not by the $\\langle J_{\\rm subs}\\rangle $ component.", "The parameters and the choices made for their variation are briefly mentioned hereafter and their consequences on the flux are discussed in sec:globalprop.", "Subhalo inner profile.", "As seen in eq:gammafluxastro, the DM density profile is the fundamental ingredient to estimate the astrophysical part of the DM annihilation flux.", "When ignoring baryonic effects, DM substructures are characterised by cuspy profiles, with two standard parametrisations being the Navarro-Frenk-White (NFW) and Einasto descriptions [120], [121].", "For a given subhalo, we calculate the normalisation and scale radius of the DM inner profile by providing the subhalo mass, ${m}_{\\rm vir}$ , and its concentration $c_{\\rm vir}({m}_{\\rm vir},R)\\equiv {r}_{\\rm vir}/{r}_{\\rm s}$ , where $R$ is the distance from the GC, and ${r}_{\\rm vir}$ , ${r}_{\\rm s}$ are the virial and scale radius of the subhalo, respectively.", "The meaning of $c_{\\rm vir}$ is further discussed in the next paragraph.", "For the Einasto profile, the shape parameter $\\alpha _{\\rm E}$ introduces an additional degree of freedom, which we fix to $\\alpha _{\\rm E}\\equiv 0.17$ .We also investigated the impact of a scattering in $\\alpha _{\\rm E}$ .", "The authors of [58] find the Aq-A subhalos equally well described by Einasto profiles with $0.16\\le \\alpha _{\\rm E} \\le 0.20$ .", "For $\\alpha _E=0.16$ , the $J$ -factors increase by $\\lesssim 20\\%$ compared to $\\alpha _E=0.17$ , and decrease by $\\lesssim 60\\%$ when choosing $\\alpha _E=0.20$ .", "The DM constraints set from the brightest subhalo change by the same amount, whereas the effect on $\\langle J_{\\rm subs}\\rangle $ is subdominant.", "An Einasto profile is the default configuration, and the effect of switching to NFW is performed in the VAR0 model (see tab:model-parameters).", "Note that recent simulations have shown that halos close to the free streaming scale [122], [123], [124], [125] could be cuspier than the NFW profile.", "Including in CLUMPY this mass-dependent inner slope—Eq.", "(2) of [124]—leads to a 13% to 43% increase of $\\langle J_{\\rm subs}\\rangle $ (comparable to the 12% to 67% increase found in [124]), but more importantly here, we checked that these boosted microhalos provide no detectable dark clumps.", "Mass-concentration parametrisation $c(m)$ .", "Once the parametrisation of a subhalo inner profile is chosen, its structural parameters (normalisation and scale radius) are fully determined from the mass $m_\\Delta $ and concentration-mass $c_\\Delta -m_\\Delta $ relation (see app:definitions for a definition of $\\Delta ={\\rm vir},\\,200\\,,500\\,\\dots $ ).", "The latter depends on the subhalo evolution in its host halo, i.e.", "its location and `evolved' mass.", "Several parametrisations, based on the results of numerical simulations have been proposed in the last few years, the most recent suggesting a flattening of the relation at low masses [126] and a higher concentration of subhalos compared to field halos.", "The latter effect was shown to yield an extra $\\sim 5$ boost factor [127], [128], [129] on $\\langle J_{\\rm subs}\\rangle $ compared to previous calculations.", "Figure: Models for the concentration c 200 c_{200} of Galactic subhalos at redshift z=0z=0.", "Beside the distance-independent parametrisation SP , shown are the values at the galactocentric distances R=[8]kpcR=[8]{kpc} (dashed-dotted lines) and R=[260]kpcR=[260]{kpc} (solid lines) from (violet and orange) and (green).Here, we consider the field halo Sánchez-Conde & Prada [126] parametrisation (SP), the distance-dependent Pieri et al.", "[67] parametrisations (P-VLII based on the Via Lactea II simulation and P-Aq based on Aquarius) and the Moliné et al.", "[129] parametrisation.As for several other refinements discussed in this section, the Pieri et al.", "and Moliné et al.", "parametrisations have been implemented in CLUMPY for the purpose of this work, but are not yet available in the public version.", "These mass-concentration prescriptions are displayed in fig:cdelta-mdelta.", "The Pieri et al.", "and Moliné et al.", "approaches account for the fact that the closer a subhalo of a given mass is to the GC, the more concentrated it is.", "The various parametrisations are compared at a distance of 8 kpc from the GC (dotted-dashed lines), where the P-VLII and Moliné et al.", "parametrisations produce similar concentrations, while P-Aq appears systematically higher.", "At large distance from the GC (solid lines), all parametrisations yield lower concentrations that become more compatible with the SP field halo distance-independent parametrisation (dashed line).", "In the following, we use the SP description as our conservative baseline and investigate the distance-dependent effects of the P-VLII and Moliné parametrisations in both the VAR6 and HIGH models.", "We will discard the P-Aq prescription in the following, but remind that using this parametrisation would result into even larger $J$ -factors than used in our optimistic model HIGH.We also investigated the P-Aq model, and found a $\\sim 60\\%$ increase for the flux from the brightest subhalos compared to the P-VLII description.", "This increase would also improve the sensitivities presented in fig:IndividualSourceSensitivity-CTA by the corresponding factor.", "Width of the mass-concentration distribution, $\\sigma _c$ .", "Rather than assuming a single concentration for a given halo mass from the mean parametrisations above, the concentration is drawn from a log-normal distribution of width $\\sigma _c$ around the values given by these parametrisations.", "This is incorporated to account for the intrinsic scatter of the $c(m)$ relation found in numerical simulations.", "We consider $\\sigma _c = 0.14$ as our default value [126], [130], and study the impact of a larger scatter in the VAR2 model where $\\sigma _c = 0.24$ [131] is used.", "Number of halos $N_{\\rm calib}$ between $10^8$ and $[10^{10}]{\\rm M_\\odot }$ .", "This number is used as a calibration for the total number of subhalos.", "$\\Lambda $ CDM simulations of MW size halos predict an overabundance of high-mass subhalos compared to the currently known satellite galaxies; this is the so-called `missing satellites' problem [132], which is linked to the `too-big-to-fail' problem [133].", "Baryonic feedback onto the cusps of DM subhalos could possibly solve this tension [134], [135].", "Indeed, hydrodynamical simulations roughly show half as many high-mass subhalos as DM only simulations.", "About $100-150$ objects are obtained above $[10^8]{\\rm M_\\odot }$ [105], [106], and we choose $N_{\\rm calib}=150$ as our default value.", "A more subhalo-rich configuration, used in models VAR4 and HIGH (see tab:model-parameters), is defined by $N_{\\rm calib}=300$ as motivated by the results of DM-only simulations [58].", "Index of the clump mass distribution, $\\alpha _m$ .", "Numerical simulations show that the DM halo mass distribution is well described as $\\mathrm {d}N/\\mathrm {d}m \\propto m^{-\\alpha _m}$ .", "For the reference model we take $\\alpha _m=1.9$ as suggested by numerical simulations of MW-like halos [58], [104] and investigate a slightly steeper mass function in the VAR1 model, where $\\alpha _m=2.0$ .", "Together with $N_{\\rm calib}$ , $m_{\\rm min}$ , and $m_{\\rm max}$ , the choice of $\\alpha _m$ determines the total number of clumps $N_{\\rm tot}$ and their total mass $M_{\\rm subs}$ (see [86] for details).", "In tab:model-parameters, we provide the resulting $N_{\\rm tot}$ and, with the knowledge of the total mass of the Galaxy, $M_{\\rm MW}$ , the global portion of DM bound into subhalos, $f_{\\rm subs}= M_{\\rm subs}{}/M_{\\rm MW}$ .", "Spatial distribution of Galactic substructures, $\\mathrm {d}P/\\mathrm {d}V=\\overline{\\rho }_{\\rm subs}{}/M_{\\rm subs}{}$ .", "The fraction of mass bound into substructures is expected to decrease towards the GC, as subhalos are tidally disrupted by the strong gradient of the gravitational potential.", "This is discussed in detail in [116], where the authors argue that this effect is the result of a selection effect of the `evolved' subhalos (suffering from tidal stripping), the `unevolved' distribution following the host smooth distribution.", "Figure: Number of subhalos within RR, normalized to N 200 N_{200}, the number of subhalos within R 200 {R}_{200}, up to R MW /R 200 =1.22R_{\\rm MW}/{R}_{200}=1.22 (R 200 =[213.5]kpc{R}_{200}=[213.5]{kpc} for our Galactic halo).", "We also show the position of the observer at R ⊙ /R 200 =0.037R_\\odot /{R}_{200}=0.037.", "The number of subhalos, N(R)N(R), results from the subhalo number density dN/dV=N tot ·dP/dV\\mathrm {d}N/\\mathrm {d}V = N_{\\rm tot}\\cdot \\mathrm {d}P/\\mathrm {d}V.fig:fsubcomp displays the cumulative number of halos as a function of the distance to the GC, where $\\overline{\\rho }_{\\rm subs}{}\\propto \\overline{\\rho }_{\\rm tot}$ is shown as dash-dotted line.", "To correctly describe $\\overline{\\rho }_{\\rm subs}$ , a generic `anti-biased' parametrisation $\\overline{\\rho }_{\\rm subs}/\\overline{\\rho }_{\\rm tot}\\propto R$ was proposed in [136], [137] (dotted line).", "This is to be compared to the result obtained from numerical simulations, namely, a fit to Aq-A1 (well described by an Einasto profile `E-AQ', long-dashed line) or to all Aquarius halos (curved power-law, cyan) halos [58], or subhalos of the Via Lactea II simulation [138], [104] (orange).", "We also show the profile for the Phoenix simulation of galaxy clusters (rescaled to ${R}_{200}$ , green line) described by an Einasto profile [103], which leads to a comparable dependence.", "Simulations produce slopes steeper than $\\overline{\\rho }_{\\rm subs}/\\overline{\\rho }_{\\rm tot}\\propto R$ , with the Aquarius and Phoenix results found to be well described by $\\overline{\\rho }_{\\rm subs}/\\overline{\\rho }_{\\rm tot}\\propto R^{1.3}$ in [116]'s model.", "In tab:model-parameters, we choose the Einasto profile fitted to the Aquarius A-1 halo as our default setup and use the Via Lactea II parametrisation in the VAR3 and HIGH models.", "tab:model-parameters also gives the local DM mass fraction under the form of subhalos in the solar neighbourhood, $f_{\\rm subs}(R_\\odot ) = \\overline{\\rho }_{\\rm subs}(R_\\odot )/\\overline{\\rho }_{\\rm tot}(R_\\odot )$ , which is dependent on the chosen $\\overline{\\rho }_{\\rm subs}$ profile.", "Flag for sub-substructures.", "The default calculation assumes no further substructures within Galactic subhalos.", "However, several levels of substructures (clumps within clumps) can be taken into account by CLUMPY for the flux calculation, which would boost the signal further.", "This is tested in model VAR5, where we assume self-similarity, with respect to the model LOW, between subhalos and sub-subhalos (following what has been shown by both the Aquarius and Via Lactea II simulations [139], [64]), i.e.", "$f_{\\rm subsub}=0.19$ , E-AQ for the spatial distribution of sub-subhalos within their host, and SP for the $c(m)$ parametrisation.", "Table: Parameters for the different models investigated in this study.", "The first seven lines correspond (from top to bottom) to: the subhalo density profile, the slope of the subhalo mass distribution, the width of the concentration distribution, the subhalo spatial distribution, the number of objects between 10 8 10^{8} and [10 10 ]M ⊙ [10^{10}]{\\rm M_\\odot }, the flag for sub-subhalos, and the mass-concentration relation.", "The columns are ordered by increasing flux of the brightest object.", "`NFW' stands for a Navarro-Frenk-White profile and `E' for an Einasto profile with α E =0.17\\alpha _{\\rm E}=0.17.", "`E-AQ' is the Einasto parametrisation fitted to the substructure distribution in Aquarius simulation , while `M-VLII' corresponds to the Via Lactea II parametrisation of Madau et al.", ".", "The mass concentration relation is `SP' for the Sánchez-Conde & Prada parametrisation , or the distance-dependent description by Moliné et al.", ", respectively by Pieri et al.", ", `P-VLII'.", "Derived parameters in the six bottom rows are the following: N tot N_{\\rm tot} is the total number of subhalos in the MW; f subs f_{\\rm subs} is the global mass fraction contained in subhalos; f subs (R ⊙ )f_{\\rm subs}(R_\\odot ) is the mass fraction contained in subhalos at the solar distance from the GC; D ˜ obs ☆ \\widetilde{D}_{\\rm obs}^{\\star }, m ˜ vir ☆ \\widetilde{m}_{\\rm vir}^{\\star }, and J ˜ ☆ \\widetilde{J}^{\\star } are the median distance from the observer, mass, and JJ-factor of the brightest subhalo from the 500 realisations of each model." ], [ "Global properties of the models", "Before performing analyses dedicated to Fermi-LAT and CTA in sec:fermi and sec:cta, we describe here the overall behaviour of the models with respect to the substructure properties described above.", "This is done by the inter-comparison of the various models (subsec:global), by comparison to the known MW satellites (subsec:comparedSph), and by comparison of the angular power spectrum to previous calculations (subsec:powspec)." ], [ "Impact of the substructure description", "Each substructure property that is varied according to tab:model-parameters essentially impacts the number of substructures and/or their associated $J-$ factors.", "Five hundred skymap realisations for each modelling (LOW, HIGH, VAR0 to VAR6) have been simulated.", "The bottom half of tab:model-parameters gives the global properties of each modelling, averaged over the 500 realisations.", "Compared to the LOW model, only the calibration number $N_{\\rm calib}$ and the slope of the mass distribution affect the total number of subhalos.", "However, the number of halos within a given $J$ -factor range will depend on all substructure-related properties and one may therefore use histograms of the subhalos $J$ -factors to assess the importance of each property.", "Figure: NO_CAPTIONFor a given model $X$ , we define $\\overline{N}_{X}(\\,>J)$ as the mean number of halos with a $J$ -factor above a certain threshold $J$ , averaged over the 500 realisations.", "This quantity is shown in the top panel of fig:Histogram1DJfactor-Nsub-fullsky for model LOW (pale blue) and HIGH (red).", "The behaviour of the histograms is well approximated by a power-law $\\propto J^{1-\\alpha }$ , with $\\alpha \\sim 2$ over a large range of $J$ values (thin solid red line).", "We also define the median $J$ -factor of the brightest halo in a given model, $\\widetilde{J}^{\\star }$ , and report these values in tab:model-parameters, along with the median mass, $\\widetilde{m}_{\\rm vir}^\\star $ , and distance from the observer, $\\widetilde{D}_{\\rm obs}^\\star $ , of this brightest halo.", "We refer the reader to app:meanmedian for a detailed discussion of the $J^{\\star }$ distribution and how $\\widetilde{J}^{\\star }$ and $\\overline{N}(\\,>J)$ are connected.", "The ratio $N_{X}(\\,>J)/N_{\\rm LOW}(\\,>J)$ is plotted in the bottom panel of fig:Histogram1DJfactor-Nsub-fullsky, where $X={\\rm VAR}i,\\;\\; i\\in [0,6]$ : Changing the subhalo inner profile (VAR0, blue), the substructure spatial distribution (VAR3, orange), the normalization of the mass distribution (VAR4, magenta), or the width of the mass-concentration description (VAR2, red) yields an increase or decrease of the number of clumps, uniformly over the entire $J-$ factor range.", "Changing the subhalo inner profile or changing the subhalo spatial distribution yields a $\\sim 40\\%$ change compared to the LOW model, while the width of the concentration distribution only affects the result by a few percents.", "The other substructure properties do not affect the number of subhalos in the same way for low and high $J$ -factor values.", "Including a boost from sub-subhalos (VAR5, light green) or having a distance-dependent concentration prescription (VAR6, black) both produce a similar effect: a larger number of halos in each $J$ -factor bin, the effect increasing with increasing $J$ (from $\\sim 50-70\\%$ to more than a factor 2).", "For the highest $J$ -decade, both distant-dependent concentration prescriptions VAR6a and VAR6b result in compatible $J$ -factors.", "This is well understood given that the high-$J$ end is populated by subhalos close to us, i.e.", "also close to the GC with $R\\approx [10]{kpc}$ .", "At these distances, P-VLII and Moliné concentrations are in agreement on a wide mass range (fig:cdelta-mdelta).", "The prescriptions then differ for less luminous subhalos, which represent subhalos at larger galactocentric distances.", "For those objects, Moliné et al.", "predict larger concentrations, which results into brighter objects and a steepening of the source count distribution.", "Steepening the slope of the mass function (VAR1, dark green) increases (decreases) the number of faint (bright) halos, the amplitude of the effect remaining $\\lesssim 50\\%$ for the $\\sim 10^3$ most luminous subhalos.", "Figure: The influence of sub-subhalos as a function of the integration angle.", "For typical bright detectable DM halos (see tab:brightestproperties), the signal with sub-substructures (green solid line) is compared to the signal without them (black solid line).", "The angular resolutions (68% containment radius; “point spread function”, PSF) for CTA at E γ ≥[30]GeVE_{\\gamma }\\ge [30]{GeV} and Fermi-LAT at E γ ≥[1]GeVE_{\\gamma }\\ge [1]{GeV} , are overlaid (grey shaded areas).", "For CTA, the sub-substructures contribution is negligible.Including sub-subhalos is only significant in the outskirts of the halos.", "This is shown in fig:subsub, where $\\mathrm {d}J/\\mathrm {d}\\Omega $ is plotted as a function from the distance to the centre for the LOW (grey solid line) and VAR5 (dashed blue and green solid lines) models.", "This finding is in agreement with [139].", "For the angular resolution of the background-dominated CTA (grey band), however, they do not play a significant role and we therefore do not include them in the remainder of this study.Assuming a spatial dependence of the sub-subhalo concentration—analogous to model VAR5 on the sub-sub level—possibly increases this contribution within the CTA resolution.", "However, the boost at this second level of substructures is very uncertain and could even be negligible for these objects [129].", "From the range of substructures properties tested as deviations from the LOW model, we find the distance-dependent concentration parametrisation and boost from sub-subhalos to have the larger effect in terms of the number of halos with the largest $J$ -factors.", "For CTA-like angular resolutions, we conclude that the mass-concentration relation is the most important substructure property to pin down in order to make reliable detectability studies.", "We use the distance-dependent concentration P-VLII by Pieri et al.", "in the HIGH model and, unless stated otherwise, the remaining of the paper will use the HIGH model as an optimistic template, while LOW remains default.", "Figure: Model HIGHFor illustration purpose, we display in fig:Skymapscompare the two corresponding differential flux skymaps computed at 4 GeV.", "The flux is obtained assuming a 200 GeV DM candidate which annihilates exclusively in the $b\\bar{b}$ channel.", "The left column shows maps of the substructures drawn by CLUMPY, while the middle column displays the total (resolved+unresolved) substructure contribution.", "As discussed above, more subhalos are resolved in model HIGH, and the flux of the unresolved component is also higher.", "The right column displays the total flux in both cases, i.e.", "including the smooth Galactic halo component, which is the dominant component towards the GC." ], [ "Comparison of the DM subhalo models to the known Milky Way satellites", "More than twenty dSph galaxies are known to orbit the Milky Way.", "Formed from the most massive DM subhalos, these objects are prime targets for indirect detection as their DM content, and therefore $J$ -factors can been inferred from stellar kinematics, e.g.", "[94], [96].", "The mass and $J$ -factors of these objects are discussed in app:mvirJ, tab:MandJ summarises their main properties.", "These values are used below for a sanity check of our models.", "First, the grey shaded area in the top panel of fig:Histogram1DJfactor-Nsub-fullsky corresponds to the cumulative histogram of $J$ -factors built from the known dSph galaxies, plus the SMC, and the LMC.", "The $J$ -factors are reported within an integration angle of $\\theta _{\\text{int}}=0.5{^{\\circ }}$ , as benchmark angular resolution of Fermi-LAT [15], [140].", "For display purposes, we have used the median values of the $J$ -factors only, but we remind the reader that these values may be very uncertain for ultra-faint dSph galaxies.", "For lower $J$ values, the number of detected dSph galaxies becomes much lower than the number of subhalos measured from the models.", "This is what one would expect given that the most numerous low-mass halo would not have retained gas and formed stars to become identified as dSph galaxies [106].", "The high-$J$ end of the histogram ($J\\gtrsim 10^{19}$ GeV$^2$ cm$^{-5}$ ) lies between the LOW and HIGH models but for the very last bin.", "The latter corresponds to the recently-discovered Triangulum II galaxy, that we have tentatively analysed here.", "Note that the authors of [141], using a non-spherical halo hypothesis have reported a median $J(0.5^\\circ )=1.6\\times 10^{20}$ GeV$^2$ cm$^{-5}$ for Triangulum II, which would bring the last bin down to lie between LOW and HIGH.When considering the credible intervals, the Triangulum II value of [141] and ours are nonetheless compatible within 1$\\sigma $ .", "This behaviour of the MW satellites gives confidence to the fact that the LOW and HIGH models do indeed encompass the uncertainties surrounding the subhalo distribution.", "Figure: NO_CAPTIONSecond, the mainly illustrative fig:Histograms2DmodelRef gives the location of the detected MW satellites (red and orange dots) in the $J-m_{\\rm vir}$ (left) and $J-D_{\\rm obs}$ planes (right) on top of the distribution of subhalos of model HIGH.", "It shows that MW satellite galaxies probe, as expected, the high-mass and high-distance ends of the subhalo population.", "The horizontal dotted lines indicate how many subhalos are expected in a given realisation, and as above, model HIGH is in excess compared to the known objects.", "They could be yet-to-discover dSph galaxies or dark halos, and the distribution of entries (shaded blue scale) shows a preference for halos slightly less massive than the known dSph galaxies.", "Finally, the brightest subhalo expected for Fermi-LAT and CTA are given by the green and yellow stars, respectively.", "The Fermi-LAT subhalo is brighter because of the larger accessible survey area (see sec:fermi and sec:cta for details)." ], [ "Subhalo angular power spectrum", "The angular power spectrum (APS) of the subhalo $\\gamma $ -ray sky maps is a powerful tool for DM analyses and provides another cross-check for our analysis.", "The APS $C_\\ell $ of an intensity map $I(\\vartheta , \\varphi )$ is defined as $C_\\ell = \\frac{1}{2\\ell + 1} \\sum \\limits _m |a_{\\ell m}|^2 \\;,$ with $a_{\\ell m}$ the coefficients of the intensity map decomposed into spherical harmonics $Y_{\\ell m}$ , $I(\\vartheta , \\varphi ) = \\sum \\limits _{\\ell = 0}^{\\ell _{\\rm max}} \\sum \\limits _{m= -\\ell }^{m= +\\ell } a_{\\ell m}\\, Y_{\\ell m}(\\vartheta , \\varphi ).$ CLUMPY's APS calculation relies on the HEALPixhttp://healpix.sourceforge.net/ package [142].", "The median and variance of $C_\\ell $ are calculated for each of the 500 ${J_{\\rm drawn}}$ maps produced for all models, with HEALPix resolution ${N_{\\rm side}}=4096$ .", "The power $C_\\ell $ caused by $J_{\\rm drawn}$ strongly depends on the number of simulated objects, and we show in the convergence study of app:APSconvergence that most of the power at all multipoles is generated by the $\\mathcal {O}(100)$ brightest subhalos.We showed in [87] that the contribution of the unresolved objects, $\\langle J_{\\rm subs}\\rangle $ , to the APS is negligible for $\\ell \\gtrsim 4$ , so this is not discussed further.", "Figure: NO_CAPTIONAs already underlined in previous analyses [144], [145], the subhalo power spectrum is Poisson-like in the range $\\ell \\lesssim 500$ , and flattens at higher multipoles due to the extended size of the brightest (high-mass) subhalos (see fig:APSicompare).", "It is possible to connect the subhalo intensity APS to the source count distributions in fig:Histogram1DJfactor-Nsub-fullsky.", "For point-like sources and at small angular scales, [144] showed that the angular power is approximately constant (Poisson-like) at all multipoles, and can be calculated by the one-subhalo term: $C_{\\ell }\\approx C^{\\rm 1 \\,sh} \\approx \\frac{1}{16\\pi ^2}\\;\\int \\limits _{L_{\\rm min}}^{L_{\\rm max}}\\int \\limits _0^{l_{\\rm max}}\\frac{L^2}{l^2}\\,\\frac{\\mathrm {d}\\overline{n}(l,\\,L)}{dL}\\,\\mathrm {d}l\\, \\mathrm {d}L \\equiv C_{\\rm P}^I,$ where $\\overline{n}(l,\\,L)$ denotes the number density of the subhalos, averaged over the solid angle $\\Omega $ .", "The coordinates $(l,\\,\\Omega )$ denote spherical coordinates with the observer at $l=0$ , and $L$ is the total luminosity of a subhalo.", "For a point-like DM halo, $L=\\frac{\\langle \\sigma v\\rangle }{2\\,m_{\\chi }^2}\\frac{\\mathrm {d}N}{\\mathrm {d}E}\\,\\mathcal {L}$ , with the luminosity $\\mathcal {L}$ as defined in [86].", "It is straightforward to show that $C_{\\rm P}^I = \\frac{1}{4\\pi } \\int _{F_{\\rm min}}^{F_{\\rm max}}F^2 \\frac{\\mathrm {d}N}{\\mathrm {d}F}\\,\\mathrm {d}F,$ where $N$ is the total number of subhalos, and $F$ their flux at the observer." ], [ "APS properties and model comparison.", "To facilitate the comparison to previous works [145], [146], [147], the APS below is given as differential intensity power at $E=[4]{GeV}$ for a $m_{\\chi }= [200]{GeV}$ , $\\chi \\chi \\rightarrow b\\bar{b}$ annihilation channel with thermal relic cross-section $\\langle \\sigma v\\rangle =[3\\cdot 10^{-26}]{cm^3\\,s^{-1}}$ (same as in fig:Skymapscompare).", "The left panel of fig:APSicompare shows the median APS of the LOW and HIGH models and their $68\\%$ confidence intervals, CI, along with the median Poisson-like $C_P^I$ given in eq:C1shfluxAndo, calculated for each model and 500 simulations.", "Here, the flux $F_{\\rm min}$ in eq:C1shfluxAndo has been set to the flux from the faintest drawn object in each realisation.", "As shown in app:APSconvergence, further lowering $F_{\\rm min}\\rightarrow 0$ adds negligible contribution to the integral.", "The flux $F_{\\rm max}$ in eq:C1shfluxAndo is $F_{\\rm max}=F^{\\star }$ , the flux from the brightest object in each skymap.", "The two approaches agree but at high-$\\ell $ where the point-like approximation is not valid anymore.", "Additionally, the medians also differ at low-$\\ell $ for the model HIGH, as the spatially isotropic distribution of the objects is violated here (see next paragraph).", "A main benefit from our approach is to properly propagate the $\\ell $ -dependent uncertainties on the APS (see left panel), which is also not limited to a single simulation and its limited mass resolution; the cosmic-variance uncertainty is of almost one order of magnitude.", "The right panel in fig:APSicompare shows the median $C_{\\ell }$ for the models LOW (black) and HIGH (red), which encompass the results based on either the Via Lactea II [145] or Aquarius simulations [146].", "The APS from Galactic subhalos obtained by [148] (green dashed line in fig:APSicompare) is also based on the Aquarius simulations.The curve for the Galactic APS from [148] has been rescaled for comparison to the other works.", "We adopted $\\mathrm {d}\\Phi ^{\\rm PP}/\\mathrm {d}E([4]{GeV},\\,m_{\\chi }=[200]{GeV},\\,\\chi \\chi \\rightarrow b\\bar{b})= [5.63\\cdot 10^{-32}]{GeV^{-3}\\,cm^{3}\\,s^{-1}}$ , compared to $\\Phi ^{\\rm PP}(E_{\\rm min} = [5]{GeV},\\,E_{\\rm max} = [10]{GeV},\\,m_{\\chi }=[100]{GeV},\\,\\chi \\chi \\rightarrow b\\bar{b})= [2.07\\cdot 10^{-31}]{GeV^{-2}\\,cm^{3}\\,s^{-1}}$ , thus rescaling by a factor $(0.56/2.07)^2\\,{GeV^{-2}}$ .", "However note that the spectral shape for the $\\chi \\chi \\rightarrow b\\bar{b}$ channel assumed by [148] slightly differs from ours.", "However, they semi-analytically computed the spectrum from a generalized version of eq:C1shlumAndo, additionally taking into account a suppression factor $|\\tilde{u}(\\ell )|^2$ due to the spatial extension of the subhalos.", "In the g15784 simulation by [147] (magenta dot-dashed line in fig:APSicompare), only subhalos with masses $>10^{8.6}\\,\\rm M_\\odot $ are resolved, which explains the lower overall power.", "Beyond the validity check, the subhalo APS is an interesting tool for the observational search strategy.", "fig:APSicompare shows that at low multipoles, a small dipole excess ($\\ell =1$ ) is visible for the model HIGH (and none for the model LOW).", "Therefore, we checked if a spatial bias exists for the location of the brightest clumps.", "In fig:Abundancedir, we show the probability distribution to find the brightest object at angular distance $\\theta $ from the direction of the GC for LOW (pale blue) and HIGH (red).", "For model LOW, only a marginal spatial preference exists to find the brightest halo in the direction of the GC.", "This is not the case for model HIGH, where the direction of the GC is clearly preferred.Note that we only consider the subhalo emission, not the DM emission from the smooth Galactic halo, which is highly peaked towards the GC.", "Therefore, this discussion does not apply for the brightness in terms of signal to background ratio.", "Cross-checking with model VAR6, we find this change to be mostly due to the distance-dependent concentration of model HIGH: the closer subhalos are to the GC, the more concentrated, hence brighter, they become.", "We come back later on the fact that the brightest objects are close-by with $D_{\\rm obs}\\approx R_\\odot \\approx \\mathcal {O}([10]{kpc})$ from both the observer and the GC." ], [ "APS and DM sensitivity.", "The APS can be compared to existing data on the anisotropy in the diffuse $\\gamma $ -ray background (DGRB, see, e.g., [47]).", "After 22 months of data-taking, Fermi-LAT reported a significant excess ($>3\\sigma $ ) of $\\gamma $ -ray anisotropy over the photon noise background in the diffuse emission at latitudes $|b|> 30{^{\\circ }}$ , in each of four different energy intervals between $[1]{GeV}\\le E \\le [50]{GeV}$ , and in the multipole range $155\\le \\ell \\le 504$ [143].", "We take their data:cleaned result, given as appropriately rescaled full-sky equivalent power, in differential form at $[4]{GeV}$ .", "It is shown in fig:APSicompare as a grey-shaded segment vertically extending over the $\\pm 1\\sigma $ uncertainty range.", "Accounting for the dominant contribution of distant unresolved blazars [149] and misaligned active galactic nuclei [150] would lower the differential intensity angular power by a factor of a few, leading to a residual anisotropy that could be attributed to DM.", "For instance, several authors have used the measurement from [143] and the blazar contribution from [149], [150] to derive upper limits on the relic cross-section $\\langle \\sigma v\\rangle $ [148], [151], [145], [152].", "A full analysis of the constraints set by the APS data is beyond the scope of this analysis.", "Nevertheless, thanks to our self-consistent derivation of the APS and source count distribution of Galactic DM subhalos, we may comment on the sensitivity of the former, and the relative merit of both approaches.", "Using the result of [143], based on 22 months of Fermi-LAT data with 1FGL point sources removed, Ando and Komatsu [148] typically find upper limits of $\\langle \\sigma v\\rangle \\sim 10^{-24}\\,\\mathrm {cm^3\\; s^{-1}}$ for $m_{\\chi }= [200]{GeV}$ and $b\\bar{b}$ (see their figure 16).", "An order-of-magnitude calculation performed from fig:APSicompare (we recall that $C_{\\ell }\\propto \\langle \\sigma v\\rangle ^2$ ) for models LOW and HIGH encompasses this value, with model HIGH yielding the lowest limit.", "This work also shows that APS-derived limits are affected by a cosmic-variance uncertainty of almost one order of magnitude.", "Alternatively, looking for dark clumps in the 3FGL catalogue (48 months of data), Schoonenberg et al.", "[81] find $\\langle \\sigma v\\rangle \\sim 10^{-25}\\,\\mathrm {cm^3\\; s^{-1}}$ (see their figure 7).", "Their underlying subhalo model is in line with our model HIGH prediction (see fig:Histogram1DJfactor-Nsub-Fermi), and also suffers from a similar variance uncertainty.", "From these crude considerations, the APS approach appears to be somewhat less favourable than dark clump searches.", "Considering many effects such as data homogeneity, masking effects, etc., the comprehensive study of [145] finds similar results.", "In the context of a large-sky survey with the CTA instrument, we present the CTA sensitivity to dark clumps in sec:cta.", "Comparing those results to anisotropy searches with CTA, as suggested by [153], is left for future work." ], [ "Revisiting dark halo searches with ", "The subhalo detection prospects for Fermi-LAT have been investigated several times [63], [64], [65], [66], [67], [79], [80], [81].", "This short section aims at comparing our work to the recently published results of [80], [81] and at commenting on some differences compared to a CTA-like survey (presented in sec:cta).", "To do so, we now move from the full sky approach of the previous section to a setup tailored to the Fermi-LAT experiment.", "Following [81], we (i) investigate the subhalo population outside the Galactic plane at $|b|>10{^{\\circ }}$ , and we adopt the same region of interest, as shown in fig:ROIskymap; (ii) limit the $J$ -factor integration angle to $\\theta _{\\text{int}}=0.8{^{\\circ }}$ , as done by [81], describing Fermi-LAT's $68\\%$ containment radius at $[1]{GeV}$ .", "Figure: Cumulative source count distribution of DM subhalos for the Fermi-LAT setup.", "The coloured bands denote the 1σ1\\sigma standard deviation around the mean N ¯\\overline{N} from the 500 simulations.", "The lower x-x-axis gives JJ-factors and the upper x-x-axis the corresponding flux for a given particle physics model, using 〈σv〉=3·10 -26 cm 3 s -1 \\langle \\sigma v\\rangle = 3\\cdot 10^{-26}\\,\\mathrm {cm^3\\; s^{-1}}.", "The Bertoni et al.", "(dotted line) and Schoonenberg et al.", "results (solid line) are also displayed.", "The vertical dashed lines show the conservative and optimistic detection thresholds chosen by .", "Taking into account an up-to-date LAT sensitivity (see text), these thresholds would move by an approximate factor 2 to the left.fig:Histogram1DJfactor-Nsub-Fermi shows the cumulative source count distributions for this setup and the subhalo models LOW (pale blue) and HIGH (red).", "The lower $x$ -axis presents the subhalo source count distribution in terms of the particle-physics independent $J$ -factor, while the upper $x$ -axis gives the corresponding integrated energy flux distribution above $[1]{GeV}$ (for $m_{\\chi }=[100]{GeV}$ , thermal annihilation cross-section, and pure annihilation into bottom quarks).", "This allows us to directly compare our modelling to the findings of Schoonenberg et al.", "[81] (black solid line)We added their distributions of point-like and extended halos which they discuss separately.", "Note that they use a simplified approach to calculate the $J$ -factors.", "Performing the full line-of-sight integration in CLUMPY, we do not find a strict difference between point-like and extended objects.", "and, with some limitations, Bertoni et al.", "[80] (black dotted line).", "Ref.", "[80] considers only subhalos at $|b|>20{^{\\circ }}$ , while we use $|b|>10{^{\\circ }}$ , meaning that, compared to our setup, the dotted line should be even higher than that shown in fig:Histogram1DJfactor-Nsub-Fermi.The integration angle used in [80] is not specified and the comparison is only valid provided that their calculation of the DM spectra is consistent with [102].", "For low $J$ values, HIGH and LOW nicely encompass the Schoonenberg et al.", "results.", "For the largest $J$ -factors, the HIGH model is consistent with [81] within uncertainties,Ref.", "[81] finds a similar sample variance of $N$ (68% CI), which is not shown in fig:Histogram1DJfactor-Nsub-Fermi.", "and in some tension with [80].", "The authors of [81] estimate the number of detectable subhalos for a specific DM particle physics model from the Fermi-LAT detection threshold in the 3FGL, for sources with a similar, relatively hard spectral shape as expected from DM annihilations.", "In particular considering the $\\chi \\chi \\rightarrow b \\bar{b}$ channel, they assume a conservative detection threshold $F_{\\rm cons,\\,b\\overline{b}}=[1.35\\cdot 10^{-12}]{erg\\,cm^{-2}\\,s^{-1}}$ , and an optimistic detection threshold $F_{\\rm opt,\\,b\\overline{b}}=[4.0\\cdot 10^{-13}]{erg\\,cm^{-2}\\,s^{-1}}$ (both fluxes integrated above $[1]{GeV}$ ).", "For $F_{\\rm opt,\\,b\\overline{b}}$ and the particle physics model chosen in fig:Histogram1DJfactor-Nsub-Fermi, model HIGH predicts $13\\pm 4$ detectable subhalos to be present in the 3FGL, while still $1\\pm 1$ halo could be found for the conservative model LOW; for $F_{\\rm cons,\\,b\\overline{b}}$ , model HIGH (LOW) yields $3\\pm 2$ ($0.1\\pm 0.4$ ) halos.", "We adopted the above flux thresholds and quote the number of detectable clumps based on the 3FGL to ease comparison with the results found by [81].", "However, after eight years of operation, and with the improved Pass8 event reconstruction [140], the LAT sensitivity to faint sources has significantly improved since the 3FGL release.", "For the background-dominated regime between $[1]{GeV}$ and $[10]{GeV}$ , the double exposure yields a factor $\\sqrt{2}$ in improved sensitivity, and the Pass8 reconstruction additionally improves the sensitivity by about $30\\%$ compared to the Pass7 analysis chain.http://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm With this, we approximate the available flux thresholds above $[1]{GeV}$ after eight years observations with Pass8 reconstruction to be $F_{\\rm 8 years} \\approx 0.5\\, F_{\\rm 3FGL}$ .", "Correspondingly, the number detectable clumps increases by factor $\\sim 0.5^{1-\\alpha }\\sim 2$ , with $1-\\alpha \\approx -1$ the slope of the source count distribution in fig:Histogram1DJfactor-Nsub-Fermi.", "Therefore, assuming $\\langle \\sigma v\\rangle \\approx [3\\cdot 10^{-26}]{cm^3\\,s^{-1}}$ and for $m_{\\chi }\\lesssim [100]{GeV}$ , DM subhalos might already have been detected by Fermi-LAT, even under conservative assumptions about the subhalo model and the detection threshold.", "Conversely, we remark that an exclusion of the DM hypothesis for most of the unidentified 3FGL objects is consistent with Fermi-LAT's finding from stacked dSph galaxy observations, $\\langle \\sigma v\\rangle <[3\\cdot 10^{-26}]{cm^3\\,s^{-1}}$ for $m_{\\chi }<[100]{GeV}$ [30], and these limits are now more stringent after four more years data taking.", "We remark that the Fermi-LAT angular resolution strongly improves at energies above $[1]{GeV}$ , reaching a $68\\%$ containment radius of $0.2{^{\\circ }}$ at $\\sim [10]{GeV}$ , and $0.1{^{\\circ }}$ above $\\sim [100]{GeV}$ [15], [140].", "Therefore, we stress that the choice of $\\theta _{\\text{int}}= 0.8{^{\\circ }}$ in this paragraph is not a description of the LAT performance, but serves for comparison of the source count distribution from [81].", "For the plausibility of dark subhalo searches, it is of interest to investigate the physical properties (mass, distance, angular extension) of the brightest subhalo.", "The median properties of the brightest object within the masked Fermi-LAT sky and for the model HIGH are presented in tab:brightestproperties.", "The properties of the brightest subhalo may depend on the angular resolution of the instrument and we find that the object with the largest overall $J$ -factor, $J(\\theta _{\\rm vir})$ , is not necessarily the same as the object with the largest $J$ -factor within some $\\theta _{\\text{int}}< \\theta _{\\rm vir}$ .", "This is accounted for in tab:brightestproperties, and the brightest object is listed separately within the integration angles $\\theta _{\\rm vir}$ , $0.8{^{\\circ }}$ and $0.1{^{\\circ }}$ .", "However, this differentiation results into barely different objects within the resolutions of Fermi-LAT (in contrast to CTA, which is discussed later).", "For Fermi-LAT, the subhalo with the median largest $J$ -factor is typically $[8^{+11}_{-6}]{kpc}$ away, and has a mass between $\\sim [10^7]{\\rm M_\\odot }-[10^9]{\\rm M_\\odot }$ .", "We finally remark that in case of potential DM subhalo candidates, alternative origins for the signal (VHE blazars, milli-second pulsars) have to be ruled out.", "Various approaches for disentangling DM and astrophysical sources are presented in [76], [75], [81], [154].", "Especially CTA will be a suitable instrument to perform dedicated follow-up observations of source candidates, and to resolve different origins by the temporal, spectral and spatial morphology of a candidate.", "Besides, CTA will be able to perform a large-area survey on its own, entering an energy regime beyond the accessibility of Fermi-LAT.", "In the following, we will now investigate whether a CTA survey can be used to search for DM subhalos." ], [ "CTA sensitivity to dark clumps", "This section presents the CTA sensitivity to dark clumps (for the sake of readability, all the technical details are postponed to app:CTAanalysisdetails).", "We first introduce the salient features of CTA and CTA's extragalactic survey (sec:nominalsensitivity), present the instrument background (sec:CTAbackground), the characteristics of the brightest clump for CTA (sec:brightestCTA), the likelihood and statistics used to draw our CIs (sec:logLTS), and then the ensuing sensitivity for CTA to these objects (sec:CTAresults).", "We underline that the knowledge of the probability distribution of the brightest clumps is mandatory to set robust constraints on DM detection.", "As discussed and shown in sec:fattails, this distribution has a long tail, so that the construction of credible intervals (CI) must rely on large samples to be correctly defined.", "For this reason, all the results presented in this section are based on $10^4$ CLUMPY runs of the models LOW and HIGH." ], [ "Observation setup and nominal sensitivity", "CTA is the next-generation ground-based $\\gamma $ -ray observatory, using the technique of imaging atmospheric Cherenkov radiation (`Imaging Atmospheric Cerenkov Telescope', IACT).", "It will feature an unprecedented resolution in energy and angular separation for $\\gamma $ -rays in the range between $30\\,\\mathrm {GeV}\\lesssim E_{\\gamma }\\lesssim 200\\,\\mathrm {TeV}$ , and, over the whole energy range, an effective collection area of about an order of magnitude larger than current IACT [82].", "Having a large field-of-view, CTA will be the first IACT to efficiently perform large-area surveys in VHE (very high energy, $\\gtrsim [100]{GeV}$ ) $\\gamma $ -rays.", "In particular, it plans to perform an extragalactic survey with an approximately uniform exposure over $25\\%$ of the sky [85].", "This survey aims at an unbiased population study of extragalactic sources, primarily to search for `dark particle accelerators' without any counterparts at other wavelengths.", "Analogously, it can be used for a search for Galactic DM annihilation in dark subhalos.", "CTA will consist of two arrays, one in the northern and one in the southern hemisphere.", "We assume, for simplicity, that most of the survey will be performed by the southern array, in circular region around the Galactic south pole ($b<-30{^{\\circ }}$ ).Depending on the final location/latitude of CTA (southern site), a substantial part of the extragalactic survey area has to be covered by CTA North.", "Moreover, some of the area we selected may be unreachable for CTA (only observable at elevations $<40{^{\\circ }}$ ), or the survey may reach smaller Galactic latitudes $|b|\\lesssim 10{^{\\circ }}$ .", "This choice covers the projected $f_{\\rm sky}=25\\%$ , as illustrated in fig:ROIskymap, and excludes the area close to the GC (see fig:Abundancedir), where, for the model HIGH, bright subhalos are found with the highest probability; therefore, scanning lower Galactic latitudes or the combination with a Galactic plane survey could in principle increase the sensitivity to dark halos.", "Around 400 to 600 hours will be available for the extragalactic survey with CTA [85].", "With an uniform distribution of this observing time over the whole survey area, a sensitivity to fluxes of about $[2.5\\cdot 10^{-12}]{cm^{-2}\\,s^{-1}}$ to $[1~\\cdot ~10^{-11}]{cm^{-2}\\,s^{-1}}$ above $100\\,\\mathrm {GeV}$ (approximately corresponding to $0.5\\%-2\\%$ the flux of the Crab nebula [155]) can be reached for a Crab-nebula-like energy spectrum [85].", "We discuss further the uniform observation strategy in the context of dark subhalos in app:obsstreg." ], [ "Diffuse $\\gamma $ -rays and residual background", "Earth-based $\\gamma $ -ray observatories suffer from a large residual background from cosmic rays.", "While sophisticated analysis techniques are able to efficiently discriminate the $\\gamma $ -ray signal from the hadronic background, it is almost impossible to separate it from cosmic electrons.", "In fig:plotrates, we show the residual background rate estimation for CTA (blue dashed-dotted line), after applying background rejection cuts optimised for an observation over $30\\,\\mathrm {min}$ , the typical observing time per field.", "This background rate is computed by MC simulations of the detector response to the cosmic-ray intensity, and represents the events passing the analysis cuts.", "In black, we show the event rate of the diffuse $\\gamma $ -rays, resulting from integrating the $\\gamma $ -ray intensity over the CTA on-axis effective area.", "We take the total $\\gamma $ -ray intensity above $|b|>20{^{\\circ }}$ , measured by Fermi-LAT up to $820\\,\\mathrm {GeV}$ [156], to compare it with the CTA residual background.", "This intensity contains all $\\gamma $ -rays outside the Galactic plane, from resolved and unresolved sources.", "Thus, this spectrum gives a rather conservatively large value for the total diffuse $\\gamma $ -ray emission outside the Galactic plane.", "From fig:plotrates (differential rates) and tab:cta-diffuse-background (integrated rates), it can be seen that the residual background outweighs the diffuse $\\gamma $ -ray emission by a factor of 1000.", "Therefore, we can safely ignore all diffuse $\\gamma $ -ray backgrounds on top of the residual cosmic-ray background, including those from Galactic and extragalactic DM." ], [ "Characteristics of the brightest halos", "Similarly to the discussion in sec:fermi for Fermi-LAT, we investigate the source count distribution for a $J$ -factor integration angle tailored to the angular resolution of the instrument, and display the result in fig:Histogram1DJfactor-Nsub-CTA for models LOW and HIGH.", "We present the distributions for the integration angles $\\theta _{\\text{int}}=0.05{^{\\circ }}$ (angular resolution of CTA at $\\gtrsim [1]{TeV}$ ), $\\theta _{\\text{int}}=0.1{^{\\circ }}$ (angular resolution of CTA at $\\lesssim [1]{TeV}$ ), and the full emission, $\\theta _{\\text{int}}=\\theta _{\\rm vir}$ .", "Comparing fig:Histogram1DJfactor-Nsub-CTA (CTA scenario) to fig:Histogram1DJfactor-Nsub-Fermi (Fermi-LAT scenario) shows that the CTA source count distributions shift to lower $J$ -factors, due to a factor $\\sim 3$ smaller survey field, and smaller integration angles.", "We also show in fig:Histogram1DJfactor-Nsub-CTA the subhalo distribution assumed by [84] for a survey field characterised by $f_{\\rm sky}=25\\%$ , based on the VL II subhalo catalogue.", "These authors consider an integration over the entire extent of the subhalos (i.e.", "$\\theta _{\\text{int}}=\\theta _{\\rm vir}$ ), dismissing only highly extended subhalos, and their distribution is in fair agreement with our model HIGH.", "This is consistent with the fact that our model HIGH approximately matches the distribution derived from VL II also for the Fermi-LAT setup (see fig:Histogram1DJfactor-Nsub-Fermi).", "However, we emphasize that assuming the whole subhalo flux, $J(\\theta _{\\rm vir})$ , originating from a point source heavily overestimates the actual CTA performance.", "An exact treatment of the sensitivity must account for the energy dependent angular resolution and the extension of the source, and is done in the next subsections.", "In fig:Histogram1DJfactor-Nsub-CTA, we show the result from this rigorous treatment.", "The dashed lines show that for the considered DM particle, the sensitivity to the full emission from the brightest, extended halo (model HIGH) roughly corresponds to the sensitivity to a point source with the smaller flux within $0.1{^{\\circ }}$ , $J(0.1{^{\\circ }})$ .", "Figure: Cumulative source count distribution of DM subhalos for the CTA setup.", "The upper x-x-axis shows the flux level for the DM particle model to which CTA is most sensitive (adopting the subhalo model HIGH; see fig:IndividualSourceSensitivity-CTA).", "The annihilation cross-section is chosen so that CTA would observe one subhalo above the flux sensitivity threshold corresponding to the chosen DM annihilation spectrum.", "Increasing (decreasing) the annihilation cross-section would result in a shift of the upper x-x-axis and the vertical dashed lines to the left (right).", "We also display the result of Brun et al.", ", who used θ int ≈θ vir \\theta _{\\text{int}}\\approx \\theta _{\\rm vir}.tab:brightestproperties lists the median properties of the brightest object for Fermi-LAT and CTA at various integration angles.", "The median $\\widetilde{J}^{\\star }(\\theta _{\\rm int})$ is obtained from $10^4$ MC simulations (see also sec:fattails).", "In terms of halo properties, the population of brightest sources resembles the one for Fermi-LAT, and consists of close and rather massive halos; we remind the reader that we have identified the distance-dependence subhalo concentration as the main driver of this behaviour.", "The high angular resolution of CTA implies that changing the integration angle more drastically affects the brightest source properties than for Fermi-LAT: the smaller the integration angle, the lighter (and closer) the average brightest object becomes (${M}_{\\rm vir}\\gtrsim [10^7]{\\rm M_\\odot }$ for $\\theta _{\\text{int}}\\lesssim 0.05{^{\\circ }}$ ).", "The subhalo source count distributions estimated so far include the most massive DM clumps, which would have formed stars and and host the dSph galaxies of the MW.", "Therefore, VHE $\\gamma $ -rays from DM annihilation in these objects will most likely be discovered by dedicated pointed VHE observations.", "Including dSph objects into the estimation of the CTA survey sensitivity to dark clumps might therefore bias the number of potentially detectable sources, even more so when deep-sky and large-area optical surveys might discover additional faint dSph galaxies.", "To study the impact of this issue, we computed the subhalo abundance in the CTA scenario (25% sky coverage), neglecting clumps heavier than $[10^6]{\\rm M_\\odot }$ or $[10^7]{\\rm M_\\odot }$ (tab:MandJ in the appendix shows that all the known dSph galaxies have a median mass larger than $[10^6]{\\rm M_\\odot }$ , and most objects are likely to have masses larger than $[10^7]{\\rm M_\\odot }$ ).", "We find that when excluding all objects above $[10^7]{\\rm M_\\odot }$ , the median $J$ -factor within $\\theta _{\\text{int}}= 0.05{^{\\circ }}$ of the brightest subhalo is not affected, with $\\log _{10}(\\widetilde{J}^{\\star }(0.05{^{\\circ }})/\\rm {GeV^2\\, cm^{-5}}) = 19.7^{+0.3}_{-0.2}\\,$ .", "Looking over the full extent of the DM subhalo, we find $\\log _{10}(\\widetilde{J}^{\\star }(\\theta _{\\rm vir})/\\rm {GeV^2\\, cm^{-5}}) = 20.0^{+0.4}_{-0.3}$ , i.e.", "a factor $\\sim 2$ decrease compared to tab:brightestproperties.", "This behaviour is understood as a lighter, but more concentrated halo is then selected as brightest object by the exclusion criterion.", "The situation changes when rejecting all objects above $[10^6]{\\rm M_\\odot }$ , where we obtain $\\log _{10}(\\widetilde{J}^{\\star }(0.05{^{\\circ }})/\\rm {GeV^2\\, cm^{-5}}) = 19.5^{+0.3}_{-0.2}\\,$ and $\\log _{10}(\\widetilde{J}^{\\star }(\\theta _{\\rm vir})/\\rm {GeV^2\\, cm^{-5}}) = 19.8^{+0.4}_{-0.3}$ .", "The median brightest dark subhalo is then expected to be a factor 2 dimmer within the central $\\theta _{\\text{int}}=0.05{^{\\circ }}$ .", "The prospects of survey discoveries are therefore only marginally affected by the distinction between dark and bright DM halos.", "This point is not considered further in the remainder of this paper, where we use the values in tab:brightestproperties (right) to characterize the brightest halo properties for a CTA extragalactic survey.", "Table: Median properties of the brightest subhalo for the survey setups tailored to the Fermi-LAT and CTA instruments, and for the subhalo model HIGH.", "The uncertainties denote the 68%68\\% CI around the median.", "For both instruments, the results for different angular resolutions are given.D obs D_{\\rm obs} is the distance from the observer, and RR the distance from the GC.", "m vir {m}_{\\rm vir} is the subhalo mass.", "r vir {r}_{\\rm vir} and r s {r}_{\\rm s} denote its virial and scale radius, c vir =r vir /r s {c}_{\\rm vir} = {r}_{\\rm vir}/{r}_{\\rm s}, and θ vir ,s =arctan(r vir ,s /D obs )\\theta _{\\rm vir,\\,s}=\\arctan (r_{\\rm vir,\\,s}/D_{\\rm obs}).", "θ h \\theta _{\\text{h}} is the radius enclosing half of the total emission, J(θ h )=0.5J(θ vir )J(\\theta _{\\text{h}}) = 0.5\\,J(\\theta _{\\rm vir}).", "For reliable medians, the values are obtained from a sample of 10 4 10^4 simulations." ], [ "Likelihood ratio and test statistic ($\\mathrm {TS}$ )", "We use the open-source CTA analysis software ctoolshttp://cta.irap.omp.eu/ctools/, based on the gammalib libraryhttp://cta.irap.omp.eu/gammalib/ [157] to compute the CTA sensitivity to the median brightest dark subhalo.", "The ctools framework allows the use of a maximum-likelihood inference of hypotheses $\\mathcal {M}$ from event data, considering all available spatial and spectral information from the data (see app:likelihood for the likelihood ${L}$ ).", "The cssens tool is used to simulate events and subsequently calculate the maximum log-likelihood ratio, with the likelihood ratio $\\lambda $ given by = L(Mbkg(bkg) |  X)L(Msig(sig)+ Mbkg(bkg) |  X)= L(Mbkg(bkg) |  X)L(Msig(sig)+ Mbkg(bkg) |  X).", "with ${X}= (N_{\\rm obs},\\,E_{\\rm {obs},\\,1\\ldots \\mathit {N}_{\\rm obs}},\\vec{k}_{\\rm {obs},\\,1\\ldots \\mathit {N}_{\\rm obs}})$ the mock data$N$ , $E$ , and $\\vec{k}$ are the number of photons, energy and direction respectively., ${\\Theta }$ the adjustable parameters in the models maximising the likelihood, and $\\widehat{{\\Theta }}$ the corresponding maximum likelihood estimators ($\\widehat{\\widehat{{\\Theta }}}_{\\rm bkg}$ under the constraint $\\mathcal {M}_{\\rm sig}=0$ ).", "For the background fit, we allow the normalisation of the rate to vary.", "The signal model $\\mathcal {M}_{\\rm sig}$ for DM consists, according to eq:flux-general, of the spatial part of our fixed $J$ -factor map (see next subsection), and a spectral part depending on the particle mass $m_{\\chi }$ and the annihilation cross-section $\\langle \\sigma v\\rangle $ .", "We scan 24 DM particle masses in $[50]{GeV}\\le m_{\\chi } \\le [100]{TeV}$ , computing for each $m_\\chi $ the $\\gamma $ -ray spectrum $\\mathrm {d}N_{\\gamma }^f/\\mathrm {d}E$ from [102].", "For each spectrum, we set the flux level to be the only free parameter, such that ${\\Theta }_{\\rm sig}=\\langle \\sigma v\\rangle $ .", "We use the logarithm of the likelihood ratio eq:lambda as the test statistic TS to exclude the signal hypothesis $\\mathcal {M}_{\\rm bkg}+\\mathcal {M}_{\\rm sig}$ (at the confidence level $1-p_{\\rm pre}$ ), namely $\\mathrm {TS}= -2\\log \\lambda .$ The $\\mathrm {TS}(p_{\\rm pre})$ values were calculated directly from a set of MC simulations, and we refer the interested reader to app:tsstatistics for the technical details and used $\\mathrm {TS}$ values.", "More importantly, we cross-checked our analysis method by calculating the sensitivity to the Sculptor dSph galaxy, and found our results consistent with Carr et al.", "(2015) [90]." ], [ "Sensitivity to dark clumps and comparison to other targets", "To calculate the CTA sensitivity to the brightest subhalo in the survey field, we build a template of the median brightest object described in sec:brightestCTA.", "We choose the $J$ -factor profile to be that of the object found to be the brightest within $\\theta _{\\text{int}}=0.05{^{\\circ }}$ (see tab:brightestproperties).", "We emphasize that the latter choice only determines the shape of the template halo, and the Likelihood-based sensitivity analysis is run over the full spatial extent of that halo.", "The instrument response, required in eq:lambda and in the likelihood function eq:likelihood-function, is taken from [83], using the publicly available results from the `Production 2' simulations.http://portal.cta-observatory.org/Pages/CTA-Performance.aspx The available instrument response data corresponds to on-axis observations of Crab-nebula-like point sources, which can be safely adopted for the highly cuspy DM template halo (with a half emission radius of $\\theta _{\\text{h}}=0.13{^{\\circ }}$ , only slightly above the CTA angular resolution).", "We use the response tables with background rejection cuts optimised for a $30\\,\\mathrm {min.", "}$ observation at a constant elevation of $70{^{\\circ }}$ with CTA (southern site),We cross-checked the analysis with the 'Production 2' rejection cuts optimized for a $5\\,\\mathrm {h}$ observation and obtained the same sensitivities.", "assuming 4 large-size telescopes, 24 medium-size telescopes and 72 small-size telescopes.", "Different observation strategies have been proposed to raster the CTA survey field with overlapping observations to obtain a preferably homogeneous exposure [85].", "We mimic the survey coverage by an on-axis observation of the template halo with a one hour exposure.", "With this choice, we calibrate our template observation setup to the optimistic benchmark performance projected for the CTA extragalactic survey, and obtain a similar sensitivity to a Crab-like point source as in [85].We obtain project a survey sensitivity to fluxes $\\gtrsim [4\\cdot 10^{-12}]{cm^{-2}\\,s^{-1}}$ above $[100]{GeV}$ (approximately $0.7\\%$ of the Crab nebula flux) for a point source with a Crab-nebula-like spectrum, requiring a test statistic of $\\mathrm {TS}= 25$ and without applying a trials correction.", "This is in an optimistic agreement with [85].", "For comparison, a point source with a spectrum $\\chi \\chi \\rightarrow \\tau ^+\\tau ^-$ , $m_{\\chi }=[500]{GeV}$ (as discussed in fig:Histogram1DJfactor-Nsub-CTA), would be detected at $F_{\\rm TS=25}=[7.8\\cdot 10^{-12}]{cm^{-2}\\,s^{-1}}$ above $[100]{GeV}$ .", "Figure: NO_CAPTIONIn fig:IndividualSourceSensitivity-CTA, we show the projected sensitivity of CTA to the WIMP DM annihilation cross-section by searching for Galactic dark subhalos in the planned extragalactic survey: In the left panel, we present the sensitivity for two benchmark annihilation channels with $100\\%$ branching ratio of $\\chi \\chi \\rightarrow b\\bar{b}$ and $\\chi \\chi \\rightarrow \\tau ^+\\tau ^-$ , respectively.", "The solid lines represent the sensitivity to the template halo with median $J$ -factor.", "The shaded bands denote the $68\\%$ ($95\\%$ ) statistical uncertainty around this median, originating from the $J$ -factor variance.", "It can be seen that the sample-to-sample variation of the sensitivity scatters over almost one order of magnitude within the 68% CI, and two orders of magnitude within the 95% CI.", "Also, the sample variance has a long tail towards low values of $\\langle \\sigma v\\rangle $ , such that despite a relative modest median sensitivity, the 95% CI reaches relatively small annihilation cross-sections.", "The shape of the $J$ -factor sample variance, which underlies the sensitivity variance, is discussed in detail in app:meanmedian.", "In the central panel, we show the impact of the model uncertainty onto the sensitivity.", "The one order of magnitude difference in the predicted fluxes between the model LOW and HIGH translates into the equivalent difference in sensitivity.", "We also calculated the sensitivity at the $1-p_{\\rm post}\\equiv 1-p_{\\rm pre}$ confidence level (see app:preposttrial), neglecting the trials penalty (dashed line), that accounts for a putative improvement of the sensitivity by more than a factor of 5.", "In the right panel, we put the sensitivity obtained from this study into broader context.", "A CTA survey search for dark subhalos provides a less powerful probe for DM annihilation than CTA pointed observations of the Galactic halo, and to a lesser extent, than the MW satellite galaxies.", "However, these other targets also suffer from systematic errors.", "fig:IndividualSourceSensitivity-CTA also shows the limits from current experiments for comparison.", "It is visible that the GC provides the best limit at high energy, whereas the Fermi-LAT experiment already reaches the CTA parameter space below TeV energies.", "Before concluding, we briefly comment on the results of Ref.", "[84], who previously discussed the CTA sensitivity to dark clumps.", "Based on the Via Lactea II simulation and a similar survey in area (1/4 of the sky, though towards the GC), they find a more favourable sensitivity than the one we obtain.", "For instance, in the $b\\bar{b}$ channel, our calculation reaches a minimum (pre-trial) $\\langle \\sigma v\\rangle \\gtrsim 2\\times 10^{-24}$ to compare to $\\langle \\sigma v\\rangle \\gtrsim 4\\times 10^{-26}$ in [84].", "Several reasons may be at the origin of this difference.", "First, our limit is based on the 95% CL whereas theirs is based on 90% CL.", "With this choice, we find a factor of 2 improvement on the pre-trial sensitivity (the post-trial sensitivity is not affected).", "Second, Ref.", "[84] models the CTA instrument characteristics starting from the H.E.S.S.", "instrumental response, assuming a factor 10 larger effective area and a factor 2 better background rejection.", "However, the improvement of the CTA performance is energy-dependent, and according to the `Production 2' simulations, the largest improvement in differential sensitivity compared to current instruments will be reached at energies above $\\sim [1]{TeV}$ .", "Because CTA will be most sensitive to WIMP masses of $m_{\\chi }\\approx [1]{TeV}$ (annihilation products below $[1]{TeV}$ ), the applied H.E.S.S.", "extrapolation most likely overestimated the CTA sensitivity to DM.", "Third, Ref.", "[84] assumes that the total subhalo $J$ -factors, $J(\\theta _{\\rm vir})$ , is enclosed in the instrumental resolution.", "This overestimates the flux of the brightest subhalo by another factor $\\sim 2$ .", "For these reasons, we are confident our analysis provides a more realistic estimate of the CTA sensitivity to dark subhalos." ], [ "Summary and discussion", "In this paper, we have revisited the detectability of dark clumps for present and future $\\gamma $ -ray instruments.", "Using the CLUMPY code, we have simulated several distributions of the subhaloe in the Galaxy, in order to critically assess the range of potential number of detectable clumps and to identify the most important parameters of such prediction.", "For each model, several hundreds of skymaps have been generated to obtain the statistical properties (mass, distance, $J$ -factor, etc.)", "for these configurations.", "The model dubbed HIGH (LOW) provides a realistic (conservative) benchmark model for the number of expected dark clumps.", "These benchmark models have been used to discuss the prospects of dark clumps detection for the Fermi-LAT and CTA instruments, and we compared our results and exclusion limits on annihilating DM to several previous calculations.", "We underline that we have carefully validated our analysis at each step of the calculation (validation and cross-checks for the distribution of $J$ and the APS from CLUMPY, and for the sensitivity calculations for CTA).", "Our findings are summarised below.", "First, for the substructure modelling, we find that: The concentration parametrisation is the main uncertainty in the determination of the number of halos with the largest $J$ -factors.", "Indeed, the brightest detectable dark clumps are found to be close by, deep in the potential well of the Galaxy, where the impact of tidal stripping on the concentration and brightness is the most critical.", "As such, our LOW model serves as a baseline for a conservative estimate, in which no tidal stripping is accounted for (subhalos assumed to be as `field' halos').", "Our HIGH model is based on [67], which agrees well, in the regime that matters for detecting dark clumps, with the very recently estimated distance-dependent concentration parametrisation of Ref. [129].", "This makes HIGH a likely realistic benchmark model, though further improvements and understanding of the stripping effect is necessary to strengthen this conclusion.", "The number of calibration subhalos $N_{\\rm calib}$ (between $10^8$ and $[10^{10}]{\\rm M_\\odot }$ ) and the boost from sub-subhalos in the external parts of the subhalos are the next impacting, though subdominant factors.", "First, moving from $N_{\\rm calib}=300$ in model HIGH to half this number in model VAR6, as hinted at by hydrodynamical simulations [105], [106], translates into a similar decrease for $J$ of the brightest object and the corresponding sensitivity.", "It is thus desirable to better constrain this number.", "Second, the boost factor is expected to be at most of a few for $10^7-10^8\\rm M_\\odot $ subhalos [129], [126].", "Compared to our conservative analysis in which this boost was discarded, this could slightly improve the prospects for Fermi-LAT, but probably not for CTA: for background-dominated instruments like CTA, the best sensitivity is achieved at the angular resolution (to decrease the background), for which the `boosted' outskirts of the subhalos are not encompassed.", "The uncertainties in the other parameters (inner profile, index of the clump mass distribution, spatial distribution, width of the mass-concentration distribution) impact the overall level of the diffuse DM emission in the Galaxy, hence changes the contrast of the subhalos.", "However, it leaves mostly unchanged the number and signal strength of the brightest clumps.", "A second aspect of our analysis, made possible by the reasonable running time of CLUMPY, is the determination of the statistical properties of the detectable clumps.", "We have shown that: The brightest clumps are typically located at $\\sim 10-20$  kpc from the GC and from us, with concentration values $\\sim 40-50$ .", "Fermi-LAT and CTA probe slightly different populations, with larger and more massive subhalos for Fermi-LAT ($M\\lesssim [10^8]{\\rm M_\\odot }$ , $\\theta _{\\text{h}}\\lesssim 0.2^{^{\\circ }}$ , and $J\\sim [3\\cdot 10^{20}]{\\rm {GeV^2\\, cm^{-5}}}$ ) than for CTA ($M\\gtrsim [10^7]{\\rm M_\\odot }$ , $\\theta _{\\text{h}}\\gtrsim 0.1^{^{\\circ }}$ , and $J\\sim [10^{19}]{\\rm {GeV^2\\, cm^{-5}}}$ ).", "These halo masses and $J$ -factors are close to the values obtained for dSph galaxies, and this raises the question whether the calculated sensitivity applies for truly dark halos or objects that could be discovered as nearby and very faint dSph galaxies in the future.", "However, we have shown that the calculated sensitivities are only mildly degraded (less than a factor 2) when rejecting masses above $M>[10^6]{\\rm M_\\odot }$ in a CTA large-sky survey scenario.", "In any case, CTA is sensitive to smaller mass ranges than Fermi-LAT.", "This difference is related to the angular resolution of the instruments ($\\sim 0.05^\\circ $ for CTA compared to $\\gtrsim 0.1^\\circ $ for Fermi-LAT), illustrating that instruments with better angular resolution are sensitive to lower mass subhalos, which are more likely to be dark.", "CTA will therefore provide a complementary view to the observation of MW satellite galaxy observations.", "The properties of the brightest clumps derived from the statistical assessment underlie a large sample variance.", "The $J$ -factor of the brightest clump varies by almost a factor 10 at 68% CI (100 at 95% CI).", "In particular, the distribution has a long tail towards large $J$ -values.", "On the one hand, this means that a dark DM subhalo might be detected even for a relatively small annihilation cross-section.", "On the other hand, the large sample variance introduces a large systematic uncertainty for the limits derived from non-observation of DM candidates in the surveys.", "Tidal effects of subhalos, modelled here via the distance-dependent concentration, leads to an anisotropy in the distribution of the directions of the brightest clumps, as seen in the small dipole power excess (compared to models using field halos for the concentration leading to an isotropic distribution).", "The impact of this effect on detectability and sensitivity is not straightforward to assess: (i) the contrast of the dark clumps w.r.t.", "the DM diffuse emission decreases towards the GC, which should mostly affect the detectability when pointing towards $\\theta \\lesssim 10^\\circ $ ; (ii) however, this should be balanced with the existence of a preferential direction to search for these dark halos (4 times more likely to lie towards the GC direction than towards the anticentre), for which the analysis could be optimised.", "In any case, the number of bright dark halos searched for in an analysis is small (by definition) and variance dominated at low$-\\ell $ .", "A dedicated study is necessary to assess the potential benefit of an optimised search.", "Finally, a last aspect of our analysis deals with the sensitivity of CTA, and to some extent of Fermi-LAT, to dark clumps, compared to previous calculations and other targets: For an observation setup such as that of Fermi-LAT, a comparison of the expected number of clumps to previous calculations confirms that Ref.", "[80] obtain a subhalo abundance slightly larger than our model HIGH.", "Our results are more in line with those of Ref.", "[81], which discusses possible explanations for the difference to the results by [80].", "However, compared to [81], we do not find two different populations and behaviours `point-like' vs `extended' source in our analysis (we perform the full integration along the line of sight, whereas [81] do not).", "As a result the shape of the number of expected dark clumps as a function of $J$ is not the same.", "The imprint of the population of DM galactic subhalos also shows on the APS, and our models LOW and HIGH encompass previously published studies.", "For a CTA large-sky survey scenario, we have based our sensitivity calculation on the projected extragalactic survey [85].", "We provide both a pre-trial and post-trial sensitivity, as scanning a quarter of the sky results in a million independent trials that must be accounted for in setting the limits.", "For pure annihilations into $b\\bar{b}$ , we find the best post-trial median sensitivity reach of $\\langle \\sigma v\\rangle \\sim 6\\times 10^{-24}~{cm^3~s^{-1}}$ at $m_{\\chi }\\sim 1$  TeV, and for the $\\tau ^+\\tau ^-$ annihilation channel, the best sensitivity of $\\langle \\sigma v\\rangle \\sim 1\\times 10^{-24}~{cm^3~s^{-1}}$ at $m_{\\chi }\\sim 500$  GeV.", "The 68% (95%) CI changes these values by a factor $\\sim 5$ ($\\sim 10$ ).", "In any case, taking the post-trial limit into account worsens the sensitivity by a factor $\\sim 5$ .", "The sensitivity reach of CTA (in model HIGH) for dark clumps in the extragalactic survey is quite competitive with dedicated dSph galaxies observations.", "Looking at the sensitivity based on the most-likely values favours the 500 hrs single bright dSph observation over that of the extragalactic survey (factor $\\sim 3$ ), but we recall that both suffer from large uncertainties.", "In particular, it has been shown that Segue 1's $J$ -factor (used in our comparison plots) could be significantly lower than previously estimated [98], [141].", "In any case, the best target remains the GC region.", "To conclude, we have shown that a CTA dark halo search provides a complementary view on a different population of subhalos compared to observing dSph galaxies.", "The planned CTA astronomical surveys will therefore nicely complement the dedicated DM searches programme (a similar analysis as done here could be performed for decaying dark matter).", "Given the unprecedented angular resolution of CTA, the search for small-scale anisotropies in its data, e.g., as proposed by [153], may provide another complementary constraint on $\\gamma $ -ray emission from Galactic dark clumps to that we have presented in this article.", "This article has gone through internal review by the CTA Consortium, and we thank Hannes Zechlin and Josep Martí for their careful reading and suggestions that helped to improve the manuscript.", "We also thank the anonymous referee for useful suggestions that have improved the quality of the paper.", "We warmly thank V. Bonnivard for helping out with the Jeans analysis of the recently discovered dSph galaxies, and L. Gérard for valuable help with the CTA analysis.", "We also thank I. Sadeh for useful discussions and comments to the manuscript, and R. Kieokaew for performing useful studies preceding this work.", "This work has been supported by the Research Training Group 1504, “Mass, Spectrum, Symmetry”, of the German Research Foundation (DFG), by the “Investissements d'avenir, Labex ENIGMASS\", and by the French ANR, Project DMAstro-LHC, ANR-12-BS05-0006.", "Some of the results in this article have been derived using the HEALPix package [142]." ], [ "Halo overdensity definition $\\Delta $ and cosmology", "At a given redshift $z$ , the enclosed mass $M_\\Delta $ in a dark matter halo is generically defined as the mass contained within a radius $R_\\Delta $ inside of which the mean interior density is $\\Delta $ times the critical density $\\rho _c$ : $M_\\Delta = \\frac{4\\pi }{3}R_\\Delta ^3\\times \\Delta \\times \\rho _c.$ The spherical top-hat collapse model provides the virial overdensity $\\Delta ={\\Delta }_{\\rm vir}$ [159] which, for the family of flat cosmologies, $\\Omega _m+\\Omega _\\Lambda =1$ (i.e., $\\Omega _K=0$ ), can be approximated by ${\\Delta }_{\\rm vir}{} \\simeq (18\\pi ^2 + 82x -39 x^2)/\\Omega _m(z)$ , with $x=\\Omega _m(z)-1$ , $\\Omega _m(z)=\\Omega _m(0)(1+z)^3/E^2(z)$ , and $E^2(z)=\\Omega _m(0)(1+z)^3+ \\Omega _\\Lambda $ [160].", "In this paper we use $\\Omega _m(0)=0.308$ and $\\Omega _\\Lambda =0.692$ [161], leading to ${\\Delta }_{\\rm vir}\\simeq 332$ .", "In principle, the virial radius ${R}_{\\rm vir}$ can be used as a criterion to identify bound objects.", "In the spherical collapse model, this radius can be interpreted as a threshold separating a region within which the material is virialized and an external region where mass is still collapsing onto the object.", "Because the details of the collapse and virialization are not-well understood, several choices have been made in the literature to identify halos in simulations, such as $\\Delta ={\\Delta }_{\\rm vir},\\,200,\\,500,\\,\\dots $ (see, e.g., [162]).", "We refer the reader to [163] for a study on whether there exists a more `universal' choice.", "CLUMPY works with $\\Delta ={\\Delta }_{\\rm vir}$ , and it uses conversion formulae from [164] to convert `200' to `${\\rm vir}$ ' quantities, whenever necessary.", "This choice can in principle impact the estimation of the halo and subhalo mass shown in several plots of this study.", "However, in CLUMPY, the mass is mostly a proxy to calculate structural parameters of subhalos, via the $c_\\Delta -M_\\Delta $ relationship: the exact definition for $\\Delta $ does not matter as long as the calculation and conversions are done consistently—the uncertainties on the signal are dominated by our ignorance on the dark matter subhalo population, not by the conversion factors.", "Then, for comparisons to the mass or $J$ -factor of real data, as discussed in app:mvirJ, the uncertainties are larger than the difference one would obtain by using different choices for $\\Delta $ .", "Finally, we could also ask how sensitive is the $c_\\Delta -M_\\Delta $ relationship to cosmological parameters.", "For instance, in sec:setsofmodels, the results based on the Via Lactea II [66], Aquarius [58], and ELVIS [165] simulations all start with different cosmologies (from WMAP 3 to WMAP 7 [166]).", "However, [129] finds a very weak dependence of $c_\\Delta -M_\\Delta $ on the cosmological parameters, within their statistical dispersion.", "We refer the reader to [167] for a pedagogical introduction to the link between various choices of $\\Delta $ , the $c_\\Delta -M_\\Delta $ relationship, and the cosmological parameters." ], [ "Mean and variance from analytical approximations", "Analytical formulae are always useful to cross-check numerical calculations and to better identify the underlying important parameters.", "Under the approximation of power-law source count distributions, we focus in this appendix on (i) the mean and median of the number of dark clumps above a given $J$ -factor, which is ultimately related to the sensitivity of an instrument to dark clumps and on (ii) the mean and variance of the angular power spectrum of the $\\gamma $ -ray signal.", "The analytical results are compared to the direct calculations from our simulations in several places in the paper." ], [ "Power-law source count distribution", "As seen in fig:Histogram1DJfactor-Nsub-fullsky, the mean number of halos $\\overline{N}(\\,>J)$ whose annihilation factor is larger than $J$ can be approximated by a power-law distribution over a large range of $J$ -factors, $\\overline{N}(\\,>J) \\equiv \\left\\langle \\; \\int \\limits _{J}^{\\infty } \\frac{\\mathrm {d}N}{\\mathrm {d}J^{\\prime }}\\,\\mathrm {d}J^{\\prime }\\,\\right\\rangle \\approx \\left(\\frac{J}{J_{\\rm lim}}\\right)^{1-\\alpha }\\;,$ where $J_{\\rm lim}$ is defined by $\\overline{N}(\\,>J_{\\rm lim})=1$ , and provided that $\\alpha > 1$ .eq:PLscd can be easily generalized including an exponential cut-off at $J_{\\rm c}$ , $\\overline{N}(\\,>J)=\\left({J}/{J_{\\rm lim}}\\right)^{1-\\alpha }\\,\\exp [-(J-J_{\\rm lim})/J_{\\rm c}]$ , obtaining the form of a Schechter luminosity function [168]." ], [ "Mean and median of the brightest DM halo", "We define the probability $P_{\\,\\ge 1}$ to obtain at least one object brighter than a given flux $J$ as $P_{\\,\\ge 1}(J)= \\sum \\limits _{N=1}^{\\infty } p[N(\\,>J)\\,|\\,\\overline{N}(\\,>J)],$ with $p(N\\,|\\,\\overline{N})$ the probability density to obtain exactly $N$ objects brighter than $J$ for an expectation value of $\\overline{N}$ .", "If $p(N\\,|\\,\\overline{N})$ follows a Poisson distribution, then the cumulative density function $P_{\\,\\ge 1}$ is given by $P_{\\,\\ge 1}(J) = 1- \\exp [- \\overline{N}(\\,>J)]\\,.$ For example, eq:PLscdcdf implies that one obtains at least one subhalo brighter than $J_{\\rm lim}$ with a chance of $1-e^{-1}=63\\%$ .", "For $\\overline{N}(>J)$ a power-law distribution, eq:PLscd, plugged into eq:PLscdcdf, gives the probability density function $\\frac{\\mathrm {d}P_{\\,\\ge 1}}{\\mathrm {d}J}(J) =\\frac{\\alpha -1}{J_{\\rm lim}}\\, \\exp \\left[-\\left(\\frac{J}{J_{\\rm lim}}\\right)^{1-\\alpha }\\right]\\,\\left(\\frac{J}{J_{\\rm lim}}\\right)^{-\\alpha }.$ The expectation value $\\overline{J}_{\\ge 1}$ is calculated from eq:PLscdpdf and the median $\\widetilde{J}_{\\,\\ge 1}$ from eq:PLscdcdf.", "If the subhalo distribution can be approximated by a power-law distribution, eq:PLscd, then $\\overline{J}_{\\ge 1}$ and $\\widetilde{J}_{\\,\\ge 1}$ are suitable approximations for the mean and median of the brightest halo: J J 1 = 0 J $\\mathrm {d}$ P 1$\\mathrm {d}$ J  $\\mathrm {d}$ J = (11-+1)Jlim, J J1 = (2)1/(1-)Jlim.", "The probability distribution eq:PLscdpdf is defined for $\\alpha >1$ and always positively skewed, even w.r.t.", "to a log-normal distribution.", "The long tail follows a power-law proportional to $J^{-\\alpha }$ , and thus the mean is only defined for $\\alpha >2$ ($\\overline{J}_{\\ge 1}\\rightarrow \\infty $ for $\\alpha \\le 2$ ).", "The mean is also always larger than the median value.", "For all our models in tab:model-parameters we obtain $\\alpha \\gtrsim 2$ , with a steepening of the $\\overline{N}(>J)$ distribution at the brightest halos (see fig:Histogram1DJfactor-Nsub-fullsky).", "Even if a mean brightest halo could be computed in all our models, the median can always be defined (even for $\\alpha \\rightarrow 2$ ) and offers a much better characterisation of the brightest object.", "Figure: Probability distribution of the brightest subhalo J ☆ J^{\\star } within the CTA survey FOV for model HIGH; J ☆ J^{\\star } is chosen here w.r.t.", "to an integration angle θ int =0.05 ∘ \\theta _{\\text{int}}=0.05{^{\\circ }}, but integrated over θ vir \\theta _{\\rm vir} (total emission).", "The histogram (red) is based on 10 4 10^{4} skymap realisations.", "The line relies on the power-law approximation dP ≥1 /dJ\\mathrm {d}P_{\\,\\ge 1}/\\mathrm {d}J from eq:PLscdpdf, with J lim =1.2·10 20 GeV 2 cm -5 J_{\\rm lim}=1.2\\cdot 10^{20}\\,\\rm {GeV^2\\, cm^{-5}} (fixed by the 37 percentile from the skymap realisations) and α=2.06\\alpha = 2.06 (obtained from a least-square fit to the histogrammed skymap realisations).", "The vertical lines give the 16, 50, and 84 percentiles—from the skymap realisations (bottom half) or from the distribution dP ≥1 /dJ\\mathrm {d}P_{\\,\\ge 1}/\\mathrm {d}J (top half)—encompassing the 68%68\\% CI.The success of this analytic approximation is illustrated in fig:brightestClump-dpdF, where the distributions from the power-law approximation (black line) and from the direct calculation over $10^4$ samples (red histogram) are compared for model HIGH.", "The two results are in excellent agreement.", "As stated in the main text, the skewness means that the limits on DM clump detection are extremely sensitive to the long tail of the distribution.", "The quantiles defining the 68% CIs are also shown: they are used for the sensitivity calculations in sec:cta.", "We remark that similar calculations as presented in this section recently have been carried out in the context of star cluster luminosities by [169]." ], [ "APS mean, median, and variance", "For a particular realisation of a point-like source count distribution, the intensity power $C_{\\rm P}^I$ defined in eq:C1shfluxAndo scatters.", "The mean number of halos $\\overline{N}(\\,>F)$ brighter than a flux $F$ behaves like $\\overline{N}(\\,>L)$ , i.e.", "it can be approximated by a power-law.", "Similarly to eq:PLscd, we define $F_{\\rm lim}$ such as $\\overline{N}(\\,>F_{\\rm lim})=1$ , and obtain $N(F_{\\rm min},\\,F^{\\star }) = \\int \\limits _{F_{\\rm min}}^{F_{\\rm max}=F^{\\star }} \\left\\langle \\frac{\\mathrm {d}N}{\\mathrm {d}F^{\\prime }} \\right\\rangle \\,\\mathrm {d}F^{\\prime } \\approx \\left(\\frac{F_{\\rm min}}{F_{\\rm lim}}\\right)^{1-\\alpha } - \\left(\\frac{F^{\\star }}{F_{\\rm lim}}\\right)^{1-\\alpha },$ by integrating $\\langle \\mathrm {d}N/\\mathrm {d}F \\rangle $ up to the flux of the brightest halo $F^{\\star }$ .", "Analogously, by using the power-law $\\langle \\mathrm {d}N/\\mathrm {d}F \\rangle = (\\alpha -1)/F_{\\rm lim} \\cdot \\left(F/F_{\\rm lim}\\right)^{-\\alpha }$ , $\\alpha >1$ , and replacing $F_{\\rm min}$ by $N$ from eq:nsubexplicitfmax, one can rewrite eq:C1shfluxAndo as $C_{\\rm P}^I(F^{\\star },\\,N) \\approx \\frac{1}{4\\pi \\beta }\\;F_{\\rm lim}^{\\;2}\\;\\left[\\left(\\frac{F^{\\star }}{F_{\\rm lim}}\\right)^{3-\\alpha }-\\left\\lbrace \\left(\\frac{F^{\\star }}{F_{\\rm lim}}\\right)^{1-\\alpha }+ N\\right\\rbrace ^{\\;-\\beta } \\right],$ with $\\beta \\equiv {(3-\\alpha )}/{(\\alpha - 1)}$ and $\\alpha \\ne 3$ .", "Thanks to this approximation, $C_{\\rm P}^I$ now became only a function of the brightest object $F^{\\star }$ and of the total number of objects $N$ .", "A further simplification arises assuming $N\\!\\rightarrow \\!\\infty $ , in which case $F_{\\rm min}\\!\\rightarrow \\!0$ and eq:APSPLapprox simplifies to $C_{\\rm P}^I(F^{\\star },\\,N) \\;\\;\\stackrel{N\\rightarrow \\infty }{\\approx }\\;\\; \\frac{1}{4\\pi \\beta }\\;F_{\\rm lim}^{\\;2}\\;\\left(\\frac{F^{\\star }}{F_{\\rm lim}}\\right)^{3-\\alpha }.$ Note that $C_{\\rm P}^I$ becomes highly sensitive to a finite $N$ ($F_{\\rm min}\\ne 0$ ) in the case $\\alpha \\rightarrow 3$ , such that the latter approximation is only applicable for $\\alpha $ sufficiently smaller than 3, and does not hold for $\\alpha \\ge 3$ .", "For the case of eq:APSPLlargeNsub being valid, the median $\\widetilde{C}_{\\rm P}^I$ is directly related to $\\widetilde{F}^{\\star }$ $\\widetilde{C}_{\\rm P}^I(F^{\\star },\\,N) \\;\\;\\stackrel{N\\rightarrow \\infty }{\\approx }\\;\\; C_{\\rm P}^I(\\widetilde{F}^{\\star }) \\approx \\frac{1}{4\\pi \\beta }\\;F_{\\rm lim}^{\\;2}\\;\\log (2)^{-\\beta }.$ It is also useful (see app:APSconvergence) to fix $N\\equiv \\overline{N}$ in eq:APSPLapprox, such that $C_{\\rm P}^I$ only becomes a function of $F^{\\star }$ .", "Then again, $\\widetilde{C}_{\\rm P}^I(F^{\\star })\\approx C_{\\rm P}^I(\\widetilde{F}^{\\star })$ holds, and one can simplify the ratio of the medians to $\\frac{\\widetilde{C}_{\\rm P}^I(F^{\\star };\\,\\overline{N})}{\\widetilde{C}_{\\rm P}^I(F^{\\star };\\,N\\rightarrow \\infty )} \\approx 1 - \\left(1 + \\frac{\\overline{N}}{\\log (2)}\\right)^{\\;-\\beta }\\,.$ Finally, one can extract the probability density function $\\mathrm {d}P/\\mathrm {d}C$ ($C\\equiv C_{\\rm P}^I$ ), $\\frac{\\mathrm {d}P}{\\mathrm {d}C}(C) = \\frac{\\mathrm {d}P_{\\,\\ge 1}}{\\mathrm {d}F}\\left(F^{\\star }(C)\\right)\\times \\left|\\frac{\\mathrm {d}F^{\\star }}{\\mathrm {d}C}(C)\\right|,$ from which one can calculate analytic expressions for the mean $\\overline{C}_{\\rm P}^I$ and standard deviation $\\sigma _{C_{\\rm P}}$ .", "The latter expression shows that the PDF of $C_{\\rm P}^I$ is proportional to the PDF of $F^{\\star }$ , as given in eq:PLscdpdf.", "This is in fact observed in our simulations (see fig:APSicompare), where the power at each $\\ell $ shows a variance skewed w.r.t a log-normal distribution, with a long tail towards high $C_{\\ell }$ values.", "We stress that the above discussion only involves the variance of the APS from the varying flux of the objects.", "For randomly distributed objects on the sphere, the APS additionally scatters because of their random positions in space.", "This variance, which decreases for larger multipoles, can, on the full sky, be estimated by [170]: $\\sigma _{C_{\\ell }} = C_{\\ell }\\,\\sqrt{\\frac{2}{2\\ell +1}}\\,.$" ], [ "APS convergence", "As described in [86], [87], CLUMPY relies on a combination of the calculation of the mean signal $\\langle J_{\\rm subs}\\rangle $ from subhalos and the calculation of individual drawn clumps $J_{\\rm drawn}$ to ensure a quick calculation of skymaps.", "A critical parameter of a CLUMPY run is the relative error $RE_{J_{\\rm drawn}}$ [86], which ultimately controls the number $N_{\\rm sub}$ of clumps to be drawn.", "In practice, a critical distance is obtained for each mass decade, by requiring the relative error of the signal integrated from $l_{\\rm crit}$ to remain lower than this user-defined precision requirement (level of fluctuation selected w.r.t.", "the mean signal).", "This reduces the number of clumps to draw in the Galaxy from $\\sim 10^{15}$ to $\\sim 10^{4}$ for angular resolutions $\\theta _{\\text{int}}\\sim 0.1{^{\\circ }}$ and $RE_{J_{\\rm drawn}}=5\\%$ .", "This appendix shows that this choice is a good compromise between speed and precision, and that it ensures convergence of our results to the expected values up to the highest multipoles the considered $\\gamma $ -ray instruments are sensitive to.", "The fair agreement with the approximate analytical further validates the CLUMPY code.", "Figure: Convergence study of the median power C ˜ ℓ \\widetilde{C}_{\\ell } (integration angle θ int =0.12 ∘ \\theta _{\\text{int}}=0.12{^{\\circ }}, i.e.", "HEALPix N side =512{N_{\\rm side}}=512 of the JJ-factor maps) as a function of RE J drawn RE_{J_{\\rm drawn}} (upper xx-axis) or, equivalently, the number of drawn clumps N ¯ sub \\overline{N}_{\\rm sub} (lower xx-axis).", "The different colours show different multipoles ℓ\\ell for the model LOW, with error bars from eq:errorofmedian.", "For comparison, we also show two analytical calculations (see app:meanmedian), based on eq:APSPLapprox (black solid line) or the further approximated eq:apsconvergencehardapprox (dotted line).fig:APSirsestudy shows the ratio $\\widetilde{C}_{\\ell }/\\widetilde{C}_{\\ell }(N_{\\rm sub}\\rightarrow \\infty )$ for DM subhalos (model LOW) at different multipoles $\\ell $ , calculated over $N_{\\rm sample}=5000$ simulations.", "The sample median was chosen because of its robustness compared to the mean, which is ill-defined for a power law index close to $\\alpha =2$ .", "The sample size $N_{\\rm sample}=5000$ was taken in order to reach at least a $5\\%$ relative accuracy of the ratio of the medians at the lowest multipoles, as calculated by the sample error of the median (shown by the coloured area) $\\Delta \\widetilde{C} = \\frac{1}{2\\,\\frac{\\mathrm {d}P}{\\mathrm {d}C}(\\widetilde{C})\\, \\sqrt{N_{\\rm sample}}}.$ The APS converges differently at different multipoles, but reaches 95% of the overall power $\\widetilde{C}_{\\ell }(N_{\\rm sub}\\rightarrow \\infty )$ at all multipoles $\\ell \\le 1024$ whenever $N_{\\rm sub} \\gtrsim 10^{4}$ drawn objects.", "All power spectra presented in this work meet this requirement.", "For comparison purpose, the black dotted line shows the expected convergence calculated from the $\\ell $ -independent power-law ($\\alpha =2.03$ ) approximation eq:apsconvergencehardapprox, which gives a fair description of the sample calculation for fixed $\\overline{N}_{\\rm sub}$ .", "This approximation reveals the interesting result $\\widetilde{C}_{\\rm P}^I(\\overline{N}_{\\rm sub}=1)/\\widetilde{C}_{\\rm P}^I(N_{\\rm sub}\\rightarrow \\infty )=0.57$ , indicating that in median, the brightest point-like object accounts for more than $50\\%$ to the overall power.", "The additional degree of freedom of a Poisson-distributed $N$ from sample to sample shifts down the median power for a low average $\\overline{N}_{\\rm sub}$ .", "This is accounted for in a more accurate numerical calculation of the median from eq:APSPLapprox, divided by eq:APSPLlargeNsubmedian, which gives an even fairer agreement with the sample median (black solid line)." ], [ "Virial mass and and $J$ -factor for Milky Way satellites", "This appendix presents the derivation of ${M}_{\\rm vir}$ and $J(0.5^\\circ )$ factors for spectroscopically confirmed satellite galaxies in the Milky Way.", "We recall that these values are used in fig:Histogram1DJfactor-Nsub-fullsky and fig:Histograms2DmodelRef for comparisons purpose with the drawn clumps of our simulations." ], [ "Jeans and Markov Chain Monte Carlo (MCMC) analysis", "Most of our values come from a reprocessing the chains from the recent analyses of classical and ultrafaint dSphs of [96] for the pre-2015 dSphs, [97] for Ret II, and [99] for Tuc 2.", "In these papers, a Jeans analysis of the light and velocity data coupled to an MCMC engine was used to recover the DM profiles, and then extract the median values and credible intervals on several quantities deriving from these profiles (e.g., mass, $J$ -factor).The robustness of the analysis has been validated on mock data [95].", "All the tools to achieve of the steps of the analysis are public, they are described in the second release of the CLUMPY code [86], [87].", "Moreover, in the last few months, and since the published results mentioned above, kinematic data of several new dSph galaxies have become publicly available.", "We therefore take the opportunity of this study to apply the same Jeans analysis on these dSphs, allowing for a more complete census for the objects shown in several plots of this paper.", "For these ultrafaint objects the number of spectroscopically measured stars is generally $\\sim 5-10$ .", "Draco II data has been taken from [171], Horologium I from [172], Pisces II from [173] and Triangulum II from [174].Thirteen member stars in Triangulum II have recently been measured by [175] but this dataset, showing distinct behaviours in the inner and outer part of the dSph galaxy, has not been used here.", "We do not consider in this study the faint and/or uncertain Boötes II [176], [177] and Hydra II [178], [173] dSph galaxies.", "We cannot exclude that several of the dSph galaxies in our selection suffer from tidal effects (see, e.g., [179] for Willman 1), but only Sagittarius dSph galaxy is undoubtedly flagged for this effect (see, e.g., [180] and references therein) and was therefore also excluded from this sample." ], [ "Tidal mass as a proxy for the virial mass", "As recalled in sec:setsofmodels, CLUMPY works with virial quantities so that comparisons must be made w.r.t.", "this choice.", "The quantity ${m}_{\\rm vir}$ depends on the definition of ${r}_{\\rm vir}$ which, strictly speaking, depends on cosmological parameters.", "However, as discussed in [162], the values of the quantities ${m}_{\\rm vir}$ and $m_{\\Delta =200\\,,500\\,\\dots }$ are tightly correlated.", "Moreover, the differences between the mass reconstructed from these various definitions is much smaller than the uncertainties we obtain from the MCMC analysis ($\\sim 2$ orders of magnitude).", "We therefore use the mass inside the virial radius ${r}_{\\rm vir}$ as a measure of the total mass inside a DM halo.", "The authors of [58] have shown the tidal radius to be a good proxy of the virial radius, this property holding even in the presence of baryons [105].", "The tidal radius is defined as [181] $r_\\text{tid} = \\left[ \\frac{m(r_\\text{tid})}{[2-\\mathrm {d}\\ln M_{\\text{MW}}/\\mathrm {d}\\ln R]\\times M_{\\text{MW}}(R)}\\right]^{(1/3)} \\times R\\;,$ where $M_{\\text{MW}}(R)$ is the mass of the MW enclosed within the galactocentric distance $R$ of the dSph, and $m(r_\\text{tid})$ is the subhalo mass inside $r_{\\rm tid}$ .", "For each model of our MCMC analysis, we compute the enclose mass within $r_{\\rm tid}$ and define it to be $m_{\\rm vir}$ of that particular model.", "We underline that integrating up to $r\\gg r_{\\rm tid}$ instead of $r_\\text{tid}$ only changes by a factor of 2 the total mass, which is much smaller than the uncertainties derived from the MCMC analysis.", "Table: The Galaxy satellites are ordered by distance and the columns correspond to (from left to right): name, distance, typical half-light radius (for dSphs) or type (for Magellanic clouds), median and 68% CI virial mass, median and 68% CI JJ-factor in an integration angle of 0.5 ∘ ^\\circ .", "In our calculations, we do not account for the uncertainties on the distance D obs D_{\\rm obs} and the half-light radius r h r_h of the dSph galaxies, and only quote the values we have actually used in the analysis.", "Objects in italic are confirmed ultrafaint dSphs while the ones denoted by the star symbol [ ☆ ^\\star ] are the most recently discovered objects, tentatively analysed here.", "The boldface objects correspond to the large and small Magellanic clouds, for which the central and Min/Max values represent the range of modelling uncertainties taken from the literature (see text)." ], [ "dSph, SMC and LMC values", "tab:MandJ gathers the distance and median (and 68% CI) of the mass $M_{\\rm vir}$ and $J(0.5^\\circ )$ for all the dSphs mentioned above.", "Given the very recent status of some of these spectroscopic measurements and the sometime intriguing kinematics they suggest (e.g., Triangulum II [175]), we urge caution in interpreting the new $J$ -factors (especially the large J-factor of Triangulum II) of all the objects denoted by the star symbol [$^\\star $ ] in the table.Near the completion of this paper, the non-spherical Jeans analysis of dSph galaxies by [141] became available on arXiv and the $J$ -factors of all dSph galaxies of tab:MandJ (including the most recently discovered) are compatible at 1$\\sigma $ with their values but for Segue 2 and UMa 1.", "Also using CLUMPY, [182] reports similar value for Triangulum II as quoted here.", "We also include estimation for the LMC and SMC in boldface.", "For these irregular objects, a spherical Jeans analysis cannot be applied, and we have taken the values from [183], [184], [185] for the LMC, and from [183], [186], [185], [187] for the SMC." ], [ "Details of the CTA analysis", "To avoid too lengthy a discussion in the core of the text, several aspects of the CTA analysis were postponed to this appendix.", "The key points developed here are related to the observation strategy (app:obsstreg), the post-trial sensitivity (app:preposttrial), the calculation of the maximum log-likelihood ratio (app:likelihood), and the robustness and values of the test statistic (app:tsstatistics) used in our analysis." ], [ "Observation strategy", "For isotropically distributed sources on the sky, one may ask if a large field survey with a shallow exposure is the most sensitive strategy to detect these objects.", "It has been shown that observing a small portion of the sky with the same available total time is not beneficial for most source populations, and the loss of chance to encounter a very bright object within a small field of view outweighs the gain in deep-exposure sensitivity [85].", "We show below that this also applies to Galactic DM subhalos isotropically distributed on large scales.", "The isotropy assumption does not strictly hold for model HIGH, but still yields a fair characterisation of the subhalo abundance.", "We assume isotropically distributed sources on the sky obeying a power-law source count distribution, eq:PLscd.", "Then one obtains for the mean number $ \\overline{N}$ of subhalos with a flux above a given threshold and on a survey area $A$ : $\\overline{N}(\\,>F,\\;A)=\\left(\\frac{F}{F_{\\rm lim,\\,fullsky}}\\right)^{1-\\alpha }\\;\\frac{A}{[4\\pi ]{sr}}\\,.$ so that $\\frac{\\overline{N}(\\,>F_1,\\;A_1)}{\\overline{N}(\\,>F_2,\\;A_2)} =\\left(\\frac{F_1}{F_2}\\right)^{1-\\alpha }\\;\\frac{A_1}{A_2}\\,.$ For a fixed total observation time $T$ available to uniformly cover an area of the total size $A$ , then an area within $A_{\\rm FOV}$ can be observed for $t=T\\times (A_{\\rm FOV}/A)$ .", "As the sensitivity to a flux $F$ goes as $1/\\sqrt{t}$ , we have $F_1/F_2=\\sqrt{t_2/t_1}$ , and $t_1/t_2=A_2/A_1$ , so that $\\frac{\\overline{N}_{\\rm detectable}(A_1,\\,T)}{\\overline{N}_{\\rm detectable}(A_2,\\,T)} = \\left(\\frac{A_1}{A_2}\\right)^{\\frac{3-\\alpha }{2}},$ with $A_1$ and $A_2$ the observed area on the sky and $\\overline{N}_{\\rm detectable}$ the mean number of detectable objects.", "Thus, for a power-law index $\\alpha <3$ , the average number of detectable subhalos is $\\overline{N}(A_1)>\\overline{N}(A_2)$ for $A_1>A_2$ independent of $T$ , and the probability of detecting an object from the population rises for increasing the survey area.", "All our subhalo models meet this requirement, with $2\\lesssim \\alpha <3$ .", "It is also useful to extract from eq:nsubfovsize the relation $\\frac{F_{\\rm lim,\\,A_1}}{F_{\\rm lim,\\,A_2}}=\\left(\\frac{A_1}{A_2}\\right)^{\\frac{1}{\\alpha -1}}.$ For a power-law distribution, according to eq:Jmean-Jlim,eq:Jmedian-Jlim, this ratio also holds for the mean and median brightest halo, $\\frac{F_{\\rm lim,\\,A_1}}{F_{\\rm lim,\\,A_2}}=\\frac{\\overline{F}_{A_1}^{\\star }}{\\overline{F}_{A_2}^{\\star }}=\\frac{\\widetilde{F}_{A_1}^{\\star }}{\\widetilde{F}_{A_2}^{\\star }}\\,.$ For a power-law index $\\alpha \\approx 2$ , the ratio of mean/median fluxes ($J$ -factors) of the brightest halo within two survey fields $A_1,\\,A_2$ is proportional to the ratio of the field sizes.", "We remark that this result also applies to the APS behaviour discussed in app:APSapprox.", "Adopting eq:APSPLlargeNsubmedian, one obtains for $\\alpha \\approx 2$ $\\widetilde{C}_{\\rm P}^I(A_1) \\approx \\left(\\frac{A_1}{A_2}\\right)^2\\;\\widetilde{C}_{\\rm P}^I(A_2) \\,.$ For $A_2=[4\\pi ]{sr}$ , it is $A_1/A_2 = f_{\\rm sky}$ , the fraction of the sky.", "Note the $f_{\\rm sky}^2$ scaling of eq:cpfovsize in the latter case." ], [ "Pre- and post-trial sensitivity", "To realistically assess a survey sensitivity, one has to account for the trials when searching for a signal from unspecified locations.", "Assuming an average $68\\%$ containment radius of the CTA point spread function of $0.05{^{\\circ }}$ , corresponding to a containment area of $[2.4\\cdot 10^{-6}]{sr}$ , scanning a quarter of the sky results in approximately $\\pi /(2.4\\cdot 10^{-6}) = 1.3\\cdot 10^6$ independent trials.By chance, this number comes close to the frequency one expects a $5\\sigma $ background up-fluctuation in $1/p=3.5\\cdot 10^6$ repetitions of a random experiment.", "In order to reject a background fluctuation in the survey search at a trials corrected confidence level $1-p_{\\rm post}$ , we calculate the required confidence level, $1-p_{\\rm pre}$ , for the template observation setup according to $p_{\\rm pre} = 1-(1-p_{\\rm post})^{1/N_{\\rm trials}}.$ Presenting our results at the $1-p_{\\rm post}=95\\%$ confidence level, with $N_{\\rm trials}= 1.3\\cdot 10^6$ , we thus require $p_{\\rm pre}=3.9\\cdot 10^{-8}$ .", "This corresponds to a Gaussian one-sided confidence level, $p=\\alpha /2$ , of $5.4\\,\\sigma $ ." ], [ "Likelihood", "The full unbinned likelihood function for a specific model $\\mathcal {M}$ considered in our study is ${L}(\\mathcal {M}\\,|\\, N_{\\rm obs},\\,E_{\\rm {obs},\\,1\\ldots \\mathit {N}_{\\rm obs}},\\vec{k}_{\\rm {obs},\\,1\\ldots \\mathit {N}_{\\rm obs}})=p(N_{\\rm obs}\\,|\\, N_{\\rm pred}(\\mathcal {M}))\\times \\!", "\\prod _1^{N_{\\rm obs}} p(E_{\\rm obs,\\,i},\\, \\vec{k}_{\\rm obs,\\,i}\\,|\\,\\mathcal {M}).$ Here, $N_{\\rm obs}$ is the total number of observed events, $E_{\\rm obs,\\,i}$ and $\\vec{k}_{\\rm obs,\\,i}$ denote the reconstructed energy and angular direction of each event.", "For Poisson statistics, $p(N_{\\rm obs}\\,|\\, N_{\\rm pred})=\\frac{N_{\\rm pred}^{\\quad N_{\\rm obs}}\\,e^{-N_{\\rm pred}}}{N_{\\rm obs}!", "}.$ The total number of predicted events, $N_{\\rm pred}$ , is calculated by $N_{\\rm pred}(\\mathcal {M}) = T_{\\rm obs}\\,\\int \\limits _{E_{\\rm min}=30~{\\rm GeV}}^{E_{\\rm max}=200~{\\rm TeV}}\\int \\limits _{\\Delta \\Omega _{\\rm obs}}\\, p(E_{\\rm obs},\\, \\vec{k}_{\\rm obs}\\,|\\,\\mathcal {M})\\;\\mathrm {d}E_{\\rm obs}\\, \\mathrm {d}\\Omega _{\\rm obs},$ with $T_{\\rm obs}$ the duration of the observation.We do not include the time coordinates of the events into the likelihood calculation, as both the event rates from background and DM annihilation are assumed to be constant in time.", "The probability $p(E_{\\rm obs},\\, \\vec{k}_{\\rm obs}\\,|\\,\\mathcal {M})$ for each event is given by the differential intensity $\\mathrm {d}\\Phi _{\\mathcal {M}}/(\\mathrm {d}E\\, \\mathrm {d}\\Omega )$ predicted by the model $\\mathcal {M}$ , integrated over the effective area, $A_{\\rm eff}$ , and convolved with the energy and angular response of the instrument, $p(E_{\\rm obs},\\, \\vec{k}_{\\rm obs}\\,|\\,\\mathcal {M}) = \\!\\!\\!\\!\\!\\!\\!\\!\\!\\int \\limits _{E,\\, \\Omega ,\\, A_{\\rm eff}(E)}\\!\\!\\!\\!\\!\\!\\!\\!\\!", "\\,p(E_{\\rm obs}\\,|\\,E,\\,\\vec{k}) \\times p(\\vec{k}_{\\rm obs}\\,|\\,E,\\,\\vec{k}) \\times \\frac{\\mathrm {d}\\Phi _{\\mathcal {M}}}{\\mathrm {d}E \\,\\mathrm {d}\\Omega }(E,\\,\\vec{k})\\;\\;\\mathrm {d}A\\,\\mathrm {d}E\\, \\mathrm {d}\\Omega ,$ with $E$ and $\\vec{k}$ the true energy and direction of the event.In general, the effective area additionally depends on the time-dependent zenith and azimuthal coordinates of the observation, i.e.", "$A_{\\rm eff} = A_{\\rm eff}(E,\\,\\vec{k},\\,t)$ , which would introduce a time dependence to $p(E_{\\rm obs},\\, \\vec{k}_{\\rm obs},\\,t\\,|\\mathcal {M})$ and to the likelihood eq:likelihood-function.", "However, for simplicity, we perform our study with a constant $A_{\\rm eff}$ .", "For computation reasons, we neglect the energy dispersion of the events, i.e.", "we set $p(E_{\\rm obs}\\,|\\,E,\\,\\vec{k})= \\delta (E-E_{\\rm obs})$ , whereas the point spread function, $ p(\\vec{k}_{\\rm obs}\\,|\\,E,\\,\\vec{k})$ , is modelled as a two-dimensional Gaussian with energy-dependent width $\\sigma (E)$ .", "For the signal model $\\mathcal {M}_{\\rm sig}$ , $\\mathrm {d}\\Phi _{\\mathcal {M}_{\\rm sig}}/(\\mathrm {d}E\\, \\mathrm {d}\\Omega )$ is the $\\gamma $ -ray intensity from DM annihilation, given by eq:flux-general.", "For the background model $\\mathcal {M}_{\\rm bkg}$ , $\\mathrm {d}\\Phi _{\\mathcal {M}_{\\rm bkg}}/(\\mathrm {d}E\\, \\mathrm {d}\\Omega )$ corresponds to the residual cosmic ray background after cuts, and we directly make use of the background rate model shown in fig:plotrates, so thatThe background rate model is taken for the chosen $A_{\\rm eff}$ and thus is also constant in time for our study.", "$p(E_{\\rm obs},\\, \\vec{k}_{\\rm obs}\\,|\\,\\mathcal {M}_{\\rm bkg}) = \\frac{\\mathrm {d}N_{\\rm bkg}}{\\mathrm {d}E_{\\rm obs}\\, \\mathrm {d}\\Omega _{\\rm obs}\\,\\mathrm {d}t}=f(E_{\\rm obs},\\, \\vec{k}_{\\rm obs}).$" ], [ "Distribution and values of the test statistic ($\\mathrm {TS}$ )", "For the test statistic $\\lambda $ defined in eq:tsdef, the hypothesis $\\mathcal {M}_{\\rm bkg}+\\mathcal {M}_{\\rm sig}$ has one more degree of freedom than the hypothesis background only.", "In the limit of a large data sample (event number) and provided the physical bound $\\langle \\sigma v\\rangle \\ge 0$ , $\\mathrm {TS}$ is expected to be distributed according to [188], [189] $p(\\mathrm {TS}) = \\frac{1}{2}\\,\\delta (\\mathrm {TS})+\\frac{1}{2}\\, \\chi ^2_{k=1}(\\mathrm {TS}).$ However, we found from $10^5$ MC simulations that for our setup, the $\\mathrm {TS}$ distribution is poorly described by eq:tspdf, and depends on the spectral shape of $\\mathcal {M}_{\\rm sig}$ .", "tab:tsdistribution shows the used test statistic ($\\mathrm {TS}$ ) values for the sensitivity calculation in sec:cta.", "We performed $N_{\\rm sim}=10^5$ MC simulations distributed over 24 spectra.", "We then merged the spectra into five groups, and calculated $\\mathrm {TS}(p_{\\rm pre}=0.05)$ separately for each group.", "Because performing $10^8$ MC simulations for $p_{\\rm pre}=3.9\\cdot 10^{-8}$ was not feasible, we fitted an exponential tail to our distributions, and computed $\\mathrm {TS}(p_{\\rm pre}=3.9\\cdot 10^{-8})$ from this extrapolation.", "tab:tsdistribution shows that the obtained $\\mathrm {TS}$ values (left and middle columns) approximately correspond to the assumption of a $\\chi ^2_{k=1}$ distribution (right columns, constant and independent of the DM spectrum), but deviate for very low and high DM masses.", "Table: Test statistic values used for the CTA sensitivity analysis in this study.", "The values given in the left and middle columns (χχ→XX\\chi \\chi \\rightarrow XX) are obtained from our MC calculation, the right columns are expected from Wilks' theorem (0.5χ k=1 2 0.5\\,\\chi ^2_{k=1}).", "The pre-trial p-value from the left block results into a post-trial p-value of 0.05 for N trials =1.3·10 6 N_{\\rm trials}= 1.3\\cdot 10^6.", "See app:preposttrial for pre- and post-trial definitions." ] ]
1606.04898
[ [ "Spectral stability of periodic waves in the generalized reduced\n Ostrovsky equation" ], [ "Abstract We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations.", "Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independently on the nonlinearity power.", "The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein--Gordon type." ], [ "Introduction", "We address the generalized reduced Ostrovsky equation written in the form $(u_t + u^p u_x)_x = u,$ where $p \\in \\mathbb {N}$ is the nonlinearity power and $u$ is a real-valued function of $(x,t)$ .", "This equation was derived in the context of long surface and internal gravity waves in a rotating fluid for $p = 1$ [22] and $p = 2$ [10].", "These two cases are the only cases, for which the reduced Ostrovsky equation is transformed to integrable semi-linear equations of the Klein–Gordon type by means of a change of coordinates [3], [9].", "We consider existence and stability of travelling periodic waves in the generalized reduced Ostrovsky equation (REF ) for any $p \\in \\mathbb {N}$ .", "The travelling $2T$ -periodic waves are given by $u(x,t) = U(x-ct)$ , where $c > 0$ is the wave speed, $U$ is the wave profile satisfying the boundary-value problem $\\frac{d}{dz} \\left[ (c - U^p) \\frac{dU}{dz} \\right] + U(z) = 0, \\quad U(-T) = U(T), \\quad U^{\\prime }(-T) = U^{\\prime }(T),$ and $z = x-ct$ is the travelling wave coordinate.", "We are looking for smooth periodic waves $U \\in H^{\\infty }_{\\rm per}(-T,T)$ satisfying (REF ).", "It is straightforward to check that periodic solutions of the second-order equation (REF ) correspond to level curves of the first-order invariant, $E = \\frac{1}{2} (c - U^p)^2 \\left( \\frac{dU}{dz} \\right)^2 + \\frac{c}{2} U^2 - \\frac{1}{p+2} U^{p+2} = {\\rm const}.$ We add a co-periodic perturbation to the travelling wave, that is, a perturbation with the same period $2T$ .", "Separating the variables, the spectral stability problem for the perturbation $v$ to $U$ is given by $\\lambda v = \\partial _z L v$ , where $L = P_0 \\left( \\partial _z^{-2} + c - U(z)^p \\right) P_0 : \\; \\dot{L}^2_{\\rm per}(-T,T) \\rightarrow \\dot{L}^2_{\\rm per}(-T,T),$ where $\\dot{L}^2_{\\rm per}(-T,T)$ denotes the space of $2T$ -periodic, square-integrable functions with zero mean and $P_0 : L^2_{\\rm per}(-T,T) \\rightarrow \\dot{L}^2_{\\rm per}(-T,T)$ is the projection operator that removes the mean value of $2T$ -periodic functions.", "Definition 1 We say that the travelling wave is spectrally stable with respect to co-periodic perturbations if the spectral problem $\\lambda v = \\partial _z L v$ with $v \\in \\dot{H}^1_{\\rm per}(-T,T)$ has no eigenvalues $\\lambda \\notin i \\mathbb {R}$ .", "Local solutions of the Cauchy problem associated with the generalized reduced Ostrovsky equation (REF ) exist in the space $\\dot{H}^s_{\\rm per}(-T,T)$ for $s > \\frac{3}{2}$ [26].", "For sufficiently large initial data, the local solutions break in finite time, similar to the inviscid Burgers equation [18], [19].", "However, if the initial data $u_0$ is small in a suitable norm, then local solutions are continued for all times in the same space, at least in the integrable cases $p = 1$ [11] and $p = 2$ [25].", "Travelling periodic waves to the generalized reduced Ostrovsky equation (REF ) were recently considered in the cases $p = 1$ and $p = 2$ .", "In these cases, travelling waves can be found in the explicit form given by the Jacobi elliptic functions after a change of coordinates [3], [9].", "Exploring this idea further, it was shown in [13], [14], [27] that the spectral stability of travelling periodic waves can be studied with the help of the eigenvalue problem $M \\psi = \\lambda \\partial _z \\psi $ , where $M$ is a second-order Schrödinger operator.", "Independently, by using higher-order conserved quantities which exist in the integrable cases $p = 1$ and $p = 2$ , it was shown in [4] that the travelling periodic waves are unconstrained minimizers of energy functions in suitable function spaces with respect to subharmonic perturbations, that is, perturbations with a multiple period to the periodic waves.", "This result yields not only spectral but also nonlinear stability of the travelling wave.", "The nonlinear stability of periodic waves was established analytically for small-amplitude waves and shown numerically for waves of arbitrary amplitude [4].", "In this paper, we give a simple argument that proves spectral stability of all smooth periodic travelling waves to the generalized reduced Ostrovsky equation (REF ) independently of the nonlinearity power $p$ and the wave amplitude.", "The spectral stability of periodic waves is defined here with respect to co-periodic perturbations in the sense of Definition REF .", "The argument is based on convexity of the energy function $H(u) = -\\frac{1}{2} \\Vert \\partial _x^{-1} u \\Vert _{L^2_{\\rm per}}^2 - \\frac{1}{(p+1)(p+2)} \\int _{-T}^T u^{p+2} dx,$ at the travelling wave profile $U$ in the energy space with fixed momentum, $X_q = \\left\\lbrace u \\in \\dot{L}^2_{\\rm per}(-T,T) \\cap L^{p+2}_{\\rm per}(-T,T) : \\quad \\Vert u \\Vert ^2_{L^2_{\\rm per}} = 2 q > 0 \\right\\rbrace .$ Note that the self-adjoint operator $L$ given by (REF ) is the Hessian operator of the extended energy function $F(u)=H(u) + c Q(u)$ , where $Q(u) = \\frac{1}{2} \\Vert u \\Vert ^2_{L^2_{\\rm per}}$ is the momentum function.", "The energy $H(u)$ and momentum $Q(u)$ , and therefore the extended energy $F(u)$ , are constants of motion, as can be seen readily by writing the evolution equation (REF ) in Hamiltonian form as $u_t = \\partial _x {\\rm grad} H(u)$ .", "Notice that the traveling wave profile $U$ is a critical point of the extended energy function $F(u)$ in the sense that the Euler–Lagrange equations for $F(u)$ are identical to the boundary-value problem (REF ) after the second-order equation is integrated twice with zero mean.", "The outline of the paper is as follows.", "Adopting the approach from [6], [7], [8], we prove in Section that the energy-to-period map $E \\mapsto 2T$ is strictly monotonically decreasing for the family of smooth periodic solutions satisfying (REF ) and (REF ).", "This result holds for every fixed $c > 0$ .", "Thanks to monotonicity of the energy-to-period map $E \\mapsto 2T$ , the inverse mapping defines the first-order invariant $E$ in terms of the half-period $T$ and the speed $c$ .", "We denote this inverse mapping by $E(T,c)$ .", "In Section , we consider continuations of the family of smooth periodic solutions with respect to parameter $c$ for every fixed $T > 0$ and prove that $E(T,c)$ is an increasing function of $c$ within a nonempty interval $(c_0(T),c_1(T))$ , where $0 < c_0(T) < c_1(T) < \\infty $ .", "We also prove that the momentum $Q(u)$ evaluated at $u = U$ is an increasing function of $c$ for every fixed $T > 0$ .", "In Section , we use the monotonicity of the mapping $E \\mapsto 2T$ for every fixed $c > 0$ and prove that the self-adjoint operator $L$ given by (REF ) has a simple negative eigenvalue, a one-dimensional kernel, and the rest of its spectrum is bounded from below by a positive number.", "Finally, in Section , we prove that the operator $L$ constrained on the space $L^2_c = \\left\\lbrace u \\in \\dot{L}^2_{\\rm per}(-T,T) : \\quad \\langle U, u \\rangle _{L^2_{\\rm per}} = 0 \\right\\rbrace $ is strictly positive except for the one-dimensional kernel induced by the translational symmetry.", "This gives convexity of $H(u)$ at $u = U$ in space of fixed $Q(u)$ given by (REF ).", "By using the standard Hamilton–Krein theorem in [15] (see also the reviews in [17] and [24]), this rules out existence of eigenvalues $\\lambda \\notin i \\mathbb {R}$ of the spectral problem $\\lambda v = \\partial _z L v$ with $v \\in \\dot{H}^1_{\\rm per}(-T,T)$ .", "All together, the existence and spectral stability of smooth periodic travelling waves of the generalized reduced Ostrovsky equation (REF ) is summarized in the following theorem.", "Theorem 1 For every $c > 0$ and $p \\in \\mathbb {N}$ , (a) there exists a smooth family of periodic solutions $U \\in \\dot{L}^2_{\\rm per}(-T,T) \\cap H^{\\infty }_{\\rm per}(-T,T)$ of equation (REF ), parameterized by the energy $E$ given in (REF ) for $E \\in (0,E_c)$ , with $E_c = \\frac{p}{2(p+2)} c^{\\frac{p+2}{p}},$ such that the energy-to-period map $E \\mapsto 2T$ is smooth and strictly monotonically decreasing.", "Moreover, there exists $T_1 \\in (0,\\pi )$ such that $T \\rightarrow \\pi c^{\\frac{1}{2}} \\quad \\mbox{\\rm as} \\quad E \\rightarrow 0 \\quad \\mbox{\\rm and} \\quad T \\rightarrow T_1 c^{\\frac{1}{2}} \\quad \\mbox{\\rm as} \\quad E \\rightarrow E_c;$ (b) for each point $U$ of the family of periodic solutions, the operator $L$ given by (REF ) has a simple negative eigenvalue, a simple zero eigenvalue associated with ${\\rm Ker}(L) = {\\rm span}\\lbrace \\partial _z U \\rbrace $ , and the rest of the spectrum is positive and bounded away from zero; (c) the spectral problem $\\lambda v = \\partial _z Lv$ with $v \\in \\dot{H}^1_{\\rm per}(-T,T)$ admits no eigenvalues $\\lambda \\notin i \\mathbb {R}$ .", "Consequently, periodic waves of the generalized reduced Ostrovsky equation (REF ) are spectrally stable with respect to co-periodic perturbations in the sense of Definition REF .", "We now compare our result to the existing literature on spectral and orbital stability of periodic waves with respect to co-periodic perturbations.", "First, in comparison with the analysis in [14], the result of Theorem REF is more general since $p \\in \\mathbb {N}$ is not restricted to the integrable cases $p = 1$ and $p = 2$ .", "On a technical level, the method of proof of Theorem REF is simple and robust, so that many unnecessary explicit computations from [14] are avoided.", "Indeed, in the transformation of the spectral problem $\\lambda v = \\partial _z L v$ to the spectral problem $M \\psi = \\lambda \\partial _z \\psi $ , where $M$ is a second-order Schrödinger operator from $H^2_{\\rm per}(-T,T) \\rightarrow L^2_{\\rm per}(-T,T)$ , the zero-mean constraint is lostNote that this transformation reflects the change of coordinates owing to which the reduced Ostrovsky equations are reduced to the semi-linear equations of the Klein–Gordon type.", "This transformation also changes the period of the travelling periodic wave.. Consequently, the operator $M$ was found in [14] to admit two negative eigenvalues in $L^2_{\\rm per}(-T,T)$ , which are computed explicitly by using eigenvalues of the Schrödinger operator with elliptic potentials.", "By adding three constraints for the spectral problem $M \\psi = \\lambda \\partial _z \\psi $ , the authors of [14] showed that the operator $M$ becomes positive on the constrained space, again by means of symbolic computations involving explicit Jacobi elliptic functions.", "All these technical details become redundant in our simple approach.", "Second, we mention another type of improvement of our method compared to the analysis of spectral stability of periodic waves in other nonlinear evolution equations [20], [21].", "By establishing first the monotonicity of the energy-to-period map $E \\mapsto 2T$ for a smooth family of periodic waves, we give a very precise count on the number of negative eigenvalues of the operator $L$ in $L^2_{\\rm per}(-T,T)$ without doing numerical approximations on solutions of the homogeneous equation $L v = 0$ .", "Indeed, the smooth family of periodic waves has a limit to zero solution, for which eigenvalues of $L$ in $L^2_{\\rm per}(-T,T)$ are found from Fourier series.", "The zero eigenvalue of $L$ is double in this limit and it splits once the amplitude of the periodic wave becomes nonzero.", "Owing to the monotonicity of the map $E \\mapsto 2T$ and continuation arguments, the negative index of the operator $L$ remains invariant along the entire family of the smooth periodic waves.", "Therefore, the negative index of the operator $L$ is found for the entire family of periodic waves by a simple argument, again avoiding cumbersome analytical or approximate numerical computations.", "Finally, we also mention that the spectral problem $\\lambda v = \\partial _z L v$ is typically difficult when it is posed in the space $L^2_{\\rm per}(-T,T)$ because the mean-zero constraint is needed on $v$ in addition to the orthogonality condition $\\langle U, v \\rangle _{L^2_{\\rm per}} = 0$ .", "The two constraints are taken into account by studying the two-parameter family of smooth periodic waves and working with a 2-by-2 matrix of projections [1], [5].", "This complication is avoided for the reduced Ostrovsky equation (REF ) because the spectral problem $\\lambda v = \\partial _z L v$ is posed in space $\\dot{L}^2_{\\rm per}(-T,T)$ and the only orthogonality condition $\\langle U, v \\rangle _{L^2_{\\rm per}} = 0$ is studied with the help of identities satisfies by the periodic wave $U$ .", "As a limitation of the results of Theorem REF we mention that the nonlinear orbital stability of travelling periodic waves cannot be established for the reduced Ostrovsky equations (REF ) by using the energy function (REF ) in space (REF ).", "This is because the local solution is defined in $\\dot{H}^s_{\\rm per}(-T,T)$ for $s > \\frac{3}{2}$ [26], whereas the energy function is defined in $\\dot{L}^2_{\\rm per}(-T,T) \\cap L^{p+2}_{\\rm per}(-T,T)$ .", "As a result, coercivity of $H(u)$ in the space of fixed momentum (REF ) only controls the $L^2$ norm of time-dependent perturbations.", "Local well-posedness in such spaces of low regularity is questionable and so is the proof of orbital stability of the travelling periodic waves in the time evolution of the reduced Ostrovsky equations (REF )." ], [ "Monotonicity of the energy-to-period map", "Traveling wave solutions of the reduced Ostrovsky equation (REF ) are solutions of the second-order differential equation (REF ) with fixed $c > 0$ and $p \\in \\mathbb {N}$ .", "The following lemma establishes a correspondence between the smooth periodic solutions of the second-order equation (REF ) and the periodic orbits around the center of an associated planar system, see Figure REF .", "For lighter notations, we replace $U(z)$ by $u(z)$ and denote the derivatives in $z$ by primes.", "Figure: Phase portraits of system () for p=2p = 2 (left) and p=1p = 1 (right).Lemma 1 For every $c>0$ and $p\\in \\mathbb {N}$ the following holds: A function $u$ is a smooth periodic solution of equation (REF ) if and only if $(u,v)=(u,u^{\\prime })$ is a periodic orbit of the planar differential system $\\left\\lbrace \\begin{array}{l}u^{\\prime } =v,\\\\[2pt]v^{\\prime } =\\dfrac{-u + pu^{p-1}v^2}{c-u^p}.\\end{array}\\right.$ The system (REF ) has a first integral given by (REF ), which we write as $E(u,v)=A(u)+B(u) v^2,$ with $A(u) = \\frac{c}{2}u^2- \\frac{1}{p+2} u^{p+2}$ and $B(u) = \\frac{1}{2}(c-u^p)^2$ .", "Every periodic orbit of system (REF ) belongs to the period annulusThe largest punctured neighbourhood of a center which consists entirely of periodic orbits is called period annulus, see [2].", "of the center at the origin of the $(u,v)$ -plane and lies inside some energy level curve of $E$ , with $E \\in (0,E_c)$ where $E_c := A(c^{1/p}) = \\frac{p}{2(p+2)} c^{\\frac{p+2}{p}}.$ .", "The assertion in $(ii)$ is proved with a straightforward calculation.", "To prove $(iii)$ , we notice that system (REF ) has no limit cycles in view of the existence of a first integral, and hence the periodic orbits form period annuli.", "A periodic orbit must surround at least one critical point.", "The unique critical point of system (REF ) is a center at the origin on the $(u,v)$ plane, corresponding to the energy level $E =0$ .", "In view of the presence of the singular line $\\lbrace u = c^{1/p}, \\quad v \\in \\mathbb {R}\\rbrace \\subset \\mathbb {R}^2$ we may conclude, applying the Poincaré-Bendixon Theorem, that the set of periodic orbits forms a punctured neighbourhood of the center, and that no other period annulus is possible.", "It remains to show $(i)$ .", "It is clear that $z \\mapsto (u,v)=(u,u^{\\prime })$ is a smooth solution of the differential system (REF ) if and only if $u$ is a smooth solution of the second-order equation (REF ) satisfying $c\\ne u(z)^p$ for all $z$ .", "We claim that $c\\ne u(z)^p$ for all $z\\in \\mathbb {R}$ for smooth periodic solutions $u$ .", "Indeed, let $p$ be odd for simplicity and recall that every periodic orbit in a planar system has exactly two turning points $(u,u^{\\prime })=(u_{\\pm },0)$ per fundamental period.", "The turning points correspond to the maximum and minimum of the periodic solution $u$ and satisfy the equation $A(u_{\\pm })=E$ .", "The graph of $A(u)$ on $\\mathbb {R}^+$ has a global maximum at $u=c^{1/p}$ with $E_c$ given in (REF ).", "The equation $A(u)=E$ has exactly two positive solutions for $E\\in (0,E_c)$ , where $u=u_+$ corresponds to the smaller one inside the period annulus.", "At $E=E_c$ , the equation $A(u) = E$ has only one positive solution given by $u_+=c^{1/p}$ .", "Now assume that for a smooth periodic solution $u$ , there exists $z_1$ such that $u(z_1)=c^{1/p}$ .", "Then, equation (REF ) implies that $u^{\\prime }(z_1)=\\pm p^{-1/2} c^{-\\frac{p-2}{2p}}$ , hence the solution $(u,u^{\\prime })(z)$ to system (REF ) tends to the points $p_{\\pm } = (c^{1/p},\\pm p^{-1/2} c^{-\\frac{p-2}{2p}})$ as $z\\rightarrow z_1$ .", "Since $E(p_{\\pm })=E_c$ and by continuity of the first integral, this orbit lies inside the $E_c$ -level set.", "For such an orbit, we have seen that its turning point is located at $u_+=c^{1/p}=u(z_1)$ .", "However, since $u^{\\prime }(z_1)\\ne 0$ , this cannot be a turning point, which leads to a contradiction.", "Hence, the assertion $(i)$ is proved.", "Remark 1 By Lemma REF , every smooth periodic solution $u$ of the differential equation (REF ) corresponds to a periodic orbit $(u,v)=(u,u^{\\prime })$ inside the period annulus of the differential system (REF ).", "Since $E$ is a first integral of (REF ), this orbit lies inside some energy level curve of $E$ , where $E \\in (0,E_c)$ .", "We denote this orbit by $\\gamma _E$ .", "The period of this orbit is given by $2T(E) = \\int _{\\gamma _E}\\frac{du}{v},$ since $\\frac{du}{dz} = v$ in view of (REF ).", "The energy levels of the first integral $E$ parameterize the set of periodic orbits inside the period annulus, and therefore this set forms a smooth family $\\lbrace \\gamma _E\\rbrace _{E\\in (0,E_c)}$ .", "In view of Lemma REF , we can therefore assert that the set of smooth periodic solutions of (REF ) forms a smooth family $\\lbrace u_E\\rbrace _{E\\in (0,E_c)}$ , which is parameterized by $E$ as well.", "Moreover, it ensures that the period $2 T(E)$ of the periodic orbit $\\gamma _E$ is equal to the period of the corresponding smooth periodic solution $u_E$ of the second-order equation (REF ).", "The main result of this section is the following proposition, from which we conclude that the energy-to-period map $E \\mapsto 2T(E)$ for the smooth periodic solutions of equation (REF ) is smooth and strictly monotonically decreasing.", "Together with the above Remark REF , this proves statement (a) of Theorem REF .", "Proposition 1 For every $c > 0$ and $p \\in \\mathbb {N}$ the function $T: (0,E_c) \\longrightarrow \\mathbb {R}^+, \\quad E \\longmapsto T(E) = \\frac{1}{2} \\int _{\\gamma _E}\\frac{du}{v},$ is strictly monotonically decreasing and satisfies $T^{\\prime }(E) = -\\frac{p}{4 (2+p) E} \\int _{\\gamma _E} \\frac{u^p}{(c-u^p)}\\frac{du}{v} < 0.$ Since $A(u) + B(u) v^2=E$ is constant along an orbit $\\gamma _E$ , we find that $2 E \\,T(E)= \\int _{\\gamma _E} B(u) v du + \\int _{\\gamma _E} A(u) \\frac{du}{v}.$ To compute the derivative of $T$ with respect to $E$ , we first resolve the singularity in the second integral in equation (REF ).", "To this end, recall that the orbit $\\gamma _E$ belongs to the level curve $\\lbrace A(u) + B(u) v^2=E\\rbrace $ and therefore $\\frac{dv}{du} = -\\frac{A^{\\prime }(u) + B^{\\prime }(u)v^2}{2B(u)v}$ along the orbit.", "Note that $B(u)$ is different from zero for $E \\in (0,E_c)$ .", "Furthermore, $BA/A^{\\prime }$ is bounded on $\\gamma _E$ .", "Using the fact that the integral of a total differential $d$ over the closed orbit $\\gamma _E$ yields zero, we find that $0 &=& \\int _{\\gamma _E} d \\left[ \\left(\\frac{2BA}{A^{\\prime }}\\right)(u) \\,v \\right]\\\\&=& \\int _{\\gamma _E}\\left(\\frac{2BA}{A^{\\prime }}\\right)^{\\prime }(u) \\,v \\,du+ \\left(\\frac{2BA}{A^{\\prime }}\\right)(u)\\, dv\\\\&=& \\int _{\\gamma _E} \\left(\\frac{2BA}{A^{\\prime }}\\right)^{\\prime }(u) \\,v \\,du- \\left(\\frac{2BA}{A^{\\prime }} \\frac{A^{\\prime }}{2B}\\right)(u) \\frac{du}{v}- \\left(\\frac{2BA}{A^{\\prime }} \\frac{B^{\\prime }}{2B}\\right)(u) \\,v \\,du\\\\&=& \\int _{\\gamma _E} \\left[\\left(\\frac{2BA}{A^{\\prime }}\\right)^{\\prime }(u) -\\left(\\frac{AB^{\\prime }}{A^{\\prime }}\\right)(u)\\right] v\\, du- A(u)\\frac{du}{v},$ where we have used relation (REF ) in the third equality.", "Denoting $G = \\left(\\frac{2BA}{A^{\\prime }} \\right)^{\\prime } - \\frac{AB^{\\prime }}{A^{\\prime }},$ this ensures that $2 E T(E) = \\int _{\\gamma _E} \\left[ B(u) + G(u) \\right] v du,$ where the integrand is no longer singular at the turning points, where the orbit $\\gamma _E$ intersects with the horizontal axis $v = 0$The idea for this approach of resolving the singularity is taken from [8], where the authors prove a more general result for polynomial systems having first integrals of the form (REF )..", "Taking now the derivative of equation (REF ) with respect to $E$ we obtain that $2 T(E) + 2 E \\,T^{\\prime }(E) = \\int _{\\gamma _E} \\frac{B(u) + G(u)}{2B(u)v} du,$ where we have used that $\\frac{\\partial v}{\\partial E} = \\frac{1}{2 B(u) v}$ in view of (REF )Note that (REF ) also follows by applying Gelfand-Leray derivatives in (REF ), see [16] Theorem 26.32, p. 526.. From (REF ), we conclude that $2 T^{\\prime }(E) &=& \\frac{1}{E} \\int _{\\gamma _E} \\left(\\frac{B+G}{2B}\\right) (u) \\frac{du}{v}- \\frac{1}{E} \\int _{\\gamma _E} \\frac{du}{v} \\\\&=& \\frac{1}{E} \\int _{\\gamma _E}\\frac{1}{2B} \\left(\\left(\\frac{2AB}{A^{\\prime }}\\right)^{\\prime } - \\frac{(AB)^{\\prime }}{A^{\\prime }}\\right) (u) \\frac{du}{v}.$ In view of the expressions for $A$ and $B$ defined in Lemma REF , further calculations show that $T^{\\prime }(E) = -\\frac{p}{4 (2+p) E} \\int _{\\gamma _E} \\frac{u^p}{(c-u^p)}\\frac{du}{v}.$ We now need to show that $T^{\\prime }(E) < 0$ for every $E \\in (0,E_c)$ .", "In view of the symmetry of the vector field with respect to the horizontal axis and taking into account (REF ), we write (REF ) in the form $\\nonumber T^{\\prime }(E) &=& -\\frac{p}{2 (2+p) E}\\int _{u_-}^{u_+} \\frac{u^p}{(c-u^p)}\\sqrt{\\frac{B(u)}{E-A(u)}} du \\\\&=& -\\frac{p}{2 \\sqrt{2} (2+p) E} \\int _{u_-}^{u_+} \\frac{u^p}{\\sqrt{E-A(u)}} du,$ where $u_{\\pm }$ denote the turning points of the orbit $\\gamma _E$ with $E=A(u_{\\pm })$ , i.e.", "the intersections of the orbit $\\gamma _E$ with the horizontal axis $v = 0$ .", "Therefore, we find that $T^{\\prime }(E) <0$ if $p$ is even.", "Now we show that the same property also holds when $p$ is odd.", "Denote $I_1(E):= \\int _{u_-}^0 \\frac{u^p}{\\sqrt{E-A(u)}} du, \\quad I_2(E):= \\int _0^{u_+} \\frac{u^p}{\\sqrt{E-A(u)}} du,$ then $T^{\\prime }(E) &= -\\frac{p}{2 \\sqrt{2}(2+p) E} \\big [ I_1(E) + I_2(E)\\big ].$ We perform the change of variables $u=u_+ x$ and find that $I_2(E) &=& \\int _0^{u_+} \\frac{u^p}{\\sqrt{A(u_+)-A(u)}} du= \\int _0^1 \\frac{ u_+^p x^p}{\\sqrt{A(u_+) - A(u_+x)}} u_+ dx\\\\&=& \\sqrt{2}u_+^p \\int _0^1\\frac{ x^p}{\\sqrt{c(1-x^2) - \\frac{2u_+^p }{p+2} (1-x^{p+2} )}} dx.$ To rewrite the first integral we change variables according to $u=-|u_-|x$ and obtain $I_1(E) &=& \\int _{-|u_-|}^0 \\frac{u^p}{\\sqrt{A(-|u_-|)-A(u)}} du= \\int _1^0 \\frac{ -|u_-|^p x^p}{\\sqrt{A(-|u_-|) - A(u_-x)}} (-|u_-|) dx\\\\&=& -\\sqrt{2}|u_-|^p \\int _0^1\\frac{ x^p}{\\sqrt{c(1-x^2) + \\frac{2|u_-|^p }{p+2} (1-x^{p+2} )}} dx.$ We claim that $|u_-| < u_+$ .", "Indeed, we have that $A(u)<A(-u)$ on $(0,c^{1/p})$ , since $A(u)-A(-u) = u^2 \\left( \\frac{c}{2} - \\frac{1}{p+2} u^p \\right)- u^2 \\left( \\frac{c}{2} + \\frac{1}{p+2} u^p \\right)=- \\frac{2}{p+2} u^{p+2} <0.$ Moreover, $A$ is monotone on $(0,c^{1/p})$ .", "Assuming to the contrary that $|u_-| \\ge u_+$ , we would have that $A(|u_-|)\\ge A(u_+)$ and hence $A(u_+)\\le A(|u_-|) < A(u_-)$ , which contradicts the fact that $A(u_+)=A(u_-)$ .", "Hence $0<|u_-|<u_+<c^{1/p}$ , which implies that $|I_1(E)|<I_2(E)$ , and therefore, $T^{\\prime }(E)<0$ also in the case when $p$ is odd.", "The proof of Proposition REF is complete.", "The following result describes the limiting points of the energy-to-period map $E \\mapsto 2T(E)$ and is proved with routine computations.", "Lemma 2 For every $c > 0$ and $p \\in \\mathbb {N}$ , let $E \\mapsto 2T(E)$ be the mapping defined by (REF ).", "Then $T(0) := \\lim _{E\\rightarrow 0} T(E)= \\pi c^{1/2},$ and there exists $T_1 \\in (0,\\pi )$ such that $T(E_c) := \\lim _{E \\rightarrow E_c} T(E)= T_1 c^{1/2},$ with $E_c$ defined in (REF ).", "We can write (REF ) in the explicit form $T(E) = \\int _{u_-}^{u_+} \\frac{\\sqrt{B(u)} du}{\\sqrt{E - A(u)}},$ where the turning points $u_{\\pm } \\gtrless 0$ are given by the roots of $A(u_{\\pm }) = E$ .", "To prove the first assertion, we use the scaling transformation $u = \\left( \\frac{2E}{c} \\right)^{1/2} x,$ to rewrite the integral in (REF ) as follows: $T(E) = c^{1/2} \\int _{v_-}^{v_+} \\frac{(1 - \\mu x^p) dx}{\\sqrt{1 - x^2 + 2 \\mu x^{p+2}/(p+2)}}, \\quad \\mu := \\frac{2^{p/2} E^{p/2}}{c^{(p+2)/2}},$ where $v_{\\pm } \\gtrless 0$ are roots of the algebraic equation $v_{\\pm }^2 = 1 + \\frac{2}{p+2} \\mu v_{\\pm }^{p+2}.$ We note that $\\mu \\rightarrow 0$ , $v_{\\pm } \\rightarrow \\pm 1$ as $E \\rightarrow 0$ , which gives the formal limit $\\int _{v_-}^{v_+} \\frac{(1 - \\mu x^p) dx}{\\sqrt{1 - x^2 + 2 \\mu x^{p+2}/(p+2)}} \\rightarrow \\int _{-1}^1 \\frac{dx}{\\sqrt{1-x^2}}= \\pi \\quad \\mbox{\\rm as} \\quad \\mu \\rightarrow 0.$ This yields the limit (REF ).", "The justification of the formal limit is performed by rescaling $[v_-,v_+]$ to $[-1,1]$ and by using Lebesgue's Dominated Convergence Theorem, since the integrand function and its limit as $\\mu \\rightarrow 0$ are absolutely integrable.", "To prove the second assertion, notice that for $E = E_c$ , the turning points $u_{\\pm }$ used in the integral (REF ) are known as $u_{\\pm } = \\pm c^{1/p} q_{\\pm }$ , where $q_+ = 1$ and $q_- > 0$ is a root of the algebraic equation $q_-^2 - \\frac{2}{p+2} (-1)^p q_-^{p+2} = \\frac{p}{p+2}.$ If $p$ is even, $q_- = 1$ , while if $p$ is odd, $q_- \\in (0,1)$ , as follows from the proof of Proposition REF .", "By splitting the integral (REF ) into two parts we integrate over $[u_-,0]$ and $[0,u_+]$ separately and use the substitution $u = \\pm c^{1/p} x$ for the two integrals.", "Since $T^{\\prime }(E)$ is bounded for every $E > 0$ from the representation (REF ), including the limit $E \\rightarrow E_c$ , we obtain that $T(E_c) := \\lim _{E \\rightarrow E_c} T(E)$ exists and is given by $T(E_c) = T_1 c^{1/2}$ , where $\\nonumber T_1 & := & \\int _{0}^{1} \\frac{(1 - x^p) dx}{\\sqrt{1 - x^2 - 2(1 - x^{p+2})/(p+2)}}\\\\& \\phantom{t} & + \\int _{0}^{q_-} \\frac{(1 - (-1)^p x^p) dx}{\\sqrt{1 - x^2 - 2(1 - (-1)^p x^{p+2})/(p+2)}}.$ Both integrals are finite and positive, from which the existence of $T_1 > 0$ is concluded.", "Since $T^{\\prime }(E) < 0$ for every $E > 0$ we have that $T_1 < \\pi $ ." ], [ "Continuation of smooth periodic waves with respect to $c$", "In Section we fixed the parameter $c > 0$ and considered a continuation of the smooth periodic wave solutions $U$ with respect to the parameter $E$ in $(0,E_c)$ , where $E = 0$ corresponds to the zero solution and $E = E_c$ corresponds to a peaked periodic wave.", "The mapping $E \\mapsto 2T(E)$ is found to be monotonically decreasing according to Proposition REF .", "Therefore, this mapping can be inverted for every fixed $c > 0$ and we denote the corresponding dependence by $E(T,c)$ .", "The range of the mapping $E \\mapsto 2T(E)$ , which was calculated in Lemma REF , specifies the domain of the function $E(T,c)$ with respect to the parameter $T$ .", "The existence interval for the smooth periodic waves between the two limiting cases (REF ) and (REF ) obtained in Lemma REF is shown in Figure REF .", "Figure: The existence region for smooth periodic waves in the (T,c)(T,c)-parameter planebetween the two limiting curves T=πc 1/2 T = \\pi c^{1/2} and T=T 1 c 1/2 T = T_1 c^{1/2} obtained in Lemma .When we fix the parameter $c > 0$ , the half-period $T$ belongs to the interval $(T_1 c^{1/2},\\pi c^{1/2})$ , which corresponds to the vertical line in Figure REF .", "When we fix the parameter $T > 0$ , the parameter $c$ belongs to the interval $(T^2/\\pi ^2,T^2/T_1^2)$ , which corresponds to the horizontal line in Figure REF .", "In this section, we will fix the period $2T$ and consider a continuation of the smooth periodic wave solutions $U$ with respect to the parameter $c$ in a subset of $\\mathbb {R}^+$ .", "The next result specifies the interval of existence for the speed $c$ .", "Lemma 3 For every $T > 0$ and $p \\in \\mathbb {N}$ , there exists a family of $2T$ -periodic solutions $U=U(z;c)$ of equation (REF ) parametrized by $c \\in (c_0(T),c_1(T))$ , where $c_0(T) := \\frac{T^2}{\\pi ^2}, \\quad c_1(T) := \\frac{T^2}{T_1^2} > c_0(T),$ with $T_1 \\in (0,\\pi )$ given in (REF ) and $U \\rightarrow 0$ as $c \\rightarrow c_0(T)$ .", "Moreover, the mapping $(c_0(T),c_1(T)) \\ni c \\mapsto U \\in \\dot{L}_{\\rm per}^2(-T,T) \\cap H^{\\infty }_{\\rm per}(-T,T)$ is $C^1$ .", "Notice that the scaling transformation $U(z;c) = c^{1/p} \\tilde{U}(\\tilde{z}), \\quad z = c^{1/2} \\tilde{z}, \\quad T = c^{1/2} \\tilde{T},$ relates $2T$ -periodic solutions $U$ of the boundary-value problem (REF ) to $2\\tilde{T}$ -periodic solution $\\tilde{U}$ of the same boundary-value problem with $c$ normalized to 1, that is, $\\frac{d}{d \\tilde{z}} \\left[ (1 - \\tilde{U}^p) \\frac{d \\tilde{U}}{d \\tilde{z}} \\right] + \\tilde{U}(\\tilde{z}) = 0, \\quad \\tilde{U}(-\\tilde{T}) = \\tilde{U}(\\tilde{T}), \\quad \\tilde{U}^{\\prime }(-\\tilde{T}) =\\tilde{U}^{\\prime }(\\tilde{T}).$ Lemma REF guarantees existence of a family $\\lbrace \\tilde{U}_E\\rbrace _{E \\in (0,E_1)}$ of $2\\tilde{T}(\\tilde{E})$ -periodic solutions of the boundary-value problem (REF ).", "In view of Lemma REF and since $T$ is fixed, we have $\\tilde{T}(\\tilde{E}) = c^{-1/2}T \\in (T_1,\\pi )$ , which implies that $c$ belongs to the interval $(c_0(T),c_1(T))$ , where $c_0(T)$ and $c_1(T)$ are given by (REF ).", "Moreover, this relation provides a one-to-one correspondence between the parameters $c$ and $\\tilde{E}$ in view of the fact that $\\tilde{T}^{\\prime }(\\tilde{E})<0$ by Proposition REF which implies that $c^{1/2} =T/\\tilde{T} (\\tilde{E})$ is monotone increasing in $\\tilde{E}$ .", "In view of the transformation (REF ), we therefore obtain existence of a family $\\lbrace U_c\\rbrace _{c \\in (c_0(T),c_1(T))}$ of $2T$ -periodic solutions of the boundary-value problem (REF ).", "The value $c_0(T)$ corresponds to the zero solution, whereas $c_1(T)$ corresponds to the peaked periodic wave.", "Recall that the mapping $E \\mapsto 2T(E)$ can be inverted for every fixed $c > 0$ , and that the corresponding dependence is denoted by $E(T,c)$ .", "The next result shows that $E(T,c)$ is a monotonically increasing function of $c\\in (c_0(T),c_1(T))$ for every fixed $T > 0$ .", "Lemma 4 For every $T > 0$ , $p \\in \\mathbb {N}$ , the mapping $(c_0(T),c_1(T)) \\ni c \\mapsto E(T,c)$ is $C^1$ and monotonically increasing.", "Using the transformation (REF ) in the boundary-value problem (REF ), we obtain that $E(T,c) = c^{\\frac{p+2}{p}} \\tilde{E},$ where $ \\tilde{E}$ is the energy level of the first integral of the second-order equation in (REF ), $\\tilde{E} = \\frac{1}{2} (1 - \\tilde{U}^p)^2 \\left(\\frac{d U}{d \\tilde{z}} \\right)^2+ \\frac{1}{2} \\tilde{U}^2 - \\frac{1}{p+2} \\tilde{U}^{p+2}.$ Now, as $T$ is fixed and $\\tilde{T} = \\tilde{T}(\\tilde{E})$ is defined by (REF ) for $c$ normalized to 1, we can define $E(T,c)$ from the root of the following equation $T = c^{\\frac{1}{2}} \\tilde{T}\\left(E(T,c) c^{-\\frac{p+2}{p}}\\right).$ Since $\\tilde{T}(0) = \\pi $ and $\\tilde{T}(E_1) = T_1$ , we have roots $E(T,c_0(T)) = 0$ and $E(T,c_1(T)) = E_c$ of the algebraic equation (REF ), with $E_c$ given by (REF ) at $c = c_1(T)$ .", "In order to continue the roots by using the implicit function theorem for every $c \\in (c_0(T),c_1(T))$ , we differentiate (REF ) with respect to $c$ at fixed $T$ and obtain $0 = \\frac{1}{2} c^{-\\frac{1}{2}} \\tilde{T}(\\tilde{E}) - \\frac{p+2}{p} Ec^{-\\frac{3p+4}{2p}} \\tilde{T}^{\\prime }(\\tilde{E}) + c^{-\\frac{p+4}{2 p}} \\tilde{T}^{\\prime }(\\tilde{E}) \\frac{\\partial E(T,c)}{\\partial c}.$ By Proposition REF , we have $\\tilde{T}^{\\prime }(\\tilde{E}) < 0$ for $\\tilde{E} \\in (0,E_1)$ , so that we can rewrite (REF ) as follows: $\\left| \\tilde{T}^{\\prime }(\\tilde{E}) \\right| \\frac{\\partial E(T,c)}{\\partial c}= \\frac{1}{2} c^{\\frac{2}{p}} \\tilde{T}(\\tilde{E}) + \\frac{p+2}{p} Ec^{-1} \\left| \\tilde{T}^{\\prime }(\\tilde{E}) \\right| > 0.$ Recall that $\\tilde{T}^{\\prime }(\\tilde{E})$ is bounded and nonzero for every $E \\in (0,E_1)$ and in the limit $E \\rightarrow E_1$ .", "By the implicit function theorem and thanks to the smoothness of all dependencies, there exists a unique, monotonically increasing $C^1$ map $(c_0(T),c_1(T)) \\ni c \\mapsto E(T,c)$ such that $E(T,c)$ is a root of equation (REF ) and $E(T,c_1(T)) = E_c$ , where $E_c$ is given by (REF ) at $c = c_1(T)$ .", "We shall now consider how the $ L^2_{\\rm per}(-T,T)$ norm of the periodic wave $U$ with fixed period $2T$ depends on the parameter $c$ .", "In order to prove that it is an increasing function of $c$ in $(c_0(T),c_1(T))$ , we obtain a number of identities satisfied by the periodic wave $U$ .", "This result will be used in the proof of Proposition REF in Section .", "Lemma 5 For every $T > 0$ , $p \\in \\mathbb {N}$ , the mapping $(c_0(T),c_1(T)) \\ni c \\mapsto \\Vert U \\Vert _{L^2_{\\rm per}(-T,T)}^2$ is $C^1$ and monotonically increasing.", "Moreover, if the operator $L$ is defined by (REF ), then $\\partial _c U \\in \\dot{L}_{\\rm per}^2(-T,T)$ satisfies $L \\partial _c U = -U$ and $\\langle \\partial _c U,U\\rangle _{L^2_{\\rm per}} > 0.$ Integrating (REF ) in $z$ with zero mean, we can write $(c-U^p) \\partial _z U + \\partial _z^{-1} U = 0.$ From here, multiplication by $\\partial _z^{-1} U$ and integration by parts yield $\\Vert \\partial _z^{-1} U \\Vert ^2_{L^2_{\\rm per}(-T,T)} = c \\Vert U \\Vert _{L^2_{\\rm per}(-T,T)}^2 - \\frac{1}{p+1} \\int _{-T}^T U^{p+2} dz.$ On the other hand, integrating (REF ) over the period $2T$ and using equations (REF ) and (REF ) yield $\\nonumber 2 E(T,c) T & = & \\frac{c}{2} \\Vert U \\Vert _{L^2_{\\rm per}(-T,T)}^2 - \\frac{1}{p+2} \\int _{-T}^T U^{p+2} dz + \\frac{1}{2}\\left\\Vert (c-U^p) \\frac{d U}{d z} \\right\\Vert ^2_{L^2_{\\rm per}(-T,T)} \\\\\\nonumber & = & \\frac{c}{2} \\Vert U \\Vert _{L^2_{\\rm per}(-T,T)}^2 - \\frac{1}{p+2} \\int _{-T}^T U^{p+2} dz + \\frac{1}{2}\\Vert \\partial _z^{-1} U \\Vert ^2_{L^2_{\\rm per}(-T,T)} \\\\& = & c \\Vert U \\Vert _{L^2_{\\rm per}(-T,T)}^2 - \\frac{(3p+4)}{2(p+1)(p+2)} \\int _{-T}^T U^{p+2} dz.$ Expressing $c \\Vert U \\Vert _{L^2_{\\rm per}(-T,T)}^2$ from equations (REF ) and (REF ), we obtain $\\Vert \\partial _z^{-1} U \\Vert ^2_{L^2_{\\rm per}} = 2 E(T,c) T + \\frac{p}{2(p+1)(p+2)} \\int _{-T}^T U^{p+2} dz.$ From the fact that $U$ is a critical point of $H(u) + c Q(u)$ given by (REF ) and (REF ) for a fixed period $2T$ , we obtain $\\frac{d \\mathcal {H}}{dc} + c \\frac{d \\mathcal {Q}}{dc} = 0,$ where $\\nonumber \\mathcal {H}(c) & = & - \\frac{1}{2} \\Vert \\partial _z^{-1} U \\Vert ^2_{L^2_{\\rm per}(-T,T)} - \\frac{1}{(p+1)(p+2)} \\int _{-T}^T U^{p+2} dz \\\\& = & -E(T,c) T - \\frac{(p+4)}{4(p+1)(p+2)}\\int _{-T}^T U^{p+2} dz$ and $\\nonumber c \\mathcal {Q}(c) & = & \\frac{c}{2} \\Vert U \\Vert ^2_{L^2_{\\rm per}(-T,T)} \\\\& = & E(T,c) T + \\frac{(3p+4)}{4(p+1)(p+2)} \\int _{-T}^T U^{p+2} dz$ are simplified with the help of equations (REF ) and (REF ) again.", "Next, we differentiate (REF ) and (REF ) in $c$ for fixed $T$ and use (REF ) to obtain a constraint $\\nonumber \\frac{d \\mathcal {H}}{dc} + c \\frac{d \\mathcal {Q}}{dc} & = & - \\frac{(p+4)}{4(p+1)(p+2)} \\frac{d}{dc} \\int _{-T}^T U^{p+2} dz - \\mathcal {Q}(c)+ \\frac{(3p+4)}{4(p+1)(p+2)} \\frac{d}{dc} \\int _{-T}^T U^{p+2} dz \\\\& = & - \\mathcal {Q}(c) + \\frac{p}{2(p+1)(p+2)} \\frac{d}{dc} \\int _{-T}^T U^{p+2} dz = 0.$ From (REF ), (REF ), (REF ), and (REF ), we finally obtain $\\nonumber c \\frac{d \\mathcal {Q}}{dc} = -\\frac{d \\mathcal {H}}{dc} & = & T \\frac{\\partial E(T,c)}{\\partial c} +\\frac{(p+4)}{4(p+1)(p+2)} \\frac{d}{dc} \\int _{-T}^T U^{p+2} dz \\\\& = & T \\frac{\\partial E(T,c)}{\\partial c} + \\frac{p+4}{2p} \\mathcal {Q}(c) > 0.$ To prove the second assertion, recall that the family of periodic waves $U(z;c)$ is $C^1$ with respect to $c$ by Lemma REF .", "Differentiating the second-order equation in (REF ) with respect to $c$ at fixed period $2T$ and integrating it twice with zero mean yields equation (REF ).", "Notice that $\\partial _c U$ is again $2T$ -periodic, since the period of $U$ is fixed independently of $c$ .", "Finally, we find that $\\langle \\partial _c U, U \\rangle _{L^2_{\\rm per}} = \\frac{1}{2} \\frac{d}{dc} \\Vert U \\Vert _{L^2_{\\rm per}}^2 >0,$ since by the first assertion, the mapping $c\\mapsto \\Vert U \\Vert _{L^2_{\\rm per}}^2$ is monotonically increasing.", "As an immediate consequence of Lemmas REF and REF , we prove the following result which will be used in the proof of Proposition REF in Section .", "Corollary 1 For every $T > 0$ , $p \\in \\mathbb {N}$ and $c\\in (c_0(T),c_1(T))$ , the periodic solution $U$ of the boundary-value problem (REF ) satisfies $\\int _{-T}^T U^{p+2} dz > 0.$ It follows from (REF ) that $\\frac{d}{dc} \\int _{-T}^T U^{p+2} dz = \\frac{2(p+1)(p+2)}{p} \\mathcal {Q}(c) > 0, \\quad c \\in (c_0(T),c_1(T)).$ On the other hand, $\\int _{-T}^T U^{p+2} dz = 0$ at $c = c_0(T)$ by Lemma REF .", "Integrating the inequality (REF ) for $c > c_0(T)$ implies positivity of $\\int _{-T}^T U^{p+2} dz$ ." ], [ "Negative index of the operator $L$", "Recall that $T(E) \\rightarrow T(0) = \\pi c^{1/2}$ and $U\\rightarrow 0$ as $E \\rightarrow 0$ in view of Lemma REF .", "In this limit, the operator given by (REF ) becomes an integral operator with constant coefficients, $L_0 = P_0( \\partial _z^{-2} + c)P_0: \\dot{L}_{per}^2(-T(0),T(0)) \\rightarrow \\dot{L}_{per}^2(-T(0),T(0)),$ whose spectrum can be computed explicitly as $\\sigma (L_0) = \\left\\lbrace c(1 - n^{-2}), \\quad n \\in \\mathbb {Z} \\backslash \\lbrace 0\\rbrace \\right\\rbrace ,$ by using Fourier series.", "For every $c > 0$ , the spectrum of $L_0$ is purely discrete and consists of double eigenvalues accumulating to the point $c$ .", "All double eigenvalues are strictly positive except for the lowest eigenvalue, which is located at the origin.", "As is shown in [4] with a perturbation argument for $p = 1$ and $p = 2$ , the spectrum of $L$ for $E$ near 0 includes a simple negative eigenvalue, a simple zero eigenvalue, and the positive spectrum is bounded away from zero.", "We will show that this conclusion remains true for the entire family of smooth periodic waves.", "Let us first prove the following.", "Lemma 6 For every $c > 0$ , $p \\in \\mathbb {N}$ , and $E \\in (0,E_c)$ , the operator $L$ given by (REF ) is self-adjoint and its spectrum includes a countable set of isolated eigenvalues below $C_-(E) := \\inf _{z \\in [-T(E),T(E)]} (c - U(z)^p) > 0.$ The self-adjoint properties of $L$ are obvious.", "For every $E \\in (0,E_c)$ , there are positive constants $C_{\\pm }(E)$ such that $C_-(E) \\le c - U(z)^p \\le C_+(E) \\quad \\mbox{\\rm for every} \\;\\; z \\in [-T(E),T(E)].$ For the rest of the proof we use the short notation $T = T(E)$ .", "The eigenvalue equation $(L-\\lambda I) v = 0$ for $v \\in \\dot{L}^2_{\\rm per}(-T,T)$ is equivalent to the spectral problem $P_0 (c-U^p-\\lambda ) P_0 v = - P_0 \\partial _z^{-2} P_0 v.$ Under the condition $\\lambda < C_-(E)$ , we have $c - U^p - \\lambda \\ge C_-(E) - \\lambda > 0$ .", "Setting $w := (c-U^p-\\lambda )^{1/2} P_0 v \\in L^2_{\\rm per}(-T,T), \\quad \\lambda < C_-(E),$ we find that $\\lambda $ is an eigenvalue of the spectral problem (REF ) if and only if 1 is an eigenvalue of the self-adjoint operator $K(\\lambda ) = - (c-U^p-\\lambda )^{-1/2} P_0 \\partial _z^{-2} P_0 (c-U^p-\\lambda )^{-1/2} : L^2_{\\rm per}(-T,T) \\rightarrow L^2_{\\rm per}(-T,T),$ that isThis reformulation can be viewed as an adjoint version of the Birmann–Schwinger principle used in analysis of isolated eigenvalues of Schrödinger operators with rapidly decaying potentials [12]., $w = K(\\lambda ) w$ .", "The operator $K(\\lambda )$ for every $\\lambda < C_-(E)$ is a compact (Hilbert–Schmidt) operator thanks to the bounds (REF ) and the compactness of $P_0 \\partial _z^{-2} P_0$ .", "Consequently, the spectrum of $K(\\lambda )$ in $L^2_{\\rm per}(-T,T)$ for every $\\lambda < C_-(E)$ is purely discrete and consists of isolated eigenvalues.", "Moreover, these eigenvalues are positive thanks to the positivity of $K(\\lambda )$ , as follows: $\\langle K(\\lambda ) w, w \\rangle _{L^2_{\\rm per}} = \\Vert P_0 \\partial _z^{-1} P_0 (c-U^p-\\lambda )^{-1/2} w \\Vert _{L^2_{\\rm per}}^2 \\ge 0, \\quad \\forall w \\in L^2_{\\rm per}(-T,T).$ We note that (a) $K(\\lambda ) \\rightarrow 0^+$ as $\\lambda \\rightarrow -\\infty $ , (b) $K^{\\prime }(\\lambda ) > 0$ for every $\\lambda < C_-(E)$ .", "Claim (a) follows from (REF ) via spectral calculus: $\\langle K(\\lambda ) w, w \\rangle _{L^2_{\\rm per}} \\sim |\\lambda |^{-1}\\Vert P_0 \\partial _z^{-1} P_0 w \\Vert _{L^2}^2 \\quad \\mbox{\\rm as} \\quad \\lambda \\rightarrow -\\infty .$ Claim (b) follows from the differentiation of $K(\\lambda )$ , $\\langle K^{\\prime }(\\lambda ) w, w \\rangle _{L^2_{\\rm per}} & = &\\frac{1}{2} \\langle \\rho (\\lambda )K(\\lambda )w,w\\rangle _{L^2_{\\rm per}}+\\frac{1}{2} \\langle K(\\lambda ) \\rho (\\lambda )w,w\\rangle _{L^2_{\\rm per}},$ where we have defined the weight function $\\rho (\\lambda ) := (c - U^p - \\lambda )^{-1}$ which is strictly positive and uniformly bounded thanks to (REF ).", "Since $K(\\lambda )$ is positive due to (REF ), both terms in the above expression are positive in view of a generalization of Sylvester's law of inertia for differential operators, see Theorem 4.2 in [23].", "Indeed, to prove that the first term is positive it suffices to show that the eigenvalues $\\mu $ of $\\rho (\\lambda ) K(\\lambda ) $ are positive.", "The corresponding spectral problem $\\rho (\\lambda ) K(\\lambda ) w = \\mu w$ is equivalent to $\\rho (\\lambda )^{1/2} K(\\lambda ) \\rho (\\lambda )^{1/2} v = \\mu v$ in view of the substitution $w=\\rho (\\lambda )^{1/2}v$ .", "By Sylvester's law, the number of negative eigenvalues of $K(\\lambda )$ is equal to the number of negative eigenvalues of the congruent operator $\\tilde{K}(\\lambda ) = \\rho (\\lambda )^{1/2} K(\\lambda ) \\rho (\\lambda )^{1/2}$ .", "Therefore, $\\rho (\\lambda )K(\\lambda )$ is positive in view of the positivity of $K(\\lambda )$ .", "The second term can be treated in the same way.", "It follows from claims (a) and (b) that positive isolated eigenvalues of $K(\\lambda )$ are monotonically increasing functions of $\\lambda $ from the zero level as $\\lambda \\rightarrow -\\infty $ .", "The location and number of crossings of these eigenvalues with the unit level gives the location and number of eigenvalues $\\lambda $ in the spectral problem (REF ).", "The compactness of $K(\\lambda )$ for $\\lambda < C_-(E)$ therefore implies that there exists a countable (finite or infinite) set of isolated eigenvalues of $L$ below $C_-(E)$ .", "Next, we inspect analytical properties of eigenvectors for isolated eigenvalues below $C_-(E) > 0$ given by (REF ).", "Lemma 7 Under the condition of Lemma REF , let $\\lambda _0 < C_-(E)$ be an eigenvalue of the operator $L$ given by (REF ).", "Then, $\\lambda _0$ is at most double and the eigenvector $v_0$ belongs to $\\dot{L}^2_{\\rm per}(-T(E),T(E)) \\cap H^{\\infty }_{\\rm per}(-T(E),T(E))$ .", "As in the proof of the previous Lemma, we use the shorthand $T = T(E)$ for lighter notation.", "The eigenvector $v_0 \\in \\dot{L}^2_{\\rm per}(-T,T)$ for the eigenvalue $\\lambda _0 < C_-(E)$ satisfies the spectral problem (REF ) written as the integral equation $P_0 \\partial _z^{-2}P_0 v_0 + P_0 (c-U^p-\\lambda _0)P_0 v_0 = 0.$ Since $U \\in H^{\\infty }_{\\rm per}(-T,T)$ and $c - U^p - \\lambda _0 \\ge C_-(E) - \\lambda _0 > 0$ , we obtain that $v_0\\in H^{2}_{\\rm per}(-T,T)$ , and by bootstrapping arguments we find that $v_0 \\in H^{\\infty }_{\\rm per}(-T,T)$ .", "Applying two derivatives to the integral equation (REF ), we obtain the equivalent differential equation for the eigenvector $v_0 \\in \\dot{L}^2_{\\rm per}(-T,T) \\cap H^{\\infty }_{\\rm per}(-T,T)$ and the eigenvalue $\\lambda _0 < C_-(E)$ : $v_0 +\\partial _z^2 \\left[ (c-U^p - \\lambda _0) v_0 \\right] = 0.$ The second-order differential equation (REF ) admits at most two linearly independent solutions in $\\dot{L}^2_{\\rm per}(-T,T)$ and so does the integral equation (REF ) for an eigenvalue $\\lambda _0 < C_-(E)$ .", "Since $L$ is self-adjoint, the eigenvalue $\\lambda _0$ is not defectiveRecall that the eigenvalue is called defective if its algebraic multiplicity exceeds its geometric multiplicity., and hence the multiplicity of $\\lambda _0$ is at most two.", "We are now ready to prove the main result of this section.", "This proves part (b) of Theorem REF .", "Proposition 2 For every $c > 0$ , $p \\in \\mathbb {N}$ , and $E \\in (0,E_c)$ , the operator $L$ given by (REF ) has exactly one simple negative eigenvalue, a simple zero eigenvalue, and the rest of the spectrum is positive and bounded away from zero.", "Thanks to Lemma REF , we only need to inspect the multiplicity of negative and zero eigenvalues of $L$ .", "By Lemma REF , the zero eigenvalue $\\lambda _0 = 0 < C_-(E)$ can be at most double.", "The first eigenvector $v_0 = \\partial _z U \\in \\dot{L}^2_{\\rm per}(-T(E),T(E)) \\cap H^{\\infty }_{\\rm per}(-T(E),T(E))$ for $\\lambda _0 = 0$ follows by the translational symmetry.", "Indeed, differentiating (REF ) with respect to $z$ , we verify that $v_0$ satisfies the differential equation (REF ) with $\\lambda _0 = 0$ and, equivalently, the integral equation (REF ) with $\\lambda _0 = 0$ .", "Another linearly independent solution $v_1 = \\partial _E U$ of the same equation (REF ) with $\\lambda _0 = 0$ is obtained by differentiating (REF ) with respect to $E$ .", "Here we understand the family $U(z;E)$ of smooth $2 T(E)$ -periodic solutions constructed in Lemma REF , where the period $2 T(E)$ is given by (REF ) and is a smooth function of $E$ .", "Now, we show that the second solution $v_1$ is not $2 T(E)$ -periodic under the condition $T^{\\prime }(E) < 0$ established in Proposition REF .", "Consequently, the zero eigenvalue $\\lambda _0 = 0$ is simple.", "For simplicity, we assume that the family $U(z;E)$ satisfies the condition $U(\\pm T(E);E) = 0$ at the end points, which can be fixed by translational symmetry.", "By differentiating the first boundary condition in (REF ) with respect to $E$ , we obtain $\\partial _E U(-T(E);E) - T^{\\prime }(E) \\partial _z U(-T(E);E) = \\partial _E U(T(E);E) + T^{\\prime }(E) \\partial _z U(T(E);E).$ Notice that $\\partial _z U(\\pm T(E);E) \\ne 0$ , since otherwise the periodic solution $U$ would be identically zero in view of (REF ) which is only possible for $E = 0$ .", "Since $T^{\\prime }(E) \\ne 0$ by Proposition REF , the solution $v_1 = \\partial _E U$ is not $2T(E)$ -periodic and therefore the zero eigenvalue $\\lambda _0 = 0$ is simple for the entire family of smooth $T(E)$ -periodic solutions.", "Next, we show that the spectrum of $L$ includes at least one negative eigenvalue.", "Indeed, from the integral version of the differential equation (REF ), $P_0 \\left(c - \\frac{1}{p+1} U^p \\right)P_0 U + P_0 \\partial _z^{-2} P_0 U = 0,$ we obtain that $L U = -\\frac{p}{p+1} P_0 U^{p+1}$ , which implies that $\\langle L U, U \\rangle _{L^2_{\\rm per}} = -\\frac{p}{p+1} \\int _{-T(E)}^{T(E)} U^{p+2} dz < 0.$ The last inequality is obvious for even $p$ .", "For odd $p$ it follows from Corollary REF for given $T(E) \\in (T_1 c^{1/2},\\pi c^{1/2})$ fixed.", "In both cases, we have shown that $L$ has at least one negative eigenvalue for every $E \\in (0,E_c)$ .", "Finally, the spectrum of $L$ includes at most one simple negative eigenvalue.", "Indeed, the family of $2 T(E)$ -periodic solutions is smooth with respect to the parameter $E \\in (0,E_c)$ and it reduces to the zero solution as $E\\rightarrow 0$ .", "It follows from the spectrum (REF ) for the operator $L_0$ at the zero solution, and the preservation of the simple zero eigenvalue with the eigenvector $\\partial _z U$ for every $E \\in (0,E_c)$ , that the splitting of a double zero eigenvalue for $E \\ne 0$ results in appearance of at most one negative eigenvalue of $L$ .", "Thus, there exists exactly one simple negative eigenvalue of $L$ for every $E \\in (0,E_c)$ ." ], [ "Applications of the Hamilton–Krein theorem", "Since $L$ has a simple zero eigenvalue in $\\dot{L}^2_{\\rm per}(-T,T)$ by Proposition REF with the eigenvector $v_0 = \\partial _z U$ , eigenvectors $v \\in \\dot{H}^1_{\\rm per}(-T,T)$ of the spectral problem $\\lambda v = \\partial _z Lv$ for nonzero eigenvalues $\\lambda $ satisfy the constraint $\\langle U, v \\rangle _{L^2_{\\rm per}} = 0$ , see definition (REF ) of the space $L^2_c$ .", "Since $\\partial _z$ is invertible in space $\\dot{L}^2_{\\rm per}(-T,T)$ and the inverse operator is bounded from $\\dot{L}^2_{\\rm per}(-T,T)$ to itself, we can rewrite the spectral problem $\\lambda v = \\partial _z Lv$ in the equivalent form $\\lambda P_0 \\partial _z^{-1} P_0 v = L v, \\quad v \\in \\dot{L}^2_{\\rm per}(-T,T).$ In this form, the Hamilton–Krein theorem from [15] applies directly in $L^2_c$ .", "According to this theorem, the number of unstable eigenvalues with $\\lambda \\notin i \\mathbb {R}$ is bounded by the number of negative eigenvalues of $L$ in the constrained space $L^2_c$ .", "Therefore, we only need to show that the operator $L$ is positive in $L^2_c$ with only a simple zero eigenvalue due to the translational invariance in order to prove part (c) of Theorem REF .", "The corresponding result is given by the following proposition.", "Proposition 3 For every $c > 0$ , $p \\in \\mathbb {N}$ , and $E \\in (0,E_c)$ , the operator $L |_{L^2_c} : L^2_c \\rightarrow L^2_c$ , where $L$ is given by (REF ), has a simple zero eigenvalue and a positive spectrum bounded away from zero.", "The proof relies on a well-known criterion (see for example Lemma 1 in [14] or Theorem 4.1 in [23]) which ensures positivity of the self-adjoint operator $L$ with properties obtained in Proposition REF , when it is restricted to a co-dimension one subspace.", "Positivity of $L |_{L^2_c} : L^2_c \\rightarrow L^2_c$ is achieved under the condition $\\langle L^{-1} U,U\\rangle _{L^2_{\\rm per}} <0.$ To show (REF ), we observe that ${\\rm Ker}(L) = {\\rm span}\\lbrace v_0\\rbrace $ , where $v_0=\\partial _z U$ and $\\langle U, v_0 \\rangle _{L^2_{\\rm per}} = 0$ implies that $U \\in {\\rm Ker}(L)^{\\perp }$ .", "By Fredholm's Alternative (see e.g.", "Theorem B.4 in [23]), $L^{-1} U$ exists in $\\dot{L}_{\\rm per}^2(-T,T)$ and can be made unique by the orthogonality condition $\\langle L^{-1} U, v_0 \\rangle _{L^2_{\\rm per}} = 0$ .", "By Lemma REF , we have the existence of $\\partial _c U \\in \\dot{L}_{\\rm per}^2(-T,T)$ such that $L \\partial _c U =-U$ , see equation (REF ).", "Moreover, $\\langle \\partial _c U, v_0 \\rangle _{L^2_{\\rm per}} =0$ , since $\\partial _c U$ and $v_0=\\partial _z U$ have opposite parity.", "Therefore, $\\partial _c U = L^{-1} U$ and we obtain $\\langle L^{-1} U,U\\rangle _{L^2_{\\rm per}} = - \\langle \\partial _c U,U \\rangle _{L^2_{\\rm per}} <0,$ where the strict negativity follows from Lemma REF .", "The proof of Theorem REF follows from the results of Propositions REF , REF , and REF ." ], [ "Acknowledgements", "A.G. is supported by the Austrian Science Fund (FWF) project J3452 “Dynamical Systems Methods in Hydrodynamics”.", "The work of D.P.", "is supported by the Ministry of Education and Science of Russian Federation (the base part of the State task No.", "2014/133, project No. 2839).", "The authors thank Todd Kapitula (Calvin College) for pointing out an error in an early version of this manuscript." ] ]
1606.05075
[ [ "Feedback Integrators" ], [ "Abstract A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves first integrals of the system.", "The idea is that given an initial point in the manifold we extend the dynamics from the manifold to its ambient Euclidean space and then modify the dynamics outside the intersection of the manifold and the level sets of the first integrals containing the initial point such that the intersection becomes a unique local attractor of the resultant dynamics.", "While the modified dynamics theoretically produces the same trajectory as the original dynamics, it yields a numerical trajectory that stably remains on the manifold and preserves the first integrals.", "The big merit of our method is that the modified dynamics can be integrated with any ordinary numerical integrator such as Euler or Runge-Kutta.", "We illustrate this method by applying it to three famous problems: the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry.", "We also carry out simulation studies to demonstrate the excellence of our method and make comparisons with the standard projection method, a splitting method and St\\\"ormer-Verlet schemes." ], [ "Introduction", "Given a dynamical system on a manifold with first integrals, it is important for a numerical integrator to preserve the manifold structure and the first integrals of the equations of motion.", "This has been the focus of much effort in the development of numerical integration schemes [2].", "In this paper we do not propose any specific numerical integration scheme, but rather propose a new paradigm of integration that can faithfully preserve conserved quantities with existing numerical integration schemes.", "The main idea in our paradigm is as follows.", "Consider a dynamical system on a manifold $M$ with first integrals $f_i : M \\rightarrow \\mathbb {R}$ , $i = 1, \\ldots , \\ell $ .", "Assume that we can embed the manifold $M$ into Euclidean space $\\mathbb {R}^n$ and extend the first integrals to a neighborhood $U$ of $M$ in $\\mathbb {R}^n$ .", "For an arbitrary point $x_0 \\in M$ , consider the set $\\Lambda = \\lbrace x \\in U \\mid x \\in M, f_i(x) = f_i(x_0), i = 1, \\ldots , \\ell \\rbrace $ which is the intersection of $M$ with all the level sets of the first integrals containing the point $x_0$ , and is an invariant set of the dynamical system.", "We then extend the dynamical system from $M$ to $U$ and then modify the dynamics outside of $\\Lambda $ such that the set $\\Lambda $ becomes a unique local attractor of the extended, modified system.", "Since the dynamics have not changed on $\\Lambda $ by the extension and modification to $U$ , both the original system on $M$ and the extended, modified system on $U$ produce the same trajectory for the initial point $x_0 \\in \\Lambda $ .", "Numerically, however, integrating the extended system has the following advantage: if the trajectory deviates from $\\Lambda $ at some numerical integration step, then it will get pushed back toward the attractor $\\Lambda $ in the extended, modified dynamics, thus remaining on the manifold $M$ and preserving all the first integrals.", "It can be rigorously shown that the discrete-time dynamical system derived from any one-step numerical integrator with uniform step size $h$ for the extended, modified continuous-time system indeed has an attractor $\\Lambda _h$ that contains the set $\\Lambda $ in its interior and converges to $\\Lambda $ as $h \\rightarrow 0+$ .", "In this paper we shall use the word, preserve, in this sense.", "It is noteworthy that the numerical integration of the extended dynamics can be carried out with any ordinary integrator and is done in one global Cartesian coordinate system on $\\mathbb {R}^n$ .", "We find conditions for applicability of this method and implement the result on the following three examples: the free rigid body dynamics, the Kepler problem, and a perturbed Kepler problem with rotational symmetry.", "We also carry out simulation studies to show the excellence of our new paradigm of integration for numerical preservation of conserved quantities in comparison with other well-known integration schemes, such as projection and splitting methods and symplectic Störmer-Verlet integrators." ], [ "Theory", "Consider a dynamical system on an open subset $U$ of $\\mathbb {R}^n$ : $\\dot{x} = X(x),$ where $X$ is a $C^1$ vector field on $U$ .", "Let us make the following assumptions: A1.", "There is a $C^2$ function $V : U \\rightarrow \\mathbb {R}$ such that $V(x) \\ge 0$ for all $x\\in U$ , $V^{-1}(0) \\ne \\emptyset $ , and $\\nabla V(x) \\cdot X (x) = 0$ for all $x \\in U$ .", "A2.", "There is a positive number $c$ such that $V^{-1}([0,c])$ is a compact subset of $U$ .", "A3.", "The set of all critical points of $V$ in $V^{-1}([0,c])$ is equal to $V^{-1}(0)$ .", "Adding the negative gradient of $V$ to (REF ), let us consider the following dynamical system on $U$ : $\\dot{x} = X(x) - \\nabla V(x).$ Since 0 is the minimum value of $V$ , $\\nabla V (x)= 0$ for all $x\\in V^{-1}(0)$ .", "Hence, the two vector fields $X$ and $X-\\nabla V$ coincide on $V^{-1}(0)$ .", "Theorem 2.1 Under assumptions A1 – A3, every trajectory of (REF ) starting from a point in $V^{-1}([0,c])$ stays in $V^{-1}([0,c])$ for all $t\\ge 0$ and asymptotically converges to the set $V^{-1}(0)$ as $t \\rightarrow \\infty $ .", "Furthermore, $V^{-1}(0)$ is an invariant set of both (REF ) and (REF ).", "Let $x(t)$ be a trajectory of (REF ) starting from a point in $V^{-1}([0,c])$ .", "By A1 $\\frac{d}{dt} V(x(t)) = \\nabla V (x(t)) \\cdot ( X(x(t))- \\nabla V(x(t)))= -|\\nabla V(x)|^2 \\le 0$ for all $t $ .", "Hence, $V^{-1}([0,c])$ is a positively invariant set of (REF ).", "From (REF ) and A3, it follows that $\\lbrace x \\in V^{-1}([0,c]) \\mid \\dot{V} (x) = 0\\rbrace = \\lbrace x \\in V^{-1}([0,c]) \\mid \\nabla V(x) = 0\\rbrace = V^{-1}(0)$ .", "Hence, by LaSalle's invariance principle [5], $x(t)$ converges asymptotically to $V^{-1}(0)$ as $t \\rightarrow \\infty $ , where A2 is used for compactness of $V^{-1}([0,c])$ .", "The invariance of $V^{-1}(0)$ follows from (REF ) and the coincidence of (REF ) and (REF ) on $V^{-1}(0)$ .", "Let us find a higher-order condition than that in assumption A3 so that A3 can be relaxed.", "For the function $V$ and the vector field $X$ in the statement of assumption A1, which are now both assumed to be of $C^\\infty $ , let $S = \\left\\lbrace x \\in U \\bigg {|} X^k \\frac{\\partial V}{\\partial x^i} = 0 \\,\\, \\forall \\,\\, k \\ge 0, 1 \\le i \\le n \\right\\rbrace ,$ where $x = (x^1, x^2, \\ldots , x^n)$ , and $X^k \\frac{\\partial V}{\\partial x^i} $ denotes the $k-$ th order directional derivative of $\\partial V/ \\partial x^i$ along $X$ , i.e., $X^0 \\frac{\\partial V}{\\partial x^i} = \\frac{\\partial V}{\\partial x^i}; \\quad X \\frac{\\partial V}{\\partial x^i} =X \\cdot \\nabla \\frac{\\partial V}{\\partial x^i}; \\quad X^k \\frac{\\partial V}{\\partial x^i} = X\\left(X^{k-1} \\frac{\\partial V}{\\partial x^i} \\right), \\,k \\ge 2.$ Consider the following assumption in place of A3: A3$^{\\prime }$ .", "$S \\cap V^{-1}([0,c]) \\subset V^{-1}(0)$ .", "The following theorem generalizes Theorem REF : Theorem 2.2 Under assumptions A1, A2 and A3$^{\\prime }$ , every trajectory of (REF ) starting in $V^{-1}([0,c])$ stays in $V^{-1}([0,c])$ for all $t\\ge 0$ and asymptotically converges to the set $V^{-1}(0)$ as $t \\rightarrow \\infty $ .", "Furthermore, $V^{-1}(0)$ is an invariant set of both (REF ) and (REF ).", "Consider the dynamics (REF ).", "It is easy to show that $V^{-1}([0,c])$ is a positively invariant set of the dynamics.", "Let $\\mathcal {M}$ be the largest invariant set in $\\mathcal {E} = \\lbrace x \\in U\\mid \\dot{V} (x)= 0\\rbrace \\cap V^{-1}([0,c])$ .", "Let $x(t)$ be an arbitrary trajectory in $\\mathcal {M}$ .", "Since $\\mathcal {E}= \\lbrace x \\in U \\mid \\nabla V (x)= 0\\rbrace \\cap V^{-1}([0,c])$ as shown in the proof of Theorem REF , the trajectory $x(t)$ satisfies $\\nabla V = 0$ , i.e., $\\frac{\\partial V}{\\partial x^i} (x(t)) =0$ for all $t \\in \\mathbb {R}$ and $ 1\\le i \\le n$ .", "Since $\\nabla V = 0$ along $x(t)$ , the trajectory $x(t)$ satisfies $\\dot{x}(t) = X(x(t))$ for all $t \\in \\mathbb {R}$ .", "By differentiating (REF ) repeatedly in $t$ and using (REF ) on each differentiation, we can show that the trajectory $x(t)$ satisfies $X^k \\frac{\\partial V}{\\partial x^i} = 0$ for all $t\\in \\mathbb {R}$ , $k \\ge 0$ and $ 1\\le i \\le n$ .", "Thus, the entire trajectory $x(t)$ is contained in the set $S$ defined in (REF ), implying $\\mathcal {M} \\subset S$ , from which and A3$^{\\prime }$ it follows $\\mathcal {M} \\subset V^{-1}(0)$ .", "Hence, by LaSalle's invariance principle, every trajectory starting in $V^{-1}([0,c])$ asymptotically converges to $\\mathcal {M}$ and thus to $V^{-1}(0)$ as $t \\rightarrow \\infty $ .", "The invariance of $V^{-1}(0)$ follows from (REF ) and the coincidence of (REF ) and (REF ) on $V^{-1}(0)$ .", "Remark 2.3 1.", "If condition (REF ) is replaced by $\\nabla V(x)\\cdot X(x) \\le 0$ in assumption A1, then Theorems REF and REF still hold provided that the invariance of $V^{-1}(0)$ is replaced by positive invariance in the statement of the theorems.", "2.Theorems REF and REF still hold with the use of the following modified dynamics $\\dot{x} = X(x) - A(x) \\nabla V(x)$ instead of (REF ), where $A(x)$ is an $n\\times n$ matrix-valued function with its symmetric part $(A(x) + A^T(x))$ positive definite at each $x\\in \\mathbb {R}^n$ .", "3.", "From the control viewpoint, the added term $-\\nabla V(x)$ in (REF ) can be regarded as a negative feedback control $ u (x) =-\\nabla V(x)$ to asymptotically stabilize the set $V^{-1}(0)$ for the control system $\\dot{x} = X(x) + u$ with control $u$ .", "Suppose that assumptions A1, A2 and A3 (or A3$^{\\prime }$ instead of A3) hold and that we want to integrate the dynamics (REF ) for an initial point $x(0) \\in V^{-1}(0)$ .", "Since $V^{-1}(0)$ is positively invariant, the trajectory must remain in $V^{-1}(0)$ for all $t\\ge 0$ .", "Recall that the two dynamics (REF ) and (REF ) coincide on $V^{-1}(0)$ , so we can integrate (REF ) instead of (REF ) for the initial condition.", "Though there is no theoretical difference between the two integrations, integrating (REF ) has a numerical advantage over integrating (REF ).", "Suppose that the trajectory numerically deviates from the positively invariant set $V^{-1}(0)$ during integration.", "Then the dynamics (REF ) will push the trajectory back toward $V^{-1}(0)$ since $V^{-1}(0)$ is the attractor of (REF ) in $V^{-1}([0,c])$ whereas the dynamics (REF ) will leave the trajectory outside of $V^{-1}(0)$ which would not happen in the exact solution.", "It is noteworthy that this integration strategy is independent of the choice of integration schemes.", "In the Appendix we show that any one-step numerical integrator, as a discrete-time dynamical system, with uniform step size $h$ for (REF ) has an attractor $\\Lambda _h$ that contains $V^{-1}(0)$ in its interior and converges to $V^{-1}(0)$ as $h\\rightarrow 0+$ .", "Let us now apply this integration strategy to numerically integrate dynamics on a manifold while preserving its first integrals and the domain manifold.", "Consider a manifold $M$ and dynamics $\\dot{x} = X(x)$ on $M$ that have $\\ell $ first integrals $f_i :M \\rightarrow \\mathbb {R}$ , $i = 1, \\ldots , \\ell $ .", "Suppose that $M$ is an embedded manifold in $\\mathbb {R}^n$ as a level set of a function $f_0: \\mathbb {R}^n \\rightarrow \\mathbb {R}^r$ for some $r$ , and that both the dynamics (REF ) and the functions $f_i$ , $i = 0, \\ldots , \\ell $ extend to an open neighborhood $U$ of $M$ in $\\mathbb {R}^n$ .", "Our goal is to numerically integrate (REF ) with an initial condition $x(0) = x_0 \\in M$ while preserving the manifold $M$ and the first integrals.", "Let $f = (f_0, f_1, \\ldots , f_\\ell ): \\mathbb {R}^n \\rightarrow \\mathbb {R}^{r+\\ell }$ and define a function $V:U \\subset \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by $V(x) = \\frac{1}{2} (f(x) - f(x_0))^T K (f(x) - f(x_0)),$ where $K$ is an $(r+\\ell ) \\times (r+\\ell )$ constant symmetric positive definite matrix.", "Notice that $V^{-1}(0) = \\lbrace x \\in U \\mid x \\in M, f_i(x) = f_i(x_0), i = 1, \\ldots , \\ell \\rbrace ,$ and that $V^{-1}(0)$ is invariant under the flow of (REF ).", "Or, more generally we can define a function $V(x)$ as $V(x) = W(f_0(x), f_1(x), \\ldots , f_\\ell (x))$ where $W: \\mathbb {R}^{r +\\ell } \\rightarrow \\mathbb {R}$ is a non-negative function that takes the value of 0 only at $(f_0(x_0), f_1(x_0), \\ldots , f_\\ell (x_0))$ .", "If the function $V$ satisfies assumptions A1, A2 and A3 (or A3$^{\\prime }$ instead of A3), then by Theorem REF (or Theorem REF ), $V^{-1}(0)$ is the local attractor of the modified dynamics $\\dot{x} = X(x) -\\nabla V(x)$ that coincide with the original dynamics (REF ) on $V^{-1}(0)$ .", "The following lemma provides a sufficient condition under which the function $V$ defined in (REF ) satisfies assumptions A2 and A3: Lemma 2.4 Consider the functions $f$ and $V$ defined in (REF ) and (REF ).", "If $V^{-1}(0)$ is compact and the Jacobian matrix $Df(x)$ of $f$ has rank $(r+\\ell )$ for all $x \\in V^{-1}(0)$ , then there is a number $c>0$ such that assumptions A2 and A3 hold.", "By compactness of $V^{-1}(0)$ and the regularity of $Df$ , there is a bounded open set $X$ such that $V^{-1}(0) \\subset X \\subset \\operatorname{cl}( X) \\subset U$ , and $Df$ (x) has rank $r+\\ell $ for all $x \\in X$ , where $\\operatorname{cl}(X)$ denotes the closure of $X$ .", "Consider now the gradient of $V$ .", "An easy calculation shows that, $\\nabla V(x) = Df(x)^TK (f(x) - f(x_0)).$ Now, since for all $x\\in X$ , $Df(x)$ is onto as a linear map, $Df(x)^{T}$ is therefore one to one.", "It follows that, for $x\\in X$ , $\\nabla V(x)=0 \\iff f(x)-f(x_{0})=0 \\iff x\\in V^{-1}(0).$ In other words, the set of all critical points of $V$ in $X$ is equal to $V^{-1}(0)$ .", "Since the boundary $\\partial X$ of $X$ , being closed and bounded, is compact and $\\partial X \\cap V^{-1}(0) = \\emptyset $ , the minimum value, denoted by $d$ , of $V$ on $\\partial X$ is positive.", "If necessary, restrict the function $V$ to $X$ , replacing its original domain $U$ with $X$ .", "Then, there is a positive number $c$ less than $d$ such that $V^{-1}([0, c]) \\subset X$ .", "Therefore, assumption A3 holds for this number $c$ .", "Since the closed set $V^{-1}([0,c])$ is contained in the bounded set $X$ , it is compact, which implies that assumption A2 holds.", "Theorem 2.5 For the functions $f$ and $V$ defined in (REF ) and (REF ), if $V$ satisfies (REF ) for all $x\\in U$ , the set $V^{-1}(0)$ is compact and the Jacobian matrix $Df(x)$ is onto for all $x \\in V^{-1}(0)$ , then there is a number $c>0$ such that every trajectory starting in $V^{-1}([0,c])$ remains in $V^{-1}([0,c])$ for all $t\\ge 0$ and asymptotically converges to $V^{-1}(0)$ as $t\\rightarrow \\infty $ .", "Theorem 2.6 For the functions $f$ and $V$ defined in (REF ) and (REF ), if $V$ satisfies (REF ) for all $x\\in U$ , the set $V^{-1}(0)$ is compact and there is an open subset $X$ of $U$ containing $V^{-1}(0)$ such that the Jacobian matrix $Df(x)$ is onto for all $x \\in X\\backslash V^{-1}(0)$ , then there is a number $c>0$ such that every trajectory starting in $V^{-1}([0,c])$ remains in $V^{-1}([0,c])$ for all $t\\ge 0$ and asymptotically converges to $V^{-1}(0)$ as $t\\rightarrow \\infty $ .", "Modify the proof of Lemma REF appropriately.", "As discussed above, we can integrate (REF ) instead of (REF ) for the initial condition $x(0) = x_0 \\in V^{-1}(0)$ , which will yield a trajectory that is expected to numerically well remain on the manifold $M$ and preserve the values of the first integrals $f_i$ , $i = 1, \\ldots , \\ell $ .", "It is noteworthy that the integration is carried out in one Cartesian coordinate system on $\\mathbb {R}^n$ rather than over local charts on the manifold $M$ which would take additional computational costs for coordinate changes between local charts.", "In the following section, we will apply this strategy to the free rigid body dynamics, the Kepler problem and a perturbed Kepler problem with rotational symmetry to integrate the dynamics preserving their first integrals and domain manifolds." ], [ "The Free Rigid Body", "Consider the free rigid body dynamics: $\\dot{R} &= R\\,\\hat{\\Omega }, \\\\\\dot{\\Omega }&= {\\mathbb {I}}^{-1} \\left((\\mathbb {I} \\Omega ) \\times \\Omega \\right), $ where $(R, \\Omega ) \\in {\\operatorname{SO}(3)} \\times \\mathbb {R}^3$ ; $\\mathbb {I}$ is the moment of inertia matrix; and $\\hat{\\Omega }= \\begin{bmatrix}0 & -\\Omega _3 & \\Omega _2 \\\\\\Omega _3 &0 & -\\Omega _1 \\\\-\\Omega _2 & \\Omega _1 & 0\\end{bmatrix}$ for $\\Omega = \\begin{bmatrix} \\Omega _1 \\\\ \\Omega _2 \\\\ \\Omega _3 \\end{bmatrix}.$ Since ${\\operatorname{SO}(3)} \\subset \\mathbb {R}^{3\\times 3}$ , from here on we assume that the rigid body dynamics are defined on the Euclidean space $\\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3$ and that the matrix $R$ denotes a $3\\times 3$ matrix, not necessarily in ${\\operatorname{SO}(3)}$ .", "This is the extension of the dynamics step.", "Define two functions $E: \\mathbb {R}^3 \\rightarrow \\mathbb {R}$ and $\\pi : \\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}^3$ by $E(\\Omega ) &= \\frac{1}{2} \\Omega ^T\\mathbb {I} \\Omega , \\\\\\pi (R, \\Omega ) &= R\\,\\mathbb {I}\\, \\Omega , $ where $E$ represents the kinetic energy of the free rigid body and $\\pi $ the spatial angular momentum vector when $R \\in \\operatorname{SO}(3)$ .", "These quantities are first integrals of (REF ).", "Choose any $R_0 \\in \\operatorname{SO}(3), \\quad \\Omega _0 \\in \\mathbb {R}^3\\backslash \\lbrace (0,0,0)\\rbrace ,$ and let $E_0 = E(\\Omega _0) >0, \\quad \\mathbb {\\pi }_0 = \\pi (R_0,\\Omega _0) \\in \\mathbb {R}^3\\backslash \\lbrace (0,0,0)\\rbrace .$ Define an open set $U$ by $U = \\lbrace (R,\\Omega ) \\in \\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3 \\mid \\det (R) >0\\rbrace $ and a function $V : U \\subset \\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}$ by $V(R,\\Omega ) = \\frac{k_0}{4}\\Vert R^TR - I\\Vert ^2 + \\frac{k_1}{2} \\left| E(\\Omega ) - E_0 \\right| ^2+ \\frac{k_2}{2}|\\pi (R,\\Omega ) - \\pi _0|^2$ for $(R,\\Omega ) \\in U \\subset \\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3$ , where $k_i >0$ , $i = 0, 1, 2$ are constants, and $\\Vert \\cdot \\Vert $ is the 2-norm defined by $\\Vert A\\Vert = \\sqrt{\\operatorname{trace}(A^TA)}$ for a matrix $A$ .", "Observe that we are endowing the space $\\mathbb {R}^{3\\times 3}\\times \\mathbb {R}^{3}$ with the standard inner product, and that the trace norm is precisely the norm induced on $\\mathbb {R}^{3\\times 3}$ by this inner product.", "We compute all gradients that follow with respect to this inner product.", "Notice that $V^{-1}(0) = \\lbrace (R,\\Omega ) \\in \\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3 \\mid R \\in \\operatorname{SO}(3), \\, E(\\Omega ) = E_0, \\, \\pi (R,\\Omega ) = \\pi _0\\rbrace .$ Lemma 3.1 The gradient $(\\nabla _R V, \\nabla _\\Omega V) \\in \\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3$ of the function $V$ (REF ) is given by $\\nabla _R V&= k_0R(R^TR - I) + k_2 (\\pi (R,\\Omega ) - \\pi _0) \\Omega ^T \\mathbb {I}, \\\\\\nabla _\\Omega V &=k_1 (E(\\Omega )-E_0) \\mathbb {I} \\Omega + k_2\\mathbb {I}R^T (\\pi (R,\\Omega ) - \\pi _0).$ Straightforward.", "The following lemma shows that the function $V$ satisfies assumption A1 stated in §.", "Lemma 3.2 The function $V$ satisfies $\\langle (\\nabla _R V, \\nabla _\\Omega V), (R\\hat{\\Omega }, \\mathbb {I} ^{-1} ( (\\mathbb {I} \\Omega ) \\times \\Omega ) )\\rangle = 0.$ One can compute $\\langle \\nabla _R V, (R\\hat{\\Omega }) \\rangle &= \\operatorname{trace} (\\hat{\\Omega }^T R^T (k_0R(R^TR - I) + k_2 (\\pi (R,\\Omega ) - \\pi _0) \\Omega ^T \\mathbb {I}) ) \\\\&= -k_0\\operatorname{trace} (\\hat{\\Omega }R^T R(R^TR - I) ) - k_2\\operatorname{trace} (\\hat{\\Omega }R^T (\\pi (R,\\Omega ) - \\pi _0) \\Omega ^T \\mathbb {I} )\\\\&= -k_2 \\Omega ^T \\mathbb {I} \\hat{\\Omega }R^T (\\pi (R,\\Omega ) - \\pi _0),$ where, in the third equality we use the fact that for $A$ symmetric and $B$ antisymmetric, $\\operatorname{trace}(AB)=0$ .", "Next, we compute, $\\langle \\nabla _\\Omega V , \\mathbb {I} ^{-1} ( (\\mathbb {I} \\Omega ) \\times \\Omega ) \\rangle &= \\langle k_1 (E(\\Omega )-E_0) \\mathbb {I} \\Omega + k_2\\mathbb {I}R^T (\\pi (R,\\Omega ) - \\pi _0), \\mathbb {I} ^{-1} ( (\\mathbb {I} \\Omega ) \\times \\Omega ) \\rangle \\\\&=k_1 (E(\\Omega )-E_0) \\langle (\\mathbb {I} \\Omega ) \\times \\Omega , \\Omega \\rangle + k_2 \\langle R^T (\\pi (R,\\Omega ) - \\pi _0), (\\mathbb {I} \\Omega ) \\times \\Omega \\rangle \\\\&= k_2 \\langle \\mathbb {I} \\Omega , \\Omega \\times R^T (\\pi (R,\\Omega ) - \\pi _0) \\rangle \\\\&= k_2 \\Omega ^T \\mathbb {I} \\hat{\\Omega }R^T (\\pi (R,\\Omega ) - \\pi _0).$ Hence, $\\langle (\\nabla _R V, \\nabla _\\Omega V), (R\\hat{\\Omega }, \\mathbb {I} ^{-1} ( (\\mathbb {I} \\Omega ) \\times \\Omega ) )\\rangle = \\langle \\nabla _R V, R\\hat{\\Omega }\\rangle + \\langle \\nabla _\\Omega V, \\mathbb {I} ^{-1} ( (\\mathbb {I} \\Omega ) \\times \\Omega ) \\rangle =0.$ The following lemma shows that the function $V$ satisfies assumptions A2 and A3 stated in §.", "Lemma 3.3 There is a number $c$ satisfying $0< c < \\min \\lbrace k_0/4, k_1|E_0|/2, k_2|\\pi _0|^2/2\\rbrace $ such that $V^{-1}([0,c])$ is a compact subset of $U$ and the set of all critical points of $V$ in $V^{-1}([0,c])$ is equal to $V^{-1}(0)$ .", "It is obvious that there is a number $c$ satisfying (REF ) such that $V^{-1}([0,c])$ becomes a compact set in $U$ .", "For such a number $c$ , the matrix $R$ is invertible for every $(R,\\Omega ) \\in V^{-1}([0,c])$ .", "Since 0 is the minimum value of $V$ , every point in $V^{-1}(0)$ is a critical point of $V$ .", "Let $(R,\\Omega ) $ be a critical point of $V$ in $V^{-1}([0,c]) \\backslash V^{-1}(0)$ .", "By Lemma REF it satisfies $k_0R(R^TR - I) + k_2 (\\pi - \\pi _0) \\Omega ^T \\mathbb {I} &=0, \\\\k_1 (E-E_0) \\mathbb {I} \\Omega + k_2\\mathbb {I}R^T (\\pi - \\pi _0) &= 0, $ where $\\pi = \\pi (R,\\Omega ), \\quad E = E(R,\\Omega ).$ Post-multiplying (REF ) by $R^T$ and pre-multiplying () by $R$ yield $k_0R(R^TR - I) R^T+ k_2 (\\pi - \\pi _0) \\pi ^T &=0, \\\\k_1 (E-E_0) \\mathbb {\\pi }+ k_2R\\mathbb {I}R^T (\\pi - \\pi _0) &= 0, $ since $\\pi =R\\mathbb {I} \\Omega $ .", "Notice that $\\Omega = 0$ would imply $V(R,\\Omega ) \\ge \\frac{k_2}{2}|\\pi _0|^2 >c$ , contradicting $(R,\\Omega ) \\in V^{-1}([0,c])$ .", "Hence, $\\Omega \\ne 0$ .", "It follows from (REF ) that if any of the three equations $R^TR - I = 0, \\quad \\pi - \\pi _0 = 0, \\quad E - E_0 = 0$ holds, then the three of them all hold.", "Thus $R^TR \\ne I, \\quad \\pi \\ne \\pi _0, \\quad E \\ne E_0$ since $(R,\\Omega ) \\notin V^{-1}(0)$ .", "Since the matrix $(\\pi - \\pi _0) \\Omega ^T \\mathbb {I}$ in (REF ) has rank 1 and the matrix $(R^TR - I) $ is symmetric, there exist a unit vector $u \\in \\mathbb {R}^3$ and a number $\\kappa \\ne 0$ such that $R^TR -I = \\kappa u u^T.$ Substitution of (REF ) into (REF ) and (REF ) yields $k_0\\kappa Ru u^T+ k_2 (\\pi -\\pi _0) \\Omega ^T \\mathbb {I}= 0,\\\\k_0\\kappa Ruu^TR^T + k_2 (\\pi -\\pi _0) \\pi ^T = 0,$ which implies $Ru \\parallel \\pi \\parallel \\pi _0, \\quad u \\parallel \\mathbb {I} \\Omega ,$ where the symbol $\\parallel $ means `is parallel to.'", "Hence, we can express $R$ and $\\pi $ as $R &= w_1 u_1^T + w_2u_2^T + a e_{\\pi _0} u^T, \\\\\\pi &= b \\pi _0,$ for some numbers $a \\ne 0$ , $b \\ne 1$ and vectors $u_1, u_2, w_1, w_2 \\in \\mathbb {R}^3$ , where $e_{\\pi _0} = \\pi _0 / |\\pi _0|$ and the vectors $u_1$ and $u_2$ can be any vectors such that $\\lbrace u_1, u_2, u\\rbrace $ becomes an orthonormal basis for $\\mathbb {R}^3$ .", "Substitution of (REF ) into (REF ) implies that $\\lbrace w_1,w_2, e_{\\pi _0}\\rbrace $ is an orthonormal basis for $\\mathbb {R}^3$ .", "Substitution of (REF ) and () into () implies $\\mathbb {I} \\Omega \\parallel \\mathbb {I} u$ , which together with $u \\parallel \\mathbb {I} \\Omega $ in (REF ), implies $u \\parallel \\mathbb {I}u$ , i.e., $u$ is an eigenvector of $\\mathbb {I}$ .", "We can now choose or re-define the unit vectors $u_1$ and $u_2$ such that they become eigenvectors of the symmetric matrix $\\mathbb {I}$ , too.", "In the orthonormal basis $\\lbrace u_1, u_2, u\\rbrace $ , we can now write the moment of inertia matrix $\\mathbb {I}$ as $\\mathbb {I} = I_1 u_1 u_1 ^T + I_2 u_2 u_2^T + I_3 uu^T,$ where $I_1, I_2, I_3$ are the eigenvalues of $\\mathbb {I}$ , which are all positive, corresponding to the eigenvectors $u_1, u_2, u$ , respectively.", "It is then easy to see that equations (REF ) imply $k_0 a^2 (a^2 -1) + k_2 |\\pi _0|^2b(b-1) &=0, \\\\k_1 \\left( \\frac{|\\pi _0|^2 b^2}{2I_3a^2 } - E_0 \\right)b + k_2 I_3 a^2 (b-1) &=0,$ where we have used $E = (1/2)\\Omega ^T\\mathbb {I} \\Omega = (1/2) \\pi ^T (R\\mathbb {I} R^T)^{-1}\\pi = |\\pi _0|^2b^2/2I_3a^2$ .", "We consider the following two separate cases: $E_ 0= |\\pi _0|^2/2I_3$ and $E_ 0\\ne |\\pi _0|^2/2I_3$ .", "Suppose $E_ 0= |\\pi _0|^2/2I_3$ .", "If $b\\le 0$ , then $V(R,\\Omega ) \\ge \\frac{k_2}{2}| \\pi - \\pi _0|^2 = \\frac{k_2}{2}(|b|+1)^2|\\pi _0|^2 > c$ by (REF ), which contradicts $(R,\\Omega ) \\in V^{-1}([0,c])$ .", "Hence, $b>0$ .", "If $b>1$ , then equation (REF ) implies $a^2 <1$ , but equation () implies $b^2 < a^2$ , implying $b^2 <1$ .", "This cannot be compatible with $b>1$ .", "Hence, $b>1$ is ruled out.", "Similarly, $0 <b<1$ can be ruled out.", "Hence, $b=1$ , which implies $\\pi = \\pi _0$ contradicting (REF ).", "Thus, when $E_ 0= |\\pi _0|^2/2I_3$ , there are no critical points of $V$ in $V^{-1}([0,c])\\backslash V^{-1}(0)$ .", "Suppose $E_ 0 \\ne |\\pi _0|^2/2I_3$ .", "We analyze equations (REF ) using a continuity argument.", "At $a^2=1$ , (REF ) implies $b= 0$ or 1, neither of which satisfies () at $a^2=1$ .", "Thus, by continuity there exists a number $\\delta $ with $0 <\\delta <1$ such that for any $a$ with $|a^2 -1|<\\delta $ there is no number $b$ satisfying both (REF ) and ().", "Hence, $ |a^2-1|\\ge \\delta $ .", "We now shrink the number $c$ such that it not only satisfies (REF ) but also $c< k_0\\delta ^2 /4$ .", "For such a number $c$ , we have $V(R,\\Omega ) \\ge \\frac{k_0}{4}\\Vert R^TR-I\\Vert ^2 =\\frac{k_0}{4}\\Vert (a^2-1)uu^T\\Vert ^2 \\ge \\frac{k_0}{4}\\delta ^2> c,$ which contradicts $(R,\\Omega ) \\in V^{-1}([0,c])$ .", "Hence, when $E_ 0\\ne |\\pi _0|^2/2I_3$ , there are no critical points of $V$ in $V^{-1}([0,c])\\backslash V^{-1}(0)$ for some $c>0$ .", "Therefore, there exists a number $c>0$ such that $V^{-1}(0)$ is the set of all critical points of $V$ in $V^{-1}([0,c])$ .", "Consider the dynamics $\\dot{R} &= R\\hat{\\Omega }-k_0R(R^TR - I) - k_2 (\\pi (R,\\Omega ) - \\pi _0) \\Omega ^T \\mathbb {I}, \\\\\\dot{\\Omega }&= \\mathbb {I} ^{-1} ( (\\mathbb {I} \\Omega ) \\times \\Omega ) - k_1 (E(\\Omega )-E_0) \\mathbb {I} \\Omega - k_2\\mathbb {I}R^T (\\pi (R,\\Omega ) - \\pi _0), $ which correspond to (REF ).", "From Theorem REF and Lemmas REF and REF comes the following theorem: Theorem 3.4 There is a number $c>0$ such that every trajectory of (REF ) starting from a point in $V^{-1}([0,c])$ stays in $V^{-1}([0,c])$ for all $t\\ge 0$ and asymptotically converges to the set $V^{-1}(0) = \\lbrace (R,\\Omega ) \\in \\mathbb {R}^{3\\times 3} \\times \\mathbb {R}^3 \\mid R \\in \\operatorname{SO}(3), \\, E(\\Omega ) = E_0, \\, \\pi (R,\\Omega ) = \\pi _0\\rbrace $ as $t \\rightarrow \\infty $ , where the function $V$ is defined in (REF ).", "Furthermore, $V^{-1}(0)$ is an invariant set of both (REF ) and (REF )." ], [ "The Kepler Problem", "The two-body dynamics in the Kepler problem are given in the usual barycentric coordinates by $\\dot{x} &= v, \\\\\\dot{v} &= -\\mu \\frac{x}{|x|^3},$ where $x \\in \\mathbb {R}^3_0:= \\mathbb {R}^3\\backslash \\lbrace (0,0,0)\\rbrace $ is the position vector, $v \\in \\mathbb {R}^3$ is the velocity vector and $\\mu $ is the gravitational parameter.", "Define two functions $L: \\mathbb {R}^3 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}^3$ and $A : \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}^3$ by $L(x,v) &= x \\times v, \\\\A(x,v) &= v \\times (x \\times v) - \\mu \\frac{x}{|x|},$ where $L$ is called the angular momentum vector and $A$ is called the Laplace-Runge-Lenz vector.", "It is known that both $L$ and $A$ are first integrals of the two-body dynamics (REF ) and they are orthogonal to each other, i.e., $L(x,v) \\perp A(x,v)$ for all $(x,v) \\in \\mathbb {R}^3_0 \\times \\mathbb {R}^3$ .", "The energy function $E(x,v) = \\frac{1}{2}|v|^2 - \\frac{\\mu }{|x|}$ satisfies $|A(x,v)|^2 = \\mu ^2 + 2E(x,v)|L(x,v)|^2$ for all $(x,v) \\in \\mathbb {R}^3_0 \\times \\mathbb {R}^3$ , implying that the energy $E$ is also a first integral of the two-body dynamics (REF ).", "It is also known that a non-degenerate elliptic Keplerian orbit is uniquely determined by a pair $(L,A)$ that satisfies $L \\perp A$ , $|L| \\ne 0$ and $|A| < \\mu $ [1].", "Fix a non-degenerate elliptic Keplerian orbit, i.e., a pair of vectors $(L_0, A_0)$ that satisfies $L_0 \\perp A_0, \\quad |L_0| \\ne 0, \\quad |A_0| < \\mu .$ Define a function $V : \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}$ by $V(x,v) = \\frac{k_1}{2}|L(x,v) - L_0|^2 + \\frac{k_2}{2} |A(x,v) - A_0|^2$ for $(x,v) \\in \\mathbb {R}^3_0 \\times \\mathbb {R}^3$ , where $k_1>0$ and $k_2>0$ .", "Notice that $V^{-1}(0) = \\lbrace (x,v) \\in \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\mid L(x,v) = L_0, A(x,v) = A_0\\rbrace ,$ which is the non-degenerate Keplerian elliptic orbit whose angular momentum vector and Laplace-Runge-Lenz vector are $L_0$ and $A_0$ , respectively.", "Lemma 3.5 The gradient $(\\nabla _x V, \\nabla _v V) \\in \\mathbb {R}^3 \\times \\mathbb {R}^3$ of the function $V$ defined in (REF ) is given by $\\nabla _x V&= k_1 v \\times \\Delta L + k_2 \\left( v \\times ( \\Delta A \\times v ) - \\frac{\\mu }{|x|} \\Delta A + \\frac{\\mu }{|x|^3} x x^T \\Delta A \\right), \\\\\\nabla _v V &=k_1 \\Delta L \\times x + k_2 ( (x \\times v) \\times \\Delta A + x \\times ( v \\times \\Delta A)),$ where $\\Delta L = L(x,v) - L_0$ and $\\Delta A = A(x,v) - A_0$ .", "The following lemma shows that the function $V$ defined in (REF ) satisfies assumptions A1 and A2 stated in §.", "Lemma 3.6 1.", "The function $V$ satisfies $\\langle (\\nabla _x V, \\nabla _vV), (v, -\\mu x /|x|^3) \\rangle =0.$ 2.", "For any number $c$ satisfying $0< c < \\min \\lbrace k_1 |L_0|^2/2, k_2 (\\mu - |A_0|)^2/2 \\rbrace ,$ the set $V^{-1} ([0,c])$ is a compact set in $\\mathbb {R}^3_0 \\times \\mathbb {R}^3$ .", "The first fact is a straightforward calculation using the previous Lemma.", "For the second, the essential idea is that the fibers of $V$ are homeomorphic to circles, corresponding to the elliptic orbits, and are therefore compact.", "For a detailed proof of the second statement, refer to Corollary 2.2 in [1].", "The following lemma shows that the function $V$ defined in (REF ) satisfies assumption A3 stated in §.", "Lemma 3.7 For any number $c$ satisfying (REF ) the set of all critical points of $V$ in $V^{-1}([0,c])$ is equal to $V^{-1}(0)$ .", "Choose an arbitrary number $c$ satisfying (REF ).", "Let $(x,v)$ be an arbitrary critical point of $V$ in $V^{-1}([0,c])$ .", "For notational convenience, let us write $L = L(x,v), \\quad A = A(x,v)$ suppressing the dependence on $(x,v)$ .", "By Lemma REF , the critical point $(x,v)$ satisfies $0&= k_1 v \\times \\Delta L + k_2 \\left( v \\times ( \\Delta A \\times v ) - \\frac{\\mu }{|x|} \\Delta A + \\frac{\\mu }{|x|^3} x x^T \\Delta A \\right), \\\\0&=k_1 \\Delta L \\times x + k_2 ( (x \\times v) \\times \\Delta A + x \\times ( v \\times \\Delta A)).", "$ If $|L| = 0$ , then $V(x,v) \\ge k_1 |L_0|^2/2 > c$ , contradicting $(x,v) \\in V^{-1}([0,c])$ .", "Hence, $|L| \\ne 0$ , which together with (REF ) implies that the three vectors $x,v, L$ form a basis for $\\mathbb {R}^3$ .", "The dot product of () with $x$ yields $0 = x \\cdot ( (x \\times v) \\times \\Delta A) = \\Delta A \\cdot (x \\times L),$ so there are numbers $a$ and $b$ such that $\\Delta A = a x + bL.$ Substitution of (REF ) into (REF ) gives $0 &= v \\times \\left( k_1 \\Delta L + k_2 \\left( aL - b v \\times L + \\frac{b\\mu }{|x|}x\\right)\\right),\\\\0 &= (k_1 \\Delta L + k_2 (2aL -b v \\times L) ) \\times x.$ It follows that there are numbers $d$ and $f$ such that $k_1 \\Delta L + k_2 \\left( aL - b v \\times L + \\frac{b\\mu }{|x|}x\\right) =& dv,\\\\k_1 \\Delta L + k_2 (2aL -b v \\times L) = &fx.", "$ From (REF ), we obtain $\\left( \\frac{bk_2 \\mu }{|x|} + f \\right)x - dv - ak_2 L = 0.$ By linear independence of $\\lbrace x,v, L\\rbrace $ , $a = 0, \\quad d = 0,\\quad f = -bk_2\\mu /|x|.$ Substitution of these into (REF ) and () gives $\\Delta A = bL, \\quad \\Delta L = \\frac{bk_2}{k_1}A,$ where we have used the definition of $A$ given in ().", "Hence, $A_0 = A - bL, \\quad L_0 = L - \\frac{bk_2}{k_1}A.$ From (REF ) and the orthogonality $A_0 \\perp L_0$ and $A \\perp L$ , it follows that $0 = A_0 \\cdot L_0 = -b \\left( |L|^2 + \\frac{k_2}{k_1}|A|^2\\right).$ Since $|L| \\ne 0$ , and recalling that $k_{1}>0$ and $k_{2}>0$ , we have $b=0$ .", "Substitution of $b=0$ into (REF ) yields $L = L_0, \\quad A = A_0,$ which implies $(x,v) \\in V^{-1}(0)$ .", "Thus, every critical point of $V$ in $V^{-1}([0,c])$ is contained in $V^{-1}(0)$ .", "Since 0 is the minimum value of $V$ , every point in $V^{-1}(0)$ is a critical point of $V$ .", "Therefore, the set of all critical points of $V$ in $V^{-1}([0,c])$ is $V^{-1}(0)$ .", "Choose a non-degenerate Keplerian elliptic orbit and let $(x_0, v_0)$ be a point on the orbit.", "Set $L_0 = L(x_0,v_0), \\quad A_0 = A(x_0, v_0)$ to be the angular momentum vector and the Laplace-Runge-Lenz vector of the orbit, respectively.", "Consider the dynamics: $\\dot{x} &= v -k_1 v \\times \\Delta L - k_2 \\left( v \\times ( \\Delta A \\times v ) - \\frac{\\mu }{|x|} \\Delta A + \\frac{\\mu }{|x|^3} x x^T \\Delta A \\right), \\\\\\dot{v} &= -\\mu \\frac{x}{|x|^3} - k_1 \\Delta L \\times x - k_2 ( (x \\times v) \\times \\Delta A + x \\times ( v \\times \\Delta A)),$ where $\\Delta L = L(x,v) - L_0$ and $\\Delta A = A(x,v) - A_0$ , which correspond to (REF ).", "From Theorem REF and Lemmas REF and REF comes the following theorem: Theorem 3.8 For any $c>0$ satisfying (REF ), every trajectory of (REF ) starting from a point in $V^{-1}([0,c])$ stays in $V^{-1}([0,c])$ for all $t\\ge 0$ and asymptotically converges to the set $V^{-1}(0) = \\lbrace (x,v) \\in \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\mid L(x,v) = L_0, A(x,v) = A_0\\rbrace $ as $t \\rightarrow \\infty $ , where the function $V$ is defined in (REF ).", "Furthermore, $V^{-1}(0)$ is an invariant set of both (REF ) and (REF )." ], [ "A Perturbed Kepler Problem with Rotational Symmetry", "Consider a perturbed Kepler problem with rotational symmetry whose equations of motion are given by $\\dot{x} &= v, \\\\\\dot{v} &= -U^\\prime (|x|) \\frac{x}{|x|},$ where $x \\in \\mathbb {R}^3_0:= \\mathbb {R}^3\\backslash \\lbrace (0,0,0)\\rbrace $ is the position vector, $v \\in \\mathbb {R}^3$ is the velocity vector, and $U : (0, \\infty ) \\rightarrow \\mathbb {R}$ is the potential function that depends only on the radial distance from the origin.", "The total energy $E: \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}$ and the angular momentum vector $L : \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}^3$ are defined by $E(x,v) &= \\frac{1}{2}|v|^2 + U(|x|), \\\\L(x,v) &= x \\times v $ and they are conserved quantities of the dynamics (REF ).", "Take any point $(x_0, v_0) \\in \\mathbb {R}^3_0 \\times \\mathbb {R}^3$ such that $x_0 \\times v_0 \\ne 0.$ Let $E_0 = E(x_0, v_0), \\quad L_0 = L(x_0, v_0) \\ne 0.$ Define a function $V: \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}$ by $V(x,v) = \\frac{k_1}{2} |E(x,v)-E_0|^2 + \\frac{k_2}{2} |L(x,v)-L_0|^2$ with $k_1>0$ and $k_2>0$ .", "Then, $V^{-1}(0) = \\lbrace (x,v) \\in \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\mid E(x,v) = E_0, L(x,v) = L_0\\rbrace .$ The gradient $(\\nabla _x V, \\nabla _v V)$ of $V$ is given by $\\nabla _x V &= k_1 \\Delta E U^\\prime (|x|) \\frac{x}{|x|} + k_2 v\\times \\Delta L,\\\\\\nabla _v V &= k_1 \\Delta E v + k_2 \\Delta L \\times x,$ where $\\Delta E = E(x,v) - E_0$ and $\\Delta L = L(x,v) - L_0$ .", "Trivially, $V$ satisfies (REF ), i.e.", "$\\langle (\\nabla _xV, \\nabla _vV), (v, -U^\\prime (|x|)x/|x|\\rangle = 0$ for all $(x,v) \\in \\mathbb {R}_0^3 \\times \\mathbb {R}^3$ .", "The modified dynamics, which correspond to (REF ), are computed as $\\dot{x} &= v - k_1 \\Delta E U^\\prime (|x|) \\frac{x}{|x|} - k_2 v\\times \\Delta L, \\\\\\dot{v} &= -U^\\prime (|x|) \\frac{x}{|x|} -k_1 \\Delta E v - k_2 \\Delta L \\times x.$ Theorem 3.9 Suppose that $V^{-1}(0)$ is compact and there is no common solution $r>0$ to the following two equations: $E_0 &= \\frac{1}{2}rU^\\prime (r) + U(r), \\\\|L_0|^2 &= r^3 U^\\prime (r).$ Then, assumptions A2 and A3 hold and there is a number $c>0$ such that every trajectory of (REF ) starting in $V^{-1}([0,c])$ remains in $V^{-1}([0,c])$ for all $t\\ge 0$ and asymptotically converges to $V^{-1}(0)$ as $t\\rightarrow \\infty $ .", "Define a function $f: \\mathbb {R}^3_0 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R} \\times \\mathbb {R}^3$ by $f(x,v) = \\begin{bmatrix}E(x,v) \\\\ L(x,v)\\end{bmatrix}.$ Then, $Df(x,v)^T = \\begin{bmatrix}U^\\prime (|x|) \\frac{x}{|x|} & \\hat{v} \\\\v & -\\hat{x}\\end{bmatrix},$ where the over-hat symbol $\\wedge $ denotes the hat map defined in (REF ).", "We want to show that the $6\\times 4$ matrix $Df(x,v)^T$ is one-to-one for all $(x,v) \\in V^{-1}(0)$ .", "Fix an arbitrary point $(x,v) \\in V^{-1}(0)$ .", "It follows $E_0 &= \\frac{1}{2}|v|^2 + U(|x|),\\\\L_0 &= x \\times v \\ne 0.", "$ Take any point $(a,w) \\in \\mathbb {R} \\times \\mathbb {R}^3$ from the kernel of $Df(x,v)^T$ .", "Then, $0& = aU^\\prime (|x|)\\frac{x}{|x|} + v \\times w, \\\\0&=a v - x \\times w. $ Suppose $a\\ne 0$ .", "Taking the inner product of (REF ) with $x$ and of () with $v$ , we obtain $0&= a U^\\prime (|x|) |x| + L_0 \\cdot w,\\\\0&= a |v|^2 + L_0 \\cdot w,$ from which it follows that $|x| U^\\prime (|x|) = |v|^2.$ Taking the inner product of () with $x$ , we get $x\\cdot v=0$ which implies $|L_0| = |x|\\cdot |v|.$ From (REF ), (REF ) and (REF ), we obtain $E_0 &= \\frac{1}{2}|x| U^\\prime (|x|) + U(|x|), \\\\|L_0|^2 &= |x|^3 U^\\prime (|x|).$ By hypothesis, there cannot be any $x \\in \\mathbb {R}_0^3$ that satisfies both (REF ) and ().", "Hence, we cannot have $a\\ne 0$ .", "Substitute $a=0$ into (REF ).", "It follows that $w$ is parallel to $x\\times v$ .", "Hence, there is a number $b$ such that $w = b L_0$ .", "Substituting this in () yields $ b x \\times L_0 = 0$ .", "Taking the cross product of this with $x$ yields $b|x|^2L_0 = 0$ since $x\\cdot L_0=0$ .", "Since $x\\ne 0$ and $L_0 \\ne 0$ , we have $b=0$ , so $w = 0$ .", "It follows that $(a,w) = (0,0)$ , which implies that $Df(x,v)^T$ is one-to-one for all $(x,v) \\in V^{-1}(0)$ .", "In other words, $Df(x,v)$ is onto for all $(x,v) \\in V^{-1}(0)$ .", "Hence, the conclusion of the theorem follows from Lemma REF , equation (REF ), and Theorem REF .", "Remark 3.10 Consider a special case in which the potential function $U(r)$ is of the form $U(r) = -\\frac{\\mu }{r} - \\frac{\\delta }{r^3},$ where $\\mu >0$ and $\\delta >0$ .", "Then equations (REF ) and () become $E_0 &= -\\frac{\\mu }{2r} + \\frac{\\delta }{2r^3}, \\\\|L_0|^2 &= \\mu r + \\frac{3\\delta }{r}.", "$ Given $E_0$ and $L_0$ , it is then easy to check if there is no common solution $r>0$ to (REF ) and ()." ], [ "The Free Rigid Body", "Consider the free rigid body dynamics in §REF with the moment of inertia matrix $\\mathbb {I} = \\operatorname{diag} (3,2,1)$ and the initial condition $R(0) = I, \\quad \\Omega (0) = (1,1,1).$ The values of the energy $E$ and the spatial angular momentum vector $\\pi = (\\pi _1, \\pi _2, \\pi _3)$ corresponding to the initial condition are $E (0)= 3, \\quad \\pi (0) = (3,2,1).$ The period $T_\\Omega $ of the trajectory of the body angular velocity vector $\\Omega (t) $ is computed approximately to be $T_\\Omega = 6.4227$ .", "We integrate the dynamics over the time interval $[0, 10^3] = [0, 155.7T_\\Omega ]$ with step size $\\Delta t = 10^{-4}$ , using the following four integration methods: a feedback integrator with the Euler scheme, a projection method with the Euler scheme, a splitting method with three rotations splitting, and the ordinary Euler method.", "The feedback integrator with the Euler scheme denotes the Euler method applied to the modified free rigid dynamics (REF ) with the following values of the parameters $k_0$ , $k_1$ , and $k_2$ $k_0 = 50, \\quad k_1 = 100, \\quad k_2 = 50.$ The projection method is the standard one explained on pp.110–111 in [2].", "In order to solve constraint equations for projection at each step of integration in the projection method, we use the Matlab command fsolve with the parameter TolFun, which is termination tolerance on the function value, set equal to $10^{-4}$ , which is the same as the integration step size $\\Delta t$ .", "The splitting method is the one explained on pp.284–285 in [2].", "The three of the projection method, the splitting method and the ordinary Euler method are applied to the original free rigid body dynamics (REF ).", "The trajectories of the body angular velocity vector $\\Omega (t)$ , the energy error $|\\Delta E (t)| = |E(t) - E(0)|$ , the error $|\\Delta \\pi (t) | = |\\pi (t) - \\pi (0)|$ in spatial angular momentum, and the deviation $\\Vert R(t)^T R(t) - I\\Vert $ of the rotation matrix $R(t)$ from ${\\operatorname{SO}(3)}$ are plotted in Figures REF , REF , REF and REF , respectively.", "In Figure REF , it is observed that the trajectories of $\\Omega (t)$ generated by the feedback integrator and the projection method maintain a periodic shape well whereas those by the splitting method and the Euler method drift away significantly from the periodic shape.", "In Figure REF , it is observed that the feedback integrator and the projection method keep the energy error sufficiently small whereas the energy errors by the other two methods increase in time.", "Although the two trajectories of energy error by the splitting method and the Euler method seem to coincide in Figure REF , an examination of the numerical data shows that the energy of the Euler method gets larger than that of the splitting method in time.", "For example, at $t=1000$ , the energy of the Euler method is bigger than that of the splitting method by $1.767\\times 10^{-3}$ .", "In Figures REF and REF , it is observed that the feedback method preserves the spatial angular momentum vector and the manifold ${\\operatorname{SO}(3)}$ sufficiently well.", "In terms of computation time, the projection method takes much more time than the others, which is due to the steps of solving the constraint equations for projection.", "The splitting method is symplectic and of order 2 whereas the other methods are of order 1.", "All of these observations lead us to the conclusion that the feedback integrator overall has produced the best outcome in the simulation of the free rigid body dynamics." ], [ "The Kepler Problem", "Consider the Kepler problem in §REF with $\\mu = 1$ and the initial condition $x(0) = (1,0,0), \\quad v(0) = (0,\\sqrt{1.8},0).$ The corresponding initial values of the angular momentum vector and the Laplace-Runge-Lenz vector are $L(0) = (0,0,\\sqrt{1.8}), \\quad A(0) = (0.8, 0,0).$ The period $T$ and the eccentricity $e$ of the Kepler orbit containing the initial point are $T = 70.2481,\\quad e = 0.8.$ We integrate the Kepler dynamics over the time interval $[0, 1000T]$ with step size $\\Delta t = 0.005$ , using the following four integration methods: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, and two Störmer-Verlet schemes.", "The feedback integrator with the Euler scheme denotes the Euler method applied to (REF ) with $k_1 = 4$ and $k_2 = 2$ .", "The standard projection method is explained on pp.110–111 in [2].", "To solve the constraint equations for projection, we use the Matlab command fsolve with the parameter TolFun set equal to $0.005$ , which is the same as the integration step size $\\Delta t$ .", "The two Störmer-Verlet schemes are those in (3.4) and (3.5) on pp.", "189–190 in [2], and we call them Störmer-Verlet-A and Störmer-Verlet-B, respectively, for convenience.", "The Störmer-Verlet schemes are symplectic methods of order 2.", "The trajectories of the planar orbit $x(t) = (x_1(t), x_2(t),0)$ , the error of the Laplace-Runge-Lenz vector, $|\\Delta A (t) | = |A(t) - A(0)|$ , and the error of the angular momentum vector, $|\\Delta L(t)| = |L(t) - L(0)|$ , are plotted in Figures REF , REF and REF .", "In Figure REF it is observed that the planar trajectories $x(t) = (x_1(t), x_2(t),0)$ generated by the feedback integrator and the projection method maintain the elliptic shape well whereas those by the Störmer-Verlet schemes precess.", "This can be also verified in Figure REF , where the feedback integrator and the projection method preserve the Laplace-Runge-Lenz vector well, but the Störmer-Verlet schemes cause the Laplace-Runge-Lenz vector to noticeably precess.", "In Figure REF , it is observed that the Störmer-Verlet schemes preserve the angular momentum vector exceptionally well in comparison with the other two methods.", "In Figures REF and REF , we can see that the precision of the feedback integrator is comparable with that of the projection method.", "However, the feedback integrator takes much less computation time than the projection method.", "The feedback integrator and the projection method used here are of order 1, whereas the Störmer-Verlet schemes are of order 2.", "All of these observations lead us to conclude that the feedback integrator has produced the best result overall." ], [ "A Perturbed Kepler Problem with Rotational Symmetry", "Consider the perturbed Kepler problem in §REF with the potential function $U$ given in (REF ) with $\\mu = 1$ and $\\delta = 0.0025$ , which is the one used in Example 4.3 on p. 111 in [2].", "We use the initial conditions $x(0) = (1-e,0,0), \\quad v(0) = (0, \\sqrt{(1+e)/(1-e)},0)$ with eccentricity $e= 0.6$ as in [2].", "The corresponding values of the energy and the angular momentum vector are $E(0) = -0.5390625, \\quad L (0) = (0,0,0.8).$ We integrate the perturbed Kepler dynamics over the time interval $[0, 200]$ with step size $\\Delta t = 0.03$ , just as on p. 111 in [2], using the following four integration methods: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, the Störmer-Verlet scheme in (3.4) on p. 189 in [2], and the Matlab command, ode45.", "The feedback integrator with the Euler scheme denotes the Euler method applied to (REF ) with $k_1 = 2$ and $k_2 = 3$ , and it is straightforward to verify that the hypotheses in Theorem REF hold true.", "The other three methods are applied to (REF ).", "The Matlab command fsolve is used in the projection method with the parameter TolFun set equal to $10^{-8}$ .", "The options of RelTol = AbsTol = $10^{-10}$ are used for the Matlab integrator, ode45, so the result generated by ode45 can be used as a reference.", "The trajectories of the planar orbit $x(t) = (x_1(t), x_2(t),0)$ , the energy error $|\\Delta E (t) | = |E(t) - E(0)|$ and the error $| \\Delta L(t)| = |L(t) - L(0)|$ in angular momentum are plotted in Figures REF , REF and REF .", "In Figure REF it is observed that the orbits generated by the feedback integrator and the Störmer-Verlet scheme are similar to that by ode45, but the orbit by the projection method precesses too much which is a very poor result.", "The projection method excels only at preserving the energy and the angular momentum as expected in view of the nature of the projection method and the small tolerance parameter value, TolFun = $10^{-8}$ , used for the Matlab command, fsolve.", "In Figure REF , it is observed that the feedback integrator is comparable with the Störmer-Verlet scheme in energy conservation.", "The feedback integrator also preserves the angular momentum well in view of the step size $\\Delta t = 0.03$ , as can be seen in Figure REF .", "The feedback integrator and the projection method used here are of order 1 whereas the Störmer-Verlet scheme is of order 2.", "From all of these observations, we conclude that the feedback integrator has produced the best result overall." ], [ "Conclusions and Future Work", "We have developed a theory to produce numerical trajectories of a dynamical system on a manifold that stably remain on the manifold and preserve first integrals of the system.", "Our theory is not a numerical integration scheme but rather a modification of the original dynamics by feedback.", "The actual numerical integration in our framework can be done with any usual integrator such as Euler and Runge-Kutta.", "Our method is successfully applied to the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry, and its excellent performance is demonstrated by simulation studies in comparison with the standard projection method, two Störmer-Verlet schemes and a splitting method via three rotations splitting.", "As future work, we plan to apply our theory to various mechanical systems with symmetry and non-holonomic systems.", "We also plan to carry out a quantitative study of the effect of the parameters in the Lyapunov function on the performance of our method." ], [ "Appendix", "We show, using results in [3], that any discrete-time dynamical system derived from a one-step numerical integration scheme with uniform step size $h$ for the modified dynamical system (REF ) has an attractor $\\Lambda _h$ that contains $V^{-1}(0)$ in its interior and converges to $V^{-1}(0)$ as $h\\rightarrow 0+$ .", "Let us first review some definitions from [3].", "Let $A$ and $B$ be nonempty, compact subsets of $\\mathbb {R}^n$ and $x$ a point in $\\mathbb {R}^n$ .", "The distance between $x$ and $A$ is defined by $\\operatorname{dist} (x,A) = \\inf \\lbrace |x- a|, a \\in A\\rbrace .$ The Hausdorff separation of $A$ from $B$ is defined by $H^*(A,B) = \\max \\lbrace \\operatorname{dist}(a,B), a \\in A\\rbrace .$ The Hausdorff distance between $A$ and $B$ is defined by $H(A,B) = \\max \\lbrace H^*(A,B), H^*(B,A)\\rbrace .$ The Hausdorff distance is a metric on the space of nonempty compact subsets of $\\mathbb {R}^n$ .", "For $r>0$ , let $S(A,r) = \\lbrace x \\in \\mathbb {R}^n \\mid \\operatorname{dist}(x,A)<r\\rbrace $ denote an $r$ -neighborhood of $A$ .", "We say that a nonempty, compact subset $\\Lambda $ of $\\mathbb {R}^n$ is uniformly stable for an autonomous dynamical system if for each $\\epsilon >0$ there exists a $\\delta = \\delta (\\epsilon ) >0$ such that $[x_0 \\in S(\\Lambda , \\delta ) \\textup { and } t \\ge 0 ] \\Rightarrow x(t;x_0) \\in S(\\Lambda , \\epsilon ),$ where $x(t;x_0)$ is the solution of the given dynamical system with initial condition $x(0) = x_0$ .", "A set $\\Lambda $ is said to be positively invariant for an autonomous dynamical system if $x(t;x_0) \\in \\Lambda $ for all $x_0 \\in \\Lambda $ and $t\\ge 0$ .", "A nonempty, compact subset $\\Lambda $ of $\\mathbb {R}^n$ is called uniformly asymptotically stable for an autonomous dynamical system if it is positively invariant and uniformly stable for the dynamical system, and additionally satisfies the following property: there is a $\\delta _0>0$ and for each $\\epsilon >0$ a time $T(\\epsilon )>0$ such that $[x_0 \\in S(\\Lambda , \\delta _0) \\textup { and } t \\ge T(\\epsilon ) ] \\Rightarrow x(t;x_0) \\in S(\\Lambda , \\epsilon ).$ Lemma 5.1 Suppose that assumptions A1, A2 and A3 (or A3$^{\\prime }$ instead of A3) stated in § hold true.", "Then, the set $V^{-1}(0)$ is uniformly asymptotically stable for the modified dynamical system (REF ).", "Since the three assumptions are satisfied, the conclusions of Theorem REF (or, REF ) hold true.", "For convenience, let $\\Lambda = V^{-1}(0)$ , which is invariant under (REF ) by Theorem REF (or, REF ).", "Let $c>0$ be the number $c$ in assumption A2.", "Using compactness of $V^{-1}([0,c])$ and continuity of $V$ , it is easy to show that for any $\\epsilon >0$ there is a $b = b(\\epsilon )>0$ such that $V^{-1}([0,b]) \\subset S(\\Lambda , \\epsilon )$ .", "It is also easy to show that for any $b>0$ there is an $\\epsilon = \\epsilon (b)>0$ such that $S(\\Lambda , \\epsilon ) \\subset V^{-1}([0,b])$ .", "Hence, we can use the family of sets $\\lbrace V^{-1}([0,b]), b>0\\rbrace $ instead of the family of open sets $\\lbrace S(\\Lambda , \\epsilon ), \\epsilon >0\\rbrace $ to show uniform stability and uniform asymptotic stability of $\\Lambda $ for (REF ).", "Let us first show uniform stability of $\\Lambda $ for (REF ).", "Given any $\\epsilon >0$ , take any $\\delta $ such that $0< \\delta \\le \\min \\lbrace \\epsilon , c\\rbrace $ .", "Then, for any $x_0 \\in V^{-1}([0,\\delta ])$ , $x(t; x_0) \\in V^{-1}([0,\\delta ]) \\subset V^{-1}([0,\\epsilon ])$ for all $t\\ge 0$ since $V$ is decreasing along the trajectory of $x(t;x_0)$ of (REF ).", "Hence, $\\Lambda $ is uniformly stable for (REF ).", "Let us now show uniform asymptotic stability of $\\Lambda $ for (REF ).", "Take any $\\delta _0$ such that $0 < \\delta _0 \\le c$ .", "By continuous dependence of $x(t;x_0)$ on initial point $x_0$ , compactness of $V^{-1}([0,\\delta _0])$ , continuity of the function $V$ , and the property that $V(x(t;x_0))$ decreases to 0 as $t\\rightarrow \\infty $ for any $x_0 \\in V^{-1}([0,c])$ , it is easy to show that for any $\\epsilon >0$ there is a time $T(\\epsilon )>0$ such that for any $x_0 \\in V^{-1}([0,\\delta _0])$ we have $x(t;x_0) \\in V^{-1}([0,\\epsilon ])$ for all $t\\ge T(\\epsilon )$ .", "Hence, $\\Lambda $ is uniformly asymptotically stable for (REF ).", "Suppose the vector field $X$ is $C^p$ and the function $V$ is $C^{p+1}$ in the modified dynamical system (REF ).", "Consider a discrete analogue of (REF ) described by any one-step numerical method of $p$ th order $x_{k+1} = x_k + hY_h(x_k)$ with uniform step size $h>0$ , where $Y_h: \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ for each $h$ .", "Theorem 5.2 Suppose that the vector field $X$ is $C^p$ and the function $V$ is $C^{p+1}$ , and that assumptions A1, A2 and A3 (or A3$^{\\prime }$ instead of A3) are satisfied.", "Then there is a number $h_2>0$ such that for each $0<h<h_2$ the discrete-time dynamical system (REF ) has a compact, uniformly asymptotically stable set $\\Lambda _h$ which contains $V^{-1}(0)$ in its interior and converges to $V^{-1}(0)$ with respect to the Hausdorff metric as $h\\rightarrow 0+$ .", "Moreover, there is a bounded, open set $U_0$ , which is independent of $h$ and contains $\\Lambda _h$ , and a time $T_0(h) = A + B p \\log \\frac{1}{h},$ where $A$ and $B$ are constants depending on the stability characteristic of $V^{-1}(0)$ , such that the iterates of (REF ) satisfy $x_k \\in \\Lambda _h$ for all $kh \\ge T_0(h)$ , $x_0 \\in U_0$ and $0 < h<h_2$ .", "We have only to show that the hypotheses in Theorem 1.1 of [3] hold.", "Since $X$ is $C^p$ and $V$ is $C^{p+1}$ , the vector field $X - \\nabla V$ of (REF ) and its derivatives of order up to $p$ are all continuous and bounded on the compact set $V^{-1}([0,c])$ .", "The set $V^{-1}(0)$ is uniformly asymptotically stable for (REF ) by Lemma REF in the above.", "Therefore, the conclusions of this theorem follow from Theorem 1.1 and Lemma 3.3 of [3].", "Refer to [3] to see how to obtain the set $U_0$ and values of the parameters $h_2$ , $A$ and $B$ that appear in the statement of the above theorem.", "The above theorem extends to multi-step numerical integrators; refer to [4] for detail.", "This research was supported in part by DGIST Research and Development Program (CPS Global Center) funded by the Ministry of Science, ICT & Future Planning, Global Research Laboratory Program (2013K1A1A2A02078326) through NRF, and Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korean government (MSIP) (No.", "B0101-15-0557, Resilient Cyber-Physical Systems Research).", "Figure: The trajectories of the body angular velocity Ω(t)=(Ω 1 (t),Ω 2 (t),Ω 3 (t))\\Omega (t) = (\\Omega _1(t), \\Omega _2(t), \\Omega _3(t)), 0≤t≤10000 \\le t \\le 1000, of the free rigid body dynamics generated by four different methods with step size Δt=10 -4 \\Delta t = 10^{-4}: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, a three rotations splitting method and the usual Euler method.Figure: The trajectories of the energy error |ΔE(t)|=|E(t)-E(0)||\\Delta E(t) | = |E(t) - E(0)|, 0≤t≤10000 \\le t \\le 1000, of the free rigid body dynamics generated by four different methods with step size Δt=10 -4 \\Delta t = 10^{-4}: a feedback integrator with the Euler scheme (∘\\circ ), the standard projection method with the Euler scheme (⋄\\diamond ), a three rotations splitting method (□\\square ) and the usual Euler method (×\\times ).Figure: The trajectories of the spatial angular momentum error |Δπ(t)|=|π(t)-π(0)||\\Delta \\pi (t) | = |\\pi (t) - \\pi (0)|, 0≤t≤10000 \\le t \\le 1000, of the free rigid body dynamics generated by four different methods with step size Δt=10 -4 \\Delta t = 10^{-4}: a feedback integrator with the Euler scheme (∘\\circ ), the standard projection method with the Euler scheme (⋄\\diamond ), a three rotations splitting method (□\\square ) and the usual Euler method (×\\times ).Figure: The trajectories of the deviation ∥R(t) T R(t)-I∥\\Vert R (t)^TR(t) - I \\Vert of the rotation matrix R(t)R(t) from SO(3){\\operatorname{SO}(3)}, 0≤t≤10000 \\le t \\le 1000, of the free rigid body dynamics generated by four different methods with step size Δt=10 -4 \\Delta t = 10^{-4}: a feedback integrator with the Euler scheme (∘\\circ ), the standard projection method with the Euler scheme (⋄\\diamond ), a three rotations splitting method (□\\square ) and the usual Euler method (×\\times ).Figure: The trajectories of the planar orbit x(t)=(x 1 (t),x 2 (t),0)x(t) = (x_1(t), x_2(t),0), 0≤t≤70,2480 \\le t \\le 70,248, in the Kepler problem generated by four different methods with step size Δt=0.005\\Delta t = 0.005: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, and two Störmer-Verlet schemes.Figure: The trajectories of the error |ΔA(t)|=|A(t)-A(0)||\\Delta A (t)| = |A(t) - A(0)|, 0≤t≤70,2480 \\le t \\le 70,248, of the Laplace-Runge-Lenz vector in the Kepler problem generated by four different methods with step size Δt=0.005\\Delta t = 0.005: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, and two Störmer-Verlet schemes.Figure: The trajectories of the angular momentum error |ΔL(t)|=|L(t)-L(0)||\\Delta L (t)| = |L(t) - L(0)|, 0≤t≤70,2480 \\le t \\le 70,248, in the Kepler problem generated by four different methods with step size Δt=0.005\\Delta t = 0.005: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, and two Störmer-Verlet schemes.Figure: The trajectories of the planar orbit x(t)=(x 1 (t),x 2 (t),0)x(t) = (x_1(t), x_2(t),0), 0≤t≤2000 \\le t \\le 200, in the perturbed Kepler problem generated by four different methods: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, a Störmer-Verlet scheme and the Matlab command ode45, where the step size Δt=0.03\\Delta t = 0.03 is used for the first three methods.Figure: The trajectories of the energy error |E(t)|=|E(t)-E(0)||E (t)| = |E(t) - E(0)|, 0≤t≤2000 \\le t \\le 200, in the perturbed Kepler problem generated by four different methods: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, a Störmer-Verlet scheme and the Matlab command ode45, where the step size Δt=0.03\\Delta t = 0.03 is used for the first three methods.Figure: The trajectories of the angular momentum error |ΔL(t)|=|L(t)-L(0)||\\Delta L (t)| = |L(t) - L(0)|, 0≤t≤2000 \\le t \\le 200, in the perturbed Kepler problem generated by four different methods: a feedback integrator with the Euler scheme, the standard projection method with the Euler scheme, and a Störmer-Verlet scheme and the Matlab command ode45, where the step size Δt=0.03\\Delta t = 0.03 is used for the first three methods." ] ]
1606.05005
[ [ "Orbifold points on Teichm\\\"uller curves and Jacobians with complex\n multiplication" ], [ "Abstract For each integer $D \\geq 5$ with $D \\equiv 0$ or $1 \\bmod 4$, the Weierstrass curve $W_D$ is an algebraic curve and a finite volume hyperbolic orbifold which admits an algebraic and isometric immersion into the moduli space of genus two Riemann surfaces.", "The Weierstrass curves are the main examples of Teichm\\\"uller curves in genus two.", "The primary goal of this paper is to determine the number and type of orbifold points on each component of $W_D$.", "Our enumeration of the orbifold points, together with work of Bainbridge and McMullen, completes the determination of the homeomorphism type of $W_D$ and gives a formula for the genus of its components.", "We use our formula to give bounds on the genus of $W_D$ and determine the Weierstrass curves of genus zero.", "We will also give several explicit descriptions of each surface labeled by an orbifold point on $W_D$." ], [ "Introduction", "Let $\\mathcal {M}_g$ be the moduli space of genus $g$ Riemann surfaces.", "The space $\\mathcal {M}_g$ can be viewed as both a complex orbifold and an algebraic variety and carries a complete Finsler Teichmüller metric.", "A Teichmüller curve is an algebraic and isometric immersion of a finite volume hyperbolic Riemann surface into moduli space: $ f\\colon \\,C=\\mathbb {H}/\\Gamma \\rightarrow \\mathcal {M}_g.", "$ The modular curve $\\mathbb {H}/\\operatorname{SL}_2(\\mathbb {Z}) \\rightarrow \\mathcal {M}_1$ is the first example of a Teichmüller curve.", "Other examples emerge from the study of polygonal billiards [29], [16] and square-tiled surfaces.", "While the Teichmüller curves in $\\mathcal {M}_2$ have been classified [19], much less is known about Teichmüller curves in $\\mathcal {M}_g$ for $g > 2$ [3], [7].", "The main source of Teichmüller curves in $\\mathcal {M}_2$ are the Weierstrass curves.", "For each integer $D \\ge 5$ with $D \\equiv 0$ or $1 \\bmod 4$ , the Weierstrass curve $W_D$ is the moduli space of Riemann surfaces whose Jacobians have real multiplication by the quadratic order $\\mathcal {O}_D=\\mathbb {Z}\\left[ \\frac{D+\\sqrt{D}}{2} \\right]$ stabilizing a holomorphic one form with double zero up to scale.", "The curve $W_D$ is a finite volume hyperbolic orbifold and the natural immersion $ W_D \\rightarrow \\mathcal {M}_2 $ is algebraic and isometric and has degree one onto its image [9], [17].", "The curve $W_D$ is a Teichmüller curve unless $D > 9$ with $D \\equiv 1 \\bmod 8$ in which case $W_D = W_D^0 \\sqcup W_D^1$ is a disjoint union of two Teichmüller curves distinguished by a spin invariant in $\\mathbb {Z}/2\\mathbb {Z}$ [18].", "A major challenge is to describe $W_D$ as an algebraic curve and as a hyperbolic orbifold.", "To date, this has been accomplished only for certain small $D$ [6], [17], [14].", "The purpose of this paper is to study the orbifold points on $W_D$ .", "Such points label surfaces with automorphisms commuting with $\\mathcal {O}_D$ .", "The first two Weierstrass curves $W_5$ and $W_8$ were studied by Veech [29] and are isomorphic to the $(2,5,\\infty )$ - and $(4,\\infty ,\\infty )$ -orbifolds.", "The surfaces with automorphisms labeled by the three orbifold points are drawn in Figure REF .", "Our primary goal is to give a formula for the number and type of orbifold points on $W_D$ (Theorem REF ).", "Together with [18] and [2], our formula completes the determination of the homeomorphism type of $W_D$ and gives a formula for the genus of $W_D$ .", "We will use our formula to give bounds for the genera of $W_D$ and $W_D^\\epsilon $ (Corollary REF ) and list the components of $\\bigcup _D W_D$ of genus zero (Corollary REF ).", "We will also give several explicit descriptions of the surfaces labeled by orbifold points on $W_D$ (Theorems REF ), giving the first examples of algebraic curves labeled by points of $W_D$ for most $D$ (Theorem REF )." ], [ "Main results.", "Our main theorem determines the number and type of orbifold points on $W_D$ : Theorem 1.1 For $D > 8$ , the orbifold points on $W_D$ all have order two, and the number of such points $e_2(W_D)$ is the weighted sum of class numbers of imaginary quadratic orders shown in Table REF .", "Table: For D>8D>8, the number of orbifold points of order two on W D W_D is given by a weighted sum of class numbers.", "The function h ˜(-D)\\widetilde{h}(-D) is defined below.", "We also give a formula for the number of orbifold points on each spin component: Theorem 1.2 Fix $D\\ge 9$ with $D \\equiv 1\\bmod 8$ .", "If $D = f^2$ is a perfect square, then all of the orbifold points on $W_D$ lie on the spin $(f+1)/2 \\bmod 2$ component: $ e_2\\left( W_D^{(f+1)/2} \\right)=\\frac{1}{2}\\widetilde{h}(-4D) \\mbox{ and } e_2\\left( W_D^{(f-1)/2} \\right)=0.$ Otherwise, $e_2(W_D^0)=e_2(W_D^1)=\\frac{1}{4}\\widetilde{h}(-4D)$ .", "When $D$ is not a square and $W_D$ is reducible, the spin components of $W_D$ have algebraic models defined over $\\mathbb {Q}(\\sqrt{D})$ and are Galois conjugate [6].", "Theorem REF confirms that the spin components have the same number and type of orbifold points.", "The class number $h(-D)$ is the order of the ideal class group $H(-D)$ for $\\mathcal {O}_{-D}$ and counts the number of elliptic curves with complex multiplication by $\\mathcal {O}_{-D}$ up to isomorphism.", "The weighted class number $ \\widetilde{h}(-D) = 2 h(-D)/\\left|\\mathcal {O}_{-D}^\\times \\right| $ appearing in Table REF is the number of elliptic curves with complex multiplication weighted by their orbifold order in $\\mathcal {M}_1$ .", "Note that $\\widetilde{h}(-D) = h(-D)$ unless $D=3$ or 4.", "The class number $h(-D)$ can be computed by enumerating integer points on a conic.", "We will give a similar method for computing $e_2(W_D)$ in Theorem REF .", "When $D$ is odd, the orbifold points on $W_D$ are labeled by elements of the group $H(-4D)/[P]$ where $[P]$ is the ideal class in $\\mathcal {O}_{-4D}$ representing the prime ideal with norm two.", "The orbifold Euler characteristics of $W_D$ and $W_D^\\epsilon $ were computed in [2] and the cusps on $W_D$ were enumerated and sorted by component in [18].", "Theorems REF and REF complete the determination of the homeomorphism type of $W_D$ and give a formula for the genera of $W_D$ and its components.", "Corollary 1.3 For any $\\epsilon > 0$ , there are constants $C_\\epsilon $ and $N_\\epsilon $ such that: $ C_\\epsilon D^{3/2+\\epsilon } > g(V) > D^{3/2}/650, $ whenever $V$ is a component of $W_D$ and $D \\ge N_\\epsilon $ .", "Modular curves of genus zero play an important role in number theory [27].", "We also determine the components of Weierstrass curves of genus zero.", "Corollary 1.4 The genus zero components of $\\bigcup _D W_D$ are the 23 components of $\\bigcup _{D \\le 41} W_D$ and the curves $W_{49}^0$ , $W_{49}^1$ and $W_{81}^1$ .", "We include a table listing the homeomorphism type of $W_D$ for $D\\le 225$ in §B.", "Figure: The first two Weierstrass curves W 5 W_5 and W 8 W_8 are isomorphic to the (2,5,∞)(2,5,\\infty ) and (4,∞,∞)(4,\\infty ,\\infty )-orbifolds.", "The point of orbifold order two is related to billiards on the LL-shaped table (left) corresponding to the golden mean γ=1+5 2\\gamma = \\frac{1+\\sqrt{5}}{2}.", "The points of orbifold order five (center) and four (right) are related to billiards on the regular pentagon and octagon." ], [ "Orbifold points on Hilbert modular surfaces.", "Theorem REF is closely related to the classification of orbifold points on Hilbert modular surfaces we prove in §.", "The Hilbert modular surface $X_D$ is the moduli space of principally polarized Abelian varieties with real multiplication by $\\mathcal {O}_D$ .", "The period map sending a Riemann surface to its Jacobian embeds $W_D$ in $X_D$ .", "Central to the story of the orbifold points on $X_D$ and $W_D$ are the moduli spaces $\\mathcal {M}_2(D_8)$ and $\\mathcal {M}_2(D_{12})$ of genus two surfaces with actions of the dihedral groups of orders 8 and 12: $ D_8 = \\left< r, J : r^2 = (Jr)^2 = J^4 = 1 \\right> \\mbox{ and } D_{12} = \\left< r, Z: r^2 = (Zr)^2 = Z^6 = 1 \\right>.", "$ The surfaces in $\\mathcal {M}_2(D_8)$ (respectively $\\mathcal {M}_2(D_{12})$ ) whose Jacobians have complex multiplication have real multiplication commuting with $J$ (respectively $Z$ ).", "The complex multiplication points on $\\mathcal {M}_2(D_8)$ and $\\mathcal {M}_2(D_{12})$ give most of the orbifold points on $\\bigcup _D X_D$ : Theorem 1.5 The orbifold points on $\\bigcup _D X_D$ which are not products of elliptic curves are the two points of order five on $X_5$ and the complex multiplication points on $\\mathcal {M}_2(D_8)$ and $\\mathcal {M}_2(D_{12})$ .", "Since the $Z$ -eigenforms on $D_{12}$ -surfaces have simple zeros and the $J$ -eigenforms on $D_8$ -surfaces have double zeros (cf.", "Proposition REF ), we have: Corollary 1.6 The orbifold points on $\\bigcup _D W_D$ are the point of order five on $W_5$ and the complex multiplication points on $\\mathcal {M}_2(D_8)$ .", "Corollary REF explains the appearance of class numbers in the formula for $e_2(W_D)$ .", "As we will see §, the involution $r$ on a $D_8$ -surface $X$ has a genus one quotient $E$ with a distinguished base point and point of order two and the family $\\mathcal {M}_2(D_8)$ is birational to the modular curve $Y_0(2)$ .", "The Jacobian $\\operatorname{Jac}(X)$ has complex multiplication by an order in $\\mathbb {Q}(\\sqrt{D},i)$ if and only if $E$ has complex multiplication by an order in $\\mathbb {Q}(\\sqrt{-D})$ .", "The formula for $e_2(W_D)$ follows by sorting the $3\\widetilde{h}(-D)$ surfaces with $D_8$ -action covering elliptic curves with complex multiplication by $\\mathcal {O}_{-D}$ by their orders for real multiplication." ], [ "The product locus $P_D$ .", "A recurring theme in the study of the Weierstrass curves is the close relationship between $W_D$ and the product locus $P_D \\subset X_D$ .", "The product locus $P_D$ consists of products of elliptic curves with real multiplication by $\\mathcal {O}_D$ and is isomorphic to a disjoint union of modular curves.", "The cusps on $W_D$ were first enumerated and sorted by spin in [18] and, for non-square $D$ , are in bijection with the cusps on $P_D$ .", "The Hilbert modular surface $X_D$ has a meromorphic modular form with a simple pole along $P_D$ and a simple zero along $W_D$ .", "This modular form can be used to give a formula for the Euler characteristic of $W_D$ and, for non-square $D$ , the Euler characteristics of $W_D$ , $X_D$ and $P_D$ satisfy ([2], Cor.", "10.4): $ \\chi (W_D) = \\chi (P_D) - 2\\chi (X_D).", "$ Our classification of the orbifold points on $X_D$ and $W_D$ in Theorem REF and Corollary REF show that all of the orbifold points of order two on $X_D$ lie on $W_D$ or $P_D$ , giving: Theorem 1.7 For non-square $D$ , the homeomorphism type of $W_D$ is determined by the homeomorphism types of $X_D$ and $P_D$ and $D \\bmod 8$ ." ], [ "The $D_8$ -family.", "A secondary goal of our analysis is to give several explicit descriptions of $D_8$ -surfaces and to characterize those with complex multiplication.", "We now outline the facts about $\\mathcal {M}_2(D_8)$ which we prove in Section ; we will outline a similar discussion for $\\mathcal {M}_2(D_{12})$ in §.", "For a genus two surface $X \\in \\mathcal {M}_2$ , the following are equivalent: Automorphisms.", "The automorphism group $\\operatorname{Aut}(X)$ admits an injective homomorphism $\\rho \\colon \\,D_8 \\rightarrow \\operatorname{Aut}(X)$ .", "Algebraic curves.", "The field of meromorphic functions $\\mathbb {C}(X)$ is isomorphic to: $ K_a = \\mathbb {C}(z,x) \\mbox{ with } z^2=(x^2-1)(x^4-ax^2+1), $ for some $a \\in \\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace $ .", "Jacobians.", "There is a number $\\tau \\in \\mathbb {H}$ such that the Jacobian $\\operatorname{Jac}(X)$ is isomorphic to the principally polarized Abelian variety: $ A_\\tau = \\mathbb {C}^2 / \\Lambda _\\tau , $ where $\\Lambda _\\tau = \\mathbb {Z}\\left< \\left( {\\begin{matrix} \\tau \\\\ \\tau +1 \\end{matrix}} \\right),\\left( {\\begin{matrix} \\tau \\\\ -\\tau -1 \\end{matrix}} \\right), \\left( {\\begin{matrix} \\tau +1 \\\\ \\tau \\end{matrix}} \\right),\\left( {\\begin{matrix} \\tau +1 \\\\ -\\tau \\end{matrix}} \\right)\\right>$ and $A_\\tau $ is polarized by the symplectic form $\\left< \\left( {\\begin{matrix} a \\\\ b \\end{matrix}} \\right),\\left( {\\begin{matrix} c \\\\ d \\end{matrix}} \\right)\\right>=\\frac{-\\operatorname{Im}(a \\overline{c}+b \\overline{d})}{2 \\operatorname{Im}(\\tau )}$ .", "Pinwheels.", "The surface $X$ is isomorphic to the surface $X_\\tau $ obtained from the polygonal pinwheel $P_\\tau $ (Figure REF ) for some $\\tau $ in the domain: $ U = \\left\\lbrace \\tau \\in \\mathbb {H}: \\tau \\ne \\frac{\\pm 1+i}{2}, \\left|\\tau \\right|^2 \\ge \\frac{1}{2} \\mbox{ and } \\left|\\operatorname{Re}{\\tau }\\right| \\le \\frac{1}{2} \\right\\rbrace .", "$ Figure: For τ\\tau in the shaded domain UU, the pinwheel P τ P_\\tau is the polygon with vertices at z=1±i 2z=\\frac{1\\pm i}{2}, -1±i 2\\frac{-1\\pm i}{2}, ±τ\\pm \\tau , and ±iτ\\pm i\\tau .", "Gluing together opposite sides on P τ P_\\tau by translation gives a genus two surface admitting an action of D 8 D_8.", "The one form ω τ \\omega _\\tau induced by dzdz is a JJ-eigenform and has a double zero.It is straightforward to identify the action of $D_8$ in most of the descriptions above.", "The field $K_a$ has automorphisms $r(z,x)=(z,-x)$ and $J(z,x)=(iz/x^3,1/x)$ .", "Multiplication by the matrices $\\left({\\begin{matrix} 1 & 0 \\\\ 0 & -1 \\end{matrix}}\\right)$ and $\\left({\\begin{matrix} 0 & 1 \\\\ -1 & 0 \\end{matrix}}\\right)$ preserve the polarized lattice $\\Lambda _\\tau $ giving automorphisms $r$ and $J$ of $A_\\tau $ .", "The surface $X_\\tau $ obtained from $P_\\tau $ has an obvious order four automorphism $J_\\tau $ and a genus two surface with an order four automorphism automatically admits a faithful action of $D_8$ (cf.", "Proposition REF ).", "The function relating the number $\\tau $ determining the polygon $P_\\tau $ and Abelian variety $A_\\tau $ to the number $a$ determining the field $K_a$ is the modular function: $ a(\\tau ) = -2+\\frac{1}{\\lambda (\\tau ) \\lambda (\\tau +1)}.", "$ The function $\\lambda (\\tau )$ is the function modular for $\\Gamma (2) = \\ker (\\operatorname{SL}_2(\\mathbb {Z}) \\rightarrow \\operatorname{SL}_2(\\mathbb {Z}/2\\mathbb {Z}))$ which covers the isomorphism $\\lambda \\colon \\,\\mathbb {H}/\\Gamma (2) \\xrightarrow{} \\mathbb {C}\\setminus \\left\\lbrace 0,1 \\right\\rbrace $ sending the cusps $\\Gamma (2) \\cdot 0$ , $\\Gamma (2) \\cdot 1$ and $\\Gamma \\cdot \\infty $ to 0, 1 and $\\infty $ respectively.", "In Sections and we will prove: Theorem 1.8 Fix $\\tau \\in U$ .", "The surface $X_\\tau $ obtained from the polygon $P_\\tau $ admits a faithful $D_8$ -action and satisfies: $ \\operatorname{Jac}(X_\\tau ) \\cong A_\\tau \\mbox{ and } \\mathbb {C}(X_\\tau ) \\cong K_{a(\\tau )}.", "$ The Jacobian $\\operatorname{Jac}(X_\\tau )$ has complex multiplication if and only if $\\tau $ is imaginary quadratic." ], [ "Enumerating orbifold points on $W_D$ .", "In addition to the formula in terms of class numbers for $e_2(W_D)$ appearing in Theorem REF , in Section we give a simple method for enumerating the orbifold points on $W_D$ .", "We define a finite set of proper pinwheel prototypes $E(D)$ consisting of triples of integers $(e,c,b)$ satisfying $D=-e^2+2bc$ along with certain additional conditions (cf.", "Equation REF ) and show: Theorem 1.9 Fix a discriminant $D \\ge 5$ .", "For any $(e,c,b) \\in E(D)$ , the surface: $ X_\\tau \\mbox{ with } \\tau = (e+\\sqrt{-D})/2c $ is labeled by an orbifold point on $W_D$ .", "For discriminants $D>8$ , the set $E(D)$ is in bijection with the points of orbifold order two on $W_D$ .", "Since the field automorphisms of $\\mathbb {C}$ permute the set of $D_8$ -surfaces with real multiplication by $\\mathcal {O}_D$ and the modular function $a \\colon \\,Y_0(2) \\rightarrow \\mathbb {C}$ is defined over $\\mathbb {Q}$ , the following is a corollary of Theorem REF : Theorem 1.10 For $D \\ge 5$ , the polynomial: $ f_D(t) = \\prod _{(e,c,b) \\in E(D)} \\left( t-a\\left( \\frac{e+\\sqrt{-D}}{2c} \\right) \\right)\\left( t-a\\left( \\frac{-e+\\sqrt{-D}}{2b} \\right) \\right), $ has rational coefficients.", "If $a$ is a root of $f_D(t)$ , then the algebraic curve with $\\mathbb {C}(X) \\cong K_a$ is labeled by an orbifold point on $W_D$ .", "For example, when $D=76$ we have: $ E(76) = \\left\\lbrace (-2,2,20), (-2,4,10), (2,4,10) \\right\\rbrace , $ and the orbifold points on $W_{76}$ label the surfaces $X_{(-1+\\sqrt{-19})/2}$ , $X_{(-1+\\sqrt{-19})/4}$ and $X_{(1+\\sqrt{-19})/4}$ .", "Setting $q=e^{2\\pi i \\tau }$ and using the $q$ -expansion: $ a(\\tau ) = -2 - 256 q - 6144 q^2 - 76800 q^3 - 671744 q^4 + \\dots $ we can approximate the coefficients for $f_{76}(t)$ to high precision to show that: $ f_{76}(t) = t^3+3t^2+3459t+6913.", "$ Table REF in Appendix C lists the polynomials $f_D(t)$ for $D \\le 56$ computed by similar means." ], [ "Outline.", "We conclude this Introduction with an outline of the proofs of our main results.", "In §, we define and study the family $\\mathcal {M}_2(D_8)$ .", "The moduli space $\\mathcal {M}_2(D_8)$ parametrizes pairs $(X,\\rho )$ where $X \\in \\mathcal {M}_2$ and $\\rho : D_8 \\rightarrow \\operatorname{Aut}(X)$ is injective.", "There, our main goal is to prove the precise relationship between the surface $X_\\tau $ , the field $K_{a(\\tau )}$ and the Abelian variety $A_\\tau $ stated in Theorem REF .", "We do so by showing, for $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ , the quotient $X/\\rho (r)$ has genus one, a distinguished base point and a distinguished point of order two (Proposition REF ), giving rise to a holomorphic map $g\\colon \\,\\mathcal {M}_2(D_8) \\rightarrow Y_0(2)$ .", "We then compute $\\mathbb {C}(X)$ and $\\operatorname{Jac}(X)$ in terms of $g(X,\\rho )$ and show that the surface $X_\\tau $ admits a $D_8$ -action $\\rho _\\tau $ with $g(X_\\tau ,\\rho _\\tau )$ equal to the genus one surface $E_\\tau = \\mathbb {C}/\\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ with distinguished base point $Z_\\tau = 0 +\\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ and point of order two $T_\\tau = 1/2+\\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ .", "In §, we define and study the Hilbert modular surface $X_D$ and its orbifold points.", "Orbifold points on $X_D$ correspond to Abelian varieties with automorphisms commuting with real multiplication.", "It is well known that there are only a few possibilities for the automorphism group of a genus two surface (cf.", "Table REF in §), that the automorphism group of $X \\in \\mathcal {M}_2$ equals the automorphism group $\\operatorname{Jac}(X)$ and that every principally polarized Abelian surface is either a product of elliptic curves or a Jacobian.", "Our classification of orbifold points on $\\bigcup _D X_D$ in Theorem REF is obtained by analyzing these possibilities.", "In §, we turn to the Weierstrass curve $W_D$ .", "Our classification of the orbifold points on $W_D$ in Corollary REF follows by analyzing which automorphism groups of genus two surfaces contain automorphisms which fix a Weierstrass point.", "We then prove the formula in Theorem REF by sorting the $D_8$ -surfaces with complex multiplication by their orders for real multiplication commuting with $J$ .", "To do so, we embed the endomorphism ring of $\\operatorname{Jac}(X)$ in the rational endomorphism ring $\\operatorname{End}(X/\\rho (r) \\times X/\\rho (r)) \\otimes \\mathbb {Q}$ , allowing us to relate the order for real multiplication on $\\operatorname{Jac}(X)$ to the order for complex multiplication on $X/\\rho (r)$ .", "We conclude § by giving a simple method for enumerating the $\\tau \\in U$ for which $X_\\tau $ is labeled by an orbifold point on $W_D$ .", "For $\\tau \\in \\mathbb {Q}(\\sqrt{-D})$ , we choose integers $e$ , $k$ , and $c$ so $\\tau = (e+k\\sqrt{-D})/(2c)$ .", "There is a rational endomorphism $T \\in \\operatorname{End}(\\operatorname{Jac}(X_\\tau ))\\otimes \\mathbb {Q}$ commuting the order four automorphism $J_\\tau $ and generated real multiplication by $\\mathcal {O}_D$ .", "By writing down how $T$ acts on $H_1(X_\\tau ,\\mathbb {Q})$ , we determine the conditions on $e$ , $k$ , and $c$ which ensure that $T$ preserves the lattice $H_1(X_\\tau ,\\mathbb {Z})$ .", "In §, we sort the orbifold points on $W_D$ by spin component when $D \\equiv 1 \\bmod 8$ .", "For such discriminants, the orbifold points on $W_D$ are labeled by elements of the ideal class group $H(-4D)$ .", "We define a spin homomorphism: $ \\epsilon _0 \\colon \\,H(-4D) \\rightarrow \\mathbb {Z}/2\\mathbb {Z}$ which is the zero map if and only if $D$ is a perfect square.", "We then relate the spin invariant of the orbifold point corresponding to the ideal class $[I]$ to the value $\\epsilon _0([I])$ to give the formula in Theorem REF .", "Finally, in §, we collect the various formulas for topological invariants of $W_D$ and bound them to give bounds on the genus of $W_D$ and its components." ], [ "Open problems.", "While the homeomorphism type of $W_D$ is now understood, describing the components of $W_D$ as Riemann surfaces remains a challenge.", "Problem 1 Describe $W_D$ as a hyperbolic orbifold and as an algebraic curve.", "Our analysis of the orbifold points on $W_D$ have given explicit descriptions of some complex multiplication points on $W_D$ .", "By the André-Oort conjecture [11], there are only finitely many complex multiplication points on $W_D$ and it would be interesting to find them.", "Problem 2 Describe the complex multiplication points on $W_D$ .", "The complex multiplication points on $\\mathcal {M}_2(D_8)$ lie on Teichmüller curves and the complex multiplication points on $\\mathcal {M}_2(D_{12})$ lie on complex geodesics in $\\mathcal {M}_2$ with infinite fundamental group.", "It would be interesting to find other examples of Shimura varieties whose complex multiplication points lie on interesting complex geodesics.", "Problem 3 Find other Shimura varieties whose complex multiplication lie on Teichmüller curves.", "The divisors supported at cusps on modular curves generate a finite subgroup of the associated Jacobian [15].", "It would be interesting to know if the same is true for Teichmüller curves.", "The first Weierstrass curve with genus one is $W_{44}$ .", "Problem 4 Compute the subgroup of $\\operatorname{Jac}(W_{44})$ generated by divisors supported at the cusps and points of order two.", "Algebraic geometers and number theorists have been interested in exhibiting explicit examples of algebraic curves whose Jacobians have endomorphisms.", "A parallel goal is to exhibit Riemann surfaces whose Jacobians have endomorphisms as polygons in the plane glued together by translations as we did for the complex multiplication points on $\\mathcal {M}_2(D_8)$ and $\\mathcal {M}_2(D_{12})$ .", "Problem 5 Exhibit surfaces whose Jacobians have complex multiplication as polygons in $\\mathbb {C}$ glued together by translation.", "There are very few Teichmüller curves $C \\rightarrow \\mathcal {M}_g$ whose images under the period mapping sending a surface to its Jacobian parametrize Shimura curves and they are classified in [21].", "The families $\\mathcal {M}_2(D_8)$ and $\\mathcal {M}_2(D_{12})$ are examples of Teichmüller curves whose images under the period mapping are dense in Shimura curves.", "Jacobians of $D_8$ - and $D_{12}$ -surfaces are dense in Shimura curves because they are characterized by their endomorphism ring.", "The family $\\mathcal {M}_2(D_8)$ (resp.", "$\\mathcal {M}_2(D_{12})$ ) is a Teichmüller curves because the map $X \\rightarrow X/\\rho (J)$ (resp.", "$X \\rightarrow X/\\rho (Z)$ ) is branched over exactly four points.", "It would be interesting to have a classification of such curves.", "Problem 6 Classify the Teichmüller curves whose images under the period mapping are dense in Shimura curves." ], [ "Notes and references.", "For a survey of results related to the Teichmüller geodesic flow, Teichmüller curves and relations to billiards see [10], [16], [12], [30].", "Background about Abelian varieties, Hilbert modular surfaces and Shimura varieties can be found in [28], [24], [4], and [23].", "The orbifold points on $X_D$ are studied in [22] and the family $\\mathcal {M}_2(D_8)$ has been studied in various settings, e.g.", "[25]." ], [ "Acknowledgments", "The author would thank C. McMullen for many helpful conversations throughout this project as well as A. Preygel and V. Gadre for several useful conversations.", "The author was partially supported in part by the National Science Foundation as a postdoctoral fellow." ], [ "The $D_8$ -family", "In this section we define and study the moduli space $\\mathcal {M}_2(D_8)$ parametrizing pairs $(X,\\rho )$ where $X \\in \\mathcal {M}_2$ and $\\rho : D_8 \\rightarrow \\operatorname{Aut}(X)$ is injective.", "In Section , we defined a domain $U$ , a polygon $P_\\tau $ and a surface $X_\\tau $ with order four automorphism $J_\\tau $ for each $\\tau \\in U$ , a field $K_a$ for $a \\in \\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace $ , an Abelian variety $A_\\tau $ for $\\tau \\in \\mathbb {H}$ and a modular function $a : \\mathbb {H}\\rightarrow \\mathbb {C}$ .", "Our main goal for this section is to prove the following proposition, establishing the claims in Theorem REF relating these different descriptions of $D_8$ -surfaces: Proposition 2.1 Fix $\\tau \\in U$ .", "There is an injective homomorphism $\\rho : D_8 \\rightarrow \\operatorname{Aut}(X_\\tau )$ with $\\rho (J) = J_\\tau $ and $X_\\tau $ satisfies: $ \\mathbb {C}(X_\\tau ) \\cong K_{a(\\tau )} \\mbox{ and } \\operatorname{Jac}(X_\\tau ) \\cong A_\\tau .", "$ For any $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ , there is a $\\tau \\in U$ so that there is an isomorphism $X \\rightarrow X_\\tau $ intertwining $\\rho (J)$ and $J_\\tau $ .", "We prove Proposition REF by studying the quotients $E_\\rho =X/\\rho (r)$ as $(X,\\rho )$ ranges in $\\mathcal {M}_2(D_8)$ .", "We show that $E_\\rho $ has genus one, a distinguished base point $Z_\\rho $ and point of order two $T_\\rho $ (cf.", "Proposition REF ), allowing us to define a holomorphic map $g : \\mathcal {M}_2(D_8) \\rightarrow Y_0(2)$ by $g(X,\\rho ) = (E_\\rho ,Z_\\rho ,T_\\rho )$ .", "We compute $\\mathbb {C}(X)$ and $\\operatorname{Jac}(X)$ in terms of $g(X,\\rho )$ (Propositions REF and REF ) and then show that $X_\\tau $ admits a $D_8$ -action $\\rho _\\tau $ with $g(X_\\tau ,\\rho _\\tau ) = (\\mathbb {C}/\\mathbb {Z}\\oplus \\tau \\mathbb {Z},0+\\mathbb {Z}\\oplus \\tau \\mathbb {Z},1/2+\\mathbb {Z}\\oplus \\tau \\mathbb {Z})$ (Proposition REF )." ], [ "Surfaces with automorphisms.", "Let $G$ be a finite group.", "We define the moduli space of $G$ -surfaces of genus $g$ to be the space: $ \\mathcal {M}_g(G) = \\left\\lbrace (X,\\rho ) : X \\in \\mathcal {M}_g \\mbox{ and } \\rho \\colon \\,G \\rightarrow \\operatorname{Aut}(X) \\mbox{ is injective.}", "\\right\\rbrace / \\sim .", "$ We will call two $G$ -surfaces $(X_1,\\rho _1)$ and $(X_2,\\rho _2)$ equivalent and write $(X_1,\\rho _1) \\sim (X_2,\\rho _2)$ if there is an isomorphism $f \\colon \\,X_1 \\rightarrow X_2$ satisfying $\\rho _1(x) = f^{-1} \\circ \\rho _2(x) \\circ f$ for each $x \\in G$ .", "The set $\\mathcal {M}_g(G)$ has a natural topology and a unique holomorphic structure so that the natural map $\\mathcal {M}_g(G) \\rightarrow \\mathcal {M}_g$ is holomorphic." ], [ "Group homomorphisms and automorphisms.", "Any injective group homomorphism $h \\colon \\,G_1 \\rightarrow G_2$ gives rise to a holomorphic map $\\mathcal {M}_g(G_2) \\rightarrow \\mathcal {M}_g(G_1)$ .", "In particular, the automorphism group $\\operatorname{Aut}(G)$ acts on $\\mathcal {M}_g(G)$ .", "The inner automorphisms of $G$ fix every point on $\\mathcal {M}_g(G)$ so the $\\operatorname{Aut}(G)$ -action factors through the outer automorphism group $\\operatorname{Out}(G) = \\operatorname{Aut}(G)/\\operatorname{Inn}(G)$ .", "Note that $\\operatorname{Out}(D_8)$ is isomorphic to $\\mathbb {Z}/2\\mathbb {Z}$ with the automorphism $\\sigma (J) = J$ and $\\sigma (r) = Jr$ representing the non-trivial outer automorphism." ], [ "Hyperelliptic involution and Weierstrass points.", "Now let $X$ be a genus two Riemann surface and $\\Omega (X)$ be the space of holomorphic one forms on $X$ .", "The canonical map $X \\rightarrow \\mathbb {P} \\Omega (X)^*$ is a degree two branched cover of the sphere branched over six points.", "The hyperelliptic involution $\\eta $ on $X$ is the Deck transformation of the canonical map and the Weierstrass points $X^W$ are the points fixed by $\\eta $ .", "Any holomorphic one form $\\omega \\in \\Omega (X)$ has either two simple zeros at points $P$ and $Q \\in X$ with $P =\\eta (Q)$ or has a double zero at a point $P \\in X^W$ ." ], [ "Automorphisms and permutations of $X^W$ .", "Since it is canonically defined, $\\eta $ is in the center of $\\operatorname{Aut}(X)$ and any $\\phi \\in \\operatorname{Aut}(X)$ induces an automorphism $\\phi ^\\eta $ of the sphere $X/\\eta $ and restricts to a permutation $\\phi |_{X^W}$ of $X^W$ .", "The conjugacy classes in the permutation group $\\operatorname{Sym}(X^W)$ are naturally labeled by partitions of six corresponding to orbit sizes in $X^W$ , and we will write $[n_1,\\dots ,n_k]$ for the conjugacy class corresponding to the partition $n_1+\\dots +n_k=6$ .", "We will denote the conjugacy class of $\\phi |_{X^W}$ in the permutation of group of $X^W$ by $[\\phi |_{X^W}]$ .", "In Table REF , we list the possibilities for $[\\phi |_{X^W}]$ and, for each possibility, we determine the possibilities for the order of $\\phi $ , the number of points in $X$ fixed by $\\phi $ and give the possible algebraic models for the pair $(X,\\phi )$ .", "The claims are elementary to prove and well-known (cf.", "[4] §11.7 or [5]).", "Most can be proved by choosing an appropriately scaled coordinate $x \\colon \\,X/\\eta \\rightarrow \\widehat{\\mathbb {C}}$ so the action $\\phi ^\\eta $ fixes $x^{-1}(0)$ and $x^{-1}(\\infty )$ .", "From Table REF , we can show: Proposition 2.2 Suppose $X \\in \\mathcal {M}_2$ has an order four automorphism $\\phi $ .", "The conjugacy class of $\\phi |_{X^W}$ is $[1,1,2,2]$ , the eigenforms for $\\phi $ have double zeros, $\\phi ^2 = \\eta $ and there is an injective group homomorphism $\\rho \\colon \\,D_8 \\rightarrow \\operatorname{Aut}(X)$ with $\\rho (J) = \\phi $ .", "From Table REF , we see that only $\\phi $ with $[\\phi |_{X^W}] = [1,1,2,2]$ have order four and that, for such automorphisms, there is a number $t \\in \\mathbb {C}$ so that $\\mathbb {C}(X)$ is isomorphic to $\\mathbb {C}(x,y)$ with: $ y^2 = x (x^4 - t x^2 + 1), $ and $\\phi (x,y) = (-x,iy)$ .", "From this algebraic model we see that $\\phi ^2(x,y) = (x,-y)$ (i.e.", "$\\phi ^2 = \\eta $ ), the eigenforms for $\\phi $ are the forms $\\frac{dx}{y} \\mathbb {C}^*$ and $x \\frac{dx}{y} \\mathbb {C}^*$ which have double zeros, and that there is an injective group homomorphism $\\rho \\colon \\,D_8 \\rightarrow \\operatorname{Aut}(X)$ satisfying $\\rho (J) = \\phi $ and $\\rho (r)(x,y) = (1/x,y/x^3)$ .", "From Table REF , it also follows easily that: Proposition 2.3 For any $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ , the $\\rho (J)$ -eigenforms have double zeros.", "For any $(X,\\rho ) \\in \\mathcal {M}_2(D_{12})$ , the $\\rho (Z)$ -eigenforms have simple zeros.", "For $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ , the automorphism $\\rho (J)$ has order four so, by Proposition REF , has eigenforms with double zeros.", "Also from Table REF we see that, for $(X,\\rho ) \\in \\mathcal {M}_2(D_{12})$ , $[\\rho (Z)|_{X^W}] = [3,3]$ so $\\rho (Z)$ fixes no Weierstrass point and has eigenforms with simple zeros.", "Table: An automorphism φ\\phi of a genus two surface XX restricts to a bijection φ| X W ∈Sym(X W )\\phi |_{X^W} \\in \\operatorname{Sym}(X^W).", "When φ| X W \\phi |_{X^W} is in one of the conjugacy classes above, the pair (X,φ)(X,\\phi ) has an algebraic model ℂ(X)≅ℂ(x,y)\\mathbb {C}(X) \\cong \\mathbb {C}(x,y) with xx, yy and φ\\phi satisfying the equations above for an appropriate choice of parameters t i ∈ℂt_i \\in \\mathbb {C}.", "The omitted conjugacy classes do not occur as restrictions of automorphisms since the φ\\phi induces automorphism of the sphere X/ηX/\\eta ." ], [ "Algebraic models for $D_8$ -surfaces.", "In Section we defined a field $K_a$ for each $a \\in \\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace $ with an explicit action of $D_8$ on $K_a$ by field automorphisms.", "Let $Y_a$ denote the genus two surface satisfying $\\mathbb {C}(Y_a) \\cong K_a$ and let $\\rho _a \\colon \\,D_8 \\rightarrow \\operatorname{Aut}(Y_a)$ be the corresponding action of $D_8$ .", "We will eventually show that the map $f \\colon \\,\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace \\rightarrow \\mathcal {M}_2(D_8)$ given by $f(a)= (Y_a,\\rho _a)$ is an isomorphism.", "To start we will show $f$ is onto: Proposition 2.4 The map $a \\mapsto (Y_a,\\rho _a)$ defines a surjective holomorphic map: $ f \\colon \\,\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace \\rightarrow \\mathcal {M}_2(D_8).", "$ In particular, $\\dim _\\mathbb {C}(\\mathcal {M}_2(D_8)) = 1$ and $\\mathcal {M}_2(D_8)$ has one irreducible component.", "Fix $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ .", "From Table REF and Proposition REF , we have that $[\\rho (r)|_{X^W}] = [2,2,2]$ , $[\\rho (J)|_{X^W}] = [1,1,2,2]$ , $\\rho (J)^2=\\eta $ and $\\operatorname{Fix}(r)$ is a single $\\eta $ -orbit consisting of two points.", "These observations allow us to choose an isomorphism $x \\colon \\,X/\\eta \\rightarrow \\widehat{\\mathbb {C}}$ so that: (1) $\\rho (r)(x) = -x$ , (2) $x(\\operatorname{Fix}(\\rho (r))) = 0$ and (3) $x(P) = 1$ for some $P \\in \\operatorname{Fix}(\\rho (J))$ .", "Since $\\rho (J r J^{-1}) = \\eta \\rho (r)$ , $J$ permutes the points in $X/\\eta $ fixed by $\\rho (r)^\\eta $ , i.e.", "$x^{-1}(\\left\\lbrace 0,\\infty \\right\\rbrace )$ , and must satisfy $\\rho (J)(x)=1/x$ .", "If $Q$ is any point in $X^W$ not fixed by $\\rho (J)$ and $t = x(Q)$ , then: $ x(X^W) = \\left\\lbrace 1,-1,t,-t,1/t,-1/t \\right\\rbrace .", "$ The field $\\mathbb {C}(X)$ is isomorphic to $\\mathbb {C}(x,y)$ where: $ y^2=(x^2-1)(x^4-ax^2+1) \\mbox{ and } a = t^2+1/t^2.", "$ Note that $a \\notin \\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace $ since the discriminant of $(x^2-1)(x^4-a x^2+1)$ is nonzer.", "By conditions (1) and (2) on the coordinate $x$ , we have $\\rho (r)(x,y) = (-x,y)$ .", "Since $\\rho (J)(x)=1/x$ , we have $\\rho (J)(x,y) = (1/x,iy/x^3)$ or $(1/x,-iy/x^3)$ .", "In the first case, the obvious isomorphism between $X$ and $Y_a$ intertwines $\\rho $ and $\\rho _a$ .", "In the second case, the composition of $\\rho (r)$ with the obvious isomorphism between $X$ and $Y_a$ intertwines $\\rho $ and $\\rho _a$ .", "In either case, $(X,\\rho )$ is in the image of $f$ ." ], [ "Genus one surfaces with distinguished points of order two.", "We now show that the quotient $(X/\\rho (r))$ for any $D_8$ -surface $(X,\\rho )$ has genus one, a distinguished base point and point of order two.", "Proposition 2.5 For $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ we define: $ E_\\rho = X/\\rho (r), \\mbox{ and } Z_\\rho = \\operatorname{Fix}(\\eta \\rho (r)) / \\rho (r), T_\\rho = \\operatorname{Fix}(\\rho (J))/\\rho (r) \\in E_\\rho .$ The quotient $E_\\rho $ has genus one and, under the group law on $E_\\rho $ with identity element $Z_\\rho $ , the point $T_\\rho $ is torsion of order two.", "By Proposition REF , it is enough to establish the claims when $(X,\\rho )$ is equivalent to $(Y_a,\\rho _a)$ .", "From the equations defining $(Y_a,\\rho _a)$ we see that $(E_{\\rho _a},Z_{\\rho _a},T_{\\rho _a})$ satisfies: $\\begin{array}{c}\\mathbb {C}(E_{\\rho _a}) = \\mathbb {C}(y,w=x^2) \\mbox{ with } y^2 = (w-1)(w^2-aw+1), \\\\Z_{\\rho _a} = w^{-1}(\\infty ) \\mbox{ and } T_{\\rho _a} = w^{-1}(1).\\end{array}$ In particular, $E_{\\rho _a}$ has genus one.", "The function $(w-1)$ vanishes to order two at $T_{\\rho _a}$ and has a pole of order two at $Z_{\\rho _a}$ , giving that $2 T_{\\rho _a} = Z_{\\rho _a}$ in the group law on $E_{\\rho _a}$ with base point $Z_{\\rho _a}$ .", "Genus one Riemann surfaces with distinguished base point and point of order two are parametrized by the modular curve: $ Y_0(2) = \\left\\lbrace (E,Z,T) : E \\in \\mathcal {M}_1 \\mbox{ and } Z, P \\in E \\mbox{ satisfies } 2 \\cdot P = 2 \\cdot Z \\right\\rbrace / \\sim .$ Two pairs $(E_1,Z_1,T_1)$ and $(E_2,Z_2,T_2)$ are equivalent if there is an isomorphism $E_1 \\rightarrow E_2$ sending $Z_1$ to $Z_2$ and $T_1$ to $T_2$ .", "The modular curve can be presented as a complex orbifold as follows.", "For $\\tau \\in \\mathbb {H}$ , let $E_\\tau = \\mathbb {C}/\\mathbb {Z}\\oplus \\tau \\mathbb {Z}, Z_\\tau = 0+\\mathbb {Z}\\oplus \\tau \\mathbb {Z}\\mbox{ and } T_\\tau = 1/2+\\mathbb {Z}\\oplus \\tau \\mathbb {Z}.$ The triples $(E_{\\tau _1},Z_{\\tau _1},T_{\\tau _1})$ and $(E_{\\tau _2},Z_{\\tau _2},T_{\\tau _2})$ are equivalent if and only if $\\tau _1$ and $\\tau _2$ are related by a Möbius transformation in the group $\\Gamma _0(2) =\\left\\lbrace \\left({\\begin{matrix} a & b \\\\ c & d \\end{matrix}}\\right) \\in \\operatorname{SL}_2(\\mathbb {Z}) : c \\equiv 0 \\bmod 2 \\right\\rbrace .", "$ The map $\\mathbb {H}\\rightarrow Y_0(2)$ given by $\\tau \\mapsto (E_\\tau ,Z_\\tau ,T_\\tau )$ descends to a bijection on $\\mathbb {H}/\\Gamma _0(2)$ , presenting $Y_0(2)$ as a complex and hyperbolic orbifold.", "A fundamental domain for $\\Gamma _0(2)$ is the convex hull of $\\left\\lbrace 0,(-1+i)/2,(1+i)/2,\\infty \\right\\rbrace $ and $Y_0(2)$ is isomorphic to the $(2,\\infty ,\\infty )$ -orbifold.", "The point of orbifold order two on $\\mathbb {H}/\\Gamma _0(2)$ is $(1+i)/2 \\cdot \\Gamma _0(2)$ and corresponds to the square torus with the points fixed by an order four automorphism distinguished.", "Proposition 2.6 Let $g \\colon \\,\\mathcal {M}_2(D_8) \\rightarrow Y_0(2)$ be the map defined by: $ g(X,\\rho ) = (E_\\rho ,Z_\\rho ,T_\\rho ).", "$ The composition $g \\circ f \\colon \\,\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace \\rightarrow Y_0(2)$ extends to a biholomorphism on $h: \\mathbb {C}\\setminus \\left\\lbrace -2 \\right\\rbrace \\rightarrow Y_0(2)$ with $h(2) = (1+i)/2\\cdot \\Gamma _0(2)$ .", "In Equation REF , we gave an explicit model for $g(f(a))$ .", "By elementary algebraic geometry, as $a$ tends to 2, $g(f(a))$ tends the square torus with the points fixed by an order four automorphism distinguished allowing us to extend the composition $g \\circ f$ to a holomorphic map $h : \\mathbb {C}\\setminus \\left\\lbrace -2 \\right\\rbrace \\rightarrow Y_0(2)$ with $h(2) = (1+i)/2 \\cdot \\Gamma _0(2)$ .", "The coarse spaces associated to $Y_0(2)$ and $\\mathbb {C}\\setminus \\left\\lbrace -2 \\right\\rbrace $ are both biholomorphic to $\\mathbb {C}^*$ .", "To show $h$ is a biholomorphism, it suffices to show $\\deg (h)=1$ .", "To do so, consider the function $j \\colon \\,Y_0(2) \\rightarrow \\mathbb {C}$ sending $(E,Z,T)$ to the $j$ -invariant of $E$ .", "The function $j$ has degree $[ \\operatorname{SL}_2(\\mathbb {Z}) : \\Gamma _0(2)] = 3$ .", "From Equation REF , it is straightforward to show: $ j\\circ h(a) = 256(a+1)^3/(a+2).", "$ Since $\\deg (j \\circ h) = \\deg (j)$ , we have $\\deg (h) = 1$ .", "Corollary 2.7 The map $f \\colon \\,\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace \\rightarrow \\mathcal {M}_2(D_8)$ defined by $f(a) = (Y_a,\\rho _a)$ is a biholomorphism.", "The map $g\\colon \\,\\mathcal {M}_2(D_8) \\rightarrow Y_0(2)$ is a biholomorphism onto its image, which is the complement of $(1+i)/2\\cdot \\Gamma _0(2)$ .", "Since $g \\circ f$ extends to a biholomorphism and $f$ is onto, both $g$ and $f$ are biholomorphisms onto their images.", "By Proposition REF , $f$ is onto and, consequently, a biholomorphism.", "This also implies that the image of $g$ is equal to the image of $g \\circ f$ , which we have shown is the complement of the point $(1+i)/2 \\cdot \\Gamma _0(2)$ ." ], [ "Outer automorphism.", "We have already seen that the non-trivial outer automorphism $\\sigma $ of $D_8$ acts on $\\mathcal {M}_2(D_8)$ .", "We will now identify the corresponding automorphisms of $\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace $ and $Y_0(2)$ intertwining $\\sigma $ with $f$ and $g$ .", "The modular curve $Y_0(2)$ has an Atkin-Lehner involution which we will denote $\\sigma _X$ .", "The involution $\\sigma _X$ is given by the formula $\\sigma _X(E,Z_E,T_E) = (F,Z_F,T_F)$ where $F$ is isomorphic to the quotient $E/T_Z$ and $T_F$ is the point of order two in the image of the two torsion on $E$ under the degree two isogeny $E \\rightarrow F = E/T_Z$ .", "In terms of our presentation of $Y_0(2)$ as a complex orbifold, $\\sigma _X$ is defined by $\\sigma _X(\\tau \\cdot \\Gamma _0(2)) = \\sigma _X(-1/2\\tau \\cdot \\Gamma _0(2))$ .", "The Riemann surface $\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace $ parametrizing algebraic models for $D_8$ -surfaces also has an involution: $ \\sigma _A(a) = (-2a+12)/(a+2).", "$ It is straightforward to show that $f$ and $g$ intertwine $\\sigma $ , $\\sigma _A$ and $\\sigma _X$ , i.e.", ": $ \\sigma \\circ f = f \\circ \\sigma _A \\mbox{ and } \\sigma _X \\circ g = g \\circ \\sigma .", "$ Proposition 2.8 Suppose $(X_1,\\rho _1)$ and $(X_2,\\rho _2)$ are $D_8$ -surfaces, and there is an isomorphism $s : X_1 \\rightarrow X_2$ intertwining $\\rho _i(J)$ .", "Then $(X_1,\\rho _1)$ is equivalent to either $(X_2,\\rho _2)$ or $\\sigma \\cdot (X_2,\\rho _2)$ .", "We will show that $s$ intertwines $\\rho _1(J^k r)$ with $\\rho _2(r)$ for some $k$ .", "If $k=0$ or $k=2$ , then either $s$ or $s \\circ \\rho _1(J)$ gives an equivalence between $(X_1,\\rho _1)$ and $(X_2,\\rho _2)$ .", "If $k=1$ or $k=3$ , then either $s$ or $s \\circ \\rho _1(J)$ gives an equivalence between $(X_1,\\rho _1)$ and $\\sigma \\cdot (X_2,\\rho _2)$ .", "Setting $\\phi = s \\circ \\rho _2(r) \\circ s^{-1}$ , our goal is to show that $\\phi = \\rho _1(J^k r)$ for some $k$ .", "For the $D_8$ -surface $f(a)=(Y_a,\\rho _a)$ , the involution $\\rho _a(r)$ interchanges the two points in $\\operatorname{Fix}(\\rho _a(J))$ .", "Since $(X_i,\\rho _i)$ are equivalent to $(Y_{a_i},\\rho _{a_i})$ for some $a_i$ by Proposition REF , the same is true for $(X_i,\\rho _i)$ .", "From $s \\circ \\rho _2(J) \\circ s^{-1} = \\rho _1(J)$ we see that $s^{-1}(\\operatorname{Fix}(\\rho _2(J)))=\\operatorname{Fix}(\\rho _1(J))$ and the composition $\\phi \\circ \\rho _1(r)$ fixes both points in $\\operatorname{Fix}(\\rho _1(J))$ .", "This in turn implies that $\\phi \\circ \\rho _1(r) = \\rho _1(J)^k$ for some $k$ ." ], [ "Fixed point of $\\sigma $ .", "The following proposition about the unique fixed point of $\\sigma $ will be useful in our discussion of surfaces obtained from pinwheels.", "Proposition 2.9 Fix $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ .", "The following are equivalent: There is an automorphism $\\phi \\in Aut(X)$ satisfying $\\phi ^2 = \\rho (J)$ , $\\sigma \\cdot (X,\\rho ) \\sim (X,\\rho )$ , $(X,\\rho )= f(6)$ , and $g(X,\\rho )= \\sqrt{-2}/2 \\cdot \\Gamma _0(2)$ .", "The proof is straightforward so we omit it." ], [ "Cusps of $\\mathcal {M}_2(D_8)/\\sigma $ .", "The coarse space associated to $\\mathcal {M}_2(D_8)/\\sigma \\cong (\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace ) / \\sigma _A$ is isomorphic to $\\mathbb {C}^*$ and has two cusps.", "The following proposition gives a geometric characterization of the difference between these two cusps and will be important in our discussion of surfaces obtained from pinwheels.", "Proposition 2.10 For a sequence of $D_8$ -surface $(X_i,\\rho _i)$ , the following are equivalent: $X_i$ tend to a stable limit with geometric genus zero as $i \\rightarrow \\infty $ , $\\mathbb {C}(X_i) \\cong K_{a_i}$ with $a_i \\rightarrow \\infty $ , and the quotients $X_i/\\rho _{i}(r)$ diverge in $\\mathcal {M}_1$ as $i \\rightarrow \\infty $ .", "The proof of this proposition is also straightforward so we omit it." ], [ "A modular function.", "In Section we defined modular functions $\\lambda (\\tau )$ and $a(\\tau )$ .", "We now show: Proposition 2.11 If $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ and $ \\tau \\in \\mathbb {H}$ satisfies $g(X,\\rho ) = (E_\\tau ,Z_\\tau ,T_\\tau )$ , then: $ \\mathbb {C}(X) \\cong K_{a(\\tau )} \\mbox{ where } a(\\tau ) = -2 + \\frac{1}{\\lambda (\\tau )\\lambda (\\tau +1)}.$ Up to precomposition by $\\tau \\mapsto -1/2\\tau $ , the function $a : \\mathbb {H}\\rightarrow \\mathbb {C}$ defined above is the unique holomorphic function satisfying (1) $a$ covers an isomorphism $\\bar{a} :Y_0(2) \\xrightarrow{} \\mathbb {C}\\setminus \\left\\lbrace -2 \\right\\rbrace $ and (2) $a\\left( (1+i)/2 \\right) = 2$ .", "The biholomorphic extension $h : \\mathbb {C}\\setminus \\left\\lbrace -2 \\right\\rbrace \\rightarrow Y_0(2)$ of $g \\circ f$ from Proposition REF has $h(2) = (1+i)/2 \\cdot \\Gamma _0(2)$ , so $h^{-1}$ also satisfies (1) and (2).", "As a consequence, $f(\\bar{a}(E_\\tau ,Z_\\tau ,T_\\tau ))$ is equal to either $f(a(\\tau ))$ or $f(a(-1/2\\tau ))=\\sigma \\cdot f(a(\\tau ))$ .", "In either case, $\\mathbb {C}(X) \\cong K_{a(\\tau )}$ .", "As we saw in the proof of Proposition REF , the function $a(\\tau )$ satisfies $ a(\\tau )^3+3 a(\\tau )^2+(3-j(\\tau )/256) a(\\tau )+1-j(\\tau )/128 = 0 $ where $j(\\tau )$ is the function, modular for $\\operatorname{SL}_2(\\mathbb {Z})$ and equal to the $j$ -invariant of $E_\\tau $ ." ], [ "Jacobians of surfaces with involutions.", "Our next goal is to compute the Jacobian of a $D_8$ -surface $(X,\\rho )$ in terms of $g(X,\\rho ) \\in Y_0(2)$ .", "We start, more generally, by describing the Jacobian of a surface $X \\in \\mathcal {M}_2$ with an involution $\\phi \\in \\operatorname{Aut}(X)$ , $\\phi \\ne \\eta $ in terms of the quotients $X/\\phi $ and $X/\\eta \\phi $ .", "From Table REF we see that there are distinct complex numbers $t_1$ and $t_2$ so that: $ \\mathbb {C}(X) \\cong \\mathbb {C}(x,y) \\mbox{ with } y^2 = (x^2-1)(x^2-t_1)(x^2-t_2) \\mbox{ and } \\phi (x,y) = (-x,y).", "$ The quotients $E = X/\\phi $ and $F = X/\\eta \\phi $ have algebraic models given by $\\begin{array}{c}\\mathbb {C}(E) \\cong \\mathbb {C}(z_E=y,w_E=x^2) \\mbox{ with } z_E^2 = (w_E-1)(w_E-t_1)(w_E-t_2) \\mbox{ and } \\\\\\mathbb {C}(F) \\cong \\mathbb {C}(z_F=xy,w_F=x^2) \\mbox{ with } z_F^2 = w_F (w_F-1)(w_F-t_1)(w_F-t_2).\\end{array}$ The genus one surfaces $E$ and $F$ have natural base points $Z_E=\\operatorname{Fix}(\\eta \\phi )/\\phi = w_E^{-1}(\\infty )$ and $Z_F=\\operatorname{Fix}(\\phi )/\\eta \\phi = w_F^{-1}(0)$ respectively.", "Also, the image of $X^W$ under the map $X \\rightarrow E \\times F$ is the set: $ \\Gamma ^W = \\left\\lbrace (w_E^{-1}(t),w_F^{-1}(t)) \\in E \\times F : t = 1,t_1 \\mbox{ or } t_2 \\right\\rbrace .", "$ The set $\\Gamma ^W$ generates a subgroup of order four of the two torsion on $E \\times F$ under the group laws with identity element $Z_E \\times Z_F$ .", "Proposition 2.12 Suppose $X$ , $\\phi $ , $E$ , $F$ and $\\Gamma ^W$ are as above.", "The Jacobian of $X$ satisfies: $ \\operatorname{Jac}(X) \\cong E \\times F / \\Gamma ^W $ and the principal polarization on $\\operatorname{Jac}(X)$ pulls back to twice the product polarization on $E \\times F$ under the quotient by $\\Gamma ^W$ map.", "Let $\\psi : X \\rightarrow E \\times F$ be the obvious map.", "By writing down explicit bases for $H_1(X,\\mathbb {Z})$ , $H_1(E,\\mathbb {Z})$ and $H_1(F,\\mathbb {Z})$ using the algebraic models defined above, it is straightforward to check that the image of $\\psi _* : H_1(X,\\mathbb {Z}) \\rightarrow H_1(E,\\mathbb {Z}) \\oplus H_1(F,\\mathbb {Z})$ has index four and that the symplectic form on $\\psi _*(H_1(X,\\mathbb {Z}))$ induced by the intersection pairing on $X$ extends to twice the ordinary symplectic form on $H_1(E,\\mathbb {Z}) \\oplus H_1(F,\\mathbb {Z})$ .", "The holomorphic map $\\psi $ factors through a map on Jacobians: $ \\psi : \\operatorname{Jac}(X) \\rightarrow \\operatorname{Jac}(E) \\times \\operatorname{Jac}(F) $ whose degree is equal to $[ H_1(E,\\mathbb {Z}) \\oplus H_1(F,\\mathbb {Z}) : \\psi _*(H_1(X,\\mathbb {Z})) ] = 4$ .", "Under the identification of $\\operatorname{Jac}(X)$ with the Picard group $\\operatorname{Pic}^0(X)$ , the two torsion in $\\operatorname{Jac}(X)$ consists of degree zero divisors of the form $[P_i - P_j]$ with $P_i \\ne P_j$ and $P_i, P_j \\in X^W$ .", "The image of the two torsion in $\\operatorname{Jac}(X)$ under $\\psi $ has image $\\Gamma ^W$ generating a subgroup of order four in $\\operatorname{Jac}(E) \\times \\operatorname{Jac}(F)$ .", "The composition of $\\psi $ with the quotient by $\\Gamma ^W$ map has degree 16, vanishes on the two-torsion and factors through multiplication by two on $\\operatorname{Jac}(X)$ to give an isomorphism." ], [ "Jacobians of $D_8$ -surfaces.", "With Proposition REF , it is now easy to compute $\\operatorname{Jac}(X)$ for a $D_8$ -surface $(X,\\rho )$ in terms of $g(X,\\rho )$ .", "In Section , we defined a principally polarized Abelian variety $A_\\tau $ for each $\\tau \\in \\mathbb {H}$ .", "Proposition 2.13 If $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ and $ \\tau \\in \\mathbb {H}$ satisfies $g(X,\\rho ) = (E_\\tau ,Z_\\tau ,T_\\tau )$ , then $\\operatorname{Jac}(X) \\cong A_\\tau $ .", "Set $\\phi = \\rho (r)$ , $E = X /\\phi $ and $F = X / \\eta \\phi $ .", "By our definition of $g$ , $E$ is isomorphic to $E_\\tau $ .", "Also, $\\eta \\phi = \\rho (J) \\circ \\phi \\circ \\rho (J^{-1})$ , so $\\rho (J)$ induces an isomorphism between $F$ and $E$ .", "In particular, $F$ is also isomorphic to $E_\\tau $ .", "The image of $\\Gamma ^W \\subset E \\times F$ under the isomorphism to $E \\times F \\rightarrow E_\\tau \\times E_\\tau $ is the graph of the induced action of $\\rho (J)$ on the two torsion of $E_\\tau $ .", "Setting $T_\\tau = 1/2 + \\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ , $Q_\\tau = \\tau /2 + \\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ and $R_\\tau = (\\tau +1)/2 + \\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ , we have: $ \\Gamma ^W = \\left\\lbrace (T_\\tau ,T_\\tau ),(Q_\\tau ,R_\\tau ),(R_\\tau ,Q_\\tau ) \\right\\rbrace .", "$ The Abelian variety $E_\\tau \\times E_\\tau /\\Gamma ^W$ , principally polarized by half the product polarization on $E_\\tau \\times E_\\tau $ , is easily checked to be equal to $A_\\tau $ .", "(Note that the lattice $\\Lambda _\\tau $ used to define $A_\\tau $ contains the vectors $\\left({\\begin{matrix} 2 \\\\0 \\end{matrix}}\\right)$ , $\\left({\\begin{matrix} 2 \\tau \\\\ 0 \\end{matrix}}\\right)$ , $\\left({\\begin{matrix} 0 \\\\ 2 \\end{matrix}}\\right)$ and $\\left({\\begin{matrix} 0 \\\\ 2 \\tau \\end{matrix}}\\right)$ )." ], [ "Pinwheels.", "In Section , we defined a domain $U$ and associated to each $\\tau \\in U$ a polygonal pinwheel $P_\\tau $ and a surface $X_\\tau $ with order four automorphism $J_\\tau $ .", "Proposition 2.14 Fix $\\tau \\in U$ .", "There is an injective homomorphism $\\rho _\\tau : D_8 \\rightarrow \\operatorname{Aut}(X_\\tau )$ with $\\rho _\\tau (J) = J_\\tau $ and $g(X_\\tau ,\\rho _\\tau ) = (E_\\tau ,Z_\\tau ,P_\\tau )$ .", "By Proposition REF , the surface $X_\\tau $ admits a faithful $D_8$ -action $\\rho _\\tau : D_8 \\rightarrow \\operatorname{Aut}(D_8)$ with $\\rho (J) = J_\\tau $ .", "Since $U$ is simply connected, we can choose $\\rho _\\tau $ so $\\tau \\mapsto (X_\\tau ,\\rho _\\tau )$ gives a holomorphic map $p : U \\rightarrow \\mathcal {M}_2(D_8)$ .", "As depicted in Figure REF , the polygons $P_\\tau $ and $P_{\\tau +1}$ differ by a Euclidean cut-and-paste operation, giving an isomorphism between $X_\\tau $ and $X_{\\tau +1}$ intertwining $J_\\tau $ and $J_{\\tau +1}$ .", "Figure: The surfaces X τ X_\\tau and X τ+1 X_{\\tau +1} are isomorphic since the polygons P τ P_\\tau and P τ+1 P_{\\tau +1} differ by a cut-and-paste operation.Also, the polygons $P_\\tau $ and $P_{-1/2\\tau }$ differ by a Euclidean similarity, giving an isomorphism between $X_\\tau $ and $X_{-1/2\\tau }$ intertwining $J_\\tau $ and $J_{-1/2\\tau }$ .", "By Proposition REF , $p$ covers a holomorphic map $\\bar{p}: U/\\sim \\rightarrow \\mathcal {M}_2(D_8)/\\sigma $ where $\\tau \\sim \\tau +1$ and $\\tau \\sim -1/2\\tau $ .", "The coarse spaces associated to both $U/\\sim $ and $\\mathcal {M}_2(D_8)/\\sigma \\cong (\\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace ) / \\sigma _A$ are biholomorphic to $\\mathbb {C}^*$ .", "Also, there is a unique point $\\tau = \\sqrt{-2}/2$ in $U$ for which $J_\\tau $ is the square of an order eight automorphism, so $ p^{-1}(\\operatorname{Fix}(\\sigma )) = \\left\\lbrace \\sqrt{-2}/2 \\right\\rbrace $ .", "In particular $\\bar{p}$ has degree one and is a biholomorphism.", "There are precisely two biholomorphisms $U/\\sim \\rightarrow \\mathcal {M}_2(D_8)/\\sigma $ satisfying $\\bar{p}(\\sqrt{-2}/2) = \\operatorname{Fix}(\\sigma )/\\sigma $ .", "They are distinguished by the geometric genus of the stable limit of $p(\\tau )$ as $\\operatorname{Im}(\\tau )$ tends to $\\infty $ .", "The geometric genus of $X_\\tau $ as $\\operatorname{Im}(\\tau )$ tends to infinity is zero.", "Another holomorphic map $p_1 : U \\rightarrow \\mathcal {M}_2(D_8)$ is given by $p_1(\\tau ) = g^{-1}(E_\\tau ,Z_\\tau ,T_\\tau )$ .", "Since $p_1$ intertwines $\\tau \\mapsto -1/2\\tau $ and the outer automorphism $\\sigma $ , and $(E_\\tau ,Z_\\tau ,T_\\tau ) = (E_{\\tau +1},Z_{\\tau +1},T_{\\tau +1})$ , $p_1$ also covers an isomorphism $\\bar{p}_1: U/\\sim \\rightarrow \\mathcal {M}_2(D_8)/\\sigma $ .", "Moreover $p_1(\\sqrt{-2}/2) = \\operatorname{Fix}(\\sigma )/\\sigma $ and the limit of $p_1(\\tau )$ diverges tends to the cusp of $\\mathcal {M}_2(D_8)/\\sigma $ with stable limit of genus zero.", "So $\\bar{p}_1 = \\bar{p}$ and the proposition follows.", "Combining the results of this section we have: [Proof of Proposition REF ] Fix $\\tau \\in U$ .", "By Proposition REF , the surface $X_\\tau $ obtained from $P_\\tau $ admits a $D_8$ -action $\\rho _\\tau $ with $g(X_\\tau ,\\rho _\\tau ) = (E_\\tau ,Z_\\tau ,P_\\tau )$ .", "By Proposition REF , $\\operatorname{Jac}(X_\\tau ) \\cong A_\\tau $ and by Proposition REF $\\mathbb {C}(X_\\tau ) \\cong K_{a(\\tau )}$ .", "Now fix any $(X,\\rho ) \\in \\mathcal {M}_2(D_8)$ .", "The domain $U$ is a fundamental domain for the group generated by $\\Gamma _0(2)$ and the Atkin-Lehner involution $\\tau \\mapsto -1/2\\tau $ .", "It follows that there is a $\\tau \\in U$ so $g(X,\\rho ) = (E_\\tau ,Z_\\tau ,T_\\tau )$ or $g(\\sigma \\cdot (X,\\rho )) = (E_\\tau ,Z_\\tau ,T_\\tau )$ .", "In either case, by Proposition REF and the fact that $g$ is an isomorphism onto its image, there is an isomorphism $X \\rightarrow X_\\tau $ intertwining $\\rho (J)$ and $J_\\tau $ ." ], [ "Orbifold points on Hilbert modular surfaces", "In this section, we discuss two dimensional Abelian varieties with real multiplication, Hilbert modular surfaces and their orbifold points.", "Our main goals are to show: (1) the Abelian variety $A_\\tau \\cong \\operatorname{Jac}(X_\\tau )$ has complex multiplication if and only if $\\tau $ is imaginary quadratic, completing the proof of Theorem REF (Proposition REF ) (2) the Jacobians of $D_8$ - and $D_{12}$ -surfaces with complex multiplication are labeled by orbifold points in $\\bigcup _{D} X_D$ (Proposition REF ) and (3) establish the characterization of orbifold points on $\\bigcup _D X_D$ of Theorem REF (Propositions REF )." ], [ "Quadratic orders.", "Each integer $D \\equiv 0$ or $1 \\bmod 4$ determines a quadratic ring: $ \\mathcal {O}_D=\\frac{\\mathbb {Z}[t]}{(t^2-D t+D(D-1)/4)}.$ The integer $D$ is called the discriminant of $\\mathcal {O}_D$ .", "We will write $\\sqrt{D}$ for the element $2t-D \\in \\mathcal {O}_D$ whose square is $D$ and define $K_D = \\mathcal {O}_D \\otimes \\mathbb {Q}$ .", "We will typically reserve the letter $D$ for positive discriminants and the letter $C$ for negative discriminants.", "For a positive discriminant $D>0$ , the ring $\\mathcal {O}_D$ is totally real, i.e.", "every homomorphism of $\\mathcal {O}_D$ into $\\mathbb {C}$ factors through $\\mathbb {R}$ .", "For such $D$ , we will denote by $\\sigma _+$ and $\\sigma _-$ the two homomorphisms $K_D \\rightarrow \\mathbb {R}$ characterized by $\\sigma _+(\\sqrt{D}) > 0 > \\sigma _-(\\sqrt{D})$ .", "We will also use $\\sigma _+$ and $\\sigma _-$ for the corresponding homomorphisms $\\operatorname{SL}_2(K_D) \\rightarrow \\operatorname{SL}_2(\\mathbb {R})$ .", "The inverse different is the fractional ideal $\\mathcal {O}_D^\\vee = \\frac{1}{\\sqrt{D}} \\mathcal {O}_D$ and is equal to the trace dual of $\\mathcal {O}_D$ ." ], [ "Unimodular modules.", "Now let $\\Lambda _D = \\mathcal {O}_D \\oplus \\mathcal {O}_D^\\vee \\subset K_D^2$ .", "The $\\mathcal {O}_D$ -module $\\Lambda _D$ has a unimodular symplectic form induced by trace: $ \\left<(x_1,y_1),(x_2,y_2) \\right> = \\operatorname{Tr}^{K_D}_\\mathbb {Q}(x_1 y_2 - x_2 y_1).", "$ Up to symplectic isomorphism of $\\mathcal {O}_D$ -modules, $\\Lambda _D$ is the unique unimodular $\\mathcal {O}_D$ -module isomorphic to $\\mathbb {Z}^4$ as an Abelian group with the property that the action of $\\mathcal {O}_D$ is self-adjoint and proper (cf.", "[20], Theorem 4.4).", "Here, self-adjoint means $\\left<\\lambda v,w \\right> = \\left<v,\\lambda w\\right>$ for each $\\lambda \\in \\mathcal {O}_D$ and proper means that the $\\mathcal {O}_D$ -module structure on $\\Lambda _D$ is faithful and does not extend to a larger ring in $K_D$ ." ], [ "Symplectic $\\mathcal {O}_D$ -module automorphisms.", "Let $\\operatorname{SL}(\\Lambda _D)$ denote the group of symplectic $\\mathcal {O}_D$ -module automorphisms of $\\Lambda _D$ .", "This group coincides with the $\\mathcal {O}_D$ -module automorphisms of $\\Lambda _D$ and equals: $\\left\\lbrace \\begin{pmatrix} a & b\\\\c & d \\end{pmatrix} : \\begin{array}{c} ad-bc=1, a, d \\in \\mathcal {O}_D, \\\\ b \\in \\sqrt{D} \\mathcal {O}_D \\mbox{ and } c \\in \\mathcal {O}_D^\\vee \\end{array}\\right\\rbrace \\subset \\operatorname{SL}_2(K_D) $ with $A = \\left({\\begin{matrix} a & b \\\\ c & d \\end{matrix}}\\right)$ acting on $\\Lambda _D$ by sending $(x,y)$ to $(ax+by,cx+dy)$ .", "The group $\\operatorname{SL}(\\Lambda _D)$ embeds in $\\operatorname{SL}_2(\\mathbb {R}) \\times \\operatorname{SL}_2(\\mathbb {R})$ via $A \\mapsto (\\sigma _+(A),\\sigma _-(A))$ and acts on $\\mathbb {H}\\times \\mathbb {H}$ by Möbius transformations: $ (\\tau _+,\\tau _-)\\cdot A = \\left(\\frac{\\sigma _+(d \\tau +b)}{\\sigma _+(c \\tau +a)},\\frac{\\sigma _-(d \\tau +b)}{\\sigma _-(c \\tau +a)}\\right) $ where $\\sigma _+(y \\tau +x) = \\sigma _+(y) \\tau _+ + \\sigma _+(x)$ and $\\sigma _-(y \\tau +x) = \\sigma _-(y) \\tau _- + \\sigma _-(x)$ .", "The following proposition characterizes the elements of $\\operatorname{SL}(\\Lambda _D)$ fixing every point in $\\mathbb {H}\\times \\mathbb {H}$ and is elementary to verify: Proposition 3.1 Let $h$ be the homomorphism $h\\colon \\,\\operatorname{SL}(\\Lambda _D) \\rightarrow \\operatorname{PSL}_2(\\mathbb {R}) \\times \\operatorname{PSL}_2(\\mathbb {R})$ given by $h(A) = (\\pm \\sigma _+(A),\\pm \\sigma _-(A))$ .", "For $A \\in \\operatorname{SL}(\\Lambda _D)$ , the following are equivalent: $A$ fixes every point in $\\mathbb {H}\\times \\mathbb {H}$ , $A^2 = 1$ , $A = \\left({\\begin{matrix} t & 0 \\\\ 0 & t \\end{matrix}}\\right)$ where $t \\in \\mathcal {O}_D$ satisfies $t^2 =1$ , and $A$ is in $\\ker (h)$ .", "The group $\\ker (h)$ is isomorphic to the Klein-four group when $D = 1$ or 4 and is cyclic of order two otherwise." ], [ "Hilbert modular surfaces.", "The group $\\operatorname{PSL}(\\Lambda _D) = \\operatorname{SL}(\\Lambda _D)/\\ker (h)$ acts faithfully and properly discontinuously on $\\mathbb {H}\\times \\mathbb {H}$ and we define $X_D$ to be the quotient: $ X_D = \\mathbb {H}\\times \\mathbb {H}/ \\operatorname{PSL}(\\Lambda _D).", "$ We will denote by $[\\tau ]$ the point in $X_D$ represented by $\\tau \\in \\mathbb {H}\\times \\mathbb {H}$ .", "The complex orbifold is a typical example of a Hilbert modular surface." ], [ "Abelian varieties with real multiplication.", "Now let $B = \\mathbb {C}^2 / \\Lambda $ be a principally polarized Abelian surface.", "The endomorphism ring $\\operatorname{End}(B)$ of $B$ is the ring of holomorphic homomorphisms from $B$ to itself.", "We will say that $B$ admits real multiplication by $\\mathcal {O}_D$ if there is a proper and self-adjoint homomorphism: $ \\iota \\colon \\,\\mathcal {O}_D \\rightarrow \\operatorname{End}(B).", "$ Self-adjoint and proper here mean that $\\iota $ turns the unimodular lattice $H_1(B,\\mathbb {Z}) = \\Lambda $ into self-adjoint and proper $\\mathcal {O}_D$ -module." ], [ "Moduli of Abelian varieties with real multiplication.", "Now let $A_2$ be the moduli space of principally polarized Abelian surfaces and set: $ A_2(\\mathcal {O}_D) = \\left\\lbrace (B,\\iota ) :\\begin{array}{c}B \\in A_2 \\mbox{ and } \\iota : \\mathcal {O}_D \\rightarrow \\operatorname{End}(B) \\\\\\mbox{ is proper and self-adjoint}\\end{array}\\right\\rbrace / \\sim .", "$ Here, two pairs $(B_1,\\iota _1)$ and $(B_2,\\iota _2)$ are equivalent, and we write $(B_1,\\iota _1) \\sim (B_2,\\iota _2)$ , if there is a polarization preserving isomorphism $B_1 \\rightarrow B_2$ intertwining $\\iota _i$ .", "As we now describe and following [20] §4 (see also [4] Chapter 9), the Hilbert modular surface $X_D$ parametrizes $A_2(\\mathcal {O}_D)$ and presents $A_2(\\mathcal {O}_D)$ as a complex orbifold.", "For $\\tau =(\\tau _+,\\tau _-) \\in \\mathbb {H}\\times \\mathbb {H}$ , define $\\phi _\\tau \\colon \\,\\Lambda _D \\rightarrow \\mathbb {C}^2$ by: $ \\phi _\\tau (x,y)= (\\sigma _+(x+y\\tau ),\\sigma _-(x+y\\tau )).", "$ The image $\\phi _\\tau (\\Lambda _D)$ is a lattice, and the complex torus $B_\\tau = \\mathbb {C}^2/\\phi _\\tau (\\Lambda _D)$ is principally polarized by the symplectic form on $\\Lambda _D$ .", "For each $x \\in \\mathcal {O}_D$ , the matrix $\\left({\\begin{matrix} \\sigma _+(x) & 0 \\\\ 0 & \\sigma _-(x) \\end{matrix}}\\right)$ preserves the lattice $\\phi _\\tau (\\Lambda _D)$ giving real multiplication by $\\mathcal {O}_D$ on $B_\\tau $ : $ \\iota _\\tau \\colon \\,\\mathcal {O}_D \\rightarrow \\operatorname{End}(B_\\tau ).", "$ For $A \\in \\operatorname{SL}(\\Lambda _D)$ , the embeddings $\\phi _{\\tau A }$ and $\\phi _\\tau $ are related by $\\phi _{ \\tau A} = C(A) \\circ \\phi _\\tau \\circ A$ where $C(A) = \\left({\\begin{matrix} \\sigma _+(a+c\\tau ) & 0 \\\\ 0 & \\sigma _-(a+c\\tau ) \\end{matrix}}\\right)$ .", "From this we see that there is a polarization preserving isomorphism between $B_\\tau $ and $B_{ \\tau A}$ that intertwines $\\iota _{\\tau }$ and $\\iota _{\\tau A}$ and that the correspondence $\\tau \\rightarrow (B_\\tau ,\\iota _\\tau )$ descends to a map $X_D \\rightarrow A_2(\\mathcal {O}_D)$ .", "This map is in fact a bijection and presents $A_2(\\mathcal {O}_D)$ as complex orbifold." ], [ "Complex multiplication.", "Now let $\\mathcal {O}$ be a degree two, totally imaginary extension of $\\mathcal {O}_D$ .", "We will say that $B \\in A_2$ admits complex multiplication by $\\mathcal {O}$ if there is a proper and Hermitian-adjoint homomorphism: $ \\iota \\colon \\,\\mathcal {O}\\rightarrow \\operatorname{End}(B).", "$ Here, Hermitian-adjoint means that the symplectic dual of $\\iota (x)$ acting on $H_1(B,\\mathbb {Z})$ is $\\iota (\\bar{x})$ where $\\bar{x}$ is the complex conjugate of $x$ and proper, as usual, means that $\\iota $ does not extend to a larger ring in $\\mathcal {O}\\otimes \\mathbb {Q}$ .", "For a one dimensional Abelian variety $E = \\mathbb {C}/\\Lambda $ in $A_1$ , we will say $E$ has complex multiplication by $\\mathcal {O}_C$ if $\\operatorname{End}(E)$ is isomorphic to $\\mathcal {O}_C$ .", "The ideal class group $H(C)$ is the set of invertible $\\mathcal {O}_C$ -ideals modulo principal ideals and is well known to be in bijection with the set of $E \\in A_1$ with $\\operatorname{End}(A) \\cong \\mathcal {O}_C$ .", "Since $\\mathcal {O}_C$ is quadratic, the invertible $\\mathcal {O}_C$ -ideals coincide with the proper $\\mathcal {O}_C$ -submodules of $\\mathcal {O}_C$ .", "The class number $h(C)$ is the order of the ideal class group $H(C)$ .", "We are now ready to determine which Jacobians of $D_8$ -surfaces have complex multiplication.", "Proposition 3.2 Fix $\\tau \\in \\mathbb {H}$ .", "The Abelian variety $A_\\tau $ has complex multiplication if and only if $\\tau $ is imaginary quadratic.", "First suppose $\\tau $ is imaginary quadratic.", "The vector space $\\Lambda _\\tau \\otimes \\mathbb {Q}$ is stabilized by the matrices $\\left({\\begin{matrix} 0 & 1 \\\\ -1 & 0 \\end{matrix}}\\right)$ and $\\left({\\begin{matrix} \\tau & 0 \\\\ 0 & \\tau \\end{matrix}}\\right)$ which together generate a Hermitian adjoint embedding $\\iota \\colon \\,\\mathbb {Q}(\\tau ,i) \\rightarrow \\operatorname{End}(A_\\tau ) \\otimes \\mathbb {Q}$ .", "The restriction of $\\iota $ to the order $\\mathcal {O}= \\iota ^{-1}(\\operatorname{End}(A_\\tau ))$ is complex multiplication by $\\mathcal {O}$ on $A_\\tau $ .", "Now suppose $\\tau $ is not imaginary quadratic.", "We have seen that there is a degree four, surjective holomorphic map $f \\colon \\,A_\\tau \\rightarrow E_\\tau \\times E_\\tau $ .", "As is well known and is implied by the stronger Proposition REF , such a map gives rise to an isomorphism between the rational endomorphism ring $\\operatorname{End}(A_\\tau ) \\otimes \\mathbb {Q}$ and $\\operatorname{End}(E_\\tau \\times E_\\tau ) \\otimes \\mathbb {Q}= M_2(\\mathbb {Q})$ .", "As a consequence, any commutative ring in $\\operatorname{End}(A_\\tau ) \\otimes \\mathbb {Q}$ has rank at most two over $\\mathbb {Q}$ and $A_\\tau $ does not have complex multiplication by any order.", "We have now proved all the claims in our Theorem from § about $D_8$ -surfaces: [Proof of Theorem REF ] The claims relating $X_\\tau $ , $K_{a(\\tau )}$ and $A_\\tau $ for $\\tau \\in U$ are established in Proposition REF .", "The characterization of when $\\operatorname{Jac}(X_\\tau )$ has complex multiplication is established in Proposition REF ." ], [ "Jacobians of $D_{12}$ -surfaces with complex multiplication.", "In Appendix we similarly define for $\\tau \\in \\mathbb {H}$ an Abelian variety $\\widetilde{A}_\\tau = \\mathbb {C}^2 / \\widetilde{\\Lambda }_\\tau $ which, for most $\\tau $ , is the Jacobian of $D_{12}$ -surface.", "A nearly identical argument shows that, for $(X,\\rho ) \\in \\mathcal {M}_2(D_{12})$ , the Jacobian $\\operatorname{Jac}(X)$ has complex multiplication if and only if $\\operatorname{Jac}(X)$ has complex multiplication by an order extending $\\rho (Z)$ , which happens if and only if $\\operatorname{Jac}(X) \\cong \\widetilde{A}_\\tau $ with $\\tau $ imaginary quadratic." ], [ "Orbifold points on Hilbert modular surfaces.", "We are now ready to study the orbifold points on $X_D$ .", "For $\\tau \\in \\mathbb {H}\\times \\mathbb {H}$ , we define the orbifold order of $[\\tau ]$ in $X_D$ to be the order of the group $\\operatorname{Stab}(\\tau ) \\subset \\operatorname{PSL}(\\Lambda _D)$ .", "We will call $[\\tau ] \\in X_D$ an orbifold point if the orbifold order of $[\\tau ]$ is greater than one.", "The following proposition gives an initial characterization of the Abelian varieties labeled by such points: Proposition 3.3 Fix $\\tau \\in \\mathbb {H}\\times \\mathbb {H}$ and an integer $n > 2$ .", "The following are equivalent: The point $\\tau $ is fixed by an $A \\in \\operatorname{SL}_2(\\Lambda _D)$ of order $n$ .", "There is an automorphism $\\phi \\in \\operatorname{Aut}(B_\\tau )$ of order $n$ that commutes with $\\iota _\\tau (\\mathcal {O}_D)$ .", "The homomorphism $\\iota _\\tau : \\mathcal {O}_D \\rightarrow \\operatorname{End}(B_\\tau )$ extends to complex multiplication by an order containing $\\mathcal {O}_D[\\zeta _n]$ where $\\zeta _n$ is a primitive $n$ th root of unity.", "First suppose $(1)$ holds with $A = \\left({\\begin{matrix} a & b \\\\ c & d \\end{matrix}}\\right)$ and $\\tau = \\tau A$ .", "We have seen that $\\phi _\\tau \\circ A$ and $\\phi _{\\tau A} = \\phi _\\tau $ differ by multiplication by the matrix $C(A)=\\left({\\begin{matrix} \\sigma _+(a+c\\tau ) & 0 \\\\ 0 & \\sigma _-(a+c\\tau ) \\end{matrix}}\\right)$ .", "It follows that $C(A)$ restricts to a symplectic automorphism of $\\phi _\\tau (\\Lambda _D)$ , giving rise to an automorphism $\\phi \\in B_\\tau $ of order $n$ which commutes with $\\iota _\\tau $ since $A$ is $\\mathcal {O}_D$ -linear.", "Now suppose $(2)$ holds.", "The homomorphism $\\iota _\\tau : \\mathcal {O}_D \\rightarrow \\operatorname{End}(B_\\tau )$ extends to $\\mathcal {O}_D[\\zeta _n]$ via $\\iota _\\tau (\\zeta _n) = \\phi $ and this extension is Hermitian-adjoint since $\\phi $ is symplectic.", "Finally, if $(3)$ holds, then the automorphism $\\iota _\\tau (\\zeta _n)$ restricts to an $\\mathcal {O}_D$ -module automorphism of $H_1(B_\\tau ,\\mathbb {Z}) = \\Lambda _D$ , giving a matrix $A \\in \\operatorname{SL}_2(\\mathcal {O}_D \\oplus \\mathcal {O}_D^\\vee )$ of order $n$ and fixing $\\tau $ .", "We can now show that the $D_8$ - and $D_{12}$ -surfaces with complex multiplication give orbifold points on Hilbert modular surfaces: Proposition 3.4 The Jacobians of $D_8$ - and $D_{12}$ -surfaces with complex multiplication are labeled by orbifold points in $\\bigcup _{D} X_D$ .", "We will show that $D_8$ Jacobians with complex multiplication are labeled by orbifold points in $\\bigcup _{D} X_D$ .", "A nearly identical argument shows the same is true of $D_{12}$ Jacobians with complex multiplication.", "By Proposition REF , any such Jacobian is isomorphic to $A_\\tau $ for some imaginary quadratic $\\tau $ .", "The order $\\mathcal {O}$ constructed in the proof of Proposition REF contains $i$ since $\\iota (i)$ is integral on $A_\\tau $ , and $(A_\\tau ,\\iota |_{\\mathcal {O}\\cap \\mathbb {R}})$ clearly satisfies condition (3) of Proposition REF .", "We conclude this section by showing most of the orbifold points on $\\bigcup _D X_D$ label Jacobians of $D_8$ - and $D_{12}$ -surfaces.", "Proposition 3.5 Fix an orbifold point $[\\tau ] \\in X_D$ .", "At least one of the following holds: $B_\\tau $ is a product of elliptic curves; $[\\tau ]$ is a point of orbifold order five on $X_5$ ; $B_\\tau $ is the Jacobian of a $D_8$ -surface with complex multiplication; or $B_\\tau $ is the Jacobian of a $D_{12}$ -surface with complex multiplication.", "By Proposition REF , the Abelian variety $B_\\tau $ labeled by $[\\tau ]$ has a symplectic automorphism $\\phi $ of order greater than two and commuting with $\\iota _\\tau $ .", "It is well known that every principally polarized two dimensional Abelian variety is either a polarized product of elliptic curves of the Jacobian of a smooth genus two Riemann surface, and that the automorphism group of a genus two Riemann surface is isomorphic to the automorphism group of its Jacobian (cf.", "[4], Chapter 11).", "If $B_\\tau $ is not a product of elliptic curves, choose $X \\in \\mathcal {M}_2$ so $\\operatorname{Jac}(X)$ is isomorphic to $B_\\tau $ and choose $\\phi _0 \\in \\operatorname{Aut}(X)$ so that an isomorphism $\\operatorname{Jac}(X) \\rightarrow B_\\tau $ intertwines $\\phi $ and $\\phi _0$ .", "From Table REF , we see that $[\\phi _0|_{X^W}]$ is in one of $[1,5]$ , $[1,1,2,2]$ , $[2,4]$ , $[1,1,4]$ , $[3,3]$ or $[6]$ .", "If $[\\phi _0|_{X^W}]=[1,5]$ , $X$ is the unique genus two surface with an order five automorphism, and $[\\tau ]$ is one of the points of order five on $X_5$ .", "If $[\\phi _0|_{X^W}] = [1,1,2,2]$ , $[2,4]$ , or $[1,1,4]$ , $B_\\tau $ is the Jacobian of $D_8$ -surface and has complex multiplication by Proposition REF .", "In the remaining cases, $B_\\tau $ is the Jacobian of $D_{12}$ -surface and, again, has complex multiplication by Proposition REF .", "We have now proved the characterization of the orbifold points on $\\bigcup _D X_D$ in Theorem REF : [Proof of Theorem REF ] By Proposition REF , the Jacobians of $D_8$ - and $D_{12}$ -surfaces with complex multiplication give orbifold points on Hilbert modular surfaces.", "By Proposition REF , these points give all of the orbifold points on $X_D$ except those which are products of elliptic curves and the points of order five on $X_5$ ." ], [ "Orbifold points on Weierstrass curves", "In this section we study the orbifold points on the Weierstrass curve $W_D$ .", "We start by recalling the definition of $W_D$ .", "An easy consequence of our classification of the orbifold points on $X_D$ in Section gives the characterization of orbifold points on $\\bigcup _D W_D$ of Corollary REF .", "We then establish the formula in Theorem REF by sorting the $D_8$ -surfaces with complex multiplication by the order for real multiplication commuting with $\\rho (J)$ .", "To do so, we relate the order for real multiplication on $(X,\\rho )$ to the order for complex multiplication on $X/\\rho (r)$ .", "Finally, we conclude this section by giving a simple method for enumerating the $\\tau \\in U$ corresponding to orbifold points on $W_D$ ." ], [ "Eigenforms for real multiplication.", "For a principally polarized Abelian variety $B$ with real multiplication $\\iota \\colon \\,\\mathcal {O}_D \\rightarrow \\operatorname{End}(B)$ , a place $\\sigma _0 \\colon \\,\\mathcal {O}_D \\rightarrow \\mathbb {R}$ distinguishes a line of $\\sigma _0$ -eigenforms on $B$ satisfying $\\iota (x)^* \\omega = \\sigma _0(x)\\omega $ for each $x \\in \\mathcal {O}_D$ .", "When $(B,\\iota ) = (B_\\tau =\\mathbb {C}^2 / \\Lambda _\\tau ,\\iota _\\tau )$ , the $\\sigma _+$ eigenforms are the multiples of $dz_1$ and the $\\sigma _-$ -eigenforms are the multiples of $dz_2$ where $z_i$ is the $i$ th-coordinate on $\\mathbb {C}^2$ .", "For a Riemann surface $X \\in \\mathcal {M}_2$ , the Abel-Jacobi map $X \\rightarrow \\operatorname{Jac}(X)$ induces an isomorphism on the space of holomorphic one forms.", "For Jacobians that admit real multiplication, a choice of real multiplication $\\iota $ on $\\operatorname{Jac}(X)$ distinguishes $\\sigma _+$ - and $\\sigma _-$ -eigenforms on $X$ .", "Conversely, a one-form up to scale $[\\omega ]$ on $X$ that happens to be stabilized by real multiplication by $\\mathcal {O}_D$ on $\\operatorname{Jac}(X)$ , there is a unique embedding $\\iota ^{[\\omega ]}_+ \\colon \\,\\mathcal {O}_D \\rightarrow \\operatorname{End}(\\operatorname{Jac}(X))$ characterized by the requirement that $\\iota ^{[\\omega ]}_+(x) \\omega = \\sigma _+(x)\\omega $ (cf.", "[20] §4)." ], [ "The Weierstrass curve.", "The Weierstrass curve $W_D$ of discriminant $D$ is the moduli space: $ W_D = \\left\\lbrace (X,[\\omega ]) : \\begin{array}{c} X \\in \\mathcal {M}_2 \\mbox{ and $\\omega $ is an eigenform for real }\\\\\\mbox{ multiplication by $\\mathcal {O}_D$ with double zero }\\end{array}\\right\\rbrace / \\sim .", "$ Here, $[\\omega ]$ is a one-form up to scale, and $(X_1,[\\omega _1])$ is equivalent to $(X_2,[\\omega _2])$ and we write $(X_1,[\\omega _1]) \\sim (X_2,[\\omega _2])$ if there is an isomorphism $\\phi \\colon \\,X_1 \\rightarrow X_2$ with $\\phi ^* \\omega _1 \\in \\mathbb {C}^* \\omega _2$ .", "The map $(X,[\\omega ]) \\mapsto (\\operatorname{Jac}(X),\\iota ^{[\\omega ]}_+)$ embeds $W_D$ in the Hilbert modular surface $X_D$ .", "The natural immersion $W_D \\rightarrow \\mathcal {M}_2$ is shown to be a finite union of Teichmüller curves in [17] (see also [9])." ], [ "Orbifold points on Weierstrass curves.", "The Weierstrass curve can be presented as a complex orbifold in several equivalent ways.", "One way is to use the $\\operatorname{SL}_2(\\mathbb {R})$ -action on the moduli space of holomorphic one forms and Veech groups as in [17].", "Another is to study the immersion $W_D \\rightarrow X_D$ and give $W_D$ the structure of a suborbifold as in [2].", "The details of these presentations are not important for our discussion; instead, we simply define the notion of orbifold order and orbifold point on $W_D$ .", "For a Riemann surface $X$ and non-zero holomorphic one form $\\omega \\in \\Omega (X)$ , let $\\operatorname{Aut}(X,\\left\\lbrace \\pm \\omega \\right\\rbrace )$ denote the subgroup of $\\operatorname{Aut}(X)$ consisting of $\\phi $ with $\\phi ^* \\omega = \\pm \\omega $ , let $\\operatorname{SO}(X,\\omega ) = \\operatorname{Aut}(X,[\\omega ])$ be the subgroup of $\\operatorname{Aut}(X)$ consisting of automorphisms $\\phi $ for which preserve $\\omega $ up to scale and set: $ \\operatorname{PSO}(X,\\omega ) = \\operatorname{SO}(X,\\omega )/\\operatorname{Aut}(X,\\left\\lbrace \\pm \\omega \\right\\rbrace ).", "$ The groups $\\operatorname{Aut}(X,\\left\\lbrace \\pm \\omega \\right\\rbrace )$ , $\\operatorname{SO}(X,\\omega )$ and $\\operatorname{PSO}(X,\\omega )$ only depend on $\\omega $ up to scale.", "For a point $(X,[\\omega ]) \\in W_D$ , we define the orbifold order of $(X,[\\omega ])$ to be the order of the group $\\operatorname{PSO}(X,\\omega )$ and we will call $(X,[\\omega ])$ an orbifold point if its orbifold order is greater than one.", "Using the characterization of orbifold points on $X_D$ in Proposition REF it is straightforward to check that $(X,[\\omega ]) \\in W_D$ is an orbifold point if and only if the pair $(\\operatorname{Jac}(X),\\iota ^{[\\omega ]}_+)$ is an orbifold point on $X_D$ ." ], [ "Orbifold points on $W_D$ and {{formula:ea26fb90-379d-4efd-9535-a78ac00b9fd7}} -surfaces with complex multiplication.", "Recall from Section that we defined a domain $U$ and associated to each $\\tau \\in U$ a polygonal pinwheel $P_\\tau $ so the surface $X_\\tau = P_\\tau /\\sim $ admits a faithful action of $D_8$ .", "Let $J_\\tau $ , as usual, denote the obvious order four automorphism of $X_\\tau $ obtained by counterclockwise rotation of $P_\\tau $ and let $\\omega _\\tau $ denote the eigenform for $J_\\tau $ obtained from $dz$ on $P_\\tau $ .", "Proposition 4.1 For $\\tau \\in U$ , the group $\\operatorname{PSO}(X_\\tau ,\\omega _\\tau )$ is cyclic of order two except when $\\tau = \\sqrt{-2}/2$ , in which case $\\operatorname{PSO}(X_\\tau ,\\omega _\\tau )$ is cyclic of order four.", "Recall from the proof of Proposition REF that $J_\\tau $ extends to a faithful action $\\rho \\colon \\,D_8 \\rightarrow \\operatorname{Aut}(X)$ with $\\rho (J)=J_\\tau $ .", "From the fact that $\\mathbb {C}(X_\\tau ) \\cong K_{a(\\tau )}$ , it is easy to verify that $\\rho $ is an isomorphism except when $(X,\\rho )$ is the fixed point of the outer automorphism $\\sigma $ , i.e.", "$\\tau = \\sqrt{-2}/2$ .", "For $\\tau \\ne \\sqrt{-2}/2$ , the group $\\operatorname{SO}(X_\\tau ,\\omega _\\tau )$ is generated by $J_\\tau $ , $\\operatorname{Aut}(X_\\tau ,\\left\\lbrace \\pm \\omega _\\tau \\right\\rbrace )$ is generated by the hyperelliptic involution $J_\\tau ^2$ and $\\operatorname{PSO}(X,\\omega _\\tau )$ is cyclic of order two.", "When $\\tau = \\sqrt{-2}/2$ , $P_\\tau $ is the regular octagon and it is easy to verify that $\\operatorname{PSO}(X_\\tau ,\\omega _\\tau )$ is cyclic of order four.", "We now show that the $D_8$ -surfaces with complex multiplication give orbifold points on Weierstrass curves: Proposition 4.2 If $\\tau \\in U$ is imaginary quadratic, then there is a discriminant $D>0$ so the surface with one form up to scale $(X_\\tau ,[\\omega _\\tau ])$ is an orbifold point on $W_D$ .", "Fix an imaginary quadratic $\\tau \\in U$ .", "As we saw in the proof of Proposition REF , the order four automorphism $J_\\tau $ on $X_\\tau $ , extends to complex multiplication by an order $\\mathcal {O}$ in $\\mathbb {Q}(\\tau ,i)$ .", "If $D$ is the discriminant of $\\mathcal {O}\\cap \\mathbb {R}$ , we see that $\\omega _\\tau $ is an eigenform for real multiplication by $\\mathcal {O}_D$ .", "Also, $\\omega _\\tau $ has a double zero, as can be seen directly by counting cone-angle around vertices of $P_\\tau $ or can be deduced from the fact that $\\omega _\\tau $ is stabilized by the order four automorphism $J_\\tau $ (cf.", "Proposition REF ).", "From this we see that $(X_\\tau ,[\\omega _\\tau ]) \\in W_D$ and is an orbifold point on $W_D$ since $\\operatorname{PSO}(X_\\tau ,\\omega _\\tau )$ has order at least two.", "We can now show that almost all of the orbifold points on $\\bigcup _D W_D$ are $D_8$ -surfaces with complex multiplication: Proposition 4.3 Suppose $(X,[\\omega ]) \\in W_D$ is an orbifold point.", "One of the following holds: $(X,[\\omega ])$ is the point of orbifold order five on $W_5$ , $(X,[\\omega ])$ is the point of orbifold order four on $W_8$ , or $(X,[\\omega ]) = (X_\\tau ,[\\omega _\\tau ])$ for some imaginary quadratic $\\tau \\in U$ with $\\tau \\ne \\sqrt{-2}/2$ .", "In particular, for $D> 8$ , all of the orbifold points on $W_D$ have orbifold order two.", "By our definition of orbifold point, $X$ has an automorphism $\\phi $ stabilizing $\\omega $ up to scale and with $\\phi ^* \\omega $ not equal to $\\pm \\omega $ .", "Such a $\\phi $ must fix the zero of $\\omega $ which is a Weierstrass point and the conjugacy class $[\\phi |_{X^W}]$ is one of $[1,5]$ , $[1,1,4]$ or $[1,1,2,2]$ .", "In the first case, $(X,[\\omega ])$ is the point of order five on $W_5$ .", "In the remaining cases, $(X,[\\omega ]) = (X_\\tau ,[\\omega _\\tau ])$ for some imaginary quadratic $\\tau \\in U$ .", "When $\\tau =\\sqrt{-2}/2$ and $(X,[\\omega ])$ is the point of orbifold order four on $W_8$ obtained from the regular octagon and when $\\tau \\ne \\sqrt{-2}/2$ , $(X,[\\omega ])$ is a point of orbifold order two for some larger discriminant.", "We have now proved the claims in Corollary REF : [Proof of Corollary REF ] By Proposition REF , the $D_8$ -surfaces with complex multiplication give orbifold points on $\\bigcup _D W_D$ .", "By Proposition REF , they give all of the orbifold points except for the point of order five on $W_5$ ." ], [ "Isogeny and endomorphism.", "In light of Proposition REF , to give a formula for the number $e_2(W_D)$ of points of orbifold order two on $W_D$ , we need to sort the $D_8$ -surfaces $(X,\\rho )$ with complex multiplication by order for real multiplication commuting with $\\rho (J)$ .", "To do so, we relate this order to the order for complex multiplication on $E_\\rho = X/\\rho (r)$ .", "We saw in Section that there is an isogeny $\\operatorname{Jac}(X) \\rightarrow E_\\rho \\times E_\\rho $ such that the polarization on $\\operatorname{Jac}(X)$ is twice the pullback of the product polarization on $E_\\rho \\times E_\\rho $ .", "The following proposition will allow us to embed the endomorphism ring of $\\operatorname{Jac}(X)$ in the rational endomorphism ring $E_\\rho \\times E_\\rho $ : Proposition 4.4 Suppose $f \\colon \\,A \\rightarrow B$ is an isogeny between principally polarized Abelian varieties with the property the polarization on $A$ is $n$ -times the polarization pulled back from $B$ .", "The ring $\\operatorname{End}(B)$ is isomorphic as an involutive algebra to the subring $R_f \\subset \\operatorname{End}(A) \\otimes \\mathbb {Q}$ given by: $ R_f = \\frac{1}{n} \\left\\lbrace \\phi \\in \\operatorname{End}(A) : \\phi \\left( \\ker (n f) \\right) \\subset \\ker (f) \\right\\rbrace .", "$ Let $f^*\\colon \\,B \\rightarrow A$ denote the isogeny dual to $f$ .", "The condition on the polarization implies that $f \\circ f^*$ and $f^* \\circ f$ are the multiplication by $n$ -maps on $B$ and $A$ respectively.", "We will show that the map: $\\psi \\colon \\,\\operatorname{End}(B) &\\rightarrow \\operatorname{End}(A) \\otimes \\mathbb {Q}\\\\\\phi &\\mapsto \\frac{1}{n} f^* \\circ \\phi \\circ f$ is an injective homomorphism and has image $\\psi (\\operatorname{End}(B)) = R_f$ .", "The map $\\psi $ is easily checked to be a homomorphism and is injective since rationally it is an isomorphism, with inverse is given by $\\psi ^{-1}(\\phi ) = \\frac{1}{n} f \\circ \\phi \\circ f^*$ .", "For an integer $k > 0$ , let $B[k]$ denote the $k$ -torsion on $B$ .", "The image of $\\operatorname{End}(B)$ is contained in $R_f$ since: $ \\ker (n f) \\xrightarrow{} B[n] \\xrightarrow{} B[n] \\xrightarrow{} \\ker (f) $ for any $\\phi \\in \\operatorname{End}(B)$ .", "To see that the image of $\\psi $ is all of $R_f$ , fix $\\phi _0 \\in \\operatorname{End}(A) \\otimes \\mathbb {Q}$ satisfying $\\phi _0(\\ker (nf)) \\subset \\ker (f)$ .", "The endomorphism $\\phi = \\frac{1}{n^2} f \\circ \\phi _0 \\circ f^*$ is integral on $B$ since: $ B[n^2] \\xrightarrow{} \\ker (nf) \\xrightarrow{} \\ker (f) \\xrightarrow{} 0.", "$ Since $\\psi (\\phi ) = \\phi _0$ , the image of $\\psi $ is all of $R_f$ .", "The isomorphism between $\\operatorname{End}(B)$ and $R_f$ has $\\psi (\\phi )^* = \\psi (\\phi ^*)$ since $(f^* \\phi f)^* = f^* \\phi ^* f$ ." ], [ "Invertible modules over finite rings.", "From Proposition REF and the fact that $\\operatorname{Jac}(X)$ admits a degree four isogeny to $E_\\rho \\times E_\\rho $ with $E_\\rho =X/\\rho (r)$ , we see that the discriminant of the order for real multiplication on $\\operatorname{Jac}(X)$ commuting with $\\rho (J)$ is, up to factors of two, equal to $-C$ where $C$ is the discriminant of $\\operatorname{End}(E_\\rho )$ .", "To determine the actual order for real multiplication, we first need to determine the possibilities for $E_\\rho [4]$ as an $\\mathcal {O}_C$ -module.", "The following proposition is well known: Proposition 4.5 Let $\\mathcal {O}$ be an imaginary quadratic order and $I$ be an $\\mathcal {O}$ -ideal which is a proper $\\mathcal {O}$ -module.", "The module $I/nI$ is isomorphic to $\\mathcal {O}/ n \\mathcal {O}$ as $\\mathcal {O}$ -modules.", "For maximal orders, see Proposition 1.4 in [26].", "For non-maximal orders, Proposition REF follows from the fact that $\\mathcal {O}$ -ideals which are proper $\\mathcal {O}$ -modules are invertible ([13], §8.1) and standard commutative algebra." ], [ "Formula for number and type of orbifold points on $W_D$ .", "We are now ready to prove our main theorem giving a formula for the number and type of orbifold points on $W_D$ : [Proof of Theorem REF ] We have already seen that, for discriminants $D>8$ , all of the orbifold points on $W_D$ have orbifold order two.", "It remains to establish the formula for the number of such points.", "Recall from Section that we constructed a holomorphic map $g: \\mathcal {M}_2(D_8) \\rightarrow Y_0(2)$ given by $g(X,\\rho ) = (E_\\rho ,Z_\\rho ,T_\\rho )$ which is an isomorphism onto the complement of the point of orbifold order two $(1+i)/2 \\cdot \\Gamma _0(2)$ .", "For each $E \\in \\mathcal {M}_1$ without automorphisms other than the elliptic involution, there are exactly three $D_8$ -surfaces $(X,\\rho )$ with $X/\\rho (r)$ isomorphic to $E$ .", "Now fix an imaginary quadratic discriminant $C< 0$ .", "Setting $E = \\mathbb {C}/ \\mathcal {O}_C$ , $Z = 0 + \\mathcal {O}_C$ , $T_1 = 1/2+\\mathcal {O}_C$ , $T_2 = (C+\\sqrt{C})/4 + \\mathcal {O}_C$ and $T_3 = T_1+T_2$ , the three $D_8$ -surfaces covering $E$ are the three surfaces $(X_i,\\rho _i) = g^{-1}(E,Z,T_i)$ .", "Using Proposition REF , it is straightforward to calculate the order for real multiplication on $\\operatorname{Jac}(X_i)$ commuting with $\\rho _i(J)$ and we do so in Table REF .", "Table: The elliptic curve E=ℂ/𝒪 C E = \\mathbb {C}/\\mathcal {O}_C is covered by three D 8 D_8-surfaces (X i ,ρ i )(X_i,\\rho _i).", "The discriminant D i D_i of the order for real multiplication on Jac(X i )\\operatorname{Jac}(X_i) commuting with JJ is computed using Proposition .For an arbitrary $E$ with $\\operatorname{End}(E) \\cong \\mathcal {O}_C$ , we have that $E[4] \\cong \\mathcal {O}_C/4\\mathcal {O}_C$ as $\\mathcal {O}_C$ -modules by Proposition REF and the orders for real multiplication commuting with $\\rho (J)$ on the $D_8$ -surfaces covering $E$ are the same as the orders when $E = \\mathbb {C}/ \\mathcal {O}_C$ .", "The formula for $e_2(W_D)$ follows easily from this and the fact that there are precisely $h(\\mathcal {O}_C)$ genus one surfaces with $\\operatorname{End}(E) \\cong \\mathcal {O}_C$ .", "The small discriminants where the $D_8$ -surfaces labeled by orbifold points on $W_D$ cover genus one surfaces with automorphisms and handled by the restriction $D>8$ and by replacing $h(\\mathcal {O}_C)$ with the reduced class number $\\widetilde{h} (\\mathcal {O}_C)$ .", "Note that the factor of two in the formula for $e_2(W_D)$ comes from the fact that the $D_8$ -surfaces $(X,\\rho )$ and $\\sigma \\cdot (X,\\rho )$ have isomorphic $J$ -eigenforms and therefore label the same orbifold point on $W_D$ ." ], [ "Enumerating orbifold points on $W_D$ .", "We conclude this section by giving a simple method for enumerating the $\\tau \\in U$ for which $(X_\\tau ,[\\omega _\\tau ]) \\in W_D$ .", "Fix a discriminant $D>0$ and a $\\tau \\in \\mathbb {Q}(\\sqrt{-D})$ .", "We can choose integers $e$ , $k$ , $b$ and $c$ so $k^2 D = -e^2+2bc$ and $\\tau = (e+k\\sqrt{-D})/(2c)$ .", "The vectors $v_1=\\left({\\begin{matrix} 1\\\\1 \\end{matrix}}\\right)$ , $v_2 = \\left({\\begin{matrix} 1 \\\\-1 \\end{matrix}}\\right)$ , $v_3=\\left({\\begin{matrix} \\tau \\\\ \\tau +1 \\end{matrix}}\\right)$ and $v_4=\\left({\\begin{matrix} \\tau \\\\ -\\tau -1 \\end{matrix}}\\right)$ generate the lattice $\\Lambda _\\tau $ .", "The lattice $\\Lambda _\\tau $ is preserved by multiplication by $\\left({\\begin{matrix} 0 & 1 \\\\ -1 & 0 \\end{matrix}}\\right)$ giving an automorphism $\\phi \\in \\operatorname{Aut}(A_\\tau )$ , and the vector space $\\Lambda _\\tau \\otimes \\mathbb {Q}$ is preserved by the matrix $\\left({\\begin{matrix} \\sqrt{-D} & 0 \\\\ 0 & \\sqrt{-D} \\end{matrix}}\\right)$ giving a rational endomorphism $T \\in \\operatorname{End}(A_\\tau \\otimes \\mathbb {Q})$ .", "Together, $\\phi $ and $T$ generate a Hermitian-adjoint homomorphism $\\iota : \\mathbb {Q}(\\sqrt{-D},i) \\rightarrow \\operatorname{End}(A_\\tau ) \\otimes \\mathbb {Q}$ .", "We want to give conditions on $e$ , $k$ , $b$ and $c$ so that the quadratic ring of discriminant $D$ in $\\mathbb {Q}(\\sqrt{-D},i)$ acts integrally on $\\Lambda _\\tau $ .", "This ring is generated by $S=(D+\\phi T)/2$ and it is straightforward to check that, in the basis $v_1,\\dots ,v_4$ for $\\Lambda _\\tau \\otimes \\mathbb {Q}$ , $S$ acts by multiplication by the matrix: $S = \\frac{1}{2k} \\begin{pmatrix}Dk+c & -e-c & 0 & 2c \\\\c+e & Dk-c & -2c & 0\\\\0 & -e-b & Dk+c & c+e \\\\b+e & 0 & -e-c & Dk-c\\end{pmatrix}.$ From this we see that $S$ is integral, and $\\operatorname{Jac}(X_\\tau ) = \\mathbb {C}^2 /\\Lambda _\\tau $ has real multiplication by an order containing $\\mathcal {O}_D$ and commuting with $\\phi $ , if and only if $k$ divides $c$ , $b$ and $e$ and $D \\equiv e/k \\equiv b/k \\equiv c/k \\bmod 2$ ." ], [ "Pinwheel prototypes.", "Motivated by this, define the set of pinwheel prototypes of discriminant $D$, which we denote by $E_0(D)$ , to be the collection of triples $(e,c,b) \\in \\mathbb {Z}^3$ satisfying: $\\begin{array}{c}D = -e^2+2bc \\mbox{ with } D \\equiv e \\equiv c \\equiv b \\bmod 2 \\mbox{ and } \\left|e\\right| \\le c \\le b \\\\\\mbox{ and if } \\left|e\\right| = c \\mbox{ or } b = c \\mbox{ then } e \\le 0.\\end{array}$ Note that if $(e,c,b) \\in E_0(D)$ , then $(fe,fc,fb) \\in E_0(f^2D)$ .", "We define $E(D)$ to be the set of proper pinwheel prototypes in $E_0(D)$ , i.e.", "those which do not arise from smaller discriminants in this way.", "Finally define $\\tau (e,c,b) = (e+\\sqrt{-D})/2c$ .", "We can now prove the following proposition, which is a more precise version of Theorem REF : Proposition 4.6 Fix a discriminant $D$ and $(e,c,b) \\in E(D)$ .", "The one form up to scale $(X_\\tau ,[\\omega _\\tau ])$ where $\\tau =(e+\\sqrt{D})/(2c)$ is an orbifold point on $W_D$ and, for $D> 8$ , the set $E(D)$ is in bijection with the orbifold points on $W_D$ : $ e_2(W_D) = \\# E(D).", "$ By our discussion above, the first two conditions on pinwheel prototypes ($D = -e^2+2bc$ and $D \\equiv e \\equiv c \\equiv b \\bmod 2$ ) are equivalent to the requirement that $(X_\\tau ,[\\omega _\\tau ])$ where $\\tau = \\tau (e,c,b)$ is an eigenform for real multiplication by an order containing $\\mathcal {O}_D$ .", "The condition that $(e,c,b)$ is proper ensures that the order for real multiplication with eigenform $\\omega _\\tau $ is $\\mathcal {O}_D$ .", "The condition that $\\left|e\\right| \\le c \\le b$ is equivalent to the condition that $\\tau (e,c,b)$ is in the domain $U$ .", "The condition that $e \\le 0$ when $\\left|e\\right| = c$ or $b=c$ ensures that $\\operatorname{Re}(\\tau (e,c,b)) \\le 0$ whenever $\\tau (e,c,b)$ is in the boundary of $U$ ." ], [ "Bounds on $e_2(W_D)$ .", "It is easy to see that the conditions defining pinwheel prototypes ensure $c \\le \\sqrt{D}$ , from which it is easy to enumerate the prototypes in $E(D)$ and prove: Proposition 4.7 For discriminants $D>8$ , the number of points of orbifold order two satisfies: $ e_2(W_D) \\le D/2.", "$ The integers $e$ and $c$ determine $b$ since $b = (D+e^2)/(2c)$ .", "From $\\left|e\\right| \\le c \\le \\sqrt{D}$ and the fact that $e$ and $c$ are congruent to $D \\bmod 2$ , we have that $e$ ranges over at most $\\sqrt{D}$ possibilities and $c$ ranges over at most $\\sqrt{D}/2$ possibilities so $e_2(W_D) < D/2$ ." ], [ "Examples.", "The sets $E(D)$ for some small discriminants $D$ are: $ \\begin{array}{c}E(5) = \\left\\lbrace (-1,1,3) \\right\\rbrace , E(8) = \\left\\lbrace (0,2,2) \\right\\rbrace , E(9) = \\left\\lbrace (-1,1,5) \\right\\rbrace , E(12) = \\left\\lbrace (-2,2,4) \\right\\rbrace , \\\\E(13) = \\left\\lbrace (-1,1,7) \\right\\rbrace , E(16) = \\left\\lbrace (0,2,4) \\right\\rbrace , \\mbox{ and } E(17) = \\left\\lbrace (-1,1,9),(-1,3,3) \\right\\rbrace .\\end{array}$" ], [ "Orbifold points by spin", "By the results in [18], the Weierstrass curve $W_D$ is usually irreducible.", "For discriminants $D > 9$ with $D \\equiv 1 \\bmod 8$ , $W_D$ has exactly two irreducible components, and they are distinguished by a spin invariant.", "Throughout this section, we assume $D$ is a such a discriminant.", "For such $D$ , the number of points of orbifold order two is given by $ e_2(W_D) = \\frac{1}{2} \\widetilde{h}(-4D) $ and by our Proof of Theorem REF in Section , an orbifold point of order two $(X,[\\omega ]) \\in W_D$ is the $\\rho (J)$ -eigenform for a faithful $D_8$ -action $\\rho $ on $X$ and the quotient $E/\\rho (r)$ corresponds to an ideal class $[I] \\in H(\\mathcal {O}_{-4D})$ .", "In this section, we define a spin homomorphism $\\epsilon _0 \\colon \\,H(-4D) \\rightarrow \\mathbb {Z}/2\\mathbb {Z}$ and relate the spin of $(X,[\\omega ])$ to $\\epsilon _0([I])$ allowing us to establish the formula in Theorem REF ." ], [ "One forms with double zero and spin structures.", "We start by recalling spin structures on Riemann surfaces following [18] (cf.", "also [1]).", "A spin structure on a symplectic vector space $V$ of dimension $2g$ over $\\mathbb {Z}/2\\mathbb {Z}$ is a quadratic form: $ q \\colon \\,V \\rightarrow \\mathbb {Z}/2\\mathbb {Z}$ satisfying $q(x+y) = q(x)+q(y)+\\left<x,y\\right>$ where $\\left<,\\right>$ is the symplectic form.", "The parity of $q$ is given by the Arf invariant: $ \\operatorname{Arf}(q) = \\sum _i q(a_i)q(b_i) \\in \\mathbb {Z}/ 2 \\mathbb {Z}$ where $a_1,b_1,\\dots ,a_g,b_g$ is a symplectic basis for $V$ .", "A one form with double zero $\\omega $ on $X \\in \\mathcal {M}_2$ determines a spin structure on $H_1(X,\\mathbb {Z}/2\\mathbb {Z})$ as follows.", "For any loop $\\gamma \\colon \\,S^1 \\rightarrow X$ whose image avoids the zero of $\\omega $ , gives a Gauss map: $ G_{\\gamma } \\colon \\,S^1 \\rightarrow S^1 \\mbox{ where } G_\\gamma (x) = \\omega (\\gamma ^{\\prime }(x))/\\left|\\gamma ^{\\prime }(x)\\right|.$ The degree of $G_\\gamma $ is invariant under homotopy that avoids the zero of $\\omega $ and changes by a multiple of two under general homotopy.", "Denoting by $[\\gamma ]$ the class in $H_1(X,\\mathbb {Z}/2/\\mathbb {Z})$ represented by $\\gamma $ , the function $ q([\\gamma ]) = 1+\\deg (G_\\gamma ) \\bmod 2 $ defines a spin structure on $X$ ." ], [ "Pinwheel spin.", "Now consider the surface $X_\\tau $ obtained from the pinwheel $P_\\tau $ and the one form $\\omega _\\tau \\in \\Omega (X_\\tau )$ obtained from $dz$ on $P_\\tau $ as usual.", "Let $b = \\tau +\\frac{1-i}{2}$ and let $x_t \\in H_1(X_\\tau ,\\mathbb {Z}/2\\mathbb {Z})$ be the class represented by the integral homology class with period $t \\in \\mathbb {Z}[i] \\oplus b \\mathbb {Z}[i]$ when integrated against $\\omega _\\tau $ .", "The classes $x_1$ , $x_i$ , $x_b$ and $x_{bi}$ form a basis for $H_1(X_\\tau ,\\mathbb {Z}/ 2 \\mathbb {Z})$ , and we have: Proposition 5.1 The spin structure $q$ on $X_\\tau $ associated to $\\omega _\\tau $ has: $ q\\left( k_1 x_1 +k_2 x_i + k_3 x_b + k_4 x_{bi} \\right) = k_1^2 + k_2^2 + k_1 k_3 + k_2 k_4 + k_3 k_4 \\bmod 2.", "$ Finding loops $\\gamma _t \\colon \\,S^1 \\rightarrow X_\\tau $ representing $x_t$ and avoiding the zero of $\\omega _\\tau $ , we can compute the degree of the Gauss map directly and show that: $ q(x_1)=q(x_i) = 1 \\mbox{ and } q(x_b)=q(x_{bi})=0.", "$ From the relation $q(x+y)=q(x)+q(y)+\\left<x,y\\right>$ , any basis $x_n$ of $H_1(X_\\tau ,\\mathbb {Z}/2\\mathbb {Z})$ has: $ q\\left(\\sum _n k_n x_n\\right) = \\sum _n k_n^2 q(x_n) + \\sum _{l<n} k_l k_n \\left< x_l, x_n \\right>.", "$ Computing the intersection pairing on the basis $\\left\\lbrace x_1,x_i,x_b,x_{bi} \\right\\rbrace $ gives the stated formula for $q$ ." ], [ "Spin components of $W_D$ .", "For $(X,[\\omega ]) \\in W_D$ , the vector space $H_1(X,\\mathbb {Z}/2\\mathbb {Z})$ is also an $\\mathcal {O}_D$ -module via $\\iota _+^{[\\omega ]} \\colon \\,\\mathcal {O}_D \\rightarrow \\operatorname{End}(\\operatorname{Jac}(X))$ .", "Let $f$ be the conductor of $\\mathcal {O}_D$ , i.e.", "the integer satisfying $D = f^2 D_0$ where $D_0$ is the maximal order in $K_D$ .", "Since $D \\equiv 1 \\bmod 8$ , the image $W$ of $(f+\\sqrt{D})/2$ acting on $H_1(X,\\mathbb {Z}/2\\mathbb {Z})$ is two dimensional, and we define the spin of $(X,[\\omega ])$ to be the parity of $q$ restricted to $W$ : $ \\epsilon (X,[\\omega ]) = \\operatorname{Arf}(q|_W).", "$ We also define: $ W_D^i = \\left\\lbrace (X,[\\omega ]) \\in W_D : \\epsilon (X,[\\omega ]) = i \\right\\rbrace , \\mbox{ for } i = 0\\mbox{ or }1.$ As was shown in [18], for discriminants $D>9$ with $D \\equiv 1 \\bmod 8$ , the components $W_D^0$ and $W_D^1$ are both non-empty and irreducible." ], [ "Ideal classes.", "Now let $I \\subset K_{-4D}$ be a fractional and proper $\\mathcal {O}_{-4D}$ -ideal, i.e.", "satisfies $\\operatorname{End}(I) = \\mathcal {O}_{-4D}$ , and let $E_I = \\mathbb {C}/I$ .", "There are three $D_8$ -surfaces $(X,\\rho )$ with $X/\\rho (r) = E_I$ .", "From Table REF , we see that exactly one of these, which we will call $(X_I,\\rho _I)$ , is labeled by an orbifold point on $W_D$ , with the others being labeled by orbifold points on $W_{16 D}$ .", "This $D_8$ -surface satisfies $g(X_I,\\rho _I) = (E_I,Z,T)$ where $Z = 0 + I$ and $T$ generates a subgroup of $E[2]$ invariant under $\\mathcal {O}_{-4D}$ ." ], [ "Spin homomorphism.", "Now let $n$ be the odd integer satisfying $\\operatorname{Nm}(I) = 2^k n$ and define: $\\epsilon _0(I) = \\frac{n-1}{2} \\bmod 2.$ We will give a formula for the spin invariant of the orbifold point on $W_D$ corresponding to $(X_I,\\rho _I)$ in terms of $\\epsilon _0(I)$ .", "To start, we show: Proposition 5.2 The number $\\epsilon _0(I)$ depends only on the ideal class of $I$ and defines a spin homomorphism: $ \\epsilon _0 \\colon \\,H(-4D) \\rightarrow \\mathbb {Z}/2\\mathbb {Z}.", "$ The spin homomorphism $\\epsilon _0$ is the zero map if and only if $D$ is a square.", "Any $x \\in \\mathcal {O}_{-4D}$ has norm $\\operatorname{Nm}(x) = x_1^2 + x_2^2 D = 2^k l$ with $l \\equiv 1 \\bmod 4$ .", "If the ideals $I$ and $J$ are in the same ideal class, they satisfy $xI = yJ$ for some $x$ and $y$ in $\\mathcal {O}_{-4D}$ and $\\epsilon _0(I) = \\epsilon _0(J)$ .", "The map $\\epsilon _0$ is a homomorphism since the norm of ideals is a homomorphism.", "Now suppose $D=f^2$ is a square.", "Any $\\mathcal {O}_{-4D}$ -ideal class has a representative of the form $I=x\\mathbb {Z}\\oplus (fi-y)\\mathbb {Z}$ with $x$ and $y$ in $\\mathbb {Z}$ .", "Since $I$ is an ideal, $x$ divides $f^2+y^2$ and since $I$ is proper $\\gcd (x,y,(f^2+y^2)/x)=1$ .", "If an odd prime $p$ divides $\\operatorname{Nm}(I)$ , then $p$ divides $x^2$ , $f^2+y^2$ , and $y$ .", "Since $f^2 \\equiv -y^2 \\bmod p$ and $p$ does not divide both $f$ and $y$ , $-1$ is a square mod $p$ , $p \\equiv 1 \\bmod 4$ and $\\epsilon _0(I)=0$ .", "If $D$ is not a square, $D = p_1^{k_1} p_2^{k_2} \\dots p_n^{k_n}$ with $p_l$ distinct odd primes and $k_1$ odd.", "By Dirichlet's theorem, there is a prime $p$ with: $p \\equiv 3 \\bmod 4$ , $p \\equiv 1 \\bmod p_l$ for $l > 1$ , and $\\left(\\frac{p}{p_1} \\right) = -1$ .", "Quadratic reciprocity gives $\\left(\\frac{-D}{p} \\right) = 1$ and guarantees a solution $x$ to $x^2 \\equiv -D \\bmod p$ .", "The ideal $I=p\\mathbb {Z}\\oplus (\\sqrt{-D}-x)\\mathbb {Z}$ is an $\\mathcal {O}_{-4D}$ -ideal, has norm $p$ and $\\epsilon _0(I)=1$ .", "Remark When $D \\equiv 1 \\bmod 8$ , the ideal $(2)$ ramifies in $\\mathcal {O}_{-4D}$ and there is a prime ideal $P$ with $P^2=(2)$ .", "Since $\\operatorname{Nm}(P)=2$ , we have $\\epsilon _0(P)=0$ .", "The ideal classes represented by $I = \\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ and $J=\\mathbb {Z}\\oplus -1/2\\tau \\mathbb {Z}$ satisfy $[I]=[P J]$ .", "This is related to the fact the polygonal pinwheels $P_\\tau $ and $P_{-1/2\\tau }$ give the same point on $W_D$ and so must have the same spin invariant.", "Proposition 5.3 Fix a $\\tau \\in U$ with $(X_\\tau ,[\\omega _\\tau ]) \\in W_D$ and let $I = \\mathbb {Z}\\oplus \\tau \\mathbb {Z}$ .", "The spin of $(X_\\tau ,[\\omega _\\tau ])$ is given by the formula: $ \\epsilon (X_\\tau ,[\\omega _\\tau ]) = \\frac{f+1}{2} +\\epsilon _0([I]) \\bmod 2.", "$ We saw in Section that $X_\\tau $ has a faithful $D_8$ -action $\\rho _\\tau $ with $g(X_\\tau ,\\rho _\\tau ) = (E_I=\\mathbb {C}/I,Z=0+I,T=1/2+I)$ .", "By our proof of Theorem REF in Section , we have that $\\operatorname{End}(E_I) = \\mathcal {O}_{-4D}$ and $T$ generates a subgroup of $E_I[2]$ invariant under $\\mathcal {O}_{-4D}$ , i.e.", "$1+\\sqrt{-D} \\in 2 I$ .", "Since $I$ is an ideal, $\\sqrt{-D} = x \\tau + y$ for some $x$ and $y \\in \\mathbb {Z}$ , $x$ divides $D+y^2$ and $I$ has the same class as $I_0=x\\mathbb {Z}\\oplus (\\sqrt{-D}-y)\\mathbb {Z}$ .", "Since $I$ is proper, $\\gcd (x,y,(D+y^2)/x)=1$ and the norm of $I_0$ is $x$ up to a factor of two.", "The condition $1 + \\sqrt{-D} \\in 2I$ implies that $x \\equiv 2 \\bmod 4$ and $\\epsilon _0(I) \\equiv \\frac{x-2}{4} \\bmod 2$ .", "To compute the spin invariant $\\epsilon (X_\\tau ,[\\omega _\\tau ])$ , we need to determine the subspace $W = \\operatorname{Im}\\left(\\frac{f+\\sqrt{D}}{2}\\right)$ of $H_1(X,\\mathbb {Z}/2\\mathbb {Z})$ and evaluate $\\operatorname{Arf}(q|_W)$ .", "The subspace $W$ is spanned by $v$ and $J_\\tau v$ where: $ v = x_{(f-i\\sqrt{-D})/2} \\equiv \\frac{2f+x}{4} x_1 +\\frac{x-2y}{4} x_i +x_{bi} \\bmod 2.", "$ Here, as above, $x_t \\in H_1(X,\\mathbb {Z}/2\\mathbb {Z})$ is the homology class represented by an integral homology class with $\\omega _\\tau $ -period $t$ .", "By Proposition REF and the fact that $q$ is $J_\\tau $ -invariant, we have: $ \\epsilon (X_\\tau ,[\\omega _\\tau ]) = q(v)^2 = \\frac{f+1}{2} + \\epsilon _0([I]) \\bmod 2.", "$ Our formula for the number of orbifold points on the spin components of $W_D$ follows readily from the previous two propositions: [Proof of Theorem REF ] When $D=f^2$ , the spin homomorphism $\\epsilon _0$ is the zero map and all of the orbifold points on $W_D$ lie on the spin $(f+1)/2 \\bmod 2$ component of $W_D$ .", "When $D$ is not a square, $\\epsilon _0$ is onto, and exactly half of the orbifold points on $W_D$ lie on each spin component.", "Remark For square discriminants $D=f^2$ , there is an elementary argument that shows $e_2\\left(W_D^{(f-1)/2}\\right)=0$ .", "For $(X_\\tau ,[\\omega _\\tau ]) \\in W_D$ , the number $\\tau $ is in $\\mathbb {Q}(i)$ and rescaling $P_\\tau $ , we can exhibit $X_\\tau $ as the quotient of a polygon $\\lambda P_\\tau $ with vertices in $\\mathbb {Z}[i]$ and area $f$ .", "The surface $X_\\tau $ is “square-tiled” and admits a degree $f$ map $\\phi \\colon \\,X_\\tau \\rightarrow \\mathbb {C}/\\mathbb {Z}[i]$ branched over a single point.", "The spin of $(X_\\tau ,[\\omega _\\tau ])$ can be determined from the number of Weierstrass points $X_\\tau ^W$ mapping to the branch locus of $\\phi $ ([18], Theorem 6.1).", "This number is, in turn, determined by $f$ , as can be seen by elementary Euclidean geometry.", "Corollary 5.4 Fix $D\\ge 9$ with $D \\equiv 1 \\bmod 8$ and conductor $f$ , a pinwheel prototype $(e,c,b) \\in E(W_D)$ and set $\\tau = \\frac{e+\\sqrt{-D}}{2c}$ .", "The surface $(X_{\\tau },[\\omega _\\tau ])$ has spin given by: $ \\epsilon (X_\\tau ,[\\omega _\\tau ]) = \\frac{c+f}{2} \\bmod 2.", "$ The surface $X_\\tau $ corresponds to the ideal class of $I = 2c \\mathbb {Z}\\oplus (-e+\\sqrt{-D}) \\mathbb {Z}$ in $H(-4D)$ and the norm of $I$ is $2c$ ." ], [ "Genus of $W_D$", "Together with [2], [18], Theorems REF and REF complete the determination of the homeomorphism type of $W_D$ , giving a formula for the genus of the irreducible components of $W_D$ .", "In this section, we will prove the following upper bound on the genus of the components of $W_D$ : Proposition 6.1 For any $\\epsilon >0$ , there are positive constants $C_\\epsilon $ and $N_\\epsilon $ such that: $C_\\epsilon D^{3/2+\\epsilon } > g(V)$ whenever $V$ is a component of $W_D$ and $D>N_\\epsilon $ .", "We will also give effective lower bounds: Proposition 6.2 Suppose $D > 0$ is a discriminant and $V$ is a component of $W_D$ .", "If $D$ is not a square, the genus of $V$ satisfies: $ g(V) \\ge D^{3/2}/600 - D/16 - D^{3/4}/2 - 75.", "$ If $D$ is a square, the genus of $V$ satisfies: $ g(V) \\ge D^{3/2}/240-7D/10-D^{3/4}/2-75.", "$ These two propositions immediately imply Corollary REF .", "Also, the following proposition shows that the components of $\\bigcup _D W_D$ with genus $g \\le 4$ are all listed in Appendix B, giving Corollary REF as an immediate consequence: Corollary 6.3 The components of $\\bigcup _D W_D$ with genus $g \\le 4$ all lie on $\\bigcup _{D \\le 121} W_D$ .", "The bounds in Proposition REF show that $g(V) > 4$ whenever $D > 7000$ for non-square $D$ and $D > 200^2$ for square $D$ .", "The remaining discriminants were checked by computer." ], [ "Orbifold Euler characteristic and genus.", "Let $\\Gamma \\subset \\operatorname{PSL}_2(\\mathbb {R})$ be a lattice and let $X = \\mathbb {H}/\\Gamma $ be the finite volume quotient.", "The homeomorphism type of $X$ is determined by the number of cusps $C(X)$ of $X$ , the number $e_n(X)$ of points of orbifold order $n$ for each $n > 1$ , and the genus $g(X)$ of $X$ .", "The orbifold Euler characteristic of $X$ is the following linear combination of these numbers: $ \\chi (X) = 2 - 2 g(X) - C(X) - \\sum _{n} (1-1/n) e_n(X) .", "$" ], [ "Euler characteristic of $X_D$ and {{formula:c9a3e7d9-a898-4ada-b357-df0ae9ea80e0}} .", "The Hilbert modular surface $X_D$ has a meromorphic modular form with a simple zero along $W_D$ and simple pole along $P_D$ .", "This gives a simple relationship between the orbifold Euler characteristics of $W_D$ , $P_D$ and $X_D$ and a modular curve $S_D$ in the boundary of $X_D$ .", "The curve $S_D$ is empty unless $D=f^2$ is a square, in which case $S_D \\cong X_1(f)$ .", "Theorem 6.4 ([2] Cor.", "10.4) The Euler characteristic of $W_D$ satisfies: $ \\chi (W_D) = \\chi (P_D) - 2 \\chi (X_D) - \\chi (S_D).", "$ For a discriminant $D$ , define: $ F(D) = \\prod _{p | f} \\left(1-\\left(\\frac{D_0}{p} \\right) p^{-2}\\right).", "$ where $f$ is the conductor of $\\mathcal {O}_D$ , $D_0=D/f^2$ is the discriminant of the maximal order in $\\mathbb {Q}(\\sqrt{D})$ and the product is over primes dividing $f$ .", "The number $F(D)$ satisfies $1 \\ge F(D) > \\zeta _\\mathbb {Q}(2)^{-1} > 6/10$ .", "For square discriminants, $\\chi (S_D) = -f^2 F(D)/12$ and the Euler characteristic of $W_D$ and its components are given by ([2] Theorem 1.4): $ \\begin{array}{c}\\chi (W_{f^2}) = -f^2(f-1)F(D)/16, \\chi (W_{f^2}^0) = -f^2(f-1) F(D)/32, \\\\\\mbox{ and }\\chi (W_{f^2}^1) = -f^2(f-3) F(D)/32.\\end{array}$ For non-square discriminants, $\\chi (S_D)=0$ and $\\chi (P_D)=-\\frac{5}{2}\\chi (X_D)$ giving $\\chi (W_D)=-\\frac{9}{2}\\chi (X_D)$ .", "The Euler characteristic $\\chi (X_D)$ can be computed from ([2] Theorem 2.12): $ \\chi (X_D) = 2 f^3 \\zeta _{D_0}(-1) F(D).", "$ Here $\\zeta _{D_0}$ is the Dedekind-zeta function and can be computed from Siegel's formula ([8], Cor.", "1.39): $ \\zeta _{D_0}(-1) = \\frac{1}{60} \\sum _{e^2 < D_0,\\, e \\equiv D_0 \\bmod 2} \\sigma \\left( \\frac{D_0-e^2}{4} \\right), $ where $\\sigma (n)$ is the sum of the divisors of $n$ .", "For reducible $W_D$ , the spin components satisfy $\\chi (W_D^0)=\\chi (W_D^1) = \\frac{1}{2}\\chi (W_D)$ ([2] Theorem 1.3).", "The well-known bound $\\sigma (n) = o(n^{1+\\epsilon })$ gives constants $C_\\epsilon $ and $N_\\epsilon $ so: $ C_\\epsilon D^{3/2+\\epsilon } > \\chi (X_D) $ whenever $D > N_\\epsilon $ .", "Using $\\sigma (n) > n+1$ and $F(D) > 6/10$ gives: $ \\chi (X_D) > D^{3/2}/300.", "$ We can now prove the upper bounds for the genus of $W_D$ : [Proof of Proposition REF ] For square discriminants $D=f^2$ , we have $\\left|\\chi (W_D)\\right| \\le f^3$ .", "For non-square discriminants, the bounds for $\\chi (X_D)$ and the formula $\\chi (W_D)=-9\\chi (X_D)/2$ gives $\\left|\\chi (W_D)\\right| = O( D^{3/2+\\epsilon })$ .", "Since $W_D$ has one or two components, $g(W_D) = O(\\left|\\chi (W_D)\\right|)$ ." ], [ "The modular curve $P_D$ .", "The modular curve $P_D$ is isomorphic to: $ \\left( \\bigsqcup _{(e,l,m)} Y_0(m) \\right) / g, $ where the union is over triples of integers $(e,l,m)$ with: $ \\begin{array}{ccc} D = e^2 + 4 l^2m, & l,m > 0, & \\mbox{and } \\gcd (e,l)=1 \\end{array}, $ and $g$ is the automorphism sending the degree $m$ isogeny $i$ on the component labeled by $(e,l,m)$ to the isogeny $i^*$ on the $(-e,l,m)$ -component (cf.", "[18] Theorem 2.1).", "The isogeny $i\\colon \\,E \\rightarrow F$ on the $(e,l,m)$ -component corresponds to the Abelian variety $B=E \\times F$ with $\\mathcal {O}_D$ generated by $\\iota \\left( \\frac{e+\\sqrt{D}}{2} \\right) = i + i^*+[e]_E$ where $[e]_E$ is the multiplication by $e$ -map on $E$ .", "In particular, the components of $P_D$ are labeled by triples $(e,l,m)$ as above subject to the additional condition $e \\ge 0$ .", "We will need the following bound on the number of such triples: Proposition 6.5 The number of components of $P_D$ satisfies $h_0(P_D) \\le D^{3/4}+150$ .", "Let $l(n)$ denote the largest integer whose square divides $n$ and let $f(n) = d(l(n))$ be the number of divisors of $l(n)$ .", "The function $f$ is multiplicative and the number of triples $(e,l,m)$ with $e$ fixed is bounded above by $f\\left( \\frac{D-e^2}{4} \\right)$ .", "There is a finite set $S$ of natural numbers $n$ for which $f(n) > n^{1/4}$ (they are all divisors of $2^{12} 3^6 5^4 7^2 11^2$ ) since $d(n)$ is $o(n^\\epsilon )$ for any $\\epsilon > 0$ and it is easy to check that $\\sum _{n \\in S} f(n) - n^{1/4} < 150$ .", "The asserted bound on $h^0(P_D)$ follows from: $ h_0(P_D) \\le \\sum _{\\begin{array}{c} e \\equiv D \\bmod 2\\\\ 0 \\le e < \\sqrt{D} \\end{array}} f\\left( \\frac{D-e^2}{4} \\right) \\le 150 + \\sum _e \\left( \\frac{D-e^2}{4} \\right)^{1/4}.", "$" ], [ "Cusps on $W_D$ and {{formula:8fd95ca2-2352-40c5-8cdc-898350c8d0e8}} .", "Let $C_1(W_D)$ and $C_2(W_D)$ be the number of one- and two-cylinder cusps on $W_D$ respectively and $C(W_D)=C_1(W_D)+C_2(W_D)$ be the total number of cusps.", "The cusps on $W_D$ were first enumerated and sorted by component in [18]: Proposition 6.6 For non-square discriminants, the number of cusps on $W_D$ is equal to the number of cusps on $P_D$ : $ C(W_D)=C_2(W_D) =C(P_D), $ and $C(W_D^0)=C(W_D^1)$ when $W_D$ is reducible.", "For square discriminants $D=f^2$ , the number of one- and two-cylinder cusps satisfy: $ C_2(W_{f^2}) < C(P_{f^2}) \\mbox{ and } C_1(W_{f^2}) < f^2/3.", "$ When $f$ is odd, $\\left|C(W_{f^2}^1)-C(W_{f^2}^0)\\right| < 7f^2/12$ .", "Except for the explicit bounds on $C_1(W_{f^2})$ and $\\left|C(W_{f^2}^1)-C(W_{f^2}^0)\\right|$ , the claims in the proposition follow from the enumeration of cusps on $P_D$ and $W_D$ in [2], §3.1.", "We now turn to the bounds on $C_1(W_{f^2})$ and $\\left|C(W_{f^2}^1)-C(W_{f^2}^0)\\right|$ .", "When $D$ is not a square, there are no one-cylinder cusps and when $D=f^2$ is a square, the one-cylinder cusps are parametrized by cyclically ordered triples $(a,b,c)$ with (cf.", "[18] Theorem A.1): $ f=a+b+c,\\, a, b,c>0 \\mbox{ and } \\gcd (a,b,c)=1.", "$ Cyclically reordering $(a,b,c)$ so $a < b$ and $a < c$ ensures that $a < f/3$ and $b < f$ , giving $C_1(W_D) < f^2/3=D/3$ .", "The difference in the number of two cylinder cusps is given by (Theorem A.4 in [18]): $ C_2(W_D^0) - C_2(W_D^1) = \\sum _{b+c=f,0<c<b} \\phi (\\gcd (b,c)), $ which is smaller than $D/4$ using $c < f/2$ and $\\phi (\\gcd (b,c)) < f/2$ .", "The bound asserted for $\\left|C(W_D^0)-C(W_D^1)\\right|$ follows.", "We are now ready to prove the lower bounds in Proposition REF : [Proof of Proposition REF ] We will handle the square and non-square discriminants separately." ], [ "Lower bounds, non-square discriminants.", "Suppose $D$ is a non-square discriminant and $D>8$ so all of the orbifold points on $W_D$ have order two.", "Using the formula $\\chi (W_D) = \\chi (P_D) - 2\\chi (X_D)$ , the equality $C(P_D)=C(W_D)$ and ignoring several terms which contribute positively to the $g(W_D)$ gives: $ g(W_D) \\ge \\chi (X_D) - h^0(P_D) - e_2(W_D)/4.", "$ Combining the bound above with $\\chi (X_D) > D^{3/2}/300$ , $h_0(P_D) < D^{3/4}+150$ , $e_2(W_D) < D/2$ (Proposition REF ) and $g(V) \\ge \\frac{1}{2} g(W_D)$ whenever $V$ is a component of $W_D$ gives the bound stated in Proposition REF ." ], [ "Lower bounds, square discriminants.", "Now suppose $D=f^2$ .", "Using the formula for $\\chi (W_D)$ in terms of $\\chi (X_D)$ , $\\chi (P_D)$ and $\\chi (S_D)$ , the bound $C_2(W_D) < C(P_D)$ and ignoring some terms which contribute positively to $g(W_D)$ gives: $ g(W_D) \\ge \\chi (X_D) - h_0(P_D) - e_2(W_D)/4 + \\chi (S_D)/2 - C_1(W_D)/2.", "$ As before we have $h_0(P_D) < D^{3/4}+150$ , $e_2(W_D) < D/2$ and $C_1(W_D) < D/3$ .", "By Theorem 2.12 and Proposition 10.5 of [2] and using $\\zeta _\\mathbb {Q}(2) > 6/10$ , we have $\\chi (X_D)+\\chi (S_D)/2 > D^{3/2}/120-D/40$ so long as $D>36$ , giving: $ g(W_D) \\ge D^{3/2}/120-2 D/5 - D^{3/4}-150.", "$ Finally, to bound $g(V)$ when $V$ is a component of $W_D$ , we bound the difference: $\\begin{array}{ll}\\left|g(W_D^0) - g(W_D^1)\\right| &\\le \\left|\\frac{\\chi (W_D^1)-\\chi (W_D^0)}{2}\\right| + \\left|\\frac{C(W_D^1)-c(W_D^0)}{2}\\right|+e_2(W_D)/4 \\\\\\end{array}.$ We have seen that $\\left|C(W_D^1) - C(W_D^0)\\right| < 7D/12$ and $e_2(W_D)/4 < D/8$ .", "Theorem 1.4 of [2] gives $\\left|\\chi (W_D^0)-\\chi (W_D^1)\\right|< D/16$ and $\\left|g(W_D^1)-g(W_D^0)\\right| < D/2$ .", "The bound asserted for $g(V)$ in Proposition REF follows." ], [ "The $D_{12}$ -family", "In this section we will describe the surfaces in $\\mathcal {M}_2(D_{12})$ .", "For a smooth surface $X \\in \\mathcal {M}_2$ , the following are equivalent: Automorphisms.", "The automorphism group $\\operatorname{Aut}(X)$ admits an injective homomorphism $\\rho : D_{12} \\rightarrow \\operatorname{Aut}(X)$ .", "Algebraic curves.", "The field of functions $\\mathbb {C}(X)$ is isomorphic to the field: $ \\widetilde{K}_a = \\mathbb {C}(z,x) \\mbox{ with } z^2=x^6-ax^3+1, $ for some $a \\in \\mathbb {C}\\setminus \\left\\lbrace \\pm 2 \\right\\rbrace $ .", "Jacobians.", "The Jacobian $\\operatorname{Jac}(X)$ is isomorphic to the principally polarized Abelian variety: $ \\widetilde{A}_\\tau = \\mathbb {C}^2 / \\widetilde{\\Lambda }_\\tau , $ where $\\widetilde{\\Lambda }_\\tau = \\mathbb {Z}\\left< \\left({\\begin{matrix} 1 \\\\ 1/\\sqrt{3} \\end{matrix}}\\right), \\left({\\begin{matrix} \\tau \\\\ \\sqrt{3} \\tau \\end{matrix}}\\right), \\left({\\begin{matrix} 1 \\\\ -1/\\sqrt{3} \\end{matrix}}\\right), \\left({\\begin{matrix} \\tau \\\\ -\\sqrt{3} \\tau \\end{matrix}}\\right) \\right>$ and is polarized by the symplectic form $\\left< \\left({\\begin{matrix} a \\\\ b \\end{matrix}}\\right), \\left({\\begin{matrix} c \\\\ d \\end{matrix}}\\right) \\right> = \\frac{-\\operatorname{Im}(a \\overline{c} + b \\overline{d})}{2 \\operatorname{Im}\\tau } $ .", "Hexagonal pinwheels.", "The surface $X$ is isomorphic to the surface $\\widetilde{X}_\\tau $ obtained by gluing the hexagonal pinwheel $H_\\tau $ (Fig.", "REF ) to $-H_\\tau $ for some $\\tau $ in the domain: $ \\widetilde{U} = \\left\\lbrace \\tau \\in \\mathbb {H}: \\tau \\ne \\zeta _{12}/\\sqrt{3} \\mbox{ or } \\zeta _{12}^5/\\sqrt{3}, \\left|\\operatorname{Re}{\\tau }\\right| \\le \\frac{1}{2} \\mbox{ and } \\left|\\tau \\right|^2 \\ge \\frac{1}{3} \\right\\rbrace .", "$ It is straightforward to identify the action of $D_{12}$ on the surfaces described above.", "The field $\\widetilde{K}_a$ has automorphisms $Z(z,x) = (-z,\\zeta _3x)$ and $r(z,x)=(z/x^3,1/x)$ .", "The polarized lattice $\\widetilde{\\Lambda }_\\tau $ is preserved by the linear transformations $r=\\left({\\begin{matrix} 1 & 0 \\\\ 0 & -1 \\end{matrix}}\\right)$ and $Z=\\frac{1}{2}\\left({\\begin{matrix} 1 & -\\sqrt{3} \\\\ \\sqrt{3} & 1 \\end{matrix}}\\right)$ .", "The surface obtained from $\\widetilde{X}_\\tau $ has an order six automorphism $Z_\\tau $ with $[Z_\\tau |_{\\widetilde{X}_\\tau ^W}] = [3,3]$ which implies that $\\widetilde{X}_\\tau $ has a faithful $D_{12}$ -action (cf.", "Table REF in §).", "The family $\\mathcal {M}_2(D_{12})$ admits an analysis similar to that of $\\mathcal {M}_2(D_8)$ .", "For $(X,\\rho ) \\in \\mathcal {M}_2(D_{12})$ , the quotient $E = X/\\rho (r)$ has genus one and a distinguished subgroup of order three in $E[3]$ .", "One can establish the precise relationship between $\\widetilde{X}_\\tau $ , $\\widetilde{A}_\\tau $ and $\\widetilde{K}_a$ by studying the corresponding map from $\\mathcal {M}_2(D_{12})$ to the modular curve $Y_0(3)$ .", "Figure: For τ\\tau in the shaded domain U ˜\\widetilde{U}, the hexagonal pinwheel H τ H_\\tau has vertices z 0 ,ζ 3 ±1 z 0 ,τ,ζ 3 ±1 τ\\left\\lbrace z_0,\\zeta _3^{\\pm 1} z_0, \\tau , \\zeta _3^{\\pm 1} \\tau \\right\\rbrace with z 0 =ζ 12 /3z_0=\\zeta _{12}/\\sqrt{3}.", "Gluing together sides on H τ H_\\tau and -H τ -H_\\tau by translation gives a genus two surface admitting an action of D 12 D_{12}.", "The one form induced by dzdz is a ZZ-eigenform." ] ]
1606.04967
[ [ "Curvature and geodesic instabilities in a geometrical approach to the\n planar three-body problem" ], [ "Abstract The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space.", "We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials.", "The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E. The geodesic flow on the full configuration space $C^3$ (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space $C^2$ and shape space $R^3$ (as well as $S^3$ and the shape sphere $S^2$ for the inverse-square potential when E = 0).", "The corresponding Riemannian submersions are described explicitly in `Hopf' coordinates which are particularly adapted to the isometries.", "For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy `pair of pants' JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points.", "We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on $C^2, R^3$ & $S^3$ with collision points removed.", "Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities.", "We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits and observe that the geodesic reformulation `regularizes' pairwise and triple collisions on $C^2$ and its quotients for arbitrary masses and allowed energies.", "For the Newtonian potential with equal masses and E=0, we find that the scalar curvature on $C^2$ is strictly negative though it could have either sign on $R^3$.", "However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time." ], [ "Introduction", "The classical three-body problem and associated questions of stability have stimulated much work in mechanics and nonlinear & chaotic dynamics [1], [2], [3], [4], [5].", "Quantum and fluid mechanical variants with potentials other than Newtonian are also of interest: e.g., the dynamics of two-electron atoms and the water molecule [6], the $N$ -vortex problem with logarithmic potentials [7], the problem of three identical bosons with inverse-square potenials (Efimov effect in cold atoms [8], [9]) and the Calogero-Moser system also with inverse-square potentials [10].", "We investigate a geometrical approach to the planar three-body problem with Newtonian and attractive inverse-square potentials.", "The inverse-square potential has some simplifying features over the Newtonian one due in part to the nature of its scaling symmetry $H(\\lambda {\\bf r}, \\lambda ^{-1} {\\bf p}) = \\lambda ^{-2} H({\\bf r}, {\\bf p})$ .", "As a consequence, the sign of energy $E$ controls asymptotic behaviour: bodies fly apart or suffer a triple collision according as $E$ is positive/negative, leaving open the special case $E=0$ [11].", "This follows from the Lagrange-Jacobi identity $\\ddot{I} = 4 E$ for the evolution of the moment of inertia $I = \\sum m_i {\\bf r}_i^2$ .", "By contrast, for the Newtonian potential, $H(\\lambda ^{-2/3} {\\bf r}, \\lambda ^{1/3} {\\bf p}) = \\lambda ^{2/3} H({\\bf r}, {\\bf p})$ leads to $\\ddot{I} = 4E - 2V$ , which is not sufficient to determine the long time behavior of $I$ when $E < 0$ .", "Our approach is based on a geometric reformulation of Newtonian trajectories.", "It is well known that trajectories of a free particle moving on a Riemannian manifold are geodesics of a mass/kinetic metric $m_{ij}$ defined by the kinetic energy $\\frac{1}{2}m_{ij}(x) \\dot{x}^i \\dot{x}^j$ .", "Indeed, geodesic flow on a compact Riemann surface of constant negative curvature is a prototypical model for chaos [6].", "In the presence of a potential $V$ , trajectories are reparametrized geodesics of the conformally related Jacobi-Maupertuis (JM) metric $g_{ij} = (E-V(x))m_{ij}$ (see Refs.", "[12], [13] and §).", "The linear stability of geodesics to perturbations is then controlled by sectional curvatures of the JM metric.", "Several authors have tried to relate the geometry of the JM metric to chaos.", "For systems with many degrees of freedom, Pettini et.", "al.", "[14], [15], [16] obtain an approximate expression for the largest Lyapunov exponent in terms of curvatures.", "In Ref.", "[17] the geometric framework is applied to investigate chaos in the Hénon-Heiles system and a suitable average sectional curvature proposed as an indicator of chaos for systems with few degrees of freedom (see also [18]).", "While negativity of curvature need not imply chaos, as the Kepler problem shows for $E > 0$ , these works suggest that chaos could arise both from negativity of curvature and from fluctuations in curvature through parametric instabilities.", "For the planar gravitational three-body problem (i.e.", "with pairwise Newtonian potentials), the JM metric on the full configuration space ${\\mathbb {R}}^6 \\cong {\\mathbb {C}}^3$ has isometries corresponding to translation and rotation invariance groups $\\bf C$ and U$(1)$ (§REF ).", "This allows one to study the reduced dynamics on the quotients: configuration space ${\\mathbb {C}}^2 \\cong {\\mathbb {C}}^3/{\\bf C}$ and shape space ${\\mathbb {R}}^3 \\cong {\\mathbb {C}}^2/{\\rm U}(1)$ [19].", "Here, collision configurations are excluded from ${\\mathbb {C}}^3$ and its quotients.", "When the Newtonian potential is replaced with the inverse-square potential, the zero-energy JM metric acquires a scaling isometry leading to additional quotients: ${\\mathbb {S}}^3 \\cong {\\mathbb {C}}^2/{\\rm scaling}$ and the shape sphere ${\\mathbb {S}}^2 \\cong {\\mathbb {R}}^3/{\\rm scaling}$ (see Fig.", "REF ).", "Since the three collision points have been removed, the (non-compact) shape sphere ${\\mathbb {S}}^2$ has the topology of a pair of pants and fundamental group given by the free group on two generators.", "As part of a series of works on the planar three-body problem, Montgomery [20] shows that for three equal masses with inverse-square potentials (sometimes referred to as a `strong' force), the curvature of the JM metric on ${\\mathbb {S}}^2$ is negative except at the two Lagrange points, where it vanishes.", "As a corollary, he shows the uniqueness of the `figure 8' solution and establishes that collision solutions are dense within bound ones.", "In Ref.", "[21], [22], he uses the geometry of the shape sphere to show that zero angular momentum negative energy solutions (other than the Lagrange homotheties) of the gravitational three-body problem have at least one syzygy (collinearity).", "In this paper, we begin by extending some of Montgomery's results on the geometry of the shape sphere to that of the configuration space ${\\mathbb {C}}^2$ (without any restriction on angular momentum) and its quotients.", "Metrics on the quotients are obtained explicitly via Riemannian submersions (§REF , §REF ) which simplify in `Hopf' coordinates, as the Killing vector fields (KVFs) point along coordinate vector fields.", "These coordinates also facilitate our explicit computation of metrics and curvatures near binary and triple collisions.", "We interpret Lagrange and Euler homotheties (`central configurations' [23]) as radial geodesics at global and local minima of the conformal factor in the JM metric for the inverse-square potential (§REF ) and thereby deduce geodesic completeness of the configuration manifold ${\\mathbb {C}}^2$ and its quotients ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ for arbitrary masses and allowed energies.", "The estimates showing completeness on ${\\mathbb {C}}^2$ are similar to those showing that the classical action (integral of Lagrangian) diverges for collisional trajectories.", "In a private communication, R Montgomery points out that this was known to Poincare and has been rediscovered several times (see for example Ref.", "[24], [25], [26]).", "Completeness establishes that the geodesic reformulation `regularizes' pairwise and triple collisions by reparametrizing time so that any collision occurs at $t = \\infty $ .", "In contrast with other regularizations [27], [28], this does not involve an extrapolation of the dynamics past a collision nor a change in dependent variables.", "Unlike for the inverse-square potential, we show that geodesics for the Newtonian potential can reach curvature singularities (binary/triple collisions) in finite geodesic time (§REF ).", "This may come as a surprise, since the Newtonian potential is less singular than the inverse-square potential and masses collide sooner under Newtonian evolution in the inverse-square potential.", "However, due to the reparametrization of time in going from trajectories to geodesics, masses can collide in finite time in the Newtonian potential while taking infinitely long to do so in the inverse-square potential.", "Indeed, for the attractive $1/r^n$ potential, the JM line-element leads to estimates $\\propto \\int _0^{\\eta _0} \\frac{d \\eta }{\\eta ^{n/2}}$ and $\\int _0^{r_0} \\frac{d r}{r^{n/2}}$ for the distances to binary and triple collisions from a nearby location (§REF ).", "These diverge for $n \\ge 2$ and are finite for $n < 2$ .", "To examine stability of geodesics, we evaluate scalar and sectional curvatures of the zero-energy, equal-mass JM metrics on ${\\mathbb {C}}^2$ and its quotients.", "For the inverse-square potential, we obtain strictly negative upper bounds for scalar curvatures on ${\\mathbb {C}}^2$ , ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ (§REF ), indicating widespread linear geodesic instability.", "Moreover, scalar curvatures are shown to be bounded below.", "In particular, they remain finite and negative at binary and triple collisions.", "O'Neill's theorem is used to determine or bound various sectional curvatures on ${\\mathbb {C}}^2$ using the more easily determined ones on its Riemannian quotients; they are found to be largely negative (§REF ).", "On the other hand, for the Newtonian potential, we find that the scalar curvature on ${\\mathbb {C}}^2$ is strictly negative, though it can have either sign on shape space ${\\mathbb {R}}^3$ (§REF ).", "Unlike for the inverse-square potential, scalar curvatures $\\rightarrow - \\infty $ at collision points.", "We also discuss the geodesic instability of Lagrange rotation and homothety solutions for equal masses (§REF ).", "We end with a cautionary remark comparing stability of geodesics to that of corresponding trajectories, simple examples are used to illustrate that the two notions of stability need not always coincide.", "In this paper we have not touched upon the interesting issues of long-term geodesic stability or chaos.", "It would be interesting to relate the local geodesic instabilities discussed here to medium- and long-time behavior.", "The dynamical consequences of sectional curvatures possessing either sign should also be of much interest." ], [ "Trajectories as geodesics of the Jacobi-Maupertuis metric", "For a system with configuration space $M$ and Lagrangian $L = (1/2) m_{ij}(x) \\dot{x}^i \\dot{x}^j$ , Lagrange's equations are equivalent to the geodesic equations with respect to the `mass' or `kinetic metric' $m_{ij}$ .", "Remarkably, this connection between trajectories and geodesics extends to a system subject to a potential $V$ .", "Indeed, this is the content of Maupertuis' principle of extremization of $\\int _{q_1}^{q_2} p dq$ holding energy fixed[12], [13].", "More precisely, the equations of motion (EOM) $m_{ki} \\ddot{x}^i(t) = - \\partial _k V - \\frac{1}{2}\\left(m_{ik,j} + m_{jk,i} - m_{ij,k} \\right) \\dot{x}^i(t) \\: \\dot{x}^j(t)$ may be regarded as reparametrized geodesic equations for the JM metric, $ds^2 = g_{ij} dx^i dx^j = (E-V) m_{ij} dx^i dx^j$ on the classically allowed `Hill' region $E - V \\ge 0$ .", "Notice that $\\sqrt{2} \\int ds = \\int p dq = \\int (L+E) dt$ so that the length of a geodesic is related to the classical action of the trajectory.", "The formula for the inverse JM metric $g^{ij} = m^{ij}/(E-V)$ may also be read off from the time-independent Hamilton-Jacobi (HJ) equation $(m^{ij}/2(E-V))\\; \\partial _i W \\partial _j W = 1$ by analogy with the rescaled kinetic metric $m^{ij}/2E$ appearing in the free particle HJ equation $(m^{ij}/2E) \\partial _i W \\partial _j W = 1$ (see p.74 of Ref.", "[11]).", "The JM metric is conformal to the kinetic metric and depends parametrically on the conserved energy $E = \\frac{1}{2}m_{ij} \\dot{x}^i \\dot{x}^j + V$ .", "The geodesic equations $\\ddot{x}^l(\\lambda ) = - \\frac{1}{2}g^{lk}\\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\\right)\\dot{x}^i(\\lambda ) \\dot{x}^j(\\lambda )$ for the JM metric reduce to (REF ) under the reparametrisation $d/d\\lambda = (1/\\sigma ) (d/dt)$ where $\\sigma = (E-V)/\\sqrt{\\cal {T}}$ .", "Here ${\\cal T} = \\frac{1}{2}g_{ij} \\dot{x}^i \\dot{x}^j$ is the conserved `kinetic energy' along geodesics and equals one-half for arc-length parametrization.", "To obtain $\\sigma $ , suppose $y^i(t)$ is a trajectory and $z^i(\\lambda )$ the corresponding geodesic.", "Then at a point $x^i = z^i(\\lambda ) = y^i(t)$ , the velocities are related by $\\sigma \\dot{z}^i = \\dot{y}^i$ leading to ${\\cal T} = \\frac{1}{2}g_{ij} \\dot{z}^i \\dot{z}^j =\\frac{E-V}{2}m_{ij} \\dot{z}^i\\dot{z}^j = \\frac{E-V}{2 \\sigma ^2} m_{ij} \\dot{y}^i\\dot{y}^j = \\left(\\frac{E-V}{ \\sigma }\\right)^2.$ This reparametrization of time may be inconsequential in some cases [e.g.", "Lagrange rotational solutions where $\\sigma $ is a constant since $V$ is constant along the trajectory (see §REF )] but may have significant effects in others [e.g.", "Lagrange homothety solutions where the exponential time-reparametrization regularizes triple collisions (see §REF )] and could even lead to a difference between linear stability of trajectories and corresponding geodesics (see §REF ).", "The curvature of the JM metric encodes information on linear stability of geodesics (see §REF ).", "For example, in the planar isotropic harmonic oscillator with potential $k r^2/2$ in plane polar coordinates, the gaussian curvature $R = 16Ek/(2E-kr^2)^3$ of the JM metric on configuration space is non-negative everywhere indicating stability.", "In the planar Kepler problem with Hamiltonian ${\\bf p}^2/2m - k/r$ , the gaussian curvature of the JM metric $ds^2 = m(E+k/r)(dr^2+r^2d\\theta ^2)$ is $R = -{Ek}/ ({m(k+Er)^3})$ .", "$R$ is everywhere negative/positive for $E$ positive/negative and vanishes identically for $E = 0$ .", "This reflects the divergence of nearby hyperbolic orbits and oscillation of nearby elliptical orbits.", "Negativity of curvature could lead to chaos, though not always, as the hyperbolic orbits of the Kepler problem show.", "As noted, chaos could also arise from curvature fluctuations [14]." ], [ "Jacobi-Maupertuis metric on configuration space and Hopf coordinates", "We consider the three-body problem with masses moving on a plane regarded as the complex plane $\\bf {C}$ .", "Its 6D configuration space (with collision points excluded) is identified with ${{\\mathbb {C}}^3}$ .", "A point on ${\\mathbb {C}}^3$ represents a triangle on the complex plane with the masses $m_{1,2,3}$ at its vertices $x_{1,2,3} \\in \\mathbf {C}$ .", "It is convenient to work in Jacobi coordinates (Fig.", "REF ) $J_1=x_2-x_1, \\quad J_2=x_3-\\frac{m_1x_1+m_2x_2}{m_1+m_2} \\quad \\text{and} \\quad J_3=\\frac{m_1x_1+m_2x_2+m_3x_3}{M_3},$ in which the kinetic energy $KE = (1/2) \\sum _i m_i |\\dot{x}_i|^2$ remains diagonal: $KE = \\frac{1}{2}\\sum _i M_i | \\dot{J}_i|^2 \\;\\; \\text{where} \\;\\;\\frac{1}{M_1}=\\frac{1}{m_1}+\\frac{1}{m_2}, \\;\\; \\frac{1}{M_2}=\\frac{1}{m_3}+\\frac{1}{m_1+m_2} \\;\\;\\text{and} \\;\\; M_3 = \\sum _i m_i.$ The KE for motion about the center of mass (CM) is $\\frac{1}{2}(M_1 |\\dot{J}_1|^2 + M_2 |\\dot{J}_2|^2)$ .", "The moment of inertia about the origin $I = \\sum _{i=1}^3 m_i |x_i|^2$ too remains diagonal in Jacobi coordinates ($I = \\sum _{i=1}^3 M_i |J_i|^2$ ), while about the CM we have $I_{\\rm CM} = M_1 |J_1|^2 + M_2 |J_2|^2$ .", "With $U = -V = \\sum _{i < j} G m_i m_j/|x_i - x_j|^2$ denoting the (negative) potential energy, the JM metric for energy $E$ on ${\\mathbb {C}}^3$ is $ds^2 = \\left( E + U \\right) \\sum _{i=1}^3 M_i |dJ_i|^2 \\quad \\text{where} \\;\\; U = \\frac{G m_1 m_2}{|J_1|^2} + \\frac{G m_2 m_3}{|J_2 - \\mu _1 J_1|^2} + \\frac{G m_3 m_1}{|J_2+\\mu _2 J_1|^2}$ and $\\mu _i= m_i/(m_1 + m_2)$ .", "Due to the inverse-square potential, $G$ does not have the usual dimensions.", "The metric is independent of the CM coordinates $J_3$ and $\\bar{J}_3$ , while $J_1,\\bar{J}_1, J_2$ and $\\bar{J}_2$ are invariant under translations $x_i \\rightarrow x_i + a$ for $a \\in \\mathbf {C}$ .", "Thus translations act as isometries of (REF ).", "Similarly, we will see that scalings (for $E =0$ ) and rotations also act as isometries.", "These isometries also act as symmetries of the Hamiltonian.", "For instance the dilatation $D = \\sum _i \\vec{x}_i \\cdot \\vec{p}_i = \\sum _i \\Re (x_i \\bar{p}_i)$ generates scale transformations $x_i \\rightarrow \\lambda x_i$ and $p_i \\rightarrow \\lambda ^{-1} p_i$ via Poisson brackets: $\\lbrace x_i, D \\rbrace = x_i$ and $\\lbrace p_i, D \\rbrace = - p_i$ .", "Since $\\lbrace H, D \\rbrace = - 2 H$ , scaling is a symmetry of the Hamiltonian only when energy vanishes.", "The study of the geometry of the JM metric is greatly facilitated by first considering the geometry of its quotients by isometries (for instance, geodesics on a quotient lift to horizontal geodesics).", "Riemannian submersions [29] provide a framework to define and compute metrics on these quotients.", "Suppose $(M,g)$ and $(N,h)$ are two Riemannian manifolds and $f: M \\rightarrow N$ a surjection.", "Then the linearization $df(p): T_p M \\rightarrow T_{f(p)} N$ is a surjection between tangent spaces.", "The vertical subspace $V(p) \\subseteq T_p M$ is defined to be the kernel of $df$ while its orthogonal complement $\\ker (df)^\\perp $ with respect to the metric $g$ is the horizontal subspace $H(p)$ .", "$f$ is a Riemannian submersion if it preserves lengths of horizontal vectors, i.e., if the isomorphism $df(p) \\colon \\ker (df(p))^{\\perp } \\rightarrow T_{f(p)} N$ is an isometry at each point.", "The Riemannian submersions we are interested in are associated to quotients of a Riemannian manifold ($M,g$ ) by the action of a suitable group of isometries $G$ .", "There is a natural surjection $f$ from $M$ to the quotient $M/G$ .", "Requiring $f$ to be a Riemannian submersion defines the quotient metric on $M/G$ : the inner product of a pair of tangent vectors $(u,v)$ to $M/G$ is defined as the inner product of any pair of horizontal preimages under the map $df$ .", "The surjection $\\left( J_1,\\bar{J}_1, J_2,\\bar{J}_2, J_3,\\bar{J}_3\\right) \\mapsto \\left( J_1,\\bar{J}_1, J_2,\\bar{J}_2\\right)$ defines a submersion from configuration space ${\\mathbb {C}}^3$ to its quotient ${\\mathbb {C}}^2$ by translations.", "The vertical and horizontal subspaces are spanned by $\\partial _{J_3}, \\partial _{\\bar{J}_3}$ and $\\partial _{ J_1}, \\partial _{\\bar{J}_1}, \\partial _{J_2}, \\partial _{\\bar{J}_2}$ respectively.", "Requiring the submersion to be Riemannian, the quotient metric on ${\\mathbb {C}}^2$ is $ds^2=( E + U ) (M_1 \\; |d J_1|^2 + M_2 \\; |d J_2|^2).$ It is convenient to define rescaled coordinates on ${\\mathbb {C}}^2$ , $z_i = \\sqrt{M_i} \\: J_i$ , in terms of which (REF ) becomes $ds^2 = (E + U) (|dz_1|^2 + |dz_2|^2)$ .", "The kinetic energy in the CM frame is $KE = (1/2) (|\\dot{z}_1|^2 + |\\dot{z}_2|^2 )$ while $I_{\\rm CM} = |z_1|^2 + |z_2|^2$ .", "We now specialize to equal masses ($m_i = m$ ) so that $M_1 = m/2$ ,$M_2={2m}/{3}$ and $\\mu _i = 1/2$ .", "The metric on ${\\mathbb {C}}^2$ is seen to be conformal to the flat Euclidean metric via the conformal factor $E + U$ : $ds^2 = \\left(E+\\frac{G m^3}{2|z_1|^2}+\\frac{2 G m^3}{3|z_2-\\frac{1}{\\sqrt{3}}z_1|^2}+\\frac{2 G m^3}{3|z_2+\\frac{1}{\\sqrt{3}}z_1|^2}\\right) \\left(|dz_1|^2+|dz_2|^2 \\right) .$ Rotations U$(1)$ act as a group of isometries of ${\\mathbb {C}}^2$ , taking $\\left(z_1,z_2\\right) \\mapsto \\left(e^{i \\theta }z_1,e^{i \\theta }z_2\\right)$ and leaving the conformal factor invariant.", "Moreover if $E = 0$ , then scaling $z_i \\mapsto \\lambda z_i$ for $\\lambda \\in {\\bf R}^+$ is also an isometry.", "Thus we may quotient the configuration manifold ${\\mathbb {C}}^2$ successively by its isometries.", "We will see that ${\\mathbb {C}}^2/$ U$(1)$ is the shape space ${\\mathbb {R}}^3$ and ${\\mathbb {C}}^2$ /scaling is ${\\mathbb {S}}^3$ .", "Furthermore the quotient of ${\\mathbb {C}}^2$ by both scaling and rotations leads to the shape sphere ${\\mathbb {S}}^2$ (see Fig.", "REF , note that collision points are excluded from ${\\mathbb {C}}^2, {\\mathbb {R}}^3, {\\mathbb {S}}^3$ and ${\\mathbb {S}}^2$ ).", "Points on shape space ${\\mathbb {R}}^3$ represent oriented congruence classes of triangles while those on the shape sphere ${\\mathbb {S}}^2$ represent oriented similarity classes of triangles.", "Each of these quotient spaces may be given a JM metric by requiring the projection maps to be Riemannian submersions.", "The geodesic dynamics on ${\\mathbb {C}}^2$ is clarified by studying its projections to these quotient manifolds.", "We will now describe these Riemannian submersions explicitly in local coordinates.", "This is greatly facilitated by choosing coordinates (unlike $z_1, z_2$ ) on ${\\mathbb {C}}^2$ in which the KVFs corresponding to the isometries point along coordinate vector fields.", "As we will see, this ensures that the vertical subspaces in the associated Riemannian submersions are spanned by coordinate vector fields.", "Thus we introduce the Hopf coordinates $(r, \\eta , \\xi _1, \\xi _2)$ on ${\\mathbb {C}}^2$ via the transformation $z_1=r e^{i ( \\xi _1+ \\xi _2)} \\sin \\eta \\quad \\text{and} \\quad z_2=r e^{i ( \\xi _1- \\xi _2)} \\cos \\eta .$ Here the radial coordinate $r = \\sqrt{|z_1|^2 + |z_2|^2} = \\sqrt{I_{\\rm CM}} \\ge 0$ is a measure of the size of the triangle with masses at its vertices.", "$\\xi _2$ determines the relative orientation of $z_1$ and $z_2$ while $\\xi _1$ fixes the orientation of the triangle as a whole.", "More precisely, $2 \\xi _2$ is the angle from the rescaled Jacobi vector $z_2$ to $z_1$ while $2 \\xi _1$ is the sum of the angles subtended by $z_1$ and $z_2$ with the horizontal axis in Fig REF .", "Thus we may take $0 \\le \\xi _1 + \\xi _2 \\le 2\\pi $ and $0 \\le \\xi _1 -\\xi _2 \\le 2 \\pi $ or equivalently, $-\\pi \\le \\xi _2\\le \\pi $ and $|\\xi _2| \\le \\xi _1 \\le 2\\pi -|\\xi _2|$ .", "Finally, $0 \\le \\eta \\le \\pi /2$ measures the relative magnitudes of $z_1$ and $z_2$ , indeed $\\tan \\eta = |z_1|/|z_2|$ .", "When $r$ is held fixed, $\\eta , \\xi _1$ and $\\xi _2$ furnish the standard Hopf coordinates parametrizing the three sphere $|z_1|^2 + |z_2|^2 = r^2$ .", "For fixed $r$ and $\\eta $ , $\\xi _1 + \\xi _2$ and $\\xi _1 - \\xi _2$ are periodic coordinates on tori.", "These tori foliate the above three-sphere as $\\eta $ ranges between 0 and $\\pi /2$ .", "Furthermore, as shown in §REF , $2 \\eta $ and $2 \\xi _2$ are polar and azimuthal angles on the two-sphere obtained as the quotient of ${\\mathbb {S}}^3$ by rotations via the Hopf map.", "Let us briefly motivate these coordinates and the identification of the above quotient spaces.", "We begin by noting that the JM metric (REF ) on ${\\mathbb {C}}^2$ is conformal to the flat Euclidean metric $|dz_1|^2 + |dz_2|^2$ .", "Recall that the cone on a Riemannian manifold $(M, ds^2_M)$ is the Cartesian product $\\mathbf {R}^+ \\times M$ with metric $dr^2 + r^2 ds_M^2$ where $r > 0$ parameterizes $\\mathbf {R}^+$ .", "In particular, Euclidean $\\mathbf {C}^2$ may be viewed as a cone on the round three sphere $\\mathbf {S}^3$ .", "If $\\bf {S}^3$ is parameterized by Hopf coordinates $\\eta , \\xi _1$ and $\\xi _2$ , then this cone structure allows us to use $r, \\eta , \\xi _1$ and $\\xi _2$ as coordinates on $\\bf {C}^2$ .", "Moreover, the Hopf map defines a Riemannian submersion from the round $\\mathbf {S}^3$ to the round two sphere $\\mathbf {S}^2$ The Hopf map $\\mathbf {S}^3 \\rightarrow \\mathbf {S}^2$ is often expressed in Cartesian coordinates.", "If $|z_1|^2 + |z_2|^2 = 1$ defines the unit-$\\mathbf {S}^3 \\subset \\mathbf {C}^2$ and $w_1^2 + w_2^2 + w_3^2 = 1/4$ defines a 2-sphere of radius $1/2$ in $\\mathbf {R}^3$ , then $w_3 = \\left(|z_2|^2-|z_1|^2\\right)/2$ and $w_1+ i w_2 = z_1 \\bar{z}_2$ .", "Using Eq.REF , we may express the Cartesian coordinates $w_i$ in terms of Hopf coordinates: $\\nonumber 2 w_3 = {r^2}\\cos 2\\eta ,\\quad 2 w_1 = r^2 \\sin (2\\eta )\\cos (2\\xi _2)\\quad \\text{and} \\quad 2 w_2 = r^2 \\sin (2\\eta )\\sin (2\\xi _2).$ .", "Indeed, if we use Hopf coordinates $\\eta , \\xi _1, \\xi _2$ on $\\bf {S}^3$ , then the Hopf map takes $(\\eta , \\xi _1, \\xi _2) \\mapsto (\\eta , \\xi _2) \\in \\bf {S}^2$ .", "In general, if $M \\rightarrow N$ is a Riemannian submersion, then there is a natural submersion from the cone on $M$ to the cone on $N$ Let $f:(M,g)\\mapsto (N,h)$ be a Riemannian submersion with local coordinates $m^i$ and $n^j$ .", "Let $(r, m^i)$ and $(r,n^j)$ be local coordinates on the cones $C(M)$ and $C(N)$ .", "Then $\\tilde{f}: (r, m) \\mapsto (r, n)$ defines a submersion from $C(M)$ to $C(N)$ .", "Consider a horizontal vector $a \\partial _r + b_i \\partial _{m_i}$ in $T_{(r,m)}C(M)$ .", "We will show that $d\\tilde{f}$ preserves its length.", "Now, if $df(b_i \\partial _{m_i}) = c_i \\partial _{n_i}$ then $d\\tilde{f}(a \\partial _r + b_i \\partial _{m_i}) = a \\partial _r + c_i \\partial _{n_i}$ .", "Since $\\partial _r \\perp \\partial _{m^i}$ , $||a \\partial _r+ b_i \\partial _{m_i}||^2 = a^2 + r^2 \\Vert b_i \\partial _{m_i}\\Vert ^2$ $= a^2 + r^2\\Vert c_i \\partial _{n_i}\\Vert ^2$ as $f$ is a Riemannian submersion.", "Moreover $a^2 + r^2 \\Vert c_i \\partial _{n_i}\\Vert ^2 = \\Vert a \\partial _r + c_i \\partial _{n_i} \\Vert ^2$ since $\\partial _r \\perp \\partial _{n^i}$ .", "Thus $\\tilde{f}$ is a Riemannian submersion..", "In particular, the Hopf map extends to a Riemannian submersion from the cone on the round $\\bf {S}^3$ to the cone on the round $\\bf {S}^2$ , i.e.", "from Euclidean $\\bf {C}^2$ to Euclidean $\\bf {R}^3$ taking $(r, \\eta , \\xi _1, \\xi _2) \\mapsto (r, \\eta , \\xi _2)$ .", "As the conformal factor is independent of rotations, the same map defines a Riemannian submersion from ${\\mathbb {C}}^2$ with the JM metric to shape space ${\\mathbb {R}}^3$ with its quotient JM metric.", "Finally, for $E=0$ , scaling ${\\vec{r}} \\rightarrow \\lambda {\\vec{r}}$ defines an isometry of the quotient JM metric on shape space ${\\mathbb {R}}^3$ .", "Quotienting by this isometry we arrive at the shape sphere ${\\mathbb {S}}^2$ with Montgomery's `pair of pants' metric.", "Alternatively, we may quotient ${\\mathbb {C}}^2$ first by the scaling isometry of its JM metric to get ${\\mathbb {S}}^3$ and then by rotations to get ${\\mathbb {S}}^2$ (see Fig.", "REF ).", "With these motivations, we express the equal-mass JM metric on ${\\mathbb {C}}^2$ in Hopf coordinates [generalization to unequal masses is obtained by replacing $Gm^3 h$ below with $\\tilde{h}(\\eta , \\xi _2)$ given in Eq.", "(REF )]: $ds^2 = \\left( E + \\frac{Gm^3 h(\\eta ,\\xi _2)}{r^2} \\right) \\left(dr^2+r^2\\left(d\\eta ^2+ d\\xi _1^2-2\\cos 2\\eta \\;d\\xi _1\\;d\\xi _2+ d\\xi _2^2\\right)\\right).$ It is convenient to write $h(\\eta ,\\xi _2) = v_1+ v_2 + v_3$ where $v_1 = r^2/(m |x_2 - x_3|^2)$ is proportional to the pairwise potential between $m_2$ and $m_3$ and cyclic permutations thereof.", "The $v_i$ are rotation and scale-invariant, and therefore functions only of $\\eta $ and $\\xi _2$ in Hopf coordinates: $v_{1,2} = \\frac{2}{\\left( 2 + \\cos 2\\eta \\mp \\sqrt{3} \\sin 2\\eta \\; \\cos 2\\xi _2\\right)} \\quad \\text{and} \\quad v_3 = \\frac{1}{2\\sin ^2\\eta }.$ Notice that $h \\rightarrow \\infty $ at pairwise collisions.", "The $v_i$ 's have the common range $1/2 \\le v_i < \\infty $ with $v_3 = 1/2$ when $m_3$ is at the CM of $m_1$ and $m_2$ etc.", "We also have $h \\ge 3$ with equality when $v_1 = v_2 = v_3$ , corresponding to Lagrange configurations with masses at vertices of an equilateral triangle.", "To see this, we compute the moment of inertia $I_{\\rm CM}$ in two ways.", "On the one hand $I_{\\rm CM}=|z_1|^2+|z_2|^2=r^2$ .", "On the other hand, for equal masses the CM lies at the centroid of the triangle defined by masses.", "Thus $I_{\\rm CM}$ is $(4m/9) \\times $ the sum of the squares of the medians, which by Apollonius' theorem is equal to $(3/4) \\times $ the sum of the squares of the sides.", "Hence $I_{\\rm CM}=\\sum _{i=1}^3 r^2/3 v_i$ .", "Comparing, we get $\\sum _{i=1}^3 1/v_i = 3$ .", "Since the arithmetic mean is bounded below by the harmonic mean, ${h}/{3} = {\\left( v_1 + v_2 + v_3 \\right)}/{3} \\ge 3 \\left( {v_1}^{-1} + {v_2}^{-1} + {v_3}^{-1} \\right)^{-1} = 1.$ Lagrange, Euler, collinear and collision configurations: The geometry of the JM metric displays interesting behavior at Lagrange and collision configurations on ${\\mathbb {C}}^2$ and its quotients.", "We identify their locations in Hopf coordinates for equal masses.", "The Jacobi vectors in Hopf coordinates are $J_1=\\sqrt{\\frac{2}{m}} r e^{i ( \\xi _1+ \\xi _2)} \\sin \\eta \\quad \\text{and} \\quad J_2=\\sqrt{\\frac{3}{2 m}}r e^{i ( \\xi _1- \\xi _2)} \\cos \\eta .$ At a Lagrange configuration, $m_{1,2,3}$ are at vertices of an equilateral triangle.", "So $|J_2| = \\sqrt{3}|J_1|/2$ (i.e.", "$\\eta = \\pi /4$ ) and $J_2$ is $\\perp $ to $J_1$ (i.e.", "$\\xi _2 = \\pm \\pi /4$ , the sign being fixed by the orientation of the triangle).", "So Lagrange configurations $L_{4,5}$ on ${\\mathbb {C}}^2$ occur when $\\eta = \\pi /4$ and $\\xi _2=\\pm \\pi /4$ with $r$ and $\\xi _1$ arbitrary.", "On quotients of ${\\mathbb {C}}^2$ , $L_{4,5}$ occur at the images under the corresponding projections.", "Since $2 \\eta $ and $2 \\xi _2$ are polar and azimuthal angles on the shape sphere, $L_{4,5}$ are at diametrically opposite equatorial locations (see Fig.", "REF ).", "Collinear configurations (syzygies) occur when $J_1$ and $J_2$ are (anti)parallel, i.e.", "when $\\xi _2 = 0$ or $\\pi /2$ , with other coordinates arbitrary.", "On the shape sphere, syzygies occur on the `great circle' through the poles corresponding to the longitudes $2\\xi _2 = 0$ and $\\pi $ .", "Collisions are special collinear configurations.", "By $C_i$ we denote a collision of particles other than the $i^{\\rm th}$ one.", "So $C_3$ corresponds to $J_1=0$ which lies at the `north pole' ($\\eta =0$ ) on ${\\mathbb {S}}^2$ .", "$m_2$ and $m_3$ collide when $J_2 = J_1/2$ so $\\eta = \\pi /3$ and $\\xi _2 = 0$ at $C_1$ .", "Similarly, at $C_2$ , $J_2 = - J_1/2$ which corresponds to $\\eta = \\pi /3$ and $\\xi _2=\\pi /2$ .", "The Euler configurations $E_i$ for equal masses are collinear configurations where mass $m_i$ is at the midpoint of the other two.", "Finally, we note that the azimuth and co-latitude ($\\theta $ and $\\phi $ ) [20] are often used as coordinates on the shape sphere, so that $L_{4,5}$ are at the poles while $C_{1,2,3}$ and $E_{1,2,3}$ lie on the equator.", "This coordinate system makes the symmetry under permutations of masses explicit, but is not convenient near any of the collisions (e.g.", "sectional curvatures can be discontinuous).", "On the other hand, our coordinates $\\eta $ and $\\xi _2$ , which are related to $\\theta $ and $\\phi $ by suitable rotations, $\\nonumber \\sin \\phi = \\cos (2\\eta -\\pi /2)\\sin (2\\xi _2), \\;\\; \\cos \\phi \\sin \\theta =\\cos (2\\eta -\\pi /2)\\cos (2\\xi _2),\\;\\; \\cos \\phi \\cos \\theta =\\sin (2\\eta -\\frac{\\pi }{2}),$ are convenient near $C_3$ but not near $E_3$ or $C_{1,2}$ (sectional curvatues can be discontinuous, see §REF ).", "The neighborhoods of the latter configurations may be studied by re-ordering the masses.", "Figure: (a) Position vectors x 1,2,3 x_{1,2,3} of masses relative to origin and Jacobi vectors J 1,2,3 J_{1,2,3}.", "(b) The shape sphere is topologically a 2-sphere with the three collision points C 1,2,3 C_{1,2,3} removed, endowed with the quotient JM metric of negative gaussian curvature.", "Coordinates and physical locations on the shape sphere are illustrated.", "2η\\eta is the polar angle (0≤η≤π/20 \\le \\eta \\le \\pi /2).", "2ξ 2 \\xi _2 is the azimuthal angle (0≤ξ 2 ≤π0\\le \\xi _2 \\le \\pi ).", "The `great circle' composed of the two longitudes ξ 2 =0\\xi _2=0 and ξ 2 =π/2\\xi _2=\\pi /2 consists of collinear configurations (syzygies) which include C 1,2,3 C_{1,2,3} and the Euler points E 1,2,3 E_{1,2,3}.", "Lagrange points L 4,5 L_{4,5} lie on the equator η=π/4\\eta = \\pi /4.", "The shape space ℝ 3 {\\mathbb {R}}^3 is a cone on the shape sphere.", "The origin r=0r=0 of shape space is the triple collision point.", "(c) Flowchart of submersions." ], [ "Quotient JM metrics on shape space, the three-sphere and the shape sphere", "Submersion from ${\\mathbb {C}}^2$ to shape space ${\\mathbb {R}}^3$ : Rotations $z_j \\rightarrow e^{i \\theta } z_j$ act as isometries of the JM metric (REF ) on ${\\mathbb {C}}^2$ .", "In the Hopf coordinates of Eq.", "(REF ), $z_1=r e^{i ( \\xi _1+ \\xi _2)} \\sin \\eta \\quad \\text{and} \\quad z_2=r e^{i ( \\xi _1- \\xi _2)} \\cos \\eta , \\quad $ rotations are generated by translations $\\xi _1 \\rightarrow \\xi _1 + \\theta $ and a discrete shift $\\xi _2 \\rightarrow \\xi _2 + \\pi $ (${\\rm mod} \\; 2\\pi $ ).", "The shift in $\\xi _2$ rotates $z_i \\mapsto -z_i$ , which is not achievable by a translation in $\\xi _1$ due to its restricted range, $|\\xi _2| \\le \\xi _1 \\le 2\\pi - |\\xi _2|$ and $-\\pi \\le \\xi _2 \\le \\pi $ .", "To quotient by this isometry, we define a submersion from ${\\mathbb {C}}^2 \\rightarrow {\\mathbb {R}}^3$ taking $(r, \\eta , \\xi _1,\\xi _2) \\mapsto (r, \\eta , \\xi _2)\\quad \\text{if} \\quad \\xi _2 \\ge 0 \\quad \\text{and} \\quad (r,\\eta , \\xi _1,\\xi _2) \\mapsto (r,\\eta , \\xi _2+ \\pi ) \\quad \\text{if} \\quad \\xi _2 < 0.$ The radial, polar and azimuthal coordinates on ${\\mathbb {R}}^3$ are given by $r$ , $2\\eta $ and $2\\xi _2$ with $m_1$ -$m_2$ collisions occurring on the ray $\\eta =0$ .", "Under the linearization of this submersion at a point $p \\in {\\mathbb {C}}^2$ , $V(p)$ is spanned by $\\partial _{\\xi _1}$ and $H(p)$ by $\\partial _r$ , $\\partial _\\eta $ and $\\cos 2 \\eta \\; \\partial _{\\xi _1} + \\partial _{\\xi _2}$ .", "These horizontal basis vectors are mapped respectively to $\\partial _r$ , $\\partial _\\eta $ and $\\partial _{\\xi _2}$ under the linearization of the map.", "Requiring lengths of horizontal vectors to be preserved we arrive at the following quotient JM metric on ${\\mathbb {R}}^3$ , conformal to the flat metric on ${\\mathbb {R}}^3$ : $ds^2= \\left( E + \\frac{Gm^3 h(\\eta ,\\xi _2)}{r^2} \\right)\\left(dr^2+r^2\\left(d\\eta ^2+\\sin ^2 2\\eta \\;d\\xi _2^2\\right)\\right).$ This metric may also be viewed as conformal to a cone on a round 2-sphere of radius one-half, since $0 \\le 2 \\eta \\le \\pi $ and $0 \\le 2 \\xi _2 \\le 2 \\pi $ are the polar and azimuthal angles.", "Submersion from shape space to the shape sphere: The group $\\bf {R}^+$ of scalings $(r, \\eta , \\xi _2) \\mapsto (\\lambda r, \\eta , \\xi _2)$ acts as an isometry of the zero-energy JM metric (REF ) on shape space ${\\mathbb {R}}^3$ .", "The orbits are radial rays emanating from the origin (and the triple collision point at the origin, which we exclude).", "The quotient space ${\\mathbb {R}}^3/{\\rm scaling}$ is the shape sphere ${\\mathbb {S}}^2$ .", "We define a submersion from shape space to the shape sphere taking $(r, \\eta , \\xi _2) \\mapsto (\\eta , \\xi _2)$ .", "Under the linearization of this map at $p \\in {\\mathbb {R}}^3$ , $V(p) = \\text{span}(\\partial _r)$ .", "Its orthogonal complement $H(p)$ is spanned by $\\partial _\\eta $ and $\\partial _{\\xi _2}$ which project to $\\partial _\\eta $ and $\\partial _{\\xi _2}$ on ${\\mathbb {S}}^2$ .", "Requiring the submersion to be Riemannian, we get the quotient `pair of pants' JM metric on the shape sphere which is conformal to the round metric on a 2-sphere of radius one-half: $ds^2 = Gm^3 h(\\eta ,\\xi _2) \\left(d\\eta ^2+\\sin ^2 2\\eta \\;d\\xi _2^2\\right).$ Submersion from ${\\mathbb {C}}^2$ to ${\\mathbb {S}}^3$ and then to ${\\mathbb {S}}^2$ : For zero energy, it is also possible to quotient the JM metric (REF ) on ${\\mathbb {C}}^2$ , first by its scaling isometries to get ${\\mathbb {S}}^3$ and then by rotations to arrive at the shape sphere.", "Interestingly, it follows from the Lagrange-Jacobi identity that when $E$ and $\\dot{I}$ vanish, $r$ is constant and the motion is confined to a 3-sphere embedded in ${\\mathbb {C}}^2$ .", "To quotient by the scaling isometries $(r, \\eta , \\xi _1, \\xi _2) \\mapsto (\\lambda r, \\eta , \\xi _1, \\xi _2)$ of ${\\mathbb {C}}^2$ , we define the submersion $(r, \\eta , \\xi _1, \\xi _2) \\mapsto (\\eta , \\xi _1, \\xi _2)$ to ${\\mathbb {S}}^3$ , with ranges of coordinates as on ${\\mathbb {C}}^2$ .", "The vertical subspace is spanned by $\\partial _r$ while $\\partial _\\eta $ , $\\partial _{\\xi _1}$ and $\\partial _{\\xi _2}$ span the horizontal subspace.", "The latter are mapped to $\\partial _\\eta $ , $\\partial _{\\xi _1}$ and $\\partial _{\\xi _2}$ on ${\\mathbb {S}}^3$ .", "The submersion is Riemannian provided we endow ${\\mathbb {S}}^3$ with the following conformally-round metric $ds^2=Gm^3 h\\left(\\eta ,\\xi _2\\right)\\left(d\\eta ^2+ d\\xi _1^2-2\\cos 2\\eta \\;d\\xi _1\\;d\\xi _2+ d\\xi _2^2\\right).$ Rotations generated by $\\xi _1 \\rightarrow \\xi _1 + \\theta $ and $\\xi _2 \\rightarrow \\xi _2 + \\pi $ (mod $2\\pi $ ) act as isometries of this metric on ${\\mathbb {S}}^3$ .", "We quotient by rotations to get the metric (REF ) on ${\\mathbb {S}}^2$ via the Riemannian submersion defined by $(\\eta ,\\xi _1,\\xi _2) \\mapsto (\\eta ,\\xi _2) \\quad \\text{if} \\quad \\xi _2 \\ge 0 \\quad \\text{and} \\quad (\\eta ,\\xi _1,\\xi _2) \\mapsto (\\eta ,\\xi _2+\\pi ) \\quad \\text{if} \\quad \\xi _2 < 0.$" ], [ "JM metric in the near-collision limit and its completeness", "The equal-mass JM metric components on configuration space ${\\mathbb {C}}^2$ and its quotients blow up at 2- and 3-body collisions.", "However, we study the geometry in the neighbourhood of collision configurations and show that the curvature remains finite in the limit.", "Remarkably, it takes infinite geodesic time for collisions to occur which we show by establishing the geodesic completeness of the JM metric on ${\\mathbb {C}}^2$ and its quotients.", "By contrast, collisions can occur in finite time for the Newtonian 3-body evolution.", "The JM geodesic flow avoids finite time collisions by reparametrizing time along Newtonian trajectories (see Eq.", "REF ).", "Thus the geodesic reformulation of the inverse-square 3-body problem `regularizes' pairwise and triple collisions." ], [ "Geometry near pairwise collisions", "For equal masses (see §REF ), the first pair of masses collide when $\\eta =0$ (with other coordinates arbitrary) while the other two binary collisions occur at $C_1$ and $C_2$ (see Fig.", "REF ).", "Triple collisions occur when $r = 0$ .", "Unlike for the Newtonian potential, sectional curvatures on coordinate 2-planes are finite at pairwise and triple collisions, though some JM metric (REF ) and Riemann tensor components blow up.", "It is therefore interesting to study the near-collision geometry of the JM metric.", "The geometry of the equal-mass JM metric in the neigbourhood of a binary collision is the same irrespective of which pair of bodies collide.", "Since Hopf coordinates are particularly convenient around $\\eta = 0$ , we focus on collisions between the first pair of masses.", "Montgomery (see eqn.", "3.10c of [20]) studied the near-collision geometry on ${\\mathbb {S}}^2$ and showed that it is geodesically complete.", "Let us briefly recall the argument.", "Expanding the equal-mass ${\\mathbb {S}}^2$ metric (REF ) around the collision point $\\eta = 0$ , we get $ds^2 \\approx \\left(\\frac{G m^3}{2\\eta ^2}\\right) \\left(d\\eta ^2+4\\eta ^2 \\;d\\xi _2^2\\right) = \\frac{G m^3}{2\\rho ^2}(d\\rho ^2 + \\rho ^2 d\\chi ^2)$ where $\\rho = 2 \\eta $ and $\\chi = 2 \\xi _2$ .", "$\\partial _{\\chi }$ is a KVF, so `radial' curves with constant $\\chi $ are geodesics.", "Approaching $\\rho = 0$ along a `radial' geodesic shows that the collision point $\\rho = 0$ is at an infinite distance $(\\sqrt{G m^3/2} \\int _{\\rho _0}^0 d\\rho /\\rho )$ from any point $(\\rho _0,\\chi )$ in its neighborhood $(0< \\rho _0 \\ll 1)$ .", "The symmetry of the metric under exchange of masses ensures that the same holds for the other two collision points: geodesics may be extended indefinitely.", "Thus the shape sphere (${\\mathbb {S}}^2$ with three collision points excluded) is geodesically complete.", "To clarify the near-collision geometry let $d \\lambda = -d\\rho /\\sqrt{2}\\rho $ or $\\lambda = - \\log (\\rho /\\rho _0)/\\sqrt{2}$ .", "This effectively stretches out the neighborhood of the collision point $\\lambda = \\infty $ .", "The asymptotic metric $ds^2 = Gm^3 \\left( d \\lambda ^2 + d\\chi ^2/2 \\right)$ for $0 \\le \\chi \\le 2\\pi $ and $\\lambda \\ge 0$ is the metric on a semi-infinite right-circular cylinder of radius $\\sqrt{Gm^3/2}$ with $\\lambda $ the coordinate along the height and $\\chi $ the azimuthal angle.", "Thus the JM metric looks like that of a semi-infinite cylinder near any of the collision points.", "More generally, for unequal masses, the near-collision metric (REF ) is $ds^2 \\approx \\frac{G m_1 m_2 M_1}{2\\eta ^2} \\left(d\\eta ^2 + 4\\eta ^2 d\\xi _2^2\\right)$ (see Eq.", "(REF -REF )) and essentially the same argument implies that the JM metric on the shape sphere is geodesically complete for arbitrary masses.", "Since ${\\mathbb {S}}^2$ arises as a Riemannian submersion of ${\\mathbb {R}}^3$ , ${\\mathbb {S}}^3$ and ${\\mathbb {C}}^2$ , the infinite distance to binary collision points on the shape sphere can be used to show that the same holds on each of the higher dimensional manifolds.", "To see this, consider the submersion from (say) ${\\mathbb {C}}^2$ to ${\\mathbb {S}}^2$ .", "Any curve $\\tilde{\\gamma }$ on ${\\mathbb {C}}^2$ maps to a curve $\\gamma $ on ${\\mathbb {S}}^2$ with $l(\\tilde{\\gamma }) \\ge l(\\gamma )$ since the lengths of horizontal vectors are preserved.", "If there was a binary collision point at finite distance on ${\\mathbb {C}}^2$ , there would have to be a geodesic of finite length ending at it.", "However, such a geodesic would project to a curve on the shape sphere of finite length ending at a collision point, contradicting its completeness.", "Thus we have shown that the JM metrics (necessarily of zero energy) on ${\\mathbb {S}}^2$ and ${\\mathbb {S}}^3$ with binary collision points removed, are geodesically complete for arbitrary masses.", "On the other hand, to examine completeness on ${\\mathbb {C}}^2$ and ${\\mathbb {R}}^3$ we must allow for triple collisions as well as non-zero energy.", "Geodesic completeness in these cases is shown in §REF .", "In the sequel we examine the near-collision geometry on ${\\mathbb {R}}^3$ , ${\\mathbb {S}}^3$ and ${\\mathbb {C}}^2$ in somewhat greater detail by Laurent expanding the JM metric components around $\\eta = 0$ and keeping only leading terms.", "Shape space geometry near binary collisions: The equal-mass shape space metric around $\\eta =0$ , in the leading order, becomes $ds^2 \\approx \\frac{G m^3}{2\\eta ^2 r^2} \\left( dr^2+ r^2 \\left(d\\eta ^2+4\\eta ^2 \\;d\\xi _2^2\\right)\\right) = G m^3 \\left( \\frac{2 dr^2}{\\rho ^2 r^2} + \\frac{d\\rho ^2}{2\\rho ^2} +\\frac{d\\chi ^2}{2} \\right),$ where $\\rho =2\\eta $ and $\\chi =2\\xi _2$ .", "We define new coordinates $\\lambda $ and $\\kappa $ by $d \\lambda = -d\\rho /\\sqrt{2}\\rho $ , $d\\kappa = dr/r$ so that $\\rho = \\rho _0 e^{-\\sqrt{2}\\lambda }$ .", "In these coordinates the collision occurs at $\\lambda = \\infty $ .", "The asymptotic metric is $ds^2 \\approx G m^3 \\left( \\frac{2}{\\rho _0^2}e^{2 \\sqrt{2}\\lambda } d\\kappa ^2 + d \\lambda ^2 + \\frac{1}{2}d\\chi ^2\\right)$ where $0 \\le \\chi \\le 2 \\pi $ (periodic), $\\lambda \\ge 0$ and $-\\infty < \\kappa < \\infty $ .", "This metric has a constant scalar curvature of $-4/Gm^3$ .", "The sectional curvature in the $\\partial _\\lambda -\\partial _\\kappa $ plane is equal to $-2/Gm^3$ , it vanishes in the other two coordinate planes.", "These values of scalar and sectional curvatures agree with the limiting values at the 1-2 collision point calculated for the full metric on shape space.", "The near-collision topology of shape space is that of the product manifold $\\bf {S}^1_\\chi \\times \\bf {R}^+_\\lambda \\times \\bf {R}_\\kappa $ .", "Near-collision geometry on ${\\mathbb {C}}^2$ : The equal-mass JM metric in leading order around $\\eta = 0$ is $ds^2 \\approx \\frac{G m^3}{2\\eta ^2 r^2} \\left( dr^2+ r^2 \\left(d\\eta ^2+d\\xi _1^2-2(1-2\\eta ^2) d\\xi _1 d\\xi _2+d\\xi _2^2\\right)\\right).$ Let us define new coordinates $\\lambda ,\\kappa ,\\xi _\\pm $ such that $d \\lambda = -d\\eta /\\sqrt{2}\\eta $ , $d\\kappa = -dr/r$ and $\\xi _\\pm =\\xi _1\\pm \\xi _2$ .", "$0 \\le \\xi _\\pm \\le 2\\pi $ are periodic coordinates parametrizing a torus.", "The asymptotic metric is $ds^2 \\approx G m^3\\left( \\frac{d\\kappa ^2}{2\\eta ^2} + d \\lambda ^2 + \\frac{1}{2\\eta ^2} d\\xi _-^2 + \\frac{1}{2} d\\xi _+^2\\right)$ where $\\eta = \\eta _0 e^{-\\sqrt{2}\\lambda }$ .", "This metric has a constant scalar curvature $-12/Gm^3$ .", "The sectional curvature of any coordinate plane containing $\\partial _{\\xi _+}$ vanishes due to the product form of the metric.", "The sectional curvatures of the remaining coordinate planes ($\\partial _\\kappa -\\partial _\\lambda , \\partial _\\kappa - \\partial _{\\xi _-}, \\partial _{\\xi _-}-\\partial _\\lambda $ ) are equal to $-2/Gm^3$ .", "The scalar and sectional curvatures (of corresponding planes) of this metric agree with the limiting values computed from the full metric on ${\\mathbb {C}}^2$ .", "Near-collision geometry on ${\\mathbb {S}}^3$ : The submersion ${\\mathbb {C}}^2 \\rightarrow {\\mathbb {S}}^3$ takes $(\\kappa , \\lambda , \\xi _\\pm ) \\mapsto ( \\lambda , \\xi _\\pm )$ .", "As the coordinate vector fields on ${\\mathbb {C}}^2$ are orthogonal, from (REF ) the asymptotic metric on ${\\mathbb {S}}^3$ near the 1-2 collision point is $ds^2 \\approx G m^3\\left( d \\lambda ^2 + \\frac{1}{2\\eta ^2} d\\xi _-^2 + \\frac{1}{2} d\\xi _+^2\\right).$ This metric has a constant scalar curvature equal to $-4/Gm^3$ .", "The sectional curvatures on the $\\lambda - \\xi _-$ coordinate 2-plane is $-2/Gm^3$ while it vanishes on the other two coordinate 2-planes." ], [ "Geometry on ${\\mathbb {R}}^3$ and {{formula:bf1c322d-3e74-4a02-8b0b-f0f5b964ee56}} near triple collisions", "We argue that the triple collision configuration (which occurs at $r=0$ on ${\\mathbb {C}}^2$ or shape space ${\\mathbb {R}}^3$ ) is at infinite distance from other configurations with respect to the equal-mass JM metrics (Eqs.", "(REF ),(REF )), which may be written in the form: $ds^2 = (G m^3 h/r^2) dr^2 + G m^3 \\: h \\: g_{i j} \\: dx^i dx^j.$ $g_{ij}$ is the positive (round) metric on $\\mathbf {S}^3$ ($x^i = (\\eta , \\xi _1, \\xi _2)$ ) or $\\mathbf {S}^2$ ($x^i = (\\eta , \\xi _2)$ ) of radius one-half: $g^{{\\mathbb {C}}^2}_{ij} = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & -\\cos 2\\eta \\\\ 0 & -\\cos 2\\eta & 1 \\end{pmatrix} \\quad \\text{and} \\quad g^{{\\mathbb {R}}^3}_{ij} = \\begin{pmatrix} 1 & 0 \\\\ 0 & \\sin 2\\eta \\end{pmatrix} .$ Together with our results on pairwise collisions (§REF ), it will follow that the manifolds are geodesically complete.", "As a consequence, the geodesic flow reformulation of the 3-body problem regularizes triple collisions.", "To show that triple collision points are at infinite distance we will use the previously obtained lower bound on the conformal factor, $h(\\xi _2, \\eta ) \\ge 3$ (see Eqn.", "REF ).", "Let $\\gamma (t)$ be a curve joining a non-collision point $\\gamma (t_0) \\equiv (r_0, x^i_0)$ and the triple collision point $\\gamma (t_1) \\equiv (r=0, x^i_1)$ .", "We show that its length $l(\\gamma )$ is infinite.", "Since $G m^3 h g_{ij}$ is a positive matrix, $l(\\gamma ) = \\int _{t_0}^{t_1} dt\\sqrt{\\frac{G m^3 h}{r^2} \\dot{r}^2 + G m^3 h g_{i j} \\dot{x}^i \\dot{x}^j} \\; \\ge \\; \\int _{t_0}^{t_1} dt \\sqrt{\\frac{G m^3 h}{r^2} \\dot{r}^2 }.$ Now using $|\\dot{r}| \\ge - \\dot{r}$ and $h \\ge 3$ , we get $l(\\gamma ) \\ge -\\sqrt{3 G m^3} \\int _{t_0}^{t_1} \\frac{\\dot{r}}{ r} dt = \\sqrt{3 G m^3}\\int _{0}^{r_0} \\frac{d r}{ r} = \\infty .$ In particular, a geodesic from a non-collision point to the triple collision point has infinite length.", "Despite appearances, the above inequality $l(\\gamma ) \\ge \\sqrt{3 G m^3} \\int _0^{r_0} dr/r$ does not imply that radial curves are always geodesics.", "This is essentially because $h$ along $\\gamma $ may be less than that on the corresponding radial curve.", "However, if $(\\eta , \\xi _1, \\xi _2)$ is an angular location where $h$ is minimal (locally), then the radial curve with those angular coordinates is indeed a geodesic because a small perturbation to the radial curve increases $h$ and consequently its length.", "The global minima of $h$ ($h = 3$ ) occur at the Lagrange configurations $L_{4,5}$ and local minima ($h = 9/2$ ) are at the Euler configurations $E_{1,2,3}$ indicating that radial curves at these angular locations are geodesics.", "In fact, the Christoffel symbols $\\Gamma ^i_{rr}$ vanish for $i = \\eta , \\xi _1, \\xi _2$ at $L_{4,5}$ and at $E_{1,2,3}$ so that radial curves $\\gamma = (r(t), x^i_0)$ satisfying $\\ddot{r} + \\Gamma ^r_{rr} \\dot{r}^2= 0$ are geodesics.", "These radial geodesics at minima of $h$ describe Lagrange and Euler homotheties (where the masses move radially inwards/outwards to/from their CM which is the center of similitude).", "These homotheties take infinite (geodesic) time to reach the triple collision.", "By contrast, the corresponding Lagrange and Euler homothety solutions to Newton's equations reach the collision point in finite time.", "This difference is due to an exponential time-reparametrization of geodesics relative to trajectories.", "In fact, if $t$ is trajectory time and $s$ arc-length along geodesics, then from § and §REF , $\\sigma = ds/dt = \\sqrt{2} (E + 3 Gm^3/r^2)$ since $h = 3$ .", "Near a triple collision (small $r$ ), $ds^2 \\approx 3 Gm^3 dr^2/r^2$ so that $s \\approx - \\frac{1}{2}\\sqrt{3 Gm^3} \\log (1-t/t_c) \\rightarrow \\infty $ as $t \\rightarrow t_c = r(0)^2/2\\sqrt{6 G m^3}$ which is the approximate time to collision.", "In fact, the exact collision time $t_c = \\sqrt{6 G m^3} \\left( -1 + \\sqrt{1+ {\\kappa r(0)^2/6 G m^3}} \\right)/\\kappa $ may be obtained by reducing Newton's equations for Lagrange homotheties to the one body problem $r^3 \\ddot{r} = - 6 G m^3$ whose conserved energy is $\\kappa = \\dot{r}^2 - 6 G m^3 / r^2$ .", "These homothety solutions illustrate how the geodesic flow reformulation regularizes the original Newtonian 3 body dynamics in the inverse-square potential.", "More generally, for unequal masses (REF )-(REF ) give the JM metric $ds^2 = \\tilde{h} dr^2/r^2 + \\tilde{g}_{i j} dx^i dx^j$ where $\\tilde{h} = \\frac{G m_1 m_2 M_1}{\\sin ^2 \\eta } + \\frac{G m_2 m_3 M_2}{\\left|\\cos \\eta - \\mu _1 \\sqrt{{M_2}/{M_1}}e^{2 i \\xi _2} \\sin \\eta \\right|^2} + \\frac{G m_1 m_3 M_2}{\\left|\\cos \\eta + \\mu _2 \\sqrt{{M_2}/{M_1}}e^{2 i \\xi _2} \\sin \\eta \\right|^2}.$ Irrespective of the masses, $\\tilde{g}_{ij}$ (REF ) is positive and $\\tilde{h}$ has a strictly positive lower bound (e.g.", "$G m_1 m_2 M_1$ ).", "Thus by the same argument as above, triple collisions are at infinite distance.", "Combining this with the corresponding results for pairwise collision points (§REF ), we conclude that the zero-energy JM metrics on ${\\mathbb {C}}^2$ and ${\\mathbb {R}}^3$ are geodesically complete for arbitrary masses.", "For non-zero energy, $ds^2 = (E + \\tilde{h}/r^2) (dr^2 + r^2 \\tilde{g}_{ij} dx^i dx^j)$ which can be approximated with the zero-energy JM metrics both near binary (say, $\\eta = 0$ ) and triple ($r = 0$ ) collisions.", "If $\\gamma $ is a curve ending at the triple collision, $l(\\gamma ) \\ge l(\\tilde{\\gamma })$ where $\\tilde{\\gamma }$ is a `tail end' of $\\gamma $ lying in a sufficiently small neighborhood of $r=0$ (i.e., $r \\ll |\\tilde{h}/E|^{1/2}$ which is guaranteed, say, if $r \\ll |G m_1 m_2 M_1/E|^{1/2}$ ).", "But then, $l(\\tilde{\\gamma })$ may be estimated using the zero-energy JM metric giving $l(\\tilde{\\gamma }) = \\infty $ .", "Thus $l(\\gamma ) = \\infty $ .", "A similar argument shows that curves ending at binary collisions have infinite length.", "Thus we conclude that the JM metrics on ${\\mathbb {C}}^2$ and ${\\mathbb {R}}^3$ are geodesically complete for arbitrary energies and masses." ], [ "Scalar curvature for equal masses and zero energy", "A geodesic through $P$ in the direction $u$ perturbed along $v$ is linearly stable/unstable [see §REF ] according as the sectional curvature $K_P(u,v)$ is positive/negative.", "The scalar curvature $R$ at $P$ is proportional to an average of sectional curvatures in planes through $P$ (§REF ).", "Thus $R$ encodes an average notion of geodesic stability.", "Here, we evaluate the scalar curvature $R$ of the equal-mass zero-energy JM metric on ${\\mathbb {C}}^2$ and its submersions to ${\\mathbb {R}}^3$ , ${\\mathbb {S}}^3$ and ${\\mathbb {S}}^2$ .", "In each case, due to the rotation and scaling isometries, $R$ is a function only of the coordinates $\\eta $ and $\\xi _2$ that parametrize the shape sphere.", "In [20] Montgomery proves that $R_{{\\mathbb {S}}^2} \\le 0$ with equality at Lagrange and collision points (see Fig.", "REF ).", "We generalize this result and prove that the scalar curvatures on ${\\mathbb {C}}^2$ , ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ are strictly negative and bounded below (see Fig.", "REF ) indicating widespread linear instability of the geodesic dynamics.", "(Note that hyperbolicity of the configuration space quotiented by translations, rotations and scaling does not extend in a simple manner to the 4-body problem [30].)", "Figure: Gaussian curvature KK (in units of 1/Gm 3 {1/Gm^3}) on 𝕊 2 {\\mathbb {S}}^2 for equal masses and E=0E=0.", "K=0K = 0 at L 4,5 L_{4,5} and C 1,2,3 C_{1,2,3}.Figure: Scalar curvatures RR on ℂ 2 {\\mathbb {C}}^2, 𝕊 3 {\\mathbb {S}}^3 and ℝ 3 {\\mathbb {R}}^3 in units of 1/Gm 3 1/{G m^3}.", "RR is strictly negative and has a global maximum at L 4,5 L_{4,5} in all cases.", "It attains a global minimum at C 1,2,3 C_{1,2,3} on ℂ 2 {\\mathbb {C}}^2 and a local maximum at collisions on ℝ 3 {\\mathbb {R}}^3 and 𝕊 3 {\\mathbb {S}}^3.", "E 1,2,3 E_{1,2,3} are saddles on ℂ 2 {\\mathbb {C}}^2 and global minima on ℝ 3 {\\mathbb {R}}^3 and 𝕊 3 {\\mathbb {S}}^3.Scalar curvature on ${\\mathbb {S}}^2$ : The quotient JM metric on ${\\mathbb {S}}^2$ (REF ) is conformal to the round (kinetic) metric on a sphere of radius $1/2$ : $ds_{{\\mathbb {S}}^2}^2 = Gm^3 \\: h(\\eta , \\xi _2) \\: ds_{\\rm kin}^2 \\quad \\text{where}\\quad ds_{\\rm kin}^2 =d\\eta ^2+ \\sin ^2 2\\eta \\: d\\xi _2^2.$ Here the conformal factor ($h = - (r^2/Gm^3) \\times $ potential energy) (REF ) is a strictly positive function on the shape sphere with double poles at collision points.", "The scalar curvature of (REF ) is $R_{{\\mathbb {S}}^2} = \\frac{1}{ G m^3 h^3 } \\left( 8 h^2 + |{\\bf \\nabla }h|^2 - h \\Delta h \\right),$ where $\\Delta $ is the Laplacian and ${\\bf \\nabla }^i h = g^{ij} \\partial _j h$ the gradient on ${\\mathbb {S}}^2$ relative to the kinetic metric: $\\Delta h = \\left(\\frac{1}{\\sin ^2 2\\eta } \\frac{\\partial ^2 h}{\\partial \\xi _2^2} + 2\\cot 2\\eta \\frac{\\partial h}{\\partial \\eta } + \\frac{\\partial ^2 h}{\\partial \\eta ^2} \\right) \\quad \\text{and} \\quad |{\\bf \\nabla }h|^2 = \\frac{1}{\\sin ^2 2\\eta } \\left(\\frac{\\partial h}{\\partial \\xi _2}\\right)^2 + \\left(\\frac{\\partial h}{\\partial \\eta }\\right)^2 .$ In fact we have an explicit formula for the scalar curvature, $R_{{\\mathbb {S}}^2} = AB/C$ where $A &=& 8 \\sin ^2\\eta \\left((\\cos 2 \\eta + 2)^2 - 3 \\sin ^2 2 \\eta \\cos ^2 2\\xi _2 \\right), \\quad C = {3 \\left(2 \\sin ^2 2 \\eta \\cos 4 \\xi _2+\\cos 4 \\eta -13\\right)^3} \\;\\; \\& \\cr B &=& \\left(-8 \\sin ^4 2 \\eta \\cos 8 \\xi _2 - 16 \\sin ^2 2 \\eta \\cos 4 \\xi _2 (\\cos 4 \\eta - 29) + 236 \\cos 4 \\eta - 3 \\cos 8 \\eta + 727 \\right).$ As shown in [20], $R_{{\\mathbb {S}}^2} \\le 0$ with equality only at Lagrange and collision points.", "Negativity of $R_{{\\mathbb {S}}^2}$ also follows from (REF ): each factor in the numerator is $\\ge 0$ (the third vanishes at $L_{4,5}$ , the second at $C_{1,2}$ and the first at $C_3$ ) while the denominator is strictly negative.", "We now use this to show that the scalar curvatures on configuration space ${\\mathbb {C}}^2$ and its quotients ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ are strictly negative.", "Scalar curvature on ${\\mathbb {C}}^2$ : The equal-mass zero-energy JM metric on ${\\mathbb {C}}^2$ from Eq.", "(REF ) is $ds^2_{{\\mathbb {C}}^2}= \\left( {G m^3}/{r^2} \\right) h(\\eta ,\\xi _2) \\left(dr^2+r^2\\left(d\\eta ^2+ d\\xi _1^2-2\\cos 2\\eta \\;d\\xi _1\\;d\\xi _2+ d\\xi _2^2\\right)\\right).$ The scalar curvature of this metric is expressible as $R_{{\\mathbb {C}}^2}= \\left( 3/2G m^3h^3 \\right) \\left(4h^2+ |{\\bf \\nabla }h|^2 -2 h \\: \\Delta h \\right),$ where $\\Delta h$ and ${\\bf \\nabla }h$ are the Laplacian and gradient with respect to the round metric on $\\bf S^2$ of radius one-half ().", "Due to the scaling and rotation isometries, $R_{{\\mathbb {C}}^2}$ is in fact a function on the shape sphere.", "The scalar curvatures on ${\\mathbb {C}}^2$ (REF ) and ${\\mathbb {S}}^2$ (REF ) are simply related: $R_{{\\mathbb {C}}^2}=3 R_{{\\mathbb {S}}^2} - \\left( 3/2 G m^3 h^3 \\right) \\left( 12 h^2 +|{\\bf \\nabla }h|^2\\right).$ This implies $R_{{\\mathbb {C}}^2} < 0$ since the second term is strictly negative everywhere as we now show.", "Notice that the second term can vanish only when $h$ is infinite, i.e., at collisions.", "Taking advantage of the fact that the geometry (on ${\\mathbb {S}}^2$ and ${\\mathbb {C}}^2$ ) in the neighborhood of all 3 collision points is the same for equal masses, it suffices to check that the second term has a strictly negative limit at $C_3$ $(\\eta = 0)$ .", "Near $\\eta =0$ , $h \\sim 1/2 \\eta ^2$ so that $R_{{\\mathbb {C}}^2} \\rightarrow - 12/Gm^3 < 0$ .", "Combining with the $r$ -independence of $R_{{\\mathbb {C}}^2}$ , we see that the scalar curvature is non-singular at binary and triple collisions.", "With a little more effort, we may obtain a non-zero upper bound for the Ricci scalar on ${\\mathbb {C}}^2$ .", "Indeed, using $R_{{\\mathbb {S}}^2} \\le 0$ and the inequality $12 h^2 + |{\\bf \\nabla }h|^2 \\ge \\zeta h^3$ proved in Appendix , we find $R_{{\\mathbb {C}}^2} < - 3 \\zeta /2 G m^3 \\quad \\text{where} \\quad \\zeta = 55/27.$ Numerically, we estimate the optimal value of $\\zeta $ to be $8/3$ .", "Scalar curvatures on ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ : Recall that the equal-mass zero-energy quotient JM metrics on shape space ${\\mathbb {R}}^3$ (REF ) and ${\\mathbb {S}}^3$ (REF ) are $ds_{{\\mathbb {R}}^3}^2 &=& \\left( Gm^3 h/r^2 \\right) \\left(dr^2+r^2\\left(d\\eta ^2+\\sin ^22\\eta \\;d\\xi _2^2\\right) \\right) \\quad \\text{and} \\quad \\cr \\cr ds_{{\\mathbb {S}}^3}^2 &=& Gm^3 h \\: \\left(d\\eta ^2+ d\\xi _1^2-2\\cos 2\\eta \\;d\\xi _1\\;d\\xi _2+ d\\xi _2^2\\right).$ The corresponding scalar curvatures are $R_{{\\mathbb {R}}^3}= \\left(16h^2+3 |{\\bf \\nabla }h|^2 - 4 h \\Delta h \\right)/2 G m^3h^3\\;\\; \\text{and} \\;\\;R_{{\\mathbb {S}}^3}= \\left(12h^2+3|{\\bf \\nabla }h|^2-4h \\Delta h\\right)/2 G m^3h^3.$ Here $\\Delta h$ and ${\\bf \\nabla }h$ are as in Eq.", "().", "The scalar curvatures are related to that on ${\\mathbb {S}}^2$ as follows $R_{{\\mathbb {R}}^3}=2 R_{{\\mathbb {S}}^2} - \\left( 16 h^2 +|{\\bf \\nabla }h|^2\\right)/{2 G m^3 h^3}\\quad \\text{and} \\quad R_{{\\mathbb {S}}^3}=2 R_{{\\mathbb {S}}^2} - \\left( 20 h^2 + |{\\bf \\nabla }h|^2\\right)/{2 G m^3 h^3}.$ As in the case of ${\\mathbb {C}}^2$ we check that the second terms in both relations are strictly negative.", "This implies both the scalar curvatures are strictly negative.", "In fact, using the inequality $12 h^2 + |{\\bf \\nabla }h|^2 > \\zeta h^3$ (see Appendix ) we find (non-optimal) non-zero upper bounds $R_{{\\mathbb {S}}^3, {\\mathbb {R}}^3} < - \\zeta /2 G m^3 \\quad \\text{where} \\quad \\zeta = 55/27.$ Moreover, we note that $R_{{\\mathbb {C}}^2} = R_{{\\mathbb {S}}^3} - \\frac{h \\Delta h}{G m^3 h^3} < R_{{\\mathbb {S}}^3} \\quad \\text{and} \\quad R_{{\\mathbb {S}}^3} = R_{{\\mathbb {R}}^3} - \\frac{4 h^2}{2 G m^3 h^3} \\le R_{{\\mathbb {R}}^3},$ with equality at collision configurations.", "Recalling that on the shape sphere, the scalar curvature vanishes at collision points (in a limiting sense) and at Lagrange points, we have the following inequalities $0 \\ge R_{{\\mathbb {S}}^2} > R_{{\\mathbb {R}}^3} \\ge R_{{\\mathbb {S}}^3} > R_{{\\mathbb {C}}^2}.$ Thus we have the remarkable result that the scalar curvatures of the JM metric on ${\\mathbb {C}}^2$ and its quotients by scaling $({\\mathbb {S}}^3)$ and rotations $({\\mathbb {R}}^3)$ are strictly negative everywhere and also strictly less than that on ${\\mathbb {S}}^2$ .", "So the full geodesic flow on ${\\mathbb {C}}^2$ is in a sense more unstable than the corresponding flow on ${\\mathbb {S}}^2$ .", "In addition to strict negativity, we may also show that the scalar curvatures are bounded below.", "For instance, from Eq.", "(REF ) $R_{{\\mathbb {S}}^2}$ can go to $- \\infty $ only when $\\Delta h \\rightarrow \\infty $ since $h \\ge 3$ .", "Now from Eq.", "(REF ) $\\Delta h$ can diverge only when $\\sin 2 \\eta = 0$ or when one of the relevant derivatives of $h$ diverges.", "From Eq.", "(REF ) this can happen only if $\\eta = 0$ (C3) or $\\eta = \\pi /2$ (E3) or when one of the $v_i \\rightarrow \\infty $ , i.e., at collisions.", "However $\\Delta h = 66$ is finite at $\\eta = \\pi /2$ and we know from §REF that $R_{{\\mathbb {S}}^2}$ is finite at collisions so that $R_{{\\mathbb {S}}^2}$ is bounded below.", "The same proof shows that scalar curvatures are bounded below on ${\\mathbb {R}}^3, {\\mathbb {S}}^3$ and ${\\mathbb {C}}^2$ as well." ], [ "Sectional curvature for three equal masses", "In §REF , we showed that the Ricci scalars $R$ on configuration space and its quotients are negative everywhere, save at Lagrange and collision points on the shape sphere where it vanishes.", "However, $R$ encodes the stability of geodesics only in an average sense.", "More precisely, a geodesic through $P$ in the direction $u$ subject to a perturbation along $v$ is linearly stable/unstable according as the sectional curvature $K_P(u,v)$ is positive/negative (see §REF ).", "Here, the sectional curvature which is a function only of the 2-plane spanned by $u$ and $v$ generalizes the Gaussian curvature to higher dimensions.", "It is defined as the ratio of the curvature biquadratic $\\mathcalligra {r} = g(R(u,v)v,u)$ to the square of the area ${\\rm Ar}(u,v)^2 = g(u,u) g(v,v) - g(u,v) g(v,u)$ of the parallelogram spanned by $u$ and $v$ .", "Here $g(u,v)$ is the Riemannian inner product and $R(u,v) = [{\\bf \\nabla }_u, {\\bf \\nabla }_v] - {\\bf \\nabla }_{[u,v]}$ the curvature tensor with components $R(e_i, e_j) e_k = R^l_{\\; k ij} e_l$ in any basis for vector fields.", "Furthermore, if $e_1, \\ldots , e_n$ are an orthonormal basis for the tangent space at $P$ , then the scalar curvature $R = \\sum _{i \\ne j} K(e_i,e_j)$ is the sum of sectional curvatures in $n \\atopwithdelims ()2$ planes through $P$ .", "It may also be regarded as an average of the curvature biquadratic $R = \\iint \\mathcalligra {r}(u,v) d\\mu _g(u) d\\mu _g(v)$ where $d\\mu _g(u) = \\exp \\left(- u^i u^j g_{ij}/2 \\right) du$ is the gaussian measure on tangent vectors with mean zero and covariance $g^{ij}$ [31].", "Thus $R$ provides an averaged notion of stability.", "To get a more precise measure of linear stability of geodesics we find the sectional curvatures in various (coordinate) tangent 2-planes of the configuration space and its quotients.", "On account of the isometries, these sectional curvatures are functions only of $\\eta $ and $\\xi _2$ [explicit expressions are omitted due to their length].", "Unlike scalar curvatures which were shown to be non-positive, we find planes in which sectional curvatures are non-positive as well as planes where they can have either sign.", "O'Neill's theorem allows us to determine or bound certain sectional curvatures on the configuration space ${\\mathbb {C}}^2$ in terms of the more easily determined curvatures on its quotients.", "Roughly, the sectional curvature of a horizontal two-plane increases under a Riemannian submersion.", "Suppose $f: (M,g) \\rightarrow (N,\\tilde{g})$ is a Riemannian submersion.", "Then O'Neill's theorem [29] states that the sectional curvature in any horizontal 2-plane at $m \\in M$ is less than or equal to that on the corresponding 2-plane at $f(m) \\in N$ : $K_N(df(X),df(Y))=K_M(X,Y)+\\frac{3}{4}\\frac{|[X,Y]^V|^2}{{\\rm Ar}(X,Y)^2}.$ Here $X$ and $Y$ are horizontal fields on $M$ spanning a non-degenerate 2-plane (${\\rm Ar}(X,Y)^2 \\ne 0$ ) and $[X,Y]^V$ is the vertical projection of their Lie bracket.", "In particular, the sectional curvatures are equal everywhere if $X$ and $Y$ are coordinate vector fields.", "We consider sectional curvatures in 6 interesting 2 planes on ${\\mathbb {C}}^2$ which are horizontal with respect to submersions to ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ .", "Under the submersion from ${\\mathbb {C}}^2$ to ${\\mathbb {R}}^3$ (§REF ), the horizontal basis vectors $\\partial _r$ , $\\partial _\\eta $ and $\\partial _\\xi \\equiv \\cos 2 \\eta \\partial _{\\xi _1} + \\partial _{\\xi _2}$ map respectively to $\\partial _r$ , $\\partial _\\eta $ and $\\partial _{\\xi _2}$ defining three pairs of corresponding 2-planes.", "Since $[\\partial _r, \\partial _\\eta ] $ and $[\\partial _r,\\partial _\\xi ]$ vanish, we have $K_{{\\mathbb {C}}^2}(\\partial _r,\\partial _\\eta ) = K_{{\\mathbb {R}}^3}(\\partial _r,\\partial _\\eta )$ and $K_{{\\mathbb {C}}^2}(\\partial _r,\\partial _\\xi ) = K_{{\\mathbb {R}}^3}(\\partial _r,\\partial _{\\xi _2})$ .", "Fig.", "REF shows that $K_{{\\mathbb {C}}^2}(\\partial _r,\\partial _\\eta )$ is mostly negative, though it is not continuous at $E_3$ , $C_1$ and $C_2$ .", "On the other hand $K_{{\\mathbb {C}}^2}(\\partial _r,\\partial _{\\xi })$ is largely negative except in a neighbourhood of $C_3$ .", "Finally, as $[\\partial _\\xi ,\\partial _\\eta ]^V = - 2 \\sin 2 \\eta \\partial _{\\xi _1} \\ne 0$ , we have $K_{{\\mathbb {C}}^2}(\\partial _\\eta ,\\partial _\\xi ) < K_{{\\mathbb {R}}^3}(\\partial _\\eta ,\\partial _{\\xi _2})$ with equality at collisions.", "Moreover the submersion from ${\\mathbb {R}}^3 \\rightarrow {\\mathbb {S}}^2$ (§REF ) implies that $K_{{\\mathbb {R}}^3}(\\partial _\\eta ,\\partial _{\\xi _2})$ coincides with $K_{{\\mathbb {S}}^2}(\\partial _\\eta ,\\partial _{\\xi _2})$ which vanishes at Lagrange and collision points and is strictly negative elsewhere (see §REF ).", "Thus $K_{{\\mathbb {C}}^2}(\\partial _\\eta ,\\partial _\\xi )$ vanishes at collision points and is strictly negative everywhere else (see Fig.", "REF ).", "In particular, Lagrange points are more unstable on the configuration space ${\\mathbb {C}}^2$ than on the shape sphere.", "Figure: Sectional curvatures on horizontal 2-planes of submersion from ℂ 2 {\\mathbb {C}}^2 to ℝ 3 {\\mathbb {R}}^3 in units of 1/Gm 3 1/{Gm^3}.Under the submersion from ${\\mathbb {C}}^2$ to ${\\mathbb {S}}^3$ (§REF ), the horizontal basis vectors $\\partial _\\eta $ , $\\partial _{\\xi _1}$ and $\\partial _{\\xi _2}$ map respectively to $\\partial _\\eta $ , $\\partial _{\\xi _1}$ and $\\partial _{\\xi _2}$ .", "The sectional curvatures on corresponding pairs of 2-planes are equal, e.g.", "$K_{{\\mathbb {C}}^2}(\\partial _\\eta ,\\partial _{\\xi _2}) =K_{{\\mathbb {S}}^3}(\\partial _\\eta ,\\partial _{\\xi _2})$ .", "As shown in Fig.", "REF , $K_{{\\mathbb {C}}^2}(\\partial _\\eta ,\\partial _{\\xi _2})$ is negative everywhere except in a neighbourhood of $E_3$ where it can have either sign.", "The qualitative behavior of the other two sectional curvatures $K_{{\\mathbb {C}}^2}(\\partial _{\\xi _1},\\partial _{\\xi _2})$ and $K_{{\\mathbb {C}}^2}(\\partial _{\\xi _1}, \\partial _\\eta )$ is similar to that of $K_{{\\mathbb {C}}^2}(\\partial _{r},\\partial _{\\xi _2})$ and $K_{{\\mathbb {C}}^2}(\\partial _{r}, \\partial _\\eta )$ discussed above.", "The approximate symmetry under $\\partial _{\\xi _1} \\leftrightarrow \\partial _r$ is not entirely surprising given that $\\partial _{\\xi _1}$ and $\\partial _{r}$ are vertical vectors in the submersions to ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ respectively.", "The remaining two coordinate 2-planes on ${\\mathbb {C}}^2$ are not horizontal under either submersion.", "We find that $K_{{\\mathbb {C}}^2}(\\partial _r, \\partial _{\\xi _1})$ is negative everywhere except at $L_{4,5}$ and $K_{{\\mathbb {C}}^2}(\\partial _r, \\partial _{\\xi _2})$ is negative except around $E_{1,2}$ .", "Figure: Sectional curvatures on horizontal 2-planes of submersion from ℂ 2 {\\mathbb {C}}^2 to 𝕊 3 {\\mathbb {S}}^3 in units of 1/Gm 3 1/{Gm^3}." ], [ "Stability tensor and linear stability of geodesics", "In this section we use the stability tensor (which provides a criterion for linear geodesic stability) to discuss the stability of Lagrange rotational and homothety solutions.", "We end with a remark on linear stability of trajectories and geodesics.", "Consider the $n$ -dimensional configuration manifold $M$ with metric $g$ .", "The geodesic deviation equation (GDE) for the evolution of the separating vector (Jacobi field) $y(t)$ between a geodesic $x(t)$ and a neighboring geodesic is [29] ${\\bf \\nabla }_{\\dot{x}}^2 y = R(\\dot{x}, y) \\dot{x} = - R(y, \\dot{x}) \\dot{x}.$ We expand the Jacobi field $y=c^k(t) e_k(t)$ in any basis $e_i(t)$ that is parallel transported along the geodesic i.e.", "${\\bf \\nabla }_{\\dot{x}}e_k=0$ [$e_i(0)$ could be taken as coordinate vector fields at $x(0)$ ].", "Taking the inner product of the GDE with $e_m$ and contracting with $g^{im}$ , we get $\\ddot{c}^i = - S^i_j c^j$ , where the `stability tensor' $S^i_k = R^i_{ jkl}\\dot{x}^j \\dot{x}^l$ .", "As $S$ is real symmetric, its eigenvectors $f_i$ can be chosen to form an orthonormal basis for $T_x M$ .", "Writing $y = d^m f_m$ , the GDE becomes $\\ddot{d}^m = -\\kappa _m d^m$ (no sum on $m$ ) where $\\kappa _m$ is the eigenvalue of $S$ corresponding to the eigenvector $f_m$ .", "The eigenvalues of $S$ (say at $t = 0$ ) control the initial evolution of the Jacobi fields in the corresponding eigendirections.", "Since $\\kappa _m = \\left( {\\rm Area}{\\langle f_k , \\dot{x} \\rangle } \\right)^2 K(f_m, \\dot{x})$ (§REF ), positive (negative) $\\kappa $ or $K$ imply local stability (instability) for the initial evolution.", "We note that calculating $S$ and its eigenvalues at a given instant (say $t=0$ ) requires no knowledge of the time evolution of $e_i(t)$ .", "So we may simply use the coordinate vector fields as the basis.", "Notice that the tangent vector to the geodesic $\\dot{x}$ is always an eigendirection of $S$ with eigenvalue zero.", "Rotational Lagrange solutions in Newtonian potential: Consider the Lagrange rotational solutions where three equal masses ($m_i = m$ ) rotate at angular speed $\\omega = \\sqrt{3 G m / a^3}$ around their CM at the vertices of an equilateral triangle of side $a$ .", "The rotational trajectory on ${\\mathbb {C}}^2$ in $r,\\eta ,\\xi _{1,2}$ coordinates is given by $x(t) = (a/\\sqrt{m}, \\pi /4, \\omega t, \\pm \\pi /4)$ with velocity vector $\\omega \\partial _{\\xi _1}$ .", "Note that trajectory and geodesic times are proportional since $\\sigma = ds/dt = (E-V)/\\sqrt{\\cal T}$ with $V(r, \\eta , \\xi _2)$ and $\\cal T$ constant along $x(t)$ .", "The stability tensor along the geodesic, $S = \\omega ^2 \\; \\text{diag} (1,-1/2, 0, -1/2 )$ is diagonal in the coordinate basis $r,\\eta , \\xi _1, \\xi _2$ .", "As always, $\\dot{x}$ is a zero-mode.", "A perturbation along $\\partial _r$ is linearly stable while those directed along $\\partial _\\eta $ or $\\partial _{\\xi _2}$ are linearly unstable.", "Note that Routh's criterion $27 (m_1 m_2 + m_2 m_3 + m_3 m_1) < M^2$ [3] predicts that Lagrange rotational solutions are linearly unstable for equal masses.", "Lagrange homotheties: For equal masses, a Lagrange homothety solution is one where the masses move radially (towards/away from their CM) while being at the vertices of equilateral triangles.", "The geodesic in Hopf coordinates takes the form $(r(t), \\eta = \\pi /4, \\xi _1, \\xi _2 = \\pm \\pi /4)$ where $\\xi _1$ is arbitrary and independent of time.", "Though an explicit expression is not needed here, $r(t)$ is the solution of $\\ddot{r} + \\Gamma ^r_{rr} \\dot{r}^2 = 0$ where $\\Gamma ^r_{rr} = - 3 G m^3/(E r^3 + 3 Gm^3 r)$ for the inverse-square potential.", "The stability tensor is diagonal: $S = \\frac{6 G m^3 \\dot{r}^2}{\\left( 3 G m^3 r + E r^3 \\right)^2}\\text{diag} \\left(0,- 3 G m^3 - 2 E r^2, - E r^2 , - 3 G m^3 - 2 E r^2 \\right).$ For a given $r$ and positive energy, perturbations along $\\partial _{\\xi _{1,2}}$ and $\\partial _\\eta $ are unstable while they are stable when $-3Gm^3/r^2 < E < - 3 Gm^3/2r^2$ .", "For intermediate (negative) energies, $\\partial _{\\eta }$ and $\\partial _{\\xi _2}$ are unstable directions while $\\partial _{\\xi _1}$ is stable.", "For the Newtonian potential, we have similar conclusions following from the corresponding stability tensor: $S = \\frac{3 G m^{5/2} \\dot{r}^2}{4 r^2 \\left( 3 G m^{5/2} + E r \\right)^2} \\text{diag} \\left(0,- 9 G m^{5/2} - 5 E r, - 2 E r, - 9 G m^{5/2} - 5 E r \\right).$ We end this section with a cautionary remark.", "For a system whose trajectories can be regarded as geodesics of the JM metric, linear stability of geodesics may not coincide with linear stability of corresponding trajectories.", "This may be due to the reparametrization of time (see §REF for examples) as well as the restriction to energy conserving perturbations in the GDE.", "We illustrate this with a 2D isotropic oscillator with spring constant $k$ .", "Here the curvature of the JM metric (see §) is $R = 2Ek/T^3$ where $T$ is the kinetic energy.", "Thus for positive $k$ , geodesics are always linearly stable while for negative $k$ they are stable/unstable according as energy is negative/positive.", "By contrast, linearizing the EOM $\\ddot{\\delta }x_i = - (k/m) \\delta x_i$ shows that trajectories are linearly stable for positive $k$ and linearly unstable for negative $k$ .", "This (possibly atypical) example illustrates the fact that geodesic stability does not necessarily imply stability of trajectories." ], [ "JM metric and its curvature on configuration and shape space", "In analogy with our geometric treatment of the planar motion of three masses subject to inverse-square potentials, we briefly discuss the gravitational analogue with Newtonian potentials.", "As before, the translation invariance of the Lagrangian $L = \\frac{1}{2}\\sum _{i=1,2,3} m_i \\dot{x}_i^2 - \\sum _{i < j} \\frac{G m_i m_j}{|x_i - x_j|}$ allows us to go from the configuration space ${\\mathbb {C}}^3$ to its quotient ${\\mathbb {C}}^2$ endowed with the JM metric $ds^2 = \\left( E + \\frac{G m_1 m_2}{|J_1|} + \\frac{G m_2 m_3}{|J_2 - \\mu _1 J_1|} + \\frac{G m_3 m_1}{|J_2+\\mu _2 J_1|} \\right) \\left( M_1 |dJ_1|^2 + M_2 |dJ_2|^2 \\right).$ The Jacobi coordinates $J_{1,2}$ , mass ratios $\\mu _{1,2}$ and reduced masses $M_{1,2}$ are as defined in Eqs.", "(REF , REF , REF ).", "In rescaled Jacobi coordinates $z_i = \\sqrt{M_i} \\: J_i$ (REF ), the JM metric on ${\\mathbb {C}}^2$ for equal masses becomes $ds^2 = \\left(E+\\frac{G m^{5/2}}{\\sqrt{2}|z_1|}+\\frac{\\sqrt{2} G m^{5/2}}{\\sqrt{3}|z_2-\\frac{1}{\\sqrt{3}}z_1|}+\\frac{\\sqrt{2} G m^{5/2}}{\\sqrt{3}|z_2+\\frac{1}{\\sqrt{3}}z_1|}\\right) \\left(|dz_1|^2+|dz_2|^2 \\right).$ Rotations $z_j \\mapsto e^{i \\theta } z_j$ continue to act as isometries corresponding to the KVF $\\partial _{\\xi _1}$ in Hopf coordinates (REF ), where the JM metric is $ds^2&=& \\left(E + {Gm^{5/2}U}/{r}\\right) \\left(dr^2+r^2\\left(d\\eta ^2+ d\\xi _1^2-2\\cos 2\\eta \\;d\\xi _1\\;d\\xi _2+ d\\xi _2^2\\right)\\right) \\quad \\text{with}\\cr U&=&\\frac{1}{\\sqrt{2}\\sin \\eta } + \\frac{\\sqrt{2}}{\\sqrt{2 + \\cos 2 \\eta - \\sqrt{3} \\sin 2 \\eta \\cos 2 \\xi _2}} + \\frac{\\sqrt{2}}{\\sqrt{2 + \\cos 2 \\eta + \\sqrt{3} \\sin 2 \\eta \\cos 2 \\xi _2 }}.$ Requiring the submersion $(r,\\eta ,\\xi _1, \\xi _2) \\mapsto (r,\\eta ,\\xi _2)$ from ${\\mathbb {C}}^2$ to its quotient by rotations to be Riemannian gives us the JM metric on shape space ${\\mathbb {R}}^3$ : $ds^2= \\left(E + {Gm^{5/2}U}/{r}\\right) \\left(dr^2+r^2\\left(d\\eta ^2+\\sin ^2 2\\eta \\;d\\xi _2^2\\right)\\right).$ Unlike for the inverse-square potential, scaling $r \\mapsto \\lambda r$ is not an isometry of the JM metric even when $E = 0$ .", "Thus we do not have a further submersion to the shape sphere.", "However, in what follows, we will consider $E=0$ , as it leads to substantially simpler curvature formulae.", "Though we do not have a submersion to the shape sphere, the quantity $U(\\eta , \\xi _2)$ in the conformal factor may be regarded as a function on a 2-sphere of radius one-half.", "This allows us to express the scalar curvatures as $R_{{\\mathbb {C}}^2} = \\frac{3}{2 G m^{5/2} r U ^3}\\left(3 U ^2+ |{\\bf \\nabla }U|^2-2 U \\Delta U \\right) \\;\\; \\text{and} \\;\\;R_{{\\mathbb {R}}^3} = \\frac{1}{4 Gm^{5/2} r U^3} \\left(30 U^2+6|{\\bf \\nabla }U|^2 -8 U \\Delta U \\right)$ where $\\Delta U$ is the Laplacian and ${\\bf \\nabla }U$ the gradient relative to the round metric on a 2-sphere of radius $1/2$ .", "Evidently, both the scalar curvatures vanish in the limit $r \\rightarrow \\infty $ of large moment of inertia $I_{\\rm CM} = r^2$ ; they are plotted in Fig.", "REF .", "Numerically, we find that for any fixed $r$ , $R_{{\\mathbb {C}}^2}$ is strictly negative and reaches its global maximum $-3/(2 Gm^{5/2} r)$ at the Lagrange configurations $L_{4,5}$ , while $R_{{\\mathbb {R}}^3}$ has a positive global maximum $1/(2 Gm^{5/2} r)$ at the same locations.", "Note that $R_{{\\mathbb {R}}^3} = 2 R_{{\\mathbb {C}}^2}/3 + (9U^2 + |{\\bf \\nabla }U|^2 )/(2 Gm^{5/2} r U^3)$ .", "As argued in Eq.", "(REF ), the second term is strictly positive and vanishes only when $r \\rightarrow \\infty $ .", "Using the negativity of $R_{{\\mathbb {C}}^2}$ , it follows that $R_{{\\mathbb {R}}^3} > R_{{\\mathbb {C}}^2}$ with $(R_{{\\mathbb {R}}^3} - R_{{\\mathbb {C}}^2})$ attaining its minimum $2/(Gm^{5/2} r)$ at $L_{4,5}$ .", "Thus in a sense, the geodesic dynamics on ${\\mathbb {C}}^2$ is more linearly unstable than on shape space.", "Like the Ricci scalars, sectional curvatures on coordinate 2-planes are $(1/r) \\times $ a function of $\\eta $ and $\\xi _2$ .", "We find that sectional curvatures are largely negative and often go to $\\pm \\infty $ at collision points (see Eq.", "(REF )).", "Figure: Ricci scalar RR for zero energy and equal masses on ℂ 2 {\\mathbb {C}}^2 and ℝ 3 {\\mathbb {R}}^3 for the Newtonian potential (in units of 1/Gm 5/2 r{1/G m^{5/2} r}).", "RR on ℂ 2 {\\mathbb {C}}^2 is strictly negative while that on ℝ 3 {\\mathbb {R}}^3 can have either sign." ], [ "Near-collision geometry and `geodesic incompleteness'", "Unlike for the inverse-square potential, the scalar curvatures on ${\\mathbb {C}}^2$ and ${\\mathbb {R}}^3$ (REF ) diverge at binary and triple collisions.", "To examine the geometry near pairwise collisions of equal masses, it suffices to study the geometry near $C_3$ ($\\eta = 0$ , $r \\ne 0$ , $\\xi _{1,2}$ arbitrary) which represents a collision of $m_1$ and $m_2$ .", "We do so by retaining only those terms in the expansion of the zero-energy metrics around $\\eta =0$ : $ds_{{\\mathbb {C}}^2}^2 &\\approx & \\left( {G m^{5/2}}/{\\sqrt{2}\\eta r} \\right) \\left( dr^2+ r^2 \\left(d\\eta ^2+d\\xi _1^2-2(1-2\\eta ^2) d\\xi _1 d\\xi _2+d\\xi _2^2\\right)\\right) \\quad \\text{and} \\cr ds_{{\\mathbb {R}}^3}^2 &\\approx & \\left( {G m^{5/2}}/{r} \\right) \\left( 1/{\\sqrt{2} \\eta }+2\\sqrt{{2}/{3}}\\right) \\left( dr^2+ r^2 \\left(d\\eta ^2+4\\eta ^2d\\xi _2^2\\right)\\right),$ that are necessary to arrive at the following curvatures to leading order in $\\eta $ : $\\text{on ${\\mathbb {C}}^2$:} && R= {-3}/{\\varrho } \\;\\; \\text{and} \\;\\;K(\\partial _\\eta ,\\partial _{r,\\xi _{1,2}}) = 2 K(\\partial _r,\\partial _{\\xi _{1,2}}) = - 2K(\\partial _{\\xi _1},\\partial _{\\xi _2}) =-{1}/{\\varrho } \\cr \\text{on ${\\mathbb {R}}^3$:} && R = {-1}/{\\varrho } \\;\\; \\text{and} \\;\\;K(\\partial _\\eta ,\\partial _r) = -2 K(\\partial _r,\\partial _{\\xi _2}) = -{1}/{\\varrho }, \\quad K(\\partial _\\eta ,\\partial _{\\xi _2}) = -\\frac{2\\sqrt{{2}/{3}}}{Gm^{5/2}}$ where $\\varrho = \\sqrt{2} G m^{5/2}\\eta r$ .", "The curvature singularity at $\\eta = 0$ is evident in the simple poles in the Ricci scalars and all but one of the sectional curvatures in coordinate planes.", "We use the near-collision JM metric of Eq.", "(REF ) to show that a pairwise collision point lies at finite geodesic distance from another point in its neighborhood.", "Thus, unlike for the inverse-square potential, the geodesic reformulation does not regularize the gravitational three-body problem.", "Consider a point $P$ near $\\eta = 0$ with coordinates $(r,\\eta _0,\\xi _1, \\xi _2)$ .", "We estimate its distance to the collision point $C_3$ $(r,0,\\xi _1, \\xi _2)$ .", "To do so, we consider a curve $\\gamma $ of constant $r$ , $\\xi _1$ and $\\xi _2$ running from $P$ to $C_3$ parametrized by $\\eta _0 \\ge \\eta \\ge 0$ .", "We will show that $\\gamma $ has finite length so that the geodesic distance to $C_3$ must be finite.", "In fact, from (REF ): $\\text{Length}(\\gamma ) = \\int _{\\eta _0}^{0} \\sqrt{\\frac{G r m^{5/2}}{\\sqrt{2}}} \\frac{d\\eta }{\\sqrt{\\eta }} = - 2 \\sqrt{\\frac{G r m^{5/2}}{\\sqrt{2}}} \\sqrt{\\eta _0} < \\infty .$ Furthermore, the image of $\\gamma $ under the Riemannian submersion to shape space ${\\mathbb {R}}^3$ is a curve of even shorter length ending at a collision point.", "Thus geodesics on ${\\mathbb {C}}^2$ and ${\\mathbb {R}}^3$ can reach binary collisions in finite time, where the scalar curvature is singular.", "It is therefore interesting to study regularizations of collisions in the three body problem and their geometric interpretation.", "Acknowledgements: We thank K G Arun, A Lakshminarayan, R Montgomery, S G Rajeev and A Thyagaraja for useful discussions and references.", "This work was supported in part by the Infosys Foundation and a Ramanujan grant of the Department of Science & Technology, Govt.", "of India." ], [ "Proof of an inequality to give an upper bound for the scalar curvature", "Here we establish a strict lower bound on the quantity that appears in the relation (REF ) between Ricci scalars on ${\\mathbb {C}}^2$ and ${\\mathbb {S}}^2$ .", "Since Montgomery has shown that $R_{{\\mathbb {S}}^2} \\le 0$ , this helps us establish strictly negative upper bounds for the scalar curvatures on ${\\mathbb {C}}^2$ , ${\\mathbb {R}}^3$ and ${\\mathbb {S}}^3$ .", "We will show here that $12 h^2 + |{\\bf \\nabla }h|^2 > \\zeta h^3 \\quad \\text{where} \\quad \\zeta = {55}/{27} \\approx 2.04.$ The best possible $\\zeta $ is estimated numerically to be $\\zeta = 8/3$ and the minimum occurs at the Euler points $E_{1,2,3}$ .", "We define the power sum symmetric functions $u_{2n} = \\sum _{i=1}^3 v_i^n$ in terms of which the pre-factor in the JM metric (REF ) is $h = v_1 + v_2 + v_3 = u_2$ .", "In [20] Montgomery shows that $|{\\bf \\nabla }h|^2 = 4s$ where the symmetric polynomial $s = (1/2) \\left(-2 u_2^2 + 4 u_2 u_4 - 3u_4^2 + 3 u_8 \\right).$ This gives $12 h^2 + |{\\bf \\nabla }h|^2 = u_2^3 \\left( 8 A + 6 B\\right) \\quad \\text{where} \\quad A = \\frac{u_2 + u_4}{ u_2^2} \\quad \\text{and} \\quad B = \\frac{u_8- u_4^2}{u_2^3} .$ We will show below that $ A \\ge 17/27$ and $B > -1/2$ , from which Eq.", "(REF ) follows (numerically we find that $B \\ge -32/81$ which leads to the above-mentioned optimal value $\\zeta = 8/3$ ).", "To prove the inequality for $B$ , we define $c = \\cos 2\\eta $ and $s = \\sin 2\\eta \\cos 2\\xi _2$ which lie in the interval $[-1,1]$ .", "Then $\\frac{u_8 - u_4^2}{u_2^3} > -\\frac{1}{2}\\quad \\Leftrightarrow \\quad u_8 - u_4^2 + \\frac{u_2^3}{2} > 0 \\quad \\Leftrightarrow \\quad \\frac{3}{8} \\left(20 - 3 (c^2 + s^2)^2 - 8 c^3 + 24 c s^2 \\right) > 0.$ For the latter to hold it is sufficient that $17 - 8 c^3 + 24 c s^2 > 0$ which is clearly true for $0 \\le c \\le 1$ .", "For $-1 \\le c < 0$ put $c = -d$ .", "Then it is enough to show that $17 + 8 d^3 - 24 d (1-d^2) > 0$ since $s^2 \\le 1 - d^2$ .", "This holds as the LHS is positive at its boundary points $d = 0, 1$ as well as at its local extremum $d = 1/2$ .", "The quantity $A$ defined in Eq.", "(REF ) is a symmetric function of $v_1, v_2$ and $v_3$ which in turn are functions of $\\eta $ and $\\xi _2$ (REF ) for $0 \\le \\eta \\le \\pi /2$ and $0 \\le \\xi _2 \\le \\pi $ .", "Since $\\sum _i 1/v_i = 3$ , we may regard $A$ as a function of any pair, say $v_1$ and $v_2$ .", "The allowed values of $\\eta $ and $\\xi _2$ define a domain $\\bar{D} = D \\amalg \\partial D$ in the $v_1$ -$v_2$ plane.", "To show that $A \\ge 17/27$ , we seek its global minimum, which must lie either at a local extremum in the interior $D$ or on the boundary $\\partial D$ .", "$\\partial D$ is defined by the curves $\\xi _2 = 0$ and $\\xi _2 = \\pi /2$ which meet at $\\eta = 0$ and $\\eta = \\pi /2$ and include the points $(v_1 = \\infty , v_2 = 2/3)$ and $(v_1 = 2/3, v_2 = \\infty )$ (see Fig.", "REF ).", "This is because, for any fixed $\\eta $ , $v_1$ and $v_2$ (REF ) are monotonic functions of $\\xi _2$ for $0 \\le \\xi _2 \\le \\pi /2$ and symmetric under reflection about $\\xi _2 = \\pi /2$ .", "Figure: The boundary ∂D\\partial D of the region DD in the v 1 v_1-v 2 v_2 plane is given by the level curves ξ 2 =0,π/2\\xi _2 = 0, \\pi /2.", "These level curves run from the collision point η=0\\eta = 0 to the Euler point η=π/2\\eta = \\pi /2, passing through the collision points at v 1 =∞v_1 = \\infty or v 2 =∞v_2 = \\infty (where η=π/3\\eta = \\pi /3).", "The level curves ξ 2 =π/8,π/4,3π/8\\xi _2 = \\pi /8, \\pi /4, 3\\pi /8 in the interior DD are also shown.", "Note that DD lies within the quadrant v 1,2 ≥1/2v_{1,2} \\ge 1/2.Along $\\partial D$ , $A = (5 \\cos 6 \\eta + 22)/27$ is independent of $\\xi _2$ and minimal at the Euler configurations $\\eta = \\pi /6$ and $\\pi /2$ with the common minimum value $17/27$ , which turns out to be the global minimum of $A$ .", "This is because its only local extremum in $D$ is at the Lagrange configuration $v_1 = v_2 = v_3 = 1$ where $A = 2/3$ .", "To see this, we note that local extrema of $A$ in $D$ must lie at the intersections of $\\partial A/\\partial v_1 = 0$ and $\\partial A/\\partial v_2 = 0$ .", "Now $\\partial A/ \\partial v_1 = (v_1 - v_3)F(v_1, v_2)/v_1^2 u_2^3$ where $F(v_1,v_2)= u_2 \\left\\lbrace v_1 + v_3 + 2 \\left( v_1^2 + v_1 v_3 + v_3^2 \\right) \\right\\rbrace - 2 (v_1 + v_3) ( u_2 + u_4).$ For $\\partial A/\\partial v_1$ to vanish, either $v_1 = v_3$ or $F(v_1,v_2) = 0$ or one of the $v_i = \\infty $ .", "The collision points $v_i = \\infty $ do not lie in $D$ .", "The conditions for $\\partial A / \\partial v_2$ to vanish are obtained via the exchange $v_1 \\leftrightarrow v_2$ .", "The intersection of the conditions $v_1 = v_3$ and $v_2 = v_3$ lies at the Lagrange configurations $v_i = 1$ where $A = 2/3$ .", "It turns out that the only intersection of $v_1 = v_3$ with $F(v_2,v_1) = 0$ or of $v_2 = v_3$ with $F(v_1, v_2) = 0$ lying in $D$ occurs at the above Lagrange configuration.", "For instance, when $v_1 = v_3 = v$ , $F(v_2,v_1) = -3 v^2 (4v-1)(v-1)/(3v - 2)^2$ vanishes when $v = 1$ or $v = 1/4$ (which violates $v \\ge 1/2$ ).", "Finally, we account for extrema lying on the zero loci of both $F(v_1,v_2)$ and $F(v_2,v_1)$ , which using $u_{-2} = 3$ , must satisfy $F(v_1,v_2) - F(v_2,v_1) = (v_1 - v_2) \\left[12 v_1 v_2 v_3 - (v_1 + v_2 + v_3) \\right] = 0.$ So either $v_1 = v_2$ or $12 v_1 v_2 v_3 = u_2$ .", "Now, we have shown above that the only extrema of $A$ on $v_1 = v_3$ in $D$ lie at the Lagrange configurations.", "Since $A$ is a symmetric function of the $v_i$ , it follows that its only extrema on $v_1 = v_2$ also lies at the Lagrange configurations.", "On the other hand, $12 v_1 v_2 v_3 - (v_1 + v_2 + v_3) \\ge 0$ for $v_i \\ge 1/2$ , with equality only at $v_i = 1/2$ which is not in $D$ .", "Thus the only extremum of $A$ in $D$ is at the Lagrange configurations (where $A = 2/3$ ) and hence its global minimum occurs on $\\partial D$ at the Euler configurations (where $A = 17/27$ )." ] ]
1606.05091
[ [ "Raman signatures of inversion symmetry breaking and structural phase\n transition in type-II Weyl semimetal MoTe2" ], [ "Abstract Transition metal dichalcogenide MoTe$_2$ is an important candidate for realizing the newly predicted type-IIWeyl fermions, for which the breaking of the inversion symmetry is a prerequisite.", "Here we present direct spectroscopic evidence for the inversion symmetry breaking in the low temperature phase of MoTe$_2$ by systematic Raman experiments and first principles calculations.", "We identify five lattice vibrational modes which are Raman active only in noncentrosymmetric structure at low temperature.", "A hysteresis is also observed in the peak intensity of inversion symmetry activated Raman modes, confirming a temperature induced structural phase transition with a concomitant change in the inversion symmetry.", "Our results provide definitive evidence for the low temperature noncentrosymmetric T$_d$ phase from vibrational spectroscopy, and suggest MoTe$_2$ as an ideal candidate for investigating the temperature induced topological phase transition." ], [ "Raman signatures of inversion symmetry breaking and structural phase transition in type-II Weyl semimetal MoTe$_2$ Kenan Zhang State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics,Tsinghua University, Beijing 100084, China Changhua Bao State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics,Tsinghua University, Beijing 100084, China Qiangqiang Gu International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Xiao Ren International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Haoxiong Zhang State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics,Tsinghua University, Beijing 100084, China Ke Deng State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics,Tsinghua University, Beijing 100084, China Yang Wu Correspondence should be sent to [email protected] and [email protected] State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics,Tsinghua University, Beijing 100084, China Tsinghua-Foxconn Nanotechnology Research Center, Tsinghua University, Beijing 100084, China Yuan Li International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing, China Ji Feng International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing, China Shuyun Zhou Correspondence should be sent to [email protected] and [email protected] State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics,Tsinghua University, Beijing 100084, China Collaborative Innovation Center of Quantum Matter, Beijing, China Transition metal dichalcogenide MoTe$_2$ is an important candidate for realizing the newly predicted type-II Weyl fermions, for which the breaking of the inversion symmetry is a prerequisite.", "Here we present direct spectroscopic evidence for the inversion symmetry breaking in the low temperature phase of MoTe$_2$ by systematic Raman experiments and first principles calculations.", "We identify five lattice vibrational modes which are Raman active only in the low temperature noncentrosymmetric structure.", "A hysteresis is also observed in the peak intensity of inversion symmetry activated Raman modes, confirming a temperature induced structural phase transition with a concomitant change in the inversion symmetry.", "Our results provide definitive evidence for the low temperature noncentrosymmetric T$_d$ phase from vibrational spectroscopy, and suggest MoTe$_2$ as an ideal candidate for investigating the temperature induced topological phase transition.", "Layered transition metal dichalcogenides (TMDs) have attracted extensive research interests due to their intriguing physical properties for both fundamental research and potential applications in electronics, optoelectronics, spintronics and valleytronics [1], [2].", "So far most of the research has been focused on semiconducting TMDs with hexagonal or trigonal (2H or 1T) structures which show strong quantum confinement effects in atomically thin films.", "In recent years, TMDs with monoclinic 1T$^\\prime $ and orthorhombic T$_d$ phase have been proposed to be important host materials for realizing novel topological quantum phenomena, e.g.", "quantum spin Hall effect [3], [4] and Weyl fermions [5].", "Weyl fermions were originally introduced in high energy physics by Hermann Weyl [6], and their condensed matter physics counterparts have not been realized until recently in Weyl semimetals in the TaAs family [7], [8], [9].", "Weyl fermions can be realized by breaking either the time reversal symmetry or inversion symmetry of a three dimensional Dirac fermion such that a pair of degenerate Dirac points separate into two bulk Weyl points with opposite chiralities, which are connected by topological Fermi arcs when projected onto the surface.", "Recently, it has been predicted that a new type of Weyl fermions can be realized in TMDs.", "Different from type-I Weyl fermions which have point-like Fermi surface and obey Lorentz invariance, the newly predicted type-II Weyl fermions emerge at the topological protected touching points of an electron and a hole pocket with strongly tilted Weyl cones [5].", "Such type-II Weyl fermions break Lorentz invariance and therefore do not have counterparts in high energy physics.", "Figure: Temperature induced phase transition in MoTe 2 _2.", "(a) Crystal structures of 1T ' ^\\prime (shadow) and T d _d (solid) phases.", "(b) Resistivity measurement shows a temperature induced phase transition.", "The inset shows a photograph of the high quality single crystal and the scale bar is 5mm.", "(c) Raman spectra at 320 K and 80 K respectively.", "The letters inside the parenthesis indicate the polarization directions for incident and scattering lights.Figure: Polarized Raman spectra measured in the high and low temperature phases.", "(a, b) Polarized Raman spectra measured at 300 K (a) and 150 K (b) respectively.", "The identified Raman peaks are labeled by lowercase (high temperature phase) and capital (Low temperature phase) letters.", "The red labels A, D, N, Q and S mark the Raman modes that are directly sensitive to the inversion symmetry breaking.", "The small peak at 83 cm -1 ^{-1} is from the instrument and not intrinsic to the sample.Type-II Weyl fermions have been first predicted in the orthorhombic (T$_d$ ) phase of WTe$_2$ with space group Pmn2$_1$ [5].", "However, it is challenging to observe the extremely small Fermi arcs in WTe$_2$ due to the small separation of the Weyl points (0.7% of the Brillouin zone).", "Weyl fermions have also been predicted in the low temperature phase of MoTe$_2$ with much larger Fermi arcs [10], [11], and signatures of the Fermi arcs have been suggested in a combined angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS) study [12] and other ARPES studies [13], [14], [15], [16], [17].", "The existence of Weyl fermions has been relied on the assumption that the low temperature phase of MoTe$_2$ is isostructural to the noncentrosymmetric T$_d$ phase of WTe$_2$ [10], [11].", "The high temperature monoclinic 1T$^\\prime $ phase with an inclined staking angle of $\\sim 93.9^\\circ $ has a centrosymmetric P2$_1$ /m space group.", "Although a temperature induced structural transition with a change in the stacking angle from $\\sim 93.9^\\circ $ to 90$^\\circ $ has been reported both crystallographyically [18], [19] and computationally [20], there are two possible space groups can be assigned to the low temperature orthorhombic phase - noncentrosymmetric Pmn2$_1$ and centrosymmetric Pnmm [18].", "Previous X-ray diffraction study was limited to resolve the subtle differences between these two space groups to provide conclusive evidence on the inversion symmetry [18].", "Recent ARPES studies have detected Fermi arcs at low temperature T$_d$ phase, but the evolution of band dispersions from high to low temperature has not been reported.", "Meanwhile the absence of Fermi arcs at high temperature 1T$^\\prime $ phase are difficult to be observed because of the reduction of Resolution.", "So ARPES is not suitable for the research of symmetry breaking during phase transition in MoTe$_2$ .", "Since the noncentrosymmetry is a prerequisite for realizing Weyl fermions for non-magnetic materials, it is critical to reveal the inversion symmetry breaking from spectroscopic measurements which are directly sensitive to the crystal symmetry.", "Results Polarized Raman spectra.", "In this paper, we provide direct experimental evidence for the inversion symmetry breaking in the low temperature phase of MoTe$_2$ and study its evolution across the temperature induced structural phase transition using Raman vibrational spectroscopy.", "Our Raman measurements reveal the emergence of five Raman and IR active modes in the low temperature phase, and they are in good agreement with first principles calculations and symmetry analysis of the T$_d$ phase.", "These peaks are however absent in the high temperature centrosymmetric 1T$^\\prime $ phase, suggesting that they are Raman signatures for the breaking of the inversion symmetry.", "A clear hysteresis is observed in the peak intensity of two A$_1$ modes - the shear mode at $\\approx $ 13 cm$^{-1}$ and the out-of-plane vibration mode at $\\approx $ 130 cm$^{-1}$ .", "Our results provide clear evidence for the lack of inversion symmetry in the low temperature T$_d$ phase from a lattice dynamics point of view, and indicate that MoTe$_2$ can be a good candidate for studying the temperature induced topological phase transition.", "Table: Comparison of the calculated and experimental Raman modes in the 1T ' ^\\prime phase and T d _d phase, in units of cm -1 ^{-1}.Figure REF (a) shows a comparison of the low temperature (solid) and high temperature (shadow) phases with corresponding space groups of Pmn2$_1$ and P2$_1/$m respectively.", "They share almost the same in-plane crystal structure with zigzag Mo metal chains and distorted Te octahedra.", "The structural phase transition is revealed by an anomaly in the temperature-dependent resistivity [21], which occurs at $\\approx $ 260 K upon warming and $\\approx $ 250 K upon cooling (Fig.", "REF (b)).", "Figure REF (c) shows the Raman spectra at 320 K and 80 K respectively on cleaved bulk single crystals.", "The polarizations for incident and scattered photons are denoted by two letters representing the crystal axes.", "For example, aa shows that both the incident and scattered photons are polarized along the a-axis direction.", "Here, we used the crystal axes of T$_d$ phase to denote the polarization directions and all single crystal samples were oriented using Laue diffraction patterns (see Supplementary Figure 1) before performing Raman characterizations.", "The comparison of Raman spectra reveals two new peaks labeled by A and D only in the low temperature T$_d$ phase, suggesting that these Raman peaks may link with the structural phase transition.", "To understand the Raman modes, we first perform group theory analysis.", "Both the 1T$^\\prime $ phase and T$_d$ phase have 12 atoms in one unit cell and correspondingly, there are a total of 36 phonon modes.", "The vibrational modes in the 1T$^\\prime $ phase decompose into 36 irreducible representations: [12A$_g$ +6B$_g$ ]+[5A$_u$ +10B$_u$ ]+[A$_u$ +2B$_u$ ], where the first, second, and third groups of irreducible representations correspond to the Raman active, infrared (IR) active, and the acoustic modes, respectively.", "Since the IR active and Raman active modes are exclusive of each other in centrosymmetric structures, the IR active modes A$_u$ and B$_u$ cannot be observed in Raman measurements.", "In the T$_d$ phase, the vibration modes decompose into 36 irreducible representations: [11A$_1$ +6A$_2$ +5B$_1$ +11B$_2$ ]+[11A$_1$ +5B$_1$ +11B$_2$ ]+[A$_1$ +B$_1$ +B$_2$ ], where A$_1$ , B$_1$ , and B$_2$ modes are both IR and Raman active while A$_2$ modes are only Raman active.", "Figure: Calculated vibrational patterns for Raman modes that are directly sensitive to the inversion symmetry breaking.", "Calculated vibrational patterns in the T d _d phase (a-e) and corresponding B u _u and A u _u modes in the 1T ' ^\\prime phases (f-j).", "A, D, Q, S modes vibrate in the bc plane and N mode in the ac plane.Figure REF shows an overview of the polarized Raman spectra measured at 300 K and 150 K. Raman selection rules (see Supplementary Table 1, and Table 2) indicate that the A$_g$ modes in 1T$^\\prime $ phase can be observed in the aa, bb, cc, and bc configurations, whereas the B$_g$ modes can be observed in the ac and ab configurations.", "To obtain all possible phonon modes at low wave number in the 1T$^\\prime $ phase, we performed Raman measurements in the cc, ac, and ab configurations at 300 K. The azimuthal dependence of the Raman peak intensities for A$_g$ and B$_g$ modes (see Supplementary Figure 2) further confirm the good alignment [22], [23].", "Eight sharp peaks of pure A$_g$ modes are detected in the cc configuration and all the six B$_g$ modes are found in the ac and ab configurations.", "The sharp peaks observed are due to improved sample quality and more peaks can be resolved clearly.", "In the T$_d$ phase, the A$_1$ modes can be observed in the aa, bb, and cc configurations, whereas the A$_2$ , B$_1$ , B$_2$ modes can only be observed in the ab, ac, and bc configurations, respectively.", "The signal leakage of A$_1$ in other polarization configurations is likely due to the imperfect cleavage of ac and bc surfaces from plate-like samples, but this does not intervene the conclusion.", "In the low temperature phase, we observe six pure A$_1$ modes in the aa configuration, five A$_2$ modes in the ab configuration, three B$_1$ modes in the ac configuration, and six B$_2$ modes in the bc configuration.", "signature of phase transition and symmetry breaking.", "The comparison of Raman modes between experimental results and theoretical calculations in Table REF shows a good agreement.", "Here we focus on Raman active modes that are sensitive to the breaking of the inversion symmetry across the phase transition.", "Owing to the crystal structure changes only slightly across the phase transition, we can track each phonon mode by comparing their vibrational pattern in these two phases.", "Since the breaking of inversion center, some Raman in-active modes that belong to A$_u$ or B$_u$ representations in the 1T$^\\prime $ phase evolve to A$_1$ , B$_1$ , or B$_2$ representations that are both IR and Raman active in the T$_d$ phase.", "Thus, the presence of these Raman modes reflects the transition into the noncentrosymmetric phase.", "Similarly, nonlinear optical method was employed to reveal the lack of inversion symmetry in few-layer MoS$_2$ and h-BN , .", "Figure: Temperature dependence of the Raman spectra for peaks A and D in the aa configuration.", "(a-d) Selected Raman spectra across the phase transition for peaks A (a, b) and D (c, d) upon cooling (a, c) and warming (b, d).", "(e, f) Temperature dependence of the Raman intensity for peaks A and D.Figure REF compiles the calculated vibrational patterns for such phonons that are directly sensitive to the inversion symmetry breaking, where the arrows scale the atomic displacements.", "The upper panels show the vibration modes labeled by A, D, N, Q, S, which indicated by red arrows in Fig.", "REF (b), where the irreducible representations in the T$_d$ phase are given in parenthesis.", "The lower panels show the corresponding vibration modes labeled by A$^\\prime $ , D$^\\prime $ , N$^\\prime $ , Q$^\\prime $ , S$^\\prime $ that belong to the A$_u$ and B$_u$ irreducible representations in 1T$^\\prime $ phase.", "These phonons have almost identical vibrational patterns as A, D, N, Q, S respectively but no Raman activity due to the centrosymmetry.", "The A and A$^\\prime $ denote interlayer shear modes along the b axis and the A peak is also observed in a previous report [26].", "The absence of the vibrational mode along the a axis (labeled as S$_2$ in Ref.", "Chen2016) in our measurements is likely due to Raman scattering matrix element effects or small Raman cross-section.", "Compared to previous work, here we present a systematic Raman characterization of the low temperature phase by distinguishing all modes that reflect the breaking of the centrosymmetry.", "The strongest Raman signals that distinguished these two phases are the interlayer shear mode A at $\\approx $ 13 cm$^{-1}$ and another out-of-plane vibration mode D at $\\approx $ 130 cm$^{-1}$ .", "Similar vibrational modes have been reported in many 2D materials, such as multilayer graphene [27], [28], [29] and TMDs e.g.", "MoS$_2$ , WSe$_2$ [30], [31], [32], [33].", "The low frequancy interlayer shear modes are sensitive to the stacking sequence, layer number, and symmetry, and can be used as a measure of interlayer coupling.", "In in-plane shear modes all atoms in the same layer all vibrates along the same direction while atoms in two adjacent layers vibrates along opposite directions.", "If there is an inversion center that lies in the layer, such shear modes have odd parity with respect to the inversion symmetry and therefore are Raman inactive.", "This is the reason why A mode is invisible in 1T$^\\prime $ MoTe$_2$ , opposite to high symmetric 2H-MoS$_2$ , MoSe$_2$ , WSe$_2$ , and 2H-MoTe$_2$ .", "However, when the crystal structure does not hold inversion symmetry, these modes are both Raman and IR active and visible in Raman spectroscopy, giving direct evidence on the breaking of centrosymmetry in the orthorhombic T$_d$ structure.", "We further track the evolution of peaks A and D that link with the inversion symmetry breaking across the phase transition.", "The evolution of the A peak at 12.5 cm$^{-1}$ , and D peak at 128.3 cm$^{-1}$ are displayed in Fig.", "REF (a-d).", "Upon warming, the intensity of the A and D peaks decreases with the sharpest decrease at $\\approx $ 260 K and eventually disappears above 300 K. Upon cooling, the A and D peaks appears at a lower temperature, and their intensities sharply increases at $\\approx $ 250 K, and reach the maximum below 200 K. The intensity of the A and D peaks as a function of temperature is shown in Fig.", "REF (e, f).", "The temperature-dependent peak position and FWHM are shown in Supplementary Figure 3.", "A discontinuity in the temperature dependent peak position can be regarded another signature of structural phase transition of MoTe$_2$ .", "The thermal hysteresis effect in the peak intensity is consistent with our transport measurement, confirming that these peaks directly indicate the structural phase transition from high temperature 1T$^\\prime $ to low temperature T$_d$ phase.", "Discussion To summarize, by performing a systematic Raman study using polarization selection rules combined with theoretical calculation, we reveal the Raman signatures of structural phase transition across the 1T$^\\prime $ to T$_d$ phase transition and provided unambiguous evidence on the absence of inversion symmetry of T$_d$ phase.", "Our work demonstrates that T$_d$ phase of MoTe$_2$ is a strong candidate for both type-II Weyl semimetal and investigating the temperature induced topological phase transition.", "Methods Sample growth and Raman measurement.", "Single crystals of MoTe$_2$ were grown by chemical vapor transport method as reported previously [12].", "Raman scattering experiments were performed in a confocal back-scattering geometry on freshly cleaved single crystal surfaces along the ab, ac, and bc planes.", "Parallel and cross polarizations between the incident and scattered lights were used.", "Raman spectra were measured using a Horiba Jobin Yvon LabRAM HR Evolution spectrometer with the $\\lambda $ = 514 nm excitation source from an Ar laser and a 1800 gr/mm grating.", "A liquid nitrogen-cooled CCD detector and BragGrate notch filters allow for measurements at low wave numbers.", "The temperature of the sample was controlled by a liquid-nitrogen flow cryostat and a heater in a chamber with a vacuum better than $5\\times 10^{-7}$ Torr.", "First-principles calculations.", "To determine the phonon frequencies, we performed first-principles calculations of the phonon modes at the zone center using the Vienna abinitio simulation package (VASP) [34] with the local density approximation (LDA) [35] and the projector augmented wave potentials (PAW) [36].", "We set a $4\\times 8\\times 2$ Monkhorst-Pack k-point mesh and 400 eV cutoff for plane waves.", "The coordinates and the cell shape in Ref.Dawson1987 have been fully relaxed until the forces acting on the atoms are all smaller than $10^{-4}$ eV/Å.", "We use the phonopy package [37], which implements the small displacement method to obtain the phonon frequencies and vibration modes at the $\\Gamma $ point.", "Data availability.", "The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.", "Acknowledgements This work is supported by the National Natural Science Foundation of China (grant No.", "11274191, 11334006), Ministry of Science and Technology of China (No.", "2015CB921001 and 2012CB932301).", "Author contributions S.Z.", "and Y.W.", "conceived the research project.", "K.Z., H.Z.", "and K.D.", "grew and characterized the samples under supervision of Y.W..", "K.Z., C.B., X.R.", "and Y.L.", "performed the Raman measurements and analyzed the data.", "Q.G.", "and J.F.", "performed the first principle calculations.", "K.Z., C.B., Q.G.", "Y.W.", "and S.Z.", "wrote the manuscript, and all authors commented on the manuscript.", "Competing financial interests The authors declare no competing financial interests." ] ]
1606.05071
[ [ "Average Shortest Path Length of Graphs of Diameter 3" ], [ "Abstract A network topology with low average shortest path length (ASPL) provides efficient data transmission while the number of nodes and the number of links incident to each node are often limited due to physical constraints.", "In this paper, we consider the construction of low ASPL graphs under these constraints by using stochastic local search (SLS) algorithms.", "Since the ASPL cannot be calculated efficiently, the ASPL is not suitable for the evaluation function of SLS algorithms.", "We first derive an equality and bounds for the ASPL of graphs of diameter 3.", "Then, we propose use the simpliest upper bound represented by the number of triangles and squares in the graph as an evaluation function for graphs of diameter 3.", "We show that the proposed evaluation function can be evaluated in O(1) time as the number of nodes and the maximum degree tend to infinity by using some data tables.", "By using the simulated annealing with the proposed evaluation function, we construct low ASPL regular graphs of diameter 3 with 10 000 nodes." ], [ "Introduction", "The network topology is significant on the performance of the interconnection network, and thus various kinds of topologies have been proposed and analyzed in many areas such as Datacenter network, High Performance Computing (HPC), Peer-to-peer system, Network-on-Chip (NoC), and so on [1], [2], [3], [4], [5], [6], [7].", "In these areas, topologies with small diameter and low average shortest path length (ASPL) are desired.", "For example, the Hypercube topologies and the de Bruijn graphs are known to be effective for NoC architectures [5] and peer-to-peer networks [1], [3], respectively.", "Some topologies based on randomness have been proposed and explored in the area of Datacenter network and HPC [2], [6], [7].", "The problem of finding the maximum graph for given maximum degree and diameter is called the degree/diameter problem, and has been studied in graph theory [8].", "However, in practice, the order (the number of nodes) is often limited due to several reasons.", "Hence, for many applications, we are given order and maximum degrees, and try to minimize the diameter and the ASPL.", "This problem is called the order/degree problem, and has not been studied sufficiently.", "Recently, the importance of the order/degree problem is pointed out [9].", "Some of the authors of [9] and their coworkers held a competition called “Graph Golf” on the order/degree problem [10].", "The stochastic local search (SLS) is a framework of approximation algorithms for general optimization problems.", "In this paper, we consider the construction of low ASPL graphs of given order and given maximum degrees by using SLS algorithms.", "In a SLS algorithm, an initial feasible solution is generated (possibly be a random solution).", "Then, the solution is iteratively replaced by one of its neighborhoods.", "For our problem, we use a simple local modification procedure called switch for defining the neighborhoods.", "In this paper, we assume that SLS algorithms use the evaluation function which represents a quality of feasible solutions [11].", "Since huge number of feasible solutions must be evaluated in SLS algorithms in general, efficient evaluation algorithms are strongly desired.", "However, the ASPL cannot be calculated efficiently at least in our knowledge, and hence is not suitable for the evaluation function.", "In this paper, we assume that for given $n$ and $d$ , the $d$ -regular random graph of order $n$ has the diameter 3 with high probability, and propose the evaluation function for graphs of diameter 3 which can be evaluated efficiently.", "More precisely, first, we derive an equality and bounds for the ASPL of graphs of diameter 3.", "On the basis of the simplest upper bound of the ASPL, we propose to use $3\\triangle +2\\Box $ as the evaluation function of SLS algorithms where $\\triangle $ and $\\Box $ denote the number of triangles and squares in a graph, respectively.", "The proposed evaluation function can be evaluated in $O(1)$ time as $n$ and $d$ tend to infinity by using some data tables.", "By using the proposed evaluation function in the iterative first improvement (IFI) and the simulated annealing (SA), we construct low ASPL regular graphs of diameter 3 with 10 000 nodes, which are the best graphs in the competition Graph Golf.", "The remainder of this paper is organized as follows.", "In Section , notions and notations used in this paper are introduced.", "In Section , the equality and bounds for the ASPL of graphs of diameter 3 are shown.", "In Section , we propose the new evaluation function, and show an efficient algorithm for calculating the new evaluation function using data tables.", "In Section , we show results of numerical experiments.", "Section  concludes this paper.", "A graph $G$ is a pair of two finite sets $V$ and $E$ where every element of $E$ is a subset of $V$ of size 2.", "Each element of $V$ is called a node.", "Each element of $E$ is called an edge.", "Let $e_{ij}:=\\lbrace i,j\\rbrace $ for $\\lbrace i,j\\rbrace \\in E$ .", "The number $n$ of nodes is called the order of $G$.", "For nodes $u,v \\in V$ , $u$ is said to be connected to $v$ if there is an edge $\\lbrace u,v\\rbrace \\in E$ .", "The degree $d_v$ of a node $v$ is the number of nodes connected to $v$ .", "A graph is $d$ -regular if the degrees of all the nodes are equal to $d$ .", "For two graphs $G=(V,E)$ and $G^{\\prime }=(V^{\\prime },E^{\\prime })$ , $G^{\\prime }$ is a subgraph of $G$ if $V^{\\prime } \\subseteq V$ and $E^{\\prime } \\subseteq E$ .", "If $G^{\\prime }$ is a subgraph of $G$ , we say $G$ contains $G^{\\prime }$ .", "We say $G$ is isomorphic to $G^{\\prime }$ if there is a bijection $f : V \\rightarrow V^{\\prime }$ that satisfies the following condition: $\\forall \\lbrace u,v\\rbrace \\subseteq V,\\hspace{20.0pt} \\lbrace u,v\\rbrace \\in E \\Longleftrightarrow \\lbrace f(u),f(v)\\rbrace \\in E^{\\prime }.$ For two graphs $G$ and $H$ , the number of $H$ in $G$ is the number of subgraphs of $G$ that are isomorphic to $H$ .", "For two nodes $s$ and $t$ , a $s$ -$t$ path or a path is a graph $P=(V,E)$ where $V = \\lbrace s=v_0,v_1,\\ldots ,v_\\ell =t\\rbrace , \\qquad E=\\bigcup _{k=0}^{\\ell -1} \\lbrace \\lbrace v_k,v_{k+1}\\rbrace \\rbrace $ for $v_0,v_1,\\ldots ,v_\\ell $ which are distinct nodes.", "Here, $\\ell $ is called the length of $s$ -$t$ path $P$ .", "If $P=(V,E)$ is a $s$ -$t$ path of length $\\ell \\ge 2$ , a graph $C=(V,E\\cup \\lbrace \\lbrace s,t\\rbrace \\rbrace )$ is called a cycle.", "The length of the cycle $C$ is $\\ell +1$ .", "We call a cycle of length 3 and 4 a triangle and a square, respectively.", "Definition 1 For $k=0,1,\\ldots $ a $k$ -multiple triangle is a graph $\\mathrm {Tri}^{(k)}=(V^{(k)},E^{(k)})$ where $V^{(k)} &:= {\\left\\lbrace \\begin{array}{ll}\\lbrace i,j\\rbrace , &\\hspace{20.0pt} k=0 \\\\V^{(k-1)}\\cup \\lbrace v_k\\rbrace , &\\hspace{20.0pt} k>0\\end{array}\\right.}", "\\\\E^{(k)} &:= {\\left\\lbrace \\begin{array}{ll}\\lbrace \\lbrace i,j\\rbrace \\rbrace , &\\hspace{20.0pt} k=0 \\\\E^{(k-1)} \\cup \\lbrace \\lbrace i,v_{k}\\rbrace ,\\lbrace v_{k},j\\rbrace \\rbrace , &\\hspace{20.0pt} k>0.\\end{array}\\right.", "}$ Such a $k$ -multiple triangle is called a $k$ -multiple triangle sharing $i,j$ when we specify the two nodes $i$ and $j$ .", "Definition 2 For $k=0,1,\\ldots $ a $k$ -multiple square is a graph $\\mathrm {Squ}^{(k)}=(V^{(k)},E^{(k)})$ where $V^{(k)} &:= {\\left\\lbrace \\begin{array}{ll}\\lbrace i,j,v_0\\rbrace , &\\hspace{20.0pt} k=0 \\\\V^{(k-1)}\\cup \\lbrace v_k\\rbrace , &\\hspace{20.0pt} k>0\\end{array}\\right.}", "\\\\E^{(k)} &:= {\\left\\lbrace \\begin{array}{ll}\\lbrace \\lbrace i,v_0\\rbrace ,\\lbrace v_0,j\\rbrace \\rbrace , &\\hspace{20.0pt} k=0 \\\\E^{(k-1)} \\cup \\lbrace \\lbrace i,v_{k}\\rbrace ,\\lbrace v_{k},j\\rbrace \\rbrace , &\\hspace{20.0pt} k>0.\\end{array}\\right.", "}$ Such a $k$ -multiple square is called a $k$ -multiple square sharing $i,j$ when we specify the two nodes $i$ and $j$ .", "Figure: Some examples of kk-multiple triangles and kk-multiple squares.Fig.", "REF shows $k$ -multiple triangles and $k$ -multiple squares sharing $i,j$ for $k=0,1,2$ .", "Note that the 1-multiple triangle and the 1-multiple square are equivalent to the triangle and the square, respectively.", "The number of $k$ -multiple triangles and the number of $k$ -multiple squares in a graph are denoted by $\\triangle ^{(k)}$ and $\\Box ^{(k)}$ , respectively.", "Let $\\triangle := \\triangle ^{(1)}$ and $\\Box :=\\Box ^{(1)}$ .", "For a graph $G=(V,E)$ and $s,t \\in V$ , a $s$ -$t$ path whose length is minimum among all $s$ -$t$ paths contained in $G$ is called the shortest path between $s$ and $t$.", "The length of the shortest path between $s$ and $t$ is called the distance between $s$ and $t$, denoted by $\\mathrm {dist}(s,t)$ .", "If $G$ contains no $s$ -$t$ paths, we define $\\mathrm {dist}(s, t) = \\infty $ .", "The diameter of $G$ is defined by $\\max _{\\lbrace i,j\\rbrace \\subseteq V, i \\ne j} \\mathrm {dist}(i,j)$ .", "Definition 3 For a graph $G=(V,E)$ , the average shortest path length of $G$ is defined by $\\mathrm {ASPL}(G) := \\frac{2}{n(n-1)}\\sum _{\\lbrace i,j\\rbrace \\subseteq V,\\, i\\ne j} \\mathrm {dist}(i,j).$ In this paper, we consider the following graph optimization problem: For given $n$ and $d$ , $\\begin{array}{ll}\\mbox{{\\rm minimize:} } & \\mathrm {ASPL}(G) \\\\\\mbox{{\\rm subject to:} }& |V(G)|=n \\\\&d_v = d, \\hspace{20.0pt} \\forall v \\in V(G).", "\\\\\\end{array}$ Here, $V(G)$ denotes the node set of $G$ .", "Note that the set of feasible solutions of this problem is the set of all $d$ -regular graphs of order $n$ .", "Figure: An image of switch for e ab e_{ab} and e cd e_{cd}." ], [ "Stochastic local search for our problem", "In this paper, we consider SLS algorithms described below.", "First, an initial $d$ -regular graph $G$ of order $n$ is chosen randomly.", "Then, $G$ is iteratively replaced by a local modification procedure called switch.", "In the switch procedure, a pair of edges $(e_{ab}, e_{cd})$ is chosen and replaced by $(e_{ac}, e_{bd})$ or $(e_{ad}, e_{bc})$ as in Fig.", "REF .", "If the new graph $G^{\\prime }$ is not $d$ -regular, i.e., at least one of the edges introduced by the switch already exists in $G$ , a new pair of edges in $G$ is chosen for the switch until a $d$ -regular graph $G^{\\prime }$ is found.", "Here, the $d$ -regular graph $G^{\\prime }$ , which can be obtained by a single switch of $G$ , is called a neighborhood of $G$ .", "Once a neighborhood $G^{\\prime }$ is found, $G^{\\prime }$ is evaluated by the evaluation function.", "According to the value of the evaluation function on $G^{\\prime }$ , it is determined whether or not the current graph $G$ is replaced by $G^{\\prime }$ .", "This updating procedure continues until further improvement is not expected.", "It is natural to define the evaluation function by the ASPL of $G^{\\prime }$ .", "In order to calculate the ASPL of a $d$ -regular graph of order $n$ , the distances of all node pairs have to be calculated, which takes $O(n^2d)$ time by using the breadth first search.", "Since it is not sufficiently efficient, low ASPL graphs cannot be obtained within a reasonable time for large $n$ and $d$ .", "In this paper, we consider an optimization of $d$ -regular graphs of diameter 3.", "The Moore bound implies that any graph of diameter 2 satisfies $n \\le d^2+1$ and any graph of diameter 3 satisfies $n \\le d^3-d^2+d+1$ [8].", "Furthermore, almost all graphs of diameter 2 satisfy $n<d^2$ [8], [12].", "Therefore, if $n\\ge d^2$ and $n$ is close to $d^2$ , then a random $d$ -regular graph of order $n$ is expected to have diameter 3 with high probability.", "For such $n$ and $d$ , if we choose a random $d$ -regular graph of order $n$ as the initial graph of a SLS algorithm, the diameter of every graph appearing in the SLS algorithm is empirically 3.", "By assuming that the diameter of graphs found by the SLS algorithm is always 3, the ASPL can be calculated in $O(d)$ timeThe ASPL of graph of diameter 3 is determined by the adjacency matrix $A$ of the graph and its square $A^2$ .", "Therefore, we can calculate the difference of the ASPLs of $G$ and $G^{\\prime }$ by updating $A$ and $A^2$ .", "The updating requires $O(d)$ time since $O(d)$ elements of $A^2$ change., which is still not sufficiently efficient for large $n$ .", "In this paper, we give an upper bound of the ASPL of graphs of diameter 3, and propose SLS algorithms which uses the upper bound as the evaluation function.", "It will be shown that the upper bound can be calculated in $O(1)$ time." ], [ "Bounds for the ASPL of graphs of diameter 3", "In this section, an equality and bounds for the ASPL of graphs of diameter 3 are shown.", "They are stated not only for $d$ -regular graphs but also for general graphs.", "Definition 4 For a graph $G=(V,E)$ , let $V_2 := \\lbrace \\lbrace i,j\\rbrace \\,:\\, i\\in V, j\\in V, i\\ne j\\rbrace $ .", "For $\\lbrace i,j\\rbrace \\in V_2$ , $W_{ij}:=(V \\backslash \\lbrace i,j\\rbrace ) \\cup \\lbrace 0\\rbrace .$ For $\\lbrace i,j\\rbrace \\in V_2$ and $k \\in W_{ij}$ , $S^{i,j}_k :={\\left\\lbrace \\begin{array}{ll}\\lbrace \\lbrace i,j\\rbrace \\,:\\, \\lbrace i,j\\rbrace \\in E \\rbrace , &\\hspace{20.0pt} k=0 \\\\\\lbrace \\lbrace i,j\\rbrace \\,:\\, \\lbrace i,k\\rbrace ,\\lbrace k,j\\rbrace \\in E \\rbrace , &\\hspace{20.0pt} k\\ne 0.\\end{array}\\right.", "}$ For $m=1,2,\\ldots ,n-1$ , $T(m) := \\sum _{\\lbrace i,j\\rbrace \\in V_2} \\sum _{\\begin{array}{c}K \\subseteq W_{ij}\\\\ |K|=m\\end{array}} \\left| \\bigcap _{k \\in K} S^{i,j}_k \\right|.$ Note that $S^{i,j}_k$ is either $\\lbrace \\lbrace i,j\\rbrace \\rbrace $ or the empty set.", "Lemma 5 For a graph $G=(V,E)$ , $n_1+n_2 = \\sum _{m=1}^{n-1} (-1)^{m-1} T(m).$ where $n_k = \\#\\lbrace \\lbrace i,j\\rbrace \\in V_2 : \\mathrm {dist}(i,j) = k \\rbrace $ for $k=1,2$ .", "For $\\lbrace i, j\\rbrace \\in V_2$ , it holds $\\lbrace i,j\\rbrace \\in \\bigcup _{k \\in W_{ij}} S^{i,j}_k \\, \\Longleftrightarrow \\, \\mathrm {dist}(i,j) \\le 2.$ We can obtain the lemma by using the inclusion-exclusion principle.", "$n_1+n_2&=& \\sum _{\\lbrace i,j\\rbrace \\in V_2} \\left| \\bigcup _{k\\in W_{ij}} S^{i,j}_k \\right| \\nonumber \\\\&=& \\sum _{\\lbrace i,j\\rbrace \\in V_2} \\sum _{m=1}^{n-1} (-1)^{m-1} \\sum _{\\begin{array}{c}K \\subseteq W_{ij}\\\\ |K|=m\\end{array}} \\left| \\bigcap _{k \\in K} S^{i,j}_k \\right| \\nonumber \\\\&=& \\sum _{m=1}^{n-1} (-1)^{m-1}T(m).", "$ Lemma 6 $T(m) = {\\left\\lbrace \\begin{array}{ll}\\frac{1}{2} \\sum _{k \\in V} d_k^2, &\\hspace{20.0pt} m=1 \\\\3\\triangle +2\\Box , &\\hspace{20.0pt} m=2 \\\\\\triangle ^{(m-1)}+\\Box ^{(m-1)}, &\\hspace{20.0pt} m \\ge 3.\\end{array}\\right.", "}$ When $m=1$ $T(1) &=& \\sum _{\\lbrace i,j\\rbrace \\in V_2} \\left| S^{i,j}_0 \\right| + \\sum _{\\begin{array}{c}\\lbrace i,j\\rbrace \\in V_2\\\\k\\in W_{ij}\\backslash \\lbrace 0\\rbrace \\end{array}} \\left|S^{i,j}_k\\right| \\\\&=& |E| + \\sum _{k \\in V} \\binom{d_k}{2}= \\frac{1}{2} \\sum _{k \\in V}d_k^2.$ Here, we used the fact $|E|=\\sum _{k\\in V} d_k/2$ .", "Assume $m\\ge 2$ .", "For $K\\subseteq W_{ij}$ and $|K|=m$ , $|\\bigcap _{k\\in K} S^{i,j}_k|=1$ if $ \\lbrace i,j\\rbrace \\in \\bigcap _{k\\in K} S^{i,j}_{k}$ and $|\\bigcap _{k\\in K} S^{i,j}_k|=0$ otherwise.", "If $0\\in K$ , (REF ) holds if and only if there is an $(m-1)$ -multiple triangle sharing $i,j$ whose node set is $\\lbrace i,j\\rbrace \\cup K\\setminus \\lbrace 0\\rbrace $ .", "Similarly, if $0\\notin K$ , the equality (REF ) holds if and only if there is an $(m-1)$ -multiple square sharing $i,j$ whose node set is $\\lbrace i,j\\rbrace \\cup K$ .", "Assume $m=2$ .", "If there is a triangle consisting of three nodes $i$ , $j$ and $k$ and of edge set $\\lbrace e_{ij},e_{jk},e_{ki}\\rbrace $ , it holds $\\lbrace i,j\\rbrace &\\in S^{i,j}_0 \\cap S^{i,j}_k,&\\lbrace j,k\\rbrace &\\in S^{j,k}_0 \\cap S^{j,k}_i,\\\\\\lbrace k,i\\rbrace &\\in S^{k,i}_0 \\cap S^{k,i}_j.$ If there is a square consisting of four nodes $a$ , $b$ , $c$ and $d$ and of edge set $\\lbrace e_{ab},e_{bc},e_{cd},e_{da}\\rbrace $ , it holds $\\lbrace a,c\\rbrace &\\in S^{a,c}_b \\cap S^{a,c}_d,&\\lbrace b,d\\rbrace &\\in S^{b,d}_a \\cap S^{b,d}_c.$ Therefore, $T(2)&= \\sum _{\\begin{array}{c}\\lbrace i,j\\rbrace \\in V_2\\\\k,l\\in W_{ij}\\end{array}} \\left|S^{i,j}_k \\cap S^{i,j}_l \\right| \\\\&= \\sum _{\\begin{array}{c}\\lbrace i,j\\rbrace \\in V_2\\\\k\\in W_{ij}\\backslash \\lbrace 0\\rbrace \\end{array}} \\left|S^{i,j}_0 \\cap S^{i,j}_k \\right|+ \\sum _{\\begin{array}{c}\\lbrace i,j\\rbrace \\in V_2\\\\k,l \\in W_{ij} \\backslash \\lbrace 0\\rbrace \\end{array}} \\left|S^{i,j}_k \\cap S^{i,j}_l \\right| \\\\&= 3\\triangle + 2\\Box .$ Assume $m \\ge 3$ .", "If there is an $(m-1)$ -multiple triangle sharing $i,j$ consisting of nodes $\\lbrace i,j\\rbrace \\cup K$ for some $K\\subseteq W_{ij}\\setminus \\lbrace 0\\rbrace $ , it holds $\\lbrace i,j\\rbrace \\in \\bigcap _{k\\in K\\cup \\lbrace 0\\rbrace } S^{i,j}_k$ and $\\left|K \\cup \\lbrace 0\\rbrace \\right| = m$ .", "Similarly, if there is an $(m-1)$ -multiple square sharing $i,j$ consisting of nodes $\\lbrace i,j\\rbrace \\cup K$ for some $K\\subseteq W_{ij}\\setminus \\lbrace 0\\rbrace $ , it holds $\\lbrace i,j\\rbrace \\in \\bigcap _{k\\in K} S^{i,j}_k$ and $\\left|K \\right|=m$ .", "Therefore, $T(m)=\\triangle ^{(m-1)}+\\Box ^{(m-1)}$ .", "Theorem 7 For a graph $G=(V,E)$ of diameter 3, $\\mathrm {ASPL}(G) &= 3-\\frac{2}{n(n-1)}\\left(|E|+\\sum _{m=1}^{n-1} (-1)^{m-1} T(m) \\right).$ $\\mathrm {ASPL}(G)$ can be represented by $\\mathrm {ASPL}(G) = \\frac{2}{n(n-1)}(n_1+2n_2+3n_3)$ where $n_k = \\#\\lbrace \\lbrace i,j\\rbrace \\in V_2 : \\mathrm {dist}(i,j) = k \\rbrace $ for $k=1,2,3$ .", "From Lemma REF and the fact of $n_1=|E|$ , $n_1+2n_2+3n_3&= 3(n_1+n_2+n_3)-n_1-(n_1+n_2) \\\\&= \\frac{3n(n-1)}{2}-|E|-\\sum _{m=1}^{n-1}(-1)^{m-1}T(m).$ Theorem 8 When $t$ is even $ \\mathrm {ASPL}(G) \\le 3-\\frac{2}{n(n-1)}\\left(|E|+\\sum _{m=1}^{t} (-1)^{m-1} T(m) \\right).$ When $t$ is odd $ \\mathrm {ASPL}(G) \\ge 3-\\frac{2}{n(n-1)}\\left(|E|+\\sum _{m=1}^{t} (-1)^{m-1} T(m) \\right).$ When $t$ is even, by applying the Bonferroni inequality to (REF ) [13], one obtains $n_1+n_2 \\ge \\sum _{m=1}^t (-1)^{m-1}T(m).$ The theorem for odd $t$ can be proved in the same way.", "By substituting $t=1,2$ into (REF ) and (REF ), respectively, the following corollaries are obtained.", "Corollary 9 For a $d$ -regular graph $G$ of order $n$ and diameter 3, $ \\mathrm {ASPL}(G) \\ge 3-\\frac{d(d+1)}{n-1}.$ Corollary 10 For a $d$ -regular graph $G$ of order $n$ and diameter 3, $ \\mathrm {ASPL}(G) \\le 3 - \\frac{d(d+1)}{n-1} + \\frac{6\\triangle +4\\Box }{n(n-1)}.$ The lower bound (REF ) agrees with the Moore bound [8]." ], [ "SLS using the upper bound as the evaluation function", "In this paper, we propose SLS using (REF ) as the evaluation function.", "More precisely, the evaluation function $g$ in our SLS is defined by $ g(G) = 3\\triangle +2\\Box .$ Since the minimization of (REF ) is equivalent to the minimization of (REF ), we can expect that the SLS algorithm using the above evaluation function finds a graph with low ASPL if all graphs appearing in the SLS algorithm have diameter 3.", "In this section, we show that a difference $g(G^{\\prime })-g(G)$ for a $d$ -regular graph $G$ of order $n$ and its neighborhood $G^{\\prime }$ can be calculated in $O(1)$ time by using three two-dimensional arrays.", "We also show that these three arrays can be updated in $O(d^2)$ time.", "In SLS algorithms, the frequency of the evaluations is much higher than the frequency of the updates.", "Therefore, our SLS is much faster than SLS using the ASPL as the evaluation function, which needs $O(d)$ time both for the evaluation and the update.", "For the current graph $G$ , the three two-dimensional arrays used for the $O(1)$ -time evaluation are defined by ${\\rm T1}[i][j] &:= {\\left\\lbrace \\begin{array}{ll}1, &\\hspace{20.0pt} \\lbrace i,j\\rbrace \\in E(G) \\\\0, &\\hspace{20.0pt} \\lbrace i,j\\rbrace \\notin E(G), \\end{array}\\right.", "}\\\\{\\rm T2}[i][j] &:= \\#\\lbrace \\text{$i$-$j$ paths of length 2 in $G$}\\rbrace , \\\\{\\rm T3}[i][j] &:= \\#\\lbrace \\text{$i$-$j$ paths of length 3 in $G$}\\rbrace .$ Here, $E(G)$ is the edge set of $G$ .", "The construction of these three arrays takes $O(nd^3)$ time since the number of paths of length at most 3 in $G$ is $O(nd^3)$ , and since $d=\\Omega (n^{1/3})$ for guaranteeing the diameter of $G$ is equal to 3.", "We first construct T1, T2 and T3 for the initial graph in the SLS.", "Then, at each step of the SLS, we evaluate a neighborhood in $O(1)$ time by using T1, T2 and T3.", "If the current graph is replaced, T1, T2 and T3 are updated in $O(d^2)$ time.", "In the following, we show algorithms for the $O(1)$ -time evaluation and the $O(d^2)$ -time update." ], [ "$O(1)$ -time evaluation with arrays", "For the neighborhood $G^{\\prime }$ , let $\\triangle ^{\\prime }$ be the number of triangles in $G^{\\prime }$ and $\\Box ^{\\prime }$ be the number of squares in $G^{\\prime }$ .", "Then, it holds $g(G^{\\prime })-g(G) = 3(\\triangle ^{\\prime }-\\triangle )+2(\\Box ^{\\prime }-\\Box ).$ We assume that a neighborhood $G^{\\prime }$ of $G$ is obtained by replacing $e_{ab},e_{cd} \\in E(G)$ with $e_{ac},e_{bd} \\notin E(G)$ .", "The switch can be divided into two steps: the removal of $e_{ab},e_{cd}$ and the addition of $e_{ac},e_{bd}$ .", "First, we consider the removal of $e_{ab}$ and $e_{cd}$ .", "Since no triangles contain both of $e_{ab}$ and $e_{cd}$ , the number of triangles decreases by ${\\rm T2}[a][b]+{\\rm T2}[c][d].$ Next, we consider the addition of $e_{ac}$ and $e_{bd}$ to the graph obtained by the removal of $e_{ab}$ and $e_{cd}$ .", "Note that the arrays T1, T2 and T3 remain unchanged at this time.", "After the removal of $e_{ab}$ and $e_{cd}$ , the addition of $e_{ac}$ increases the number of triangles by ${\\rm T2}[a][c] - {\\rm T1}[a][d]-{\\rm T1}[b][c].$ Here, the terms ${\\rm T1}[a][d]$ and ${\\rm T1}[b][c]$ take account of non-existent paths $a$ -$d$ -$c$ and $a$ -$b$ -$c$ , respectively.", "Similarly, the addition of $e_{bd}$ increases the number of triangles by ${\\rm T2}[b][d] - {\\rm T1}[a][d]-{\\rm T1}[b][c].$ Hence, from (REF ), (REF ) and (REF ), one obtains $\\triangle ^{\\prime } - \\triangle = &-{\\rm T2}[a][b]-{\\rm T2}[c][d]+{\\rm T2}[a][c]+{\\rm T2}[b][d]\\\\&- 2 ({\\rm T1}[a][d]+{\\rm T1}[b][c]).$ Next, we consider $\\Box ^{\\prime }-\\Box $ .", "Since the number of squares that contain both of $e_{ab}$ and $e_{cd}$ is ${\\rm T1}[a][d]{\\rm T1}[b][c]$ (Recall that $e_{ac}$ and $e_{bd}$ are not in $E(G)$ ), the removal of $e_{ab}$ and $e_{cd}$ decreases the number of squares by ${\\rm T3}[a][b]+{\\rm T3}[c][d] - {\\rm T1}[a][d]{\\rm T1}[b][c].$ After the removal of $e_{ab}$ and $e_{cd}$ , the addition of $e_{ac}$ increases the number of squares by $ {\\rm T3}[a][c] - {\\rm T2}[a][d] - {\\rm T2}[b][c].$ Here, the terms ${\\rm T2}[a][d]$ and ${\\rm T2}[b][c]$ take account of non-existent paths $a$ -$*$ -$d$ -$c$ and $a$ -$b$ -$*$ -$c$ , respectively, where $*$ represents an arbitrary node (A non-existent path $a$ -$b$ -$d$ -$c$ is not counted since $e_{bd}\\notin E(G)$ ).", "Similarly, after the addition of $e_{ac}$ , the addition of $e_{bd}$ increases the number of squares by $ {\\rm T3}[b][d] - {\\rm T2}[a][d] - {\\rm T2}[b][c] + {\\rm T1}[a][d]{\\rm T1}[b][c].$ Here, the term ${\\rm T1}[a][d]{\\rm T1}[b][c]$ takes account of the path $b$ -$c$ -$a$ -$d$ in $G^{\\prime }$ .", "Hence, from (REF ), (REF ) and (REF ), one obtains $\\Box ^{\\prime }-\\Box = &-{\\rm T3}[a][b]-{\\rm T3}[c][d]+{\\rm T3}[a][c]+{\\rm T3}[b][d]\\\\&-2({\\rm T2}[a][d]+{\\rm T2}[b][c]-{\\rm T1}[a][d]{\\rm T1}[b][c]).$ We conclude that we can calculate $g(G^{\\prime })-g(G)$ in $O(1)$ time if T1, T2 and T3 for $G$ are given." ], [ "Array update", "When $e_{ab}$ is removed, T2 can be updated by decrementing ${\\rm T2}[t][s]$ and ${\\rm T2}[s][t]$ for all $s$ -$t$ paths of length 2 using $e_{ab}$ .", "Since the number of these paths is $O(d)$ , this calculation can be done in $O(d)$ time.", "Also T3 can be updated by decrementing ${\\rm T3}[s][t]$ and ${\\rm T3}[t][s]$ for all $s$ -$t$ paths of length 3 using $e_{ab}$ .", "The number of these paths is $O(d^2)$ and thus this calculation can be done in $O(d^2)$ time.", "When edges are added, the arrays can be updated in $O(d^2)$ time in the same way." ], [ "Accuracy of approximations", "In this section, we show by numerical experiments that the upper bounds and lower bounds obtained by Theorem REF are also good approximations for the ASPL.", "Table REF shows the relative error $(\\widetilde{\\mathrm {ASPL}}(G)-\\mathrm {ASPL}(G))/\\mathrm {ASPL}(G)$ of the approximations where $\\widetilde{\\mathrm {ASPL}}(G)$ denotes the bounds obtained by Theorem REF for $t=1,2,3$ , and where $G$ denotes a random $d$ -regular graph of order $n$ where $(n,d)=(4096,60),(4096,64),(10\\,000,60),(10\\,000,64)$ .", "The diameters of these graphs are 3.", "Table: The relative ASPL errors of approximations.Table REF implies that the approximations are more accurate as $t$ increases.", "Furthermore, Table REF also implies that the approximations are more accurate for sparse graphs.", "It can be easily understood since $\\triangle ,\\Box ,\\triangle ^{(2)},\\Box ^{(2)},\\ldots $ are small when a graph is sparse." ], [ "Construction by Iterative First Improvement", "The Iterative First Improvement (IFI) algorithm is one of the simplest SLS algorithms.", "The IFI algorithm evaluates neighborhoods one after another.", "When the IFI finds the neighborhood whose value of the evaluation function is lower than that of the current solution, it replaces the current solution by the neighborhood and then starts to evaluate the next neighborhood.", "The evaluation and replacement continues until the IFI algorithm finds the local optimum.", "We consider the IFI algorithm with the $O(1)$ -time evaluation function proposed in Section .", "In order to improve the efficiency of the IFI algorithm, we modify the IFI algorithm as follows: At every 50 replacements, the set of edges are sorted by the number of triangles and squares that contain the edge.", "More precisely, all edges are sorted in descending order of $3\\triangle _e + 2\\Box _e$ where $\\triangle _e$ and $\\Box _e$ denote the number of triangles and squares that contain the edge $e$ , respectively.", "After the sort, all pairs of edges are chosen in the order of $(e_1,e_2), (e_1,e_3), \\cdots , (e_1,e_{|E|}), (e_2,e_3), (e_2,e_4),\\cdots $ .", "This modification is useful for finding efficiently a neighborhood improving the value of the evaluation function.", "Table REF shows the ASPL, the Moore bound (REF ), denoted by $L$ , and the ASPL gap $(\\mathrm {ASPL}(G)-L)/L$ for random $d$ -regular graph $G$ of order $n$ .", "From these random graphs, we applied the above modified IFI algorithm, and obtained local optimals.", "Table REF shows the ASPL, the ASPL gap for the local optimals and the required time for the modified IFI.", "These numerical experiments are performed on MacBook Pro, Intel Core i7 2.6GHz.", "The modified IFI found a local optimal within two days.", "The obtained local optimals have smaller ASPL gap than the original random graphs.", "Especially, for $(n,d)=(10\\,000,60), (10\\,000,64)$ , the ASPL gaps are significantly improved.", "Table: ASPL of Random GraphsTable: Results of the modified IFI" ], [ "Construction by Simulated Annealing", "The Simulated Annealing (SA) is one of the most successful SLS algorithms.", "The SA in our numerical experiments works as follows.", "At each step, a neighborhood is chosen uniformly at random and evaluated.", "If the value $E^{\\prime }$ of evaluation function for the neighborhood is lower than that $E$ for the current solution, then the current solution is replaced by the neighborhood.", "Also, even if the neighborhood has the higher evaluation value than the current solution, i.e., $E^{\\prime }>E$ , the current graph is also replaced by the neighborhood with probability $P$ where $P = \\exp \\left\\lbrace -\\frac{E^{\\prime }-E}{T}\\right\\rbrace .$ Here, $T$ is a parameter called the temperature which decreases at each step of SA.", "The annealing schedule, which determines the temperature at each step, is crucial to the SA.", "In the numerical experiments, we set $T(k):=11/ \\ln (k+1)$ where $k$ denotes the number of graphs which are evaluated until that time.", "Table REF shows the ASPL of graphs constructed by applying the SA 60 days, and then applying the modified IFI.", "They are the best graphs obtained in the Graph Golf [10].", "Table: Results of the SA" ], [ "Conclusion", "We derived the equality and bounds for the ASPL of graphs of diameter 3 by using the number of triangles, squares, and some other structures in a graph.", "By numerical experiments, we confirmed that the upper and lower bounds obtained are also accurate approximation for the ASPL.", "On the basis of one of our bounds, we propose to use $3\\triangle + 2\\Box $ as the evaluation function of SLS for the ASPL minimization problem where $n$ is at least and close to $d^2$ , so that the random graph has diameter 3 with high probability.", "We show that the proposed SLS algorithms requires $O(1)$ time for the evaluation and $O(d^2)$ time for the update.", "We construct low ASPL graphs of diameter 3 by IFI and SA using our evaluation function.", "While the upper bound evaluated by the proposed algorithm is represented only by the number of triangles and squares, the other bounds represented by the number of more complicated structures such as $\\triangle ^{(2)},\\Box ^{(2)}$ are confirmed to be more accurate.", "Thus, if there exists $O(1)$ -time evaluation algorithm that calculates one of these bounds, then we can use the bound as the evaluation function of SLS.", "It would construct graphs with lower ASPL.", "Finally, we note that our bounds of ASPL for graphs of diameter 3 can be generalized to larger diameter in a similar way.", "The construction of low ASPL graphs of larger diameter by similar algorithm is an interesting future work." ], [ "Acknowledgment", "This work was supported by MEXT KAKENHI Grant Number 24106008." ] ]
1606.05119
[ [ "Logarithmic Time One-Against-Some" ], [ "Abstract We create a new online reduction of multiclass classification to binary classification for which training and prediction time scale logarithmically with the number of classes.", "Compared to previous approaches, we obtain substantially better statistical performance for two reasons: First, we prove a tighter and more complete boosting theorem, and second we translate the results more directly into an algorithm.", "We show that several simple techniques give rise to an algorithm that can compete with one-against-all in both space and predictive power while offering exponential improvements in speed when the number of classes is large." ], [ "Introduction", "Can we effectively predict one of $K$ classes in polylogarithmic time in $K$ ?", "This question gives rise to the area of extreme multiclass classification [1], [3], [4], [8], [18], [20], [27], in which $K$ is very large.", "If efficiency is not a concern, the most common and generally effective representation for multiclass prediction is a one-against-all (OAA) structure.", "Here, inference consists of computing a score for each class and returning the class with the maximum score.", "An attractive strategy for picking one of $K$ items efficiently is to use a tree; unfortunately, this often comes at the cost of increased error.", "A general replacement for the one-against-all approach must satisfy a difficult set of desiderata.", "[noitemsep,nolistsep] High accuracy: The approach should provide accuracy competitive with OAA, a remarkably strong baseline[21] which is the standard “output layer” of many learning systems such as winners of the ImageNet contest [12], [23].", "High speed at training time and test time: A multiclass classifier must spend at least $\\Omega (\\log K)$ time [8]) so this is a natural benchmark to optimize against.", "Online operation: Many learning algorithms use either online updates or mini-batch updates.", "Approaches satisfying this constraint can be easily composed into an end-to-end learning system for solving complex problems like image recognition.", "For algorithms which operate in batch fashion, online components can be easily used.", "Linear space: In order to have a drop-in replacement for OAA, an approach must not take much more space than OAA.", "Memory is at a premium when $K$ is very large, especially for models trained on GPUs, or deployed to small devices.", "We use an OAA-like structure to make a final prediction, but instead of scoring every class, we only score a small subset of $O(\\log K)$ classes.", "We call this “one-against-some” (OAS).", "How can you efficiently determine what classes should be scored?", "We use a dynamically built tree to efficiently whittle down the set of candidate classes.", "The goal of the tree is to maximize the recall of the candidate set so we call this approach “The Recall Tree.” Figure REF depicts the inference procedure for the Recall Tree: an example is routed through a tree until termination, and then the set of eligible classes compete to predict the label.", "We use this inference procedure at training time, to facilitate end-to-end joint optimization of the predictors at each internal node in the tree (the “routers”), the tree structure, and the final OAS predictors.", "The Recall Tree achieves good accuracy, improving on previous online approaches [8] and sometimes surpassing the OAA baseline.", "The algorithm requires only $\\mathrm {poly}(\\log K)$ time during training and testing.", "In practice, the computational benefits are substantial when $K\\ge 1000$ .Our implementation of baseline approaches, including OAA, involve vectorized computations that increase throughput by a factor of 10 to 20, making them much more difficult to outpace than naïve implementations.", "The Recall Tree constructs a tree and learns parameters in a fully online manner as a reduction allowing composition with systems trained via online updates.", "All of this requires only a factor of 2 more space than OAA approaches.", "Our contributions are the following: [noitemsep,nolistsep] We propose a new online tree construction algorithm which jointly optimizes the construction of the tree, the routers and the underlying OAS predictors (see section REF ).", "We analyze elements of the algorithm, including a new boosting bound (see section REF ) on multiclass classification performance and a representational trick which allows the algorithm to perform well if either a tree representation does well or a OAA representation does well as discussed in section REF .", "We experiment with the new algorithm, both to analyze its performance relative to baselines and understand the impact of design decisions via ablation experiments.", "The net effect is a theoretically motivated algorithm which empirically performs well providing a plausible replacement for the standard one-against-all approach in the large $K$ setting.", "Figure: Left: Pseudocode for prediction where f n (x)f_\\mathrm {n}(x) evaluates the node's route, Predict y (x)\\mbox{Predict}_y(x) evaluates a per-class regressor, 𝐫𝐞𝐜𝐚𝐥𝐥 ^(∘)\\widehat{\\mathbf {recall}} (\\circ ) is an empirical bound on the recall of a node (∘)(\\circ ) (see section ), and x∧{(n:1)}x \\wedge \\lbrace (n:1) \\rbrace indicates the addition of a sparse feature with index nn and value 1.", "Right: An example is routed through the tree to a leaf node associated with a set of eligible classes." ], [ "Prior Work", "The LOMTree[7], [8] is the closest prior work.", "It misses on space requirements: up to a factor of 64 more space than OAA was used experimentally.", "Despite working with radically less space we show the Recall Tree typically provides better predictive performance.", "The key differences here are algorithmic: A tighter reduction at internal nodes and the one-against-some approach yields generally better performance despite much tighter resource constraints.", "Boosting trees [13] for multiclass learning [24] on a generalized notion of entropy are known to results in low 0/1 loss.", "Relative to these works we show how to efficiently achieve weak learning by reduction to binary classification making this approach empirically practical.", "We also address a structural issue in the multiclass analysis (see section REF ).", "Other approaches such as hierarchical softmax (HSM) and the the Filter Tree [3] use a fixed tree structure [18].", "In domains in which there is no prespecified tree hierarchy, using a random tree structure can lead to considerable underperformance as shown previously [1], [8].", "Most other approaches in extreme classification either do not work online [17], [20] or only focus on speeding up either prediction time or training time but not both.", "Most of the works that enjoy sublinear inference time (but (super)linear training time) are based on tree decomposition approaches.", "In [17] the authors try to add tree structure learning to HSM via iteratively clustering the classes.", "While the end result is a classifier whose inference time scales logarithmically with the number of classes, the clustering steps are batch and scale poorly with the number of classes.", "Similar remarks apply to [1] where the authors propose to learn a tree by solving an eigenvalue problem after (OAA) training.", "The work of [27] is similar in spirit to ours, as the authors propose to learn a label filter to reduce the number of candidate classes in an OAA approach.", "However they learn the tree after training the underlying OAA predictors while here we learn and, more crucially, use the tree during training of the OAS predictors.", "Among the approaches that speed up training time we distinguish exact ones [9], [25] that have only been proposed for particular loss functions and approximate ones such as negative sampling as used e.g.", "in [26].", "Though these techniques do not address inference time, separate procedures for speeding up inference (given a trained model) have been proposed [22].", "However, such two step procedures can lead to substantially suboptimal results." ], [ "The Recall Tree Algorithm", "Here we present a concrete description of the Recall Tree and defer all theoretical results that motivate our decisions to the next section.", "The Recall Tree maintains one predictor for each class and a tree whose purpose is to eliminate predictors from consideration.", "We refer to the per-class predictors as one-against-some (OAS) predictors.", "The tree creates a high recall set of candidate classes and then leverages the OAS predictors to achieve precision.", "Crucially, the leaves of the tree do not partition the set of classes: classes can (and do) have support at multiple leaves.", "Figure: Learning procedure.Figure REF outlines the learning procedures, which we now describe in more detail.", "Each node in the tree maintains a set of statistics.", "First, each node $n$ maintains a router, denoted $f_\\mathrm {n}$ , that maps an example to either a left or right child.", "This router is implemented as a binary classifier.", "Second, each node maintains a histogram of the labels of all training examples that have been routed to, or through, that node.", "This histogram is used in two ways: (1) the most frequent classes form the competitor set for the OAS predictors; (2) the histogram is used to decide whether the statistics at each node can be trusted.", "This is a crucial issue with trees because a child node sees fewer data than its parent.", "Therefore we do not simply rely on the empirical recall (i.e.", "the observed fraction of labels that fall into the most frequent $F$ labels at this node) of a node since such estimate can have considerable variance at deep nodes.", "Instead, we use a lower bound of the true recall which we compute via an empirical Bernstein inequality (see Section REF ).", "Learning the predictors for each class In Figure REF $\\mathbf {update\\_predictors}$ updates the candidate set predictors using the standard OAA strategy restricted to the set of eligible classes.", "If the true label is not in the $F$ most frequent classes at this node then no update occurs.", "Learning the set of candidates in each node In Figure REF $\\mathbf {update\\_candidates}$ updates the count of the true label at this node.", "At each node, the most frequent $F$ labels are the candidate set.", "Learning the routers at each node In Figure REF $\\mathbf {update\\_router}$ updates the router at a node by optimizing the decrease in the entropy of the label distribution (the label entropy) due to routing.", "This is in accordance with our theory (Section REF ).", "The label entropy for a node is estimated using the empirical counts of each class label entering the node.", "These counts are reliable as $\\mathbf {update\\_router}$ is only called for the root or nodes whose true recall bound is better than their children.", "The expected label entropy after routing is estimated by averaging the estimated label entropy of each child node, weighted by the fraction of examples routing left or right.", "Finally, we compute the advantage of routing left vs. right by taking the difference of the expected label entropies for routing left vs. right.", "The sign of this difference determines the binary label for updating the router.", "Tree depth control We calculate a lower bound $\\widehat{\\mathbf {recall}} (\\mathrm {n})$ on the true recall of node $n$ (Section REF ), halting descent as in Figure REF .", "As we descend the tree, the bound first increases (empirical recall increases) then declines (variance increases).", "We also limit the maximum depth $d$ of the tree.", "This parameter is typically not operative but adds an additional safety check and sees some use on datasets where multipasses are employed." ], [ "Theoretical Motivation", "Online construction of an optimal logarithmic time predictor for multiclass classification given an arbitrary fixed representation at each node appears deeply intractable.", "A primary difficulty is that decisions have to be hard since we cannot afford to maintain a distribution over all class labels.", "Choosing a classifier so as to minimize error rate has been considered for cryptographic primitives [5] so it is plausibly hard on averager rather than merely hard in the worst case.", "Furthermore, the joint optimization of all predictors does not nicely decompose into independent problems.", "Solving the above problems requires an implausible break-through in complexity theory which we do not achieve here.", "Instead, we use learning theory to assist the design by analyzing various simplifications of the problem." ], [ "One-Against-Some Prediction and Recall", "For binary classification, branching programs [15] result in exponentially more succinct representations than decision trees [13] by joining nodes to create directed acyclic graphs.", "The key observation is that nodes in the same level with a similar distribution over class labels can be joined into one node, implying that the number of nodes at one level is only $\\theta (1/\\gamma )$ where $\\gamma $ is the weak learning parameter rather than exponential in the depth.", "This approach generally fails in the multiclass setting because covering the simplex of multiclass label distributions requires $(K-1)^{\\theta (1/\\gamma )}$ nodes.", "One easy special case exists.", "When the distribution over class labels is skewed so one label is the majority class, learning a minimum entropy classifier is equivalent to predicting whether the class is the majority or not.", "There are only $K$ possible OAS predictors of this sort so maintaining one for each class label is computationally tractable.", "Using OAS classifiers creates a limited branching program structure over predictions.", "Aside from the space savings generated, this also implies that nodes deep in the tree use many more labeled examples than are otherwise available.", "In finite sample regimes, which are not covered by these boosting analyses, having more labeled samples implies a higher quality predictor as per standard sample complexity analysis.", "A fundamental issue with a tree-structured prediction is that the number of labeled examples incident on the root is much larger than the number of labeled examples incident on a leaf.", "This potentially leads to: (1) underfitting toward the leaves; and (2) insufficient representation complexity toward the root.", "Optimizing recall, rather than accuracy, ameliorates this drawback.", "Instead of halting at a leaf, we can halt at an internal node $n$ for which the top $F$ most frequent labels contain the true answer with a sufficiently high probability.", "When $F = O(\\log K)$ this does not compromise the goal of achieving logarithmic time classification.", "Nevertheless, as data gets divided down the branches of the tree, empirical estimates for the “top $F$ most frequent labels” suffer from a substantial missing mass problem [11].", "Thus, instead of computing empirical recall to determine when to halt descent, we use an empirical Bernstein (lower) bound [16], which is summarized by the following proposition.", "Proposition 1 For all learning problems $D$ and all nodes $n$ in a fixed tree there exists a constant $\\lambda > 0$ such that with probability $1-\\delta $ : $r_n \\ge \\hat{r}_n - \\sqrt{\\frac{\\lambda \\hat{r}_n (1 - \\hat{r}_n)}{m_n}} - \\frac{\\lambda }{m_n}$ where $\\hat{r}_n$ is the empirical frequency amongst $m_n$ events that the true label is in the top $F$ labels and $r_n$ is the expected value in the population limit.", "Reducing the depth of the tree by using a bound on $r_n$ and joining labeled examples from many leaves in a one-against-some approach both relieves data sparsity problems and allows greater error tolerance by the root node." ], [ "Path Features", "The relative representational power of different solutions is an important consideration.", "Are OAA types of representations inherently more or less powerful than a tree based representation?", "Figure REF shows two learning problems illustrating two extremes under the assumption of a linear representation.", "Linear OAA: If all the class parameter vectors happen to have the same magnitude then OAA classification is equivalent to finding the nearest neighbor amongst a set of vectors (one per class) which partition the space into a Voronoi diagram as in REF on the left.", "The general case, with unequal vectors corresponds to a weighted Voronoi diagram where the magnitude of two vectors sharing a border determines the edge of the partition.", "No weighted Voronoi diagram can account for the partition on the right.", "Trees: If the partition of a space can be represented by a sequence of conditional splits, then a tree can represent the solution accurately as in REF on the right.", "On the other hand, extra work is generally required to represent a Voronoi diagram as on the left.", "In general, the number of edges in a multidimensional Voronoi diagram may grow at least quadratically in the number of points implying that the number of nodes required for a tree to faithfully represent a Voronoi diagram is at least $\\Theta (n^2)$ .", "Based on this, neither tree-based nor OAA style prediction is inherently more powerful, with the best solution being problem dependent.", "Since we are interested in starting with a tree-based approach and ending with a OAS classifier there is a simple representational trick which provides the best of both worlds.", "We can add features which record the path through the tree.", "To be precise, let $T$ be a tree and $\\mbox{path}_T(x)$ be a vector with one dimension per node in $T$ which is set to 1 if $x$ traverses the node and 0 otherwise.", "The following proposition holds.", "Proposition For any learning problem $D$ for which a tree $T$ achieves error rate $\\epsilon $ , $\\mbox{OAA}(x,\\mbox{path}_T(x))$ with a linear representation can achieve error rate $\\epsilon $ .", "Linear representations are special, because they are tractably analyzed and because they are the fundamental building blocks around which many more complex representations are built.", "Hence, this representational change eases prediction in many common settings.", "A linear OAA classifier is defined by a matrix $w_{iy}$ where $i$ ranges over the input and $y$ ranges over the labels.", "Let $w_{iy} = 0$ by default and 1 when $i$ corresponds to a leaf for which the tree predicts $y$ .", "Under this representation, the prediction of $\\mbox{OAA}(x,\\mbox{path}_{T}(x))$ is identical to $T(x)$ , and hence achieves the same error rate." ], [ "Optimization Objective", "The Shannon Entropy of class labels is optimized in the router of figure REF .", "Why?", "Since the Recall Tree jointly optimizes over many base learning algorithms, the systemic properties of the joint optimization are important to consider.", "A theory of decision tree learning as boosting [13] provides a way to understand these joint properties in a population limit (or equivalently on a training set iterated until convergence).", "In essence, the analysis shows that each level of the decision tree boosts the accuracy of the resulting tree with this conclusion holding for several common objectives.", "In boosting for multiclass classification [7], [8], [24], it is important to achieve a weak dependence on the number of class labels.", "Shannon Entropy is particularly well-suited to this goal, because it has only a logarithmic dependence on the number of class labels.", "Let $\\pi _{i|n}$ be the probability that the correct label is $i$ , conditioned on the corresponding example reaching node $n$ .", "Then $H_{n}=\\sum _{i=1}^{K}\\pi _{i|n}\\log _2 \\frac{1}{\\pi _{i|n}}$ is the Shannon entropy of class labels reaching node $n$ .", "For this section, we consider a simplified algorithm which neglects concerns of finite sample analysis, how optimization is done, and the leaf predictors.", "What's left is the value of optimizing the router objective.", "We consider an algorithm which recursively splits the leaf with the largest fraction $f$ of all examples starting at the root and reaching the leaf.", "The leaf is split into two new leaves to the left $l$ and right $r$ .", "If $f_l$ and $f_r$ are the fraction of examples going left and right, the split criterion minimizes the expectation over the leaves of the average class entropy, $f_l H_l+f_r H_r$ .", "This might be achieved by $\\mathbf {update\\_router}$ in Figure REF or by any other means.", "With this criterion we are in a position to directly optimize information boosting.", "Definition 1 ($\\gamma $ -Weak Learning Assumption) For all distributions $n(x,y)$ a learning algorithm using examples $(x,y)^*$ IID from $n$ finds a binary classifier $c:X\\rightarrow \\lbrace l,r\\rbrace $ satisfying $f_l H_l + f_r H_r\\le H_n-\\gamma \\,\\,\\,\\,.$ This approach is similar to previous [24] except that we boost in an additive rather than a multiplicative sense.", "This is good because it suppresses an implicit dependence on $K$ (since for any nontrivial $\\gamma $ there exists a $K$ such that with a uniform distribution $U$ , $H_U(1-\\gamma ) > 1$ ), yeilding a strictly stronger result.", "As long as Weak Learning occurs, we can prove the following theorem.", "Theorem 2 If $\\gamma $ Weak Learning holds for every node in the tree and nodes with the largest fraction of examples are split first, then after $t>2$ splits the multiclass error rate $\\epsilon $ of the tree is bounded by: $ \\epsilon \\le H_1 - \\gamma (1 + \\ln t ) $ where $H_1$ is the entropy of the marginal distribution of class labels.", "The proof in appendix  reuses techniques from [8], [13] but has a tighter result.", "The most important observation from the theorem is that as $t$ (the number of splits) increases, the error rate is increasingly bounded.", "This rate depends on $\\ln t$ agreeing with the intuition that boosting happens level by level in the tree.", "The dependence on the initial entropy $H_1$ shows that skewed marginal class distributions are inherently easier to learn than uniform marginal class distributions, as might be expected.", "These results are similar to previous results [7], [8], [13], [24] with advantages.", "We handle multiclass rather than binary classification [13], we bound error rates instead of entropy [7], [8], and we use additive rather than multiplicative weak learning [24]." ], [ "Empirical Results", "We study several questions empirically.", "What is the benefit of using one-against-some on a recall set?", "What is the benefit of path features?", "Is the online nature of the Recall Tree useful on nonstationary problems?", "How does the Recall Tree compare to one-against-all statistically and computationally?", "How does the Recall Tree compare to LOMTree statistically and computationally?", "Throughout this section we conduct experiments using learning with a linear representation." ], [ "Datasets", "Table REF overviews the data sets used for experimentation.", "These include the largest datasets where published results are available for LOMTree (Aloi, Imagenet, ODP), plus an additional language modeling data set (LTCB).", "Implementations of the learning algorithms, and scripts to reproduce the data sets and experimental results, are available at (url redacted).", "Additional details about the datasets can be found in Appendix ." ], [ "Comparison with other Algorithms", "In our first set of experiments, we compare Recall Tree with a strong computational baseline and a strong statistical baseline.", "The computational baseline is LOMTree, the only other online logarithmic-time multiclass algorithm we know of.", "The statistical baseline is OAA, whose statistical performance we want to match (or even exceed), and whose linear computational dependence on the number of classes we want to avoid.", "Details regarding the experimental methodology are in Appendix .", "Results are summarized in Figure REF .", "Figure: Empirical comparison of statistical (left) and computational(right) performance of Recall Tree against two strong competitors: OAA(statistically good) and LOMTree (computationally good).", "Recall Tree has poly (log)\\mathrm {poly} (log) dependence upon number of classes (like LOMTree)but can surpass OAA statistically." ], [ "Comparison with LOMTree", "The Recall Tree uses a factor of 32 less state than the LOMTree which makes a dramatic difference in feasibility for large scale applications.", "Given this state reduction, the default expectation is worse prediction performance by the Recall Tree.", "Instead, we observe superior or onpar statistical performance despite the state constraint.", "This typically comes with an additional computational cost since the Recall Tree evaluates a number of per-class predictors." ], [ "Comparison with OAA", "On one dataset (Aloi) prediction performance is superior to OAA while on the others it is somewhat worse.", "Computationally OAA has favorable constant factors since it is highly amenable to vectorization.", "Conversely, the conditional execution pattern of the Recall Tree frustrates vectorization even with example mini-batching.", "Thus on ALOI although Recall Tree does on average 50 hyperplane evaluations per example while OAA does 1000, OAA is actually faster: larger numbers of classes are required to experience the asymptotic benefits.", "For ODP with $\\sim {}10^5$ classes, with negative gradient subsampling and using 24 cores in parallel, OAA is about the same wall clock time to train as Recall Tree on a single core.While not yet implemented, Recall Tree can presumably also leverage multicore for acceleration.", "Negative gradient sampling does not improve inference times, which are roughly 300 times slower for OAA than Recall Tree on ODP." ], [ "Online Operation", "In this experiment we leverage the online nature of the algorithm to exploit nonstationarity in the data to improve results.", "This is not something that is easily done with batch oriented algorithms, or with algorithms that post-process a trained predictor to accelerate inference.", "Figure: Test error on ALOI for various candidate set sizes, with or without path features (all other parameters held fixed).", "Using multiple predictors per leaf and including path features improves performance.We consider two versions of LTCB.", "In both versions the task is to predict the next word given the previous 6 tokens.", "The difference is that in one version, the Wikipedia dump is processed in the original order (“in-order”); whereas in the other version the training data is permuted prior to input to the learning algorithm (“permuted”).", "We assess progressive validation loss [6] on the sequence.", "The result in Figure REF confirms the Recall Tree is able to take advantage of the sequentially revealed data; in particular, the far-right difference in accuracies is significant at a factor $P<0.0001$ according to an $N-1$ Chi-squared test." ], [ "Path Features and Multiple Predictors", "Two differences between Recall Tree and LOMTree are the use of multiple predictors at each tree node and the augmentation of the example with path features.", "In this experiment we explore the impact of these design choices using the ALOI dataset.", "Figure REF shows the effect of these two aspects on statistical performance.", "As the candidate set size is increased, test error decreases, but with diminishing returns.", "Disabling path features degrades performance, and the effect is more pronounced as the candidate set size increases.", "This is expected, as a larger candidate set size decreases the difficulty of obtaining good recall (i.e., a good tree) but increases the difficulty of obtaining good precision (i.e., good class predictors), and path features are only applicable to the latter.", "All differences here are significant at a $P<0.0001$ according to an $N-1$ Chi-squared test, except for when the candidate set size is 2, where there is no significant difference." ], [ "The Empirical Bernstein Bound", "Is the empirical Bernstein bound used helpful?", "To test this we trained on the LTCB dataset with a multiplier on the bound of either 0 (i.e.", "just using empirical recall directly) or 1.", "The results are stark: with a multiplier of 1, the test error was $78\\%$ while with a multiplier of 0 the test error was $91\\%$ .", "Clearly, in the small samples per class regime this form of direct regularization is extraordinarily helpful." ], [ "Conclusion", "In this work we proposed the Recall Tree, a reduction of multiclass to binary classification, which operates online and scales logarithmically with the number of classes.", "Unlike the LOMTree [8], we share classifiers among the nodes of the tree which alleviates data sparsity at deep levels while greatly reducing the required state.", "We also use a tighter analysis which is more closely followed in the implementation.", "These features allow us to reduce the statistical gap with OAA while still operating many orders of magnitude faster for large $K$ multiclass datasets.", "In the future we plan to investigate multiway splits in the tree since $O(\\log K)$ -way splits does not affect our $O(\\textrm {poly}\\log K)$ running time and they might reduce contention in the root and nodes high in the tree.", "Acknowledgements We would like to thank an anonymous reviewer for NIPS who spotted an error in the theorem proof and made several good suggestions for improving it." ], [ "Proof of theorem ", "For the fixed tree at timestep $t$ (there have been $t-1$ previous splits) with a fixed partition function in the nodes, the weighted entropy of class labels is $W_{t}=\\sum _{\\lbrace n\\in \\mbox{Leaves}\\rbrace }f_{n}H_{n}.$ When we split the $t$ th node, the weak learning assumption implies entropy decreases by $\\gamma $ according to: $H_{n}\\ge \\left(\\frac{f_{l}}{f_{n}}H_{l}+\\frac{f_{r}}{f_{n}}H_{r}\\right)+ \\gamma $ where $\\gamma $ is the advantage of the weak learner.", "Hence, $W_{t}-W_{t+1}=f_{n}H_{n}-f_{l}H_{l}-f_{r}H_{r}\\ge f_{n} \\gamma \\,\\,\\,\\,.$ We can bound $f_{n}$ according to $\\max _n f_n \\ge \\frac{1}{t}$ which implies $W_{t}-W_{t+1}\\ge \\frac{\\gamma }{t}$ .", "This can be solved recursively to get: $W_{t+1} & \\le W_{1} - \\gamma \\sum _{i=1}^t \\frac{1}{i}\\\\& \\le W_{1} - \\gamma \\left(1+\\int _{i=1}^t \\frac{1}{i} di \\right)\\\\& = W_{1} - \\gamma (1+ \\ln t)\\\\& = H_{1} - \\gamma (1+ \\ln t)\\\\$ where the second inequality follows from bounding each term of the sum with successive integrals, and $H_1$ is the marginal Shannon entropy of the class labels.", "To finish the proof, we bound the multiclass loss in terms of the average entropy.", "For any leaf node $n$ we can assign the most likely label, $y = \\arg \\max _i \\pi _{ni}$ so the error rate is $\\epsilon _n = 1-\\pi _{ny}$ .", "$W_{t+1}& = \\sum _{\\lbrace n\\in \\mbox{Leaves}\\rbrace }f_{n}\\sum _i \\pi _{ni} \\ln \\frac{1}{\\pi _{ni}} \\\\& \\ge \\sum _{\\lbrace n\\in \\mbox{Leaves}\\rbrace }f_{n} \\sum _i \\pi _{ni} \\ln \\frac{1}{\\pi _{ny}} \\\\& = \\sum _{\\lbrace n\\in \\mbox{Leaves}\\rbrace }f_{n} \\ln \\frac{1}{1-\\epsilon _n}\\\\& \\ge \\sum _{\\lbrace n\\in \\mbox{Leaves}\\rbrace }f_{n} \\epsilon _n = \\\\& = \\epsilon $ Putting these inequalities together we have: $ \\epsilon \\le H_1 - \\gamma (1 + \\ln t ) $" ], [ "Datasets", "ALOI [10] is a color image collection of one-thousand small objects recorded for scientific purposes [10].", "We use the same train-test split and representation as Choromanska et.", "al. [8].", "Imagenet consists of features extracted from intermediate layers of a convolutional neural network trained on the ILVSRC2012 challenge dataset.", "This dataset was originally developed to study transfer learning in visual tasks [19]; more details are at http://www.di.ens.fr/willow/research/cnn/.", "We utilize a predictor linear in this representation.", "LTCB is the Large Text Compression Benchmark, consisting of the first billion bytes of a particular Wikipedia dump [14].", "Originally developed to study text compression, it is now commonly used as a language modeling benchmark where the task is to predict the next word in the sequence.", "We limit the vocabulary to 80000 words plus a single out-of-vocabulary indicator; utilize a model linear in the 6 previous unigrams, the previous bigram, and the previous trigram; and utilize a 90-10 train-test split on entire Wikipedia articles.", "ODP[2] is a multiclass dataset derived from the Open Directory Project.", "We utilize the same train-test split and labels from [8].", "Specifically there is a fixed train-test split of 2:1, the representation of a document is a bag of words, and the class label is the most specific category associated with each document." ], [ "Default Performance Methodology", "Hyperparameter selection can be computationally burdensome for large data sets, which is relevant to any claims of decreased training times.", "Therefore we report results using the default values indicated in Table REF .", "For the larger data sets (Imagenet, ODP), we do a single pass over the training data; for the smaller data set (ALOI), we do multiple passes over the training data, monitoring a 10% held-out portion of the training data to determine when to stop optimizing." ], [ "Tuned Performance Methodology", "For tuned performance, we use random search over hyperparameters, taking the best result over 59 probes.", "For the smaller data set (ALOI), we optimize validation error on a 10% held-out subset of the training data.", "For the larger data sets (Imagenet, ODP), we optimize progressive validation loss on the initial 10% of the training data.", "After determining hyperparameters we retrain with the entire training set and report the resulting test error.", "When available we report published LOMtree results, although they utilized a different method for optimizing hyperparameters." ], [ "Timing Measurements", "All timings are taken from the same 24 core xeon server machine.", "Furthermore, all algorithms are implemented in the Vowpal Wabbit toolkit and therefore share file formats, parser, and binary classification base learner implying differences are attributable to the different reductions.", "Our baseline OAA implementation is mature and highly tuned: it always exploits vectorization, and furthermore can optionally utilize multicore training and negative gradient subsampling to accelerate training.", "For the larger datasets these latter features were necessary to complete the experiments: estimated unaccelerated training times are given, along with wall clock times in parenthesis." ] ]
1606.04988
[ [ "Characterization of Three High Efficiency and Blue Sensitive Silicon\n Photomultipliers" ], [ "Abstract We report about the optical and electrical characterization of three high efficiency and blue sensitive Silicon photomultipliers from FBK, Hamamatsu, and SensL.", "Key features of the tested devices when operated at 90% breakdown probability are peak photon detection efficiencies between 40% and 55%, temperature dependencies of gain and PDE that are less than 1%/$^{\\circ}$C, dark rates of $\\sim$50\\,kHz/mm$^{2}$ at room temperature, afterpulsing of about 2%, and direct optical crosstalk between 6% and 20%.", "The characteristics of all three devices impressively demonstrate how the Silicon-photomultiplier technology has improved over the past ten years.", "It is further demonstrated how the voltage and temperature characteristics of a number of quantities can be parameterized on the basis of physical models.", "The models provide a deeper understanding of the device characteristics over a wide bias and temperature range.", "They also serve as examples how producers could provide the characteristics of their SiPMs to users.", "A standardized parameterization of SiPMs would enable users to find the optimal SiPM for their application and the operating point of SiPMs without having to perform measurements thus significantly reducing design and development cycles." ], [ "Introduction", "Silicon photomultipliers (SiPMs) have attracted significant attention over the past few years.", "They are becoming increasingly popular in scientific and industrial applications, which require fast, highly-efficient, single-photon-resolving photon detectors.", "Some prominent applications are in the fields of high-energy physics, astroparticle physics, and medical imaging (s. e.g.", "[1], [2], [3], [4]).", "Reasons for the popularity of SiPMs are their high photon-detection efficiencies, mechanical and electrical robustness, low mass, low power, low bias voltages.", "Another reason for the increasing popularity of SiPMs is that in recent years, they have been subject to many improvements.", "In particular, recent developments have successfully addressed nuisances such as high optical crosstalk, high afterpulsing, and high dark rates, but they have also improved the photon detection efficiency, which previously limited the usefulness of SiPMs in several applications.", "We are interested in SiPMs because we aim to use them in Cherenkov telescopes to detect gamma rays from astrophysical sources.", "Cherenkov telescopes image the Cherenkov light emitted from relativistic particle showers that are initiated by cosmic rays and gamma rays in the atmosphere [5].", "An in-depth understanding of photon detectors down to the level of device physics is key in the pursuit of minimizing the systematic uncertainties present in Cherenkov telescope data.", "In this paper we present an in-depth and comparative study of three recent, blue-sensitive SiPMs from FBK, SensL, and Hamamatsu, which demonstrate impressive performance improvements compared to devices from only a few years ago, e.g.", "[6].", "Beside the three tested devices many more devices exist from other vendors, which could not be tested due to a lack of time and resources.", "Along with our results we give a detailed description of our test setups and discuss the measurement procedures and resulting systematic uncertainties.", "We, furthermore, parameterize the overvoltage and temperature dependencies of most parameters.", "Where possible we use a physics-motivated model for the parameterization, which allows us to gain further insight into the device physics of SiPMs.", "We hope that the parameterizations we use will help to further standardize the measurement and parameterization of SiPM characteristics.", "Figure: Full scale pictures of the three tested SiPMs.Figure: Close-up pictures of the cells of the three tested SiPMs.", "The scale indicated by the black line in the images represents 20 μ\\mu m." ], [ "Device descriptions", "SiPMs are semiconductor-based photon detectors that consist of a matrix of elementary cells, which are avalanche photodiodes operating in Geiger mode.", "In the conventional SiPM, which is the type of SiPMs tested here, each cell is connected to a series resistor that limits the current flowing during the breakdown and thus ensures that the avalanche current is eventually quenched.", "Furthermore, all cells are connected to one common output.", "For a review of the history of SiPMs and their basic functionality, the reader is referred to [7] and references therein.", "The three tested devices are a NUV-HD SiPM from FBK, a S13360-3050CS MPPC from Hamamatsu, and a MicroFJ-SMTPA-30035-E46 SiPM from SensL.", "A picture of each SiPM is shown in Fig.", "REF .", "All three devices are based on a p-on-n structure, which means that the avalanche structure consists of a p-implant in an n-doped substrate.", "In this configuration the electric field directs electrons produced by blue photons just below the surface into the high-field region, which is also why the sensitivity of all three devices peaks at wavelengths in the blue or near UV." ], [ "FBK NUV-HD", "The FBK device is fabricated with NUV-HD technology [8].", "The device investigated in this study has a custom geometry, which fits the requirements for the Cherenkov Telescope Array (CTA) [9] project.", "Unlike the other two devices, the NUV-HD does not have an epoxy, silicone resin, or similar protective coating.", "The dimensions of the FBK SiPM are $(6.8\\times 6.8)$  mm$^2$ with a micro-cell pitch of $30\\,\\mu $ m. One SiPM has a total of 40,394 cells.", "The chip came glued onto a PCB carrier and is wire bonded.", "Fig.", "REF shows a picture of four cells taken under a microscope.", "Clearly visible are the quench resistors (red) and the metal line that connects the output of all cells." ], [ "Hamamatsu LCT5", "The SiPM from Hamamatsu is a S13360-3050CS MPPC [10].", "It is fabricated using their latest technology, which is also called LCT5 because it is the fifth iteration of a low-cross-talk development.", "The dimensions of the tested device are $(3\\times 3)$  mm$^2$ with a cell pitch of $50\\,\\mu $ m (s. Fig.", "REF ) and a total of 3,600 cells.", "The device is mounted onto a ceramic chip carrier and coated with UV-transparent silicon resin.", "Electrical contacts between the chip and the pins of the carrier are made with wire bonds.", "Hamamatsu produces the same type of SiPM also with through-silicon-via (TSV) technology, which allows several chips to be packed into large matrices with minimal dead space." ], [ "SensL J-Series", "The device from SensL is a pre-production J-Series SiPM [11].", "The dimensions of the active area are $(3.07\\times 3.07)$  mm$^2$ and the cell pitch is about $41\\,\\mu $ m resulting in a total of 5,676 cells.", "The SiPM is embedded in a 4-side tileable, chip scale package with TSV that is reflow soldered onto a PCB.", "The SiPM came surface mounted on an evaluation board (MicroF-SMTPA).", "A unique feature of SensL SiPMs is the presence of fast and slow readout terminals.", "The fast terminal capacitively couples directly to the cells, whereas the slow output is the conventional readout via the quench resistor.", "We used the signal from the slow terminal for our measurements.", "The photon detection efficiency (PDE) quantifies the absolute efficiency of a photon detector to absorb a photon and produce a measurable signal at its output.", "The PDE of SiPMs is determined by several factors of which the three most important are the geometrical efficiency, the quantum efficiency, and the probability to produce a Geiger breakdown, hereafter breakdown probability.", "The breakdown probability is also referred to as triggering probability.", "We measure the PDE as a function of wavelength in three steps.", "In the first step, the PDE is measured at four wavelengths.", "In the second step, the relative spectral response is measured between 200 nm and 1000 nm.", "In the last step, the spectral response is scaled to match the four PDE points and thus arrive at the PDE for all wavelengths between 200 nm and 1000 nm.", "In the following we walk in detail through each of these steps.", "All PDE and spectral response measurements are carried out at room temperature (23$^{\\circ }$ C-25$^{\\circ }$ C)." ], [ "Concept of measuring the PDE", "The PDE at four different wavelengths is measured with the SiPM being biased above breakdown and illuminated with fast light flashes of known intensity, and from the response of the SiPM the PDE is calculated.", "For the measurement we use the same procedure that is described in [12].", "A pulsed LED flashes fast light pulses into an integrating sphere with two exit ports, which acts as an optical splitter.", "The measurement of the splitting ratio is detailed in section REF A calibrated PiN diode is mounted to one exit port, and the SiPM under test is mounted to the other port.", "The response of both sensors is recorded for each flash.", "After 10,000 flashes, the average number of photons at the position of the SiPM is calculated from the average PiN-diode signal, the quantum efficiency of the PiN Diode, and the splitting ratio of the integrating sphere.", "The PDE of the SiPM then follows from the ratio of the average number of photons detected by the SiPM and the calculated average number of photons at the SiPM position.", "The average number of photons and dark counts detected by the SiPM $\\overline{N}_{\\mbox{\\footnotesize Ph+DC}}$ in each flash is calculated under the assumption that the number of photons and dark counts in each flash follows a Poisson distribution.", "By counting the flashes $N_0$ for which the SiPM did not detect a photon, the average number of detected photons and dark counts is $\\overline{N}_{\\mbox{\\small Ph+DC}} = -\\ln \\left( \\frac{N_0}{N_{\\mbox{\\small total}}} \\right)\\,,$ where $N_{\\mbox{\\small total}}$ is the number flashes.", "The contribution from dark counts is determined by triggering the read out $N_{\\mbox{\\small total}}$ times without flashing the LED.", "As in the previous case, the number of times the SiPM did not record a signal ($N_0^{\\mbox{\\tiny DC}}$ ) is counted.", "The dark-count-subtracted average number of photons detected by the SiPM is then $\\overline{N}_{\\mbox{\\small Ph}} = \\ln \\left( \\frac{ N_0^{\\mbox{\\tiny DC}} }{N_0} \\right)\\,.$ The described procedure is commonly used to calculate the mean number of photons detected by SiPMs because it is immune to afterpulsing and optical crosstalk." ], [ "PDE measurement setup", "The setup of our PDE measurement is sketched in Fig.", "REF .", "An LED pulses 20 ns-long flashes of light at 200 Hz into a UV-transparent liquid fiber that guides the light into a hollow cylinder made out of spectralon.The same integrating sphere that was also used in [12].", "The entry port and the two exit ports of the integrating sphere are all oriented perpendicular to each other.", "Attached to each exit port is an aluminum cylinder with the inside of the cylinder covered with black felt.", "Each cylinder is closed with a black plastic cap that has a hole in its center.", "A calibrated PiN diode is mounted to the cap with the larger hole ($\\sim 10\\,$ mm diameter), and the SiPM is mounted to the cap with the smaller hole ($\\sim 1\\,$ mm diameter).", "Each SiPM is held in place with an adapter that is custom designed and 3D-printed for each device.", "The adapter ensures that only the active area of the SiPM is illuminated by the light that exits the integrating sphere through the end-cap of the aluminum cylinder.", "The diameter of the light beam is about 1 mm.", "Four different LEDs fitted with narrow bandpass optical filters are used in the PDE measurement.", "The spectra of the four LEDs after the filter are shown in Fig.", "REF .", "The full width at half maximum (FWHM) of each spectrum is $\\sim 10$  nm.", "The PiN diode used in this study is a Hamamatsu S3590-08.", "The noise of the PiN-diode is minimized by reverse biasing the diode at 70 V thus decreasing the internal capacitance of the diode.", "The diode signal is first amplified with a Cremat 110 charge-sensitive preamplifier and then further amplified and shaped with an ORTEC Model 410 linear amplifier.", "The best signal-to-noise ratio is achieved with $2\\,\\mu $ s differentiating and integrating shaping time constants.", "The noise performance of the PiN-diode signal chain is limited by the capacitance of the diode and the intrinsic noise of the preamplifier and is about 300 equivalent noise charge (ENC).", "After amplification the signal is recorded with an Alazar ATS 9870 8 bit, 1 GS/s digitizer.", "The SiPM signal is amplified with a Mini-Circuits 500-NL amplifier and then shaped with a simple variable parallel RC circuit that differentiates the signal (C) and provides pole-zero cancellation (R).", "After shaping, the typical full width of the SiPM signal is less than 10 ns.", "The signal is further amplified with a LeCroy Model 612A amplifier before being digitized with the ATS 9870 digitizer.", "A switchable attenuator before the LeCroy amplifier is used to adjust the single photoelectron amplitude at the input of the digitizer to $\\sim 30$  mV.", "Figure: Pulse-height distributions of Hamamatsu SiPM signals recorded in a PDE measurement.", "See text for details on the signal extraction.", "A total of 10,000 flashes contribute to eachdistribution.", "The blue distribution is from signals recorded when the SiPM is flashed with the LED.", "The red distribution is from signals recorded when the LED is not flashing.", "Events to theleft side of the dashed vertical line can be identified as those in which the SiPM did not generate a signal.The LED signal is extracted from the recorded trace of the SiPM by sliding a window of three samples (3 ns) through the trace starting before the LED signal is expected in the trace and stopping 250 ns later.", "At each position the sum of the three samples is calculated, and at the end of the scan, the maximum sum is filled into a histogram.", "To extract the dark count rate, the procedure is repeated by starting 300 ns before the LED signal and sliding the three-sample window for another 250 ns through the trace stopping before the LED signal is expected in the trace.", "The maximum of the sliding window is again filled into a histogram.", "Fig.", "REF shows the two resulting histograms for a typical measurement.", "Entries to the left of the dashed vertical line correspond to events during which the SiPM did not generate a signal within the 250 ns.", "The integral of these events are $N_0^{\\mbox{\\tiny DC}}$ (red histogram) and $N_0$ (blue histogram), respectively.", "Note the good separation between the noise peak on the left and the first peak on the right side of the vertical line, which is necessary to keep the systematic uncertainties on the measured mean number of detected photons low.", "In all measurements the number of events in the minimum, where the dashed vertical line is placed, is 1% or less than the number of events in the maximum of the peak to the left.", "In that way the systematic uncertainty in the reconstructed mean number of photons is kept below 1%.", "The PiN diode signal is extracted by fitting a template pulse shape to the trace and recording the amplitude of the fitted pulse.", "The template pulse shape is obtained by averaging over 1000 pulses.", "The average number of photons at the PiN-diode position is calculated from the PiN-diode signals by taking the full LED spectrum and the wavelength-dependent quantum efficiency (QE) of the PiN diode into account." ], [ "Calibration of the PDE setup", "Before a PDE value can be calculated, the PiN diode, the integrating sphere, and the PiN diode signal chain need to be calibrated.", "The Hamamatsu S3590-08 PiN diode has been calibrated by Hamamatsu, with a systematic uncertainty of 2-3% between 250 nm and 800 nm and up to 5% outside of that range [13].", "For the measurement of the splitting ratio of the integrating sphere, S3590-08 PiN diodes are placed at the end cap of each aluminum cylinder.", "An LED connected to a constant current source then shines into the entrance port of the integrating sphere.", "After one hour the LED has stabilized such that its intensity does not vary by more than 0.1% over the course of one calibration measurement.", "The currents of both PiN diodes are simultaneously recorded with two Keithley 6847 picoammeters.", "The photo current measured at the SiPM position (where the intensity is lowest) is at least 1000 times the PiN-diode dark current.", "In a series of measurements the PiN diodes are swapped.", "The splitting ratio is first calculated by using the currents that were measured with the same diode at the two exit ports.", "The ratio is then calculated a second time by using the currents that were measured with the two diodes simultaneously.", "In the final calculation, the currents are corrected for the small differences in the quantum efficiencies of the two PiN-diodes.", "All measurements of the splitting ratio agree within 2%.", "The ratio was, furthermore, measured with all four LEDs used in the PDE measurements and found to vary within 1%.", "Figure: Pulse-height distributions recorded with the calibrated PiN diode attached to the integrating sphere.", "Shown are distributions for all four LEDs, the 241 ^{241}Am source, and thepedestal.", "The fit of the 59 keV bin with a Gaussian function is also shown.The PiN-diode signal chain is calibrated in photoelectrons by attaching a $^{241}$ Am source to the diode and recording the signals of 59.54 keV gamma rays.", "Using a Fano factor of 3.62 eV/eh-pair it can be shown that the gamma rays produce on average 16448 eh-pairs in the diode [14].", "A typical $^{241}$ Am spectrum recorded with our setup is shown in Fig.", "REF together with pulse-height distributions for each of the four LEDs.", "The linearity of the PiN-diode signal chain is better than 3% down to signal amplitudes that are $\\sim 10$ % of an average 59 keV signal.", "We estimate that the relative systematic uncertainty of our PDE measurements is 5%.", "The relative systematic uncertainty is dominated by systematic uncertainties of the PiN diode's QE (3%), uncertainties in the ratio of the spectralon cylinder (1%), and the signal extraction of the SiPM (1%) and PiN diode (3%)." ], [ "PDE measurements", "The PDE of all three devices is shown as a function of bias for all four wavelengths in Fig.", "REF .", "Each of the bias-dependent PDE curves is well described by an exponential function of the form $PDE(U) = PDE_{\\mbox{\\footnotesize max}}\\left[1-e^{-\\left( U - U_{\\mbox{\\tiny BD}}\\right)/a} \\right]\\,$ with fit probabilities that are in all but one case better than 60%.", "The good agreement indicates that the chosen analytical function is an appropriate empirical model of the data.", "The breakdown voltage $U_{\\mbox{\\tiny BD}}$ is determined from the best fit of the 400 nm data and fixed in the fits of the data for the remaining wavelengths.", "The reasons for fixing the breakdown voltage are twofold.", "Firstly, the uncertainty of the best fit breakdown voltage is smallest in the fits of the 400 nm data, and secondly, the breakdown voltage does not depend on photon wavelength.", "We note that the breakdown voltages obtained here are in agreement with the dedicated breakdown-voltage measurements presented later.", "The dashed vertical lines in Fig.", "REF denote the bias at which each device reaches 90% of the maximum PDE at 400 nm as inferred from the fit of the data.", "For the remainder of this paper we refer to this bias voltage as the operating point of an SiPM and mark it accordingly in all figures with a downward pointing arrow.", "Note that the bias where the PDE reaches 90% of its maximum depends on wavelength as will be discussed next.", "The term in the square brackets in Equation REF has to be interpreted as the breakdown probability, because the breakdown probability is the only contribution to the PDE that depends on bias, so long as the active volume of a cell is fully depleted (which can be safely assumed for the tested devices).", "After rewriting the exponent in units of relative overvoltage $U_{\\mbox{\\tiny rel}} = \\frac{U - U_{\\mbox{\\tiny BD}}}{U_{\\mbox{\\tiny BD}}}\\,,$ which can in fact also be interpreted as the relative electric field strength above the critical electric field strength, the breakdown probability becomes $P_{\\mbox{\\footnotesize BD}}(U_{\\mbox{\\tiny rel}}) = 1-e^{-U_{\\mbox{\\tiny rel}}/\\alpha }\\,.$ It is interesting to note that one parameter, $\\alpha =a/U_{\\mbox{\\tiny BD}}$ , is sufficient to properly describe the electric field/bias dependence of the breakdown probability.", "The parameter $\\alpha $ depends, of course, on the geometry of the avalanche region, where in the avalanche region a photon is absorbed, on the impact ionization factors of electrons and holes, and other factors and is thus device and wavelength specific.", "A small $\\alpha $ value means that the breakdown probability rises quickly with bias as opposed to a slow rise if $\\alpha $ is large.", "We discuss the interpretation of $\\alpha $ in more detail in the following.", "Figure: Breakdown probability as a function of relative overvoltage above breakdown for all three SiPMs and for all four wavelengths.", "The corresponding α\\alpha values are listed in Table.Fig.", "REF shows the breakdown probability as a function of relative overvoltage / relative excess electric field for all three SiPMs and all four tested wavelengths.", "The corresponding values for $\\alpha $ are listed in Table REF .", "All three devices have in common that $\\alpha $ increases with increasing wavelength.", "This behaviour can be explained with the absorption length of photons, which increases with wavelength.", "For photons absorbed close to the surface of the SiPM (blue photons), it is the photoelectron that drifts into the avalanche region in p-on-n devices.", "For photons absorbed below the avalanche region (redder photons), it is the hole that drifts upward into the avalanche region and initiates a breakdown.", "Because holes have always lower ionization factors than electrons, the breakdown probability for hole-dominated breakdowns is lower than for electron-dominated ones.", "The ionization factors for electrons and holes grow rapidly with bias, therefore, the breakdown probability also increases until saturation is reached.", "Even though the ionization factor of holes increases faster with bias than the one for electrons it never becomes larger than the ionization factor of electrons.", "Thus the breakdown probability is always less for longer wavelengths than for shorter wavelengths and saturation is reached at a higher bias.", "The Hamamatsu SiPM has the lowest $\\alpha $ of the three devices at all wavelengths, while the FBK device features the largest $\\alpha $ values.", "These differences can be qualitatively attributed to differences in the location of the avalanche region (how close it is to the surface), spatial extent of the avalanche region, the geometry of the avalanche region, and variations of it when the bias is being changed.", "It is evident that all three devices can be operated at a breakdown probability of 90% or more—at least in the blue.", "This is a significant improvement compared to a few years ago when most devices could only operate at a maximum overvoltage of 5%-10%, and, therefore, yielded much lower breakdown probabilities [6].", "Table: α\\alpha Values of the Fit Results in Fig.", ".Figure: Sketch of the spectral response setup." ], [ "Concept of the spectral response measurement", "For the spectral response measurement, we use the setup that is sketched in Fig.", "REF .", "The SiPM is biased at the voltage that yields a 90% breakdown probability at 400 nm as defined in the previous section.", "The SiPM is measured first and then replaced with the reference detector instead of measuring both sensors simultaneously like in the PDE measurement.", "Doing the spectral response measurement in this way eliminates optical elements that split the light between the two sensors and therefore would have to be calibrated.", "A main source of systematic uncertainties is thus eliminated.", "Any variability of the light source is monitored and recorded with a permanently installed PiN diode.", "Further corrections that are applied in the data analysis are a) subtraction of dark currents of all sensors and b) subtraction of stray light transmitted through the monochromator, which affects measurements mainly below 350 nm.", "The intensity of the light source is adjusted throughout a measurement by controlling the slits of the monochromator such that the SiPM current is within 50 to 75 times the dark current of the SiPM.", "Keeping the current of the SiPM quasi-constant guarantees that the fraction of SiPM cells that are in recovery remains about the same, and thus the geometrical efficiency of the SiPM also remains constant.", "The current limits are such that only a small fraction of the cells of an SiPM ($<$ 1%) are always in recovery and, therefore, saturation effects of the SiPM are avoided.", "The light spot at the position of the SiPM is larger than the sensor itself.", "Each spectral response measurement is cross-checked by increasing the current limits to be between 100 and 150 times the dark current and making sure that the residuals between the two measurements remain less than 2%.", "The spectral response measurement is a relative one and is converted into an absolute PDE measurement by fitting it to the PDE measurements presented earlier.", "Corrections for optical crosstalk and afterpulsing, therefore, do not have to be applied to the spectral response measurements." ], [ "Setup of the spectral response measurement", "The light source in the spectral response measurement is a 300 W UV-enhanced Xenon arc lamp (PE300BUV from Cermax).", "The light of the lamp is air-coupled into a Czerny-Turner single-grating monochromator Digikröm DK 240 1/4$\\lambda $ from Spectral Products.", "The grating of the monochromator that is used for all measurements has 1200 grooves per millimeter and a 300 nm blaze wavelength.", "The output of the monochromator is coupled into a dark box where the light beam is further conditioned before it illuminates the monitoring diode and the SiPM or reference sensor.", "Inside the dark box the light first passes an adjustable aperture followed by a lens with a focal length of 35 mm.", "The beam is then split by a polka dot beamsplitter.", "The reflected part of the beam illuminates the monitoring diode—an unbiased Hamamatsu S3590-08 PiN diode.", "The size of the beam spot matches the size of the monitoring diode.", "The transmitted part of the beam passes through an optical long-pass filter that is mounted onto a filter wheel, followed by an optional broadband polarizer (UBB01A from Moxtek) before the beam illuminates either the SiPM or the reference sensor.", "The beam spot is larger than the size of the reference sensor or the SiPM.", "The reference sensor is a UV-enhanced, Si-diode from Hamamatsu (type S1227-1010BQ, calibrated by Hamamatsu).", "All optical elements are UV transparent down to 200 nm.", "A total of three long-pass filters with cut-off wavelengths at 280 nm, 400 nm, and 750 nm are mounted into a computer-controlled filter wheel.", "The 280 nm filter is used to quantify stray light with wavelengths above the cut-off wavelength that gets transmitted through the monochromator and affects measurements below 270 nm.", "The 400 nm filter is used to quantify the stray-light component that affects measurements between 270 nm and 350 nm.", "The 400 nm filter is also used to suppress higher-order diffraction above 430 nm.", "The 700 nm filter suppresses higher-order diffraction above 770 nm.", "The current of the monitoring diode is recorded with a Keithley 6845 picoammeter, and the currents of the reference sensor and the SiPM are measured with a Keithley 6847 picoammeter.", "The readings of both instruments are transfered via serial link to a computer, which also controls the monochromator and the filter wheel.", "For the spectral response measurement, the SiPM is fixed on a rotary mount that allows making spectral response measurements as a function of the angle of incidence between 0 degrees (normal incidence) and 90 degrees.", "The SiPM is biased with the internal voltage source of the Keithley 6847 picoammeter.", "In the measurement the monochromator output is changed between 200 nm and 1000 nm and for each wavelength the exit and entrance slits of the monochromator are adjusted to keep the SiPM current within the previously discussed limit of 50-75 times the SiPM's dark current.", "The long-pass filters are inserted at the above mentioned wavelengths.", "The SiPM is then swapped out with the calibrated Si-diode, and the photocurrent of the diode is recorded at the same wavelengths and with the same monochromator slit settings used in the SiPM measurement.", "The spectral response $S$ at a given wavelength is calculated as $S = \\frac{I_{\\mbox{\\tiny SiPM}}}{I_{\\mbox{\\tiny Si-Diode}}}\\cdot \\frac{I_{\\mbox{\\tiny Mon.", "Si-Diode}}}{I_{\\mbox{\\tiny Mon.", "SiPM}}}\\cdot QE_{\\mbox{\\tiny Si-Diode}}\\,,$ where $I_{\\mbox{\\tiny SiPM}}$ and $I_{\\mbox{\\tiny Si-Diode}}$ are the dark and the stray-light corrected currents of the SiPM and the calibrated Si-diode, respectively.", "The factor in the middle is the ratio of the dark-current-subtracted currents of the monitoring diode that corrects for fluctuations of the Xe lamp.", "The last factor $QE_{\\mbox{\\tiny Si-Diode}}$ is the quantum efficiency of the reference sensor.", "The systematic uncertainties between 300 nm and 800 nm are dominated by uncertainties in the wavelength-dependent response of the calibrated Si-diode ($\\sim $ 3%) and variations in the SiPM photocurrent that cause the fraction of recovering SiPM cells to vary accordingly ($\\sim $ 1%).", "Below 300 nm the systematic uncertainties are dominated by residuals in the stray-light correction when the PDE of the SiPM drops below 10%.", "They reach 100% when the PDE of the SiPM drops below a couple of percent.", "Above 800 nm the uncertainties are dominated by the uncertainty in the QE of the reference sensor, which is $\\sim 4\\%$ .", "Figure: PDE vs. wavelength for all three devices between 200 nm and 1000 nm.", "The bias voltage for each device results in a 90% breakdown probability at 400 nm, the operating point ofeach SiPM." ], [ "Wavelength dependent PDE", "The spectral response measurement is a relative one and converted into an absolute PDE measurement by fitting it to the previously discussed PDE measurements at four wavelengths.", "The fit is done by invoking a scaling factor that minimizes $\\chi ^2$ between the four PDE points and the spectral response measurements.", "In the fit it is taken into account that the spectral response of the SiPM varies across the spectra of the LEDs that have been used in the PDE measurements.", "In order to find the correct wavelength that corresponds to the measured PDE, an LED spectrum is weighted with the spectral response of the SiPM, and the mean wavelength of the weighted spectrum is used as the wavelength of the PDE measurement.", "The correction, however, is small, and the shift with respect to the mean LED wavelength is $<1\\,$ nm.", "Afterpulsing and optical crosstalk do not affect the outcome of the scaling because both result in a wavelength-independent factor that gets marginalized in the fit.", "The spectral response measurements scaled to absolute PDE are shown in Fig.", "REF .", "Also shown are the four PDE measurements for each device to which the spectral response measurements have been scaled.", "The FBK device has the highest peak PDE of the three tested SiPMs with 56% at 395 nm, even though it has the smallest pitch between cells.", "The oscillations in the PDE are due to interference caused by the thin passivation layer and the lack of a coating on top of the device like in the other two devices.", "In a previous study we tested an NUV-HD device with coating that shows a comparable PDE down to 300 nm.", "Below 300 nm the FBK device presented here has a better efficiency because it is not coated with silicon resin.", "The full width at half maximum (FWHM) of the FBK PDE extends from 280 nm to 560 nm.", "The Hamamatsu device has a peak PDE of 52% at 455 nm and a FWHM of the PDE response that extends from 310 nm to 700 nm, which is significantly more red sensitive than the FBK SiPM.", "The SensL device has a peak PDE of 41% at 420 nm and a FWHM of the PDE response from 310 nm to 560 nm, which is similar to the response of the FBK SiPM.", "Compared to similar SiPMs from only a few years ago [6], all three devices are testaments to the major improvements that have been made in increasing the PDE and shifting the response of SiPMs to shorter wavelengths.", "Figure: Response as a function of angle of incidence relative to normal incidence with light polarized perpendicular to the plane of incidence.Figure: Response as a function of angle of incidence relative to normal incidence with light polarized parallel to the plane of incidence." ], [ "Dependence of SiPM response on angle of incidence", "The dependence of the PDE on the angle of incidence was tested for light polarized in the plane of incidence (parallel polarization) and perpendicular to the plane of incidence for angles incidence angles of $20^{\\circ }$ , $40^{\\circ }$ , $50^{\\circ }$ , $60^{\\circ }$ , and $70^{\\circ }$ .", "For this measurement a broadband polarizer UBB01A from Moxtek was inserted after the beam splitter.", "Fig.", "REF shows the response of the three SiPMs relative to normal incidence for polarization perpendicular to the plane of incidence and in Fig.", "REF for light polarized parallel to the plane of incidence.", "The measurements are corrected for the change in the projected area of the light beam onto the SiPM with different angle of incidence.", "We estimate a maximum uncertainty on the angle of incidence of $2^{\\circ }$ , which translates into a maximum systematic uncertainty of 10% on the measurements done at $70^{\\circ }$ and less at smaller angles.", "The response to different angles of incidence depends to a large fraction on the coating of the chip and also how the chip is packaged.", "In order to reduce effects from stray light that reflects off the chip carrier into the edges of the chip or light that directly enters through the edges of the chip under larger angles, the boundaries of the Hamamatsu and the SensL SiPM were covered with thin copper tape.", "Unfortunately, the FBK SiPM could not be taped because the chip is not protected, thus edge effects are included in the measurement.", "The response of all devices is relatively constant up to angles of $60^{\\circ }$ , when the response is still about 80% and better than 90% for perpendicular and parallel polarized light, respectively.", "At larger angles the sensitivity starts to quickly drop.", "Note that there is a steep increase in sensitivity of the SensL device to parallel polarized light between 300 nm and 400 nm for larger angles of incidence.", "Figure: Sketch of the basic measurement setup" ], [ "IV curves", "For the measurement of the electrical characteristics, the SiPMs are placed in a thermal chamber, and their performance is measured between -40$^{\\circ }$ C and 40$^{\\circ }$ C in steps of 20$^{\\circ }$ C. Fig.", "REF shows a sketch of the setup.", "In this section the IV-curve measurements are discussed.", "For each measurement, an SiPM is connected to a Keithley 6847 picoammeter that biases the SiPM and records the current.", "The measurements are done in DC mode as opposed to a pulsed mode, which is acceptable given the small amount of power dissipated by the SiPM ($<20$  mW when biased in the forward direction and $<1$  nW when biased in reverse).", "From the IV-curves the average value of the quench resistor and the breakdown voltage are derived." ], [ "Quench resistor values", "The quench resistor values are derived from the linear part of the forward biased IV curves (see Fig.", "REF ), i.e.", "in the regime where the resistance of the pn-junction of a cell becomes negligible, and the total resistance is dominated by that of the quench resistor.", "The inverse of the slope of the IV curve yields the resistance of all quench resistors of the SiPM connected in parallel.", "Multiplying the total parallel resistance with the number of cells of an SiPM thus gives the average value of a quench resistor, which is shown in Fig.", "REF as a function of temperature for all three SiPMs.", "Figure: Average quench-resistor values for all three SiPMs at five different temperatures.", "The change in resistance with temperature shown in each figure is determined from a fit of thedata points with a linear function.The figure also gives the temperature coefficients of the quench resistors, which are determined by fitting a linear function to the data points, which is a good approximation for the Hamamatsu and SensL data.", "For the FBK SiPM, the quench resistor values fluctuate significantly.", "In particular the value at 40$^{\\circ }$ C is higher than one would expect by extrapolating the quench resistor values from lower temperatures.", "We can not exclude that a contamination of the uncoated device during handling or residual humidity is responsible for these effects.", "The quench resistors of the Hamamatsu device have the smallest relative dependence on temperature with $2\\cdot 10^{-3}$ , followed by $3\\cdot 10^{-3}$ for the SensL device, and $5\\cdot 10^{-3}$ for the FBK device.", "The temperature coefficient and the absolute value of the quench resistor determine the maximum temperature and bias at which a device can be operated before a breakdown cannot be reliably quenched anymore.", "It, furthermore, determines how the recovery time of a cell changes with temperature.", "The temperature coefficients of all three SiPMs, however, are too small to have any practical impact on the maximum operating temperature or cell recovery times." ], [ "Breakdown voltages", "The second characteristic derived from the IV-curves is the breakdown voltage.", "We took a close look at three different proposed methods [15], [16], [17] to extract the breakdown voltage, and we compare them with the classical method that uses gain vs. bias measurements.", "Based on our findings we propose yet another method that is based on [15], [16] and yields breakdown voltages within $\\pm 0.2\\%$ of the breakdown voltage derived from gain bias measurements.", "It has been noted, based on empirical evidence, that the IV curve of single SiPM cells (also called SPADs) can be described by a parabola above breakdown [18].", "Here we give a physical explanation why a parabola is in fact expected for the IV curve just above breakdown.", "Biased just above breakdown, the current is proportional to the product of gain $G = C\\cdot \\Delta U = C\\cdot U_{\\mbox{\\footnotesize BD}}\\cdot U_{\\mbox{\\footnotesize rel}}$ and breakdown probability $1-\\exp \\left( -U_{\\mbox{\\footnotesize rel}}/\\alpha \\right)$ , where $C$ is the effective capacitance of one SiPM cellThe cell capacitance is determined from gain vs. bias measurements and is discussed later.", "and $\\Delta U = U - U_{\\mbox{\\footnotesize BD}}$ .", "The proportionality constant is the sum of the dark current $I_{\\mbox{\\footnotesize DC}}$ and the current due to external light sources $I_{\\mbox{\\footnotesize ext}}$ multiplied by the optical crosstalk probability $P_{\\mbox{\\footnotesize OC}}$ and afterpulsing probability $P_{\\mbox{\\footnotesize AP}}$ .", "The total current above breakdown is then $I(U_{\\mbox{\\footnotesize rel}}) &= &\\left[I_{\\mbox{\\footnotesize DC}}(U_{\\mbox{\\footnotesize rel}}) + I_{\\mbox{\\footnotesize ext}}\\right]\\nonumber \\\\&& \\cdot \\left[1+P_{\\mbox{\\footnotesize OC}}(U_{\\mbox{\\footnotesize rel}})+P_{\\mbox{\\footnotesize AP}}(U_{\\mbox{\\footnotesize rel}})\\right]\\nonumber \\\\&& \\cdot C\\cdot U_{\\mbox{\\footnotesize BD}} \\cdot U_{\\mbox{\\footnotesize rel}}\\cdot \\left[1-e^{\\left( -U_{\\mbox{\\footnotesize rel}}/\\alpha \\right)}\\right].$ The dark current changes much less with bias than the breakdown probability, and the gain and can thus be assumed constant if only a small range around the breakdown voltage is considered.", "The impact of a varying dark current is further suppressed by illuminating the SiPM with an external light source that produces a current that is ten times or more than the SiPM dark current.An external light source that produces a current 100 times the dark current will not affect the response of the SiPM (see spectral response measurement section).", "In fact, for this method to also work at low temperatures when the dark current becomes too low to provide a large enough primary signal, an external light source is needed.", "Optical crosstalk and afterpulsing are only a few percent around the breakdown voltage and can, therefore, be neglected.", "With these simplifications the total current becomes $I(U_{\\mbox{\\footnotesize rel}})&\\approx &\\left[I_{\\mbox{\\footnotesize DC}} + I_{\\mbox{\\footnotesize ext}}\\right]\\cdot C\\cdot U_{\\mbox{\\footnotesize BD}}\\nonumber \\\\&& \\cdot U_{\\mbox{\\footnotesize rel}}\\cdot \\left[1-e^{\\left( -U_{\\mbox{\\footnotesize rel}}/\\alpha \\right)}\\right]\\,.$ Doing a series expansion of the exponential function to second order in $U_{\\mbox{\\footnotesize rel}}/\\alpha $ gives $I(U_{\\mbox{\\footnotesize rel}})&\\approx &\\left[I_{\\mbox{\\footnotesize DC}} + I_{\\mbox{\\footnotesize ext}}\\right] \\cdot C \\cdot U_{\\mbox{\\footnotesize BD}}\\nonumber \\\\&&\\cdot \\left[ U_{\\mbox{\\footnotesize rel}}^2/\\alpha + U_{\\mbox{\\footnotesize rel}}^3/2\\alpha ^2 + \\ldots \\right]\\,.$ Thus in leading order the current above breakdown is indeed proportional to $\\Delta U^2$ as long as $U_{\\mbox{\\footnotesize rel}}/\\alpha <1$ , which is the case for overvoltages that are less than 5%-10% (see Table REF ).", "To obtain the breakdown voltage from an IV curve, [16] proposes using the voltage where $($ d$I/$ d$U)/I$ is maximal, whereas [15] proposes using the maximum of $\\mbox{d}\\ln \\left(I(U)\\right)/\\mbox{d}U$ .", "Both methods are equivalent because if applied to Eqn.", "REF both yield $\\frac{\\mbox{d}I/\\mbox{d}U}{I}=\\frac{\\mbox{d}\\ln \\left(I(U)\\right)}{\\mbox{d}U}=\\frac{2 + f(y)}{U-U_{\\mbox{\\footnotesize BD}}}\\,.$ The function $f(y)=(y+1-\\exp (y))/(\\exp (y)-1)$ , with $y=U_{\\mbox{\\footnotesize rel}}/\\alpha $ , is about -0.2 for values of $y$ that are typical for the tested devices.", "We verified that processing our IV measurements in both ways does indeed yield identical results.", "Fig.", "REF shows the outcome when they are processed according to $\\mbox{d}\\ln \\left(I(U)\\right)/\\mbox{d}U$ .", "In all of these measurements the SiPMs were illuminated with a dimmed 400 nm LED.", "The peak positions shown in Fig.", "REF are systematically above the breakdown voltage derived from gain vs. bias measurements by about 0.7%, which is not acceptable in some applications.", "In an effort to obtain a better estimate of the breakdown voltage, we fit each curve in Fig.", "REF with Eqn.", "REF .", "The results of the fits are shown in Fig.", "REF on top of the data.", "The breakdown voltages extracted from the fit are shown together with those from the gain measurements in Fig.", "REF .", "Differences between the fitting method and the gain method are less than $\\pm 0.2$  %, which is significantly better than the 0.7% offset observed in the peak-derivative method.", "Some of the remaining offset can be explained with systematic uncertainties in the calibration of the signal chain that is used in the gain vs. bias measurements.", "An obvious outlier is the result obtained for the Hamamatsu SiPM where all breakdown voltages derived from the IV curve have a relative offset of 0.4% from the gain vs. bias derived breakdown voltages, which is too large an offset to be explained by calibration uncertainties.", "The measurement of the breakdown voltage done by Hamamatsu agrees with the one from our gain vs. bias measurement.", "We cannot exclude with certainty that variations of the cell capacitance with bias might be a possible cause for the discrepancy in the breakdown voltage measurements.", "But we note that the gain vs. bias curves in Figure REF are linear down to 1 Volt overvoltage.", "Thus any significant change in the cell capacitance must happen around the breakdown voltage and thus invalidate the model of the IV curve (Equation REF ) and the gain method, which both assume a constant cell capacitance.", "An additional benefit of the fit is that it also extracts values for $\\alpha $ .", "For all three devices the fit produces $\\alpha $ -values at room temperature that are consistent with those listed in Table REF .", "The data seem to indicate a weak increase of $\\alpha $ with temperature but the uncertainties are too large to make a more quantitative statement.", "The last method we investigated to extract the breakdown voltage from the IV curve is to use the maximum of the second derivative of the logarithm of the current [17].", "The estimated breakdown voltages are shown in Fig.", "REF as open squares and yield a similarly good estimate of the breakdown voltage as our fitting method.", "For the Hamamatsu SiPM the position of the maximum of the second derivative gives slightly better results, but it is still offset from the breakdown voltage obtained with the gain bias method.", "The breakdown voltages in Fig.", "REF change proportionally with temperature for all three devices.", "The temperature coefficients of the breakdown voltage are given in the same figure.", "The relative change in breakdown voltage with temperature is about the same for all three devices, namely $10^{-3}$ per one degree Celsius.", "Figure: Derivative of the logarithm of the current around the breakdown voltage.", "The solid lines are fits to the curves from which the breakdown voltage is determined.Figure: Breakdown voltage derived from the derivative of the IV-curves (solid dots), the second derivative of the IV-curves (empty squares), and gain measurements (triangles)." ], [ "IV curve simulations in the breakdown region", "We simulated IV curves for two reasons.", "First we want to explain why the position of the maximum in the derivative of the logarithm of the IV curve does not match with the breakdown voltage derived from the gain measurement.", "The second reason is that we want to validate the other two methods to derive the breakdown voltage.", "The model of the simulated IV curve is based on Equation REF extended by the fraction of the dark current, which does not get amplified.", "The additional term allows one to simulate the IV curve below the breakdown voltage.", "As before, contributions from optical crosstalk and afterpulsing have again been neglected.", "Equation REF is a model of the absolute current, whereas relevant for the derivation of the breakdown voltage is only the relative change of the current, see Eqn.", "REF .", "Therefore, only the relative current versus bias curve is simulated: $\\hspace{-8.5pt}I_{\\mbox{\\tiny rel}}(U_{\\mbox{\\tiny rel}})=\\frac{I(U_{\\mbox{\\tiny rel}})}{I_{\\mbox{\\tiny ampl}}} =h + U_{\\mbox{\\tiny rel}}\\cdot G\\cdot \\left[1-e^{\\left( -U_{\\mbox{\\tiny rel}}/\\alpha \\right)}\\right]\\,.$ Where the normalization $I_{\\mbox{\\tiny ampl}}$ is the part of $I_{\\mbox{\\tiny DC}}+I_{\\mbox{\\tiny ext}}$ that makes it into the avalanche region and gets amplified.", "Note that in Eqn.", "REF and subsequent equations $I_{\\mbox{\\tiny DC}}+I_{\\mbox{\\tiny ext}}$ implicitly denote only the amplified part of the total dark and external generated current.", "$G$ becomes the product of the cell capacitance and the breakdown voltage and is $6.4\\cdot 10^6$ , $3.5\\cdot 10^7$ , and $2.5\\cdot 10^7$ for the FBK, Hamamatsu, and SensL device, respectively.", "Note that we restrict ourselves to measurements done at 20$^{\\circ }$ C. The quantity $h$ is the ratio of the unamplified and amplified part of $I_{\\mbox{\\tiny DC}}+I_{\\mbox{\\tiny ext}}$ .", "The value for $h$ is adjusted in the model until the simulated ratio of the currents at 10% overvoltage and before breakdown matches the data and typically assumes values of 1000 or more.", "Cell-to-cell variations of the breakdown voltage are included by simulating 10,000 cells each with a different breakdown voltage that is randomly picked from a normal distribution with a mean of zero and a standard deviation that is a free parameter in the simulation.", "The simulated IV curve is the sum of the currents of all 10,000 cells.", "The last parameter in the simulation is $\\alpha $ .", "A small $\\alpha $ is expected if the majority of the dark current enters the multiplication region from the front, such as photoelectrons generated by blue photons, and a large $\\alpha $ is expected if the dark current is generated behind the avalanche region, e.g.", "in the bulk.", "Increasing $\\alpha $ in the model shifts the position of the maximum of the derivative of the logarithm of the IV curve towards higher relative overvoltages and can thus be used to tune the simulations to get a match with the data.", "A good agreement with measurements is achieved if $\\alpha $ is 0.015, 0.05, and 0.1 for the FBK, Hamamatsu, and SensL devices, respectively.", "The agreement remains good if $\\alpha $ is varied within the range of values listed for each device in Table REF .", "The width of the peak of the derivative of the logarithm of the IV curve is tuned by changing the standard deviation of the cell-to-cell variations of the breakdown voltage.", "A value of 0.001 reproduces the FWHM of the measurements of all three SiPMs.", "We remark that we did not perform a rigorous tuning of the model parameters.", "Therefore, we cannot exclude that a completely different set of model parameters with different physics implications can equally well reproduce the data.", "However, we are confident that the model and its parameterization is good enough to discuss the validity of the different methods to extract the breakdown voltage.", "The simulations confirm that the peak position of the derivative of the logarithm of the IV curve is systematically above the breakdown voltage.", "We also find that fitting the derivative reproduces the true breakdown voltage within 0.1%.", "The maximum of the second derivative also lies within 0.1% of the breakdown voltage.", "Our fitting method and the second-derivative method to extract the breakdown voltage, therefore, seem to be on solid footing.", "However, we emphasize that the breakdown voltages extracted from the IV curves of the Hamamatsu SiPM are inconsistent with the ones from the gain vs. bias measurements on the level of 0.4% (200 mV) for which we do not have an explanation.", "Figure: Snapshot of an SiPM trace recorded with 1GS/s and 8 bit resolution after amplification (red).", "The remaining two curves show the trace at two different stages of its processing toreduce the signal widths.", "See text for details." ], [ "Signal trace analysis", "In the remainder of the paper, we discuss the analysis of SiPM signals recorded with the Alazar ATS 9870 digitizer after amplifying the signal with a Mini-Circuits ZFL 500LN+ amplifier and a LeCroy Model 612A amplifier (see Fig.", "REF ).", "For the absolute calibration of the gain measurement, the SiPM signals were recorded in parallel with a Tektronix TDS 3054C oscilloscope after amplification of the SiPM signals with the Mini-Circuits ZFL 500LN+ preamplifier.", "The SiPM signals need to be processed to eliminate the long tails of the individual signals.", "Fig.", "REF shows an example of a recorded SiPM trace before (red) and after (blue) processing.", "Long tails are a general feature of SiPMs with surface areas larger than 1 mm$^2$ because their terminal capacitance increases with sensor area which, combined with a 50 Ohm input impedance preamplifier, results in long tails.", "Long tails are also the result of cell recovery times that are less than a few hundred nanoseconds long.", "To process the signals, we follow a two-step procedure similar to the approach used in [19].", "In the first step, a copy of the original trace is shifted by three nanoseconds and subtracted from the original trace.", "This step results in a significant shortening of individual SiPM signals down to a full width of about 9 ns.", "An example of the outcome of this processing step is shown as the green trace in Fig.", "REF .", "A small remaining undershoot is subtracted from the trace by applying a background-subtraction algorithm that is implemented in the ROOT analysis framework [20].", "The final result is shown as the blue trace in the figure.", "Figure: The histogram shows the time difference between two consecutive SiPM signals on the x-axis and the amplitude of the second signal on the y-axis.", "Note the logarithmic scale of thex-axis.", "The colors represent the number of events in each bin on a logarithmic scale.", "Several populations can be identified and are correspondingly labeled.The general procedure of the signal trace analysis is to record randomly triggered 10 ms long signal traces until enough pulses are accumulated to reconstruct all parameters of interest with high enough precision.", "The measurement of the afterpulsing is typically the bottleneck and defines how many traces need to be recorded.", "At low temperatures a dimmed LED is used to increase the SiPM signal rate and thus speed up the afterpulsing measurement.", "Measurements of the dark rate are made with the LED turned off.", "After a trace is processed, all SiPM signals with an amplitude of at least 0.5 photoelectrons (p.e.)", "are identified, i.e.", "signals with at least half the amplitude that is generated when one cell of an SiPM breaks down.", "The amplitudes and times of the identified signals are then used to extract the SiPM parameters (similar to how it is described in [19]).", "An illustrative example of the type of information that can be extracted from the amplitudes and times is given in Fig.", "REF .", "It is a two-dimensional histogram that has the time difference between two consecutive signals on the x-axis and the amplitude of the second signal in units of p.e.", "on the y-axis.", "The color gives the number of events per bin.", "In this figure only signal pairs have been selected in which the first signal has an amplitude of one photoelectron.", "A number of different populations can easily be identified.", "The biggest population is made up by signals in which only one cell of the SiPM fires.", "That population peaks at a time difference of $\\approx 10\\,\\mu $ s, which is the expected average time difference between two uncorrelated dark count signals, i.e.", "the inverse of the dark count rate for that specific device and temperature.", "The bands above that population are from signals where one cell fires due to an uncorrelated dark count, and one or two additional cells fire in coincidence due to direct optical crosstalk.", "To the left of the main blob is a smaller population that is due to delayed optical crosstalk signals.", "The amplitudes of the delayed optical-crosstalk signals to the very left depend on the time when the signal appears because there is significant overlap with the preceding signal, and the signal-extraction algorithm is not able to properly handle the overlap.The width of one signal is 9 ns after a trace is processed.", "Also visible are afterpulsing events that generate a second signal from the same cell before it is fully recharged.", "The solid black line shows a fit to the afterpulsing events in the dashed box and is used to measure the recovery time of one cell." ], [ "Gain, Cell Capacitance and Breakdown Voltage", "The first information extracted from the signal amplitudes is the signal charge in units of electrons, which is commonly referred to as the gain of an SiPM.", "The amplitudes of signals between 0.5 and 1.5 p.e.", "are averaged and then converted into signal charge.", "For this conversion, a separate calibration of the entire signal chain was performed for each SiPM.", "In the first step of the calibration, the average single p.e.", "amplitude was read off a Tektronix TDS 3054C oscilloscope at a temperature of $-20^{\\circ }$ C and at two different bias voltages after amplification of the raw signals with a Mini-Circuits ZFL-500LN+.", "The uncertainty in reading the amplitude off the oscilloscope is 0.2% and dominates the uncertainty of the absolute gain and breakdown voltage measurement.", "In the second step, the signal amplitudes are divided by the gain of the amplifier (30 dB).", "In the third step, the calibrated amplitudes are multiplied with the integral of the normalized raw signal shape,The signal shape was normalized to a peak amplitude of one.", "thus obtaining two absolute gain measurements.", "These two absolute gain measurements and the average single-cell amplitudes that were extracted from the processed traces at the same bias and temperature are then used to define a linear transformation from processed signal amplitude to absolute charge.", "An example of a calibrated gain measurement is shown in Fig.", "REF .", "The solid lines are linear fits to the data.", "A closer inspection of the data points reveals small residuals with respect to the fits, which can be attributed to non-linearities in the front-end amplifier of the digitizer.", "The linear dependence of the gain on bias can be explained in the small-signal model of SiPMs where the cell of an SiPM is represented by a capacitance $C_{\\mbox{\\footnotesize cell}}$ that is discharged to the breakdown voltage in a breakdown.", "The total charge $G$ of the signal is then $G = C_{\\mbox{\\footnotesize cell}} \\cdot \\left(U - U_{\\mbox{\\footnotesize BD}}\\right)\\,.$ If $G$ is given in units of electrons, it is usually referred to as the gain of the device, which is the definition of $G$ we adopt in this paper.", "Based on Equation REF the breakdown voltage can be measured from the gain vs. bias curve as the voltage where the gain is zero.", "The determined breakdown voltage is shown in Fig.", "REF together with those extracted from the IV-curves.", "Figure: Cell capacitance.The cell capacitance $C_{\\mbox{\\small cell}}$ is given by the slope of the gain vs. bias measurement and is shown in Fig.", "REF .", "For the Hamamatsu and the FBK SiPM the cell capacitance remains constant, whereas a 5% change is seen in the SensL SiPM between $-40^{\\circ }$ C and $40^{\\circ }$ C. The gain vs. bias curves are well described by linear functions, and aside from the residuals that can be attributed to the digitizer, no further deviation from linearity is observed that would point to a dependence of the cell capacitance on bias for any of the tested devices." ], [ "Dark count rates", "The dark count rates are measured by counting all signals with an amplitude larger than 0.5 p.e.", "and dividing that number by the total duration of all analyzed traces.", "Included in this measurement are, therefore, thermal generated dark counts as well as delayed optical crosstalk and afterpulsing.", "However, the latter two contribute only minor to the total dark count rate as they are less than 2% at 90% breakdown probability.", "Two pulses have to be at least $\\approx 3$  ns apart in order to be identified as separate signals.", "Figure: Dark count rates.", "The arrow marks the nominal operating bias of each device.Fig.", "REF shows the dark count rates per one square millimeter sensor area for all temperatures and for all three devices.", "The solid lines are fits to the data with the function $DC\\left(U_{\\mbox{\\footnotesize rel}}\\right) = e^{a+b\\cdot U_{\\mbox{\\tiny rel}}} \\cdot \\left[1-e^{\\left(-U_{\\mbox{\\footnotesize rel}}/\\alpha \\right)}\\right]\\,,$ where the last term is the breakdown probability and is only used in the fit of the dark rate measurement of the Hamamatsu SiPM.", "For the SensL and FBK SiPMs the dark-rate measurements start at an overvoltage where the breakdown probability is already close to 90% (check the position of the arrow).", "The turnover in the data for the FBK device occurs because the small cell capacitance results in signals too small to be reliably detected with our signal chain at low overvoltages.", "The results of the fits are shown in Table REF .", "The $\\alpha $ values extracted for the Hamamatsu SiPM are consistent with the $\\alpha $ extracted from the PDE measurements (s. Table REF ) for short photon wavelengths, which indicates that the majority of the dark noise enters the avalanche region from the surface of the device.", "Table: Best Fit Values Obtained From the Fit of the Dark Rate Measurements Shown in Fig.", ".Figure: Relative change in dark count rates.The rates in Fig.", "REF are shown versus relative overvoltage.", "For a fixed relative overvoltage, any change in the dark rate with temperature can be attributed to changes in the thermal generation of charge carriers.", "Fig.", "REF shows how the dark count rate changes with temperature for a fixed overvoltage relative to the dark count rate at 40$^{\\circ }$ C and averaged over the operating voltage range at 40$^{\\circ }$ C. The relative change in dark count rate with temperature for all three devices is well described by $e^{a+b\\cdot T}$ .", "The change in temperature needed to change the dark count rate by a factor of two is stated in the inserts in the figure." ], [ "Optical crosstalk", "Optical crosstalk (OC) is the correlated firing of cells due to photons emitted in the breakdown of one cell.", "Any of these photons can initiate the breakdown of a neighboring cell.", "Two types of optical crosstalk can be distinguished.", "Direct OC is due to crosstalk photons that get absorbed in the active volume of a neighboring cell and cause the breakdown of that cell, which happens quasi-simultaneous to the first one.", "Delayed OC is due to crosstalk photons that convert in the non-depleted bulk.", "In this case the generated charge carrier has to first diffuse into the active volume of the cell [19], [21], [22].", "The diffusion process introduces a measurable time delay between the breakdown of the first cell and the breakdown of the second cell.", "Measurements of direct OC are presented in this section and the delayed OC measurements are discussed together with afterpulsing measurements in the next section.", "Figure: Example of a pulse-height distribution of signals from the SensL device.", "The vertical line at 1.5 p.e.", "marks the boundary between signals in which only one cell fired (left) and more than one(right).", "The small peak at the left is due to afterpulsing events that can also be identified in Fig.", ".", "Only signals with an amplitude of at least 0.5 p.e.", "are used in theoptical crosstalk analysis.Direct OC is extracted from the pulse-height distribution of the SiPM signals.", "Fig.", "REF shows an example of such a distribution where events can be clearly identified that are due to 1, 2, or 3 cells firing simultaneously.", "The small peak on the left is due to afterpulses, which are the same events that are also marked as afterpulses in Fig.", "REF .", "The OC probability is determined by counting all events with an amplitude larger than 1.5 p.e.", "and dividing that number by the total number of events with an amplitude larger than 0.5 p.e.", "Figure: Direct optical crosstalk.", "The arrow marks the nominal operating bias of each device.Fig.", "REF shows the direct OC for all three SiPMs as a function of relative overvoltage.", "At their respective operating voltages, marked by the arrow, the FBK device has the highest OC at 23% followed by the SensL and the Hamamatsu SiPM, which has the lowest OC (6%).", "The OC of the Hamamatsu device does not depend on temperature, whereas the SensL OC increases with temperature; both behaviors can be explained with a constant and increasing cell capacitance, respectively, as will be detailed later.", "The OC measured for the FBK device on the other hand shows a clear offset of the curves that is about $\\pm $ 5 %.", "Upon further investigation we came to the conclusion that the offset is a systematic effect due to the partial overlap of the individual peaks in the pulse-height distribution of the FBK device.", "The same effect also explains the small offset of the OC measurement at 40$^{\\circ }$ C for the SensL and the Hamamatsu device.", "We note that the FBK device is by far the largest of the three tested devices, which is why the absolute dark count rates are also highest and the probability of overlapping pulses is, therefore, more frequent than in the other two devices.", "We also remark that optical crosstalk increases with the size of the device, and our measurements are not corrected for that effect.", "The overvoltage dependence of the OC can be understood in the following way.", "The number of photons emitted in the breakdown of one cell is $f\\cdot C_{\\mbox{\\footnotesize eff}}\\cdot \\Delta U$ , where $f$ is about $3\\cdot 10^{-5}$ photons per electron in the avalanche [21] and $C_{\\mbox{\\footnotesize eff}}\\cdot \\Delta U$ is the gain in units of electrons.", "Each emitted photon has a probability $\\gamma $ to absorb in the active volume of a neighboring cell and generate a charge carrier.", "The likelihood of that charge carrier to initiate a breakdown is given by the breakdown probability $1-\\exp \\left(-U_{\\mbox{rel}}/\\alpha \\right)$ .", "Combining all factors, the OC as a function of relative overvoltage becomes $OC(U_{\\mbox{\\footnotesize rel}}) = f\\cdot C_{\\mbox{\\footnotesize eff}}\\cdot U_{\\mbox{\\footnotesize rel}}\\cdot U_{\\mbox{\\footnotesize BD}}\\cdot \\gamma \\cdot \\left[1-e^{\\left(-U_{\\mbox{\\footnotesize rel}}/\\alpha \\right)}\\right]\\,.$ The probability $\\gamma $ is thus a device-specific number that quantifies how well a given structure suppresses OC and is hereafter referred to as optical crosstalk efficiency.", "While our specific parameterization of the OC is different, it is conceptually equivalent to the one used in [23].", "The measured OC curves are fit with the above function, and the best fit $\\gamma $ and $\\alpha $ values are listed in Table REF .", "All OC curves including the FBK curve are well described by the fit function.", "For the Hamamatsu and the FBK device the small $\\alpha $ value indicates that the majority of optical crosstalk photons enter the avalanche region from the surface whereas the large $\\alpha $ value for the SensL device indicates that the majority of the crosstalk photons enter the cell from below.", "With a $\\gamma $ of 0.08, the OC efficiency is lowest for the Hamamatsu device, which has filled trenches between cells to prevent photons from crossing into a neighboring cell.", "For the SensL device, which does not have trenches, the OC efficiency is twice as high.", "We note that the $\\gamma $ values of 0.5 and the $\\alpha $ values for the FBK device are likely affected by the above mentioned systematic effects caused by the reduced separability of the peaks in the pulse-height distribution and thus should be interpreted with caution.", "Table: Best Fit Values Obtained From Fitting the Direct Optical Crosstalk Measurements Shown in Fig.", ".", "The Last Column Shows the Probability That a Photon Emitted in aBreakdown Results in a Breakdown of a Neighboring Cell." ], [ "Afterpulsing and Delayed Optical Crosstalk", "Afterpulsing and delayed OC both produce signals that are correlated in time with respect to a previous SiPM signal.", "Both effects are quantified by selecting SiPM signals with amplitudes between 0.5 and 1.5 p.e.", "and recording the time to the next signal.", "Fig.", "REF shows an example of the distribution of the time differences.", "The main peak is due to uncorrelated, Poisson-distributed dark-noise counts.", "The position of the peak is at the average time difference between two dark counts, which is equal to the inverse of the dark-count rate.", "We note that the binning of the histograms is logarithmic, and as a result of the binning, the Poisson distribution takes the form $a\\cdot t \\cdot \\exp (-t/\\tau )$ instead of a pure exponential function.", "The main peak is well fit with a Poisson distribution, and the residuals due to delayed optical crosstalk and afterpulses at small time differences are clearly visible.", "For the extraction of the delayed OC and afterpulsing probabilities, however, we histogram not only the time difference between the first and the next pulse, but all following pulses up to a time difference of 100$\\,\\mu $ s. In this way we eliminate the need to consider cases in which an afterpulse or delayed OC signal is missed because of an earlier dark count.", "Fig.", "REF shows the corresponding pulse-height distribution.", "The Poisson-distributed dark counts follow a line through the origin now.", "The fit of the distribution with a line was performed between 10 $\\mu $ s and 100 $\\mu $ s. The figure to the right shows the residuals between the data and the fit, which are due to delayed OC and afterpulses.", "The residuals consist of two components.", "The left component is due to delayed OC, and the right is due to afterpulses.", "The two components are better visible in the amplitude vs. time distribution shown in Fig.", "REF .", "Delayed OC produces signals with amplitudes of 1 p.e.", "or larger, whereas afterpulses have amplitudes between 0 and 1.", "For the measurement of the afterpulsing probability, we select all the events in the residual distribution that are to the right-hand side of the time when the amplitude of afterpulses reaches 0.5 p.e.", "Residuals with shorter time delays are assumed to be due to delayed OC.", "The vertical lines in Fig.", "REF give an example of where the boundary between the two components is placed for the SensL SiPM.", "The dividing time delay is 50 ns for the FBK, 17 ns for the Hamamatsu, and 20 ns for the SensL device.", "The method is robust but does not provide a clean separation between the two components.", "A more rigorous approach would also include the amplitude information, which allows a clear separation between the two components (see Fig.", "REF ).", "Such an approach would also allow extracting the trapping times of the afterpulses.", "We did not implement such an analysis because our method to extract the amplitudes and times becomes increasingly inefficient if two pulses are separated by less than 10 ns.", "This inefficiency introduces a considerable systematic effect and results in an underestimation of the delayed optical crosstalk, which dominates the uncertainty in our measurements.", "Figure: Delayed optical crosstalk.", "The arrow marks the nominal operating bias of each device.Figure: Afterpulsing.", "The arrow marks the nominal operating bias of each device.Figures REF and REF show the delayed OC and afterpulsing probabilities, respectively.", "At their respective operating voltages all devices have a probability for delayed OC of about 2%.", "The afterpulsing probability is less than 2% for the Hamamatsu SiPM and less than 1% for the FBK and SensL SiPM.", "Again we note that the delayed OC has to be understood as a lower limit due to the inefficiencies of extracting pulses with time differences that are less than 10 ns.", "The afterpulsing probabilities on the other hand are likely overestimated by about 20% because of the hard cut that is applied in the residuals to divide the two components.", "The best separation between the two components is achieved in the measurement of the Hamamatsu device and is thus the least affected by an overspill of OC events.", "From the point of view of judging the performance of the three SiPMs in an application, the afterpulsing and delayed OC probabilities at the operating voltages are sufficiently low that it is in fact not necessary to perform a more detailed analysis of, for example, the afterpulsing trapping time constants.", "The overvoltage dependence of the delayed OC can be expected to be described in the same way as the direct OC, i.e., with Equation REF .", "Fits to the Hamamatsu data are shown in the Figure REF .", "However, due to the inefficiency in our pulse-extraction algorithm, we could not extract meaningful parameters from the fit, which is also reflected by a poor probability of the fit.", "The afterpulsing vs. overvoltage data are fit with the function $AP(U_{\\mbox{\\footnotesize rel}}) = A\\cdot e^{\\left(U_{\\mbox{\\footnotesize rel}}/\\delta \\right)} \\cdot \\left[1-e^{\\left(-U_{\\mbox{\\footnotesize rel}}/\\alpha \\right)}\\right]\\,,$ where $A$ is a normalization, and the second term describes the bias dependence of the afterpulsing probability.", "The last term has to be understood as an effective breakdown probability because it averages over all possible times when afterpulses can happen during the recovery of a cell.", "Because individual trapping times are exponentially distributed, the majority of the trapped charges are released shortly after the breakdown of a cell has stopped.", "This means that the breakdown probability is small at the time when most afterpulse are released and $\\alpha $ , therefore, expected to be large.", "The afterpulsing as function of bias does not show a dependence on temperature for the Hamamatsu SiPM.", "We note that trapping time constants decrease exponentially with increasing temperature.", "It is thus expected that afterpulsing decreases with increasing temperature because more trapped carriers are released before the cell recovers to a meaningful breakdown probability.", "The expected temperature behavior is observed in the FBK device but not in the SensL device.", "We cannot rule out that the observed behaviour is due to a contamination of afterpulses with delayed optical crosstalk events.", "For the FBK, Hamamatsu, and SensL SiPMs, the fit values averaged over all temperatures for $\\alpha $ are 80, 80, and 100, respectively.", "For $\\delta $ they are 0.2, 0.09, and 0.15, respectively.", "The uncertainties are fairly large and hide any temperature dependencies." ], [ "Cell Recovery Times", "The last quantity measured is the cell recovery time.", "Cell recovery times can be measured by flashing an SiPM with two fast consecutive pulses and recording how the second SiPM signal amplitude changes as a function of the time difference between the two pulses.", "The recovery time can also be measured by analyzing the amplitude vs. time characteristics of afterpulses, which is expected to be described with $A(t) = A_0 \\left[1-e^{t/\\tau }\\right]\\,,$ where $\\tau $ is the time constant of the recovery time.", "We measured the recovery time using the latter method.", "The black dots in Fig.", "REF are afterpulses selected to be fit with the above function, which is shown as the solid black line in the figure.", "The measured recovery time constants are shown in Fig.", "REF for all devices.", "At the operating voltages, the time constants are in good agreement with the product of the cell capacitance and quench resistors.", "An expected trend that is observed for all devices is the decrease of the recovery time with increasing temperature, which is due to the decreasing value of the quench resistor.", "(s. Fig.", "REF ).", "Figure: Recovery times.", "The arrow marks the nominal operating bias of each device." ], [ "Discussion", "In this paper we presented the characterization of three recent, blue-sensitive SiPMs from FBK, Hamamatsu, and SensL.", "All three devices show superior performance in terms of their optical and electrical characteristics with respect to past generations of SiPMs.", "The very good performance of the three devices motivated us to investigate how to best parameterize SiPM characteristics as a function of bias and temperature.", "We believe that standardizing the parameterization of SiPMs will become increasingly important as the community of SiPM users is constantly growing, and not everyone has in-house capabilities to perform in-depth device studies.", "Furthermore, the optimal operating point of an SiPM varies from application to application, which requires knowledge of SiPM parameters over a wide range of temperature and bias.", "With a standardized SiPM parameterization at hand, the user can focus on the application and with the help of the model find the optimal SiPM and its operating point.", "We have found parameterizations of the breakdown probability, optical crosstalk, dark rate, and afterpulsing as a function of temperature and bias that can be applied to all three tested SiPMs.", "The parameterization also allows extraction of physical parameters like the location of the high-field region using $\\alpha $ in the breakdown probability, or the optical crosstalk efficiency factor $\\gamma $ .", "The choice of comparing device characteristics at the bias where the PDE at 400 nm reaches 90% breakdown probability is driven by our ultimate desire to obtain SiPMs with the highest optical efficiency and, at the same time, sufficiently low nuisance parameters.", "If one has to select one of the three devices for an application, detailed end-to-end simulations are needed that find the bias that results in the best compromise between PDE and nuisance parameters.", "Such a study is not within the scope of this paper.", "Instead we discuss how well the tested devices match the requirements for Cherenkov telescopes when the SiPMs are operated at 90% breakdown probability, and we point out the remaining shortcomings that prevent the tested devices from being perfect photon detectors for Cherenkov telescopes when operated at that bias.", "Reduced optical crosstalk, afterpulsing, and dark-count rates allow the operation of all three devices at much higher relative overvoltages, thus yielding breakdown probabilities of more than 90% for blue photons.", "Not only does a 90% breakdown probability provide a significant boost in PDE, but it also reduces the sensitivity of gain and PDE on temperature changes.", "Using that one degree change in temperature shifts the breakdown voltage by 0.1% for all three devices; the gain of an SiPM changes by 1%/$^{\\circ }$ C if it is operated at 10% overvoltage.", "If a device is operated at 20% overvoltage, the gain changes by only 0.5%/$^{\\circ }$ C. The three tested devices operate in between these limits.", "The temperature dependence of the PDE is even smaller because the breakdown probability is in saturation.", "With our parameterization of the breakdown probability it can be calculated that the relative PDE changes between 0.2%/$^{\\circ }$ C and 0.3%/$^{\\circ }$ C for the three tested devices if they are operated at 90% breakdown probability.", "These values are on par with typical values for bialkali photomultiplier tubes [24].", "Measures to temperature-stabilize SiPMs in applications or to correct data offline is, therefore, not necessary anymore, or the requirements to temperature-stabilize devices can be much more relaxed.", "The peak PDE of the three devices ranges between 40% and 50%, which, again, is a huge improvement compared to the PDEs of devices available just 10 years ago.", "Being able to operate at 90% breakdown probability is certainly one main reason for the high PDEs, but it is worth noting that the spectral response has shifted considerably into the blue/UV region.", "Considering that the maximum achievable geometrical fill factor is probably around 80%, the maximum possible PDE that can be expected for SiPMs is around 65% assuming a 90% breakdown probability and a 90% quantum efficiency.", "In fact, FBK recently presented results of SiPMs with a peak PDE of more than 60% PDE [25].", "Enhancing the blue efficiency of SiPMs further and shifting their peak efficiency toward lower wavelengths is likely to be realized by thinning the passivation layer and the first implant, which will be technological challenges.", "Optical crosstalk, dark rates, and afterpulsing are also much reduced in comparison to older devices.", "Dark rates are typically a few ten kHz/mm$^2$ , whereas early devices typically had rates of one MHz/mm$^2$ .", "Optical crosstalk has been lowered by reducing cell capacitances, introducing trenches between cells, and optimizing the layout of structures.", "Each tested devices has successfully implemented one or more of the aforementioned measures, and direct optical crosstalk ranges between 6% and 20% at 90% breakdown probability.", "Delayed optical crosstalk and afterpulsing are two more nuisance parameters that could be considerably improved, with typical values being $\\sim 2$ %.", "Parameters that are well within the requirements are cell recovery time and gain.", "A lower gain and a shorter cell recovery time in future devices is perfectly acceptable.", "A lower gain would reduce power dissipated by the SiPM, which is a plus when SiPMs are used in environments with intense photon backgrounds.", "Given all of these improvements, only a short list of desirable changes remain: The sensitivity should be highest between 250 nm and 550 nm if possible with a flat response.", "Above 550 nm the sensitivity should cut off sharply.", "Such a spectral response would maximize the detection of Cherenkov light and at the same time efficiently reject ambient light coming from the night sky, which dominates at long wavelengths.", "Of the three tested devices, the FBK device comes closest to the ideal response, but improvements would still be desirable to further suppress the response at long wavelengths.", "Direct optical crosstalk is one of the main factors limiting the lowest achievable trigger threshold.", "The majority of trigger concepts used in Cherenkov telescopes employ an n-fold coincidence of neighboring camera pixels.", "In the coincidence, each pixel has to have a signal above a certain threshold.", "How low that threshold can be set depends ideally only on the maximum acceptable trigger rate due to statistical up-fluctuations in the ambient light.", "For most operating or planned Cherenkov telescopes, a direct optical crosstalk of 3% would double that trigger rate which would be acceptable.", "It is of course desirable to minimize optical crosstalk as much as possible.", "With 6% optical crosstalk, the Hamamatsu device is not far from an optimal value.", "Afterpulsing and delayed optical crosstalk add to the effective dark-count rate and contaminate the extracted Cherenkov signal by introducing a positive bias.", "With about 2% afterpulsing and delayed optical crosstalk, respectively, all three devices have acceptable values that can be dealt with at the stage of signal extraction.", "However, keeping both effects below 1% would simplify the data analysis and reduce systematic uncertainties in the energy scale of Cherenkov telescopes.", "The cost of SiPMs is still a dominant contribution to the total per channel costs (readout electronics and photosensor).", "Considerable efforts have been made in the past to reduce the cost of the readout electronics, and it is not unreasonable to assume that with new concepts costs of $5 per readout channel can be realized in the future.", "SiPMs would have to cost about $$0.1/\\mbox{mm}^{2}$ to contribute equally to the per channel costs.", "All these items are major technological challenges, but it is not evident that fundamental physical limitations preclude one from surmounting them.", "Therefore, we are confident that new and improved devices will become available in the future." ], [ "Acknowledgment", "We are grateful to FBK, Hamamatsu, and SensL for providing us with samples of their latest developments.", "We thank J. Biteau for useful discussions and input that improved the extraction of the breakdown voltage from the IV curves.", "in our procedure to fit the IV curves This research was in part supported by the National Science Foundation under grant no.", "PHYS-1505228." ] ]
1606.05186
[ [ "New Insights into Properties of Large-N Holographic Thermal QCD at\n Finite Gauge Coupling at (the Non-Conformal/Next-to) Leading Order in N" ], [ "Abstract In the context of [1]'s string theoretic dual of large-N thermal QCD-like theories at finite gauge/string coupling (as part of the `MQGP' limit of [2]), we discuss the following.", "First, up to LO in N, using the results of [3], we show that the local T^3 of [2] is the T^2-invariant sLag of [3] in a resolved conifold.", "This, together with the results of [4], shows that for a (predominantly resolved or deformed) resolved warped deformed conifold, the local T^3 of [2] in the MQGP limit, is the T^2-invariant sLag of [3] justifying the construction of the delocalized SYZ type IIA mirror of the type IIB background of [1].", "Then, using the prescription of [5], we obtain the temperature dependence of the thermal (and electrical) conductivity working up to leading order in N (the number of D3-branes), and upon comparison with [6] show that the results mimic a 1+1-dimensional Luttinger liquid with impurities.", "Further, including sub-leading non-conformal terms in the metric determined by M (the number of fractional D-branes = the number of colors = 3 in the IR after the end of a Seiberg duality cascade), by looking at respectively the scalar, vector and tensor modes of metric perturbations and using [7]'s prescription of constructing appropriate gauge-invariant perturbations, we obtain respectively the speed of sound, the diffusion constant and the shear viscosity \\eta (and \\eta/s) including the non-conformal O((g_s M^2) (g_s N_f)/N<<1)-corrections, N_f being the number of flavor D7-branes." ], [ "Introduction", "The AdS/CFT correspondence or in general the gauge/gravity duality has proved to be a very useful tool in understanding the properties of super Yang-Mills theory at large t'Hooft coupling.", "According to the correspondence, physics of $\\mathcal {N}=4$ $SU(N)$ SYM theory in the large $N$ limit can be obtained from type $IIB$ superstring theory on $AdS_5\\times S^5$ geometry, where $AdS_5$ is the five dimensional anti-de Sitter space and $S^5$ is the five sphere.", "The $\\mathcal {N}=4$ $SU(N)$ SYM theory is a conformal field theory which means its gauge coupling does not run with the energy scale.", "On the other hand QCD is non-conformal.", "QCD with $SU(N_c)$ gauge group, where $N_c$ is the number of quark colors, is an asymptotically free theory so that the gauge coupling is scale dependent and vanishes logarithmically with large characteristic momentum or with short distance.", "So to deal with QCD-like theories using Gauge/Gravity duality we need to generalize the AdS/CFT correspondence and incorporate a running coupling in the theory.", "Building up on the Klebanov-Witten [8], Klebanov-Nekrasov [9] and Klebanov-Tseytlin [10] models, a logarithmic RG flow just like QCD was obtained in the non-conformal Klebanov-Strassler model [11] by considering $M$ fractional $D3$ branes along with $N$ $D3$ branes in a conifold geometry wherein the IR geometry was modified resulting in a deformed conifold.", "So far we have not talked about the temperature at all.", "In fact the AdS/CFT correspondence mentioned above is valid at zero temperature.", "At finite temperature the situation is different on the gravity side of the correspondence.", "On the other hand the field theory in question i.e.", "thermal QCD, is an IR-confining theory at $T=0$ and becomes non-confining at $T\\gg \\Lambda _{QCD}$ , where $\\Lambda _{QCD}$ is the only scale that we have here.", "It possesses a phase transition from confining phase to a non-confining phase at $T=T_c\\approx C \\Lambda _{QCD}$ , where $C=\\mathcal {O}(1)$ .", "At sufficiently high temperature i.e.", "at $T\\gg \\Lambda _{QCD} $ , the interaction strength $\\lambda (T)\\ll 1$ and hence the theory is weakly coupled.", "However the thermal gauge theory we want to understand is not in the weak coupling regime.", "In particular, to explore the physics of QCD at $T\\approx T_c$ , we have to take a look at the strongly coupled regime where $\\lambda \\approx 1$ .", "So we cannot apply perturbative methods any more.", "In lattice gauge theory using numerical simulations the equilibrium properties of the strongly coupled hot QCD can be explored.", "But interesting non-equilibrium properties such as hydrodynamic behavior or the real time dynamics cannot be seen from the equilibrium correlation functions.", "So the lack of non-perturbative methods to study hot QCD forces us to look for either a different theory/model or a different limit of a known theory/model.", "At finite temperature the equilibrium or non-equilibrium properties of the Euclidean theory are studied requiring time to have periodicity $\\beta \\sim \\frac{1}{T}$ .", "Thus, at non-zero temperature, the Euclidean space-time looks like a cylinder with the topology $\\mathbb {R}^3\\times S^1$ .", "The AdS/CFT correspondence tells us that at $T=0$ the $4d$ SYM theory defined on $\\mathbb {R}^4$ is dual to string theory on $5d$ AdS space with $\\mathbb {R}^4$ as the boundary of the same.", "So at zero temperature we can think of the field theory as living on the boundary of AdS space.", "However, the prime interest is to investigate the finite temperature aspects of the dual field theory from the physics of supergravity.", "Hence at finite temperature, the space-time of the gravitational description somehow has to be changed such that one gets a geometry of the boundary which is equivalent to $\\mathbb {R}^3\\times S^1$ and not $\\mathbb {R}^4$ .", "In other words one needs to find some bulk geometry which has a boundary with the topology $\\mathbb {R}^3\\times S^1$ .", "One possible answer is the AdS-BH space-time with the following metric sometimes called black-brane metric given as: $ds^2=a(r)\\Biggl (-g(r)dt^2+d\\vec{x}^2\\Biggr )+b(r)dr^2$ with Minkowskian signature.", "Here $r$ is the radial coordinate and $g(r)$ , dependent on the horizon radius $r_h$ , is a `black-hole function'.", "By construction, the time coordinate is defined to be periodic with period $\\beta $ which is inverse of temperature and is related to the horizon radius $r_h$ .", "Now, let us go back to the Klebanov-Strasslar model where the temperature is turned on in the field theory side effected by introducing a black hole in the dual geometry.", "Interestingly the KS background with a black hole has the geometry equivalent to the AdS-BH spacetime in the large $r$ limit.", "Moreover the embedding of $D7$ -branes in KS model via the holomorphic Ouyang embedding [12] and finally the M-theory uplift of the whole set up keeps the background geometry as required provided we consider some limiting values of the parameters in the theory.", "The details about this, based on [2], [12], will be reviewed in Section 2.", "This paper, apart from providing important evidence validating construction in [3] of delocalized Strominger-Yau-Zaslow (SYZ) mirror of [2]'s type IIB holographic dual of large $N$ thermal QCD, we believe, fills in a pair of important gaps in the literature pertaining to a top-down holographic study of large-$N$ thermal QCD.", "First, all such large-$N$ holographic models cater to the large t'Hooft-coupling limit while keeping the gauge coupling vanishingly small.", "However, in systems such as sQGP, it is believed that not only should the t'Hooft coupling be large, but even the gauge/string coupling should also be finite [1].", "A finite gauge coupling would imply a finite string coupling which necessitates addressing the limit from an M-theory point of view.", "Also, for a realistic thermal QCD computation, the number of colors should be set to three.", "This can be realized in the IR after the end of a Seiberg duality cascade and in the MQGP limit of (REF ).", "This study was initiated in [3], [7] wherein a large-$N$ limit, referred to as the `MQGP limit' (REF ), was defined in which the gauge coupling was kept to be slightly less than unity and hence finite.", "By studying some transport coefficients in this paper, we obtain even at the leading order in $N$ , a remarkable result that holographic large-$N$ thermal QCD at finite gauge coupling for $\\mu _{\\rm Ouyang}\\equiv $ (Ouyang embedding parameter)$\\sim r_h^\\alpha ,\\ \\alpha \\le 0$ mimics qualitatively $D=1+1$ Luttinger liquid with impurities close to `$\\frac{1}{3}$ -doping'; for $\\alpha =\\frac{5}{2}$ one is able to reproduce the expected linear large-$T$ variation of DC electrical conductivity characteristic of most strongly coupled gauge theories with five-dimensional gravity duals with a black hole [13].", "Second, in the context of top-down holographic models of large-$N$ thermal QCD at finite gauge coupling, there are no previous results that we are aware of pertaining to evaluation of the non-conformal corrections to hydrodynamical quantities such as the shear viscosity $\\eta $ (as well as the shear-viscosity-entropy-density ratio $\\frac{\\eta }{s}$ ), shear mode diffusion constant $D$ and the speed of sound $v_s$ .", "These non-conformal corrections at finite gauge coupling, determined for the first time in this paper in the given context, are particularly relevant in the IR and in fact also encode the scale-dependence of aforementioned physical quantities, and hence are extremely important to be determined for making direct contact with sQGP.", "The main non-trivial insight gained via such computations is the realization that at NLO in $N$ there is a partial universality in these corrections determined by $N_f$ and $M$ apart from $N$ .", "The following is a section-wise description of the sets of issues addressed and the new insights obtained in this paper.", "Sec.", "3 - Identification of sLag in a (predominantly) resolved conifold up to LO in $N$: Up to leading order in $N$ and in the UV-IR interpolating region/UV, using the results of [6], we show that the local $T^3$ of [3] is a $T^2$ -invariant special Lagrangian three-cycle in a resolved conifold.", "This, together with the results of [7], shows that for a (predominantly resolved or deformed) resolved warped deformed conifold, the local $T^3$ of [3] in the MQGP limit of [3], is the $T^2$ -invariant special Lagrangian three-cycle of [6], justifying the construction in [3] of the delocalized SYZ type IIA mirror of the type IIB background of [2].", "This was a crucial step missing in [3], [7] in construction of the delocalized SYZ mirror of the top-down type IIB holographic dual of large-$N$ thermal QCD of [2], at finite gauge coupling.", "Transport Coefficients up to (N)LO in $N$ : We study some transport coefficients of large-$N$ thermal QCD leading to evaluation of various transport coefficients up to (next-to) leading order in $N$ .", "This boils down to evaluating various retarded Green's functions, but computed from the gravity dual as prescribed in [14].", "In order to study the transport phenomenon from the gravity picture, we need to consider a perturbation of the given modified OKS-BH metric - the type IIB string dual of large-$N$ thermal QCD as given in [2].", "In response to this perturbation the BH will emit gravitational waves with a long period of damping oscillation.", "The modes associated with this kind of gravitational radiation are called quasinormal modes.", "Quasinormal modes are the solutions to the linearized EOMs that one gets by considering fluctuations of gravitational background satisfying specific boundary conditions both at the black hole horizon and at the boundary.", "At the horizon, the quasinormal modes satisfy a pure incoming-wave boundary condition and at the spatial infinity the perturbative field or some gauge invariant combinations of the fields vanishes, that means it follows the Dirichlet boundary condition at infinity.", "It was shown in [15] that the quasinormal frequency associated with the quasinormal modes defined above in an asymptotically AdS spacetime exactly matches with the pole of the two point correlation function involving operators in the field theory dual to different metric perturbations.", "Hence evaluating the quasinormal frequency $\\omega $ as a function of the special momentum $q$ , gives the thermodynamic and hydrodynamic behavior of the plasma.", "Analogous to [11], the non-conformality in [2] is introduced via $M$ number of fractional $D3$ -branes, the latter appearing explicitly in $B_2, H_3$ and after construction of a delocalized SYZ type IIA mirror (resulting in mixing of $B_2$ with the metric components after taking a triple T-dual of [2]) as well as its local M-theory uplift, also in the metric.", "In the context of a (local) M-theory uplift of a top-down holographic thermal QCD dual such as that of [2] at finite gauge coupling, to the best of our knowledge, we estimate for the first time, the non-conformal corrections appearing at the NLO in $N$ to the speed of sound $v_s$ , shear mode diffusion constant $D$ , the shear viscosity $\\eta $ and the shear viscosity - entropy density ratio $\\frac{\\eta }{s}$ .", "The main new insight gained by this set of results is that the non-conformal corrections in all the aforementioned quantities are found to display a partial universality in the sense that at the NLO in $N$ the same are always determined by $\\left(\\frac{(g_s M^2)(g_s N_f)}{N}\\right)$ , $N_f$ being the number of flavor $D7$ -branes.", "Thus, we see that the same are determined by the product of the very small $\\frac{g_sM^2}{N}\\ll 1$ - part of the MQGP limit (REF ) - and the finite $g_s N_f\\sim {\\cal O}(1)$ (also part of (REF )).", "Of course, the leading order conformal contributions though at vanishing string coupling and large t'Hooft coupling were (in)directly known in the literature.", "It is interesting to see the conformal limit of our results at finite $g_s$ obtained by turning off of $M$ - which encodes the non-conformal contributions - reduce to the known conformal results for vanishing $g_s$ .", "Sec.", "4 - (Thermal and electrical) Conductivity, Wiedemann-Franz law and $D=1+1$ Luttinger liquid at LO in $N$: As a thermal gradient corresponding to a gauge field fluctuation also turns on vector modes of metric fluctuations, we consider turning on simultaneously gauge and vector modes of metric fluctuations, and evaluate the thermal ($\\kappa _T$ ) and electrical ($\\sigma $ ) conductivities, and the Wiedemann-Franz law ($\\frac{\\kappa _T}{T\\sigma }$ ).", "The new insight gained is that for $\\mu _{\\rm Ouyang}\\equiv $ (Ouyang embedding parameter)$\\sim r_h^\\alpha ,\\ \\alpha \\le 0$ , the temperature dependence of $\\kappa _T, \\sigma $ and the consequent deviation from the Wiedemann-Franz law, all point to the remarkable similarity with $D=1+1$ Luttinger liquid with impurities at `$\\frac{1}{3}$ -doping'; for $\\alpha =\\frac{5}{2}$ one is able to reproduce the expected linear large-$T$ variation of DC electrical conductivity for most strongly coupled gauge theories with five-dimensional gravity duals with a black hole [13].", "Sec.", "5 - Speed of sound: For the metric fluctuations in the sound channel the corresponding quasinormal frequency is given by $w=\\pm v_s q-i\\Gamma _s q^2$ with $v_s$ defined as the speed of sound and $\\Gamma _s$ as the damping constant of the sound mode.", "Again for the sound channel the pole of the correlations of longitudinal momentum density gives the same dispersion relation in the conformal limit.", "From the knowledge of quasinormal modes associated with the scalar modes of metric perturbations, we have computed the next-to-leading order correction to the speed of sound ($v_s$ ) at finite gauge coupling (part of the MQGP limit).", "Up to LO in $N$ , we calculate $v_s$ using four routes: (i) (subsection 5.1.1) the poles appearing in the common denominator of the solutions to the individual scalar modes of metric perturbations (the pure gauge solutions and the incoming-wave solutions), (ii) (subsection 5.1.2) the poles appearing in the coefficient of the asymptotic value of the time-time component of the scalar metric perturbation in the on-shell surface action, (iii) (subsection 5.2.1) the dispersion relation obtained via a Dirichlet boundary condition imposed on an appropriate gauge-invariant combination of perturbations - using the prescription of [5] - at the asymptotic boundary, and (iv) (subsection 5.2.2) the pole structure of the retarded Green's function calculated from the on-shell surface action written out in terms of the same single gauge invariant function.", "The third approach (of solving a single second-order differential equation for a single gauge-invariant perturbation using the prescription of [5]) is then extended to include the non-conformal corrections to the metric and obtain for the first time in the context of a top-down large-$N$ holographic thermal QCD at finite gauge coupling uplifted to M theory, an estimate of the non-conformal corrections to $v_s$ up to NLO in $N$ .", "Sec.", "6 - Shear mode diffusion constant: The quasinormal frequency for the vector modes of black brane metric fluctuation reads $\\omega =-iD q^2$ , where $D$ is the shear mode diffusion constant .", "This dispersion relation also follows from the pole structure of the correlations of transverse momentum density.", "From the knowledge of quasinormal modes associated with the vector modes of metric perturbations obtained by imposing Dirichlet boundary condition at the asymptotic boundary, on an appropriate gauge-invariant perturbation constructed using the prescription of [5], we have computed for the first time in the context of the same top-down large-$N$ holographic thermal QCD at finite gauge coupling uplifted to M theory, the non-conformal corrections to the shear mode diffusion constant up to NLO in $N$ .", "Sec.", "7 - Shear viscosity(-to-entropy density ratio): We have also evaluated for the first time in the context of the aforementioned M-theory uplift corresponding to finite $g_s$ , the non-conformal temperature-dependent correction at the NLO in $N$ , to the shear viscosity $\\eta $ and shear viscosity - entropy density ratio $\\frac{\\eta }{s}$ from the two point energy-momentum tensor correlation function corresponding to the tensor mode of metric perturbation.", "The results for the NLO (in $N$ ) corrections are particularly important as they suggest a scale dependance to the above mentioned quantities and hence leads to a non-conformal nature of the field theory in the IR.", "We have commented on this issue even in section 8.", "The paper is organized as follows.", "First, we briefly review the supergravity dual background of large $N$ strongly coupled QCD like theories.", "The whole discussion, for the sake of clear understanding of the reader is presented stepwise through first three subsections in section 2.", "In 2.1, the type $IIB$ supergravity background of [2] dual to large-$N$ thermal gauge theory which is UV complete and closely resembles thermal QCD, is briefly reviewed.", "In 2.2, the 'MQGP Limit' of [3] and its motivation, in particular to address the properties of strongly coupled QGP medium, is briefly reviewed.", "In 2.3, using the 'MQGP Limit' we review briefly the delocalized SYZ type $IIA$ mirror via three T dualities along a $T^2$ -invariant special lagrangian $T^3$ fibered over a large base in a predominantly warped resolved conifold - this serves as a precursor to the material of Sec.", "3.", "In the same sub-section, we discuss it's local uplift to M-theory, where in the large $r$ limit the spacetime is given by $AdS_5\\times M_6$ .", "In 2.4, following [14] we review the recipe to calculate two-point correlation function with Minkowskian signature.", "In 2.5, following [5] the gauge invariant variables for vector, scalar and tensor modes of background metric perturbations are discussed - this will be useful to obtain the results of (sub-)sections 5.3, 6 and 7.", "In section 3, we show that in the MQGP limit of [3], the local $T^3$ of [3] is the $T^2$ -invariant special Lagrangian three-cycle of a resolved conifold as given in [6].", "This together with the result reviewd in 2.3, shows that in the MQGP limit, the local $T^3$ of [3] is the $T^2$ -invariant sLag of [6] for both, a predominantly resolved (resolution $>$ deformation) or predominantly deformed (deformation $>$ resolution), resolved warped deformed conifold.", "This is important for SYZ mirror construction to work.", "In section 4, we compute the temperature dependance of thermal (electrical) conductivity via Kubo formula at finite temperature and finite baryon density up to LO in $N$ .", "The same and deviations from the Wiedemann-Franz formula, upon comparison with [4], mimic remarkably a $D=1+1$ Luttinger liquid with impurities.", "In section 5, through four subsections we present the calculation of speed of sound both at leading order (and NLO) in $N$ in the 'MQGP Limit' in four different ways.", "We then show that the leading order result as obtained from the quasinormal modes of scalar metric perturbation is consistent with that obtained from the two point correlation function.", "In section 6, we evaluate the NLO correction to the shear mode diffusion constant again from the quasinormal modes of the vector metric perturbations.", "Section 7, is devoted to the NLO correction to the shear viscosity $\\eta $ and shear viscosity - entropy density ratio $\\frac{\\eta }{s}$ .", "Section 8 has a summary of the main results of the paper.", "The technical details of sections 3 - 7 are relegated to eight appendices." ], [ "The Background", "In this section, via five sub-sections we will: provide a short review of the type IIB background of [2] which is supposed to provide a UV complete holographic dual of large-$N$ thermal QCD, as well as their precursors in subsection 2.1, discuss the 'MQGP' limit of [3] and the motivation for considering the same in subsection 2.2, briefly review issues pertaining to construction of delocalized S(trominger) Y(au) Z(aslow) mirror and approximate supersymmetry in subsection 2.3, review the recipe of [14] to evaluate Minkowskian-signature space correlators in subsection 2.4, briefly discuss the vector, tensor and scalar modes of metric perturbations and construction of gauge-invariant variables in subsection 2.5" ], [ "Type IIB Dual of Large-$N$ Thermal QCD", "In this subsection, we will discuss a UV complete holographic dual of large-$N$ thermal QCD as given in Dasgupta-Mia et al [2].", "As partly mentioned in Sec.", "1, this was inspired by the zero-temperature Klebanov-Witten model [8], the non-conformal Klebanov-Tseytlin model [10], its IR completion as given in the Klebanov-Strassler model [11] and Ouyang's inclusion [12] of flavor in the same See [19] for earlier attempts at studying back-reacted $D3/D7$ geometry at zero temperature; we thank L. Zayas for bringing [19], [18] to our attention., as well as the non-zero temperature/non-extremal version of [16] (the solution however was not regular as the non-extremality/black hole function and the ten-dimensional warp factor vanished simultaneously at the horizon radius), [17] (valid only at large temperatures) of the Klebanov-Tseytlin model and [18] (addressing the IR), in the absence of flavors.", "(a) Brane construction In order to include fundamental quarks at non-zero temperature in the context of type IIB string theory, to the best of our knowledge, the following model proposed in [2] is the closest to a UV complete holographic dual of large-$N$ thermal QCD.", "The KS model (after a duality cascade) and QCD have similar IR behavior: $SU(M)$ gauge group and IR confinement.", "However, they differ drastically in the UV as the former yields a logarithmically divergent gauge coupling (in the UV) - Landau pole.", "This necessitates modification of the UV sector of the KS model apart from inclusion of non-extremality factors.", "With this in mind and building up on all of the above, the type IIB holographic dual of [2] was constructed.", "The setup of [2] is summarized below.", "From a gauge-theory perspective, the authors of [2] considered $N$ black $D3$ -branes placed at the tip of six-dimensional conifold, $M\\ D5$ -branes wrapping the vanishing two-cycle and $M\\ \\overline{D5}$ -branes distributed along the resolved two-cycle and placed at the outer boundary of the IR-UV interpolating region/inner boundary of the UV region.", "More specifically, the $M\\ \\overline{D5}$ are distributed around the antipodal point relative to the location of $M\\ D5$ branes on the blown-up $S^2$ .", "If the $D5/\\overline{D5}$ separation is given by ${\\cal R}_{D5/\\overline{D5}}$ , then this provides the boundary common to the outer UV-IR interpolating region and the inner UV region.", "The region $r>{\\cal R}_{D5/\\overline{D5}}$ is the UV.", "In other words, the radial space, in [2] is divided into the IR, the IR-UV interpolating region and the UV.", "To summarize the above: $r_0$ ($T=0$ ), $r_h$ ($T>0$ )$<r<|\\mu _{\\rm Ouyang}|^{\\frac{2}{3}}(T=0), {\\cal R}_{D5/\\overline{D5}}(T>0)$ : the IR/IR-UV interpolating regions with $r\\sim \\Lambda $ : deep IR where the $SU(M)$ gauge theory confines $r>|\\mu _{\\rm Ouyang}|^{\\frac{2}{3}} (T=0), {\\cal R}_{D5/\\overline{D5}}(T>0)$ : the UV region.", "$N_f\\ D7$ -branes, via Ouyang embedding, are holomorphically embedded in the UV (asymptotically $AdS_5\\times T^{1,1}$ ), the IR-UV interpolating region and dipping into the (confining) IR (up to a certain minimum value of $r$ corresponding to the lightest quark) and $N_f\\ \\overline{D7}$ -branes present in the UV and the UV-IR interpolating (not the confining IR).", "This is to ensure turning off of three-form fluxes, constancy of the axion-dilaton modulus and hence conformality and absence of Landau poles in the UV.", "The resultant ten-dimensional geometry hence involves a resolved warped deformed conifold.", "Back-reactions are included, e.g., in the ten-dimensional warp factor.", "Of course, the gravity dual, as in the Klebanov-Strassler construct, at the end of the Seiberg-duality cascade will have no $D3$ -branes and the $D5$ -branes are smeared/dissolved over the blown-up $S^3$ and thus replaced by fluxes in the IR.", "The delocalized S(trominger) Y(au) Z(aslow) type IIA mirror of the aforementioned type IIB background of [2] and its M-theory uplift had been obtained in [3], [7], and newer aspects of the same will be looked into in this paper.", "(b) Seiberg duality cascade, IR confining $SU(M)$ gauge theory at finite temperature and $N_c = N_{\\rm eff}(r) + M_{\\rm eff}(r)$ IR Confinement after Seiberg Duality Cascade: Footnote numbered 3 shows that one effectively adds on to the number of $D3$ -branes in the UV and hence, one has $SU(N+M)\\times SU(N+M)$ color gauge group (implying an asymptotic $AdS_5$ ) and $SU(N_f)\\times SU(N_f)$ flavor gauge group, in the UV: $r\\ge {\\cal R}_{D5/\\overline{D5}}$ .", "It is expected that there will be a partial Higgsing of $SU(N+M)\\times SU(N+M)$ to $SU(N+M)\\times SU(N)$ at $r={\\cal R}_{D5/\\overline{D5}}$ [20].", "The two gauge couplings, $g_{SU(N+M)}$ and $g_{SU(N)}$ flow logarithmically and oppositely in the IR: $4\\pi ^2\\left(\\frac{1}{g_{SU(N+M)}^2} + \\frac{1}{g_{SU(N)}^2}\\right)e^\\phi \\sim \\pi ;\\ 4\\pi ^2\\left(\\frac{1}{g_{SU(N+M)}^2} - \\frac{1}{g_{SU(N)}^2}\\right)e^\\phi \\sim \\frac{1}{2\\pi \\alpha ^\\prime }\\int _{S^2}B_2.$ Had it not been for $\\int _{S^2}B_2$ , in the UV, one could have set $g_{SU(M+N)}^2=g_{SU(N)}^2=g_{YM}^2\\sim g_s\\equiv $ constant (implying conformality) which is the reason for inclusion of $M$ $\\overline{D5}$ -branes at the common boundary of the UV-IR interpolating and the UV regions, to annul this contribution.", "In fact, the running also receives a contribution from the $N_f$ flavor $D7$ -branes which needs to be annulled via $N_f\\ \\overline{D7}$ -branes.", "The gauge coupling $g_{SU(N+M)}$ flows towards strong coupling and the $SU(N)$ gauge coupling flows towards weak coupling.", "Upon application of Seiberg duality, $SU(N+M)_{\\rm strong}\\stackrel{\\rm Seiberg\\ Dual}{\\longrightarrow }SU(N-(M - N_f))_{\\rm weak}$ in the IR; assuming after repeated Seiberg dualities or duality cascade, $N$ decreases to 0 and there is a finite $M$ , one will be left with $SU(M)$ gauge theory with $N_f$ flavors that confines in the IR - the finite temperature version of the same is what was looked at by [2].", "Obtaining $N_c=3$ , and Color-Flavor Enhancement of Length Scale in the IR: So, in the IR, at the end of the duality cascade, what gets identified with the number of colors $N_c$ is $M$ , which in the `MQGP limit' to be discussed below, can be tuned to equal 3.", "One can identify $N_c$ with $N_{\\rm eff}(r) + M_{\\rm eff}(r)$ , where $N_{\\rm eff}(r) = \\int _{\\rm Base\\ of\\ Resolved\\ Warped\\ Deformed\\ Conifold}F_5$ and $M_{\\rm eff} = \\int _{S^3}\\tilde{F}_3$ (the $S^3$ being dual to $\\ e_\\psi \\wedge \\left(\\sin \\theta _1 d\\theta _1\\wedge d\\phi _1 - B_1\\sin \\theta _2\\wedge d\\phi _2\\right)$ , wherein $B_1$ is an asymmetry factor defined in [2], and $e_\\psi \\equiv d\\psi + {\\rm cos}~\\theta _1~d\\phi _1 + {\\rm cos}~\\theta _2~d\\phi _2$ ) where $\\tilde{F}_3 (\\equiv F_3 - \\tau H_3)\\propto M(r)\\equiv 1 - \\frac{e^{\\alpha (r-{\\cal R}_{D5/\\overline{D5}})}}{1 + e^{\\alpha (r-{\\cal R}_{D5/\\overline{D5}})}}, \\alpha \\gg 1$ [21].", "The effective number $N_{\\rm eff}$ of $D3$ -branes varies between $N\\gg 1$ in the UV and 0 in the deep IR, and the effective number $M_{\\rm eff}$ of $D5$ -branes varies between 0 in the UV and $M$ in the deep IR (i.e., at the end of the duality cacade in the IR).", "Hence, the number of colors $N_c$ varies between $M$ in the deep IR and a large value [even in the MQGP limit of (REF ) (for a large value of $N$ )] in the UV.", "Hence, at very low energies, the number of colors $N_c$ can be approximated by $M$ , which in the MQGP limit is taken to be finite and can hence be taken to be equal to three.", "However, in this discussion, the low energy or the IR is relative to the string scale.", "But these energies which are much less than the string scale, can still be much larger than $T_c$ .", "Therefore, for all practical purposes, as regard the energy scales relevant to QCD, the number of colors can be tuned to three.", "In the IR in the MQGP limit, with the inclusion of terms higher order in $g_s N_f$ in the RR and NS-NS three-form fluxes and the NLO terms in the angular part of the metric, there occurs an IR color-flavor enhancement of the length scale as compared to a Planckian length scale in KS for ${\\cal O}(1)$ $M$ , thereby showing that quantum corrections will be suppressed.", "Using [2]: $& & N_{\\rm eff}(r) = N\\left[ 1 + \\frac{3 g_s M_{\\rm eff}^2}{2\\pi N}\\left(\\log r + \\frac{3 g_s N_f^{\\rm eff}}{2\\pi }\\left(\\log r\\right)^2\\right)\\right],\\nonumber \\\\& & M_{\\rm eff}(r) = M + \\frac{3g_s N_f M}{2\\pi }\\log r + \\sum _{m\\ge 1}\\sum _{n\\ge 1} N_f^m M^n f_{mn}(r),\\nonumber \\\\& & N^{\\rm eff}_f(r) = N_f + \\sum _{m\\ge 1}\\sum _{n\\ge 0} N_f^m M^n g_{mn}(r).$ it was argued in [22] that the length scale of the OKS-BH metric in the IR will be given by: $& & L_{\\rm OKS-BH}\\sim \\sqrt{M}N_f^{\\frac{3}{4}}\\sqrt{\\left(\\sum _{m\\ge 0}\\sum _{n\\ge 0}N_f^mM^nf_{mn}(\\Lambda )\\right)}\\left(\\sum _{l\\ge 0}\\sum _{p\\ge 0}N_f^lM^p g_{lp}(\\Lambda )\\right)^{\\frac{1}{4}}g_s^{\\frac{1}{4}}\\sqrt{\\alpha ^\\prime }\\nonumber \\\\& & \\equiv N_f^{\\frac{3}{4}}\\left.\\sqrt{\\left(\\sum _{m\\ge 0}\\sum _{n\\ge 0}N_f^mM^nf_{mn}(\\Lambda )\\right)}\\left(\\sum _{l\\ge 0}\\sum _{p\\ge 0}N_f^lM^p g_{lp}(\\Lambda )\\right)^{\\frac{1}{4}} L_{\\rm KS}\\right|_{\\Lambda :\\log \\Lambda {<}{\\frac{2\\pi }{3g_sN_f}}},$ which implies that in the IR, relative to KS, there is a color-flavor enhancement of the length scale in the OKS-BH metric.", "Hence, in the IR, even for $N_c^{\\rm IR}=M=3$ and $N_f=6$ upon inclusion of of $n,m>1$ terms in $M_{\\rm eff}$ and $N_f^{\\rm eff}$ in (REF ), $L_{\\rm OKS-BH}\\gg L_{\\rm KS}(\\sim L_{\\rm Planck})$ in the MQGP limit involving $g_s\\stackrel{\\sim }{<}1$ , implying that the stringy corrections are suppressed and one can trust supergravity calculations.", "As a reminder one will generate higher powers of $M$ and $N_f$ in the double summation in $M_{\\rm eff}$ in (REF ), e.g., from the terms higher order in $g_s N_f$ in the RR and NS-NS three-form fluxes that become relevant for the aforementioned values of $g_s, N_f$ .", "Further, the global flavor group in the UV-IR interpolating and UV regions, due to presence of $N_f$ $D7$ and $N_f\\ \\overline{D7}$ -branes, is $SU(N_f)\\times SU(N_f)$ , which is broken in the IR to $SU(N_f)$ as the IR has only $N_f$ $D7$ -branes.", "Hence, the following features of the type IIB model of [2] make it an ideal holographic dual of thermal QCD: the theory having quarks transforming in the fundamental representation, is UV conformal and IR confining with the required chiral symmetry breaking in the IR and restoration at high temperatures the theory is UV complete with the gauge coupling remaining finite in the UV (absence of Landau poles) the theory is not just defined for large temperatures but for low and high temperatures (as will become evident in Sec.", "3) with the inclusion of a finite baryon chemical potential, the theory provides a lattice-compatible QCD confinement-deconfinement temperature $T_c$ for the right number of light quark flavors and masses, and is also thermodynamically stable; given the IR proximity of the value of the lattice-compatible $T_c$ , after the end of the Seiberg duality cascade, the number of quark flavors approximately equals $M$ which in the `MQGP' limit of (REF ) can be tuned to equal 3 in the MQGP limit (REF ) which requires considering a finite gauge coupling and hence string coupling, the theory was shown in [3] to be holographically renormalizable from an M-theory perspective with the M-theory uplift also being thermodynamically stable.", "(d) Supergravity solution on resolved warped deformed conifold The working metric is given by : $ds^2 = \\frac{1}{\\sqrt{h}}\\left(-g_1 dt^2+dx_1^2+dx_2^2+dx_3^2\\right)+\\sqrt{h}\\biggl [g_2^{-1}dr^2+r^2 d{\\cal M}_5^2\\biggr ].$ $g_i$ 's are black hole functions in modified OKS(Ouyang-Klebanov-Strassler)-BH (Black Hole) background and are assumed to be: $ g_{1,2}(r,\\theta _1,\\theta _2)= 1-\\frac{r_h^4}{r^4} + {\\cal O}\\left(\\frac{g_sM^2}{N}\\right)$ where $r_h$ is the horizon, and the ($\\theta _1, \\theta _2$ ) dependence come from the ${\\cal O}\\left(\\frac{g_sM^2}{N}\\right)$ corrections.", "The $h_i$ 's are expected to receive corrections of ${\\cal O}\\left(\\frac{g_sM^2}{N}\\right)$ [20].", "We assume the same to also be true of the `black hole functions' $g_{1,2}$ .", "The compact five dimensional metric in (REF ), is given as: $& & d{\\cal M}_5^2 = h_1 (d\\psi + {\\rm cos}~\\theta _1~d\\phi _1 + {\\rm cos}~\\theta _2~d\\phi _2)^2 +h_2 (d\\theta _1^2 + {\\rm sin}^2 \\theta _1 ~d\\phi _1^2) + \\nonumber \\\\&& + h_4 (h_3 d\\theta _2^2 + {\\rm sin}^2 \\theta _2 ~d\\phi _2^2) + h_5~{\\rm cos}~\\psi \\left(d\\theta _1 d\\theta _2 -{\\rm sin}~\\theta _1 {\\rm sin}~\\theta _2 d\\phi _1 d\\phi _2\\right) + \\nonumber \\\\&& + h_5 ~{\\rm sin}~\\psi \\left({\\rm sin}~\\theta _1~d\\theta _2 d\\phi _1 +{\\rm sin}~\\theta _2~d\\theta _1 d\\phi _2\\right),$ $r\\gg a, h_5\\sim \\frac{({\\rm deformation\\ parameter})^2}{r^3}\\ll 1$ for $r \\gg ({\\rm deformation\\ parameter})^{\\frac{2}{3}}$ , i.e.", "in the UV/IR-UV interpolating region.", "The $h_i$ 's appearing in internal metric as well as $M, N_f$ are not constant and up to linear order depend on $g_s, M, N_f$ are given as below: $& & \\hspace{-32.52127pt}h_1 = \\frac{1}{9} + {\\cal O}\\left(\\frac{g_sM^2}{N}\\right),\\ h_2 = \\frac{1}{6} + {\\cal O}\\left(\\frac{g_sM^2}{N}\\right),\\ h_4 = h_2 + \\frac{a^2}{r^2},\\nonumber \\\\& & h_3 = 1 + {\\cal O}\\left(\\frac{g_sM^2}{N}\\right),\\ h_5\\ne 0,\\ L=\\left(4\\pi g_s N\\right)^{\\frac{1}{4}}.$ One sees from (REF ) and (REF ) that one has a non-extremal resolved warped deformed conifold involving an $S^2$ -blowup (as $h_4 - h_2 = \\frac{a^2}{r^2}$ ), an $S^3$ -blowup (as $h_5\\ne 0$ ) and squashing of an $S^2$ (as $h_3$ is not strictly unity).", "The horizon (being at a finite $r=r_h$ ) is warped squashed $S^2\\times S^3$ .", "In the deep IR, in principle one ends up with a warped squashed $S^2(a)\\times S^3(\\epsilon ),\\ \\epsilon $ being the deformation parameter.", "Assuming $\\epsilon ^{\\frac{2}{3}}>a$ and given that $a={\\cal O}\\left(\\frac{g_s M^2}{N}\\right)r_h$ [20], in the IR and in the MQGP limit, $N_{\\rm eff}(r\\in {\\rm IR})=\\int _{{\\rm warped\\ squashed}\\ S^2(a)\\times S^3(\\epsilon )}F_5(r\\in {\\rm IR})\\ll M = \\int _{S^3(\\epsilon )}F_3(r\\in {\\rm IR})$ ; we have a confining $SU(M)$ gauge theory in the IR.", "The warp factor that includes the back-reaction, in the IR is given as: $&& \\hspace{-32.52127pt}h =\\frac{L^4}{r^4}\\Bigg [1+\\frac{3g_sM_{\\rm eff}^2}{2\\pi N}{\\rm log}r\\left\\lbrace 1+\\frac{3g_sN^{\\rm eff}_f}{2\\pi }\\left({\\rm log}r+\\frac{1}{2}\\right)+\\frac{g_sN^{\\rm eff}_f}{4\\pi }{\\rm log}\\left({\\rm sin}\\frac{\\theta _1}{2}{\\rm sin}\\frac{\\theta _2}{2}\\right)\\right\\rbrace \\Biggr ],$ where, in principle, $M_{\\rm eff}/N_f^{\\rm eff}$ are not necessarily the same as $M/N_f$ ; we however will assume that up to ${\\cal O}\\left(\\frac{g_sM^2}{N}\\right)$ , they are.", "Proper UV behavior requires [20]: $& & h = \\frac{L^4}{r^4}\\left[1 + \\sum _{i=1}\\frac{{\\cal H}_i\\left(\\phi _{1,2},\\theta _{1,2},\\psi \\right)}{r^i}\\right],\\ {\\rm large}\\ r;\\nonumber \\\\& & h = \\frac{L^4}{r^4}\\left[1 + \\sum _{i,j; (i,j)\\ne (0,0)}\\frac{h_{ij}\\left(\\phi _{1,2},\\theta _{1,2},\\psi \\right)\\log ^ir}{r^j}\\right],\\ {\\rm small}\\ r.$ In the IR, up to ${\\cal O}(g_s N_f)$ and setting $h_5=0$ , the three-forms are as given in [2]: $& & \\hspace{-28.90755pt}(a) {\\widetilde{F}}_3 = 2M { A_1} \\left(1 + \\frac{3g_sN_f}{2\\pi }~{\\rm log}~r\\right) ~e_\\psi \\wedge \\frac{1}{2}\\left({\\rm sin}~\\theta _1~ d\\theta _1 \\wedge d\\phi _1-{ B_1}~{\\rm sin}~\\theta _2~ d\\theta _2 \\wedge d\\phi _2\\right)\\nonumber \\\\&& \\hspace{-21.68121pt}-\\frac{3g_s MN_f}{4\\pi } { A_2}~\\frac{dr}{r}\\wedge e_\\psi \\wedge \\left({\\rm cot}~\\frac{\\theta _2}{2}~{\\rm sin}~\\theta _2 ~d\\phi _2- { B_2}~ {\\rm cot}~\\frac{\\theta _1}{2}~{\\rm sin}~\\theta _1 ~d\\phi _1\\right)\\nonumber \\\\&& \\hspace{-21.68121pt}-\\frac{3g_s MN_f}{8\\pi }{ A_3} ~{\\rm sin}~\\theta _1 ~{\\rm sin}~\\theta _2 \\left({\\rm cot}~\\frac{\\theta _2}{2}~d\\theta _1 +{ B_3}~ {\\rm cot}~\\frac{\\theta _1}{2}~d\\theta _2\\right)\\wedge d\\phi _1 \\wedge d\\phi _2, \\nonumber \\\\& & \\hspace{-28.90755pt}(b) H_3 = {6g_s { A_4} M}\\Biggl (1+\\frac{9g_s N_f}{4\\pi }~{\\rm log}~r+\\frac{g_s N_f}{2\\pi }~{\\rm log}~{\\rm sin}\\frac{\\theta _1}{2}~{\\rm sin}\\frac{\\theta _2}{2}\\Biggr )\\frac{dr}{r}\\nonumber \\\\&& \\hspace{-21.68121pt}\\wedge \\frac{1}{2}\\Biggl ({\\rm sin}~\\theta _1~ d\\theta _1 \\wedge d\\phi _1- { B_4}~{\\rm sin}~\\theta _2~ d\\theta _2 \\wedge d\\phi _2\\Biggr )+ \\frac{3g^2_s M N_f}{8\\pi } { A_5} \\Biggl (\\frac{dr}{r}\\wedge e_\\psi -\\frac{1}{2}de_\\psi \\Biggr )\\nonumber \\\\&& \\wedge \\Biggl ({\\rm cot}~\\frac{\\theta _2}{2}~d\\theta _2-{ B_5}~{\\rm cot}~\\frac{\\theta _1}{2} ~d\\theta _1\\Biggr ).", "\\nonumber \\\\$ The asymmetry factors in (REF ) are given by: $ A_i=1 +{\\cal O}\\left(\\frac{a^2}{r^2}\\ {\\rm or}\\ \\frac{a^2\\log r}{r}\\ {\\rm or}\\ \\frac{a^2\\log r}{r^2}\\right) + {\\cal O}\\left(\\frac{{\\rm deformation\\ parameter }^2}{r^3}\\right),$ $ B_i = 1 + {\\cal O}\\left(\\frac{a^2\\log r}{r}\\ {\\rm or}\\ \\frac{a^2\\log r}{r^2}\\ {\\rm or}\\ \\frac{a^2\\log r}{r^3}\\right)+{\\cal O}\\left(\\frac{({\\rm deformation\\ parameter})^2}{r^3}\\right)$ .", "As in the UV, $\\frac{({\\rm deformation\\ parameter})^2}{r^3}\\ll \\frac{({\\rm resolution\\ parameter})^2}{r^2}$ , we will assume the same three-form fluxes for $h_5\\ne 0$ .", "Further, to ensure UV conformality, it is important to ensure that the axion-dilaton modulus approaches a constant implying a vanishing beta function in the UV.", "This was discussed in detail in appendix B of [22], wherein in particular, assuming an F-theory uplift involving, locally, an elliptically fibered $K3$ , it was shown that UV conformality and the Ouyang embedding are mutually consistent." ], [ "The `MQGP Limit'", "In [3], we had considered the following two limits: $& & \\hspace{-12.28577pt}(i) {\\rm weak}(g_s){\\rm coupling-large\\ t^{\\prime }Hooft\\ coupling\\ limit}:\\nonumber \\\\& & \\hspace{-12.28577pt}g_s\\ll 1, g_sN_f\\ll 1, \\frac{g_sM^2}{N}\\ll 1, g_sM\\gg 1, g_sN\\gg 1\\nonumber \\\\& & \\hspace{-12.28577pt}{\\rm effected\\ by}: g_s\\sim \\epsilon ^{d}, M\\sim \\left({\\cal O}(1)\\epsilon \\right)^{-\\frac{3d}{2}}, N\\sim \\left({\\cal O}(1)\\epsilon \\right)^{-19d}, \\epsilon \\ll 1, d>0$ (the limit in the first line though not its realization in the second line, considered in [2]); $& & \\hspace{-12.28577pt}(ii) {\\rm MQGP\\ limit}: \\frac{g_sM^2}{N}\\ll 1, g_sN\\gg 1, {\\rm finite}\\ g_s, M\\ \\nonumber \\\\& & \\hspace{-12.28577pt}{\\rm effected\\ by}: g_s\\sim \\epsilon ^d, M\\sim \\left({\\cal O}(1)\\epsilon \\right)^{-\\frac{3d}{2}}, N\\sim \\left({\\cal O}(1)\\epsilon \\right)^{-39d}, \\epsilon \\lesssim 1, d>0.$ Let us enumerate the motivation for considering the MQGP limit which was discussed in detail in [22].", "There are principally two.", "Unlike the AdS/CFT limit wherein $g_{\\rm YM}\\rightarrow 0, N\\rightarrow \\infty $ such that $g_{\\rm YM}^2N$ is large, for strongly coupled thermal systems like sQGP, what is relevant is $g_{\\rm YM}\\sim {\\cal O}(1)$ and $N_c=3$ .", "From the discussion in the previous paragraphs specially the one in point (c) of sub-section 2.1, one sees that in the IR after the Seiberg duality cascade, effectively $N_c=M$ which in the MQGP limit of (REF ) can be tuned to 3.", "Further, in the same limit, the string coupling $g_s\\stackrel{<}{\\sim }1$ .", "The finiteness of the string coupling necessitates addressing the same from an M theory perspective.", "This is the reason for coining the name: `MQGP limit'.", "In fact this is the reason why one is required to first construct a type IIA mirror, which was done in [3] a la delocalized Strominger-Yau-Zaslow mirror symmetry, and then take its M-theory uplift.", "From the perspective of calculational simplification in supergravity, the following are examples of the same and constitute therefore the second set of reasons for looking at the MQGP limit of (REF ): In the UV-IR interpolating region and the UV, $(M_{\\rm eff}, N_{\\rm eff}, N_f^{\\rm eff})\\stackrel{\\rm MQGP}{\\approx }(M, N, N_f)$ Asymmetry Factors $A_i, B_j$ (in three-form fluxes)$\\stackrel{MQGP}{\\rightarrow }1$ in the UV-IR interpolating region and the UV.", "Simplification of ten-dimensional warp factor and non-extremality function in MQGP limit With ${\\cal R}_{D5/\\overline{D5}}$ denoting the boundary common to the UV-IR interpolating region and the UV region, $\\tilde{F}_{lmn}, H_{lmn}=0$ for $r\\ge {\\cal R}_{D5/\\overline{D5}}$ is required to ensure conformality in the UV.", "Near the $\\theta _1=\\theta _2=0$ -branch, assuming: $\\theta _{1,2}\\rightarrow 0$ as $\\epsilon ^{\\gamma _\\theta >0}$ and $r\\rightarrow {\\cal R}_{\\rm UV}\\rightarrow \\infty $ as $\\epsilon ^{-\\gamma _r <0}, \\lim _{r\\rightarrow \\infty }\\tilde{F}_{lmn}=0$ and $\\lim _{r\\rightarrow \\infty }H_{lmn}=0$ for all components except $H_{\\theta _1\\theta _2\\phi _{1,2}}$ ; in the MQGP limit and near $\\theta _{1,2}=\\pi /0$ -branch, $H_{\\theta _1\\theta _2\\phi _{1,2}}=0/\\left.\\frac{3 g_s^2MN_f}{8\\pi }\\right|_{N_f=2,g_s=0.6, M=\\left({\\cal O}(1)g_s\\right)^{-\\frac{3}{2}}}\\ll 1.$ So, the UV nature too is captured near $\\theta _{1,2}=0$ -branch in the MQGP limit.", "This mimics addition of $\\overline{D5}$ -branes in [2] to ensure cancellation of $\\tilde{F}_3$ ." ], [ "Approximate Supersymmetry, Construction of the Delocalized SYZ IIA Mirror and Its M-Theory Uplift in the MQGP Limit", "A central issue to [3], [7] has been implementation of delocalized mirror symmetry via the Strominger Yau Zaslow prescription according to which the mirror of a Calabi-Yau can be constructed via three T dualities along a special Lagrangian $T^3$ fibered over a large base in the Calabi-Yau.", "This sub-section is a quick review of precisely this.", "To implement the quantum mirror symmetry a la S(trominger)Y(au)Z(aslow) [23], one needs a special Lagrangian (sLag) $T^3$ fibered over a large base (to nullify contributions from open-string disc instantons with boundaries as non-contractible one-cycles in the sLag).", "Defining delocalized T-duality coordinates, $(\\phi _1,\\phi _2,\\psi )\\rightarrow (x,y,z)$ valued in $T^3(x,y,z)$ [3]: $x = \\sqrt{h_2}h^{\\frac{1}{4}}sin\\langle \\theta _1\\rangle \\langle r\\rangle \\phi _1,\\ y = \\sqrt{h_4}h^{\\frac{1}{4}}sin\\langle \\theta _2\\rangle \\langle r\\rangle \\phi _2,\\ z=\\sqrt{h_1}\\langle r\\rangle h^{\\frac{1}{4}}\\psi ,$ using the results of [6] it was shown in [7] that the following conditions are satisfied: $& & i^* J \\approx 0,\\nonumber \\\\& & \\Im m\\left( i^*\\Omega \\right) \\approx 0,\\nonumber \\\\& & \\Re e\\left(i^*\\Omega \\right)\\sim {\\rm volume \\ form}\\left(T^3(x,y,z)\\right),$ for the $T^2$ -invariant sLag of [6] for a deformed conifold.", "It will be shown in the Section 3 that (REF ) is also satisfied for the $T^2$ -invariant sLag of [6] for a resolved conifold, implying thus: $\\left.i^* J\\right|_{RC/DC}\\approx 0, \\Im m\\left.\\left( i^*\\Omega \\right)\\right|_{RC/DC} \\approx 0, \\Re e\\left.\\left(i^*\\Omega \\right)\\right|_{RC/DC}\\sim {\\rm volume \\ form}\\left(T^3(x,y,z)\\right)$ .", "Hence, if the resolved warped deformed conifold is predominantly either resolved or deformed, the local $T^3$ of (REF ) is the required sLag to effect SYZ mirror construction.", "Interestingly, in the `delocalized limit' [24] $\\psi =\\langle \\psi \\rangle $ , under the coordinate transformation: $\\left(\\begin{array}{c} sin\\theta _2 d\\phi _2 \\\\ d\\theta _2\\end{array} \\right)\\rightarrow \\left(\\begin{array}{cc} cos\\langle \\psi \\rangle & sin\\langle \\psi \\rangle \\\\- sin\\langle \\psi \\rangle & cos\\langle \\psi \\rangle \\end{array}\\right)\\left(\\begin{array}{c}sin\\theta _2 d\\phi _2\\\\d\\theta _2\\end{array}\\right),$ and $\\psi \\rightarrow \\psi - \\cos \\langle {\\bar{\\theta }}_2\\rangle \\phi _2 + \\cos \\langle \\theta _2\\rangle \\phi _2 - \\tan \\langle \\psi \\rangle ln\\sin {\\bar{\\theta }}_2$ , the $h_5$ term becomes $h_5\\left[d\\theta _1 d\\theta _2 - sin\\theta _1 sin\\theta _2 d\\phi _1d\\phi _2\\right]$ , $e_\\psi \\rightarrow e_\\psi $ , i.e., one introduces an isometry along $\\psi $ in addition to the isometries along $\\phi _{1,2}$ .", "This clearly is not valid globally - the deformed conifold does not possess a third global isometry.", "To enable use of SYZ-mirror duality via three T dualities, one also needs to ensure a large base (implying large complex structures of the aforementioned two two-tori) of the $T^3(x,y,z)$ fibration.", "This is effected via [25]: $& & d\\psi \\rightarrow d\\psi + f_1(\\theta _1)\\cos \\theta _1 d\\theta _1 + f_2(\\theta _2)\\cos \\theta _2d\\theta _2,\\nonumber \\\\& & d\\phi _{1,2}\\rightarrow d\\phi _{1,2} - f_{1,2}(\\theta _{1,2})d\\theta _{1,2},$ for appropriately chosen large values of $f_{1,2}(\\theta _{1,2})$ .", "The three-form fluxes remain invariant.", "The fact that one can choose such large values of $f_{1,2}(\\theta _{1,2})$ , was justified in [3].", "The guiding principle is that one requires the metric obtained after SYZ-mirror transformation applied to the non-Kähler resolved warped deformed conifold is like a non-Kähler warped resolved conifold at least locally.", "Then $G^{IIA}_{\\theta _1\\theta _2}$ needs to vanish [3].", "This is shown to be true anywhere in the UV in Appendix C. The mirror type IIA metric after performing three T-dualities, first along $x$ , then along $y$ and finally along $z$ , utilizing the results of [24] was worked out in [3].", "We can get a one-form type IIA potential from the triple T-dual (along $x, y, z$ ) of the type IIB $F_{1,3,5}$ in [3] and using which the following $D=11$ metric was obtained in [3] ($u\\equiv \\frac{r_h}{r}$ ): $& &\\hspace{-43.36243pt}ds^2_{11} = e^{-\\frac{2\\phi ^{IIA}}{3}} \\left[g_{tt}dt^2 + g_{\\mathbb {R}^3}\\left(dx^2 + dy^2 + dz^2\\right) + g_{uu}du^2 + ds^2_{IIA}({\\theta _{1,2},\\phi _{1,2},\\psi })\\right] \\nonumber \\\\& & \\hspace{-43.36243pt}+ e^{\\frac{4{\\phi }^{IIA}}{3}}\\Bigl (dx_{11} + A^{F_1}+A^{F_3}+A^{F_5}\\Bigr )^2 \\equiv \\ {\\rm Black}\\ M3-{\\rm Brane}+{\\cal O}\\left(\\left[\\frac{g_s M^2 \\log N}{N}\\right] \\left(g_sM\\right)N_f\\right),\\nonumber \\\\& & {\\rm where}:\\nonumber \\\\& & g_{uu}=\\frac{3^{2/3}(2\\sqrt{\\pi g_s N})}{u^2(1-u^4)}\\left(1 - \\frac{3 g_s^2 M^2 N_f \\log (N) \\log \\left(\\frac{r_h}{u}\\right)}{32 \\pi ^2 N}\\right)\\nonumber \\\\& & g_{tt} = \\frac{3^{2/3}(u^4-1)r_h^2}{u^2(2\\sqrt{\\pi g_{s}N})} \\left(\\frac{3 g_s^2 M^2 N_f \\log (N) \\log \\left(\\frac{r_h}{u}\\right)}{32 \\pi ^2 N}+1\\right)\\nonumber \\\\& & g_{\\mathbb {R}^3} = \\frac{3^{2/3}r_h^2}{u^2(2\\sqrt{\\pi g_{s}N})} \\left(\\frac{3 g_s^2 M^2 N_f \\log (N) \\log \\left(\\frac{r_h}{u}\\right)}{32 \\pi ^2 N}+1\\right).$ Further, in the UV: $& & G_{{x}{x}}^M = G_{{y}{y}}^M = G_{{z}{z}}^M = \\frac{3^{2/3}r_h^2}{g_{s}^{2/3}u^2(2\\sqrt{\\pi g_{s}N})} \\left(\\frac{3 g_s^2 M^2 N_f \\log (N) \\log \\left(\\frac{r_h}{u}\\right)}{32 \\pi ^2 N}+1\\right)\\nonumber \\\\& & G_{\\phi _1r}^M \\sim \\nonumber \\\\& & \\frac{2 {g_s}^{4/3} {N_f}^2 {\\sin ^2\\phi _1} 2\\sin \\left(\\frac{\\psi }{2}\\right) \\sin ^2({\\theta _1}) \\left(9 \\sin ^2({\\theta _1})+6 \\cos ^2({\\theta _1})+4 \\cos ({\\theta _1})\\right) \\@root 4 \\of {{g_s} N \\left(1-\\frac{3 {g_s}^2 M^2 {N_f} \\log (N)\\log (r)}{16 \\pi ^2 N}\\right)} }{3^{5/6} \\pi ^{7/4} (\\cos (2 {\\theta _1})-5)^2}\\nonumber \\\\& & \\times \\left(9 {h_5} \\sin ({\\theta _1})+4 \\cos ^2({\\theta _1}) \\csc ({\\theta _2})-2\\cos ({\\theta _1}) \\cot ({\\theta _2})+6 \\sin ^2({\\theta _1}) \\csc ({\\theta _2})\\right)\\nonumber \\\\& & G_{11\\ r}^M\\sim \\frac{3^{\\frac{3}{2}}g_s^{\\frac{4}{3}} N_f \\sin \\phi _1\\left( - 8 \\cos \\theta _1 + 3( - 5 +\\cos (2\\theta _1))\\right)\\sin \\theta _1}{\\pi \\left( - 5 + \\cos (2\\theta _1)\\right)}.$ In [22], we showed explicitly the existence of a local type IIA $SU(3)$ structure and a local $G_2$ structure of the M-theory uplift around $\\theta _1\\sim \\frac{1}{N^{\\frac{1}{5}}}, \\theta _2\\sim \\frac{1}{N^{\\frac{3}{10}}}$ .", "Near this coordinate patch, one sees that: $& & G_{\\phi _1r}\\sim \\frac{10 g_s^{\\frac{19}{12}}\\sin ^2\\phi _1 \\sin \\left(\\frac{\\psi }{2}\\right)N^{\\frac{3}{20}}}{2 3^{\\frac{5}{6}}\\pi ^{\\frac{7}{4}}} + {\\cal O}\\left(\\frac{1}{N^{\\frac{1}{4}}}\\right)\\ll 1\\ {\\rm for}\\ \\psi \\sim \\frac{1}{N^{\\alpha \\gg \\frac{3}{20}}},\\nonumber \\\\& & G_{11\\ r}\\sim {\\cal O}\\left(\\frac{1}{N^{\\frac{1}{5}}}\\right).$ Thus, in the MQGP limit, around $\\theta _1\\sim \\frac{1}{N^{\\frac{1}{5}}}, \\theta _2\\sim \\frac{1}{N^{\\frac{3}{10}}}$ , the five-dimensional $M_5(t,x_{1,2,3},u)$ decouples from $M_6(\\theta _{1,2},\\phi _{1,2},\\psi ,x_{10})$ .", "As in Klebanov-Strassler construction, a single T-duality along a direction orthogonal to the $D3$ -brane world volume, e.g., $z$ of (REF ), yields $D4$ branes straddling a pair of $NS5$ -branes consisting of world-volume coordinates $(\\theta _1,x)$ and $(\\theta _2,y)$ .", "Further, T-dualizing along $x$ and then $y$ would yield a Taub-NUT space from each of the two $NS5$ -branes [26].", "The $D7$ -branes yield $D6$ -branes which get uplifted to Kaluza-Klein monopoles in M-theory [27] which too involve Taub-NUT spaces.", "Globally, probably the eleven-dimensional uplift would involve a seven-fold of $G_2$ -structure, analogous to the uplift of $D5$ -branes wrapping a two-cycle in a resolved warped conifold [28].", "Now, in the delocalized limit of [24], in [7], e.g., $\\left.\\int _{C_4(\\theta _{1,2},\\phi _{1/2},x_{10})}G_4\\right|_{\\phi _{2/1}=\\langle \\phi _{2/1}\\rangle ,\\psi =\\langle \\psi \\rangle ,\\langle r\\rangle }$ was estimated to be very large.", "There is a two-fold reason for the same.", "First, using the local $T^3$ -coordinates of (REF ), this large flux is estimated in the MQGP limit to be $\\left(g_s N\\right)^{\\frac{1}{4}}$ (as, using (REF ), $G_{\\phi _1\\ {\\rm or}\\ \\phi _2\\ {\\rm or}\\ \\psi \\bullet \\bullet \\bullet }\\sim \\left(g_s N\\right)^{\\frac{1}{4}}G_{x\\ {\\rm or}\\ y\\ {\\rm or}\\ z\\bullet \\bullet \\bullet }$ where the bullets denote directions other than $\\phi _1,\\phi _2,\\psi $ ).", "This in the MQGP limit, is large.", "The second is the following.", "Now, $G_4 = H\\wedge (A^{F_1+F-3+F_5} - dx_{10})$ [3] where $A^{F_1+F_3+F_5}$ is the type IIA one-form gauge field obtained after SYZ mirror construction via triple T dualities on the type IIB $F_{1,3,5}$ .", "As the $S^2(\\theta _1,\\phi _1)$ is a vanishing two-sphere, to obtain a finite $\\int _{S^2(\\theta _1,\\phi _1)}B_2$ - that appears in the RG equation (REF ) - one requires a large $B_2$ (From [2] one sees that such a large contribution to $B_2$ is obtained near the $\\theta _1=\\theta _2=0$ branch.)", "Therefore, this too contributes to a large $G_4$ via a large $H$ .", "Locally, the uplift (REF ) can hence be thought of as black $M3$ -brane metric, which in the UV, can be thought of as black $M5$ -branes wrapping a two cycle homologous to: $n_1 S^2(\\theta _1,x_{10}) + n_2 S^2(\\theta _2,\\phi _{1/2}) + m_1 S^2(\\theta _1,\\phi _{1/2}) + m_2 S^2(\\theta _2,x_{10})$ for some large $n_{1,2},m_{1,2}\\in \\mathbb {Z}$ [7].", "In the large-$r$ limit, the $D=11$ space-time is a warped product of $AdS_5(\\mathbb {R}^{1,3}\\times \\mathbb {R}_{>0})$ and ${\\cal M}_6(\\theta _{1,2},\\phi _{1,2},\\psi ,x_{10})$ $\\hspace{-28.90755pt}\\begin{array}{cc}&{\\cal M}_6(\\theta _{1,2},\\phi _{1,2},\\psi ,x_{10}) \\longleftarrow S^1(x_{10}) \\\\&\\downarrow \\\\{\\cal M}_3(\\phi _1,\\phi _2,\\psi ) \\hspace{-28.90755pt}& \\longrightarrow {\\cal M}_5(\\theta _{1,2},\\phi _{1,2},\\psi ) \\\\&\\downarrow \\\\& \\hspace{65.04256pt}{\\cal B}_2(\\theta _1,\\theta _2) \\longleftarrow [0,1]_{\\theta _1} \\\\& \\downarrow \\\\& [0,1]_{\\theta _2}\\end{array}.$ The $D=11$ SUGRA EOMs/Bianchi identity [29] were shown in [7] to be satisfied near the $\\theta _{1,2}=0,\\pi $ -branches in the MQGP limit." ], [ "Recipe to find Minkowski Correlators", "Following [14] we briefly review the prescription to find the thermal correlator in Minkowski signature.", "According to AdS/CFT correspondence, there exists an operator $\\mathcal {O}$ in the field theory side dual to a field $\\phi $ defined in the bulk of AdS geometry such that on the boundary of the anti-de Sitter space $\\phi $ tends to a value $\\phi _{0}$ which acts as a source for the operator $\\mathcal {O}$ .", "we are interested in calculating the retarded Green's function $G^R$ of the operator $\\mathcal {O}$ in Minkowski space.", "Our working background (Type $IIB$ or it's M-theory uplift) can be expressed as the following $5d$ metric, $ds^2=-g_{tt}(u)dt^2+g_{xx}(u)\\Biggl (dx^2+dy^2+dz^2\\Biggr )+g_{uu}(u)du^2.$ Here $u$ is the radial coordinate defined as $u=r_h/r$ so that $u=0$ is the boundary and $u=1$ is the horizon of the AdS space.", "A solution of the linearized field equation for any field $\\phi (u,x)$ choosing $q^\\mu =(w,q,0,0)$ is given as, $\\phi (u,x)=\\int \\frac{{d^4}q}{(2\\pi )^4}e^{-i wt + i qx}f_{q}(u)\\phi _{0}(q)$ where $ f_{q}(u)$ is normalized to 1 at the boundary and satisfies the incoming wave boundary condition at $u=1$ , and $\\phi _{0}(q)$ is determined by, $\\phi (u=0,x)=\\int \\frac{{d^4}q}{(2\\pi )^4}e^{-i wt + i qx}\\phi _{0}(q).$ If the kinetic term for $\\phi (u,x)$ is given by: $\\frac{1}{2}\\int {d^4}x du A(u)\\left(\\partial _u\\phi (x,u)\\right)^2$ , then using the equation of motion for $\\phi $ it is possible to reduce an on-shell action to the surface terms as, $S=\\int \\frac{{d^4}q}{(2\\pi )^4}\\phi _{0}(-q)\\mathcal {F}(q,u)\\phi _{0}(q)|^{u=1}_{u=0}$ where the function $\\mathcal {F}(q,u) = A(u) f_{\\pm q}(u)\\partial _{u}f_{\\pm q}(u).$ Finally, the retarded Green's function is given by the formula: $G^{R}(q)=-2\\mathcal {F}(q,u)|_{u=0}.$ The different retarded Green's functions are defined as $G^{R\\ T}_{\\mu \\nu ,\\rho \\sigma }(q)=-i\\int {d^4}x e^{-i wt + i qx}\\theta (t) \\langle [T_{\\mu \\nu }(x),T_{\\rho \\sigma }(0)]\\rangle ,$ with $\\langle [T_{\\mu \\nu },T_{\\rho \\sigma }]\\rangle \\sim \\frac{\\delta ^2S}{\\delta h_{\\mu \\nu }\\delta h_{\\rho \\sigma }}$ and $G^{R\\ J}_{\\mu \\nu }(q)=-i\\int {d^4}x e^{-i wt + i qx}\\theta (t) \\langle [J_{\\mu }(x),J_{\\nu }(0)]\\rangle $ with $\\langle [J_{\\mu }(x),J_{\\nu }(0)]\\rangle \\sim \\frac{\\delta ^2S}{\\delta A_\\mu \\delta A_\\nu }$ , as the energy-momentum tensor $T_{\\mu \\nu }(x)$ and the current $J_{\\mu }(x)$ couple respectively to the metric and gauge field." ], [ "Perturbations of the background and the gauge invariant combinations", "In the background of section 2 (Type $IIB$ and it's M-theory uplift), we consider a small linear fluctuation of the black brane metric of (REF ) as: $g_{\\mu \\nu }=g^0_{\\mu \\nu }+h_{\\mu \\nu },$ where $g^0_{\\mu \\nu }$ denotes the background metric.", "The inverse metric is defined as(up to second order in perturbation) $g^{\\mu \\nu }=g^{(0)\\mu \\nu }-h^{\\mu \\nu }+h^{\\mu l}h_{l}^{~\\nu }.$ For the evaluation of the temperature dependance of thermal conductivity in section 6 we consider the fluctuation gauge field $A_{\\mu }$ also along with the metric perturbation given as: $A_{\\mu }=A^0_{\\mu }+\\mathcal {A}_{\\mu },$ where in this case we consider the coupling of gauge field fluctuation $\\mathcal {A}_{\\mu }$ with the background metric perturbation.", "Assuming the momenta to be along the $x$ -direction, the metric and the gauge field fluctuations can be written as the following Fourier decomposed form: $h_{\\mu \\nu }( x,t,u)=\\int \\frac{d^{4}q}{(2\\pi )^4 }e^{-iwt+iqx} h_{\\mu \\nu }(q,w,u)\\nonumber \\\\\\mathcal {A_{\\mu }}(x,t,u)=\\int \\frac{d^{4}q}{(2\\pi )^4 }e^{-iwt+iqx}\\mathcal {A_{\\mu }}(q,w,u).$ We will work in the gauges where $h_{u\\mu }$ and $\\mathcal {A}_{u}$ are both zero for all $\\mu $ including $u$ .", "Based on the the spin of different metric perturbations under $SO(2)$ rotation in $(y,z)$ plane, the same can be classified into three types as follows: vector modes: $h_{x y}, h_{t y}\\ne 0$ or $h_{x z}, h_{t z}\\ne 0$ , with all other $h_{\\mu \\nu }=0$ .", "Scalar modes: $h_{x x}=h_{y y}=h_{z z}=h_{tt}\\ne 0$ , $h_{x t}\\ne 0$ , with all other $h_{\\mu \\nu }=0$ .", "Tensor modes: $h_{y z}\\ne 0$ , with all other $h_{\\mu \\nu }=0$ .", "The most important step to calculate the two point correlation function as discussed in section 2 is to solve the linearized equation of motion for the field in question.", "In this paper the EOMs for the scalar and vector type metric perturbations are all coupled to each other and hence they are not easy to solve.", "However following [5] one can construct a particular combination of different perturbations which is gauge invariant and all the coupled EOMs can be replaced by a single equation involving the gauge invariant variable.", "This combination which is invariant under diffeomorphisms: $h_{\\mu \\nu }\\rightarrow h_{\\mu \\nu }-\\nabla _{(\\mu }\\xi _{\\nu )}$ is given as [5]: ${\\rm Vector\\ type}: Z_v = q H_{ty} + w H_{xy}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $\\nonumber {\\rm Scalar\\ type}: Z_s=-q^2(1-u^4)H_{tt}+2wqH_{xt}+w^2H_{xx}~~~~~~~~~~~~~~~~~~\\\\~~~~~~~~~~~~~~~~~~~~~+q^2(1-u^4)\\left(1+\\frac{g_{xx}(-4u^3)}{g^{\\prime }_{xx}(1-u^4)}-\\frac{w^2}{q^2(1-u^4)}\\right)H_{yy}$ ${\\rm Tensor\\ type}: Z_t=H_{yz},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ where $H_{tt}=-g^{tt}h_{tt}$ , $H_{xx}=g^{xx}h_{xx}$ , $H_{yy}=g^{xx}h_{yy}$ , $H_{xt}=g^{xx}h_{xt}$ , $H_{xy}=g^{xx}h_{xy}$ .", "The two second order differential equations corresponding to the EOMs of $Z_v$ , $Z_s$ and $Z_t$ are solved and the required quasinormal modes are obtained by imposing Dirichlet boundary conditions at $u=0$ [5]." ], [ "The Local $T^3$ is a {{formula:00bda248-7bcf-4319-b88d-9cf7bac9085a}} -Invariant sLag", "In [7], we had shown that the local $T^3$ of [3] used for constructing a delocalized SYZ type IIA mirror of the type IIB string theory construct of [2] dual to large-$N$ thermal QCD, in the MQGP limit of [3], is the $T^2$ -invariant special Lagrangian (sLag) three-cycle of [6] of a deformed conifold.", "In this section, we show that the same $T^3$ is also a the $T^2$ -invariant sLag of [6] of a resolved conifold.", "The two results together, hence show the existence of a sLag in the MQGP limit in a predominantly resolved (resolution $>$ deformation) or deformed (deformation $>$ resolution) resolved warped deformed conifold of [2].", "From [6] we note that the following is a $T^2$ -invariant special Lagrangian three-cycle in a resolved conifold: $& & \\frac{K^\\prime }{2}\\left(|x|^2-|y|^2\\right) + 4 a^2\\frac{|\\lambda _2|^2}{|\\lambda _1|^2+|\\lambda _2|^2} = c_1,\\nonumber \\\\& & \\frac{K^\\prime }{2}\\left(|v|^2-|u|^2\\right) + 4 a^2\\frac{|\\lambda _2|^2}{|\\lambda _1|^2+|\\lambda _2|^2} = c_2,\\nonumber \\\\& & \\Im m\\left(xy\\right) = c_3,$ wherein one uses the following complex structure for a resolved conifold [30]: $x & =& \\left( 9 a^2 r^4 + r ^6 \\right) ^{1/4} e^{i/2(\\psi -\\phi _1-\\phi _2)}\\,\\sin \\frac{\\theta _1}{2}\\,\\sin \\frac{\\theta _2}{2} \\nonumber \\\\y & =& \\left( 9 a^2 r^4 + r ^6 \\right) ^{1/4} e^{i/2(\\psi +\\phi _1+\\phi _2)}\\,\\cos \\frac{\\theta _1}{2}\\,\\cos \\frac{\\theta _2}{2} \\nonumber \\\\u & =& \\left( 9 a^2 r^4 + r ^6 \\right) ^{1/4} e^{i/2(\\psi +\\phi _1-\\phi _2)}\\,\\cos \\frac{\\theta _1}{2}\\,\\sin \\frac{\\theta _2}{2} \\nonumber \\\\v & =& \\left( 9 a^2 r^4 + r ^6 \\right) ^{1/4} e^{i/2(\\psi -\\phi _1+\\phi _2)}\\,\\sin \\frac{\\theta _1}{2}\\,\\cos \\frac{\\theta _2}{2}\\,$ $[\\lambda _1:\\lambda _2]$ being the homogeneous coordinates of the blown-up $\\mathbb {CP}^1=S^2$ ; $\\frac{\\lambda _2}{\\lambda _1}=\\frac{x}{-u}=\\frac{v}{-y}=-e^{-i\\phi _1}\\tan \\frac{\\theta _1}{2}$ .", "In (REF ), $\\gamma (r^2)\\equiv r^2 K^\\prime (r^2)= - 2 a^2 + 4 a^4 N^{-\\frac{1}{3}}(r^2) + N^{\\frac{1}{3}}(r^2)$ , where $N(r^2)\\equiv \\frac{1}{2}\\left(r^4 - 16 a^6 + \\sqrt{r^8 - 32 a^6 r^4}\\right)$ .", "Defining $\\rho \\equiv \\sqrt{\\frac{3}{2}}\\sqrt{\\gamma }$ , upon inversion yields: $r\\approx \\left(\\frac{2}{3}\\right)^{\\frac{3}{4}}\\left(3 a^2 + \\rho ^2\\right)^{\\frac{3}{4}}$ and $K^\\prime (r^2)=\\frac{\\sqrt{\\frac{3}{2}}\\rho ^2}{\\left(3 a^2 + \\rho ^2\\right)^{\\frac{3}{2}}}$ .", "The system of equations (REF ) are solved in appendix A to yield (REF ) - (REF ), which provides an embedding $\\rho =\\rho (\\psi )$ and hence $\\theta _{1,2}=\\theta _{1,2}(\\psi )$ .", "As $\\theta _1, \\theta _2\\rightarrow 0$ as $\\frac{1}{N^{\\frac{1}{5}}},\\frac{1}{N^{\\frac{3}{10}}}$ (whereat an explicit local $SU(3)$ -structure of the type IIA mirror and an explicit local $G_2$ -structure of the M-theory uplift was obtained in [22]) and in the UV-IR interpolating region/UV: $r\\rightarrow {\\cal R}_0 > r_0$ , therefore in this domain of $(\\theta _1,\\theta _2,r)$ choose $c_{1,2}$ as given in (REF ).", "The large-$N$ [as given in (REF )] small-$\\psi $ [as given in (REF ) for an appropriate $\\psi =\\langle \\psi \\rangle $ determined by (REF ) which is solved to yield (REF )] expansion, is discussed in appendix A.", "Hence, using (REF ) of appendix A, the embedding of $\\theta _2$ near $(\\theta _1\\sim \\frac{1}{N^{\\frac{1}{5}}},\\theta _2\\sim \\frac{1}{N^{\\frac{3}{10}}})$ , and in the UV-IR interpolating region/UV: $r={\\cal R}_0\\gg a, r_h)$ is given as: $& & \\theta _2\\approx \\sqrt{\\frac{0.324 {\\rho _0}^2 {\\sin \\psi }^{2/3}}{{c_3}^{2/3}}-\\frac{0.096 {\\rho _0}^6 {\\sin \\psi }^2}{{c_3}^2}+0.5}\\nonumber \\\\& & +\\frac{a^2 \\left(0.193{c_3}^{4/3} {\\rho _0}^2 {\\sin \\psi }^{4/3}+0.096 {\\rho _0}^6 {\\sin \\psi }^{8/3}\\right)}{{c_3}^{8/3} \\sqrt{\\frac{0.324 {\\rho _0}^2{\\sin \\psi }^{2/3}}{{c_3}^{2/3}}-\\frac{0.096 {\\rho _0}^6 {\\sin \\psi }^2}{{c_3}^2}+0.5}} + {\\cal O}\\left(a^3\\right).$ Similarly, using (REF ) of appendix A, yields: $& &\\hspace{-43.36243pt}\\theta _1\\approx \\frac{a^2 \\left(-2.89 {c_3}^{2/3} {\\rho _0}^2 {\\sin \\psi }^{4/3}-8.25 {c_3}^{4/3} {\\sin \\psi }^{2/3}+2.31 {\\rho _0}^4{\\sin \\psi }^2\\right)}{{c_3}^2 \\sqrt{\\frac{52.49 {\\rho _0}^2 {\\sin \\psi }^{2/3}}{{c_3}^{2/3}}-\\frac{14.6969 {\\rho _0}^6{\\sin \\psi }^2}{{c_3}^2}+81.", "}}\\nonumber \\\\& & \\hspace{-43.36243pt}+\\sqrt{\\frac{0.32 {\\rho _0}^2 {\\sin \\psi }^{2/3}}{{c_3}^{2/3}}-\\frac{0.091 {\\rho _0}^6{\\sin \\psi }^2}{{c_3}^2}+0.5}+O\\left(a^3\\right).$ As is evident from equations (REF ) and (REF ) of appendix A, the numerical factors are rather cumbersome to be retained as such if one is interested in eventually numerically verifying that (REF ) indeed satisfies (REF ).", "This is the reason why the exact numerical factors in (REF ) and (REF ) of appendix A have been replaced by corresponding decimals to arrive at(REF ) and (REF ).", "It is for the same reason that decimals also appear in (REF ) - (REF ).", "The fundamental two-form is found to be [30] $J &=& -\\frac{\\rho }{3}\\,d\\rho \\wedge (d\\psi +\\cos \\theta _1\\,d\\phi _1+\\cos \\theta _2\\,d\\phi _2)\\nonumber \\\\& & -\\frac{\\rho ^2}{6}\\,\\sin \\theta _1\\,d\\phi _1\\wedge d\\theta _1- \\frac{\\rho ^2+6a^2}{6}\\, \\sin \\theta _2\\,d\\phi _2\\wedge d\\theta _2$ and is closed, and the holomorphic three-form is [30]: $& & \\hspace{-36.135pt}\\Omega = \\frac{\\rho (\\rho ^2+6a^2)}{6\\sqrt{\\rho ^2+9a^2}}\\,(\\cos \\psi -i\\sin \\psi )\\,d\\rho \\wedge \\Big [\\sin \\theta _1\\,d\\theta _2\\wedge d\\phi _1 - \\sin \\theta _2\\,d\\theta _1\\wedge d\\phi _2+i(d\\theta _1\\wedge d\\theta _2-\\sin \\theta _1\\sin \\theta _2\\,d\\phi _1\\wedge d\\phi _2)\\Big ] \\nonumber \\\\&& \\hspace{-36.135pt}+ \\, \\frac{\\rho ^2}{18}\\,\\sqrt{\\rho ^2+9a^2}\\,(\\cos \\psi -i\\sin \\psi )\\Big [d\\theta _1\\wedge d\\theta _2\\wedge (d\\psi +\\cos \\theta _1\\,d\\phi _1+\\cos \\theta _2\\,d\\phi _2) -\\sin \\theta _1\\sin \\theta _2\\,d\\phi _1\\wedge d\\phi _2\\wedge d\\psi \\nonumber \\\\& & \\hspace{-36.135pt}\\qquad -i(\\sin \\theta _1\\,d\\theta _2\\wedge d\\phi _1 - \\sin \\theta _2\\,d\\theta _1\\wedge d\\phi _2)\\wedge d\\psi -i(\\sin \\theta _1\\cos \\theta _2\\,d\\theta _2+\\cos \\theta _1\\sin \\theta _2\\,d\\theta _1)\\wedge d\\phi _1\\wedge d\\phi _2\\Big ]\\,.$ So, writing $c_1=\\alpha _{c_1}\\rho _0^2,$ and to LO in $N,\\ d\\phi _1=\\frac{\\beta _{\\phi _1}dx}{\\left(g_sN\\right)^{\\frac{1}{4}}\\frac{1}{N^{\\frac{1}{5}}}},\\ d\\phi _2=\\frac{\\beta _{\\phi _2}dy}{\\left(g_sN\\right)^{\\frac{1}{4}}\\frac{1}{N^{\\frac{3}{10}}}},\\ d\\psi =\\frac{\\beta _{\\psi }dz}{\\left(g_sN\\right)^{\\frac{1}{4}}}$ [3], one obtains (REF ) of Appendix A.", "This implies: $& & i^*J\\approx \\left(\\frac{0.07\\alpha _{c_1}^2\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _1}}{\\sqrt{g_s}\\sin \\psi N^{\\frac{3}{10}}} - \\frac{0.007\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _1}}{\\sin \\psi \\sqrt{g_s}N^{\\frac{3}{10}}} - \\frac{0.84\\beta _{\\psi }\\beta _{\\phi _1}}{N^{\\frac{3}{10}}\\sqrt{g_s}\\sin ^{\\frac{5}{3}}\\psi }\\right)dz\\wedge dx\\nonumber \\\\& & + \\left(\\frac{0.07\\alpha _{c_1}^2\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _2}}{\\sqrt{g_s}\\sin \\psi N^{\\frac{3}{10}}} - \\frac{0.03\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _2}}{\\sin \\psi \\sqrt{g_s}} - \\frac{0.84\\beta _{\\psi }\\beta _{\\phi _2}}{N^{\\frac{3}{10}}\\sqrt{g_s}\\sin ^{\\frac{5}{3}}\\psi }\\right)dz\\wedge dy.$ Choosing: $\\sin ^{\\frac{2}{3}}\\langle \\psi \\rangle = \\frac{\\alpha }{\\rho _0^2}$ and $\\alpha _{c_1}: \\frac{0.07\\alpha _{c_1}^2}{3} - 0.03 - \\frac{0.84}{\\alpha } = 0,$ i.e., $\\alpha _{c_1}=6.54\\sqrt{0.03+\\frac{0.84}{\\alpha }},$ one obtains: $i^*J\\sim 0.02\\frac{\\beta _{\\psi }\\beta _{\\phi _1}}{N^{\\frac{3}{10}}}dz\\wedge dx\\approx 0.$ Further, the three-forms relevant to evaluation of $i^*\\Omega $ - using (REF ) - are collected in (REF ).", "Using (REF ), choose $\\alpha $ : $\\frac{1}{6}\\left(\\frac{2\\alpha _{c_1}^2}{3^3} - \\frac{2^{\\frac{4}{3}}}{3\\alpha }\\right) +\\frac{1}{36}\\left( N^{\\frac{1}{10}}0.21 + \\frac{2^{\\frac{1}{6}}}{3^{\\frac{3}{2}}N^{\\frac{1}{10}}}\\right)\\ll 1,$ i.e., $0.016 +\\frac{0.304}{\\alpha }+0.006 \\@root 10 \\of {N}+\\frac{0.006}{\\@root 10 \\of {N}}\\ll 1.$ Now, (REF ) will be satisfied by any $\\alpha \\gg 1$ for a reasonably large $N$ but less than $10^{10}$ .", "Hence, $& & i^*J\\approx 0;\\nonumber \\\\& & \\Im m (i^*\\Omega )=0,\\nonumber \\\\& & \\Re e (i^*\\Omega )\\sim {\\rm vol}\\left(T^3\\right),$ implying thus: $\\left.i^* J\\right|_{RC/DC}\\approx 0, \\Im m\\left.\\left( i^*\\Omega \\right)\\right|_{RC/DC} \\approx 0, \\Re e\\left.\\left(i^*\\Omega \\right)\\right|_{RC/DC}\\sim {\\rm volume \\ form}\\left(T^3(x,y,z)\\right)$ .", "Hence, if the resolved warped deformed conifold is predominantly either resolved (resolution $>$ deformation) or deformed (deformation $>$ resolution), the local $T^3$ of (REF ) is the required sLag to effect the delocalized SYZ mirror of the type IIB background of [2], carried out in [7]." ], [ "Thermal (Electrical) Conductivity, Deviation from the Wiedemann-Franz Law and $D=1+1$ Luttinger Liquids up to LO in {{formula:cbd1a363-d487-422d-a99f-00c97a332f0a}}", "In this section we compute the temperature dependance of thermal ($\\kappa _T$ ) and electrical ($\\sigma $ ) conductivities for a gauge theory at finite temperature and density, and hence explore deviation from the Wiedemann-Franz law.", "Remarkably, we find that the results qualitatively mimic those of a $D=1+1$ Luttinger liquid with impurities.", "Finite temperature in the gauge theory is effected by placing a black hole in the dual bulk gravitational background.", "To get the finite density in the boundary we consider the theory at non-zero chemical potential.", "The embedding of $N_f$ $D7$ branes in the background geometry introduces $N_f$ no of flavor fields, all in fundamental representation of the gauge group $U(N_f)$ .", "The $U(1)_B$ subgroup of $U(N_f)$ is identified as the baryon number.", "Hence the $D7$ brane puts the boundary field theory at finite baryon density or equivalently at finite chemical potential $\\mu _C$ .", "In the supergravity description one have a $U(1)$ gauge field $A_{\\mu }$ in the worldvolume of the $D7$ brane, dual to the current operator $j_{\\mu }$ in the boundary.", "The nonzero time component $A_t$ of the gauge field has to be turned on to get a finite baryon density $<j_t>$ in the field theory.", "Here we will consider the $5d$ Einstein-Hilbert action and the $D7$ brane DBI action together, of course after integrating over the three angular directions of the later.", "The $D7$ brane DBI action in presence of a $U(1)$ gauge field is given as: $S_{D7}= T_{D7}\\int d^8\\xi e^{-\\Phi }\\sqrt{-\\det (g+B+\\hat{F})}$ where $g$ is the induced metric on $D7$ brane and $\\hat{F}$ is the gauge field strength with the only nonzero component given by $\\hat{F_{rt}}=\\frac{ce^{\\Phi }}{\\sqrt{c^2e^{2\\Phi }+r^{9/2}}}$ [22], where $\\Phi $ is the dilaton and $c$ is a constant.", "Now the finite chemical potential or equivalently the finite charge density will mix the heat (energy) current and the electric current together.", "According to the AdS/CFT correspondence for every operator in the boundary field theory, there is a bulk field in the dual gravity theory.", "Heat current is sourced by the energy momentum tensor $T_{\\mu \\nu }$ in the boundary and the corresponding field in the gravitational description is the bulk metric $g_{\\mu \\nu }$ .", "Similarly, as already mentioned, the electric current sourced by the current operator $j_{\\mu }$ corresponds to the $U(1)$ bulk gauge field $A_{\\mu }$ .", "Hence for the computation of thermal conductivity we consider the following linear fluctuations of both the background metric $g^{(0)}_{\\mu \\nu }$ and the gauge field $A^{(0)}_{\\mu }$ as, $g_{\\mu \\nu }=g^{(0)}_{\\mu \\nu }+h_{\\mu \\nu } ~~~~~~~~~~~~~~~~ A_{\\mu }=A^{(0)}_{\\mu }+\\mathcal {A_{\\mu }},$ where $h_{\\mu \\nu }$ and $\\mathcal {A_{\\mu }}$ represents the metric and the gauge field fluctuations respectively.", "Considering the $y$ -component of the gauge field as the only perturbation, it can be shown that only the $(ty)$ and the $(xy)$ component of the metric gets perturbed.", "Assuming that the above perturbations depends only on the radial coordinate $u$ , time $t$ and spatial coordinate $x$ , can be decomposed as the following way, $h_{ty}=g^{(0)}_{xx}H_{ty}(u)e^{-iwt+iqx} ~~~~~~~~~~~~~ h_{xy}=g^{(0)}_{xx}H_{xy}(u)e^{-iwt+iqx} ~~~~~~~~~~~~~ \\mathcal {A}_{y}=\\phi (u)e^{-iwt+iqx}$ Now including the above fluctuations in the DBI action, we perform the three angular integrations on $\\phi _1$ , $\\phi _2$ and $\\theta _2$ .", "The integration over two of the three angular variable namely $\\phi _1$ and $\\phi _2$ gives constant factors.", "To perform the $\\theta _2$ integration, we first expand the DBI action in (REF ) (Taking into account the fluctuations) up to quadratic order in fluctuating fields to get, $\\hspace{-28.90755pt}\\sqrt{-\\det (g+h+B+\\hat{F}+F)}=\\sqrt{a}_1 \\left(1 + \\frac{a_2 H^2_{ty}(u) + a_3 H_{ty}(u) \\phi ^{^{\\prime }}(u) + a_4 H^2_{xy}(u) + a_5 \\phi ^{2}(u) + a_6 \\phi ^{^{\\prime }}(u)^2}{2 a_1}\\right).\\nonumber \\\\& &$ where $h$ and $F$ represents the fluctuations of the two fields.", "The coefficients $a_1,a_2,a_3,a_4,a_5,a_6$ are given as, $& & a_1 = -\\frac{\\cot ^2(\\frac{\\theta _2}{2}) \\csc ^4(\\frac{\\theta _2}{2})}{1296 \\left(c^2 e^{2 \\Phi }+r^{9/2}\\right)}r^{9/2}\\left\\lbrace \\Biggl (r^3+2 (5 \\mu _{\\rm Ouyang} ^2-2 r^3) \\cos \\theta _2+14 \\mu _{\\rm Ouyang}^2+3 r^3 \\cos 2 \\theta _2\\Biggr )\\right\\rbrace \\nonumber \\\\& & \\times \\Biggl \\lbrace \\left(8 \\mu _{\\rm Ouyang}^2-4 r^3\\right) \\cos \\theta _2+r^3 (\\cos 2\\theta _2+3)\\Biggr \\rbrace \\nonumber \\\\& & a_2 = a_1\\frac{r^{-\\frac{1}{2}}\\left(c^2 e^{2 \\Phi }+r^{9/2}\\right)}{\\left(r^4-r_h^4\\right)}e^{2i(qx-tw)}\\nonumber \\\\& & a_3 = 2a_1\\frac{ ce^{\\Phi }\\sqrt{\\left(c^2 e^{2 \\Phi }+r^{9/2}\\right)}}{r^{\\frac{9}{2}}}e^{2i(qx-tw)}\\nonumber \\\\& & a_4 = a_1e^{2i(qx-tw)}\\nonumber \\\\& & a_5 =4 a_1\\frac{(g_s N\\pi )}{r^{\\frac{9}{2}}\\left(r^4-r_h^4\\right)} \\left\\lbrace iw^2 c^2 e^{2 \\Phi }+\\sqrt{r} \\left(iq^2 \\left(r_h^4-r^4\\right)+iw^2r^4\\right)\\right\\rbrace e^{2i(qx-tw)}\\nonumber \\\\& & a_6 = a_1\\frac{\\left(c^2 e^{2 \\Phi }+r^{9/2}\\right)}{r^{\\frac{9}{2}}}\\left(1 - \\frac{r_h^4}{r^4}\\right)e^{2i(qx-tw)},$ where the coordinate $r$ is related to $u$ as $u=\\frac{r_h}{r}$ .", "Upon changing the variable from $r$ to $u$ to the above mentioned variables one see that the coefficients $a_2,a_3,a_4,a_5,a_6$ each after the division by $a_1$ are independent of $\\theta _2$ and only depends on $u$ .", "The integration of $\\sqrt{a_1}$ over $\\theta _2$ gives some function of $u$ say $\\mathcal {M}(u)$ given by, $\\mathcal {M}(u)=\\sqrt{\\mu _{\\rm Ouyang}}\\left(\\frac{r_h}{u}\\right)^{9/2}\\sqrt{\\frac{1}{c^2e^{2\\Phi }+\\left(\\frac{r_h}{u}\\right)^{9/2}}}$ In this way reducing the dimension from eight to five the DBI action takes the following form: $& & \\hspace{-21.68121pt}S_{D7}=\\left(\\frac{a_{DBI}T_{D_7}}{g_s}\\right)\\int du~ d^4x~ \\mathcal {M}(u)\\left(1+\\frac{a_2 H^2_{ty}(u) + a_3 H_{ty}(u) \\phi ^{^{\\prime }}(u) + a_4 H^2_{xy}(u) + a_5 \\phi ^{2}(u) + a_6 \\phi ^{^{\\prime }}(u)^2}{2 a_1}\\right)\\nonumber \\\\& &$ where $a_{DBI}$ includes all the constant terms resulting after the angular integrations; $T_{D7}$ is the tension on the $D7$ brane.", "We will henceforth be working in a hydrodynamical approximation wherein we will approximate the plane-wave exponentials by unity.", "Finally taking into account the Einstein-Hilbert action given as, $S_{EH}=a_{EH}\\int du d^4x\\sqrt{-g_{(5)}}(R-2\\lambda ),$ where $g_{(5)}$ is the determinant of the $5d$ metric, the total action is given by: $S_{tot}=S_{\\rm EH}+S_{D7}$ .", "The type $IIB$ metric satisfying the above action $S_{tot}$ has the form: $ds^2=g_{tt}dt^2+g_{xx}(dx^2+dy^2+dz^2)+g_{uu}du^2,$ where the different background metric components, in the UV (as the gauge fluctuation will be solved for, near the UV $u=0$ ) and to LO in $N$ , are given as, $\\nonumber g_{tt}=\\frac{\\left(u^4-1\\right)r_h^2}{2 u^2 \\sqrt{\\pi g_s N }}\\\\\\nonumber g_{xx}=g_{yy}=g_{zz}=\\frac{r_h^2}{2 u^2 \\sqrt{\\pi g_s N }}\\\\g_{uu}=\\frac{2 u^2 \\sqrt{\\pi g_s N }}{\\left(1-u^4\\right)r_h^2}.$ Now from the total action defined above, we can write down the EOMs in the hydrodynamical limit for $H_{ty}$ , $H_{xy}$  and $\\phi $ and they are given as: $H_{ty}(u)$ EOM: $& & a_{EH}\\sqrt{-g_{(5)}}\\left(\\mathcal {R}^{(1)}_{ty}-\\frac{2}{3}\\lambda g_{xx}H_{ty}(u)\\right)+\\left(\\frac{a_{DBI}T_{D_7}}{g_s}\\right)\\mathcal {M}(u)\\left(\\frac{a_2(u)}{a_1(u)}H_{ty}(u)+ \\frac{a_3(u)}{2a_1(u)}\\phi ^{^{\\prime }}(u)\\right)=0;\\nonumber \\\\& &$ $H_{xy}(u)$ EOM: $& & a_{EH}\\sqrt{-g_{(5)}}\\left(\\mathcal {R}^{(1)}_{xy}-\\frac{2}{3}\\lambda g_{xx}H_{xy}(u)\\right)+\\left(\\frac{a_{DBI}T_{D_7}}{g_s}\\right)\\mathcal {M}(u)\\left(\\frac{a_4}{a_1(u)}H_{xy}(u)\\right)=0;\\nonumber \\\\& &$ $\\phi (u)$ EOM: $& & \\hspace{-21.68121pt}\\frac{d}{du}\\left(\\frac{a_3(u)}{2 a_1(u)}\\mathcal {M}(u)H_{ty}(u)\\right) + \\frac{d}{du}\\left(\\frac{a_6(u)}{a_1(u)}\\mathcal {M}(u)\\right)\\phi ^{^{\\prime }}(u)+\\left(\\frac{a_6(u)}{a_1(u)}\\mathcal {M}(u)\\right)\\phi ^{^{\\prime \\prime }}(u)-\\left(\\frac{a_5(u)}{a_1(u)}\\mathcal {M}(u)\\right)\\phi (u)=0,\\nonumber \\\\& &$ where $\\mathcal {R}^{(1)}_{\\mu \\nu }$ is the linear ordered perturbation of the Ricci tensor.", "Now substituting the exact form of $\\mathcal {M}(u)$ as well as all of the six coefficients $a_1(u), a_2(u), a_3(u), a_4, a_5(u), a_6(u)$ , the above three equations regarding $H_{ty},H_{xy}$ and $\\phi $ can be rewritten as (REF ) - (REF ) in appendix B which also contains their solutions.", "As the pre-factor multiplying $\\frac{\\phi ^{\\prime }(u)}{\\phi (u)}$ from (REF ), the $A(u)$ in (REF ) - the coefficient of the kinetic term of $\\phi (u)$ - that will appear in the current-current correlator is $\\left(\\frac{\\sqrt{\\mu }r_h^{\\frac{13}{4}}}{72u^{\\frac{17}{4}}}\\right)$ , to obtain a finite $\\left\\lbrace \\lim _{{u}\\rightarrow 0}\\frac{1}{u^{\\frac{17}{4}}}\\frac{\\phi ^{\\prime }(u)}{\\phi (u)}\\right\\rbrace $ , one needs $\\phi (u)\\sim e^{{\\rm constant}\\ u^{\\frac{21}{4}}}$ .", "Expanding (REF ) about $u=0$ : $\\phi (u\\sim 0;q=0) & = & \\frac{\\left(i g_s N \\pi \\right)^{7/8}\\omega ^{7/4}c_2^\\Phi u^{\\frac{21}{4}}}{33^{\\frac{3}{4}}r_h^{\\frac{7}{2}}} -\\frac{4cg_sc_1^\\Phi \\Gamma (\\frac{13}{24})u^{\\frac{13}{4}}}{63r_h^{\\frac{9}{4}}\\Gamma (\\frac{37}{24})}+\\frac{4cg_sc_1^\\Phi \\Gamma (-\\frac{1}{3})u^{\\frac{13}{4}}}{63r_h^{\\frac{9}{4}}\\Gamma (\\frac{2}{3})}\\nonumber \\\\& & + c_1^\\Phi + \\frac{c}{r_h^{\\frac{9}{4}}}{\\cal O}(u^6).$ Now, in terms of a dimensionless ratio: $\\kappa \\equiv \\frac{C}{r_h^{\\frac{9}{4}}}$ and choosing $C$ to be $m_{\\rho }$ and $r_h$ in units of GeV implying $\\kappa \\ll 1$ [22].", "Therefore, $& & \\phi (u\\sim 0;q=0)=\\frac{(0.08+0.39\\ i)\\left(g_s N\\right)^{\\frac{7}{8}}w^{\\frac{7}{4}}c_2^\\Phi u^{\\frac{21}{4}}}{r_h^{\\frac{7}{2}}}+c_1^\\Phi +\\frac{c}{r_h^{\\frac{9}{4}}}{\\cal O}(u^6)\\nonumber \\\\& & \\approx c_1^\\Phi e^{\\frac{(0.08+0.39\\ i)g_s^{\\frac{7}{8}}N^{\\frac{7}{8}}w^{\\frac{7}{4}}c_2^\\Phi u^{\\frac{21}{4}}}{c_1r_h^{\\frac{7}{2}}}}.$ Analogous to (the reason given in) Sec.", "3, in (REF ), we use decimals.", "At this point we require to calculate some of the thermodynamic parameters like pressure, energy density, entropy density etc.", "In particular, pressure and energy density follows from the thermodynamic relations as given by $s=\\frac{\\partial P}{\\partial T}$ and $\\epsilon =-P+Ts+\\mu _{C}n_q$ , where $s$ is called the entropy density and is given as, $& & s = \\mathcal {O}(1)r_h^3 = \\mathcal {O}(1)\\pi ^3\\left(4\\pi g_s N\\right)^{3/2}T^3.$ Now the density of Gibbs potential $\\Omega $ which is equal to the pressure with a minus sign can be used to find the charge density $n_q$ using the relation $n_q=\\frac{\\partial \\Omega }{\\partial \\mu _C}$ , where $\\mu _c$ being the chemical potential is given by $& & \\mu _C = \\frac{\\left(2 {\\kappa } {g_s}\\right)^{\\frac{4}{9}} {r_h} \\Gamma \\left(\\frac{5}{18}\\right) \\Gamma \\left(\\frac{11}{9}\\right)}{\\@root 18 \\of {\\pi } (2\\pi -{g_s} {N_f} \\log |\\mu _{\\rm Ouyang}|)^{4/9}}-{r_h}\\ _2F_1 \\left(\\frac{11}{9};-\\frac{(2 \\pi -{g_s} {N_f} \\log |\\mu _{\\rm Ouyang}| )^2}{4{\\kappa }^2 {g_s}^2 \\pi ^2}\\right)\\nonumber \\\\& & = \\frac{36 \\pi {\\kappa } {g_s} {r_h} \\Gamma \\left(\\frac{11}{9}\\right)}{5 \\Gamma \\left(\\frac{2}{9}\\right) (2 \\pi -{g_s} {N_f}\\log |\\mu _{\\rm Ouyang}| )} + {\\cal O}\\left(\\kappa ^{\\frac{19}{9}}\\right),$ from which we get $T=\\left(\\frac{8}{5}\\right)^{4/5}\\left(\\frac{g_s^{\\frac{3}{10}}C^{\\frac{4}{5}}}{\\left(2\\pi - g_s N_f \\log |\\mu _{\\rm Ouyang}|\\right)^{4/5}}\\right)\\left(\\frac{\\mu ^{-\\frac{4}{5}}_C}{2\\pi ^{\\frac{7}{10}}\\sqrt{N}}\\right).$ Substituting the above result for $T$ in the expression for Gibbs potential and differentiating w.r.t $\\mu _C$ we get charge density as $n_q=\\left(\\frac{8}{5}\\right)^{\\frac{16}{5}}\\left(\\frac{2}{5}\\right)\\left(\\frac{g_s^{\\frac{27}{10}}\\pi ^{\\frac{17}{10}}C^{\\frac{16}{5}}\\mu _C^{-\\frac{21}{5}}}{\\sqrt{N}\\left(2\\pi - g_s N_f \\log |\\mu _{\\rm Ouyang}|\\right)^{16/5}}\\right).$ Hence, $& & \\hspace{-21.68121pt}\\kappa _T = \\frac{\\left(\\epsilon +P\\right)^2\\sigma }{n_q^2T} = -\\left(\\frac{\\epsilon +P}{n_q}\\right)^2\\left(\\frac{\\sqrt{|\\mu _{\\rm Ouyang}|}r_h^{\\frac{13}{4}}}{72 T u^{\\frac{17}{4}}}\\right)\\lim _{{\\omega }\\rightarrow 0}\\frac{1}{{\\omega }}\\Im m\\left.\\frac{\\phi ^\\prime (u)}{\\phi (u)}\\right|_{u=0}\\nonumber \\\\&& \\hspace{-21.68121pt}= \\frac{9}{200\\sqrt{2}}\\frac{g_s^{\\frac{3}{4}}C^2}{ N^{\\frac{5}{4}}\\pi ^{\\frac{7}{4}}\\left(2\\pi - g_s N_f \\log |\\mu _{\\rm Ouyang}|\\right)^2T^{\\frac{7}{2}}}\\sqrt{|\\mu _{\\rm Ouyang}|}\\left(T\\pi \\sqrt{4\\pi g_s N}\\right)^{13/4}\\lim _{{\\omega }\\rightarrow 0}\\frac{(0.39i)g_s^{7/8}N^{7/8}{\\omega }^{3/4}c_2^\\Phi }{\\left(T\\pi \\sqrt{4\\pi g_s N}\\right)^{7/2}},\\nonumber \\\\& &$ which for $c_2^\\Phi \\sim -i {\\omega }^{-\\frac{3}{4}}$ implies: $& & \\sigma =(0.39)\\frac{\\sqrt{|\\mu _{\\rm Ouyang}|}\\left(g_s N\\right)^{\\frac{3}{4}}}{2^{\\frac{1}{4}}T^{\\frac{1}{4}}\\pi ^{\\frac{3}{8}}},\\nonumber \\\\& & \\kappa _T = \\frac{9\\times 0.39}{200\\times 2^{3/4}}\\frac{\\sqrt{|\\mu _{\\rm Ouyang}|}g_s^{\\frac{3}{2}}C^2}{\\sqrt{N}\\pi ^{\\frac{17}{8}}T^{\\frac{15}{4}}\\left(2\\pi - g_s N_f \\log |\\mu _{\\rm Ouyang}|\\right)^2};\\nonumber \\\\& & {\\rm Wiedemann-Franz\\ law}: \\frac{\\kappa _T}{\\sigma T}=\\frac{9}{200\\sqrt{2}}\\frac{g_s^{\\frac{3}{4}}C^2}{N^{\\frac{5}{4}}\\pi ^{\\frac{7}{2}}\\left(2\\pi - g_s N_f \\log |\\mu _{\\rm Ouyang}|\\right)^2T^{\\frac{9}{2}}}.$ (a) Assuming the Ouyang embedding parameter to depend on the temperature via the horizon radius as $|\\mu _{\\rm Ouyang}|\\sim r_h^\\alpha ,\\ \\alpha \\le 0$ .", "Then, the temperature dependence of $\\sigma , \\kappa _T$ and the temperature dependences of the Wiedemann-Franz law in (REF ), upon comparison with Table 2 of [4], qualitatively mimic a $D=1+1$ Luttinger liquid with impurities/doping (close to `$\\frac{1}{3}$ -filling') in the following sense.", "With $v_i, K_i, i=c$ (harge), $s$ (pin) being the parameters appearing in the Luttinger liquid Hamiltonian as $\\sum _{i=c,s}v_i\\left[K_i \\left(\\partial _x\\theta _i\\right)^2 + \\frac{1}{K_i}\\left(\\partial _x\\phi _i\\right)^2\\right]$ wherein the spin ($s$ ) and charge ($c$ ) densities are $\\phi _{s,c}$ and their canonically conjugate fields are $\\partial _x\\theta _i$ , $n_s=0,1$ for even and odd $n_c$ respectively where $n_{c,s}$ along with $g,a$ appear in the Umklapp scattering Hamiltonian $\\frac{g}{\\left(2\\pi a\\right)^{n_c}}\\int \\left(e^{i\\sqrt{2}\\left(n_c\\phi _c + n_s\\phi _s\\right) - i \\Delta k x} + {\\rm h.c.}\\right)$ , $D$ as a parameter appearing in the two-point correlation function of the impurity field $\\eta (x)$ via $\\langle \\eta (x)\\eta (x^\\prime )\\rangle = D \\delta (x-x^\\prime )$ with $\\eta (x)$ appearing in the back-scattering Hamiltonian due to disorder $\\frac{1}{\\pi a}\\int dx \\eta (x) \\left[e^{i\\sqrt{2}\\phi _c}\\cos \\left(\\sqrt{2}\\phi _s\\right) + {\\rm h.c.}\\right]$ , the authors of [4] define the following dimensionless parameters: $\\tilde{D} \\equiv \\frac{\\rm Impurity\\ scattering\\ rate}{\\rm Umklapp\\ scattering\\ rate}=\\frac{D a^{2n_c-3}}{g^2\\left(\\frac{a T}{v_c}\\right)^\\gamma }, \\tilde{\\delta }\\equiv \\frac{\\delta }{\\tilde{D}^{\\frac{1}{\\gamma }}}$ where $\\delta \\equiv \\frac{v_c\\Delta k}{\\pi T}, \\gamma \\equiv (n_c^2-1)K_c + (n_s^2-1)K_s - 1$ and dimensionless temperature: $\\tilde{T}\\equiv \\frac{T}{T_D}$ where $T_D\\equiv \\frac{v_c}{a}\\left(\\frac{D a^{2n_c-3}}{g^2}\\right)^{\\frac{1}{\\gamma }}$ .", "One then notes that for $\\tilde{\\delta }=10, 20$ and for $T>T_D$ , $\\frac{d\\sigma }{dT}, \\frac{d\\kappa _T}{dT}, \\frac{d\\left(\\frac{\\kappa _T}{T \\sigma }\\right)}{dT} < 0$ which is also reflected in (REF ).", "In $\\alpha ^\\prime =1$ -units $[T] = [C^{\\frac{4}{9}}]$ , where $[..]$ denotes that canonical dimension.", "To ensure a constant finite value of $\\frac{\\kappa _T}{T\\sigma }$ for small temperatures as per [4], we assume, in the MQGP limit, for $T : \\frac{T}{C^{\\frac{4}{9}}}<1$ , i.e., $T\\sim C^{\\frac{4}{9}}\\epsilon ^{\\alpha _T>0}, 0<\\epsilon <1$ and $N\\sim \\beta _N\\epsilon ^{-\\alpha _N}$ , so that if $0<\\frac{9\\alpha _T}{2} - \\frac{5\\alpha _N}{4}\\ll 1$ then $\\lim _{T\\rightarrow 0}\\left(\\frac{\\kappa _T}{T\\sigma }\\sim \\frac{g_s^{\\frac{3}{4}}C^2}{N^{\\frac{5}{4}}T^{\\frac{9}{2}}\\left(2\\pi - g_s N_f\\log \\mu \\right)^2}\\right)\\sim \\frac{g_s^{\\frac{3}{4}}}{\\epsilon ^{-\\frac{5\\alpha _N}{4} + \\frac{9\\alpha _T}{2}}\\left(2\\pi - g_sN_f\\left\\lbrace \\frac{\\alpha _N}{4} + \\alpha _T\\right\\rbrace \\log \\epsilon + \\frac{g_sN_f}{4}\\log (\\beta _Ng_s)\\right)^2}\\ne 0$ .", "(b) For $\\alpha $ (figuring in $|\\mu _{\\rm Ouyang}|\\sim r_h^\\alpha $ )$>0$ , interestingly for the specific choice of $\\alpha =\\frac{5}{2}$ one reproduces the large-$T$ (as $T>C^{\\frac{4}{9}}=m_\\rho =760 MeV$ ([22])$>T_c=175 MeV$ , is considered large) linear behavior of DC electrical conductivity $\\sigma \\sim T$ characteristic of most strongly coupled gauge theories with a five-dimensional gravity dual with a black hole [13].", "As $\\frac{C^2}{T^{\\frac{9}{2}}}$ is dimensionless, this yields dimensionally $\\kappa _T\\sim ({\\rm temperature})^2$ , though $\\kappa _T\\sim T^{\\frac{5}{2}}$ in the aforementioned large-$T$ limit." ], [ "Scalar Metric Perturbation Modes and Speed of Sound in MQGP Limit", "In this section, by considering scalar modes of metric perturbations, we will evaluate the speed of sound, first up to leading order in $N$ four ways: (i) (subsection 5.1.1) the poles appearing in the common denominator of the solutions to the individual scalar modes of metric perturbations (the pure gauge solutions and the incoming-wave solutions); (ii) (subsection 5.1.2) the poles appearing in the coefficient of the asymptotic value of the square of the time-time component of the scalar metric perturbation in the on-shell surface action; (iii) (subsection 5.2.1) the dispersion relation obtained via a Dirichlet boundary condition imposed on an appropriate single gauge-invariant metric perturbation - using the prescription of [5] - at the asymptotic boundary; (iv) (subsection 5.2.2) the poles appearing in the coefficient of the asymptotic value of the square of the time-time component of the scalar metric perturbation in the on-shell surface action written out in terms of the same single gauge-invariant metric perturbation.", "The third approach is then extended to include the non-conformal corrections to the metric and obtain an estimate of the corrections to $v_s$ up to NLO in $N$ .", "Having reduced the $D=11$ metric as given in (REF ) to $M_5(\\mathbb {R}^{1,3},u)$ , up to leading order in $N$ and considering the non-zero scalar modes of metric perturbations defined in subsection 2.4, we get a set of seven differential equations from the Einstein's equation.", "Defining the dimensionless energy and momentum, $\\omega _3=\\frac{w}{\\pi T},\\ q_3=\\frac{q}{\\pi T},$ the set of seven equations are given as: $& & H_{tt}^{\\prime \\prime } + \\frac{1}{u} \\left(-\\frac{6}{g} + 5\\right) H_{tt}^\\prime +H_s^{\\prime \\prime } + \\frac{1}{u} \\left(-\\frac{2}{g} + 1\\right) H_s^\\prime = 0,\\nonumber \\\\& & H_{tt}^{\\prime \\prime } + \\frac{2}{u} \\left(-\\frac{3}{g} + 1\\right) H_{tt}^\\prime + \\frac{1}{u} \\left(-\\frac{2}{g} + 1\\right) H_s^\\prime - \\frac{q_3^2}{g} H_{tt}+ \\frac{\\omega _3^2}{g^2} H_s + 2 \\frac{q_3 \\omega _3}{g^2} H_{{x}t} = 0,\\nonumber \\\\& & H_s^{\\prime \\prime } - \\frac{3}{u} H_{tt}^\\prime - \\frac{2}{u} \\left(1 + \\frac{2}{g}\\right) H_s^\\prime - \\frac{q_3^2}{g} H_{tt} + \\frac{\\omega _3^2}{g^2} H_s -\\frac{4q_3^2}{g} H_{{y}{y}} +\\frac{2 \\omega _3 q_3}{g^2} H_{{x}t} = 0,\\nonumber \\\\& & H_{{y}{y}}^{\\prime \\prime } - \\frac{H_{tt}^\\prime }{u} - \\frac{H_s^\\prime }{u} + \\frac{1}{u} \\left(-\\frac{4}{g} + 1\\right) H_{{y}{y}}^\\prime + \\frac{1}{g^2} \\left(\\omega _3^2 - g q_3^2\\right) H_{{y}{y}} = 0,\\nonumber \\\\& & H_{{x}t}^{\\prime \\prime } - \\frac{3}{u} H_{{x}t}^\\prime + \\frac{2 q_3 \\omega _3}{g} H_{{y}{y}} = 0,\\nonumber \\\\& & q_3 \\left(-g H_{tt}^\\prime + 2 u^3 H_{tt}\\right) - 2 q_3 g H_{{y}{y}}^\\prime + \\omega _3 H_{{x}t}^\\prime = 0,\\nonumber \\\\& & \\omega _3 \\left(g H_s^\\prime + 2 u^3 H_s\\right) +q_3 \\left(g H_{{x}t}^\\prime + 4 u^3 H_{{x}t}\\right) = 0$ where we define $H_{tt}=\\left(\\frac{g_s^{2/3}u^2L^2}{r_h^2g}\\right)h_{tt}$ , $H_{{x}{x}}=\\left(\\frac{g_s^{2/3}u^2L^2}{r_h^2}\\right)h_{{x}{x}}$ , $H_{{y}{y}}=H_{{z}{z}}=\\left(\\frac{g_s^{2/3}u^2L^2}{r_h^2g}\\right)h_{{y}{y}}$ , and $H_s=H_{{x}{x}}+2H_{{y}{y}}$ .", "The above system of equations can be reduced to the following linearly independent set of four equations $H^\\prime _{{x}{x}} &=&\\frac{3\\omega _3^2-2q_3^2u^4}{q_3^2\\left(u^4-3\\right)}H^\\prime _{tt}+\\frac{2 u \\left(q_3^4 \\left(1-u^4\\right)^2-\\omega _3^2 \\left(-2 u^6+6 u^2+\\omega _3^2\\right)\\right)}{q_3^2 \\left(u^4-3\\right)\\left(1-u^4\\right)^2}H_{{y}{y}}\\nonumber \\\\& & +\\frac{u \\omega _3^2 \\left(q_3^2 \\left(u^4-1\\right)+2 u^6-6 u^2-\\omega _3^2\\right)}{q_3^2 \\left(1-u^4\\right)^2 \\left(u^4-3\\right)}H_{{x}{x}}+\\frac{2 u w3 \\left(q3^2 \\left(u^4-1\\right)+2 u^6-6 u^2-w3^2\\right)}{q3 \\left(1-u^4\\right)^2 \\left(u^4-3\\right)}H_{{x}t}\\nonumber \\\\& & +\\frac{u \\left(q_3^2 \\left(u^4-1\\right)+2 u^6-6 u^2-\\omega _3^2\\right)}{\\left(u^4-3\\right) \\left(u^4-1\\right)}H_{tt}\\nonumber \\\\H^\\prime _{{y}{y}}&=&-\\frac{q3^2 \\left(u^4-3\\right)+3 \\omega _3^2}{2 q_3^2 \\left(u^4-3\\right)}H^\\prime _{tt}+\\frac{u \\omega _3^2 \\left(q3^2 \\left(u^4-1\\right)-2 u^6+6 u^2+\\omega _3^2\\right)}{q_3^2 \\left(1-u^4\\right)^2 \\left(u^4-3\\right)}H_{{y}{y}}\\nonumber \\\\& & +\\frac{u \\omega _3^2 \\left(-2 u^6+6 u^2+\\omega _3^2\\right)}{2 q_3^2 \\left(1-u^4\\right)^2 \\left(u^4-3\\right)}H_{{x}{x}}+\\frac{u \\omega _3 \\left(-2 u^6+6 u^2+\\omega _3^2\\right)}{q_3 \\left(1-u^4\\right)^2 \\left(u^4-3\\right)}H_{{x}t}\\nonumber \\\\& & +\\frac{u \\left(q_3^2 \\left(u^4-1\\right)+2 u^6-6 u^2-\\omega _3^2\\right)}{\\left(u^4-3\\right) \\left(u^4-1\\right)}H_{tt}\\nonumber \\\\H_{{x}t}^\\prime &=&\\frac{3 \\left(u^4-1\\right) \\omega _3}{q_3 \\left(u^4-3\\right)}H_{tt}^\\prime -\\frac{2 u \\omega _3 \\left(q_3^2 \\left(u^4-1\\right)-2 u^6+6 u^2+\\omega _3^2\\right)}{q_3 \\left(u^4-3\\right) \\left(u^4-1\\right)}H_{{y}{y}}\\nonumber \\\\& & -\\frac{u \\omega _3 \\left(-2 u^6+6 u^2+\\omega _3^2\\right)}{q_3 \\left(u^4-3\\right) \\left(u^4-1\\right)}H_{{x}{x}}+\\frac{2 u \\left(2 u^6-6 u^2-\\omega _3^2\\right)}{\\left(u^4-3\\right) \\left(u^4-1\\right)}H_{{x}t}\\nonumber \\\\& & -\\frac{u q_3 \\omega _3}{u^4-3}H_{tt}\\nonumber \\\\H_{tt}^{\\prime \\prime }&=&\\frac{u^8+2 u^4+9}{u \\left(u^4-3\\right) \\left(u^4-1\\right)}H_{tt}^\\prime -\\frac{2 \\left(q_3^2 \\left(u^4+1\\right)+2 \\omega _3^2\\right)}{\\left(u^4-3\\right) \\left(u^4-1\\right)}H_{{y}{y}}\\nonumber \\\\ & & -\\frac{2 \\omega _3^2}{\\left(u^4-3\\right) \\left(u^4-1\\right)}H_{{x}{x}}-\\frac{4 q_3 \\omega _3}{\\left(u^4-3\\right) \\left(u^4-1\\right)}H_{{x}t} -\\frac{2 q_3^2}{u^4-3}H_{tt}.$ To solve the system of equation (REF ) we look for the behavior of the solution near $u=1$ .", "Hence for time being we reconsider equation (REF ) and write them as the following system of six first order differential equations $& & H^\\prime _{tt}=\\frac{1}{g}P_{tt}\\nonumber \\\\& & H^\\prime _{yy}=-\\frac{1}{2 g}P_{tt}+\\frac{ u^3}{ g}H_{tt}+\\frac{ \\omega _3}{2 q_3g }P_{xt}\\nonumber \\\\& & H^\\prime _{s} =-\\frac{2 u^3 }{g}H_s-\\frac{4 q_3 u^3}{w_{3} g}H_{xt}-\\frac{q_3 }{\\omega _3}P_{xt}\\nonumber \\\\& & H^\\prime _{xt}=P_{xt}\\nonumber \\\\& & P^\\prime _{xt}=\\frac{3}{u}P_{xt}-\\frac{2q_3 \\omega _3 }{g}H_{yy}\\nonumber \\\\& & P^\\prime _{xt}=-\\frac{2(u^4-2)}{u g}P_{tt}+q_3^2H_{tt}-\\frac{q_3(u^4+1)}{u \\omega _3}P_{xt}-\\frac{2u^2+2u^6+\\omega _3^2}{g}\\left(H_s+\\frac{2q_3}{\\omega _3}H_{xt}\\right).$ In matrix form the above equation can be written as $X^\\prime =A(u)X$ where $A$ is a $6\\times 6$ matrix and is singular for all values of $u$ .", "Equation (REF ) can be solved by substituting the ansatz $X=(1-u)^r F(u)$ into the same, where the exponent $r$ can be evaluated from the eigenvalues of the matrix $(1-u)A(u)$ near $u=1$ .", "They are given by $r_1=r_2=0,r_3=-1/2,r_4=i\\omega _3/4,r_5=-i\\omega _3/4$ and $r_6=1/2$ .", "Two of the eigenvalues namely $r=\\mp i\\omega _3/4$ represent the incoming/outgiong wave." ], [ "The Longer Route up to Leading Order in $N$ - Via Solutions of EOMs", "In this subsection, we describe the evaluation of $v_s$ , first from the solutions to the EOMs for the scalar metric perturbation modes and then putting the same result on a firmer footing, from a two-point correlation function of energy momentum tensor: $\\langle T_{00}T_{00}\\rangle $ .", "We limit ourselves, in this subsection, to the leading order in $N$ ." ], [ "From the Pole Structure of Solutions to $H_{ab}(u)$", "Based on [31], we give below a discussion on three gauge transformations that preserve $h_{\\mu u}=0$ , for the black $M3$ -brane metric (REF ) having integrated out the $M_6$ in the (asymptotic) $AdS_5\\times M_6$ in the MQGP limit of [3].", "This is then utilized to obtain solutions to the scalar metric perturbation modes' equations of motion (REF ) near $u=0$ and thereafter the speed of sound.", "We verify the result for the speed of sound by also calculating the same from an two-point energy-momentum correlation function.", "Demanding that infinitesimal diffeomorphism: $x^\\mu \\rightarrow x^\\mu + \\xi ^\\mu , g_{\\mu \\nu }\\rightarrow g_{\\mu \\nu } - \\nabla _{(\\mu }\\xi _{\\nu )}$ preserves the gauge condition $h_{\\mu u}=0$ implies imposing [31]: $\\partial _{(\\mu }\\xi _{u)} - 2 \\Gamma ^\\rho _{\\mu u}\\xi _\\rho = 0,$ wherein $\\Gamma ^\\rho _{\\mu u}$ is calculated w.r.t.", "$g_{\\mu \\nu } = g_{\\mu \\nu }^{(0)} + h_{\\mu \\nu }$ .", "There are three residual gauge transformations under which the system of differential equations (REF ) remains invariant.", "They are given in (REF ), (REF ) and (REF ).", "Choosing $C_{u}, \\tilde{C}_{{x},u}: \\left(C_{u},\\frac{\\tilde{C}_{t,{x}}}{i}\\right)\\frac{g_s^{\\frac{2}{3}}}{L^2}=1$ , the non-zero pure gauge solutions gauge equivalent to $H_{ab}=0$ ($H_{ab}=0, \\xi _a=0$ ), near $u=0$ , are given by: $& & H_{xx}^{(I)}(0) = - 2 q_3, H_{xx}^{(III)}(0) = 2;\\nonumber \\\\& & H_{tt}^{(II)}(0) = 2 \\omega _3;\\nonumber \\\\& & H_{xt}^{(I)}(0) = \\omega _3, H_{xt}^{(II)}(0) = q_3.$ Writing $H^{\\rm inc}_{ab}(u)$ as the incoming solution to the differential equations, the general solution can be written as the following form, $H_{ab}(u) = a H^{(I)}_{ab}(u) + b H^{(II)}_{ab}(u) + c H^{(III)}_{ab}(u) + d H^{\\rm inc}_{ab}(u).$ To determine $H^{\\rm inc}_{ab}(u)$ , we Solve (REF ) near the horizon $u=1$ (this enables solving the fourth, fifth and sixth equations of (REF ) independent of the first, second, third and seventh equations), where we have already shown that the same is a regular singular point with exponent of the indicial equation corresponding to the incoming solution given by $-\\frac{i\\omega _3}{4}$ , implying that $H_{ab}^{\\rm inc}(u)=(1-u)^{-\\frac{i\\omega _3}{4}}{\\cal H}_{ab}(u)$ .", "Making double perturbative ansatze: ${\\cal H}_{ab}(u) = \\sum _{m=0}^\\infty \\sum _{n=0}^\\infty {\\cal H}_{ab}^{(m,n)}(u)q_3^m\\omega _3^n$ , one obtains near u=0 the solutions given in (REF ).", "Upon using $H_{tt}(0)=H_{t}^{(0)}, H_{xt}(0)=H_{xt}^{(0)}, H_s(0)=H_s^{(0)}$ and solving for $a, b, c$ and $d$ , the following is the common denominator: $\\Omega (\\omega _3,q_3) & \\equiv & \\alpha _{yy}^{(0,0)} + \\alpha _{yy}^{(1,0)} q_3 + C_{1yy}^{(2,0)} q_3^2 +\\alpha _{yy}^{(1,0)} \\omega _3 + \\left(-\\frac{i}{4} + C_{2yy}^{(1,1)} - \\frac{2}{9} C_{1yy}^{(1,1)} e^3\\right) q_3 \\omega _3\\nonumber \\\\& & + \\left(C_{1yy}^{(0,2)} +C_{2yy}^{(0,2)} + \\frac{i}{4} \\Sigma _{2yy}^{(0,1)}\\right) \\omega _3^2,$ where $\\alpha _{yy}^{(m,n)}, C_{ayy}^{(m,n)}, a,b=1,2$ are constants appearing in the solutions to ${\\cal H}_{ab}^{(m,n)}(u)$ in (REF ).", "Now, (REF ) can be solved for $\\omega _3$ and the solution is given in (REF ) in appendix E. Assuming $\\alpha _{yy}^{(0,0)}\\ll 1, |\\Sigma _{2\\ yy}^{(0,1)}|\\gg 1(i \\Sigma _{2\\ yy}^{(0,1)}\\in \\mathbb {R}): \\alpha _{yy}^{(0,0)}\\Sigma _{2\\ yy}^{(0,1)}<1; \\alpha _{yy}^{(1,0)} = - |\\alpha _{yy}^{(1,0)}|$ , consistent with the constraints such as (REF ) and (REF ) of appendix E, implies the roots (REF ) and (REF ) as given in appendix E. In the same appendix, it is shown that: $\\omega _3\\approx \\pm q_3\\left(1 + i \\frac{\\alpha _{yy}^{(00)}\\Sigma _{2\\ yy}^{(0,1)}}{2\\left(\\alpha _{yy}^{(1,0)}\\right)^2}\\right)\\equiv \\pm v_s q_3.$ One can show that one can consistently choose $ \\frac{\\alpha _{yy}^{(00)}\\left(i\\Sigma _{2\\ yy}^{(0,1)}\\right)}{2\\left(\\alpha _{yy}^{(1,0)}\\right)^2} = \\frac{1}{\\sqrt{3}} - 1$ to yield $v_s=\\frac{1}{\\sqrt{3}}$ ." ], [ "Via Two-Point Correlation Function $\\langle T_{00} T_{00}\\rangle $ using ON-Shell Reduction of Action and LO EOM's Solutions", "To put the results of 5.1.1 on a sound footing, we will now looking at the evaluation of the two-point correlation function $\\langle T_{00}T_{00}\\rangle $ from the on-shell action having dimensionally reduced $M$ theory on $M_5\\times M_6$ in the MQGP limit to $M_5$ , which asmptotically is $AdS_5$ .", "On-shellness dictates that: $R^{(0)}=\\frac{10}{3}\\Lambda $ under the metric perturbation given in (REF ).", "The pure gravitational part of the $5d$ action along with the Gibbons-Hawking York surface term [32] and a counter term (required to regularize the action) is given by: $\\int _0^1 du \\int {d^4}x \\sqrt{-g}(R - 2 \\Lambda )+\\int {d^4}x\\sqrt{-g_B}~2K+a\\int {d^4}x \\sqrt{-g_B}$ where $\\Lambda $ is a cosmological constant term, $g_B^{\\mu \\nu }$ is the pull-back metric on the boundary of AdS space and $K$ is the extrinsic curvature.", "For the given metric in this paper the cosmological constant is $\\lambda =-\\frac{6 g_s^{2/3}}{L^2}$ , also we choose $a=-\\frac{6 g^{1/3}_s}{L}$ to make the action in equation (REF ) finite.", "On-shell, the bilinear part of the above action, in the limit $q_3\\rightarrow 0, \\omega _3\\rightarrow 0$ , reduces to the following surface term: $& & \\int {d^4}x\\Biggl [\\frac{1}{4}\\Biggl (H_{tt}^2+8H_{{x}t}^2+2H_{{x}{x}}H_{tt}+4H_{{y}{y}}H_{tt}+4H_{{x}{x}}H_{{y}{y}}-H_{{x}{x}}^2\\Biggr )\\nonumber \\\\& & -\\frac{1}{2\\epsilon ^3}\\Biggl (H_{{x}t}^2+H_{{y}{y}}^2+H_{{x}{x}}H_{tt}+H_{{y}{y}}H_{tt}+2H_{{x}{x}}H_{{y}{y}}\\Biggr )^\\prime ~\\Biggr ].$ The equations of motion imply that $H_{tt}^\\prime (u=0) = H_s^\\prime (u=0) = H_{xt}^\\prime (u=0) = H_{yy}^\\prime (u=0) = 0$ , and we will further assume that $& & \\left(\\begin{array}{c}H_{yy}(u=0)\\\\ H_{s}(u=0)\\end{array}\\right) = \\left(\\begin{array}{cc} -\\beta _{yt} & -\\beta _{yx} \\\\ -\\beta _{st} & -\\beta _{sx} \\end{array}\\right)\\left(\\begin{array}{c} H_{tt}(u=0)\\\\ H_{xx}(u=0)\\end{array}\\right).$ So, the relevant two-point correlation function involving $T_{00}$ will require finding out the coefficient of $\\left(H^{(0)}_{t}\\right)^2$ upon substitution of (REF ) - (REF ) and (REF ) along with the values of $a, b, c, d$ with the common denominator $\\Omega (\\omega _3,q_3)$ of (REF ).", "As the generic form of this two-point function in the hydrodynamical limit [33] : $\\omega _3\\rightarrow 0, q_3\\rightarrow 0: \\frac{\\omega _3}{q_3}=\\alpha \\equiv $ constant - is expected to be of the form: $\\frac{q_3^2}{\\omega _3^2 - v_s^2 q_3^2}$ , we isolate these terms and work up to leading order in $\\Sigma _{2yy}^{(0,1)}$ .", "We find from (REF ) the following coefficients of $(H_t^{(0)})^2$ coming from the $H^2$ -like terms and $HH^{\\prime }$ -like terms: $& & \\hspace{-50.58878pt}H^2\\ {\\rm terms}:\\nonumber \\\\& &\\hspace{-50.58878pt}-\\frac{i \\Sigma _{2yy}^{01} \\Biggr (\\alpha ^4 \\left(\\beta _{st}^2+\\beta _{st} (2-8 \\beta _{yt})+12 \\beta _{yt}^2-1\\right)-\\alpha ^2\\left(\\beta _{st}^2+\\beta _{st} (56 \\beta _{yt}+2)+12 \\beta _{yt}^2-1\\right)-32 \\beta _{yt} (\\pi \\beta _{yt}-2)\\Biggl )}{16 \\left(\\alpha ^2-1\\right)};\\nonumber \\\\& & \\hspace{-50.58878pt}\\left.\\frac{\\left(HH^\\prime \\right)^{{\\cal O}(u^0)}}{u^3}\\right|_{u=\\epsilon } =\\nonumber \\\\& & \\hspace{-50.58878pt}-\\frac{i \\alpha ^2 \\Sigma _{2yy}^{01}\\beta _{yt} \\Biggr (\\alpha ^2 ((8+\\pi ) \\beta _{st}-2 ((\\pi -6) \\beta _{yt}+1))+(16+\\pi ) \\beta _{st}-12\\beta _{yt}+\\pi ^2 \\beta _{yt}+14 \\pi \\beta _{yt}-2 \\pi -22\\Biggl )}{16 \\left(\\alpha ^2-1\\right)} ;\\nonumber \\\\& & \\hspace{-50.58878pt}\\left.\\frac{\\left(HH^\\prime \\right)^{{\\cal O}(u)}}{u^3}\\right|_{u=\\epsilon }=-\\frac{1}{16 \\left(\\alpha ^2-1\\right)}\\Biggl \\lbrace i \\alpha ^2 \\Sigma _{2yy}^{01} \\Biggl (\\alpha ^2 \\beta _{yt} ((16+\\pi ) \\beta _{st}-(\\pi -20)\\beta _{yt}+2)-2 \\alpha (2 \\beta _{st}+\\pi \\beta _{yt}-2)\\nonumber \\\\& &\\hspace{-50.58878pt}+\\beta _{yt} \\left((\\pi -24)\\beta _{st}+\\left(-20-3 \\pi +\\pi ^2\\right) \\beta _{yt}-2 \\pi +6\\right)\\Biggr )\\Biggr \\rbrace ;\\nonumber \\\\& &\\hspace{-50.58878pt}\\left.\\frac{\\left(HH^\\prime \\right)^{{\\cal O}(u^2)}}{u^3}\\right|_{u=\\epsilon }-\\frac{1}{32 \\left(\\alpha ^2-1\\right)}\\Biggr \\lbrace i \\alpha ^2 \\Sigma _{2yy}^{01}\\beta _{yt} \\Biggl (2 \\alpha ^2 ((\\pi -24) \\beta _{st}+10\\beta _{yt})-(\\pi -36) \\alpha (2 \\beta _{st}+\\pi \\beta _{yt}-2)\\nonumber \\\\& &\\hspace{-50.58878pt}+2 \\left((\\pi -24) \\beta _{st}+\\left(-10-24 \\pi +\\pi ^2\\right) \\beta _{yt}-2 \\pi +48\\right)\\Biggr )\\Biggl \\rbrace ;\\nonumber \\\\& &\\hspace{-50.58878pt}\\left.\\frac{\\left(HH^\\prime \\right)^{{\\cal O}(u^3)}}{u^3}\\right|_{u=\\epsilon }=\\nonumber \\\\& & \\hspace{-50.58878pt}\\frac{1}{160 \\left(\\alpha ^2-1\\right)^2}\\Biggl \\lbrace i \\Sigma _{2yy}^{01} \\Biggl (5 \\alpha ^6 \\left(24 \\beta _{st}^2+ \\beta _{st} (8-2 (\\pi -2) \\beta _{yt})+\\beta _{yt} (2-\\pi \\beta _{yt})\\right)-40 \\alpha ^5 \\beta _{yt} (2 \\beta _{st}+\\pi \\beta _{yt}-2)\\nonumber \\\\& & \\hspace{-50.58878pt}+2 \\alpha ^4 \\left(60 \\beta _{st}^2+ \\beta _{st} (6 (15 \\pi -8) \\beta _{yt}-200)-5 \\left((\\pi -11) \\pi \\beta _{yt}^2+(22-4 \\pi ) \\beta _{yt}+4\\right)\\right)+40 \\alpha ^3\\beta _{yt} (2 \\beta _{st}+\\pi \\beta _{yt}-2)\\nonumber \\\\& & \\hspace{-50.58878pt}+\\alpha ^2 \\left(2 \\beta _{st} ((38+35 \\pi )\\beta _{yt}-60)+\\pi (70 \\pi -233) \\beta _{yt}^2+(466-280 \\pi )\\beta _{yt}+280\\right)+128 \\beta _{yt} (\\pi \\beta _{yt}-2)\\Biggr )\\Biggr \\rbrace .$ From (REF ), we see that for $\\beta _{yt}=0,\\beta _{st}=1$ , the first line in (REF ) yields a contribution: $i\\alpha ^2\\Sigma _{2yy}^{01}\\frac{q_3^2}{\\left(\\omega _3^2 - v_s^2 q_3^2\\right)}$ and from the second line in (REF ), there is no required contribution from $\\frac{\\left(HH^\\prime \\right)^{{\\cal O}(u^{0,1,2})}}{u^3}$ and $\\frac{\\left(HH^\\prime \\right)^{{\\cal O}(u^3)}}{u^3}$ terms yield: $i \\alpha ^2\\Sigma _{2yy}^{01}\\frac{q_3^2}{\\left(\\omega _3^2 - v_s^2 q_3^2\\right)}$ ." ], [ "The Shorter Route - Use of Gauge-Invariant Variable", "In this subsection, we carry on the same calculation as we did in the last subsection for the speed of sound up to leading order in $N$ via a different approach.", "This time following [5], we first obtain the EOM for appropriate gauge-invariant variable corresponding to the non zero scalar modes of metric perturbations as defined in (REF ) and then compute the quasinormal modes, hence the speed of sound $v_s$ by solving that EOM in the hydrodynamic approximation.", "we have also calculate the two point correlation function of energy momentum tensor using the above solution for the gauge invariant variable.", "Latter following the same approach we compute the next to leading order correction to speed of sound by using the metric components as given in (REF ) corrected up to NLO in $N$ ." ], [ "From the solution of Gauge Invariant Variable up to Leading Order in $N$", "Going back to (REF ) we see that the four linearly independent equations using the following gauge invariant combination of perturbations namely, $Z_s(u)=2 q_3 \\omega _3 H_{{x}t}+\\omega _3^2 H_{{x}{x}}+H_{{y}{y}}\\left[q_3^2\\left(u^4+1\\right)-\\omega _3^2\\right]-q_3^2 \\left(1-u^4\\right) H_{tt},$ can be written as a single second order differential equation involving $Z_s(u)$ : $& & Z_s^{\\prime \\prime }(u)-\\frac{q_3^2 \\left(7 u^8-8 u^4+9\\right)-3 \\left(u^4+3\\right) \\omega _3^2}{u \\left(u^4-1\\right) \\left(q_3^2 \\left(u^4-3\\right)+3\\omega _3^2\\right)}Z_s^\\prime (u)\\nonumber \\\\& & +\\frac{q_3^4 \\left(u^8-4 u^4+3\\right)+2 q_3^2 \\left(8 u^{10}-8 u^6+2 u^4 \\omega _3^2-3 \\omega _3^2\\right)+3 \\omega _3^4}{\\left(1-u^4\\right)^2\\left(q_3^2 \\left(u^4-3\\right)+3 \\omega _3^2\\right)}Z_s(u)=0.$ The above equation can be solve by considering an ansatz $Z_s(u)=(1-u)^rF(u)$ where $F(u)$ is regular near the horizon $u=1$ .", "We have already obtained the value of exponent $r$ at the end of section 5 and it is given by $\\pm \\frac{i\\omega _3}{4}$ .", "we choose the negative sign here as it represents an incoming wave.", "The evaluation of the function $F(u)$ can be done perturbatively using hydrodynamic approximation, given as: $\\omega _3\\ll 1$ , $q_3\\ll 1$ .", "For analytic solution the momentum has to be light-like, means $\\omega _3$ and $q_3$ would be of the same order.", "Hence we can rescale $\\omega _3$ and $q_3$ by a same parameter $\\lambda $ as: $\\omega _3\\rightarrow \\lambda \\omega _3$ , $q_3\\rightarrow \\lambda q_3$ and expand equation (REF ) to first order in $\\lambda $ , where the limit $\\lambda \\ll 1$ ensure that we are working in the hydrodynamic regime.", "We choose the following series expansion of $F(u)$ for small frequency and momentum as: $F(u)=F_0(u)+\\omega _3F_1(u)+\\mathcal {O}(\\omega _3^2,q_3^2,\\omega _3q_3).$ Plugging in the equation (REF ) into the equation (REF ) one can get an equation involving $F_0(u)$ only: $u \\left(u^4-1\\right) \\left(q_3^2 \\left(u^4-3\\right)+3 \\omega _3^2\\right) F_0^{\\prime \\prime }+ \\left(q_3^2 \\left(-7 u^8+8u^4-9\\right)+3 \\left(u^4+3\\right) \\omega _3^2\\right)F_0^\\prime +16 q_3^2 u^7 F_0=0.\\nonumber \\\\& &$ A solution to the above equation is given by, $F_0(u)=\\frac{c_1 \\left(q_3^2 \\left(u^4+1\\right)-3 \\omega _3^2\\right)}{17 q_3^2-3 \\omega _3^2}+\\frac{c_2 \\left(q_3^2 \\left(u^4+1\\right)-3\\omega _3^2\\right) \\left(-\\frac{2 q_3^2-3 \\omega _3^2}{q_3^2 \\left(u^4+1\\right)-3 \\omega _3^2}-\\frac{1}{4} \\log \\left(u^4-1\\right)\\right)}{17 q_3^2-3 \\omega _3^2}.\\nonumber \\\\& &$ For the regularity of $F_0(u)$ near the horizon $u=1$ , we choose the constant $c_2$ to be equal to zero.", "Using this solution for $F_0(u)$ , another equation for $F_1(u)$ can be found from (REF ), $& & u \\left(u^4-1\\right)\\left\\lbrace 17 q_3^4\\left(u^4-3\\right)-3 q_3^2\\omega _3^2 \\left(u^4-20\\right) -9 \\omega _3^4\\right\\rbrace F_1^{\\prime \\prime }\\nonumber \\\\& & + \\left\\lbrace -17 q_3^4 \\left(7 u^8-8u^4+9\\right)+3 q_3^2 \\omega _3^2\\left(7 u^8+9 u^4+60\\right) -9 \\omega _3^4 \\left(u^4+3\\right)\\right\\rbrace F_1^{\\prime }\\nonumber \\\\& & +16 u^7q_3^2 \\left(17q_3^2-3 \\omega _3^2\\right) F_1+16 i u^7 q_3^2\\left(2 q_3^2-3 \\omega _3^2\\right)c_1=0.$ A general solution is given as: $& & F_1(u)=-\\frac{c_1 i \\left(2 q_3^2-3 \\omega _3^2\\right)}{17 q_3^2-3 \\omega _3^2}+\\frac{c_2 \\left(q_3^2 \\left(u^4+1\\right)-3w3^2\\right)}{17 q_3^2-3 \\omega _3^2}\\nonumber \\\\& & +\\frac{c_3 \\left(q_3^2 \\left(u^4+1\\right)-3 \\omega _3^2\\right) \\left(-\\frac{2 q_3^2-3\\omega _3^2}{q_3^2 \\left(u^4+1\\right)-3 \\omega _3^2}-\\frac{1}{4} \\log \\left(u^4-1\\right)\\right)}{17 q_3^2-3 \\omega _3^2}.$ Again demanding the regularity of the above solution near the horizon, we put $c_3$ to zero.", "Also imposing a boundary condition $F_1(u=1)=0$ , we determine the constant $c_2$ to be equal to $ic_1$ .", "With this the final expression of $Z_s(u)$ is given as: $Z_s(u)=c_1(1-u^4)^{-i\\omega _3/4}\\left(\\frac{q_3^2 \\left(u^4+1\\right)-3 \\omega _3^2}{17 q_3^2-3 \\omega _3^2}-\\frac{i q_3^2\\omega _3 \\left(1-u^4\\right)}{17 q_3^2-3 \\omega _3^2}\\right).$ Imposing the Dirichlet boundary condition $Z(u=0)=0$ we get the quasinormal frequency, $\\omega _3=\\pm \\frac{q_3}{\\sqrt{3}}-\\frac{i q_3^2}{6}+\\mathcal {O}.$ Using (REF ), we get the following dispersion relation: $w=\\pm \\frac{q}{\\sqrt{3}}-\\frac{i q^2}{6\\pi T}.$ Comparing this with the dispersion relation corresponding to the sound wave mode, $w=\\pm q v_s - i\\Gamma _s q^2$ where $v_s$ is the speed of sound and $\\Gamma _s$ is the attenuation constant, we get their exact values." ], [ "Via Two-Point Correlation Function $\\langle T_{00} T_{00}\\rangle $ - Using the Solution of EOM involving Gauge Invariant Variable", "The relevant part of the bilinear surface term of the full action (REF ) as given in (REF ) can be rewritten in terms of the gauge invariant variable $Z_s(u)$ as: $S^{(2)}_{\\epsilon }=\\lim _{u\\rightarrow 0}\\int \\frac{dw dq}{2\\pi ^2} A(\\omega _3,q_3,u)Z_s^\\prime (u,q)Z_s(u,-q).$ Using the equations of motion (REF ) along with (REF ), we find the function $A(\\omega _3,q_3,u)$ as: $A(\\omega _3,q_3,u)=\\frac{3}{u^3 \\left(q_3^2 \\left(u^4-3\\right)+3 \\omega _3^2\\right)^2}$ For the computation of two point function we need the solution of equation (REF ) as given in equation (REF ), where the constant $c_1$ is determined by the boundary condition $Z_s(u=0)=-H_{tt}^0 q_3^2+2 H_{{x}t}^0 q_3 \\omega _3+H_{{x}{x}}^0 \\omega _3^2+H_{{y}{y}}^0 \\left(q_3^2-\\omega _3^2\\right),$ where we define $H_{ab}(u=0)=H_{ab}^0$ .", "Now putting the above expression of $A(\\omega _3,q_3,u)$ and the solution $Z_s(u)$ back in equation (REF ) one get the two point correlator $G_{tt,tt}$ as: $\\nonumber G_{tt,tt}=\\frac{\\delta ^2S_{\\epsilon }^{(2)}}{\\delta H_{tt}^{(0)}(k)\\delta H_{tt}^{(0)}(-k)}\\\\=\\frac{8 q^6}{3 \\left(q^2-3 w^2\\right) \\left(q^2-w^2\\right)^2}$ Hence the pole structure of the Green's function gives the correct value of the speed of sound wave, $v_s=\\frac{1}{\\sqrt{3}}$ propagating through hot plasma.", "The above value of speed of sound also matches exactly with the value that we have already got from the solution of hydrodynamic equations, thus provides a quantitative checks of the validity of Gauge/Gravity duality." ], [ "From the solution of Gauge Invariant Variable - Going up to NLO in $N$ in the MQGP Limit", "Considering the Next-to-Leading Order corrections in $N$ of the metric components as given in (REF ), and using the gauge invariant combination given in (REF ), (REF ) and (REF ), the Einstein equation can be expressed in terms of a single equation of the form $Z_i^{\\prime \\prime }(u) = m_i(u) Z_i^\\prime (u) + l_i(u) Z_i(u)$ , where, $i=s({\\rm calar})$ , $v({\\rm ector})$ , $t({\\rm ensor })$ .", "In 5.3.1, we first evaluate $v_s$ including the non-conformal contribution to the M-theory metric evaluated at a finite $r$ and large $N$ , i.e., $\\log \\left(\\frac{r}{\\sqrt{\\alpha ^\\prime }}\\right)<\\log N$ , thereby dropping $\\log r \\log N$ as compared to $\\left(\\log N\\right)^2$ .", "Then, in 5.3.2, we attempt a full-blown non-conformal estimate of $v_s$ up to NLO in $N$ by working at an $r: \\log \\left(\\frac{r}{\\sqrt{\\alpha ^\\prime }}\\right)\\sim \\log N$ .", "It turns out, unlike the former, the horizon becomes an irregular singular point for the latter.", "We set $\\alpha ^\\prime $ to unity throughout.", "Given that in both, 5.3.1 and 5.3.2, we are interested in numerics, exact numerical factors in all expressions will be replaced by their decimal equivalents." ], [ "Dropping $\\log r \\log N$ As Compared to {{formula:8788f7ff-c922-4149-b167-0ea347d7ff32}}", "Including the NLO terms, the EOM for the gauge invariant variable $Z_s(u)$ - given by (REF ) - can be rewritten as: $(u-1)^2Z_s^{\\prime \\prime }(u) + (u-1)P(u-1) Z_s^\\prime (u) + Q(u-1) Z_s(u) = 0,$ in which $P(u-1) = \\sum _{n=0}^\\infty p_n(u-1)^n$ and $Q(u-1) = \\sum _{m=0}^\\infty q_n (u-1)^n$ wherein, up to ${\\cal O}\\left(\\frac{1}{N}\\right)$ , $p_n, q_n$ are worked out in (REF ).", "The horizon $u=1$ being a regular singular point of (REF ), the Frobenius method then dictates that the incoming-wave solution is given by: $Z_s(u) = \\left(1 - u \\right)^{\\frac{3 {g_s}^2 M^2 {N_f} \\log (N) \\left(8 {q_3}^2 {\\omega _3}^2 \\log (N)+\\left({\\omega _3}^2+4\\right) \\left(10 {q_3}^2-27{\\omega _3}^2\\right)\\right)}{2048 \\pi ^2 N q_3^2 \\omega _3 \\left(-1\\right)^{3/2}}-\\frac{i {\\omega _3}}{4}}\\left(1 + \\sum _{m=1}a_m (u - 1)^m\\right),$ where $a_{1,2}$ are given in (REF ).", "Following [5], imposing Dirichlet boundary condition $Z_s(u=0)=0$ and going up to second order in powers of $(u-1)$ in (REF ) and considering in the hydrodynamical limit $\\omega _3^nq_3^m:m+n=2$ one obtains: $\\omega _3 = -\\frac{2 {q_3}}{\\sqrt{3}}-\\frac{9 i {q_3}^2}{32},$ which yields a result for the speed of sound similar to, though not identical to, (REF ) for $n=0,1$ .", "To get the LO or conformal result for the speed of sound $v_s = \\frac{1}{\\sqrt{3}}$ , let us go to the fourth order in (REF ).", "For this, up to ${\\cal O}\\left(\\frac{1}{N}\\right)$ , $p_n, q_n$ are worked out in (REF ).", "We will not quote the expressions for $a_3$ and $a_4$ because they are too cumbersome.", "Substituting the expressions for $a_{1,2,3,4}$ into $Z_s(u)$ and implementing the Dirichlet boundary condition: $Z_s(u=0)=0$ , in the hydrodynamical limit, going up to ${\\cal O}(\\omega _3^4)$ one sees that one can write the Dirichlet boundary condition as a quartic: $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where $a, b, c, d, f, g$ are given in (REF ).", "One of the four roots yields: $\\omega _3 \\approx 0.46 q_3 - 0.31 i q_3^2,$ with no ${\\cal O}\\left(\\frac{1}{N}\\right)$ -corrections!", "The coefficient of $q_3$ is not too different from the conformal value of $\\frac{1}{\\sqrt{3}}\\approx 0.58$ .", "We expect the leading order term in the coefficient of $q_3$ to converge to $\\frac{1}{\\sqrt{3}}$ .", "Also, the coefficient of $q^2$ term turns out to be $\\frac{0.31}{\\pi }$ which is not terribly far from the conformal result of $\\frac{0.17}{\\pi }$ .", "We are certain that the inclusion of higher order terms in (REF ) will ensure that we get a perfect match with the conformal result.", "The reason we do obtain the NLO non-conformal contribution to $v_s$ is that at the very outset, we have neglected the non-conformal $\\log r$ -contributions by working at a large but finite $r$ , but such that $\\frac{\\frac{r}{\\sqrt{\\alpha ^\\prime }}}{N}\\ll 1.$ We will see how to obtain the non-conformal contribution with the inclusion of the same in 5.3.2 below." ], [ "Retaining $\\log r \\log N$ and {{formula:d9b478d4-948d-4284-b608-7fb154997dcd}} Terms", "Constructing a $Z_s(u)$ given by (REF ) and retaining the non-conformal $\\log r \\log N$ -contribution as well as $\\left(\\log N\\right)^2$ terms, one sees one obtains (REF ) as the equation of motion for $Z_s(u)$ .", "The horizon $u=1$ due to inclusion of the non-conformal corrections to the metric, becomes an irregular singular point.", "One then tries the ansatz: $Z_s(u) = e^{S(u)}$ near $u=1$ [34].", "Assuming that $\\left(S^{\\prime }\\right)^2\\gg S^{\\prime \\prime }(u)$ near $u=1$ the differential equation (REF ), which could written as $Z_s^{\\prime \\prime }(u) = m(u)Z_s^\\prime + l(u) Z_s(u)$ can be approximated by: $\\left(S^\\prime \\right)^2 - m(u) S^\\prime (u) - l(u) \\approx 0.$ A solution to (REF ) is given in (REF ) in Appendix D. Taking first the MQGP limit, integrating with respect to $u$ , the solution (REF ) will reflect the singular nature of $Z_s(u)$ 's equation of motion (REF ) via $Z_s(u)\\sim \\left(1 - u \\right)^{-\\frac{1}{2} + \\frac{15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log \\left(\\frac{1}{N}\\right)}{256 \\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)}}F(u),$ where $F(u)$ is regular in $u$ and its equation of motion, around $u=0$ , is given by (REF ) whose solution is given in (REF ).", "One notes from (REF ) that $F(u\\sim 0) = c_1$ .", "This needs to be improved upon by including the sub-leading terms in $u$ in $F^{\\prime }(u)$ which is discussed in detail in Appendix D. For $Z_s(u=0)=0$ to obtain $\\omega = \\omega (q)$ to determine the speed of sound, one requires $F(u=0)=0$ .", "From (REF ), this can be effected by requiring $\\hspace{-36.135pt}\\frac{225 {g_s}^4 {N_f}^2 {\\omega _3}^2 \\log ^2(N) M^4+4800 {g_s}^2 N{N_f} \\pi ^2 \\left(4 {q_3}^2-5 {\\omega _3}^2\\right) \\log (N) M^2+139264 N^2 \\pi ^4 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}{128 N \\pi ^2\\left(15 {g_s}^2 {N_f} \\left(8 {q_3}^2-11 {\\omega _3}^2\\right) \\log (N) M^2+896 N \\pi ^2 \\left(2 {q_3}^2-3{\\omega _3}^2\\right)\\right)} = - n\\in \\mathbb {Z}^-$ or $\\omega = {q_3} \\left(\\frac{\\sqrt{14 {n}+17}}{\\sqrt{21 {n}+\\frac{51}{2}}}+\\frac{5 (2 {n}+5) {g_s}^2 M^2 {N_f} \\log N}{128 \\pi ^2 \\sqrt{14 {n}+17} \\sqrt{84 {n}+102} N}\\right),$ implying the following estimate of the speed of sound: $v_s\\approx \\frac{\\sqrt{14 {n}+17}}{\\sqrt{21 {n}+\\frac{51}{2}}}+\\frac{5 (2 {n}+5) {g_s}^2 M^2 {N_f} \\log N}{128 \\pi ^2 \\sqrt{14 {n}+17} \\sqrt{84 {n}+102} N}.$ Given that (REF ) is an approximate solution to (REF ), one expects to obtain an expression for $v_s$ from an $M3$ -brane upliftFor a $p$ -brane solution, to LO in $N$ , one expects $v_s=\\frac{1}{\\sqrt{p}}$ [35]., to be of the form $v_s \\approx \\frac{{\\cal O}(1)}{\\sqrt{3}} + {\\cal O}\\left(\\frac{g_sM^2}{N}\\right)$ , and (REF ) is exactly of this form for $n=0,1$ ." ], [ "Vector Mode Perturbations and Shear Mode Diffusion Constant up to NLO in $N$ in the MQGP Limit", "The equations of motion for the vector perturbation modes up next-to-leading order in $N$ , can be reduced to the following single equation of motion in terms of a gauge-invariant variable $Z_v(u)$ (given by (REF )): $Z_v^{\\prime \\prime }(u) - m_v(u) Z_v^\\prime (u) - l_v(u) Z_v(u) = 0,$ where $m_v(u), l_v(u)$ are given in (REF ).", "The horizon $u=1$ is a regular singular point of (REF ) and the root of the indicial equation corresponding to the incoming-wave solution is given by: $-\\frac{i {\\omega _3}}{4} + \\frac{3 i {g_s}^2 M^2 {N_f} {\\omega _3} \\log ^2(N)}{256 \\pi ^2 N}.$ (a) Using the Frobenius method, taking the solution about $u=1$ to be: $Z_v(u) = (1 - u)^{-\\frac{i {\\omega _3}}{4} + \\frac{3 i {g_s}^2 M^2 {N_f} {\\omega _3} \\log ^2(N)}{256 \\pi ^2 N}}\\left(1 + \\sum _{n=1}^\\infty a_n (u - 1)^n\\right),$ by truncating the infinite series in (REF ) to ${\\cal O}((u-1)^2)$ one obtains in (REF ) of Appendix F, values for $a_1, a_2$ .", "The Dirichlet boundary condition $Z(u=0)=0$ in the hydrodynamical limit retaining therefore terms only up to ${\\cal O}(\\omega _3^mq_3^n):\\ m+n=4$ , reduces to: $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where $a, b, c, d, f, g$ are given in (REF ).", "Analogous to 5.3, once again as we are interested in numerics, exact numerical factors in all expressions will be replaced by their decimal equivalents for most part of this section.", "One of the four roots of $Z_v(u=0)=0$ is: $\\omega _3 = -8.18 i + \\frac{0.14 i g_s^2 M^2 N_f(\\log N)^2}{N} + \\left(-0.005 i - \\frac{0.002 i g_s^2 M^2 N_f (\\log N)^2}{N}\\right)q_3^2 + {\\cal O}(q_3^3).$ (b) Using the Frobenius method and going up to ${\\cal O}((u-1)^3)$ in (REF ), one obtains in (REF ) values of $a_3$ .", "The Dirichlet condition $Z_v(u=0)=0$ reduces to $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where $a, b, c, d, f, g$ are given in (REF ).", "One of the four roots of the quartic in $\\omega _3$ is: $\\omega _3 = \\left(- 0.73 i + \\frac{0.003 i g_s^2 M^2 N_f (\\log N)^2}{N}\\right)q_3^2 + {\\cal O}(q_3^3).$ The leading order coefficient of $q_3^2$ is not terribly far off the correct value $-\\frac{i}{4}$ already at the third order in the infinite series (REF ).", "(c) Let us look at (REF ) up to the fourth order.", "One finds in (REF ) the value of $a_4$ .", "In the hydrodynamical limit the Dirichlet boundary condition $Z_v(u=0)=0$ reduces to $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where $a, b, c, d, f, g$ are given in (REF ).", "Incredibly, one of the roots of the quartic equation in $\\omega _3$ is: $& & \\omega _3 = \\left( - \\frac{i}{4} + \\frac{3 i g_s^2 M^2 N_f \\log N\\left(5 + 2 \\log N\\right)}{512 \\pi ^2 N}\\right)q_3^2 + {\\cal O}\\left(q_3^3\\right).$ Hence, the leading order (in $N$ ) yields a diffusion constant of the shear mode $D = \\frac{1}{4\\pi T}$ , exactly the conformal result!", "Including the non-conformal corrections which appear at NLO in $N$ , one obtains: $D = \\frac{1}{\\pi T}\\left(\\frac{1}{4} - \\frac{3 g_s^2 M^2 N_f \\log N\\left(5 + 2 \\log N\\right)}{512 \\pi ^2 N}\\right).$ We conjecture that all terms in (REF ) at fifth order and higher, do not contribute to the Dirichlet boundary condition up to the required order in the hydrodynamical limit.", "Figure: DT c D T_c vs. T T c \\frac{T}{T_c} for T≥T c T\\ge T_cThe variation of the shear mode diffusion constant with temperature is shown in Figure 1 for $N_f=3$ , $M=3$ , $g_s=0.9$ , $N=100$ .", "As the lowest order conformal result we obtain exactly $\\frac{1}{4\\pi T}$ as obtained in [36], for the black brane metric of the form (REF )." ], [ "NLO Corrections in N to $\\eta $ and {{formula:51625f23-9c60-49db-8bf3-6572c031d450}}", "Using the non-conformal $M_5(\\mathbb {R}^{1,3},u)$ metric components of (REF ), we now evaluate the non-conformal $\\frac{g_s M^2}{N}$ -corrections to the shear viscosity $\\eta $ by considering the EOM for the tensor mode of metric fluctuations up to NLO in $N$ , and also estimate the same for the shear viscosity - entropy density ratio $\\frac{\\eta }{s}$ .", "The EOM for the tensor mode of metric fluctuation, using (REF ), is given as under: $& & Z_t^{\\prime \\prime }(u) + Z_t^\\prime (u) \\left(-\\frac{15 {g_s}^2 M^2 {N_f} \\log {N}}{64 \\pi ^2 N u}+\\frac{u^4+3}{u \\left(u^4-1\\right)}\\right)+Z_t(u) \\Biggl [\\frac{{q_3}^2\\left(u^4-1\\right)+{\\omega _3}^2}{\\left(u^4-1\\right)^2}\\nonumber \\\\& & -\\frac{3 \\left({q_3}^2 u^4-{q_3}^2+{\\omega _3}^2\\right) \\left({g_s}^2 M^2 {N_f} \\log ^2{N}+2 {g_s}^2 M^2 {N_f} \\log {N} \\log \\left(\\frac{2 \\pi ^{3/2} \\sqrt{{g_s}} T}{u}\\right)\\right)}{32 \\pi ^2 N \\left(u^4-1\\right)^2}\\Biggr ]=0.\\nonumber \\\\& &$ Realizing the horizon is a regular singular point, one makes the following double perturbative ansatz in $\\omega _3$ and $q_3$ :z $\\hspace{-28.90755pt}Z_t(u) = \\left(1 - u\\right)^{-i {\\omega _3} \\left(\\frac{1}{4}-\\frac{3 {g_s}^2 M^2 {N_f} \\log (N) \\log r_h}{128 \\pi ^2 N}\\right)}\\left(z_{00}(u) + \\omega _3 z_{01}(u) + q_3 z_{10}(u) + {\\cal O}(q_3^m\\omega _3^n)_{m+n\\ge 2}\\right).$ Using equations (REF ) - (REF ) in Appendix G, $& & Z_t(u) = -\\frac{i}{3072 \\pi ^2 N}(1-u)^{-i {\\omega _3} \\left(\\frac{1}{4}-\\frac{3 {g_s}^2 M^2 {N_f} \\log (N) \\log r_h}{128 \\pi ^2 N}\\right)}\\nonumber \\\\& & \\times \\Biggl (-3 {g_s}^2 M^2 {N_f} u \\log {N} \\left(4 c_2 \\left(2 u^2+3 u+6\\right) {\\omega _3} \\log \\left(2\\pi ^{3/2} \\sqrt{{g_s}} T\\right)+15 i c_5 {q_3} u^3 (1-4 \\log (u))\\right)\\nonumber \\\\& & -6 c_2 {g_s}^2 M^2 {N_f} u \\left(2 u^2+3 u+6\\right) {\\omega _3} \\left(\\log {N}\\right)^2 +128 \\pi ^2 N \\biggl [6 i \\left(c_5 {q_3} u^4+4 c_3 {q_3}+4 c_4 {\\omega _3}\\right)\\nonumber \\\\& & +c_2 \\left(2 u^3 {\\omega _3}+3 u^2 {\\omega _3}+6 u{\\omega _3}+24 i\\right)\\biggr ]\\Biggr ).$ Setting $q_3=0$ one obtains (REF ) wherein the ${\\cal O}(u^3\\omega _3)$ term, without worrying about overall numerical multiplicative constants, is given by: $\\frac{i}{4}-\\frac{3 i {g_s}^2 M^2 {N_f} \\log (N) \\log r_h}{128 \\pi ^2 N}$ Using arguments of [7], setting $\\kappa _{11}^2=1$ , the coefficient of the kinetic term of $Z_t(u)$ near $u=0$ and near $\\theta _1=\\frac{\\alpha _{\\theta _1}}{N^{\\frac{1}{5}}}$ (whereat an explicit local $SU(3)$ -structure of the type IIA mirror and an explicit local $G_2$ -structure of the M-theory uplift was obtained in [22]) is $\\frac{r_h^4}{g_s^2u^3}\\int d\\theta _1\\cot ^3\\theta _1 \\sin \\theta _1 f_1(\\theta _1)\\delta \\left(\\theta _1 - \\frac{\\alpha _\\theta }{N^{\\frac{1}{5}}}\\right)\\sim \\frac{r_h^4}{g_s^2u^3}\\frac{N^{\\frac{2}{5}}}{\\alpha _N\\alpha _{\\theta _1}^2},$ where $f_1(\\theta _1) = \\frac{\\cot \\theta _1}{\\alpha _N},\\ f_2(\\theta _2) = -\\alpha _N\\cot \\theta _2$ [22].", "The exact result for the temperature, assuming the resolution to be larger than the deformation in the resolved warped deformed conifold in the type IIB background of [2] Refer to appendix H for details pertaining to this assumption.", "in the MQGP limit, and utilizing the IR-valued warp factor $h(r,\\theta _1,\\theta _2)$ , is given as under: $& & T = \\frac{\\partial _rG^{\\cal M}_{00}}{4\\pi \\sqrt{G^{\\cal M}_{00}G^{\\cal M}_{rr}}}\\nonumber \\\\& & = {r_h} \\left[\\frac{1}{2 \\pi ^{3/2} \\sqrt{{g_s} N}}-\\frac{3 {g_s}^{\\frac{3}{2}} M^2 {N_f} \\log ({r_h}) \\left(-\\log {N}+12 \\log ({r_h})+\\frac{8 \\pi }{g_s N_f} +6-\\log (16)\\right)}{64 \\pi ^{7/2} N^{3/2}} \\right]\\nonumber \\\\& & + a^2 \\left(\\frac{3}{4 \\pi ^{3/2} \\sqrt{{g_s}} \\sqrt{N} {r_h}}-\\frac{9 {g_s}^{3/2} M^2 {N_f} \\log ({r_h})\\left(\\frac{8 \\pi }{{g_s} {N_f}}-\\log (N)+12 \\log ({r_h})+6-2 \\log (4)\\right)}{128 \\pi ^{7/2} N^{3/2}{r_h}}\\right).\\nonumber \\\\& &$ Now, we will assume a non-zero `bare' resolution parameter $\\sim \\alpha $ in the resolution parameter $a(r_h)$ and substitute $a = \\left(\\alpha + \\gamma \\frac{g_s M^2}{N} + \\beta \\frac{g_s M^2}{N}\\log r_h\\right)r_h$ [20] into (REF ).", "One can hence calculate the shear viscosity $\\eta $ : $& &\\hspace{-57.81621pt}\\eta = \\Upsilon \\frac{N^{\\frac{2}{5}}}{g_s^2\\alpha _N\\alpha _{\\theta _1}^2}\\lim _{\\omega _3\\rightarrow 0}\\left(\\frac{1}{\\omega _3 T}\\lim _{u\\rightarrow 0}\\left[\\frac{r_h^4}{u^3}\\Im m \\left(\\frac{Z_t^\\prime (u)}{Z_t(u)}\\right)\\right]\\right).$ where $\\Upsilon $ is an overall multiplicative constant.", "For the purpose of comparison of $\\frac{\\eta }{s}$ with lattice/RHIC data for QGP and consequently be able to express $r_h$ in terms of $\\tilde{t}\\equiv \\frac{T}{T_c}-1$ , let us revisit our calculation of $T_c$ as given in [22] with the difference that unlike [22] wherein we had assumed a constant dilaton profile, this time around we will take: $& (a) & T>0:\\nonumber \\\\& & e^{-\\Phi } = \\frac{1}{g_s} - \\frac{N_f}{8\\pi }\\log (r^6 + a^2 r^4) - \\frac{N_f}{2\\pi }\\log \\left(\\sin \\frac{\\theta _1}{2}\\sin \\frac{\\theta _2}{2}\\right),\\ r<{\\cal R}_{D5/\\overline{D5}},\\nonumber \\\\& & e^{-\\Phi } = \\frac{1}{g_s},\\ r>{\\cal R}_{D5/\\overline{D5}};\\nonumber \\\\& (b) & T=0:\\nonumber \\\\& & e^{-\\Phi } = \\frac{1}{g_s} - \\frac{3 N_f}{4\\pi }\\log r - \\frac{N_f}{2\\pi }\\log \\left(\\sin \\frac{\\theta _1}{2}\\sin \\frac{\\theta _2}{2}\\right),\\ r<\\left|\\mu _{\\rm Ouyang}\\right|^{\\frac{2}{3}},\\nonumber \\\\& & e^{-\\Phi } = \\frac{1}{g_s},\\ r>\\left|\\mu _{\\rm Ouyang}\\right|^{\\frac{2}{3}}.$ Hence, setting the Newtonian constant to unity, performing a large-N expansion and then a large $r_\\Lambda $ -expansion, one obtains for the thermal background ($r_h=0$ ) for which $r\\in [r_0,r_\\Lambda ]$ where $r_0$ and $r_\\Lambda $ are respectively the IR and UV cut-offs: $& & V_1 = -\\frac{1}{2}\\int _{r=r_0}^{r_\\Lambda } d^5x\\sqrt{-g}e^{-2\\Phi }\\left(R - 2\\Lambda \\right) - \\int _{r=r_\\Lambda }\\sqrt{-h}e^{-2\\Phi }K\\nonumber \\\\& & = \\frac{3 {r_\\Lambda }^4}{2{N^{5/4}} \\sqrt{2} \\pi ^{5/4} {g_s}^{13/4}} -\\frac{1}{32 \\left(\\sqrt{2} \\pi ^{9/4} {g_s}^{13/4}\\right)}\\Biggl \\lbrace \\frac{1}{{N^{5/4}}} \\Biggl (4\\pi {g_s}^2 \\log ^2 N \\left({|\\mu _{\\rm Ouyang}|^{\\frac{8}{3}}}-{r_0}^4\\right)\\nonumber \\\\& & - {g_s} \\log N\\Biggl [\\left({r_0}^4-{|\\mu _{\\rm Ouyang}|^{\\frac{8}{3}}}\\right) (-12 {g_s} {N_f} \\log ({|\\mu _{\\rm Ouyang}|^{\\frac{2}{3}}})+{g_s} {N_f} (3+16 \\pi \\log (4))+16 \\pi )\\nonumber \\\\& & -12{g_s} {N_f} {r_0}^4 \\log \\left(\\frac{{r_0}}{{|\\mu _{\\rm Ouyang}|^{\\frac{2}{3}}}}\\right)\\Biggr ]-16 \\pi \\left({|\\mu _{\\rm Ouyang}|^{\\frac{8}{3}}}-2{r_h}^4\\right)\\Biggr )\\Biggr \\rbrace \\nonumber \\\\& & + {\\cal O}\\left(\\frac{1}{{r_\\Lambda }^2}\\right).$ Similarly, for the black hole background, for which $r\\in [r_h,r_\\Lambda ]$ one obtains: $& & V_2 = -\\frac{1}{2}\\int _{r=r_h}^{r_\\Lambda } d^5x\\sqrt{-g}e^{-2\\Phi }\\left(R - 2\\Lambda \\right) - \\int _{r=r_\\Lambda }\\sqrt{-h}e^{-2\\Phi }K\\nonumber \\\\& & = \\frac{3 \\left(\\frac{1}{N}\\right)^{5/4} {r_\\Lambda }^4}{2 \\sqrt{2} \\pi ^{5/4} {g_s}^{13/4}}+\\frac{9 a^2 \\left(\\frac{1}{N}\\right)^{5/4}{r_\\Lambda }^2}{4 \\sqrt{2} \\pi ^{5/4} {g_s}^{13/4}}\\nonumber \\\\& & +\\frac{1}{32 \\sqrt{2} \\pi ^{9/4} {g_s}^{13/4}{{\\cal R}^2_{D5/\\overline{D5}}}}\\Biggl \\lbrace \\frac{1}{N^{5/4}} \\Biggl [6 \\pi a^2 {g_s}^2 \\log ^2\\left(\\frac{1}{N}\\right) \\left({r_h}^4-{{\\cal R}^4_{D5/\\overline{D5}}}\\right)\\nonumber \\\\& & - 3 a^2 {g_s} \\log N \\Biggl (6 {g_s} {N_f}\\left({r_h}^4-{{\\cal R}^4_{D5/\\overline{D5}}}\\right) \\log ({{\\cal R}_{D5/\\overline{D5}}})\\nonumber \\\\& & +\\left({{\\cal R}^2_{D5/\\overline{D5}}}-{r_h}^2\\right) \\Biggl [{g_s} {N_f} \\left({{\\cal R}^2_{D5/\\overline{D5}}} (8 \\pi \\log (4)-9)+{r_h}^2 (8 \\pi \\log (4)-3)\\right)\\nonumber \\\\& & +8 \\pi \\left({{\\cal R}^2_{D5/\\overline{D5}}}+{r_h}^2\\right)\\Biggr ]\\Biggr )+8 \\pi \\left(3 a^2\\left({{\\cal R}^4_{D5/\\overline{D5}}}-{r_h}^4\\right)-4 {{\\cal R}^2_{D5/\\overline{D5}}} {r_h}^4\\right)\\Biggr ]\\Biggr \\rbrace \\nonumber \\\\& & + {\\cal O}\\left(\\frac{1}{{r_\\Lambda ^2}}\\right).$ Now, in the $r_\\Lambda \\rightarrow \\infty $ -limit, realizing: $& & \\left.\\sqrt{-h^{\\rm Thermal}}\\right|_{r=r_\\Lambda } = \\frac{{r_\\Lambda }^4}{4 \\pi {g_s} N},\\nonumber \\\\& & \\left.\\sqrt{-h^{\\rm BH}}\\right|_{r=r_\\Lambda } = \\frac{{r_\\Lambda }^4-3 a^2 {r_\\Lambda }^2}{8 \\sqrt{2} \\pi ^{3/4} {g_s}^{3/4} N^{3/4}},$ one sees that the required counter term required to be added to $V_2-V_1$ (required later) is: $& & \\int _{r=r_\\Lambda }\\left(-\\frac{3 \\@root 4 \\of {\\frac{1}{N}} \\left(\\@root 4 \\of {\\pi } \\@root 4 \\of {{g_s}} \\@root 4 \\of {N} \\sqrt{-h^{\\rm Thermal}}-2 \\sqrt{2}\\sqrt{-h^{\\rm BH}}\\right)}{\\sqrt{2 \\pi } {g_s}^{5/2} \\@root 4 \\of {N}}\\right).$ Therefore, $& & \\hspace{-21.68121pt}(V_2 - V_1)^{\\rm UV-finite} = \\nonumber \\\\& & \\frac{1}{32 \\sqrt{2} \\pi ^{9/4} {g_s}^{13/4}}\\left(\\frac{1}{N}\\right)^{5/4} \\Biggl (3 {g_s}^2 {\\log N} {N_f} \\left(9 a^4-{r_h}^4\\right) (2 \\log (a)+\\log (3))\\nonumber \\\\& & -9 a^4\\left({g_s}^2 {\\log N} (2 \\pi {\\log N}+{N_f} (8 \\pi \\log (4)-9))+8 \\pi {g_s} {\\log N}-8 \\pi \\right)-18 a^2{g_s}^2 {\\log N} {N_f} {r_h}^2\\nonumber \\\\& & +4 \\pi {g_s}^2 {\\log N}^2 \\left(|\\mu _{\\rm Ouyang}|^{8/3}-{r_0}^4\\right)-{g_s} {\\log N}\\Biggl [\\left({r_0}^4-|\\mu _{\\rm Ouyang}|^{8/3}\\right) (-8 {g_s} {N_f} \\log (|\\mu _{\\rm Ouyang}|)\\nonumber \\\\& & +{g_s} {N_f} (3+16 \\pi \\log (4))+16 \\pi ) -12{g_s} {N_f} {r_0}^4 \\log \\left(\\frac{{r_0}}{|\\mu _{\\rm Ouyang}|^{2/3}}\\right)\\Biggr ]-16 \\pi \\left(|\\mu _{\\rm Ouyang}|^{8/3}-2{r_h}^4\\right)\\nonumber \\\\& & -{r_h}^4 \\left({g_s}^2 {\\log N} ({N_f} (3-8 \\pi \\log (4))-2 \\pi {\\log N}) -8 \\pi {g_s}{\\log N}+40 \\pi \\right)\\Biggr ).$ Now assuming ${\\cal R}_{D5/\\overline{D5}} = \\sqrt{3}a$ (to be justified via a finite temperature $0^{++}$ glueball mass calculation via the WKB quantization method in [39]), $|\\mu _{\\rm Ouyang}|^{\\frac{2}{3}}=\\delta r_0$ and assuming an IR-valued $r_h, r_0$ , $(V_2 - V_1)^{\\rm UV-finite} =0$ yields: $& & \\hspace{-43.36243pt}2 \\pi {g_s}^2 {\\log N}^2 \\left(\\left(1-9 \\alpha ^4\\right) {rh}^4+2 \\left(\\delta ^{8/3}-1\\right) {r_0}^4\\right)+6 {g_s}^2{\\log N} {N_f} \\left(\\left(9 \\alpha ^4-1\\right) {r_h}^4 \\log ({rh})-2 \\left(\\delta ^{8/3}-1\\right) {r0}^4 \\log ({r_0})\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}-8 \\pi \\left(\\left(1-9 \\alpha ^4\\right) {r_h}^4+2 \\delta ^{8/3} {r0}^4\\right) = 0,$ whose solution is given by: $& & \\hspace{-28.90755pt}\\frac{\\@root 4 \\of {\\frac{2}{3}} \\@root 4 \\of {-\\frac{\\left(9 \\alpha ^4-1\\right) {rh}^4 \\left(\\pi \\left({g_s}^2 {\\log N}^2-4\\right)-3{g_s}^2 {\\log N} {N_f} \\log ({rh})\\right)}{\\left(\\delta ^{8/3}-1\\right) {g_s}^2 {\\log N}{N_f}}}}{\\@root 4 \\of {{\\cal PL}\\left(-\\frac{\\left(9 \\alpha ^4-1\\right) {rh}^4 \\left(2^{\\frac{1}{2} \\left(\\delta ^{8/3}-1\\right)} \\exp \\left(\\frac{\\pi \\left(\\delta ^{8/3} \\left(4-{g_s}^2 {\\log N}^2\\right)+{g_s}^2 {\\log N}^2\\right)}{3 {g_s}^2 {\\log N}{N_f}}\\right)\\right)^{\\frac{4}{\\delta ^{8/3}-1}} \\left(\\pi \\left({g_s}^2 {\\log N}^2-4\\right)-3 {g_s}^2 {\\log N} {N_f}\\log ({rh})\\right)}{6 \\left(\\delta ^{8/3}-1\\right) {g_s}^2 {\\log N} {N_f}}\\right)}},\\nonumber \\\\& &$ where ${\\cal PL}$ is the `ProductLog' function.", "This yields: $r_0 = r_h\\@root 4 \\of {\\left|\\frac{9 \\alpha ^4-1}{2(\\delta ^{\\frac{8}{3}}-1)}\\right|} + {\\cal O}\\left(\\frac{1}{\\log N}\\right).$ For subsequent calculations and comparison with RHIC data, we will be defining: $\\tilde{t}\\equiv \\frac{T}{T_c}-1$ .", "Now, as we will show in [39], the lightest $0^{++}$ scalar glueball mass is given by: $m_{\\rm glueball} \\approx \\frac{4 r_0}{L^2}.$ Now, lattice calculations for $0^{++}$ scalar glueball masses at zero temperature (zero temperature being relevant as the IR cut-off is required at zero temperature whereas $r_h$ provides the same for the black hole background) [40], yield the lightest mass to be around $1,700 MeV$ .", "From (REF ), replacing $\\frac{r_0}{L^2}$ by $\\frac{m_{\\rm glueball}}{4}$ we obtain: $T_c = \\left.\\frac{m_{\\rm glueball}\\left(1 + \\frac{3\\alpha ^2}{2}\\right)}{2^{\\frac{7}{4}}\\pi \\@root 4 \\of {\\left|\\frac{9 \\alpha ^4-1}{2(\\delta ^{\\frac{8}{3}}-1)}\\right|}}\\right|_{\\alpha =0.6,\\delta =1.02}=179 MeV.$ As the expressions in the following will become very cumbersome to deal with and to type, specially for the purpose of comparison with RHIC QGP data, we will henceforth deal only with numerical expressions setting $g_s=0.9, N=100, M=3, N_f=2, \\alpha =0.6, \\delta =1.02$ .", "We now discuss the $\\frac{1}{N}$ -corrections to the entropy density $s$ by estimating the same from the $D=11$ supergravity action result of [3], and hence work out the $\\frac{1}{N}$ corrections to $\\frac{\\eta }{s}$ .", "The UV-finite part of the $D=11$ supergravity action, given by the Gibbons-Hawking-York (GHY) surface action $S_{\\rm GHY}$ from [3] (without worrying about overall multiplicative constants) is [3]: $\\left.\\int _{r={\\cal R}_{\\rm UV}\\equiv {\\rm UV\\ cut-off}}K\\sqrt{h}\\right|_{\\theta _{1,2}\\sim 0}\\sim \\frac{\\cot ^2\\theta _1 f_2(\\theta _2)}{g_s^{\\frac{11}{4}}N^{\\frac{3}{4}}\\left(\\sin ^2\\theta _1 + \\sin ^2\\theta _2\\right)}\\left(\\frac{1}{T}\\right),$ ($K$ being the extrinsic curvature and $h$ being the determinant of the pull-back of the $D=11$ metric on to $r={\\cal R}_{\\rm UV}$ ).", "Further, assume that what appears in (REF ) is $f_1(\\theta _1)$ .", "Now, unlike the scaling given in (REF ) used in [3], we will be using: $\\theta _{1,2}\\rightarrow 0$ as $\\theta _1=\\frac{\\alpha _{\\theta _1}}{N^{\\frac{1}{5}}},\\ \\theta _2=\\frac{\\alpha _{\\theta _2}}{N^{\\frac{3}{10}}} (N\\sim 10^8)$ - as used in [22] (to discuss a local $SU(3)$ structure of the type IIA delocalized SYZ mirror and a local $G_2$ structure of its M-theory uplift), as well as this paper.", "This can be used to evaluate $ S^{\\rm UV-finite}_{GHY}$ and the entropy density: $s = - T \\frac{\\partial S^{\\rm UV-finite}_{\\rm GHY}}{\\partial T} - S^{\\rm UV-finite}_{\\rm GHY}$ .", "This yields: $& & \\hspace{-57.81621pt}\\frac{\\eta }{s} = {\\cal O}(1)\\times \\nonumber \\\\& & \\hspace{-57.81621pt}\\left[\\frac{\\frac{1}{4 \\pi }-0.00051 \\log ({r_h})}{1 - 0.064 \\gamma + 0.004 \\gamma ^2 + \\sum _{n=1}^4 a_n(\\beta ,\\gamma )\\log ^n r_h + \\frac{\\sum _{n=0}^4 b_n(\\beta ,\\gamma )\\log ^n r_h}{\\sum _{n=0}^2c_n(\\beta ,\\gamma )\\log ^nr_h}}\\right],$ where $a_n,b_n,c_n$ are known functions of $\\beta $ and $\\gamma $ , and there is freedom to choose the ${\\cal O}(1)$ constant.", "We will impose two conditions, as per RHIC QGP data, on $\\beta $ and $\\gamma $ and the ${\\cal O}(1)$ constant: $\\left.\\frac{\\eta }{s}\\right|_{T=T_c}=0.1,$ and $\\left.\\frac{d\\left(\\frac{\\eta }{s}\\right)}{d\\tilde{t}}\\right|_{\\tilde{t}>0}>0$ .", "Numerically, one sees that setting $(\\beta ,\\gamma )=(4,4)$ and consequently $r_h=\\frac{35546.9 ({\\tilde{t}}+1)}{{\\cal PL}(2706.3 ({\\tilde{t}}+1))}$ where ${\\cal P L}$ is the “ProductLog\" function, and the ${\\cal O}(1)$ constant equal to $5.8$ , fits the bill.", "Hence, $& & \\hspace{-50.58878pt}\\frac{\\eta }{s} = 5.8\\Biggl [\\frac{{9.18\\times 10^{-8}} \\log ^3({r_h})-1.6\\times 10^{-5} \\log ^2({r_h})+2.7\\times 10^{-4} \\log ({r_h})+1.7\\times 10^{-3}}{-{2.5\\times 10^{-7}} \\log ^6({r_h})+\\frac{9 \\log ^5({r_h})}{10^6}-\\frac{\\log ^4({r_h})}{10^4}+3.1\\times 10^{-4}\\log ^3({r_h})+0.002 \\log ^2({r_h})+3.6\\times 10^{-3} \\log ({r_h})+0.047}\\Biggr ].\\nonumber \\\\& &$ The graphical variation of $\\frac{\\eta }{s}\\left(N_f=3,M=3,g_s=0.9,N=100\\right)$ vs. $\\tilde{t}=\\frac{T - T_c}{T_c}$ is shown in the following graph in Figure 2, and the RHIC data plot from [41]One of us (KS) thanks R. Lacey to permit us to reproduce the graph in Figure 3 from their paper [41]., is shown in Figure 3.", "Figure: η s\\frac{\\eta }{s} vs. T-T c T c \\frac{T-T_c}{T_c} for T≥T c T\\ge T_c assuming η s T=T c =0.1\\left.\\frac{\\eta }{s}\\right|_{T=T_c}=0.1Figure: η s\\frac{\\eta }{s} vs. T-T c T c \\frac{T-T_c}{T_c} reproduced from .We draw a third graph in which the plots of Figures 2 and 3 are drawn on the same graph.", "Figure: Combined plots of Figures 2 and 3: the graph in red is from Figure 2 (our calculations) and the set of three points with error bars are from Figure 3 (RHIC QGP data from ).The combined plots in Figure 4 make the comparison of our results with those of RHIC data in [41], very clear.", "We conclude the following: $\\frac{\\eta }{s}(T=T_c)= 0.1$ , and $\\left.\\frac{d\\left(\\frac{\\eta }{s}\\right)}{dT}\\right|_{T>T_c}>0$ - this is clear from Figure 2.", "The numerical values, unlike [41], remain close to the value at $T=T_c$ .", "In other words, unlike Figure 3 quoted from [41], in Figure 4, $\\frac{\\eta }{s}$ is found to be a much more slowly varying function of $\\tilde{t}=\\frac{T - T_c}{T_c}$ .", "Also, $\\frac{d^2\\left(\\frac{\\eta }{s}\\right)}{d\\tilde{t}^2}<0$ in Figure 2 and $\\frac{d^2\\left(\\frac{\\eta }{s}\\right)}{d\\tilde{t}^2}>0$ in Figure 3.", "The error bars appearing in Figure 3 as shown more clearly in Figure 4, for $\\frac{T - T_c}{T_c}\\in [0,1.1]$ - the range covered in [41] - permit our deviations from [41] at least for $\\frac{T - T_c}{T_c}\\in [0,0.6]$ ." ], [ "Summary and New Insights into (Transport) Properties of Large $N$ Thermal QCD at Finite Gauge Coupling", "A realistic computation pertaining to thermal QCD systems such as sQGP, require a finite gauge coupling and not just a large t'Hooft coupling [1], and the number of colors $N_c$ equal to three.", "Such computations, missing in the literature, were initiated in [3], [7].", "Further, computations quantifying the IR non-conformality in physical quantities pertaining to large-$N$ thermal QCD at finite gauge coupling that appear at the NLO in $N$ in the corresponding holographic description in string [2]/M-theory [3], have been missing thus far in the literature.", "In this paper, at finite gauge coupling with $N_c=M_{r\\in {\\rm IR}}=3$ as part of the MQGP limit (REF ), we have addressed a Math issue and obtained new insights into some transport properties at LO in $N$ , and non-conformal corrections appearing at the NLO in $N$ in a variety of hydrodynamical quantities crucial to characterizing thermal QCD - like systems at finite gauge coupling such as sQGP.", "In this paper we have discussed identification of the local $T^3$ of [3] (used for construction of the delocalized SYZ type IIA mirror in [3] of [2]'s type IIB holographic dual of large-$N$ thermal QCD) with a special Lagrangian three-cycle, as well as a number of different issues relevant to the transport properties of large-$N$ thermal QCD at finite gauge coupling often inclusive of the non-conformal $\\frac{1}{N}$ temperature-dependent corrections, in the context of gauge/gravity duality.", "For the latter set of issues, the calculations from the gravitational description involves scalar, vector and tensor modes of the asymptotically $AdS_5$ metric perturbations.", "In particular, solving the Einstein's equation involving gauge invariant combination of different perturbations we obtain the quasinormal frequencies.", "The speed of sound follows from the quasinormal frequency corresponding to scalar modes of metric perturbations while the diffusion constant of the shear mode is obtained from the quasinormal frequency corresponding to the vector modes of metric perturbation.", "Before summarizing our main results, we would first summarize the assumptions made to arrive at the results.", "The three-form fluxes of (REF ) obtained in [2] for a resolved warped conifold, is also valid in the UV-IR interpolating region and the UV for a resolved warped deformed conifold which is predominantly (warped and) resolved.", "The reason, as stated below (REF ), has to do with the fact that the corrections due to the resolution parameter $a$ appear as $\\frac{a^2}{r^2}$ whereas those due to the deformation parameter $\\epsilon $ appear as $\\frac{\\epsilon ^2}{r^3}$ .", "In the UV-IR interpolating region, assuming $a,\\epsilon ^{\\frac{3}{2}}<r_h$ and in the UV, the latter is sub-dominant as compared to the former.", "To ensure UV conformality for $r>{\\cal R}_{D5/\\overline{D5}}$ , as explained in the last paragraph of 2.2, one requires a vanishing $D5$ -brane and $D7$ -brane charges in the UV.", "This requires that $F_3$ and $B_3$ vanish in the UV which, as explained in 2, is effected by including $M\\ \\overline{D5}$ -branes which like $M\\ D5$ -branes, wrap the vanishing $S^2$ , but are distributed at the antipodal points of the resolved $S^2$ relative to the $D5$ -branes at $r={\\cal R}_{D5/\\overline{D5}}$ .", "This is implemented in our calculations by assuming that $r\\rightarrow r_\\Lambda $ (UV cut-off)$\\rightarrow \\infty $ as $\\epsilon ^{-\\gamma _r}$ near the coordinate patch $\\theta _{1,2}=0$ effected by $\\theta _{1,2}\\rightarrow \\epsilon ^{\\gamma _{\\theta _{1,2}}}$ for $\\epsilon \\ll 1$ .", "In the MQGP limit of (REF ), one can then show that $\\lim _{r\\rightarrow \\infty }(F_3,H_3)=0$ .", "One can do a better job, as suggested in [21], by working with the following ansatze for ${\\widetilde{F}}_3$ : $&& \\hspace{-21.68121pt}{\\widetilde{F}}_3 = \\left({a}_o - {3 \\over 2\\pi r^{g_sN_f}} \\right)\\sum _\\alpha {2M(r)c_\\alpha \\over r^{\\epsilon _{(\\alpha )}}}\\left({\\rm sin}~\\theta _1~ d\\theta _1 \\wedge d\\phi _1-\\sum _\\alpha {f_\\alpha \\over r^{\\epsilon _{(\\alpha )}}}~{\\rm sin}~\\theta _2~ d\\theta _2 \\wedge d\\phi _2\\right)\\nonumber \\\\&&\\hspace{-21.68121pt}\\wedge ~ {e_\\psi \\over 2}-\\sum _\\alpha {3g_s M(r)N_f d_\\alpha \\over 4\\pi r^{\\epsilon _{(\\alpha )}}}~{dr}\\wedge e_\\psi \\wedge \\left({\\rm cot}~{\\theta _2 \\over 2}~{\\rm sin}~\\theta _2 ~d\\phi _2- \\sum _\\alpha {g_\\alpha \\over r^{\\epsilon _{(\\alpha )}}}~{\\rm cot}~{\\theta _1 \\over 2}~{\\rm sin}~\\theta _1 ~d\\phi _1\\right)\\nonumber \\\\&& \\hspace{-21.68121pt}-\\sum _\\alpha {3g_s M(r) N_f e_\\alpha \\over 8\\pi r^{\\epsilon _{(\\alpha )}}}~{\\rm sin}~\\theta _1 ~{\\rm sin}~\\theta _2 \\left({\\rm cot}~{\\theta _2 \\over 2}~d\\theta _1 +\\sum _\\alpha {h_\\alpha \\over r^{\\epsilon _{(\\alpha )}}}~{\\rm cot}~{\\theta _1 \\over 2}~d\\theta _2\\right)\\wedge d\\phi _1 \\wedge d\\phi _2$ where $M(r)\\equiv 1 - \\frac{e^{\\alpha (r-{\\cal R}_{D5/\\overline{D5}})}}{1 + e^{\\alpha (r-{\\cal R}_{D5/\\overline{D5}})}}, \\alpha \\gg 1$ , and $a_o = 1 + {3\\over 2\\pi }$ and ($c_\\alpha , \\epsilon _{\\alpha }, h_\\alpha $ ) are constants.", "Further investigation using (REF ) however will be deferred to a later work.", "Further, given that the number $N_f$ of flavor $D7$ -branes appears in the expression of the dilaton in (REF ), to ensure a constant axion-dilaton modulus in the UV required for UV conformality, as explained in 2, one adds an equal number of $\\overline{D7}$ -branes in the UV and the UV-IR interpolating region and not the IR.", "This is implemented in our calculations by assuming that $\\frac{3\\gamma _r}{4}=\\gamma _\\theta $ ensuring that $\\lim _{r\\rightarrow \\infty }e^{-\\Phi }(\\theta _{1,2}\\rightarrow 0)=\\frac{1}{g_s}$ .", "The functions $h_i$ of (REF ) appearing in the resolved warped deformed conifold metric (REF ) along with (REF ) are assumed to receive corrections of ${\\cal O}\\left(\\frac{g_s M^2}{N}\\right)$ .", "The $D=10$ warp factor $h(r,\\theta _{1,2},\\phi _{1,2})$ , as stated in (REF ) and (REF ), is assumed to be: $&& \\hspace{-32.52127pt}h =\\frac{L^4}{r^4}\\Bigg [1+\\frac{3g_sM_{\\rm eff}^2}{2\\pi N}{\\rm log}r\\left\\lbrace 1+\\frac{3g_sN^{\\rm eff}_f}{2\\pi }\\left({\\rm log}r+\\frac{1}{2}\\right)+\\frac{g_sN^{\\rm eff}_f}{4\\pi }{\\rm log}\\left({\\rm sin}\\frac{\\theta _1}{2}{\\rm sin}\\frac{\\theta _2}{2}\\right)\\right\\rbrace \\Biggr ],\\ {\\rm in\\ the\\ IR};\\nonumber \\\\& & \\hspace{-32.52127pt}h = \\frac{L^4}{r^4}\\left[1 + \\sum _{i=1}\\frac{{\\cal H}_i\\left(\\phi _{1,2},\\theta _{1,2},\\psi \\right)}{r^i}\\right],\\ {\\rm in\\ the\\ UV},$ where, in principle, $M_{\\rm eff}/N_f^{\\rm eff}$ are not necessarily the same as $M/N_f$ ; we however assume that up to ${\\cal O}\\left(\\frac{g_sM^2}{N}\\right)$ , they are.", "We also assume that ${\\cal H}_i\\left(\\phi _{1,2},\\theta _{1,2},\\psi \\right)={\\cal O}\\left(\\frac{g_s M^2}{N}\\right)$ .", "The following provides a summary of the new results obtained in this paper as well as the new insights into the Physics of strongly coupled thermal QCD laboratories like sQGP gained therefrom.", "Legitimacy of the local $T^3$ of [3] for effecting delocalized SYZ mirror transformation: In the MQGP limit, in the UV (as well as the UV-IR interpolating) region(s): $r\\gg r_h$ , we have shown that the local $T^3$ defined in [3] is the same as the $T^2$ -invariant special Lagrangian (sLag) three-cycle of [6] in a resolved conifold.", "Earlier in [7], it was already shown that in the MQGP limit the aforementioned $T^3$ is also the $T^2$ -invariant sLag of [6] in a deformed conifold.", "Together, the new insight gained is that the local $T^3$ defined in [3] used for constructing the type IIA delocalized Strominger-Yau-Zaslow(SYZ) mirror of [2]'s type IIB holographic dual of large-$N$ thermal QCD, in the MQGP limit, is shown to be a bonafide special Lagrangian three-cycle necessary to construct the required type IIA SYZ mirror.", "This is valid for both, a predominantly resolved (resolution $>$ deformation - this paper) or a predominantly deformed (deformation $>$ resolution - [7]) resolved warped deformed conifold.", "Though we limited ourselves to the LO in $N$ for this purpose, but the same can also be shown to be true at the NLO in $N$ - the computations will become extremely cumbersome though.", "This was crucial in justifying the construction of the SYZ type IIA mirror in [3] of the [2]'s type IIB holographic dual of large-$N$ thermal QCD.", "$\\kappa _T, \\sigma $ , Wiedemann-Franz law at LO in $N$ and $D=1+1$ Luttinger Liquid with impurities: As gauge fluctuations are tied to vector modes of metric fluctuations, by solving the coupled set of equations for both, we obtained the temperature dependence of the thermal and electrical conductivities as well as looked at whether the Wiedemann-Franz law was satisfied.", "This revealed a remarkable insight into the properties of large-$N$ thermal QCD at finite gauge coupling namely that the type IIB holographic dual of large-$N$ thermal QCD with a temperature-dependent Ouyang embedding parameter: $|\\mu _{\\rm Ouyang}|\\sim r_h^{\\alpha \\le 0}$ , effectively qualitatively mimicked a $D=1+1$ Luttinger liquid with impurities/doping.", "It will be extremely interesting to explore this unexpected duality, further.", "For $\\alpha =\\frac{5}{2}$ , one is able to reproduce the usual linear large-temperature dependence of DC electrical conductivity for most strongly coupled systems with five-dimensional gravity duals with a black hole [13].", "The non-conformal/NLO-in-$N$ corrections to Transport Coefficients: For ease of readability and convenience of the reader, the main results pertaining to obtaining the non-conformal temperature-dependent ${\\cal O}\\left(\\frac{(g_sM^2)(g_sN_f)}{N}\\right)$ corrections to $v_s$ (the speed of sound), $D$ (shear mode diffusion constant ), $\\eta $ (shear viscosity) and $\\frac{\\eta }{s}$ (shear-viscosity-entropy density ratio) are summarized in Table 1 below.", "Table: Summary of local non-nonformal 𝒪g s M 2 N{\\cal O}\\left(\\frac{g_sM^2}{N}\\right) corrections to v s ,D,η sv_s, D, \\frac{\\eta }{s}We showed that in the spirit of gauge/gravity duality, the leading order result of speed of sound from the quasinormal modes can be reproduced from (a) the pole of the common denominator that appears in the solutions to the scalar modes of metric perturbations, (b) the pole of the retarded Green's function corresponding to the energy momentum tensor two-point correlation function $\\langle T_{00}T_{00}\\rangle $ using the on-shell surface action written in terms of the metric perturbation modes, (c) imposing Dirichlet boundary condition on the solution to the EOM of an appropriate single gauge-invariant perturbation and (d) $\\langle T_{00}T_{00}\\rangle $ -computation using the on-shell surface action written in terms of this gauge-invariant perturbation.", "The leading order result for the diffusion constant of the shear mode as well as the ratio of shear viscosity-to-entropy density ratio were already discussed in [7].", "The non-trivial insight thus gained at LO in $N$ into the transport properties of holographic large-$N$ thermal QCD at finite gauge coupling is that the LO-in-$N$ conformal result for finite $g_s$ as obtained in this paper, matches the LO-in-$N$ conformal result for vanishing $g_s$ as is expected/known in the literature for a $p$ -brane for $p=3$ .", "The non-conformal corrections in all the aforementioned quantities, start appearing at ${\\cal O}\\left(\\frac{(g_s M^2)(g_s N_f)}{N}\\right)$ , $N_f$ being the number of flavor $D7$ -branes.", "Thus, at NLO in $N$ , the new insight gained is that there is a partial universality in the non-conformal corrections to the transport coefficients in the sense that the same are determined by the product of the very small $\\frac{g_sM^2}{N}\\ll 1$ - part of the MQGP limit (REF ) - and the finite $g_s N_f\\sim {\\cal O}(1)$ (also part of (REF )).", "The NLO-corrected results in this paper reflect the non-conformality of the field theory in the IR.", "As discussed in section 2 that in the Klebanov-Strassler backgroud [11] the number of $D3$ branes $N$ decreases with decreasing (the non-compact radial coordinate) $r$ , which according to AdS/CFT dictionary, behaves as an energy scale.", "This decrease in $N$ is due to a series of repeated Seiberg dualities, where in the extreme IR, at the end of this duality cascade the number of fractional $D3$ branes $M$ which is taken to be finite in the 'MQGP Limit' gets identified with the number of colors in the theory.", "In other words, the number of $D3$ branes $N$ exhibits a scale dependance due to the duality cascade.", "Hence from the NLO-corrected expressions of the shear mode diffusion constant and the viscosity, we conclude that these quantities also exhibit a scale dependance through $N$ ; the appearance of $M$ in the NLO-in-$N$ corrections to the transport coefficients appearing as $\\frac{(g_s M^2)(g_s N_f)}{N}$ signals the non-conformality of the field theory in the IR.", "This is because of the following reason.", "In the KS picture the presence of finite number $M$ of fractional $D3$ branes makes the field theory non-conformal in the IR while in the UV the presence of $\\overline{D5}$ branes cancels the effects of the $D5$ -branes and restore the conformality in the UV.", "Now at large $r$ the effective number $N_{\\rm eff}$ of (REF ), is so large that the NLO term can be neglected and we will be left with the leading order conformal results.", "But in the IR region the NLO terms have to be considered due to small value of $N_{\\rm eff}$ - this is rather nicely captured, e.g., by the non-conformal/NLO corrections to $\\eta $ (See e.g.", "Table 1.)", "We compared our results for $\\frac{\\eta }{s}$ with the QGP-related RHIC data for $T\\ge T_c$ in Section 7.", "Let us also make some remarks as regard comparison of some of our results with some well-cited bottom-up holographic QCD models like [42] (as well as references therein) and the more recent [43] based on the Veneziano's QCD model.", "As regard the speed of sound, like [42], for $T>T_c$ (which is the temperature range in which we calculated the speed of sound in Section 5) the speed of sound approaches a constant value; the difference however is that the NLO non-conformal corrections in our results pushes the value to slightly above $\\frac{1}{\\sqrt{3}}$ - our LO result and the saturation value in [42].", "Upon comparison with some of the results of [44] which works with the finite temperature version of [43], one sees that the authors of the same work in the limit: $N_f\\rightarrow \\infty , N_c\\rightarrow \\infty : \\frac{N_f}{N_c}\\equiv $ fixed and $g_{\\rm YM}^2 N_c\\equiv $ fixed, which is very different from the MQGP limit of (REF ).", "A similarity however pertaining to the QCD phase diagram in the same and our results of [22] is that $\\mu _C(T=T_c)$ (for $N_f=2$ )$\\approx 0$ ." ], [ "Acknowledgements", "One of us (KS) is supported by a senior research fellowship (SRF) from the Ministry of Human Resource and Development (MHRD).", "One of us (AM) would like to thank N. Evans for useful discussions and MPI (Werner Heisenberg Institute) for Physics, Munich, and AEI (MPI for Gravitational Physics), Golm, for their hospitality where part of this work was completed.", "Some of the results of this paper were presented in seminars given by one of us (AM) at U. Southampton, MPI Munich and AEI Golm.", "We also thank P. Pandey for participating in, as part of his Masters project, the material discussed in Sec.", "4." ], [ "Details of Local $T^3$ Being a {{formula:8548b84a-438f-413e-89f8-9ab0eccd85d5}} -Invariant sLag in a Resolved Conifold in the MQGP Limit", "The system of equations (REF ) are solved to yield (REF ).", "$& & \\hspace{-50.58878pt}\\cos ^2\\frac{\\theta _1}{2} = \\nonumber \\\\& & \\hspace{-50.58878pt}\\frac{4 a^2 \\sqrt{3 a^2+\\rho ^2} \\left(-3 {c_1}-3 {c_2}+8 \\rho ^2\\right)+\\rho ^2 \\left(-4 {c_1} \\sqrt{3 a^2+\\rho ^2}-4 {c_2} \\sqrt{3 a^2+\\rho ^2}+\\sqrt{6} \\sqrt{9 a^2 \\rho ^4+\\rho ^6}\\right)+96 a^4 \\sqrt{3 a^2+\\rho ^2}}{32 a^2 \\rho ^2 \\sqrt{3 a^2+\\rho ^2}+2 \\sqrt{6} \\rho ^2 \\sqrt{9 a^2 \\rho ^4+\\rho ^6}+96 a^4 \\sqrt{3 a^2+\\rho ^2}},\\nonumber \\\\& & \\hspace{-50.58878pt}\\cos ^2\\frac{\\theta _2}{2} = \\frac{\\rho ^2 \\left(-4 {c_1} \\sqrt{3 a^2+\\rho ^2}+4 {c_2} \\sqrt{3 a^2+\\rho ^2}+\\sqrt{6} \\sqrt{9 a^2 \\rho ^4+\\rho ^6}\\right)-12 a^2 \\sqrt{3 a^2+\\rho ^2}({c_1}-{c_2})}{2 \\sqrt{6} \\rho ^2 \\sqrt{9 a^2 \\rho ^4+\\rho ^6}}.$ Substituting (REF ) into the third equation of (REF ), one obtains: $& & \\frac{1}{9} \\left(a^2 {\\sin \\psi }^2 \\left(-4 \\left(16 \\sqrt{6}-27\\right) a^2 ({c_1}+{c_2})+\\left(8 \\sqrt{6}-27\\right) {c_1}^2+16 \\sqrt{6} {c_1}{c_2}+\\left(8 \\sqrt{6}-27\\right) {c_2}^2\\right)-9 {c_3}^2\\right)\\nonumber \\\\& & -\\frac{{\\sin \\psi }^2 \\left(32 \\sqrt{6} a^2 ({c_1}-{c_2})^2({c_1}+{c_2})-4 \\sqrt{6} \\left({c_1}^2-{c_2}^2\\right)^2\\right)}{9 \\sqrt{6} \\rho ^2} +\\frac{9}{16} a^2 \\rho ^4 {\\sin \\psi }^2\\nonumber \\\\& & -\\frac{1}{3} \\rho ^2{\\sin \\psi }^2 \\left(8 a^4-4 a^2 ({c_1}+{c_2})+{c_1}^2+{c_2}^2\\right) +\\frac{\\rho ^6 {\\sin \\psi }^2}{16} + {\\cal O}\\left(\\frac{a^2c_1^4\\sin ^2\\psi }{\\rho ^4}\\right) = 0.$ This obtains: $\\rho ^2 = -3 a^2+\\frac{4\\ 2^{2/3} {\\sin \\psi }^{2/3} \\left({c_1}^2+{c_2}^2\\right)}{9 {c_3}^{2/3}}+\\frac{2 \\@root 3 \\of {2} {c_3}^{2/3}}{{\\sin \\psi }^{2/3}} + {\\cal O}\\left(\\frac{a^2c_1^2\\sin ^{\\frac{4}{3}}\\psi }{c_3^{\\frac{4}{3}}}\\right).$ As $\\theta _1,\\theta _2\\rightarrow 0$ as $\\frac{1}{N^{\\frac{1}{5}}},\\frac{1}{N^{\\frac{3}{10}}}$ (whereat an explicit local $SU(3)$ -structure of the type IIA mirror and an explicit local $G_2$ -structure of the M-theory uplift was obtained in [22]) and in the UV-IR interpolating region/UV: $r\\rightarrow {\\cal R}_0$ , therefore in this domain of $(\\theta _1,\\theta _2,r)$ choose: $c_1\\sim {\\cal R}_0^2,\\ c_2\\sim \\frac{{\\cal R}_0^2}{N^{\\frac{2}{5}}}.$ Hence, making a large-$N$ expansion: $& & \\hspace{-86.72377pt}\\cos ^2\\theta _2 = \\nonumber \\\\& & \\hspace{-86.72377pt}\\frac{1}{4\\left(-3 a^2+\\frac{4\\ 2^{2/3} {\\rho _0}^4 {\\sin \\psi }^{2/3}}{9 {c_3}^{2/3}}+\\frac{2 \\@root 3 \\of {2} {c_3}^{2/3}}{{\\sin \\psi }^{2/3}}\\right)\\sqrt{\\frac{19683 a^6 {c_3}^2 {\\sin \\psi }^2+54\\ 2^{2/3} {c_3}^{4/3} {\\rho _0}^4 {\\sin \\psi }^{8/3} \\left(16 {\\rho _0}^4-81a^4\\right)-243 \\@root 3 \\of {2} {c_3}^{8/3} {\\sin \\psi }^{4/3} \\left(81 a^4-16 {\\rho _0}^4\\right)+5832 {c_3}^4+128 {\\rho _0}^{12}{\\sin \\psi }^4}{{c_3}^2 {\\sin \\psi }^2}}}\\nonumber \\\\& & \\hspace{-86.72377pt}\\times \\Biggl \\lbrace 9 \\sqrt{3} \\Biggl (\\frac{1}{243 {c_3}^{2/3} {\\sin \\psi }^{2/3}}\\Biggl \\lbrace 2 \\left(-27 a^2 {c_3}^{2/3} {\\sin \\psi }^{2/3}+18 \\@root 3 \\of {2} {c_3}^{4/3}+4\\ 2^{2/3} {\\rho _0}^4{\\sin \\psi }^{4/3}\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}\\Biggl [\\sqrt{3} \\sqrt{\\frac{19683 a^6 {c_3}^2 {\\sin \\psi }^2+54\\ 2^{2/3} {c_3}^{4/3} {\\rho _0}^4 {\\sin \\psi }^{8/3}\\left(16 {\\rho _0}^4-81 a^4\\right)-243 \\@root 3 \\of {2} {c_3}^{8/3} {\\sin \\psi }^{4/3} \\left(81 a^4-16 {\\rho _0}^4\\right)+5832 {c_3}^4+128{\\rho _0}^{12} {\\sin \\psi }^4}{{c_3}^2 {\\sin \\psi }^2}}\\nonumber \\\\& & \\hspace{-57.81621pt}- 18\\ 2^{2/3} {\\rho _0}^2 \\sqrt{\\frac{2 \\@root 3 \\of {2} {\\rho _0}^4{\\sin \\psi }^{2/3}}{{c_3}^{2/3}}+\\frac{9 {c_3}^{2/3}}{{\\sin \\psi }^{2/3}}}\\Biggr ]\\Biggr \\rbrace -4\\ 2^{2/3} a^2{\\rho _0}^2 \\sqrt{\\frac{2 \\@root 3 \\of {2} {\\rho _0}^4 {\\sin \\psi }^{2/3}}{{c_3}^{2/3}}+\\frac{9 {c_3}^{2/3}}{{\\sin \\psi }^{2/3}}}\\Biggr )\\Biggr \\rbrace + {\\cal O}\\left(\\frac{1}{N^{\\frac{2}{5}}}\\right).$ Making subsequently a small-$\\psi $ expansion: $\\cos ^2\\theta _2 = \\frac{0.003 {\\sin \\psi }^2 \\left(32 {\\rho _0}^6-405 a^4 {\\rho _0}^2\\right)}{{c_3}^2}-\\frac{0.386 a^2 {\\rho _0}^2{\\sin \\psi }^{4/3}}{{c_3}^{4/3}}-\\frac{0.324 {\\rho _0}^2 {\\sin \\psi }^{2/3}}{{c_3}^{2/3}}+0.5+ {\\cal O}\\left(\\frac{a^2\\rho _0^6\\sin ^{\\frac{8}{3}}\\psi }{c_3^{\\frac{8}{3}}}\\right).$ For (REF ) to be a valid embedding near $\\theta _1=\\frac{1}{N^{\\frac{1}{5}}}, \\theta _2=\\frac{1}{N^{\\frac{3}{10}}}$ (whereat an explicit local $SU(3)$ -structure of the type IIA mirror and an explicit local $G_2$ -structure of the M-theory uplift was obtained in [22]) in the UV, $\\psi $ is near $\\langle \\psi \\rangle $ determined by: $0.5 - 0.32 \\xi + \\frac{a^2(-0.386 - 0.82 a^4 + 0.216\\rho _0^4)\\xi ^2}{\\rho _0^4} + \\left(0.091 - \\frac{1.148 a^4}{c_3\\rho _0^4}\\right) = 1$ where $\\xi \\equiv \\frac{\\rho _0^2\\sin ^{\\frac{2}{3}}\\langle \\psi \\rangle }{c_3^{\\frac{2}{3}}}$ .", "This is solved to yield: $\\xi = 2.416 - 0.993 (a \\rho _0)^2 + 0.304 (a \\rho _0)^4 - 0.034 (a \\rho _0)^6 + {\\cal O}\\left((a \\rho _0)^8\\right).$ One sees that (REF ) can be satisfied by requiring: $\\xi \\sim 1.9;\\ a^2 \\rho _0^2 \\sim 0.8,$ implying: $\\sin \\langle \\psi \\rangle \\sim \\frac{1.9^{\\frac{3}{2}}c_3}{\\rho _0^3},\\ a\\sim \\frac{0.9}{\\rho _0}.$ Similarly, $& & \\cos ^2\\theta _1 = \\frac{1}{2} + \\frac{\\left(3 \\sqrt{6}-16\\right) a^2 {\\sin \\psi }^{4/3} \\left(4 a^2-{\\rho _0}^2\\right)}{12\\ 2^{2/3} {c_3}^{4/3}}+\\frac{{\\sin \\psi }^{2/3} \\left(4a^2-{\\rho _0}^2\\right)}{2^{5/6} \\sqrt{3} {c_3}^{2/3}}\\nonumber \\\\& & -\\frac{{\\sin \\psi }^2 \\left(4 a^2-{\\rho _0}^2\\right) \\left(3 \\left(576-391\\sqrt{6}\\right) a^4+32 \\sqrt{6} {\\rho _0}^4\\right)}{864 {c_3}^2} + {\\cal O}\\left({\\sin \\psi }^{7/3}\\right).$ From (REF ), (REF ) and (REF ): $& & \\rho d\\rho \\approx \\left(- \\frac{2^{\\frac{4}{3}}}{3\\sin ^{\\frac{5}{3}}\\psi } + \\frac{2^{\\frac{5}{3}}(c_1^2+c_2^2)}{3^3c_3^{\\frac{2}{3}}}\\sin ^{\\frac{1}{3}}\\psi \\right)d\\psi ;\\nonumber \\\\& & -\\sin 2\\theta _1d\\theta _1\\approx -2\\sin \\theta _1d\\theta _1\\approx \\left(-\\frac{2}{3}\\frac{\\rho _0^2}{2^{\\frac{5}{6}}\\sqrt{3}\\sin ^{\\frac{1}{3}}\\psi c_3^{\\frac{2}{3}}} + \\frac{4a^2(16 - 3\\sqrt{6})\\rho _0^2\\sin ^{\\frac{1}{3}}\\psi }{36 2^{\\frac{2}{3}}c_3^{\\frac{4}{3}}}\\right)d\\psi \\nonumber \\\\& & \\approx -2\\left(\\frac{2^{\\frac{1}{6}}}{3^{\\frac{3}{2}}\\sin \\psi } + {\\cal O}\\left(\\frac{a^2}{\\sin \\psi \\rho _0^2}\\right)\\right)d\\psi ;\\nonumber \\\\& & -\\sin 2\\theta _2d\\theta _2\\approx -2\\sin \\theta _2d\\theta _2\\approx \\left(\\frac{0.64}{\\rho }_0^2{3c_3^{\\frac{2}{3}}\\sin ^{\\frac{1}{3}}\\psi } - \\frac{1.56\\rho _0^2\\sin ^{\\frac{1}{3}}\\psi a^2}{3c_3^{\\frac{4}{3}}}\\right)d\\psi \\nonumber \\\\& & -2\\left(\\frac{0.64}{3\\sin \\psi } + {\\cal O}\\left(\\frac{a^2}{\\sin \\psi \\rho _0^2}\\right)\\right)d\\psi .$ So, writing $c_1=\\alpha _{c_1}\\rho _0^2, d\\phi _1=\\frac{\\beta _{\\phi _1}dx}{\\left(g_sN\\right)^{\\frac{1}{4}}\\frac{1}{N^{\\frac{1}{5}}}} d\\phi _2=\\frac{\\beta _{\\phi _2}dy}{\\left(g_sN\\right)^{\\frac{1}{4}}\\frac{1}{N^{\\frac{3}{10}}}}, d\\psi =\\frac{\\beta _{\\psi }dz}{\\left(g_sN\\right)^{\\frac{1}{4}}}$ : $& & \\frac{\\rho }{3}d\\rho \\wedge \\cos \\theta _1d\\phi _1\\approx \\left(\\frac{0.07\\alpha _{c_1}^2\\rho _0^2}{\\sin \\psi } - \\frac{0.84}{\\sin ^{\\frac{5}{3}}\\psi }\\right)N^{\\frac{3}{10}}\\beta _{\\psi }\\beta _{\\phi _1}\\frac{dz\\wedge dx}{\\sqrt{g_s}};\\nonumber \\\\& & \\frac{\\rho }{3}d\\rho \\wedge \\cos \\theta _2d\\phi _2\\approx \\left(\\frac{0.07\\alpha _{c_1}^2\\rho _0^2}{\\sin \\psi } - \\frac{0.84}{\\sin ^{\\frac{5}{3}}\\psi }\\right)\\beta _{\\psi }\\beta _{\\phi _2}\\frac{dz\\wedge dy}{\\sqrt{g_s}};\\nonumber \\\\& & \\frac{\\rho ^2}{6}\\sin \\theta _1d\\phi _1\\wedge d\\theta _1\\approx -\\frac{0.007\\rho ^2_\\Lambda \\beta _{\\psi }\\beta _{\\phi _1}}{\\sin \\psi }\\frac{dz\\wedge dx}{\\sqrt{g_s}};\\nonumber \\\\& & \\frac{(\\rho ^2+6a^2)}{6}\\sin \\theta _2d\\phi _2\\wedge d\\phi _2\\sim -\\frac{\\rho _0^2}{6}\\frac{0.64}{3\\sin \\psi }d\\phi _2\\wedge d\\psi \\nonumber \\\\& & =-\\frac{0.03\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _2}}{\\sin \\psi }\\frac{dz\\wedge dy}{\\sqrt{g_s}}.$ This implies: $& & i^*J\\approx \\left(\\frac{0.07\\alpha _{c_1}^2\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _1}}{\\sqrt{g_s}\\sin \\psi N^{\\frac{3}{10}}} - \\frac{0.007\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _1}}{\\sin \\psi \\sqrt{g_s}N^{\\frac{3}{10}}} - \\frac{0.84\\beta _{\\psi }\\beta _{\\phi _1}}{N^{\\frac{3}{10}}\\sqrt{g_s}\\sin ^{\\frac{5}{3}}\\psi }\\right)dz\\wedge dx\\nonumber \\\\& & + \\left(\\frac{0.07\\alpha _{c_1}^2\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _2}}{\\sqrt{g_s}\\sin \\psi N^{\\frac{3}{10}}} - \\frac{0.03\\rho _0^2\\beta _{\\psi }\\beta _{\\phi _2}}{\\sin \\psi \\sqrt{g_s}} - \\frac{0.84\\beta _{\\psi }\\beta _{\\phi _2}}{N^{\\frac{3}{10}}\\sqrt{g_s}\\sin ^{\\frac{5}{3}}\\psi }\\right)dz\\wedge dy.$ Further, $& & -i\\frac{\\rho ^2}{6}d\\rho (\\cos \\psi - i \\sin \\psi )\\wedge \\sin \\theta _1d\\phi _1\\wedge \\sin \\theta _2d\\phi _2\\approx -i\\frac{\\rho _0^6\\cos \\psi }{18\\left(g_sN\\right)^{\\frac{3}{4}}}\\frac{\\left(\\frac{2\\alpha _{c_1}^2}{3^3} - \\frac{2^{\\frac{4}{3}}}{3\\alpha }\\right)}{\\alpha ^{\\frac{3}{2}}}\\beta _{\\psi }\\beta _{\\phi _1}\\beta _{\\phi _2}dz\\wedge dx\\wedge dy;\\nonumber \\\\& & -\\frac{\\rho ^3}{18}(\\cos \\psi - i \\sin \\psi )\\sin \\theta _1d\\phi _1\\wedge \\sin \\theta _2d\\phi _2\\wedge d\\psi \\approx -\\frac{\\rho _0^3\\cos \\psi \\beta _{\\psi }\\beta _{\\phi _1}\\beta _{\\phi _2}}{\\left(g_sN\\right)^{\\frac{3}{4}}}dx\\wedge dy\\wedge dz;\\nonumber \\\\& & -i\\frac{\\rho ^3}{18}(\\cos \\psi - i\\sin \\psi )(\\sin \\theta _1\\cos \\theta _2d\\theta _2 + \\sin \\theta _2\\cos \\theta _1d\\theta _1)\\wedge d\\phi _1\\wedge d\\phi _2\\nonumber \\\\& & \\approx -i\\frac{\\rho _0^6\\cos \\psi }{36\\left(g_sN\\right)^{\\frac{3}{4}}}\\frac{\\beta _{\\psi }\\beta _{\\phi _1}\\beta _{\\phi _2}}{\\alpha ^{\\frac{3}{2}}}\\left(N^{\\frac{1}{10}}0.21 + \\frac{2^{\\frac{1}{6}}}{3^{\\frac{3}{2}}N^{\\frac{1}{10}}}\\right)dz\\wedge dx\\wedge dy.$" ], [ "EOMs for (Vector Mode) Metric and Gauge Fluctuations, and Their Solutions near $u=0$", "(A)  EOM for $H_{ty}$ $& & \\frac{e^{2 i (q x-t w)}\\sqrt{\\mu } \\sqrt{c^2 e^{2 \\Phi }+\\left(\\frac{r_h}{u}\\right)^{9/2}} r_h^2 \\left(H_{ty}(u) \\sqrt{c^2 e^{2 \\Phi }+\\left(\\frac{r_h}{u}\\right)^{9/2}}-ce^{\\Phi } u^4 \\phi ^{\\prime }(u)+c e^{\\Phi } \\phi ^{\\prime }(u)\\right)}{36\\sqrt{c^2 e^{2 \\Phi }u^4+\\left(\\frac{r_h^9}{u}\\right)^{1/2}} r_h \\left(u^4-1\\right)}-\\nonumber \\\\& & \\frac{e^{i (q x-t w)}\\sqrt{\\frac{r_h^8}{L^6 u^{10} g_s^{10/3}}} \\Biggl (g_1(u) r_h^2 \\left(3 H_{ty}^{\\prime }(u)-u H_{ty}^{\\prime \\prime }(u)\\right)+L^4 q^2 u H_{ty}(u)+L^4 q u wH_{xy}(u)\\Biggr )}{2 L^4 u}=0.\\nonumber \\\\& &$ (B)  EOM for $H_{xy}$ $& & \\hspace{-50.58878pt}\\frac{e^{i (q x-2 t w)}}{36 L^4 u^4 g_1(u)}\\Biggl [L^4 {H_{xy}}(u)\\Biggl \\lbrace \\sqrt{\\mu } {r_h}^3 g_1(u) e^{i q x} \\sqrt{\\frac{{r_h}}{u}} \\sqrt{\\frac{{r_h}^4}{c^2 e^{2 \\Phi } u^4+\\left(\\frac{r_h^9}{u}\\right)^{1/2}}}+18 u^4 w^2 e^{i t w} \\sqrt{\\frac{r_h^8}{L^6 u^{10}g_s^{10/3}}}\\Biggr \\rbrace \\nonumber \\\\& & \\hspace{-50.58878pt}+18 L^4 q u^4 w {H_{ty}}(u) e^{i t w} \\sqrt{\\frac{r_h^8}{L^6 u^{10} g_s^{10/3}}}+18 u^3 g_1(u) r_h^2 e^{i t w} \\Biggl (u g_1(u){H_{xy}}^{\\prime \\prime }(u)+\\left(g_1(u)-4\\right) {H_{xy}}^{\\prime }(u)\\Biggr ) \\sqrt{\\frac{r_h^8}{L^6 u^{10} g_s^{10/3}}}\\Biggr ].\\nonumber \\\\& &$ (C) For $\\mu =u$ and $\\nu =y$ $-a_{EH}\\frac{i e^{i (q x-t w)} \\sqrt{\\frac{r_h^8}{L^6 u^{10} g_s^{10/3}}} \\left(q g_1(u) H_{xy}^{\\prime }(u)+w H_{ty}^{\\prime }(u)\\right)}{2 g_1(u)}=0.$ (D)  EOM for $\\phi $ : $& & \\hspace{-50.58878pt}\\Biggl (\\frac{c \\sqrt{\\mu } r_h^5 \\left(\\frac{{r_h}}{u}\\right)^{11/4} \\sqrt{c^2 e^{2 \\Phi }+\\left(\\frac{{r_h}}{u}\\right)^{9/2}} \\left(c^6 e^{6 \\Phi } u^{13}+3 c^4 e^{4 \\Phi } r_h^4 u^9 \\sqrt{\\frac{{r_h}}{u}}+3 c^2 e^{2 \\Phi } {r_h}^9 u^4+{r_h}^{13}\\sqrt{\\frac{{r_h}}{u}}\\right) e^{\\Phi +2 i q x-2 i t w}}{18 \\left(c^2 e^{2 \\Phi } u^4+{r_h}^4 \\sqrt{\\frac{{r_h}}{u}}\\right)^5\\left(\\frac{r_h^5}{c^2 e^{2 \\Phi } u^5 \\sqrt{\\frac{{r_h}}{u}}+{r_h}^5}\\right)^{3/2}}\\Biggr )H_{ty}\\nonumber \\\\& & \\hspace{-50.58878pt}-\\Biggl (\\frac{c \\sqrt{\\mu } {r_h}^6 \\left(\\frac{{r_h}}{u}\\right)^{7/4} \\sqrt{c^2 e^{2 \\phi }+\\left(\\frac{{r_h}}{u}\\right)^{9/2}} \\left(c^6 e^{6 \\Phi } u^{13}+3 c^4 e^{4 \\Phi } {r_h}^4 u^9 \\sqrt{\\frac{{r_h}}{u}}+3 c^2 e^{2 \\Phi } {r_h}^9 u^4+{r_h}^{13}\\sqrt{\\frac{{r_h}}{u}}\\right) e^{\\Phi +2 i q x-2 i t w}}{36 \\left(c^2 e^{2 \\Phi } u^4+{r_h}^4 \\sqrt{\\frac{{r_h}}{u}}\\right)^5\\left(\\frac{{r_h}^5}{c^2 e^{2 \\Phi } u^5 \\sqrt{\\frac{{r_h}}{u}}+{r_h}^5}\\right)^{3/2}}\\Biggr )H^{^{\\prime }}_{ty}\\nonumber \\\\& & \\hspace{-50.58878pt}+\\Biggl (\\frac{\\sqrt{\\mu } {r_h}^6 \\left(u^4-1\\right) \\left(\\frac{{r_h}}{u}\\right)^{7/4} \\left(c^6 e^{6 \\Phi } u^{13}+3 c^4 e^{4 \\Phi } {r_h}^4 u^9\\sqrt{\\frac{{r_h}}{u}}+3 c^2 e^{2 \\Phi } {r_h}^9 u^4+{r_h}^{13} \\sqrt{\\frac{{r_h}}{u}}\\right) e^{2 i (q x-t w)}}{36 \\left(c^2 e^{2 \\Phi } u^5+{r_h}^4 u \\sqrt{\\frac{{r_h}}{u}}\\right)^4 \\left(\\frac{{r_h}^5}{c^2 e^{2 \\Phi } u^5\\sqrt{\\frac{{r_h}}{u}}+{r_h}^5}\\right)^{3/2}}\\Biggr )\\phi ^{^{\\prime \\prime }}\\nonumber \\\\& & \\hspace{-50.58878pt}-\\Biggl [\\frac{\\sqrt{\\mu } {r_h}^6 \\left(\\frac{{r_h}}{u}\\right)^{7/4}e^{2 i (q x-t w)}}{144 u^5 \\left(c^2 e^{2 \\Phi } u^4+{r_h}^4 \\sqrt{\\frac{{r_h}}{u}}\\right)^4\\left(\\frac{{r_h}^5}{c^2 e^{2 \\Phi } u^5 \\sqrt{\\frac{{r_h}}{u}}+{r_h}^5}\\right)^{3/2}} \\nonumber \\\\& & \\hspace{-28.90755pt}\\left\\lbrace -8 c^6 e^{6 \\Phi } u^{13} \\left(u^4+1\\right)-3 c^4 e^{4 \\Phi } u^9\\left(5 u^4+11\\right) \\sqrt{\\frac{r_h^9}{u}}-6c^2e^{2 \\Phi }{r_h}^9 u^4 \\left(u^4+7\\right)+ \\left(u^4-17\\right)\\sqrt{\\frac{r_h^{27}}{u}}\\right\\rbrace \\Biggr ]\\phi ^{^{\\prime }}\\nonumber \\\\& & \\hspace{-50.58878pt}+\\Biggl (\\frac{\\pi {g_s} \\sqrt{\\mu } N u \\left(\\frac{{r_h}}{u}\\right)^{3/4} \\sqrt{\\frac{{r_h}^5}{c^2 e^{2 \\Phi } u^5\\sqrt{\\frac{{r_h}}{u}}+{r_h}^5}} e^{2 i (q x-t w)} \\left({i w}^2 \\left(c^2 e^{2 \\Phi } u^4+{r_h}^4\\sqrt{\\frac{{r_h}}{u}}\\right)+{iq}^2 {r_h}^4 \\left(u^4-1\\right) \\sqrt{\\frac{{r_h}}{u}}\\right)}{9 {r_h}^6 \\left(u^4-1\\right)}\\Biggr )\\phi = 0.\\nonumber \\\\& &$ The $H_{ty}(u)$ EOM, setting $q=0$ and near $u=0$ is: $& & \\frac{1}{36} \\sqrt{\\mu } {r_h}^{13/4} u^{7/4} {H_{ty}}(u)-\\frac{{r_h}^6 \\left(u {H_{ty}}^{\\prime \\prime }(u)-3 {H_{ty}}^{\\prime }(u)\\right)}{2 {g_s}^{5/3} L^7}=0,$ whose solution is given by: $& & H_{ty}(u) = \\frac{4 2^{2/11} c_1 {g_s}^{40/33} L^{56/11} \\mu ^{4/11} u^2 \\Gamma \\left(-\\frac{5}{11}\\right) I_{-\\frac{16}{11}}\\left(\\frac{4 \\sqrt{2} {g_s}^{5/6}L^{7/2} \\@root 4 \\of {\\mu } u^{11/8}}{33 {r_h}^{11/8}}\\right)}{33 33^{5/11} {r_h}^2}\\nonumber \\\\& & -\\frac{64 (-1)^{5/11} 2^{2/11} c_2 {g_s}^{40/33} L^{56/11} \\mu ^{4/11} u^2 \\Gamma \\left(\\frac{16}{11}\\right) I_{\\frac{16}{11}}\\left(\\frac{4 \\sqrt{2} {g_s}^{5/6} L^{7/2} \\@root 4 \\of {\\mu } u^{11/8}}{33{r_h}^{11/8}}\\right)}{363 33^{5/11} {r_h}^2}\\nonumber \\\\& & = \\kappa _1 + \\kappa _2 u^{\\frac{11}{4}} + \\gamma \\kappa _2 u^4 + ....,$ where: $& & \\kappa _1 \\equiv c_1;\\nonumber \\\\& & \\kappa _2 \\equiv -\\frac{64 \\sqrt{2} \\pi ^{7/4} c_1 {g_s}^{41/12} \\sqrt{\\mu } N^{7/4}}{495 {r_h}^{11/4}};\\nonumber \\\\& & \\gamma \\equiv \\frac{320 (-1)^{5/11} 2^{21/22} \\pi ^{35/44} c_2 {g_s}^{205/132} \\mu ^{5/22} N^{35/44} \\Gamma \\left(\\frac{16}{11}\\right)}{121 33^{10/11} c_1 {r_h}^{5/4}\\Gamma \\left(\\frac{27}{11}\\right)}.$ The $H_{xy}$ EOM near $u=0$ is: $& & 18 r_h^2 \\left(u {H_{xy}}^{\\prime \\prime }(u)-3 {H_{xy}}^{\\prime }(u)\\right) \\sqrt{\\frac{r_h^8}{L^6 g_s^{10/3}}}+L^4 \\sqrt{\\mu } {r_h}^{13/4} u^{7/4} {H_{xy}}(u)=0,$ whose solution is given by: $& & H_{xy}(u)=\\frac{64 2^{2/11} c_2 L^{56/11} \\mu ^{4/11} {r_h}^{26/11} u^2 \\Gamma \\left(\\frac{16}{11}\\right) g_s^{40/33} J_{\\frac{16}{11}}\\left(\\frac{4 \\sqrt{2} L^{7/2}\\@root 4 \\of {\\mu } {r_h}^{13/8} u^{11/8} g_s^{5/6}}{33 r_h^3}\\right)}{363 33^{5/11} r_h^{48/11}}\\nonumber \\\\& & +\\frac{4 2^{2/11} c_1 L^{56/11} \\mu ^{4/11} {r_h}^{26/11}u^2 \\Gamma \\left(-\\frac{5}{11}\\right) g_s^{40/33} J_{-\\frac{16}{11}}\\left(\\frac{4 \\sqrt{2} L^{7/2} \\@root 4 \\of {\\mu } {r_h}^{13/8} u^{11/8} g_s^{5/6}}{33r_h^3}\\right)}{33 33^{5/11} r_h^{48/11}}.$ Substituting (REF ), the $\\phi (u)$ EOM near $u=0$ can be approximated by: $\\hspace{-43.36243pt}-4 c {g_s} {r_h}^4 u {H_{ty}}^{\\prime }(u)+8 c {g_s} {r_h}^4 {H_{ty}}(u)+16 \\pi {g_s} i N {r_h}^{9/4} u^{11/4} \\left(q^2-w^2\\right) \\phi (u)+\\frac{17 {r_h}^{25/4} \\phi ^{\\prime }(u)}{u^{9/4}}-\\frac{4 {r_h}^{25/4} \\phi ^{\\prime \\prime }(u)}{u^{5/4}}=0,\\nonumber \\\\& &$ or $& & \\hspace{-43.36243pt}\\frac{{r_h}^{7/4} \\left(c {g_s} u^{9/4} \\left(-8 \\gamma {k_2} u^4+8 {k_1}-3 {k_2} u^{11/4}\\right)-4 {r_h}^{9/4} u \\phi ^{\\prime \\prime }(u)+17{r_h}^{9/4} \\phi ^{\\prime }(u)\\right)+16 \\pi {g_s} i N u^5 \\left(q^2-w^2\\right) \\phi (u)}{u^{9/4}}=0,\\nonumber \\\\& &$ whose solution is given by: $& &\\hspace{-101.17755pt}\\phi (u) = -\\frac{1}{3024 \\pi ^{9/16} {r_h}^{9/4} \\left(-w^2\\right)^{23/16} \\left(\\frac{u^3 \\sqrt{-{g_s} i N w^2}}{{r_h}^2}\\right)^{7/8} \\Gamma \\left(\\frac{2}{3}\\right) \\Gamma \\left(\\frac{4}{3}\\right) \\Gamma \\left(\\frac{37}{24}\\right) \\Gamma \\left(\\frac{53}{24}\\right)}\\nonumber \\\\& & \\hspace{-101.17755pt}\\times \\Biggl \\lbrace {g_s}^{7/16} u^{13/4} \\Biggl (\\frac{1}{{r_h}^4}\\Biggl \\lbrace u^{19/8} \\Gamma \\left(\\frac{2}{3}\\right) \\Biggl [192 3^{7/8} c \\gamma {g_s}^{9/16} {k_2} \\@root 8 \\of {\\pi }{r_h}^4 u^{13/8} \\left(-w^2\\right)^{23/16} I_{\\frac{7}{8}}\\left(\\frac{2 \\sqrt{\\pi } u^3 \\sqrt{-{g_s} i N w^2}}{3 {r_h}^2}\\right) \\Gamma \\left(\\frac{1}{3}\\right) \\Gamma \\left(\\frac{37}{24}\\right) \\Gamma \\left(\\frac{15}{8}\\right) \\Gamma \\left(\\frac{53}{24}\\right)\\nonumber \\\\& &\\hspace{-101.17755pt}\\,_1F_2\\left(\\frac{1}{3};\\frac{1}{8},\\frac{4}{3};-\\frac{{g_s} i N \\pi u^6 w^2}{9 {r_h}^4}\\right)-\\frac{1}{\\@root 4 \\of {\\frac{u^3 \\sqrt{-{g_s} i N w^2}}{{r_h}^2}}}\\Biggl \\lbrace w^2 \\Gamma \\left(\\frac{4}{3}\\right)\\Biggl (64\\@root 8 \\of {3} c {g_s}^{25/16} i {k_1} N \\pi \\left(-w^2\\right)^{23/16} I_{-\\frac{7}{8}}\\left(\\frac{2 \\sqrt{\\pi } u^3 \\sqrt{-{g_s} i N w^2}}{3{r_h}^2}\\right) \\Gamma \\left(\\frac{1}{8}\\right) \\Gamma \\left(\\frac{13}{24}\\right) \\Gamma \\left(\\frac{53}{24}\\right)\\nonumber \\\\& & \\hspace{-101.17755pt}\\times \\,_1F_2\\left(\\frac{13}{24};\\frac{37}{24},\\frac{15}{8};-\\frac{{g_s} i N \\pi u^6 w^2}{9 {r_h}^4}\\right) u^{29/8}\\nonumber \\\\& &\\hspace{-101.17755pt}+\\Gamma \\left(\\frac{37}{24}\\right)\\Biggl [9 {r_h}^{5/2} \\Gamma \\left(\\frac{53}{24}\\right) \\Biggl (\\frac{1}{u^3}\\Biggl \\lbrace \\left(-w^2\\right)^{7/16} I_{-\\frac{7}{8}}\\left(\\frac{2 \\sqrt{\\pi } u^3 \\sqrt{-{g_s}i N w^2}}{3 {r_h}^2}\\right) \\Gamma \\left(\\frac{1}{8}\\right)\\nonumber \\\\& & \\hspace{-86.72377pt}\\times \\Biggl [21 \\@root 8 \\of {3} c {g_s}^{9/16} {k_2} u^{27/8}-24 c {g_s}^{9/16} {k_2}\\@root 16 \\of {\\pi } \\@root 8 \\of {\\frac{\\sqrt{{g_s}} \\sqrt{i} \\sqrt{N} u^3 \\sqrt{-w^2}}{{r_h}^2}} I_{-\\frac{1}{8}}\\left(\\frac{2 \\sqrt{{g_s}} \\sqrt{i}\\sqrt{N} \\sqrt{\\pi } u^3 \\sqrt{-w^2}}{3 {r_h}^2}\\right)\\nonumber \\\\& & \\hspace{-101.17755pt}\\Gamma \\left(\\frac{15}{8}\\right) u^{27/8}-112 \\@root 8 \\of {3} i^{7/16} N^{7/16} \\pi \\sqrt{{r_h}}\\left(-w^2\\right)^{7/16} \\left(\\frac{u^3 \\sqrt{-{g_s} i N w^2}}{{r_h}^2}\\right)^{9/8} c_1\\Biggr ] {r_h}^{3/2}\\Biggr \\rbrace +8 \\@root 8 \\of {{g_s}}\\@root 8 \\of {3 \\pi } \\@root 8 \\of {\\frac{u^3 \\sqrt{-{g_s} i N w^2}}{{r_h}^2}} I_{\\frac{7}{8}}\\left(\\frac{2 \\sqrt{\\pi } u^3 \\sqrt{-{g_s} i N w^2}}{3{r_h}^2}\\right) \\Gamma \\left(\\frac{15}{8}\\right)\\nonumber \\\\& & \\hspace{-101.17755pt}\\left(14 i^{9/16} N^{9/16} (-\\pi )^{7/8} \\left(-{g_s} i N w^2\\right)^{3/8} c_2 w^2+3^{3/4} c{g_s}^{7/16} {k_2} {r_h}^{3/2} u^{3/8} \\left(-w^2\\right)^{7/16} \\@root 8 \\of {\\frac{u^3 \\sqrt{-{g_s} i N w^2}}{{r_h}^2}} \\Gamma \\left(\\frac{1}{8}\\right) \\left(;\\frac{9}{8};-\\frac{{g_s} i N \\pi u^6 w^2}{9 {r_h}^4}\\right)\\right)\\Biggr )\\nonumber \\\\& & \\hspace{-86.72377pt}-64 \\@root 8 \\of {3} c \\gamma {g_s}^{25/16}i {k_2} N \\pi u^{61/8} \\left(-w^2\\right)^{23/16} I_{-\\frac{7}{8}}\\left(\\frac{2 \\sqrt{\\pi } u^3 \\sqrt{-{g_s} i N w^2}}{3 {r_h}^2}\\right) \\Gamma \\left(\\frac{1}{8}\\right) \\Gamma \\left(\\frac{29}{24}\\right) \\, _1F_2\\left(\\frac{29}{24};\\frac{15}{8},\\frac{53}{24};-\\frac{{g_s} i N \\pi u^6 w^2}{9{r_h}^4}\\right)\\Biggr ]\\Biggr )\\Biggr \\rbrace \\Biggr ]\\Biggr \\rbrace \\nonumber \\\\& & \\hspace{-101.17755pt}-192 3^{7/8} c {g_s}^{9/16} {k_1}\\@root 8 \\of {\\pi } \\left(-w^2\\right)^{23/16} I_{\\frac{7}{8}}\\left(\\frac{2 \\sqrt{\\pi } u^3 \\sqrt{-{g_s} i N w^2}}{3 {r_h}^2}\\right) \\Gamma \\left(-\\frac{1}{3}\\right) \\Gamma \\left(\\frac{4}{3}\\right) \\Gamma \\left(\\frac{37}{24}\\right) \\Gamma \\left(\\frac{15}{8}\\right) \\Gamma \\left(\\frac{53}{24}\\right) \\, _1F_2\\left(-\\frac{1}{3};\\frac{1}{8},\\frac{2}{3};-\\frac{{g_s} i N \\pi u^6 w^2}{9 {r_h}^4}\\right)\\Biggr )\\Biggr \\rbrace .$" ], [ "Frobenius Solution of EOM of Gauge-Invariant $Z_s(u)$ for Scalar Modes of Metric Fluctuations for\n({{formula:e0912bb8-288e-4fd8-b284-684bb0184505}} ) {{formula:1f8a6312-4270-4e2b-a51a-a97612e4c7e8}}", "The $Z_s(u)$ EOM can be rewritten as: $(u-1)^2Z_s^{\\prime \\prime }(u) + (u-1)P(u-1) Z_s^\\prime (u) + Q(u-1) Z_s(u) = 0,$ in which $P(u-1) = \\sum _{n=0}^\\infty p_n(u-1)^n$ and $Q(u-1) = \\sum _{m=0}^\\infty q_n (u-1)^n$ wherein, up to ${\\cal O}\\left(\\frac{1}{N}\\right)$ : $& & p_0 = 1,\\nonumber \\\\& & p_1 = \\frac{3 {g_s}^2 M^2 {N_f} \\log (N) \\left(28 {q_3}^4+36 {q_3}^2 {\\omega _3}^2-81 {\\omega _3}^4\\right)}{64 \\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)^2}+\\frac{10 {q_3}^2+9 {\\omega _3}^2}{4 {q_3}^2-6 {\\omega _3}^2},\\nonumber \\\\& & p_2 = \\frac{3 {g_s}^2 M^2 {N_f} \\log (N) \\left(712 {q_3}^6-948 {q_3}^4 {\\omega _3}^2-162 {q_3}^2 {\\omega _3}^4+405 {\\omega _3}^6\\right)}{64 \\pi ^2 N \\left(2{q_3}^2-3 {\\omega _3}^2\\right)^3}+\\frac{364 {q_3}^4-420 {q_3}^2 {\\omega _3}^2+99 {\\omega _3}^4}{4 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^2};\\nonumber \\\\& & q_0 = \\frac{3 {g_s}^2 M^2 {N_f} \\log (N) \\left(\\left({\\omega _3}^2+4\\right) \\left(27 {\\omega _3}^2-10 {q_3}^2\\right)-8 {q_3}^2 {\\omega _3}^2 \\log (N)\\right)}{4096 \\pi ^2 N {q_3}^2}+\\frac{{\\omega _3}^2}{16},\\nonumber \\\\& & q_1 = \\frac{1}{4096 \\pi ^2 N {q_3}^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^2}\\nonumber \\\\& & \\times \\Biggl \\lbrace 3 {g_s}^2 M^2 {N_f} \\log (N) \\biggl (-8 \\log (N) \\left(4 {q_3}^2-3 {\\omega _3}^2\\right) \\left(2 {q_3}^3-3 {q_3} {\\omega _3}^2\\right)^2 \\nonumber \\\\& & -96{q_3}^8+8 {q_3}^6 \\left(51 {\\omega _3}^2-52\\right)-36 {q_3}^4 {\\omega _3}^2 \\left(9 {\\omega _3}^2+52\\right)-54 {q_3}^2 {\\omega _3}^4 \\left(9{\\omega _3}^2-28\\right)+81 {\\omega _3}^6 \\left(7 {\\omega _3}^2+20\\right)\\biggr )\\Biggr \\rbrace \\nonumber \\\\& & +\\frac{8{q_3}^4-2 {q_3}^2 \\left(9 {\\omega _3}^2+32\\right)+9 {\\omega _3}^4}{32 {q_3}^2-48 {\\omega _3}^2},\\nonumber \\\\& & q_2 = \\frac{-96 {q_3}^6+52 {q_3}^4 \\left(7 {\\omega _3}^2-64\\right)+{q_3}^2 \\left(3456 {\\omega _3}^2-444 {\\omega _3}^4\\right)+171 {\\omega _3}^6}{64 \\left(2{q_3}^2-3 {\\omega _3}^2\\right)^2}\\nonumber \\\\& & -\\frac{3 {g_s}^2 M^2 {N_f} \\log (N) }{16384 \\pi ^2 N {q_3}^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^3}\\nonumber \\\\& & \\times \\Biggl [-8 {q_3}^2 \\log (N) \\left(24 {q_3}^2-19 {\\omega _3}^2\\right) \\left(2{q_3}^2-3 {\\omega _3}^2\\right)^3+1920 {q_3}^{10}-16 {q_3}^8 \\left(157 {\\omega _3}^2-6268\\right)\\nonumber \\\\& & -96 {q_3}^6 {\\omega _3}^2 \\left(351{\\omega _3}^2+1444\\right)+216 {q_3}^4 {\\omega _3}^4 \\left(559 {\\omega _3}^2+76\\right)-864 {q_3}^2 {\\omega _3}^6 \\left(173 {\\omega _3}^2+53\\right)+243 {\\omega _3}^8\\left(265 {\\omega _3}^2+308\\right)\\Biggr ].$ The Frobenius method then dictates that the solution is given by: $Z_s(u) = \\left(1 - u \\right)^{\\frac{3 {g_s}^2 M^2 {N_f} {q_3} {\\omega _3}^2 \\log (N) \\left(8 {q_3}^2 {\\omega _3}^2 \\log (N)+\\left({\\omega _3}^2+4\\right) \\left(10 {q_3}^2-27{\\omega _3}^2\\right)\\right)}{2048 \\pi ^2 N \\left(-{q_3}^2 {\\omega _3}^2\\right)^{3/2}}-\\frac{i {\\omega _3}}{4}}\\left(1 + \\sum _{m=1}a_m (u - 1)^m\\right),$ where $& & a_1 = \\frac{8 i {q_3}^4+2 {q_3}^2 \\left(-9 i {\\omega _3}^2+10 {\\omega _3}-32 i\\right)+9 (2+i {\\omega _3}) {\\omega _3}^3}{8 ({\\omega _3}+2 i) \\left(3 {\\omega _3}^2-2{q_3}^2\\right)}\\nonumber \\\\& & -\\frac{1}{4096 \\pi ^2 N{q_3}^2 {\\omega _3} ({\\omega _3}+2 i)^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^2}\\nonumber \\\\& & \\times \\Biggl \\lbrace 3 i {g_s}^2 M^2 {N_f} \\log (N) \\biggl (({\\omega _3}+2 i) \\biggl (-32 {q_3}^8 ({\\omega _3}+10 i)-8 {q_3}^6 \\left(27{\\omega _3}^3-146 i {\\omega _3}^2+364 {\\omega _3}-520 i\\right)\\nonumber \\\\& & +12 {q_3}^4 {\\omega _3}^2 \\left(141 {\\omega _3}^3-486 i {\\omega _3}^2+356 {\\omega _3}-1336 i\\right)-54{q_3}^2 {\\omega _3}^4 \\left(59 {\\omega _3}^3-130 i {\\omega _3}^2+44 {\\omega _3}-200 i\\right)\\nonumber \\\\& & +81 {\\omega _3}^6 \\left(23 {\\omega _3}^3-18 i {\\omega _3}^2+4 {\\omega _3}+72i\\right)\\biggr )-8 {q_3}^2 {\\omega _3} \\log (N) \\biggl [16 {q_3}^6 ({\\omega _3}+4 i)-4 {q_3}^4 {\\omega _3} \\left(15 {\\omega _3}^2+60 i {\\omega _3}-52\\right)\\nonumber \\\\& & +24{q_3}^2 {\\omega _3}^3 \\left(3 {\\omega _3}^2+12 i {\\omega _3}-10\\right)-27 {\\omega _3}^5 \\left({\\omega _3}^2+4 i {\\omega _3}+4\\right)\\biggr ]\\biggr )\\Biggr \\rbrace ,\\nonumber \\\\& & a_2 = \\frac{1}{128 ({\\omega _3}+2 i) ({\\omega _3}+4 i) \\left(3 {\\omega _3}^2-2 {q_3}^2\\right)}\\nonumber \\\\& & \\times {32 {q_3}^6-32 {q_3}^4 {\\omega _3} (3 {\\omega _3}+8 i)+2 {q_3}^2 \\left(45 {\\omega _3}^4+98 i {\\omega _3}^3+624 {\\omega _3}^2+32 i {\\omega _3}+3072\\right)-3{\\omega _3}^3 \\left(9 {\\omega _3}^3+2 i {\\omega _3}^2+48 {\\omega _3}+32 i\\right)}\\nonumber \\\\& & -\\frac{1}{32768 \\pi ^2 N {q_3}^2 {\\omega _3} ({\\omega _3}+2 i)^2({\\omega _3}+4 i)^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^2}\\nonumber \\\\& & \\times \\Biggl \\lbrace 3{g_s}^2 M^2 {N_f} \\log (N) \\Biggl [(-{\\omega _3}-2 i) \\biggl (128 {q_3}^{10} \\left({\\omega _3}^2+19 i {\\omega _3}-30\\right)+256 {q_3}^8 \\left(3{\\omega _3}^4-18 i {\\omega _3}^3+33 {\\omega _3}^2-76 i {\\omega _3}+160\\right)\\nonumber \\\\& & -8 {q_3}^6 \\left(927 {\\omega _3}^6-104 i {\\omega _3}^5+17784 {\\omega _3}^4+11808 i{\\omega _3}^3+70256 {\\omega _3}^2+42368 i {\\omega _3}+93440\\right)\\nonumber \\\\& & +12 {q_3}^4 {\\omega _3}^2 \\left(1485 {\\omega _3}^6+128 i {\\omega _3}^5+31912 {\\omega _3}^4+14176 i{\\omega _3}^3+96656 {\\omega _3}^2-68992 i {\\omega _3}+262912\\right)\\nonumber \\\\& & -18 {q_3}^2 {\\omega _3}^4 \\left(945 {\\omega _3}^6-1388 i {\\omega _3}^5+27920 {\\omega _3}^4+13664 i{\\omega _3}^3+54736 {\\omega _3}^2-80768 i {\\omega _3}+171776\\right)\\nonumber \\\\& & +81 {\\omega _3}^6 \\left(69 {\\omega _3}^6-416 i {\\omega _3}^5+3960 {\\omega _3}^4+4128 i {\\omega _3}^3+5520{\\omega _3}^2+4736 i {\\omega _3}+768\\right)\\biggr )-8 {q_3}^2 {\\omega _3} \\log (N)\\nonumber \\\\& & \\times \\biggl (64 {q_3}^8 \\left({\\omega _3}^2+9 i {\\omega _3}-16\\right)-32 {q_3}^6{\\omega _3} \\left(9 {\\omega _3}^3+89 i {\\omega _3}^2-240 {\\omega _3}-192 i\\right)\\nonumber \\\\& & +4 {q_3}^4 {\\omega _3} \\left(117 {\\omega _3}^5+1198 i {\\omega _3}^4-3612 {\\omega _3}^3-1624i {\\omega _3}^2-8064 {\\omega _3}-9344 i\\right)\\nonumber \\\\& & -12 {q_3}^2 {\\omega _3}^3 ({\\omega _3}+4 i)^2 \\left(27 {\\omega _3}^3+52 i {\\omega _3}^2+116 {\\omega _3}+296 i\\right)\\nonumber \\\\& & +9{\\omega _3}^5 ({\\omega _3}+2 i)^2 \\left(9 {\\omega _3}^3+46 i {\\omega _3}^2+64 {\\omega _3}+32 i\\right)\\biggr )\\Biggr ]\\Biggr \\rbrace .$ As stated in 5.3.1, imposing Dirichlet boundary condition $Z_s(u=0)=0$ and going up to second order in powers of $(u-1)$ in (REF ) and considering in the hydrodynamical limit $\\omega _3^nq_3^m:m+n=2$ one obtains: $\\omega _3 = -\\frac{2 {q_3}}{\\sqrt{3}}-\\frac{9 i {q_3}^2}{32},$ which yields a result for the speed of sound similar to (REF ) for $n=0,1$ .", "To get the LO or conformal result for the speed of sound $v_s = \\frac{1}{\\sqrt{3}}$ , let us go to the fourth order in (REF ).", "For this, up to ${\\cal O}\\left(\\frac{1}{N}\\right)$ , we will need: $& & \\hspace{-36.135pt}p_3 = \\frac{3 {g_s}^2 M^2 {N_f} \\log (N) \\left(6256 {q_3}^8-9600 {q_3}^6 {\\omega _3}^2-4104 {q_3}^4 {\\omega _3}^4+12960 {q_3}^2 {\\omega _3}^6-5589{\\omega _3}^8\\right)}{64 \\pi ^2 N \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^4}\\nonumber \\\\& & \\hspace{-36.135pt}+\\frac{3880 {q_3}^6-4788 {q_3}^4 {\\omega _3}^2+270 {q_3}^2 {\\omega _3}^4+729{\\omega _3}^6}{8 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^3},\\nonumber \\\\& &\\hspace{-36.135pt}p_4 = \\frac{3 {g_s}^2 M^2 {N_f} \\log (N) }{64 \\pi ^2 N \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^5}\\nonumber \\\\& &\\hspace{-36.135pt}\\times \\left(53536 {q_3}^{10}-110256 {q_3}^8 {\\omega _3}^2-2736 {q_3}^6 {\\omega _3}^4+168264 {q_3}^4{\\omega _3}^6-156006 {q_3}^2 {\\omega _3}^8+47385 {\\omega _3}^{10}\\right)\\nonumber \\\\& &\\hspace{-36.135pt}+\\frac{3 \\left(17360{q_3}^8-32992 {q_3}^6 {\\omega _3}^2+19320 {q_3}^4 {\\omega _3}^4-5112 {q_3}^2 {\\omega _3}^6+1485 {\\omega _3}^8\\right)}{16 \\left(2 {q_3}^2-3{\\omega _3}^2\\right)^4};\\nonumber \\\\& &\\hspace{-36.135pt}q_3 = \\frac{3 {g_s}^2 M^2 {N_f} \\log (N) }{4096 \\pi ^2 N {q_3}^2 \\left(2 {q_3}^2-3{\\omega _3}^2\\right)^4}\\nonumber \\\\& &\\hspace{-36.135pt}\\times \\biggl [-40 {q_3}^2 \\log (N) \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^4 \\left({q_3}^2-{\\omega _3}^2\\right)-3552{q_3}^{12}+416 {q_3}^{10} \\left(31 {\\omega _3}^2-648\\right)+96 {q_3}^8 {\\omega _3}^2 \\left(466 {\\omega _3}^2+5695\\right)\\nonumber \\\\& & \\hspace{-36.135pt}-288 {q_3}^6 {\\omega _3}^4\\left(1107 {\\omega _3}^2+818\\right)+54 {q_3}^4 {\\omega _3}^6 \\left(12319 {\\omega _3}^2+840\\right)-162 {q_3}^2 {\\omega _3}^8 \\left(3779{\\omega _3}^2+1632\\right)\\nonumber \\\\& & \\hspace{-36.135pt}+729 {\\omega _3}^{10} \\left(293 {\\omega _3}^2+250\\right)\\biggr ]\\nonumber \\\\& &\\hspace{-36.135pt}+\\frac{40 {q_3}^8-4 {q_3}^6 \\left(55 {\\omega _3}^2+1488\\right)+6 {q_3}^4 {\\omega _3}^2 \\left(75 {\\omega _3}^2+1696\\right)-9{q_3}^2 {\\omega _3}^4 \\left(45 {\\omega _3}^2+464\\right)+135 {\\omega _3}^8}{16 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^3},\\nonumber \\\\& &\\hspace{-36.135pt}q_4 = \\nonumber \\\\& & \\hspace{-36.135pt}\\frac{-640 {q_3}^{10}+48 {q_3}^8 \\left(99 {\\omega _3}^2-12352\\right)-288 {q_3}^6 {\\omega _3}^2 \\left(49 {\\omega _3}^2-4400\\right)+216 {q_3}^4 {\\omega _3}^4\\left(97 {\\omega _3}^2-3680\\right)-1728 {q_3}^2 {\\omega _3}^6 \\left(9 {\\omega _3}^2-70\\right)+4617 {\\omega _3}^{10}}{256 \\left(2 {q_3}^2-3{\\omega _3}^2\\right)^4}\\nonumber \\\\& &\\hspace{-36.135pt}-\\frac{1}{65536 \\pi ^2 N {q_3}^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^5}\\nonumber \\\\& & \\hspace{-36.135pt}\\times \\Biggl \\lbrace 3 {g_s}^2 M^2 {N_f} \\log (N) \\biggl (3 \\biggl [112128 {q_3}^{14}-64 {q_3}^{12} \\left(6133 {\\omega _3}^2-190428\\right)-192{q_3}^{10} {\\omega _3}^2 \\left(15391 {\\omega _3}^2+151076\\right)\\nonumber \\\\& & \\hspace{-36.135pt}+720 {q_3}^8 {\\omega _3}^4 \\left(30785 {\\omega _3}^2+17332\\right)-28800 {q_3}^6 {\\omega _3}^6\\left(2079 {\\omega _3}^2-443\\right)+324 {q_3}^4 {\\omega _3}^8 \\left(252677 {\\omega _3}^2+420\\right)-972 {q_3}^2 {\\omega _3}^{10} \\left(58571{\\omega _3}^2+17828\\right)\\nonumber \\\\& & \\hspace{-36.135pt}+729 {\\omega _3}^{12} \\left(22027 {\\omega _3}^2+12156\\right)\\biggr ]-8 {q_3}^2 \\log (N) \\left(40 {q_3}^2-57 {\\omega _3}^2\\right) \\left(2{q_3}^2-3 {\\omega _3}^2\\right)^5\\biggr )\\Biggr \\rbrace .$ We will not quote the expressions for $a_3$ and $a_4$ because they are too cumbersome.", "Substituting the expressions for $a_{1,2,3,4}$ into $Z(u)$ and implementing the Dirichlet boundary condition: $Z_s(u=0)=0$ , in the hydrodynamical limit, going up to ${\\cal O}(\\omega _3^4)$ one sees that one can write the Dirichlet boundary condition as a quartic: $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where: $& & a = -\\frac{(17978967982080+432082299912192 i) {g_s}^2 M^2 {N_f} \\log (N)}{N},\\nonumber \\\\& & b = -16384 {q_3}^2 \\left(55717134336 \\pi ^2-\\frac{(8065585152-2189804544 i) {g_s}^2 M^2 {N_f} \\log (N)}{N}\\right),\\nonumber \\\\& & c = \\frac{(6351753314304+163465918414848 i) {g_s}^2 M^2 {N_f} {q_3}^2 \\log (N)}{N},\\nonumber \\\\& & f = 196608 {q_3}^4 \\left(981467136 \\pi ^2-\\frac{(25958400-36690432 i) {g_s}^2 M^2 {N_f} \\log (N)}{N}\\right),\\nonumber \\\\& & g = -\\frac{(842551787520+22613002813440 i) {g_s}^2 M^2 {N_f} {q_3}^4 \\log (N)}{N}.$" ], [ "Frobenius Solution of EOM of Gauge-Invariant $Z_s(u)$ for Scalar Modes of Metric Fluctuations for {{formula:3c849f1c-a92d-40b0-bbe3-34e66a991a3f}}", "Constructing a $Z_s(u)$ which is invariant under diffeomorphisms: $h_{\\mu \\nu }\\rightarrow h_{\\mu \\nu } - \\nabla _{(\\mu }\\xi _{\\nu )}$ , one sees one obtain the following equation of motion for $Z_s(u)$ : $& & Z_s^{\\prime \\prime }(u) = \\Biggl [\\frac{q_3^2 \\left(7 u^8-8 u^4+9\\right)-3 \\left(u^4+3\\right) \\omega _3^2}{u \\left(u^4-1\\right) \\left(q_3^2 \\left(u^4-3\\right)+3\\omega _3^2\\right)}\\nonumber \\\\& & -\\frac{1}{64 \\pi ^2 N u \\left(u^8-4 u^4+3\\right)\\left(q_3^2 \\left(u^4-3\\right)+3 \\omega _3^2\\right)^2}\\Biggl \\lbrace -3 {g_s}^2 M^2 {N_f} \\log N\\nonumber \\\\& & \\times \\Biggl (q_3^4 \\left(5 u^{16}-98 u^{12}+372 u^8-414 u^4+135\\right)+2q_3^2 \\left(32 u^{12}-183 u^8+306 u^4-135\\right) \\omega _3^2\\nonumber \\\\& & +3 \\left(u^8-66 u^4+45\\right) \\omega _3^4\\Biggr )\\Biggr \\rbrace \\Biggr ]Z_s^\\prime (u)\\nonumber \\\\& & + \\Biggl [\\frac{1}{128 \\pi ^2 Nq_3^2 \\left(u^4-3\\right) \\left(u^4-1\\right)^3 \\left(q_3^2 \\left(u^4-3\\right)+3 \\omega _3^2\\right)^2}\\Biggl \\lbrace -3 {g_s}^2 M^2 {N_f} \\log N\\nonumber \\\\& & \\times \\Biggl (30 q_3^6 u^{22}-542 q_3^6 u^{18}-7 q_3^6 u^{16} \\omega _3^2+2540q_3^6 u^{14}+46 q_3^6 u^{12} \\omega _3^2-4764 q_3^6 u^{10}-84 q_3^6 u^8 \\omega _3^2\\nonumber \\\\& & +4086 q_3^6 u^6+18 q_3^6 u^4\\omega _3^2-1350 q_3^6 u^2+27 q_3^6 \\omega _3^2+318 q_3^4 u^{18} \\omega _3^2-2464 q_3^4 u^{14} \\omega _3^2\\nonumber \\\\& & -49 q_3^4 u^{12}\\omega _3^4+6972 q_3^4 u^{10} \\omega _3^2+189 q_3^4 u^8 \\omega _3^4-8496 q_3^4 u^6 \\omega _3^2-99 q_3^4 u^4 \\omega _3^4+3510q_3^4 u^2 \\omega _3^2-81 q_3^4 \\omega _3^4\\nonumber \\\\& & +114 q_3^2 u^{14} \\omega _3^4-2262 q_3^2 u^{10} \\omega _3^4-105 q_3^2 u^8\\omega _3^6+5598 q_3^2 u^6 \\omega _3^4+144 q_3^2 u^4 \\omega _3^6-2970 q_3^2 u^2 \\omega _3^4+81 q_3^2 \\omega _3^6\\nonumber \\\\& & -8 \\left(u^8-4u^4+3\\right) \\left(q_3^2 \\left(u^4-1\\right)+\\omega _3^2\\right) \\left(q_3^3 \\left(u^4-3\\right)+3 q_3 \\omega _3^2\\right)^2 \\log \\left(\\frac{{r_h}}{u}\\right)\\nonumber \\\\& & +18 u^{10} \\omega _3^6-1188 u^6 \\omega _3^6-63 u^4 \\omega _3^8+810 u^2 \\omega _3^6-27 \\omega _3^8\\Biggr )\\Biggr \\rbrace \\nonumber \\\\& & -\\frac{q_3^4 \\left(u^8-4 u^4+3\\right)+2q_3^2 \\left(8 u^{10}-8 u^6+2 u^4 \\omega _3^2-3 \\omega _3^2\\right)+3 \\omega _3^4}{\\left(u^4-1\\right)^2 \\left(q_3^2 \\left(u^4-3\\right)+3\\omega _3^2\\right)}\\Biggr ]Z_s(u).$ The horizon $u=1$ due to inclusion of the non-conformal corrections to the metric, ceases to be an irregular singular point.", "One then tries the ansatz: $Z_s(u) = e^{S(u)}$ near $u=1$ .", "Assuming that $\\left(S^{\\prime }\\right)^2\\gg S^{\\prime \\prime }(u)$ near $u=1$ the differential equation (REF ), which could written as $Z_s^{\\prime \\prime }(u) = m(u)Z_s^\\prime + l(u) Z_s(u)$ can be approximated by: $\\left(S^\\prime \\right)^2 - m(u) S^\\prime (u) - l(u) \\approx 0.$ A solution to (REF ) is: $& & S(u) = \\frac{1}{2}\\left(m(u) - \\sqrt{m^2(u) + 4 l(u)}\\right)\\nonumber \\\\& & = -\\frac{\\sqrt{\\frac{15}{2}} \\sqrt{\\frac{{g_s}^2 M^2 {N_f} {\\omega _3}^2 \\left({\\omega _3}^2+4\\right) \\log \\left(\\frac{1}{N}\\right)}{N {q_3}^2}}}{64 \\pi (u-1)^{3/2}} + \\frac{\\frac{15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log \\left(\\frac{1}{N}\\right)}{256 \\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)}-\\frac{1}{2}}{u-1}\\nonumber \\\\& & +\\frac{3 {g_s}^2 M^2 {N_f} \\log \\left(\\frac{1}{N}\\right) \\left(112 {q_3}^4+214{q_3}^2 {\\omega _3}^2-369 {\\omega _3}^4\\right)+128 \\pi ^2 N \\left(-20 {q_3}^4+12 {q_3}^2 {\\omega _3}^2+27 {\\omega _3}^4\\right)}{512 \\pi ^2 N \\left(2{q_3}^2-3 {\\omega _3}^2\\right)^2}\\nonumber \\\\& & +\\frac{1}{1024 \\sqrt{30} \\pi ^3N^2 {q_3}^2 \\sqrt{u-1} \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^2 \\sqrt{\\frac{{g_s}^2 M^2 {N_f} {\\omega _3}^2 \\left({\\omega _3}^2+4\\right) \\log \\left(\\frac{1}{N}\\right)}{N {q_3}^2}}}\\nonumber \\\\& & \\times \\Biggl \\lbrace -225 {g_s}^4 M^4 {N_f}^2 {q_3}^2 {\\omega _3}^4 \\log ^2\\left(\\frac{1}{N}\\right)+12 \\pi ^2 {g_s}^2 M^2N {N_f} \\log \\left(\\frac{1}{N}\\right) \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)\\nonumber \\\\& & \\times \\left(-80 {q_3}^4 \\left({\\omega _3}^2-4\\right)+2 {q_3}^2 {\\omega _3}^2\\left(57 {\\omega _3}^2-412\\right)+64 \\left(2 {q_3}^4 {\\omega _3}^2-3 {q_3}^2 {\\omega _3}^4\\right) \\log ({r_h})+9 {\\omega _3}^4\\left({\\omega _3}^2+164\\right)\\right)\\nonumber \\\\& & +4096 \\pi ^4 N^2 {q_3}^2 \\left({\\omega _3}^2-4\\right) \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^2\\Biggr \\rbrace + {\\cal O}\\left(\\sqrt{u-1}\\right).$ Taking first the MQGP limit, the first term in the RHS of (REF ) can be dropped.", "After integrating with respect to $u$ , the solution (REF ) to equation (REF ) will reflect the singular nature of $Z(u)$ 's equation of motion (REF ) via $Z_s(u)\\sim \\left(1 - u \\right)^{-\\frac{1}{2} + \\frac{15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log \\left(\\frac{1}{N}\\right)}{256 \\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)}}F(u),$ where $F(u)$ is regular in $u$ and its equation of motion, around $u=0$ , is given by: $& & 256 F^{\\prime \\prime }(u)+\\frac{F^{\\prime }(u) \\left(\\frac{60 {g_s}^2 M^2 {N_f} \\log \\left(\\frac{1}{N}\\right)}{\\pi ^2 N}-768\\right)}{u}\\nonumber \\\\& & -\\frac{3 F(u) \\left(64 \\pi ^2 N-5{g_s}^2 M^2 {N_f} \\log \\left(\\frac{1}{N}\\right)\\right) \\left(15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log (N)+128 \\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)\\right)}{64 \\pi ^4 N^2 u \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)} = 0.\\nonumber \\\\& &$ The solution to (REF ) is given by: $& & F(u) = 2^{-\\frac{105 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}-28} 3^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{128 \\pi ^2 N}+2} N^{-\\frac{15 {g_s}^2 M^2{N_f} \\log (N)}{64 \\pi ^2 N}-4} \\pi ^{-\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{32 \\pi ^2 N}-8}\\nonumber \\\\& & \\times \\left(5 {g_s}^2 M^2 {N_f} \\log (N)+64 \\pi ^2N\\right)^{2-\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{128 \\pi ^2 N}} u^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{128 \\pi ^2 N}+2} \\left(2 {q_3}^2-3{\\omega _3}^2\\right)^{-\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{128 \\pi ^2 N}-2}\\nonumber \\\\& & \\times \\left(15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log (N)+128 \\pi ^2 N \\left(2{q_3}^2-3 {\\omega _3}^2\\right)\\right)^{2-\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{128 \\pi ^2 N}}\\nonumber \\\\& & \\times \\Biggl (c_1 \\left(5 {g_s}^2 M^2 {N_f} \\log (N)+64\\pi ^2 N\\right)^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}} \\Gamma \\left(-\\frac{15 {g_s}^2 {N_f} \\log (N) M^2}{64 N \\pi ^2}-3\\right)\\nonumber \\\\& & \\times \\left(15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log (N)+128 \\pi ^2 N \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)\\right)^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}}\\nonumber \\\\& & \\times I_{-\\frac{15 {g_s}^2 {N_f} \\log (N) M^2}{64 N \\pi ^2}-4}\\left(\\frac{\\sqrt{3} \\sqrt{u} \\sqrt{\\left(5 {g_s}^2 {N_f} \\log (N)M^2+64 N \\pi ^2\\right) \\left(15 {g_s}^2 M^2 {N_f} \\log (N) {\\omega _3}^2+128 N \\pi ^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)\\right)}}{64 \\pi ^2\\sqrt{N^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}}\\right)\\nonumber \\\\& & +c_2 N^{\\frac{15 {g_s}^2 M^2 {N_f} (2 \\log (N)+i \\pi )}{64 \\pi ^2 N}} \\Gamma \\left(\\frac{15{g_s}^2 {N_f} \\log (N) M^2}{64 N \\pi ^2}+5\\right) \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}}\\nonumber \\\\& & \\times \\left(\\left(5 {g_s}^2 M^2 {N_f} \\log (N)+64 \\pi ^2 N\\right) \\left(15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log (N)+128 \\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)\\right)\\right)^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}}\\nonumber \\\\& & \\times \\left(N^2 \\left(2 {q_3}^2-3{\\omega _3}^2\\right)\\right)^{-\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}}\\nonumber \\\\& & \\times I_{\\frac{15 {g_s}^2 {N_f} \\log (N) M^2}{64 N \\pi ^2}+4}\\left(\\frac{\\sqrt{3} \\sqrt{u} \\sqrt{\\left(5 {g_s}^2 {N_f} \\log (N) M^2+64 N \\pi ^2\\right) \\left(15 {g_s}^2 M^2 {N_f} \\log (N){\\omega _3}^2+128 N \\pi ^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)\\right)}}{64 \\pi ^2 \\sqrt{N^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}}\\right)\\Biggr ).\\nonumber \\\\& &$ One notes from (REF ) that $F(u\\sim 0) = c_1$ .", "This needs to be improved upon by including the sub-leading terms in $u$ in $F^{\\prime }(u)$ in (REF ), implying that we should look at: $& & \\hspace{-86.72377pt}256 F^{\\prime \\prime }(u)\\nonumber \\\\& & \\hspace{-86.72377pt}+F^{\\prime }(u) \\left(\\frac{120 {g_s}^2 M^2 {N_f} \\log N\\left(2 {q_3}^2-3 {\\omega _3}^2\\right)+30 {g_s}^2 M^2{N_f} {\\omega _3}^2 \\log (N)+1792 \\pi ^2 N \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}{\\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)}+\\frac{\\frac{-60 {g_s}^2 M^2 {N_f} \\log N}{\\pi ^2 N}-768}{u}\\right)\\nonumber \\\\& &\\hspace{-86.72377pt}-\\frac{3 F(u) \\left(64 \\pi ^2 N+5{g_s}^2 M^2 {N_f} \\log N\\right) \\left(15 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log (N)+128 \\pi ^2 N \\left(2{q_3}^2-3 {\\omega _3}^2\\right)\\right)}{64 \\pi ^4 N^2 u \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)} = 0.$ The solution to (REF ) near $u=0$ , is given as under: $& & N^{\\frac{15 {g_s}^2 M^2 {N_f} u \\left(11 {\\omega _3}^2-8 {q_3}^2\\right)}{128 \\pi ^2 N \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}}\\nonumber \\\\& & \\times \\Biggl [u^4\\Biggl (\\frac{c_1 \\Gamma \\left(-\\frac{15 {g_s}^2 {N_f} \\log (N) M^2}{64 N \\pi ^2}-4\\right)}{\\Gamma \\left(-\\frac{3 \\left(5 {g_s}^2 {N_f}\\log (N) M^2+64 N \\pi ^2\\right) \\left(15 {g_s}^2 {N_f} \\left(16 {q_3}^2-23 {\\omega _3}^2\\right) \\log (N) M^2+1664 N \\pi ^2 \\left(2{q_3}^2-3 {\\omega _3}^2\\right)\\right)}{128 N \\pi ^2 \\left(15 {g_s}^2 {N_f} \\left(8 {q_3}^2-11 {\\omega _3}^2\\right) \\log (N) M^2+896 N \\pi ^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)\\right)}\\right)}\\nonumber \\\\& & +c_2 L_{-\\frac{225 {g_s}^4 M^4 {N_f}^2 {\\omega _3}^2 \\log ^2(N)+4800 \\pi ^2{g_s}^2 M^2 N {N_f} \\log (N) \\left(4 {q_3}^2-5 {\\omega _3}^2\\right)+139264 \\pi ^4 N^2 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}{128 \\pi ^2 N\\left(15 {g_s}^2 M^2 {N_f} \\log (N) \\left(8 {q_3}^2-11 {\\omega _3}^2\\right)+896 \\pi ^2 N \\left(2 {q_3}^2-3{\\omega _3}^2\\right)\\right)}}^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}+4}(0)\\Biggr )\\nonumber \\\\& & +\\frac{1}{\\Gamma \\left(\\frac{225 {g_s}^4 {N_f}^2 {\\omega _3}^2 \\log ^2(N) M^4+4800 {g_s}^2 N{N_f} \\pi ^2 \\left(4 {q_3}^2-5 {\\omega _3}^2\\right) \\log (N) M^2+139264 N^2 \\pi ^4 \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}{128 N \\pi ^2\\left(15 {g_s}^2 {N_f} \\left(8 {q_3}^2-11 {\\omega _3}^2\\right) \\log (N) M^2+896 N \\pi ^2 \\left(2 {q_3}^2-3{\\omega _3}^2\\right)\\right)}\\right)}\\nonumber \\\\& & \\times \\Biggl \\lbrace c_1 2^{\\frac{105 {g_s}^2 M^2 {N_f}\\log (N)}{64 \\pi ^2 N}+28} \\pi ^{\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{32 \\pi ^2 N}+8} u^{-\\frac{15 {g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}} \\Gamma \\left(\\frac{15 {g_s}^2 {N_f} \\log (N) M^2}{64 N \\pi ^2}+4\\right)\\nonumber \\\\& & \\left(\\frac{15 {g_s}^2 M^2 {N_f} \\log (N) \\left(8{q_3}^2-11 {\\omega _3}^2\\right)+896 \\pi ^2 N \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}{N \\left(2 {q_3}^2-3 {\\omega _3}^2\\right)}\\right)^{-\\frac{15{g_s}^2 M^2 {N_f} \\log (N)}{64 \\pi ^2 N}-4}\\Biggr \\rbrace \\Biggr ].\\nonumber \\\\& &$" ], [ "Gauge Transformations Preserving $h_{m\\ \\mu }=0$", "There are three gauge transformations that preserve $h_{\\mu u}=0$ , for the black $M3$ -brane metric having integrated out the $M_6$ in the (asymptotic) $AdS_5\\times M_6$ in the MQGP limit.", "They are given below: Set I: The Gauge transformations are generated by $&& \\xi _{{x}}= \\frac{C_{{x}}(t,{x})}{u^2}+\\xi ^{(1)}_{{x}}(u,t,{x})\\nonumber \\\\&& \\xi _{t}=\\xi ^{(1)}_{t}(u,t,{x})$ The Gauge Solutions for the above kind of transformations are given as: $&& H^{\\rm Gauge(I)}_{tt}=\\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[\\frac{2 \\omega _3 u^2}{g_1}\\tilde{\\xi }^{(1)}_{t}-q_3 \\tilde{C}_{{x}} H_{t t}-\\frac{ 2 \\omega _3 \\tilde{C}_{{x}}}{g_1} H_{t {x}}\\right]\\nonumber \\\\&& H^{\\rm Gauge(I)}_{{x} {x}}=\\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[ q_3 \\tilde{C}_{{x}} H_{{x} {x}}-2 q_3 u^2 \\tilde{\\xi }^{(1)}_{{x}}- { 2 q_3 \\tilde{C}_{{x}}} \\right]\\nonumber \\\\&&H^{\\rm Gauge(I)}_{t {x}}=\\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[\\omega _3 \\tilde{C}_{{x}}+ u^2 \\omega _3 \\tilde{\\xi }^{(1)}_{x} - u^2 q_3 \\tilde{\\xi }^{(1)}_{t}-\\omega _3 \\tilde{C}_{{x}}H_{{x} {x}}\\right]\\nonumber \\\\&& H^{\\rm gauge (I)}_{a a}= \\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[ -q_3 \\tilde{C}_{{x}} H_{aa}\\right]$ where $H_{aa}=H_{{y} {y}}+H_{{z} {z}}, \\tilde{C}_{{x}}\\equiv \\frac{C_{{x}}}{\\pi T}, \\tilde{\\xi }^{(1)}_t\\equiv \\frac{\\xi ^{(1)}_t}{\\pi T}$ .", "Set II: The Gauge transformations are generated by $&& \\xi _{t}=- \\frac{ g_1 C_t(t,{x})}{u^2}+\\xi ^{(1)}_{t}(u,t,{x})\\nonumber \\\\&& \\xi _{{x}}=\\xi ^{(1)}_{{x}}(u,t,{x})$ The Gauge Solutions for the above kind of transformations are given as: $&& H^{\\rm Gauge(II)}_{tt}=\\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[-2 \\omega _3 \\tilde{C}_{t}+\\frac{2 \\omega _3 u^2}{g_1}\\tilde{\\xi }^{(1)}_{t}-\\omega _3 \\tilde{C}_{t} H_{tt} \\right]\\nonumber \\\\&& H^{\\rm Gauge(II)}_{{x} {x}}=\\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[ -2 q_3 u^2 \\tilde{\\xi }^{(1)}_{{x}}+ 2 q_3 \\tilde{C}_{t} H_{t {x}}+ \\omega _3 \\tilde{C}_t H_{{x} {x}} \\right]\\nonumber \\\\&&H^{\\rm Gauge(II)}_{t {x}}=\\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[q_3 g_1 \\tilde{C}_{t}- u^2 q_3 \\tilde{\\xi }^{(1)}_{t}+ u^2 \\omega _3 \\tilde{\\xi }^{(1)}_{{x}}+ q_3 g_1 \\tilde{C}_{t}H_{tt}\\right]\\nonumber \\\\&& H^{\\rm gauge (II)}_{a a}= \\frac{g^{\\frac{2}{3}}_{s}}{i L^2}\\left[ \\omega _3 \\tilde{C}_{t} H_{aa}\\right]$ where $H_{aa}=H_{{y} {y}}+H_{{z} {z}}, \\tilde{\\xi }^{(1)}_{{x}}\\equiv \\frac{\\xi ^{(1)}_{{x}}}{\\pi T}$ .", "Set III: Writing $\\xi _u^{(0)}=\\frac{C_u(t,{x})}{u\\sqrt{g}}, \\xi ^{(0)}_t = - \\partial _t C_u(t,{x})\\psi (u), \\xi ^{(0)}_{{x}} = - \\partial _{{x}}C_u(t,{x})\\chi (u)$ , and demanding the solutions to be well behave at $u=0$ , one obtains: $& & \\xi _u^{(0)} = \\frac{C_u(t,{x})}{u\\sqrt{g}};\\nonumber \\\\& & \\xi ^{(0)}_t = - \\left(\\frac{1}{2} - \\frac{u^4}{3}\\right)\\sqrt{g} \\partial _t C_u(t,{x});\\nonumber \\\\& & \\xi ^{(0)}_{{x}} = - \\partial _{{x}}C_u(t,{x}) \\frac{F(\\sin ^{-1}u|1)}{u} = - \\partial _{{x}}C_u(t,{x})\\left(1 + \\frac{u^4}{10} + {\\cal O}(u^8)\\right).$ This yields the following: $& & H^{\\rm Gauge(III)}_{tt}=2\\frac{g_s^{\\frac{2}{3}}u^2\\omega _3^2C_u(t,{x})}{L^2 \\sqrt{1-u^4}}\\left(\\frac{1}{2} - \\frac{u^4}{3}\\right) - 2\\frac{g_s^{\\frac{2}{3}}}{L^2\\sqrt{1-u^4}}C_u(t,{x})(1+u^4) \\nonumber \\\\& & + \\frac{\\omega _3q_3g_s^{\\frac{2}{3}}}{L^2} H_{tt} C_u F\\left(\\sin ^{-1}u|1\\right)u - \\omega _3^2g_s^{\\frac{2}{3}}H_{tt}\\frac{\\left(\\frac{1}{2} - \\frac{u^4}{3}\\right)}{\\sqrt{1-u^4}} + \\frac{g_s^{\\frac{2}{3}}C_u\\left(u H_{t{x}}^{\\prime } - 2 H_{t{x}}\\right)}{2L^2\\sqrt{1-u^4}};\\nonumber \\\\& & H^{\\rm Gauge(III)}_{{x}t}=-\\frac{\\omega _3q_3g_s^{\\frac{2}{3}}}{L^2}\\sqrt{1-u^4}C_u u^2\\left(\\frac{1}{2} - \\frac{7u^4}{30}\\right);\\nonumber \\\\& & H^{\\rm Gauge(III)}_{{x} {x}}=-2\\frac{q_3^2g_s^{\\frac{2}{3}}}{L^2}C_u\\left(1+\\frac{u^4}{10}\\right) + 2\\frac{\\sqrt{g}g_s^{\\frac{2}{3}}}{L^2}C_u(1+u^4);\\nonumber \\\\& & H^{\\rm Gauge(III)}_{{y}{y}}=\\frac{g_s^{\\frac{2}{3}}\\sqrt{1-u^4}}{L^2}\\left(-H_{{x}{x}} + u H_{{x}{x}}^{\\prime }\\right).$" ], [ "Pole Structure of $\\Omega (\\omega _3,q_3)$", "The equation (REF ) can be solved for $\\omega _3$ and the solution is given by: $& & \\omega _3 = -\\frac{2 \\left({\\alpha _{yy}^{(1,0)}}+\\sqrt{{\\alpha _{yy}^{(1,0)}}^2-{\\alpha _{yy}^{(0,0)}} (4 {C_{1yy}^{(0,2)}}+4{C_{2yy}^{(0,2)}}+i {\\Sigma _{2yy}^{(0,1)}})}\\right)}{4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2\\ yy}^{(0,1)}}} \\nonumber \\\\& & + \\frac{{q_3} \\left(\\frac{{\\alpha _{yy}^{(1,0)}} \\left(72 {C_{1yy}^{(0,2)}}+8 e^3 {C_{1yy}^{(1,1)}}+72 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+18 i {\\Sigma _{2\\ yy}^{(0,1)}}+9i\\right)}{\\sqrt{{\\alpha _{yy}^{(1,0)}}^2-{\\alpha _{yy}^{(0,0)}} (4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{02}}+i{\\Sigma _{2\\ yy}^{(0,1)}})}}+8 e^3 {C_{1yy}^{(1,1)}}-36 {C_{2yy}^{(1,1)}}+9 i\\right)}{18(4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2\\ yy}^{(0,1)}})}\\nonumber \\\\& & + {\\cal O}(q_3^2);\\nonumber \\\\& & -\\frac{2 \\left({\\alpha _{yy}^{(1,0)}}-\\sqrt{{\\alpha _{yy}^{(1,0)}}^2-{\\alpha _{yy}^{(0,0)}} (4 {C_{1yy}^{(0,2)}}+4{C_{2yy}^{(0,2)}}+i {\\Sigma _{2yy}^{(0,1)}})}\\right)}{4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2\\ yy}^{(0,1)}}} \\nonumber \\\\& & + \\frac{{q_3} \\left(-\\frac{{\\alpha _{yy}^{(1,0)}} \\left(72 {C_{1yy}^{(0,2)}}+8 e^3 {C_{1yy}^{(1,1)}}+72 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+18 i {\\Sigma _{2\\ yy}^{(0,1)}}+9i\\right)}{\\sqrt{{\\alpha _{yy}^{(1,0)}}^2-{\\alpha _{yy}^{(0,0)}} (4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{02}}+i{\\Sigma _{2\\ yy}^{(0,1)}})}}+8 e^3 {C_{1yy}^{(1,1)}}-36 {C_{2yy}^{(1,1)}}+9 i\\right)}{18(4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2\\ yy}^{(0,1)}})}\\nonumber \\\\& & + {\\cal O}(q_3^2)$ Assuming $\\alpha _{yy}^{(0,0)}\\ll 1, |\\Sigma _{2\\ yy}^{(0,1)}|\\gg 1(i \\Sigma _{2\\ yy}^{(0,1)}\\in \\mathbb {R}): \\alpha _{yy}^{(0,0)}\\Sigma _{2\\ yy}^{(0,1)}<1; \\alpha _{yy}^{(1,0)} = - |\\alpha _{yy}^{(1,0)}|$ , consistent with the constraints (REF ,), (REF ) implies the following.", "Root 1: $& & \\hspace{-43.36243pt}\\omega _3 = \\left(-\\frac{2 \\left(\\sqrt{{\\alpha _{yy}^{(1,0)}}^2}+{\\alpha _{yy}^{(1,0)}}\\right)}{4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2yy}^{(0,1)}}}-\\frac{{\\alpha _{yy}^{(0,0)}}\\sqrt{{\\alpha _{yy}^{(1,0)}}^2} (-4 {C_{1yy}^{(0,2)}}-4 {C_{2yy}^{(0,2)}}-i {\\Sigma _{2yy}^{(0,1)}})}{{\\alpha _{yy}^{(1,0)}}^2 (4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i{\\Sigma _{2yy}^{(0,1)}})}+O\\left({\\alpha _{yy}^{(0,0)}}^2\\right)\\right)+\\nonumber \\\\& & \\hspace{-72.26999pt}q_3 \\Biggl [\\frac{\\sqrt{{\\alpha _{yy}^{(1,0)}}^2} \\left(8 e^3 {C_{1yy}^{(1,1)}}-36{C_{2yy}^{(1,1)}}+9 i\\right)+{\\alpha _{yy}^{(1,0)}} \\left(72 {C_{1yy}^{(0,2)}}+72 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+18 i {\\Sigma _{2yy}^{(0,1)}}+8 {C_{1yy}^{(1,1)}} e^3+9i\\right)}{18 \\sqrt{{\\alpha _{yy}^{(1,0)}}^2} (4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2yy}^{(0,1)}})}+\\nonumber \\\\& & \\hspace{-72.26999pt}\\frac{1}{18 (4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i{\\Sigma _{2yy}^{(0,1)}})} \\Biggl (\\frac{\\left(8 e^3 {C_{1yy}^{(1,1)}}-36{C_{2yy}^{(1,1)}}+9 i\\right) (-4 {C_{1yy}^{(0,2)}}-4 {C_{2yy}^{(0,2)}}-i {\\Sigma _{2yy}^{(0,1)}})}{2 {\\alpha _{yy}^{(1,0)}}^2}\\nonumber \\\\& & \\hspace{-86.72377pt}-\\frac{\\left(\\sqrt{{\\alpha _{yy}^{(1,0)}}^2}\\left(8 e^3 {C_{1yy}^{(1,1)}}-36 {C_{2yy}^{(1,1)}}+9 i\\right)+{\\alpha _{yy}^{(1,0)}} \\left(72 {C_{1yy}^{(0,2)}}+72 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+18 i{\\Sigma _{2yy}^{(0,1)}}+8 {C_{1yy}^{(1,1)}} e^3+9 i\\right)\\right)}{2 {\\alpha _{yy}^{(1,0)}}^2\\sqrt{{\\alpha _{yy}^{(1,0)}}^2}}\\Biggr ) \\nonumber \\\\& & \\hspace{-72.26999pt}\\times (-4 {C_{1yy}^{(0,2)}}-4 {C_{2yy}^{(0,2)}}-i {\\Sigma _{2yy}^{(0,1)}}){\\alpha _{yy}^{(0,0)}} +O\\left({\\alpha _{yy}^{(0,0)}}^2\\right)\\Biggr ]+O\\left(q_3^2\\right).$ The expansion (REF ) implies: $& & \\hspace{-28.90755pt}\\omega _3= -\\frac{{\\alpha _{yy}^{(0,0)}}}{{\\alpha _{yy}^{(1,0)}}} + \\nonumber \\\\& & \\hspace{-28.90755pt}q_3\\Biggl [ -\\frac{72 {\\alpha _{yy}^{(0,0)}} {C_{1yy}^{(0,2)}}+8 e^3 {\\alpha _{yy}^{(0,0)}} {C_{1yy}^{(1,1)}}+72 {\\alpha _{yy}^{(0,0)}} {C_{2yy}^{(0,2)}}-36 {\\alpha _{yy}^{(0,0)}} {C_{2yy}^{(1,1)}}+18i {\\alpha _{yy}^{(0,0)}} {\\Sigma _{2yy}^{(0,1)}}+9 i {\\alpha _{yy}^{(0,0)}}+36 {\\alpha _{yy}^{(1,0)}}^2}{36 {\\alpha _{yy}^{(1,0)}}^2}\\Biggr ]\\nonumber \\\\& & \\hspace{-28.90755pt}\\approx - q_3\\left(1 + i \\frac{\\alpha _{yy}^{(00)}\\Sigma _{2\\ yy}^{(0,1)}}{2\\left(\\alpha _{yy}^{(1,0)}\\right)^2}\\right).$ Root 2: $&& \\hspace{-43.36243pt}\\omega _3 = \\left(\\frac{2 \\left(\\sqrt{{\\alpha _{yy}^{(1,0)}}^2}-{\\alpha _{yy}^{(1,0)}}\\right)}{4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2yy}^{(0,1)}}}-\\frac{{\\alpha _{yy}^{(0,0)}}\\sqrt{{\\alpha _{yy}^{(1,0)}}^2}}{{\\alpha _{yy}^{(1,0)}}^2}+O\\left({\\alpha _{yy}^{(0,0)}}^2\\right)\\right)+ \\nonumber \\\\& & \\hspace{-43.36243pt}q_3 \\left(\\frac{\\sqrt{{\\alpha _{yy}^{(1,0)}}^2} \\left(8 e^3{C_{1yy}^{(1,1)}}-36 {C_{2yy}^{(1,1)}}+9 i\\right)-{\\alpha _{yy}^{(1,0)}} \\left(72 {C_{1yy}^{(0,2)}}+72 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+18 i {\\Sigma _{2yy}^{(0,1)}}+8{C_{1yy}^{(1,1)}} e^3+9 i\\right)}{18 \\sqrt{{\\alpha _{yy}^{(1,0)}}^2} (4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i{\\Sigma _{2yy}^{(0,1)}})}\\right.\\nonumber \\\\& & \\hspace{-43.36243pt}\\left.-\\frac{\\left(\\sqrt{{\\alpha _{yy}^{(1,0)}}^2} \\left(72 {C_{1yy}^{(0,2)}}+72 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+18 i {\\Sigma _{2yy}^{(0,1)}}+8{C_{1yy}^{(1,1)}} e^3+9 i\\right)\\right) {\\alpha _{yy}^{(0,0)}}}{36 {\\alpha _{yy}^{(1,0)}}^3}\\right.\\nonumber \\\\& &\\hspace{-43.36243pt}\\left.+O\\left({\\alpha _{yy}^{(0,0)}}^2\\right)\\right)+O\\left(q_3^2\\right).$ The expansion (REF ) implies: $& & \\omega _3 = \\sqrt{{\\alpha _{yy}^{(1,0)}}^2} \\left(-\\frac{{\\alpha _{yy}^{(0,0)}}}{{\\alpha _{yy}^{(1,0)}}^2}+\\frac{4}{4 {C_{1yy}^{(0,2)}}+4 {C_{2yy}^{(0,2)}}+i {\\Sigma _{2yy}^{(0,1)}}}\\right)\\nonumber \\\\& & + q_3\\Biggl [\\frac{ \\left(36 {C_{1yy}^{(0,2)}}+8 e^3 {C_{1yy}^{(1,1)}}+36 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+9 i {\\Sigma _{2yy}^{(0,1)}}+9 i\\right)}{9 (4 {C_{1yy}^{(0,2)}}+4{C_{2yy}^{(0,2)}}+i {\\Sigma _{2yy}^{(0,1)}})} \\nonumber \\\\& & + \\frac{\\alpha _{yy}^{(0,0)} \\left(72 {C_{1yy}^{(0,2)}}+8 e^3 {C_{1yy}^{(1,1)}}+72 {C_{2yy}^{(0,2)}}-36 {C_{2yy}^{(1,1)}}+18 i {\\Sigma _{2yy}^{(0,1)}}+9 i\\right)}{36{\\alpha _{yy}^{(1,0)}}^2} \\Biggr ]\\nonumber \\\\& & \\approx q_3\\left(1 + i \\frac{\\alpha _{yy}^{(00)}\\Sigma _{2\\ yy}^{(0,1)}}{2\\left(\\alpha _{yy}^{(1,0)}\\right)^2}\\right).$" ], [ "Solutions to ${\\cal H}_{ab}(u)$", "Making double perturbative ansatze: ${\\cal H}_{ab}(u) = \\sum _{m=0}^\\infty \\sum _{n=0}^\\infty {\\cal H}_{ab}^{(m,n)}(u)q_3^m\\omega _3^n$ , one obtains near $u=0$ the solutions to the scalar modes' EOMs (REF ): $& & {\\cal H}_{yy}^{(0,0)}(u) = \\alpha _{yy}^{(0,0)} + \\beta _{yy}^{(0,0)} u^4,\\nonumber \\\\& & {\\cal H}_{yy}^{(1,0)}(u) = \\alpha _{yy}^{(1,0)} + \\beta _{yy}^{(1,0)} u^4,\\nonumber \\\\& & {\\cal H}_{yy}^{(0,1)}(u) = \\Sigma _{2\\ yy}^{(0,1)} + \\frac{i}{16} \\alpha _{yy}^{(0,0)}u^2\\nonumber \\\\& & {\\rm where:}\\Sigma _{2\\ yy}^{(0,1)}\\equiv \\frac{1}{8}\\left(- 195 i - 60 \\pi - 35\\beta _{yy}^{(0,0)}(13 i + 4\\pi ) + 8 C_{2yy}^{(0,1)}\\right),\\nonumber \\\\& & {\\cal H}_{yy}^{(1,1)} = -\\frac{i}{4} - \\frac{2}{9} e^3 C_{1yy}^{(1,1)} + C_{2yy}^{(1,1)} +i \\frac{u}{4},\\nonumber \\\\& & {\\cal H}_{yy}^{(2,0)}(u) = C_{1yy}^{(2,0)} + C_{2yy}^{(2,0)} u^4,\\nonumber \\\\& & {\\cal H}_{yy}^{(0,2)}(u) = i \\frac{\\Sigma _{2\\ yy}^{(0,1)}}{4} + C_{1yy}^{(0,2)} + C_{2yy}^{(0,2)} - \\frac{i}{4} \\Sigma _{2\\ yy}^{(0,1)} u;\\nonumber \\\\& & {\\cal H}_{xt}^{(0,0)}(u) = \\alpha _{xt}{(0,0)},\\nonumber \\\\& & {\\cal H}_{xt}^{(1,0)}(u) = \\alpha _{xt}^{(1,0)} + \\beta _{xt}^{(1,0)} u^4,\\nonumber \\\\& & {\\cal H}_{xt}^{(0,1)}(u) = C_{2xt}^{(0,1)} + \\frac{i}{4} \\alpha _{xt}^{(0,0)} u,\\nonumber \\\\& & {\\cal H}_{xt}^{(1,1)}(u) = C_{2xt}^{(1,1)} + \\frac{1}{96} \\left[12 i \\left(\\alpha _{xt}^{(1,1)} + 5 \\beta _{xt}^{(1,1)}\\right) u +6 \\left(3 i \\alpha _{xt}^{(1,1)} + 4 \\alpha _{yy} + 5 i \\beta _{xt}^{(1,1)} - 4 \\beta _{yy}\\right) u^2\\right],\\nonumber \\\\& & {\\cal H}_{xt}^{(2,0)}(u) = \\alpha _{xt}^{(2,0)} + \\beta _{xt}^{(2,0)} u^4,\\nonumber \\\\& & {\\cal H}_{xt}^{(0,2)}(u) = \\frac{i}{4} C_{2xt}^{(0,1)} u + \\frac{1}{4} u^4 C_{1xt}^{(0,2)} + C_{2xt}^{(0,2)};\\nonumber \\\\& & {\\cal H}_{tt}^{(0,0)}(u) = \\left(\\frac{4}{3} \\beta _{yy}^{(0,0)} -i C_{1tt}^{(0,0)}\\right) + \\left(-4 \\beta _{yy}^{(0,0)}/3 - \\frac{i}{2} C_{1tt}^{(0,0)}\\right) u^4 - \\frac{3i}{8}C_{1tt}^{(0,0)}u^8,\\nonumber \\\\& & {\\cal H}_{tt}^{(0,1)}(u) =\\alpha _{tt}^{(0,1)} + \\frac{i}{12} \\left(6 \\alpha _{yy}^{(0,0)} + 4 \\beta _{yy}^{(0,0)} - 3 i C_{1tt}^{(0,0)}\\right) u +\\frac{i}{24} \\left(3 \\alpha + 4 \\beta - 3 i C_{tt}^{(0,0)}\\right) u^2,\\nonumber \\\\& & {\\cal H}_{tt}^{(1,0)}(u) = \\left(\\frac{4}{3} \\beta _{yy}^{(1,0)} -i C_{1tt}^{(1,0)}\\right) + \\left(-\\frac{4}{3} \\beta _{yy}^{(1,0)} - (i/2) C_{1yy}^{(1,0)}\\right) u^4,\\nonumber \\\\& & {\\cal H}_{tt}^{(1,1)}(u) =C_{1tt}^{(1,1)} + 1/12 i \\left(-6 + 6 \\alpha _{yy}^{(1,0)} + 4 \\beta _{yy}^{(1,0)} - 3 i C1tt^{(1,0)}\\right) u +1/24 i \\left(6 \\alpha _{yy}^{(1,0)} + 4 \\beta _{yy}^{(1,0)} - 3 i C_{1tt}^{(1,0)}\\right) u^2,\\nonumber \\\\& & {\\cal H}_{tt}^{(0,2)}(u) =\\frac{i}{192} \\left(12 \\alpha _{tt}^{(0,1)} \\pi + 12 \\alpha _{xt}^{(1,0)} \\pi +6 i \\alpha _{yy}^{(0,0)} \\pi + 4 i \\beta _{yy}^{(0,0)} \\pi + 3 C_{1tt}^{(0,0)} \\pi -24 \\pi \\Sigma _{2\\ yy}^{(0,1)} - 192 C_{1tt}^{(0,2)}\\right)\\nonumber \\\\& & -\\frac{i}{8} \\left(2 \\alpha _{tt}^{(0,1)} + 2 \\alpha _{xt}^{(1,0)} - \\alpha _{xt}^{(1,1)} - 5 \\beta _{xt}^{(1,1)}-12 \\Sigma _{2\\ yy}^{(0,1)}\\right) u \\nonumber \\\\& &-\\frac{i}{96} \\left(12 \\alpha _{tt}^{(0,1)} + 12 \\alpha _{xt}^{(1,0)} + 6 i \\alpha _{yy}^{(0,0)} +4 i \\beta _{yy}^{(0,0)} + 3 C_{1tt}^{(0,0)} - 24 \\Sigma _{2\\ yy}^{(0,1)}\\right) u^2,\\nonumber \\\\& & {\\cal H}_{tt}^{(2,0)}(u) = \\left(\\frac{4}{3} C_{2yy}^{(2,0)} -i C_{1tt}^{(2,0)}\\right) + \\left(-\\frac{4}{3} C_{2yy}^{(2,0)} - 1/2 i C_{1tt}^{(2,0)}\\right) u^4;\\nonumber \\\\$ $&& {\\cal H}_s^{(0,0)}(u) = \\frac{C_{1s}^{(0,0)}}{2}u^2 + C_{2s}^{(0,0)},\\nonumber \\\\& & {\\cal H}_s^{(1,0)}(u) = \\frac{C_{1s}^{(1,0)}}{2} u^2 + C_{2s}^{(1,0)},\\nonumber \\\\& & {\\cal H}_s^{(0,1)}(u) = -\\frac{(2 + 2 i) C_{1s}^{(0,1)}}{\\pi },\\nonumber \\\\& & {\\cal H}_s^{(2,0)}(u) = \\frac{C_{1s}^{(2,0)}}{2} u^2 + C_{2s}^{(2,0)},\\nonumber \\\\& & {\\cal H}_s^{(0,2)}(u) = \\Sigma _{s}^{(0,2)} u + \\frac{C_{1s}^{(0,2)}}{2} u^2 + C_{2s}^{(0,2)}.$ such that: $& & 171 i + 2 i \\alpha _{yy}^{(0,0)} + 319 i \\beta _{yy}^{(0,0)} + 24 C_{1yy}^{(0,0)} = 0;\\nonumber \\\\& & 3 \\alpha _{yy}^{(0,0)} + 4 \\beta _{yy}^{(0,0)} - 3 i C_{1tt}^{(0,0)} - 3 C_{2s}^{(0,0)} = 0$ For consistency checks, we have ensured that (REF ) obtained from the fourth, fifth and the sixth equations of (REF ), also solve the first, second, third and seventh equations near $u=0$ and up to ${\\cal O}(q_3^m\\omega _3^n):m+n=2$ by imposing suitable additional constraints on the constants appearing in (REF )." ], [ "Frobenius Solution to EOM of Gauge-Invariant $Z_v(u)$ for Vector Modes of Metric Fluctuations", "The equations of motion for the vector perturbation modes up next-to-leading order in $N$ , can be reduced to the following single equation of motion in terms of a gauge-invariant variable $Z_v(u)$ : $Z_v^{\\prime \\prime }(u) - m(u) Z_v^\\prime (u) - l(u) Z_v(u) = 0,$ where $& & m(u)\\equiv \\frac{15 {g_s}^2 M^2 {N_f} \\left(u^4-1\\right) \\log (N) \\left({q_3}^2 \\left(u^4-1\\right)+{\\omega _3}^2\\right)+64 \\pi ^2 N \\left(3 {q_3}^2\\left(u^4-1\\right)^2-\\left(u^4+3\\right) {\\omega _3}^2\\right)}{64 \\pi ^2 N u \\left(u^4-1\\right) \\left({q_3}^2 \\left(u^4-1\\right)+{\\omega _3}^2\\right)},\\nonumber \\\\& & l(u)\\equiv -\\frac{\\left({q_3}^2 \\left(u^4-1\\right)+{\\omega _3}^2\\right) \\left(32 \\pi ^2 N-3 {g_s}^2 M^2 {N_f} \\log ^2(N)\\right)}{32 \\pi ^2 N \\left(u^4-1\\right)^2}.$ The horizon $u=1$ is a regular singular point of (REF ) and the root of the indicial equation corresponding to the incoming-wave solution is given by: $-\\frac{i {\\omega _3}}{4} + \\frac{3 i {g_s}^2 M^2 {N_f} {\\omega _3} \\log ^2(N)}{256 \\pi ^2 N}.$ (a) Using the Frobenius method, taking the solution about $u=1$ to be: $Z_v(u) = (1 - u)^{-\\frac{i {\\omega _3}}{4} + \\frac{3 i {g_s}^2 M^2 {N_f} {\\omega _3} \\log ^2(N)}{256 \\pi ^2 N}}\\left(1 + \\sum _{n=1}^\\infty a_n (u - 1)^n\\right),$ by truncating the infinite series in (REF ) to ${\\cal O}((u-1)^2)$ one obtains: $& & \\hspace{-57.81621pt}a_1 = \\frac{1}{512 \\pi ^2 N {\\omega _3} ({\\omega _3}+2 i)^2}\\Biggl \\lbrace 3 i {g_s}^2 M^2 {N_f} \\log (N)\\left(20 (2-i {\\omega _3}) {\\omega _3}^2-\\log (N) \\left(3 {\\omega _3}^2 \\left({\\omega _3}^2+4 i {\\omega _3}+4\\right)-4{q_3}^2 \\left({\\omega _3}^2+4 i {\\omega _3}-8\\right)\\right)\\right)\\Biggr \\rbrace \\nonumber \\\\& & \\hspace{-57.81621pt}+\\frac{4 {q_3}^2 (4-i {\\omega _3})+3 (2+i{\\omega _3}) {\\omega _3}^2}{8 {\\omega _3} ({\\omega _3}+2 i)} + {\\cal O}\\left(\\frac{1}{N^2}\\right),\\nonumber \\\\& & \\hspace{-57.81621pt}a_2 = -\\frac{1}{\\Sigma }\\Biggl \\lbrace 4 \\Biggl (405 i {g_s}^8 M^8 {N_f}^4 {\\omega _3}^5 \\log ^7(N)+8640 i \\pi ^2 {g_s}^6 M^6 N {N_f}^3 {\\omega _3}^2 \\log ^5(N) \\left(16 {q_3}^2({\\omega _3}+2 i)+(-13 {\\omega _3}+4 i) {\\omega _3}^2\\right)\\nonumber \\\\& &\\hspace{-57.81621pt}+368640 \\pi ^4 {g_s}^4 M^4 N^2 {N_f}^2 {\\omega _3} \\log ^3(N) \\left(4 {q_3}^2 \\left(-3 i{\\omega _3}^2+12 {\\omega _3}+16 i\\right)+i {\\omega _3}^2 \\left(9 {\\omega _3}^2+8 i {\\omega _3}+16\\right)\\right)\\nonumber \\\\& &\\hspace{-36.135pt}+7864320 \\pi ^6 {g_s}^2 M^2 N^3 {N_f} {\\omega _3} \\log (N) \\left(4 i {q_3}^2 \\left({\\omega _3}^2+6 i {\\omega _3}-16\\right)+{\\omega _3}^2 \\left(-3 i {\\omega _3}^2+4 {\\omega _3}-16 i\\right)\\right)\\nonumber \\\\& &\\hspace{-57.81621pt}-49152 \\pi ^4 {g_s}^2M^2 N^2 {N_f} {\\omega _3} \\log ^2(N) \\Biggl (75 {g_s}^2 M^2 {N_f} ({\\omega _3}+4 i) {\\omega _3}^2+8 \\pi ^2 N \\Biggl [32 {q_3}^4 ({\\omega _3}+6 i)-48{q_3}^2 \\left({\\omega _3}^3+12 {\\omega _3}+16 i\\right)\\nonumber \\\\& &\\hspace{-57.81621pt}+{\\omega _3}^2 \\left(18 {\\omega _3}^3-111 i {\\omega _3}^2+200 {\\omega _3}+16 i\\right)\\Biggr ]\\Biggr )\\nonumber \\\\& & \\hspace{-57.81621pt}+108 {g_s}^6M^6 {N_f}^3 {\\omega _3} \\log ^6(N) \\left(-75 {g_s}^2 M^2 {N_f} {\\omega _3}^3+2 \\pi ^2 N \\left(128 i {q_3}^4-24 {q_3}^2 {\\omega _3}^2 ({\\omega _3}+4i)+{\\omega _3}^4 (19 {\\omega _3}+22 i)\\right)\\right)\\nonumber \\\\& &\\hspace{-57.81621pt}+2304 \\pi ^2 {g_s}^4 M^4 N {N_f}^2 \\log ^4(N)\\nonumber \\\\& &\\hspace{-57.81621pt}\\times \\left(150 {g_s}^2 M^2 {N_f} ({\\omega _3}+2 i){\\omega _3}^3+\\pi ^2 N \\left(256 {q_3}^4 \\left({\\omega _3}^2+3 i {\\omega _3}+4\\right)-24 {q_3}^2 {\\omega _3}^2 \\left(15 {\\omega _3}^2-8 i{\\omega _3}+96\\right)+{\\omega _3}^4 \\left(125 {\\omega _3}^2-636 i {\\omega _3}+208\\right)\\right)\\right)\\nonumber \\\\& &\\hspace{-57.81621pt}+4194304 \\pi ^8 N^4 {\\omega _3} \\left(16 {q_3}^4 ({\\omega _3}+8i)-24 {q_3}^2 \\left({\\omega _3}^3+24 {\\omega _3}+64 i\\right)+{\\omega _3}^2 \\left(9 {\\omega _3}^3-74 i {\\omega _3}^2+200 {\\omega _3}+32i\\right)\\right)\\Biggr )\\Biggr \\rbrace + {\\cal O}\\left(\\frac{1}{N^2}\\right),$ where: $& & \\Sigma \\equiv {\\omega _3}^2 \\Biggl [-81 {g_s}^8 M^8 {N_f}^4 {\\omega _3}^4 \\log ^8(N)+41472 i \\pi ^2 {g_s}^6 M^6 N {N_f}^3 {\\omega _3}^3 \\log ^6(N)+294912 \\pi ^4 {g_s}^4 M^4 N^2 {N_f}^2 (26-3 i {\\omega _3}) {\\omega _3}^2 \\log ^4(N)\\nonumber \\\\& & -201326592 \\pi ^6 {g_s}^2 M^2 N^3 {N_f} {\\omega _3}({\\omega _3}+3 i) \\log ^2(N)+2147483648 \\pi ^8 N^4 \\left({\\omega _3}^2+6 i {\\omega _3}-8\\right)\\Biggr ].$ The Dirichlet boundary condition $Z_v(u=0)=0$ in the hydrodynamical limit retaining therefore terms only up to ${\\cal O}(\\omega _3^mq_3^n):\\ m+n=4$ , reduces to: $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where: $& & a = 3\\left(96\\pi ^2 + \\frac{13 g_s^2 M^2 N_f (\\log N)^2}{N}\\right),\\nonumber \\\\& & b = 2 i \\left(1664\\pi ^2 + 39 g_s^2 M^2 N_f \\frac{(\\log N)^2}{N}\\right),\\nonumber \\\\& & c = 128 \\pi ^2\\left(-70 + 3 q_3^2\\right) + 78 g_s^2 M^2 N_f\\left(-2 + q_3^2\\right)\\frac{(\\log N)^2}{N},\\nonumber \\\\& & f = 8 i\\left(64\\pi ^2(-16 + 7 q_3^2) - 6 g_s^2 M^2 N_f \\frac{(\\log N)^2}{N}q_3^2\\right),\\nonumber \\\\& & g = 16 q_3^2\\left(64\\pi ^2(-4 + q_3^2) + 3 g_s^2 M^2 N_f(4 - 3 q_3^2)\\frac{(\\log N)^2}{N}\\right),$ One of the four roots of $Z(u=0)=0$ is: $\\omega _3 = -8.18 i + \\frac{0.14 i g_s^2 M^2 N_f(\\log N)^2}{N} + \\left(-0.005 i - \\frac{0.002 i g_s^2 M^2 N_f (\\log N)^2}{N}\\right)q_3^2 + {\\cal O}(q_3^3).$ (b) Using the Frobenius method and going up to ${\\cal O}((u-1)^3)$ in (REF ), one obtains: $& & \\hspace{-43.36243pt}a_3 = \\frac{1}{65536 \\pi ^2 N {\\omega _3}^2 ({\\omega _3}+2 i)^2 ({\\omega _3}+4 i)^2 ({\\omega _3}+6 i)^2}\\Biggl \\lbrace i {g_s}^2 M^2 {N_f} \\log (N) \\Biggl (20 i {\\omega _3} \\left({\\omega _3}^3+12 i {\\omega _3}^2-44 {\\omega _3}-48 i\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}\\times \\left(48 {q_3}^4\\left({\\omega _3}^2+12 i {\\omega _3}-48\\right)-8 {q_3}^2 \\left(9 {\\omega _3}^4+48 i {\\omega _3}^3+60 {\\omega _3}^2+1472 i {\\omega _3}-3840\\right)+{\\omega _3}^2 \\left(27{\\omega _3}^4-42 i {\\omega _3}^3+1288 {\\omega _3}^2+2464 i {\\omega _3}-2048\\right)\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}-\\log (N) \\Biggl [64 {q_3}^6 {\\omega _3} \\left(3 {\\omega _3}^4+72 i{\\omega _3}^3-652 {\\omega _3}^2-2400 i {\\omega _3}+2880\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}-48 {q_3}^4 \\left(9 {\\omega _3}^7+156 i {\\omega _3}^6-668 {\\omega _3}^5+3072 i {\\omega _3}^4-37024{\\omega _3}^3-124416 i {\\omega _3}^2+160768 {\\omega _3}+49152 i\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}+4 {q_3}^2 {\\omega _3} \\left(81 {\\omega _3}^8+852 i {\\omega _3}^7+4324 {\\omega _3}^6+85824 i{\\omega _3}^5-444320 {\\omega _3}^4-1143552 i {\\omega _3}^3+1270784 {\\omega _3}^2-454656 i {\\omega _3}+1769472\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}-{\\omega _3}^3 \\left(81 {\\omega _3}^8+288 i{\\omega _3}^7+13136 {\\omega _3}^6+103296 i {\\omega _3}^5-183440 {\\omega _3}^4+289152 i {\\omega _3}^3-925696 {\\omega _3}^2-436224 i{\\omega _3}+221184\\right)\\Biggr ]\\Biggr )\\Biggr \\rbrace \\nonumber \\\\& & \\hspace{-43.36243pt}+\\frac{1}{3072 {\\omega _3}^2 \\left({\\omega _3}^3+12 i {\\omega _3}^2-44 {\\omega _3}-48 i\\right)}\\Biggl \\lbrace 64 i {q_3}^6 {\\omega _3}({\\omega _3}+12 i)\\nonumber \\\\& & \\hspace{-43.36243pt}+48 {q_3}^4 \\left(-3 i {\\omega _3}^4+6 {\\omega _3}^3-208 i {\\omega _3}^2+960 {\\omega _3}+512 i\\right)+4 {q_3}^2 {\\omega _3} \\left(27 i{\\omega _3}^5+222 {\\omega _3}^4+2272 i {\\omega _3}^3-7200 {\\omega _3}^2+4736 i {\\omega _3}-36864\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}+{\\omega _3}^3 \\left(-27 i {\\omega _3}^5-504 {\\omega _3}^4-932 i{\\omega _3}^3-5424 {\\omega _3}^2-4544 i {\\omega _3}+4608\\right)\\Biggr \\rbrace + {\\cal O}\\left(\\frac{1}{N^2}\\right).$ The Dirichlet condition $Z_v(u=0)=0$ reduces to $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where $& & a = -\\frac{957 {g_s}^2 M^2 {N_f} \\log ^2(N)}{N}-63264 \\pi ^2,\\nonumber \\\\& & b = -48 i \\left(\\frac{27 {g_s}^2 M^2 {N_f} \\log ^2(N)}{N}+2240 \\pi ^2\\right),\\nonumber \\\\& & c = 8 \\left(\\frac{15 {g_s}^2 M^2 {N_f} {q_3}^2 \\log ^2(N)}{N}+32 \\pi ^2 \\left(127 {q_3}^2+288\\right)\\right)\\nonumber \\\\& & f = 576 i {q_3}^2 \\left(64 \\pi ^2-\\frac{3 {g_s}^2 M^2 {N_f} \\log ^2(N)}{N}\\right),\\nonumber \\\\& & g = 384 {q_3}^4 \\left(32 \\pi ^2-\\frac{3 {g_s}^2 M^2 {N_f} \\log ^2(N)}{N}\\right).$ One of the four roots of the quartic in $\\omega _3$ is: $\\omega _3 = \\left(- 0.73 i + \\frac{0.003 i g_s^2 M^2 N_f (\\log N)^2}{N}\\right)q_3^2 + {\\cal O}(q_3^3).$ The leading order coefficient of $q_3^2$ is not terribly far off the correct value $-\\frac{i}{4}$ already at the third order in the infinite series (REF ).", "(c) Let us look at (REF ) up to the fourth order.", "One finds: $& & a_4 = \\frac{1}{98304 {\\omega _3}^4 \\left({\\omega _3}^4+20 i {\\omega _3}^3-140 {\\omega _3}^2-400 i{\\omega _3}+384\\right)}\\nonumber \\\\& & \\times \\Biggl \\lbrace 256 {q_3}^8 {\\omega _3}^3 ({\\omega _3}+16 i)-768 {q_3}^6 \\left({\\omega _3}^6+4 i {\\omega _3}^5+136 {\\omega _3}^4+832 i {\\omega _3}^3+256 {\\omega _3}^2+7168 i{\\omega _3}-12288\\right)\\nonumber \\\\& & +32 {q_3}^4 {\\omega _3}^2 \\left(27 {\\omega _3}^6-222 i {\\omega _3}^5+4880 {\\omega _3}^4+18176 i {\\omega _3}^3+110464 {\\omega _3}^2+652288 i{\\omega _3}-675840\\right)\\nonumber \\\\& & -16 {q_3}^2 {\\omega _3}^3 \\left(27 {\\omega _3}^7-558 i {\\omega _3}^6+3320 {\\omega _3}^5-9232 i {\\omega _3}^4+198656 {\\omega _3}^3+888320 i{\\omega _3}^2-774144 {\\omega _3}+589824 i\\right)\\nonumber \\\\& & +3 {\\omega _3}^5 \\left(27 {\\omega _3}^7-900 i {\\omega _3}^6-1316 {\\omega _3}^5-53104 i {\\omega _3}^4+108800 {\\omega _3}^3+147200i {\\omega _3}^2-487424 {\\omega _3}-344064 i\\right)\\Biggr \\rbrace \\nonumber \\\\& & -\\frac{1}{524288 \\pi ^2 N {\\omega _3}^4\\left({\\omega _3}^2+6 i {\\omega _3}-8\\right)^2 \\left({\\omega _3}^2+14 i {\\omega _3}-48\\right)^2}\\nonumber \\\\& & \\times \\Biggl \\lbrace -{g_s}^2 M^2 {N_f} \\log N \\biggl (-\\log N \\Biggl [256 {q_3}^8 {\\omega _3}^3\\left({\\omega _3}^5+37 i {\\omega _3}^4-530 {\\omega _3}^3-3500 i {\\omega _3}^2+10368 {\\omega _3}+10752 i\\right)\\nonumber \\\\& & -768 {q_3}^6 \\Biggl ({\\omega _3}^{10}+28 i {\\omega _3}^9-222{\\omega _3}^8+848 i {\\omega _3}^7-24192 {\\omega _3}^6-153184 i {\\omega _3}^5+399360 {\\omega _3}^4\\nonumber \\\\& & +133120 i {\\omega _3}^3+1531904 {\\omega _3}^2+3293184 i{\\omega _3}-2359296\\Biggr )\\nonumber \\\\& & +16 {q_3}^4 {\\omega _3}^2 \\Biggl (54 {\\omega _3}^{10}+1017 i {\\omega _3}^9+2420 {\\omega _3}^8+195388 i {\\omega _3}^7-1954848 {\\omega _3}^6-8216832i {\\omega _3}^5+5373440 {\\omega _3}^4\\nonumber \\\\& & -87731200 i {\\omega _3}^3+345751552 {\\omega _3}^2+510885888 i {\\omega _3}-259522560\\Biggr )\\nonumber \\\\& & -8 {q_3}^2 {\\omega _3}^3 \\Biggl (54{\\omega _3}^{11}+513 i {\\omega _3}^{10}+14300 {\\omega _3}^9+252484 i {\\omega _3}^8-1373088 {\\omega _3}^7-588832 i {\\omega _3}^6\\nonumber \\\\& & -30598656 {\\omega _3}^5-183382016 i{\\omega _3}^4+519692288 {\\omega _3}^3+707788800 i {\\omega _3}^2-297271296 {\\omega _3}+113246208 i\\Biggr )\\nonumber \\\\& & +3 {\\omega _3}^5 \\Biggl (27 {\\omega _3}^{11}+11672 {\\omega _3}^9+105584 i{\\omega _3}^8+196016 {\\omega _3}^7+6136320 i {\\omega _3}^6-29371904 {\\omega _3}^5-60586752 i {\\omega _3}^4\\nonumber \\\\& & +67778560 {\\omega _3}^3+79093760 i {\\omega _3}^2-93585408{\\omega _3}-33030144 i\\Biggr )\\Biggr ]+20 i \\left({\\omega _3}^4+20 i {\\omega _3}^3-140 {\\omega _3}^2-400 i {\\omega _3}+384\\right)\\nonumber \\\\& & \\times {\\omega _3}^2 \\biggl [64 {q_3}^6 {\\omega _3}\\left({\\omega _3}^2+18 i {\\omega _3}-96\\right)-16 {q_3}^4 \\left(9 {\\omega _3}^5+84 i {\\omega _3}^4+192 {\\omega _3}^3+6496 i {\\omega _3}^2-23296 {\\omega _3}-9216 i\\right)\\nonumber \\\\& & +4{q_3}^2 {\\omega _3} \\left(27 {\\omega _3}^6+12 i {\\omega _3}^5+2756 {\\omega _3}^4+22208 i {\\omega _3}^3-71680 {\\omega _3}^2+27136 i {\\omega _3}-270336\\right)\\nonumber \\\\& & +{\\omega _3}^4\\left(-27 {\\omega _3}^5+234 i {\\omega _3}^4-3704 {\\omega _3}^3-4224 i {\\omega _3}^2+1408 {\\omega _3}+52224 i\\right)\\biggr ]\\biggr )\\Biggr \\rbrace + {\\cal O}\\left(\\frac{1}{N^2}\\right).$ In the hydrodynamical limit the Dirichlet boundary condition $Z_v(u=0)=0$ reduces to $a \\omega _3^4 + b \\omega _3^3 + c \\omega _3^2 + f \\omega _3 + g = 0$ where $& & a = 9849372385059274752 i \\pi ^2 + {\\cal O}\\left(q_3^2\\right),\\nonumber \\\\& & b = \\frac{19237055439568896 {q_3}^2 \\left(3 {g_s}^2 {\\log N} (2 {\\log N}+5) M^2 {N_f}-128 \\pi ^2 N\\right)}{N},\\nonumber \\\\& & c = {\\cal O}\\left(q_3^4\\right),\\nonumber \\\\& & f = {\\cal O}\\left(q_3^6\\right),\\nonumber \\\\& & f = {\\cal O}\\left(q_3^6\\right).$" ], [ "$Z_t(u)$ from Tensor Mode of Metric Fluctuations", "The EOM for the tensor metric perturbation mode $Z_t(u)$ , inclusive of the non-conformal corrections in the metric (REF ) was written out in equation (REF ).", "Realizing that $u=1$ is a regular singular point of (REF ), using the Frobenius method we made a double perturbative ansatz (REF ) for the analytic part of the solution.", "Substituting (REF ) into (REF ), setting the coefficient of $\\omega _3$ to zero one gets: $& & {z_{00}}(u) \\left(-6 {g_s}^2 M^2 {N_f} \\log (N) \\log \\left(2 \\pi ^{3/2} \\sqrt{{g_s}} T\\right)-3 {g_s}^2 M^2 {N_f} \\log ^2(N)+64 \\pi ^2N\\right)\\nonumber \\\\& & \\times \\left(64 \\pi ^2 N \\left(u^2+2 u+3\\right)+15 {g_s}^2 M^2 {N_f} \\left(u^3+u^2+u+1\\right) \\log \\left({N}\\right)\\right)\\nonumber \\\\& & -128 i \\pi ^2 N\\Biggl [2 \\left(z_{01}^{\\prime }(u) \\left(-15 {g_s}^2 M^2 {N_f} \\left(u^4-1\\right) \\log \\left({N}\\right)+64 \\pi ^2 N \\left(u^4+3\\right)\\right)+64 \\pi ^2N u \\left(u^4-1\\right) z_{01}^{\\prime \\prime }(u)\\right)\\nonumber \\\\& & -i u \\left(u^3+u^2+u+1\\right) {z_{00}}^{\\prime }(u) \\left(-6 {g_s}^2 M^2 {N_f} \\log (N) \\log \\left(2 \\pi ^{3/2}\\sqrt{{g_s}} T\\right)-3 {g_s}^2 M^2 {N_f} \\log ^2(N)+64 \\pi ^2 N\\right)\\Biggr ].\\nonumber \\\\& &$ By setting the coefficient of $q_3$ to zero: ${z_{10}}^{\\prime }(u) \\left(-15 {g_s}^2 M^2 {N_f} \\left(u^4-1\\right) \\log \\left({N}\\right)+64 \\pi ^2 N \\left(u^4+3\\right)\\right)+64 \\pi ^2 N u\\left(u^4-1\\right) {z_{10}}^{\\prime \\prime }(u) = 0,\\nonumber \\\\& &$ which solves to yield: $& & z_{10}(u) = c_2-\\frac{1}{\\left(64 \\pi ^2 N+15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)\\right) \\left(128 \\pi ^2 N+15{g_s}^2 M^2 {N_f} \\log \\left({N}\\right)\\right)}\\nonumber \\\\& & \\times \\Biggl \\lbrace 16 \\pi ^2 c_1 N u^{1+\\frac{15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{64 \\pi ^2 N}} \\Biggl (2 u \\left(64 \\pi ^2 N+15 {g_s}^2 M^2{N_f} \\log \\left({N}\\right)\\right)\\nonumber \\\\& & \\times \\, _2F_1\\left(1,1+\\frac{15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{128 N \\pi ^2};2+\\frac{15{g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{128 N \\pi ^2};-u^2\\right)+\\left(128 \\pi ^2 N+15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)\\right)\\nonumber \\\\& & \\times \\, _2F_1\\left(1,1+\\frac{15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{64 N \\pi ^2};2+\\frac{15 {g_s}^2 M^2{N_f} \\log \\left({N}\\right)}{64 N \\pi ^2};-u\\right)\\nonumber \\\\& & -\\left(15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)-128 \\pi ^2 N\\right) \\,_2F_1\\left(1,1+\\frac{15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{64 N \\pi ^2};2+\\frac{15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{64 N \\pi ^2};u\\right)\\Biggr )\\Biggr \\rbrace \\nonumber \\\\& & = u^{\\frac{15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{64 \\pi ^2 N}} \\Bigg (\\frac{64 N \\pi ^2 c_1 u^4}{256 N \\pi ^2+15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}\\nonumber \\\\& & +\\frac{64 N \\pi ^2 c_1 u^8}{512 N \\pi ^2+15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}+\\frac{64 N \\pi ^2 c_1 u^{12}}{768 N\\pi ^2+15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}+O\\left(u^{13}\\right)\\Biggr )+c_2\\nonumber \\\\$ Setting $c_1=0$ for convenience, one obtains: $& & z_{01}(u)= c_4+\\frac{1}{3072}\\Biggl \\lbrace u \\Biggl [\\frac{6 i c_2 {g_s}^2 M^2 {N_f} \\left(3 u^3+4 u^2+6 u+12\\right) \\log (N) \\log \\left(2 \\pi ^{3/2} \\sqrt{{g_s}} T\\right)}{\\pi ^2N}\\nonumber \\\\& & +\\frac{3 i c_2 {g_s}^2 M^2 {N_f} \\left(3 u^3+4 u^2+6 u+12\\right) \\log ^2(N)}{\\pi ^2 N}+64 \\Biggl (\\frac{48 c_3 u^{3+\\frac{15 {g_s}^2 M^2 {N_f}\\log \\left({N}\\right)}{64 \\pi ^2 N}}}{4+\\frac{15 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right)}{64 \\pi ^2 N}}\\nonumber \\\\& & -i c_2 \\left(3 u^3+4 u^2+6u+12\\right)\\Biggr )\\Biggr ]\\Biggr \\rbrace \\nonumber \\\\& & = - \\left(\\frac{c_5 u^4}{4}+c_3\\right)\\frac{15 \\left(c_5 {g_s}^2 M^2 {N_f} u^4 \\log \\left({N}\\right) (4 \\log (u)-1)\\right)}{1024 \\pi ^2N}+ {\\cal O}\\left(\\frac{1}{N^2}\\right)\\nonumber \\\\& & = u^3 \\left(\\frac{i c_2 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right) \\left(\\log \\left({N}\\right)+2 \\log \\left(2 \\pi ^{3/2} \\sqrt{{g_s}}T\\right)\\right)}{256 \\pi ^2 N}-\\frac{i c_2}{12}\\right)\\nonumber \\\\& & +u^2 \\left(\\frac{3 i c_2 {g_s}^2 M^2 {N_f} \\log \\left({N}\\right) \\left(\\log \\left({N}\\right)+2 \\log \\left(2 \\pi ^{3/2} \\sqrt{{g_s}} T\\right)\\right)}{512 \\pi ^2 N}-\\frac{i c_2}{8}\\right)\\nonumber \\\\& & +u \\left(\\frac{3 i c_2 {g_s}^2 M^2{N_f} \\log \\left({N}\\right) \\left(\\log \\left({N}\\right)+2 \\log \\left(2 \\pi ^{3/2} \\sqrt{{g_s}} T\\right)\\right)}{256 \\pi ^2 N}-\\frac{ic_2}{4}\\right)+c_4 + {\\cal O}\\left(\\frac{u^4}{N},\\frac{1}{N^2}\\right).\\nonumber \\\\& &$ Similarly, $& & z_{10}(u) = \\frac{15 c_5 {g_s}^2 M^2 {N_f} u^4 \\log \\left({N}\\right) (4 \\log (u)-1)}{1024 \\pi ^2 N}+\\frac{c_5 u^4}{4}+c_3.$ The constant (in $\\omega _3, q_3$ ) yields: $& & \\frac{{z_{00}}^{\\prime }(u) \\left(-15 {g_s}^2 M^2 {N_f} \\left(u^4-1\\right) \\log \\left({N}\\right)+64 \\pi ^2 N \\left(u^4+3\\right)\\right)}{64 \\pi ^2 N u\\left(u^4-1\\right)}+{z_{00}}^{\\prime \\prime }(u) = 0,$ which is identical in form to the EOM of $z_{10}(u)$ .", "Setting $q_3=0$ in (REF ), one obtains: $& &\\hspace{-43.36243pt}\\frac{Z_t^\\prime (u)}{Z_t(u)} = \\left(\\frac{i c_4 {\\omega _3}^2}{4 \\left(c_4 {\\omega _3}+c_2\\right)}+\\frac{3 i c_4 {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\log \\left({N}\\right) \\left(\\log \\left({N}\\right)+2 \\log \\left(2 \\pi ^{3/2} \\sqrt{{g_s}} T\\right)\\right)}{256 \\pi ^2 N \\left(c_4{\\omega _3}+c_2\\right)}+{\\cal O}\\left(\\frac{1}{N^2}\\right)\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}+u \\Biggl (\\frac{{\\omega _3}^2 \\left(c_2^2+4 i c_4 c_2+4 i {\\omega _3} c_4^2\\right)}{16\\left(c_2+{\\omega _3} c_4\\right){}^2}\\nonumber \\\\& & \\hspace{-43.36243pt}-\\frac{3 i {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\left(-i c_2^2+2 c_4 c_2+2 {\\omega _3} c_4^2\\right) \\log \\left({N}\\right) \\left(\\log \\left({N}\\right)+2 \\log \\left(2 \\sqrt{{g_s}} \\pi ^{3/2} T\\right)\\right)}{512 \\pi ^2 \\left(c_2+{\\omega _3}c_4\\right){}^2 N}\\nonumber \\\\& &\\hspace{-43.36243pt}+ {\\cal O}\\left(\\frac{1}{N^2}\\right)\\Biggr )+u^2 \\Biggl \\lbrace \\frac{{\\omega _3}^2 \\left((i {\\omega _3}+6) c_2^3+2 (3 {\\omega _3}+8 i) c_4 c_2^2+32 i{\\omega _3} c_4^2 c_2+16 i {\\omega _3}^2 c_4^3\\right)}{64 \\left(c_2+{\\omega _3} c_4\\right){}^3}\\nonumber \\\\& &\\hspace{-43.36243pt}-\\frac{3 i {g_s}^2 M^2 {N_f} {\\omega _3}^2 \\left(3 ({\\omega _3}-4 i)c_2^3+4 (4-3 i {\\omega _3}) c_4 c_2^2+32 {\\omega _3} c_4^2 c_2+16 {\\omega _3}^2 c_4^3\\right) \\log \\left({N}\\right) \\left(\\log \\left({N}\\right)+2 \\log \\left(2 \\sqrt{{g_s}} \\pi ^{3/2} T\\right)\\right)}{4096 \\pi ^2 \\left(c_2+{\\omega _3} c_4\\right){}^3 N}\\nonumber \\\\& &\\hspace{-43.36243pt}+{\\cal O}\\left(\\frac{1}{N^2}\\right)\\Biggr \\rbrace +u^3\\nonumber \\\\& & \\hspace{-43.36243pt}\\times \\Biggl (\\frac{1}{768 \\left(c_2+{\\omega _3} c_4\\right){}^4}\\Biggl \\lbrace {\\omega _3} \\Biggl [\\left(-3 {\\omega _3}^3+24 i {\\omega _3}^2+88 {\\omega _3}+192 i\\right) c_2^4+8 {\\omega _3} \\left(3 i {\\omega _3}^2+22 {\\omega _3}+96 i\\right) c_4c_2^3\\nonumber \\\\& & \\hspace{-43.36243pt}+8 {\\omega _3}^2 (11 {\\omega _3}+144 i) c_4^2 c_2^2+768 i {\\omega _3}^3 c_4^3 c_2+192 i {\\omega _3}^4 c_4^4\\Biggr ]\\Biggr \\rbrace \\nonumber \\\\& & \\hspace{-43.36243pt}-\\frac{1}{4096 \\pi ^2 \\left(c_2+{\\omega _3} c_4\\right){}^4N}\\Biggl \\lbrace i{g_s}^2 M^2 {N_f} {\\omega _3}\\Biggl [\\left(3 i {\\omega _3}^3+18 {\\omega _3}^2-44 i {\\omega _3}+48\\right) c_2^4+2 {\\omega _3} \\left(9 {\\omega _3}^2-44 i{\\omega _3}+96\\right) c_4 c_2^3\\nonumber \\\\& & \\hspace{-43.36243pt}+4 (72-11 i {\\omega _3}) {\\omega _3}^2 c_4^2 c_2^2+192 {\\omega _3}^3 c_4^3 c_2+48 {\\omega _3}^4 c_4^4\\Biggr ] \\log \\left({N}\\right)\\left(\\log \\left({N}\\right)+2 \\log \\left(2 \\sqrt{{g_s}} \\pi ^{3/2} T\\right)\\right)\\Biggr \\rbrace \\nonumber \\\\& & \\hspace{-43.36243pt}+{\\cal O}\\left(\\frac{1}{N^2}\\right)\\Biggr )+{\\cal O}\\left(u^4\\right).\\nonumber \\\\& &$" ], [ "$T$ in the {{formula:e01934b1-e924-4424-ab20-3056f4761b35}} Uplift involving a Deformed/Resolved Conifold", "In this appendix we show that it is only when the resolution is larger than the deformation that the temperature from the local $D=11$ uplift of the type IIB background of [2] goes like the horizon radius; when the deformation is larger than the resolution the temperature goes like the reciprocal of the horizon radius.", "Using the exact expression for the deformed conifold metric component $g_{rr}$ [37] for appropriately redefined radial coordinate, and the type IIA dilaton $\\phi ^{\\rm IIA}$ , one obtains in the MQGP limit the following expression for the $D=11$ metric component $G^{\\cal M}_{rr}$ : $& & G^{\\cal M}_{rr} = e^{-\\frac{2\\phi ^{\\rm IIA}}{3}} g_{rr}\\sqrt{h} \\nonumber \\\\& & = \\frac{3 r^8 \\@root 3 \\of {1-\\frac{9 \\varepsilon ^4}{r^6}} \\left(-3 {N_f} \\log \\left(9 a^2 r^4+r^6\\right)+\\frac{24 \\pi }{{g_s}}+6{N_f} \\log (N)\\right)^{2/3} }{4 2^{5/6} \\pi ^{7/6} \\left(r^4-{r_h}^4\\right) \\left(\\varepsilon ^4 \\left(\\sinh \\left(2 \\cosh ^{-1}\\left(\\frac{r^3}{\\varepsilon ^2}\\right)\\right)-2 \\cosh ^{-1}\\left(\\frac{r^3}{\\varepsilon ^2}\\right)\\right)\\right)^{2/3}}\\nonumber \\\\& & \\times \\sqrt{\\frac{{g_s} \\left(18 {g_s}^2 M^2 {N_f} \\log ^2(r)+3 {g_s} M^2 \\log (r) \\left({g_s} {N_f} \\log \\left(\\frac{1}{4 \\sqrt{N}}\\right)+3 {g_s} {N_f}+4 \\pi \\right)+8 \\pi ^2N\\right)}{r^4}},\\nonumber \\\\& &$ where $\\varepsilon $ is the deformation parameter ($h_5\\equiv \\frac{\\varepsilon ^2}{r^3}$ in (REF )).", "Using (REF ) one obtains the following expression for the temperature: $&& \\hspace{-43.36243pt}T = \\frac{\\partial _rG^{\\cal M}_{00}}{4\\pi \\sqrt{G^{\\cal M}_{00}G^{\\cal M}_{rr}}}\\nonumber \\\\& & \\hspace{-43.36243pt}= \\frac{2^{2/3} \\@root 3 \\of {\\varepsilon ^4 \\left(\\sinh \\left(2 \\cosh ^{-1}\\left(\\frac{{r_h}^3}{\\varepsilon ^2}\\right)\\right)-2 \\cosh ^{-1}\\left(\\frac{{r_h}^3}{\\varepsilon ^2}\\right)\\right)}}{\\sqrt{3 \\pi } {r_h} \\sqrt{{g_s} \\left(18 {g_s}^2 M^2{N_f} \\log ^2({r_h})+3 {g_s} M^2 \\log ({r_h}) \\left({g_s} {N_f} \\log \\left(\\frac{1}{4\\sqrt{N}}\\right)+3 {g_s} {N_f}+4 \\pi \\right)+8 \\pi ^2 N\\right)}}\\nonumber \\\\& & \\hspace{-43.36243pt}= \\frac{\\@root 3 \\of {\\varepsilon ^4 \\left(\\sinh \\left(2 \\cosh ^{-1}\\left(\\frac{{r_h}^3}{\\varepsilon ^2}\\right)\\right)-2 \\cosh ^{-1}\\left(\\frac{{r_h}^3}{\\varepsilon ^2}\\right)\\right)}}{2^{5/6} \\sqrt{3} \\pi ^{3/2} \\sqrt{{g_s}} \\sqrt{N}{r_h}}\\nonumber \\\\& & \\hspace{-43.36243pt}-\\frac{\\sqrt{3} \\sqrt{{g_s}} M^2 \\log ({r_h}) \\@root 3 \\of {\\varepsilon ^4 \\left(\\sinh \\left(2 \\cosh ^{-1}\\left(\\frac{{r_h}^3}{\\varepsilon ^2}\\right)\\right)-2 \\cosh ^{-1}\\left(\\frac{{r_h}^3}{\\varepsilon ^2}\\right)\\right)} }{32 N^{3/2} \\left(2^{5/6} \\pi ^{7/2}{r_h}\\right)}\\nonumber \\\\& & \\hspace{-43.36243pt}\\times \\left({g_s}{N_f} \\log \\left(\\frac{1}{N}\\right)+12 {g_s} {N_f} \\log ({r_h})+6 {g_s} {N_f}-2 {g_s}{N_f} \\log (4)+8 \\pi \\right) + {\\cal O}\\left(\\left(\\frac{1}{N}\\right)^{5/2}\\right)\\nonumber \\\\& & \\hspace{-43.36243pt}= \\frac{1}{6 \\pi ^{3/2} \\sqrt{{g_s}} \\sqrt{N} {r_h}}\\Biggl \\lbrace \\@root 6 \\of {2} \\sqrt{3} \\@root 3 \\of {-\\varepsilon ^4 \\left(\\sinh \\left(\\frac{\\varepsilon ^4}{2 {r_h}^6}+4 \\log (d)-2 \\log \\left(2{r_h}^3\\right)\\right)-4 \\log (d)+2 \\log \\left(2 {r_h}^3\\right)\\right)}\\nonumber \\\\& & \\hspace{-43.36243pt}-\\frac{3 \\sqrt{3} \\varepsilon ^{4/3} {g_s} M^2\\log ({r_h}) \\@root 3 \\of {\\frac{\\varepsilon ^4}{{r_h}^6}-2 \\sinh \\left(\\frac{\\varepsilon ^4}{2 {r_h}^6}+4 \\log (d)-2 \\log \\left(2{r_h}^3\\right)\\right)+8 \\log (d)-4 \\log \\left(2 {r_h}^3\\right)} }{32 \\@root 6 \\of {2} \\pi ^2 N}\\Biggr \\rbrace \\nonumber \\\\& & \\hspace{-43.36243pt}\\times \\left(-{g_s} {N_f} \\log (N)+12 {g_s}{N_f} \\log ({r_h})+6 {g_s} {N_f}-2 {g_s} {N_f} \\log (4)+8 \\pi \\right).$ When the resolution is larger than the deformation, using the exact expression for the $g_{rr}$ component of the resolved conifold [38] with appropriate redefinition of the radial coordinate, the $D=11$ component $G^{\\cal M}_{rr}$ is given as under: $& & \\hspace{-43.36243pt}G^{\\cal M}_{rr} = \\frac{r^4 \\left(6 a^2+r^2\\right) \\@root 3 \\of {1-\\frac{9 \\varepsilon ^4}{r^6}} \\left(-3 {N_f} \\log \\left(9 a^2 r^4+r^6\\right)+\\frac{24\\pi }{{g_s}}+6 {N_f} \\log (N)\\right)^{2/3} }{4 \\sqrt{2} \\pi ^{7/6} \\left(9 a^2+r^2\\right) \\left(r^4-{r_h}^4\\right)}\\nonumber \\\\& & \\hspace{-43.36243pt}\\times \\sqrt{\\frac{{g_s} \\left(18 {g_s}^2 M^2 {N_f} \\log ^2(r)+3{g_s} M^2 \\log (r) \\left({g_s} {N_f} \\log \\left(\\frac{1}{4 \\sqrt{N}}\\right)+3 {g_s} {N_f}+4 \\pi \\right)+8 \\pi ^2 N\\right)}{r^4}}.$ Using (REF ), one obtains the following expression for the temperature: $& & T = \\frac{\\partial _rG^{\\cal M}_{00}}{4\\pi \\sqrt{G^{\\cal M}_{00}G^{\\cal M}_{rr}}}\\nonumber \\\\& & = {r_h} \\left[\\frac{1}{2 \\pi ^{3/2} \\sqrt{{g_s} N}}-\\frac{3 {g_s}^{\\frac{3}{2}} M^2 {N_f} \\log ({r_h}) \\left(-\\log {N}+12 \\log ({r_h})+\\frac{8 \\pi }{g_sN_f} +6-\\log (16)\\right)}{64 \\pi ^{7/2} N^{3/2}} \\right]\\nonumber \\\\& & + a^2 \\left(\\frac{3}{4 \\pi ^{3/2} \\sqrt{{g_s}} \\sqrt{N} {r_h}}-\\frac{9 {g_s}^{3/2} M^2 {N_f} \\log ({r_h})\\left(\\frac{8 \\pi }{{g_s} {N_f}}-\\log (N)+12 \\log ({r_h})+6-2 \\log (4)\\right)}{128 \\pi ^{7/2} N^{3/2}{r_h}}\\right).\\nonumber \\\\& &$" ] ]
1606.04949
[ [ "Pruning Random Forests for Prediction on a Budget" ], [ "Abstract We propose to prune a random forest (RF) for resource-constrained prediction.", "We first construct a RF and then prune it to optimize expected feature cost & accuracy.", "We pose pruning RFs as a novel 0-1 integer program with linear constraints that encourages feature re-use.", "We establish total unimodularity of the constraint set to prove that the corresponding LP relaxation solves the original integer program.", "We then exploit connections to combinatorial optimization and develop an efficient primal-dual algorithm, scalable to large datasets.", "In contrast to our bottom-up approach, which benefits from good RF initialization, conventional methods are top-down acquiring features based on their utility value and is generally intractable, requiring heuristics.", "Empirically, our pruning algorithm outperforms existing state-of-the-art resource-constrained algorithms." ], [ "Introduction", "Many modern classification systems, including internet applications (such as web-search engines, recommendation systems, and spam filtering) and security & surveillance applications (such as wide-area surveillance and classification on large video corpora), face the challenge of prediction-time budget constraints [26].", "Prediction-time budgets can arise due to monetary costs associated with acquiring information or computation time (or delay) involved in extracting features and running the algorithm.", "We seek to learn a classifier by training on fully annotated training datasets that maintains high-accuracy while meeting average resource constraints during prediction-time.", "We consider a system that adaptively acquires features as needed depending on the instance(example) for high classification accuracy with reduced feature acquisition cost.", "We propose a two-stage algorithm.", "In the first stage, we train a random forest (RF) of trees using an impurity function such as entropy or more specialized cost-adaptive impurity [17].", "Our second stage takes a RF as input and attempts to jointly prune each tree in the forest to meet global resource constraints.", "During prediction-time, an example is routed through all the trees in the ensemble to the corresponding leaf nodes and the final prediction is based on a majority vote.", "The total feature cost for a test example is the sum of acquisition costs of unique featuresWhen an example arrives at an internal node, the feature associated with the node is used to direct the example.", "If the feature has never been acquired for the example an acquisition cost is incurred.", "Otherwise, no acquisition cost is incurred as we assume that feature values are stored once computed.", "acquired for the example in the entire ensemble of trees in the forest.", "For time-sensitive cases such as web-search we parallelize the implementation by creating parallel jobs across all features and trees.", "We can then terminate jobs based on what features are returned.", "We derive an efficient scheme to learn a globally optimal pruning of a RF minimizing the empirical error and incurred average costs.", "We formulate the pruning problem as a 0-1 integer linear program that incorporates feature-reuse constraints.", "By establishing total unimodularity of the constraint set, we show that solving the linear program relaxation of the integer program yields the optimal solution to the integer program resulting in a polynomial time algorithm for optimal pruning.", "We develop a primal-dual algorithm by leveraging results from network-flow theory for scaling the linear program to large datasets.", "Empirically, this pruning outperforms state-of-the-art resource efficient algorithms on benchmarked datasets.", "Table: Typical feature usage in a 40 tree RF before and after pruning (our algorithm) on the MiniBooNE dataset.", "Columns 2-4 list percentage of test examples that do not use the feature, use it 1 to 7 times, and use it greater than 7 times, respectively.", "Before pruning, 91% examples use the feature only a few (1 to 7) times, paying a significant cost for its acquisition; after pruning, 68% of the total examples no longer use this feature, reducing cost with minimal error increase.", "Column 5 is the average feature cost (the average number of unique features used by test examples).", "Column 6 is the test error of RFs.", "Overall, pruning dramatically reduces average feature cost while maintaining the same error level.Our approach is motivated by the following considerations: (i) RFs are scalable to large datasets and produce flexible decision boundaries yielding high prediction-time accuracy.", "The sequential feature usage of decision trees lends itself to adaptive feature acquisition.", "(ii) RF feature usage is superfluous, utilizing features with introduced randomness to increase diversity and generalization.", "Pruning can yield significant cost reduction with negligible performance loss by selectively pruning features sparsely used across trees, leading to cost reduction with minimal accuracy degradation (due to majority vote).", "See Table REF .", "(iii) Optimal pruning encourages examples to use features either a large number of times, allowing for complex decision boundaries in the space of those features, or not to use them at all, avoiding incurring the cost of acquisition.", "It enforces the fact that once a feature is acquired for an example, repeated use incurs no additional acquisition cost.", "Intuitively, features should be repeatedly used to increase discriminative ability without incurring further cost.", "(iv) Resource constrained prediction has been conventionally viewed as a top-down (tree-growing) approach, wherein new features are acquired based on their utility value.", "This is often an intractable problem with combinatorial (feature subsets) and continuous components (classifiers) requiring several relaxations and heuristics.", "In contrast, ours is a bottom-up approach that starts with good initialization (RF) and prunes to realize optimal cost-accuracy tradeoff.", "Indeed, while we do not pursue it, our approach can also be used in conjunction with existing approaches.", "Related Work: Learning decision rules to minimize error subject to a budget constraint during prediction-time is an area of recent interest, with many approaches proposed to solve the prediction-time budget constrained problem [16], [23], [22], [24], [9], [27], [21], [25], [12].", "These approaches focus on learning complex adaptive decision functions and can be viewed as orthogonal to our work.", "Conceptually, these are top-down “growing” methods as we described earlier (see (iv)).", "Our approach is bottom-up that seeks to prune complex classifiers to tradeoff cost vs. accuracy.", "Our work is based on RF classifiers [3].", "Traditionally, feature cost is not incorporated when constructing RFs, however recent work has involved approximation of budget constraints to learn budgeted RFs [17].", "The tree-growing algorithm in [17] does not take feature re-use into account.", "Rather than attempting to approximate the budget constraint during tree construction, our work focuses on pruning ensembles of trees subject to a budget constraint.", "Methods such as traditional ensemble learning and budgeted random forests can be viewed as complementary.", "Decision tree pruning has been studied extensively to improve generalization performance, we are not aware of any existing pruning method that takes into account the feature costs.", "A popular method for pruning to reduce generalization error is Cost-Complexity Pruning (CCP), introduced by Breiman et al.", "[4].", "CCP trades-off classification ability for tree size, however it does not account for feature costs.", "As pointed out by Li et al.", "[15], CCP has undesirable “jumps\" in the sequence of pruned tree sizes.", "To alleviate this, they proposed a Dynamic-Program-based Pruning (DPP) method for binary trees.", "The DPP algorithm is able to obtain optimally pruned trees of all sizes; however, it faces the curse of dimensionality when pruning an ensemble of decision trees and taking feature cost into account.", "[28], [20] proposed to solve the pruning problem as a 0-1 integer program; again, their formulations do not account for feature costs that we focus on in this paper.", "The coupling nature of feature usage makes our problem much harder.", "In general pruning RFs is not a focus of attention as it is assumed that overfitting can be avoided by constructing an ensemble of trees.", "While this is true, it often leads to extremely large prediction-time costs.", "Kulkarni and Sinha [11] provide a survey of methods to prune RFs in order to reduce ensemble size.", "However, these methods do not explicitly account for feature costs." ], [ "Learning with Resource Constraints", "In this paper, we consider solving the Lagrangian relaxed problem of learning under prediction-time resource constraints, also known as the error-cost tradeoff problem: $\\min _{f \\in \\mathcal {F}}E_{(x,y)\\sim \\mathcal {P}}\\left[err\\left(y,f(x)\\right)\\right]+ \\lambda E_{x\\sim \\mathcal {P}_x}\\left[C\\left(f,x\\right)\\right],$ where example/label pairs $(x,y)$ are drawn from a distribution $\\mathcal {P}$ ; $err(y,\\hat{y})$ is the error function; $C(f,x)$ is the cost of evaluating the classifier $f$ on example $x$ ; $\\lambda $ is a tradeoff parameter.", "A larger $\\lambda $ places a larger penalty on cost, pushing the classifier to have smaller cost.", "By adjusting $\\lambda $ we can obtain a classifier satisfying the budget constraint.", "The family of classifiers $\\mathcal {F}$ in our setting is the space of RFs, and each RF $f$ is composed of $T$ decision trees $\\mathcal {T}_1,\\dots ,\\mathcal {T}_T$ .", "Our approach: Rather than attempting to construct the optimal ensemble by solving Eqn.", "(REF ) directly, we instead propose a two-step algorithm that first constructs an ensemble with low prediction error, then prunes it by solving Eqn.", "(REF ) to produce a pruned ensemble given the input ensemble.", "By adopting this two-step strategy, we obtain an ensemble with low expected cost while simultaneously preserving the low prediction error.", "There are many existing methods to construct RFs, however the focus of this paper is on the second step, where we propose a novel approach to prune RFs to solve the tradeoff problem Eqn.", "(REF ).", "Our pruning algorithm is capable of taking any RF as input, offering the flexibility to incorporate any state-of-the-art RF algorithm." ], [ "Pruning with Costs", "In this section, we treat the error-cost tradeoff problem Eqn.", "(REF ) as an RF pruning problem.", "Our key contribution is to formulate pruning as a 0-1 integer program with totally unimodular constraints.", "We first define notations used throughout the paper.", "A training sample $S=\\lbrace (\\mathbf {x}^{(i)},y^{(i)}):{i=1,\\dots ,N}\\rbrace $ is generated i.i.d.", "from an unknown distribution, where $\\mathbf {x}^{(i)} \\in \\Re ^K$ is the feature vector with a cost assigned to each of the $K$ features and $y^{(i)}$ is the label for the $i^{\\mbox{th}}$ example.", "In the case of multi-class classification $y \\in \\lbrace 1,\\dots ,M\\rbrace $ , where $M$ is the number of classes.", "Given a decision tree $\\mathcal {T}$ , we index the nodes as $h\\in \\lbrace 1,\\dots ,|\\mathcal {T}|\\rbrace $ , where node 1 represents the root node.", "Let $\\tilde{\\mathcal {T}}$ denote the set of leaf nodes of tree $\\mathcal {T}$ .", "Finally, the corresponding definitions for $\\mathcal {T}$ can be extended to an ensemble of $T$ decision trees $\\lbrace \\mathcal {T}_t :t=1,\\dots ,T\\rbrace $ by adding an subscript $t$ .", "Pruning Parametrization: In order to model ensemble pruning as an optimization problem, we parametrize the space of all prunings of an ensemble.", "The process of pruning a decision tree $\\mathcal {T}$ at an internal node $h$ involves collapsing the subtree of $\\mathcal {T}$ rooted at $h$ , making $h$ a leaf node.", "We say a pruned tree $\\mathcal {T}^{(p)}$ is a valid pruned tree of $\\mathcal {T}$ if (1) $\\mathcal {T}^{(p)}$ is a subtree of $\\mathcal {T}$ containing root node 1 and (2) for any $h\\ne 1$ contained in $\\mathcal {T}^{(p)}$ , the sibling nodes (the set of nodes that share the same immediate parent node as $h$ in $\\mathcal {T}$ ) must also be contained in $\\mathcal {T}^{(p)}$ .", "Specifying a pruning is equivalent to specifying the nodes that are leaves in the pruned tree.", "We therefore introduce the following binary variable for each node $h\\in \\mathcal {T}$ $z_h=\\left\\lbrace \\begin{array}{rl}1 & \\text{if node } h \\text{ is a leaf in the pruned tree} ,\\\\0 & \\text{otherwise}.\\end{array} \\right.$ We call the set $\\lbrace z_h, \\forall h \\in \\mathcal {T}\\rbrace $ the node variables as they are associated with each node in the tree.", "Consider any root-to-leaf path in a tree $\\mathcal {T}$ , there should be exactly one node in the path that is a leaf node in the pruned tree.", "Let $p(h)$ denote the set of predecessor nodes, the set of nodes (including $h$ ) that lie on the path from the root node to $h$ .", "The set of valid pruned trees can be represented as the set of node variables satisfying the following set of constraints: $\\sum _{u\\in p(h)} z_u=1 \\quad \\forall h \\in \\tilde{\\mathcal {T}}$ .", "Given a valid pruning for a tree, we now seek to parameterize the error of the pruning.", "Pruning error: As in most supervised empirical risk minimization problems, we aim to minimize the error on training data as a surrogate to minimizing the expected error.", "In a decision tree $\\mathcal {T}$ , each node $h$ is associated with a predicted label corresponding to the majority label among the training examples that fall into the node $h$ .", "Let $S_h$ denote the subset of examples in $S$ routed to or through node $h$ on $\\mathcal {T}$ and let $\\text{Pred}_h$ denote the predicted label at $h$ .", "The number of misclassified examples at $h$ is therefore $e_h=\\sum _{i\\in S_h} {1}_{\\left[ {y^{(i)}\\ne \\text{Pred}_h} \\right] }$ .", "We can thus estimate the error of tree $\\mathcal {T}$ in terms of the number of misclassified examples in the leaf nodes: $\\frac{1}{N}\\sum _{h\\in \\tilde{\\mathcal {T}}}e_h$ , where $N=|S|$ is the total number of examples.", "Our goal is to minimize the expected test error of the trees in the random forest, which we empirically approximate based on the aggregated probability distribution in Step (REF ) of Algorithm REF with $\\frac{1}{TN}\\sum _{t=1}^{T}\\sum _{h\\in \\tilde{\\mathcal {T}_t}}e_h$ .", "We can express this error in terms of the node variables: $\\frac{1}{TN}\\sum _{t=1}^{T}\\sum _{h\\in \\mathcal {T}_t}e_h z_h$ .", "Pruning cost: Assume the acquisition cost for the $K$ features, $\\lbrace c_k:k=1,\\dots ,K\\rbrace $ , are given.", "The feature acquisition cost incurred by an example is the sum of the acquisition costs of unique features acquired in the process of running the example through the forest.", "This cost structure arises due to the assumption that an acquired feature is cached and subsequent usage by the same example incurs no additional cost.", "Formally, the feature cost of classifying an example $i$ on the ensemble $\\mathcal {T}_{[T]}$ is given by $C_{\\text{feature}}(\\mathcal {T}_{[T]},\\mathbf {x}^{(i)}) =\\sum _{k=1}^{K}c_k w_{k,i}$ , where the binary variables $w_{k,i}$ serve as the indicators: $w_{k,i}=\\left\\lbrace \\begin{array}{rl}1 & \\text{ if feature } k \\text{ is used by }\\mathbf {x}^{(i)} \\text{ in any } \\mathcal {T}_t, t=1,\\dots ,T\\\\0 & \\text{ otherwise}.\\end{array} \\right.$ The expected feature cost of a test example can be approximated as $\\frac{1}{N}\\sum _{i=1}^{N}\\sum _{k=1}^{K}c_k w_{k,i}$ .", "In some scenarios, it is useful to account for computation cost along with feature acquisition cost during prediction-time.", "In an ensemble, this corresponds to the expected number of Boolean operations required running a test through the trees, which is equal to the expected depth of the trees.", "This can be modeled as $\\frac{1}{N}\\sum _{t=1}^{T}\\sum _{h\\in \\mathcal {T}_t} |S_h| d_h z_h$ , where $d_h$ is the depth of node $h$ .", "Putting it together: Having modeled the pruning constraints, prediction performance and costs, we formulate the problem of pruning using the relationship between the node variables $z_h$ 's and feature usage variables $w_{k,i}$ 's.", "Given a tree $\\mathcal {T}$ , feature $k$ , and example $\\mathbf {x}^{(i)}$ , let $u_{k,i}$ be the first node associated with feature $k$ on the root-to-leaf path the example follows in $\\mathcal {T}$ .", "Feature $k$ is used by $\\mathbf {x}^{(i)}$ if and only if none of the nodes between the root and $u_{k,i}$ is a leaf.", "We represent this by the constraint $w_{k,i}+\\sum _{h\\in p(u_{k,i})} z_h = 1$ for every feature $k$ used by example $x^{(i)}$ in $\\mathcal {T}$ .", "Recall $w_{k,i}$ indicates whether or not feature $k$ is used by example $i$ and $p(u_{k,i})$ denotes the set of predecessor nodes of $u_{k,i}$ .", "Intuitively, this constraint says that either the tree is pruned along the path followed by example $i$ before feature $k$ is acquired, in which case $z_h=1$ for some node $h\\in p(u_{k,i})$ and $w_{k,i}=0$ ; or $w_{k,i}=1$ , indicating that feature $k$ is acquired for example $i$ .", "We extend the notations to ensemble pruning with tree index $t$ : $z^{(t)}_h$ indicates whether node $h$ in $\\mathcal {T}_t$ is a leaf after pruning; $w^{(t)}_{k,i}$ indicates whether feature $k$ is used by the $i^{\\mbox{th}}$ example in $\\mathcal {T}_t$ ; $w_{k,i}$ indicates whether feature $k$ is used by the $i^{\\mbox{th}}$ example in any of the $T$ trees $\\mathcal {T}_1,\\dots ,\\mathcal {T}_T$ ; $u_{t,k,i}$ is the first node associated with feature $k$ on the root-to-leaf path the example follows in $\\mathcal {T}_t$ ; $K_{t,i}$ denotes the set of features the $i^{\\mbox{th}}$ example uses on tree $\\mathcal {T}_t$ .", "We arrive at the following integer program.", "$\\hspace{-14.22636pt}\\begin{array}{rlll}\\displaystyle \\min _{\\begin{array}{c}z^{(t)}_h, w^{(t)}_{k,i}, w_{k,i} \\in \\lbrace 0,1\\rbrace \\end{array}} & \\multicolumn{2}{l}{\\overbrace{\\frac{1}{NT}\\displaystyle \\sum _{t=1}^{T}\\sum _{h\\in \\mathcal {T}_t} e^{(t)}_h z^{(t)}_h}^{{\\text{error}}} +\\lambda \\left( \\overbrace{\\sum _{k=1}^{K}c_k(\\frac{1}{N}\\sum _{i=1}^{N}w_{k,i})}^{{\\text{feature acquisition cost}}} + \\overbrace{\\displaystyle \\frac{1}{N}\\sum _{t=1}^{T}\\sum _{h\\in \\mathcal {T}_t} |S_h| d_h z_h}^{{\\text{computational cost}}} \\right) } \\quad \\textbf {(IP)}\\\\\\textrm {s.t.}", "& \\sum _{u\\in p(h)} z^{(t)}_u=1, & \\forall h \\in \\tilde{\\mathcal {T}}_t, \\forall t \\in [T], \\qquad \\quad \\text{(feasible prunings)} \\\\& w^{(t)}_{k,i}+ \\sum _{h\\in p(u_{t,k,i})} z^{(t)}_h=1 , & \\forall k\\in K_{t,i},\\forall i\\in S, \\forall t \\in [T], \\text{ (feature usage/ tree)}\\\\& w^{(t)}_{k,i} \\le w_{k,i}, & \\forall k\\in [K], \\forall i\\in S, \\forall t\\in [T].", "\\text{ (global feature usage)}\\end{array}$ Totally Unimodular constraints: Even though integer programs are NP-hard to solve in general, we show that (IP) can be solved exactly by solving its LP relaxation.", "We prove this in two steps: first, we examine the special structure of the equality constraints; then we examine the inequality constraint that couples the trees.", "Recall that a network matrix is one with each column having exactly one element equal to 1, one element equal to -1 and the remaining elements being 0.", "A network matrix defines a directed graph with the nodes in the rows and arcs in the columns.", "We have the following lemma.", "Lemma 3.1 The equality constraints in (IP) can be turned into an equivalent network matrix form for each tree.", "Proof We observe the first constraint $\\sum _{u\\in p(h)} z^{(t)}_u=1$ requires the sum of the node variables along a path to be 1.", "The second constraints $w^{(t)}_{k,i}+ \\sum _{h\\in p(u_{t,k,i})} z^{(t)}_h=1$ has a similar sum except the variable $w^{(t)}_{k,i}$ .", "Imagine $w^{(t)}_{k,i}$ as yet another node variable for a fictitious child node of $u_{t,k,i}$ and the two equations are essentially equivalent.", "The rest of proof follows directly from the construction in Proposition 3 of [20].", "Figure REF illustrates such a construction.", "The nodes are numbered 1 to 5.", "The subscripts at node 1 and 3 are the feature index used in the nodes.", "Since the equality constraints in (IP) can be separated based on the trees, we consider only one tree and one example being routed to node 4 on the tree for simplicity.", "The equality constraints can be organized in the matrix form as shown in the middle of Figure REF .", "Through row operations, the constraint matrix can be transformed to an equivalent network matrix.", "Such transformation always works as long as the leaf nodes are arranged in a pre-order manner.", "Next, we deal with the inequality constraints and obtain our main result.", "every tree node/.style=minimum width=1em,draw,circle, blank/.style=draw=none, edge from parent/.style= draw, edge from parent path=() – (), level distance=1cm every tree node/.style=minimum width=1em,draw,circle, blank/.style=draw=none, edge from parent/.style= draw, edge from parent path=() – (), level distance=1cm Figure: A decision tree example with node numbers and associated feature in subscripts together with the constraint matrix and its equivalent network matrix form.Theorem 3.2 The LP relaxation of (IP), where the 0-1 integer constraints are relaxed to interval constraints $[0,1]$ for all integer variables, has integral optimal solutions.", "Due to space limit the proof can be found in the Appendix.", "The main idea is to show the constraints are still totally unimodular even after adding the coupling constraints and the LP relaxed polyhedron has only integral extreme points [19].", "As a result, solving the LP relaxation results in the optimal solution to the integer program (IP), allowing for polynomial time optimization.", "The nice result of totally unimodular constraints is due to our specific formulation.", "See Appendix for an alternative formulation that does not have such a property.", "[hb] BudgetPrune [1]  During Training: input - ensemble($\\mathcal {T}_1,\\dots ,\\mathcal {T}_T$ ), training/validation data with labels, $\\lambda $   initialize dual variables $\\beta _{k,i}^{(t)} \\leftarrow 0$ .", "update $z^{(t)}_h,w^{(t)}_{k,i}$ for each tree $t$ (shortest-path algo).", "$w_{k,i}=0$ if $\\mu _{k,i}>0$ , $w_{k,i}=1$ if $\\mu _{k,i}<0$ .", "$\\beta _{k,i}^{(t)} \\leftarrow [\\beta _{k,i}^{(t)}+ \\gamma (w_{k,i}^{(t)}-w_{k,i})]_+$ for step size $\\gamma $ , where $[\\cdot ]_+=\\max \\lbrace 0,\\cdot \\rbrace $ .", "go to Step 2 until duality gap is small enough.", "During Prediction: input - test exmaple $\\mathbf {x}$   Run $\\mathbf {x}$ on each tree to leaf, obtain the probability distribution over label classes $\\mathbf {p}_t$ at leaf.", "Aggregate $\\mathbf {p}=\\frac{1}{T}\\sum _{t=1}^{T}\\mathbf {p}_t$ .", "Predict the class with the highest probability in $\\mathbf {p}$ .", "A Primal-Dual Algorithm Even though we can solve (IP) via its LP relaxation, the resulting LP can be too large in practical applications for any general-purpose LP solver.", "In particular, the number of variables and constraints is roughly $O(T \\times |\\mathcal {T}_{\\text{max}}|+N \\times T \\times K_{\\text{max}})$ , where $T$ is the number of trees; $|\\mathcal {T}_{\\text{max}}|$ is the maximum number of nodes in a tree; $N$ is the number of examples; $K_{\\text{max}}$ is the maximum number of features an example uses in a tree.", "The runtime of the LP thus scales $O(T^{3})$ with the number of trees in the ensemble, limiting the application to only small ensembles.", "In this section we propose a primal-dual approach that effectively decomposes the optimization into many sub-problems.", "Each sub-problem corresponds to a tree in the ensemble and can be solved efficiently as a shortest path problem.", "The runtime per iteration is $O(\\frac{T}{p}(|\\mathcal {T}_{\\text{max}}|+N \\times K_{\\text{max}})\\log (|\\mathcal {T}_{\\text{max}}|+N \\times K_{\\text{max}}))$ , where $p$ is the number of processors.", "We can thus massively parallelize the optimization and scale to much larger ensembles as the runtime depends only linearly on $\\frac{T}{p}$ .", "To this end, we assign dual variables $\\beta _{k,i}^{(t)}$ for the inequality constraints $ w^{(t)}_{k,i} \\le w_{k,i}$ and derive the dual problem.", "$\\hspace{-8.5359pt}\\begin{array}{rlll}\\displaystyle \\max _{\\beta _{k,i}^{(t)}\\ge 0} \\min _{\\begin{array}{c}z^{(t)}_h\\in [0,1] \\\\ w^{(t)}_{k,i}\\in [0,1] \\\\w_{k,i}\\in [0,1]\\end{array}} & \\multicolumn{2}{l}{\\frac{1}{NT}\\displaystyle \\sum _{t=1}^{T}\\sum _{h\\in \\mathcal {T}_t} \\hat{e}^{(t)}_h z^{(t)}_h +\\lambda \\left( \\sum _{k=1}^{K}c_k(\\frac{1}{N}\\sum _{i=1}^{N}w_{k,i}) \\right) + \\sum _{t=1}^T \\sum _{i=1}^N \\sum _{k\\in K_{t,i}} \\beta _{k,i}^{(t)}(w_{k,i}^{(t)}-w_{k,i})}\\\\\\textrm {s.t.}", "& \\displaystyle \\sum _{u\\in p(h)} z^{(t)}_u=1, & \\forall h \\in \\tilde{\\mathcal {T}}_t, \\forall t \\in [T], \\\\& w^{(t)}_{k,i}+ \\displaystyle \\sum _{h\\in p(u_{t,k,i})} z^{(t)}_h=1 , & \\forall k\\in K_{t,i},\\forall i\\in S, \\forall t \\in [T], \\\\\\end{array}$ where for simplicity we have combined coefficients of $z_h^{(t)}$ in the objective of (IP) to $\\hat{e}_h^{(t)}$ .", "The primal-dual algorithm is summarized in Algorithm REF .", "It alternates between updating the primal and the dual variables.", "The key is to observe that given dual variables, the primal problem (inner minimization) can be decomposed for each tree in the ensemble and solved in parallel as shortest path problems due to Lemma REF .", "(See also Appendix).", "The primal variables $w_{k,i}$ can be solved in closed form: simply compute $\\mu _{k,i}=\\lambda c_k / N-\\sum _{t\\in T_{k,i}} \\beta _{k,i}^{(t)}$ , where $T_{k,i}$ is the set of trees in which example $i$ encounters feature $k$ .", "So $w_{k,i}$ should be set to 0 if $\\mu _{k,i}>0$ and $w_{k,i}=1$ if $\\mu _{k,i}<0$ .", "Note that our prediction rule aggregates the leaf distributions from all trees instead of just their predicted labels.", "In the case where the leaves are pure (each leaf contains only one class of examples), this prediction rule coincides with the majority vote rule commonly used in random forests.", "Whenever the leaves contain mixed classes, this rule takes into account the prediction confidence of each tree in contrast to majority voting.", "Empirically, this rule consistently gives lower prediction error than majority voting with pruned trees.", "Experiments We test our pruning algorithm BudgetPrune on four benchmark datasets used for prediction-time budget algorithms.", "The first two datasets have unknown feature acquisition costs so we assign costs to be 1 for all features; the aim is to show that BudgetPrune successfully selects a sparse subset of features on average to classify each example with high accuracy.", "In contrast to traditional sparse feature selection, our algorithm allows adaptivity, meaning different examples use different subsets of features.", "The last two datasets have real feature acquisition costs measured in terms of CPU time.", "BudgetPrune achieves high prediction accuracy spending much less CPU time in feature acquisition.", "For each dataset we first train a RF and apply BudgetPrune on it using different $\\lambda $ 's to obtain various points on the accuracy-cost tradeoff curve.", "We use in-bag data to estimate error probability at each node and the validation data for the feature cost variables $w_{k,i}$ 's.", "We implement BudgetPrune using CPLEX [1] network flow solver for the primal update step.", "The running time is significantly reduced (from hours down to minutes) compared to directly solving the LP relaxation of (IP) using standard solvers such as Gurobi [10].", "Futhermore, the standard solvers simply break trying to solve the larger experiments whereas BudgetPrune handles them with ease.", "We run the experiments for 10 times and report the means and standard deviations.", "Competing methods: We compare against four other approaches.", "(i) BudgetRF[17]: the recursive node splitting process for each tree is stopped as soon as node impurity (entropy or Pairs) falls below a threshold.", "The threshold is a measure of impurity tolerated in the leaf nodes.", "This can be considered as a naive pruning method as it reduces feature acquisition cost while maintaining low impurity in the leaves.", "Figure: Comparison of BudgetPrune against CCP, BudgetRF with early stopping, GreedyPrune and GreedyMiser on 4 real world datasets.", "BudgetPrune (red) outperforms competing state-of-art methods.", "GreedyMiser dominates ASTC , CSTC  and DAG  significantly on all datasets.", "We omit them in the plots to clearly depict the differences between competing methods.", "(ii) Cost-Complexity Pruning (CCP) [4]: it iteratively prunes subtrees such that the resulting tree has low error and small size.", "We perform CCP on individual trees to different levels to obtain various points on the accuracy-cost tradeoff curve.", "CCP does not take into account feature costs.", "(iii) GreedyPrune: is a greedy global feature pruning strategy that we propose; at each iteration it attempts to remove all nodes corresponding to one feature from the RF such that the resulting pruned RF has the lowest training error and average feature cost.", "The process terminates in at most K iterations, where K is the number of features.", "The idea is to reduce feature costs by successively removing features that result in large cost reduction yet small accuracy loss.", "We also compare against the state-of-the-art methods in budgeted learning (iv) GreedyMiser [27]: it is a modification of gradient boosted regression tree [8] to incorporate feature cost.", "Specifically, each weak learner (a low-depth decision tree) is built to minimize squared loss with respect to current gradient at the training examples plus feature acquisition cost.", "To build each weak learner the feature costs are set to zero for those features already used in previous weak learners.", "Other prediction-time budget algorithms such as ASTC [12], CSTC [26] and cost-weighted $l$ -1 classifiers are shown to perform strictly worse than GreedyMiser by a significant amount [12], [17] so we omit them in our plots.", "Since only the feature acquisition costs are standardized, for fair comparison we do not include the computation cost term in the objective of (IP) and focus instead on feature acquisition costs.", "MiniBooNE Particle Identification and Forest Covertype Datasets:[7] Feature costs are uniform in both datasets.", "Our base RF consists of 40 trees using entropy split criteria and choosing from the full set of features at each split.", "As shown in (a) and (b) of Figure REF , BudgetPrune (in red) achieves the best accuracy-cost tradeoff.", "The advantage of BudgetPrune is particularly large in (b).", "GreedyMiser has lower accuracy in the high budget region compared to BudgetPrune in (a) and significantly lower accuracy in (b).", "The gap between BudgetPrune and other pruning methods is small in (a) but much larger in (b).", "This indicates large gains from globally encouraging feature sharing in the case of (b) compared to (a).", "In both datasets, BudgetPrune successfully prunes away large number of features while maintaining high accuracy.", "For example in (a), using only 18 unique features on average instead of 40, we can get essentially the same accuracy as the original RF.", "Yahoo!", "Learning to Rank:[6] This ranking dataset consists of 473134 web documents and 19944 queries.", "Each example in the dataset contains features of a query-document pair together with the relevance rank of the document to the query.", "There are $141397/146769/184968$ examples in the training/validation/test sets.", "There are 519 features for each example; each feature is associated with an acquisition cost in the set $\\lbrace 1,5,20,50,100,150,200\\rbrace $ , which represents the units of CPU time required to extract the feature and is provided by a Yahoo!", "employee.", "The labels are binarized so that the document is either relevant or not relevant to the query.", "The task is to learn a model that takes a new query and its associated set of documents to produce an accurate ranking using as little feature cost as possible.", "As in [17], we use the Average Precision@5 as the performance metric, which gives a high reward for ranking the relevant documents on top.", "Our base RF consists of 140 trees using cost weighted entropy split criteria as in [17] and choosing from a random subset of 400 features at each split.", "As shown in (c) of Figure REF , BudgetPrune achieves similar ranking accuracy as GreedyMiser using only 30% of its cost.", "Scene15 [13]: This scene recognition dataset contains 4485 images from 15 scene classes (labels).", "Following [27] we divide it into $1500/300/2685$ examples for training/validation/test sets.", "We use a diverse set of visual descriptors and object detectors from the Object Bank [14].", "We treat each individual detector as an independent descriptor so we have a total of 184 visual descriptors.", "The acquisition costs of these visual descriptors range from 0.0374 to 9.2820.", "For each descriptor we train 15 one-vs-rest kernel SVMs and use the output (margins) as features.", "Once any feature corresponding to a visual descriptor is used for a test example, an acquisition cost of the visual descriptor is incurred and subsequent usage of features from the same group is free for the test example.", "Our base RF consists of 500 trees using entropy split criteria and choosing from a random subset of 20 features at each split.", "As shown in (d) of Figure REF , BudgetPrune and GreedyPrune significantly outperform other competing methods.", "BudgetPrune has the same accuracy at the cost of 9 as at the full cost of 32.", "BudgetPrune and GreedyPrune perform similarly, indicating the greedy approach happen to solve the global optimization in this particular initial RF.", "Discussion & Concluding Comments We have empirically evaluated several resource constrained learning algorithms including BudgetPrune and its variations on benchmarked datasets here and in the Appendix.", "We highlight key features of our approach below.", "(i) State-of-the-art Methods.", "Recent work has established that GreedyMiser and BudgetRF are among the state-of-the-art methods dominating a number of other methods [12], [26], [25] on these benchmarked datasets.", "GreedyMiser requires building class-specific ensembles and tends to perform poorly and is increasingly difficult to tune in multi-class settings.", "RF, by its nature, can handle multi-class settings efficiently.", "On the other hand, as we described earlier, [12], [25], [26] are fundamentally \"tree-growing\" approaches, namely they are top-down methods acquiring features sequentially based on a surrogate utility value.", "This is a fundamentally combinatorial problem that is known to be NP hard [5], [26] and thus requires a number of relaxations and heuristics with no guarantees on performance.", "In contrast our pruning strategy is initialized to realize good performance (RF initialization) and we are able to globally optimize cost-accuracy objective.", "(ii) Variations on Pruning.", "By explicitly modeling feature costs, BudgetPrune outperforms other pruning methods such as early stopping of BudgetRF and CCP that do not consider costs.", "GreedyPrune performs well validating our intuition (see Table.", "1) that pruning sparsely occurring feature nodes utilized by large fraction of examples can improve test-time cost-accuracy tradeoff.", "Nevertheless, the BudgetPrune outperforms GreedyPrune, which is indicative of the fact that apart from obvious high-budget regimes, node-pruning must account for how removal of one node may have an adverse impact on another downstream one.", "(iii) Sensitivity to Impurity, Feature Costs, & other inputs.", "We explore these issues in Appendix.", "We experiment BudgetPrune with different impurity functions such as entropy and Pairs [17] criteria.", "Pairs-impurity tends to build RFs with lower cost but also lower accuracy compared to entropy and so has poorer performance.", "We also explored how non-uniform costs can impact cost-accuracy tradeoff.", "An elegant approach has been suggested by [2], who propose an adversarial feature cost proportional to feature utility value.", "We find that BudgetPrune is robust with such costs.", "Other RF parameters including number of trees and feature subset size at each split do impact cost-accuracy tradeoff in obvious ways with more trees and moderate feature subset size improving prediction accuracy while incurring higher cost.", "To conclude, our proposed formulation possesses 1) elegant theoretical properties, 2) an algorithm scalable to large problems and 3) superior empirical performance.", "Acknowledgment We thank Dr Kilian Weinberger for helpful discussions and Dr David Castanon for the insights on the primal dual algorithm.", "Appendix A Naive Pruning Formulation The nice property of totally unimodular constraints in Theorem 3.2 is due to our specific formulation.", "Here we present an alternative integer program formulation and show its deficiency.", "Recall we defined the following node variables $z_h=\\left\\lbrace \\begin{array}{rl}1 & \\text{if node } h \\text{ is a leaf in the pruned tree} ,\\\\0 & \\text{otherwise}.\\end{array} \\right.$ and indicator variables of feature usage: $w_{k,i}=\\left\\lbrace \\begin{array}{rl}1 & \\text{ if feature } k \\text{ is used by }\\mathbf {x}^{(i)} \\text{ in any } \\mathcal {T}_t, t=1,\\dots ,T\\\\0 & \\text{ otherwise}.\\end{array} \\right.$ First, note that if $z_h=1$ for some node $h$ , then the examples that are routed to $h$ must have used all the features in the predecessor nodes $p(h)$ , excluding $h$ .", "We use $k\\sim p(h)$ to denote feature $k$ is used in any predecessor of $h$ , excluding $h$ .", "Then for each feature $k$ and example $i$ , we must have $w_{k,i}\\ge z_h$ for all nodes $h$ such that $i\\in S_h$ and $k\\sim p(h)$ .", "Combining these constraints with the pruning constraints we formulate pruning as a 0-1 integer program for an individual tree: $\\begin{array}{rlll}\\displaystyle \\min _{\\begin{array}{c}z_h \\in \\lbrace 0,1\\rbrace \\\\w_{k,i}\\in \\lbrace 0,1\\rbrace \\end{array}} & \\multicolumn{2}{l}{\\frac{1}{N}\\displaystyle \\sum _{h\\in \\mathcal {N}} e_h z_h +\\lambda \\sum _{k=1}^{K}c_k(\\frac{1}{N}\\sum _{i=1}^{N}w_{k,i})} \\\\\\textrm {s.t.}", "& z_h+ \\sum _{u\\in p(h)} z_u=1 & \\forall h \\in \\tilde{\\mathcal {T}}, \\\\& w_{k,i}\\ge z_h & \\forall h:i\\in S_h \\wedge k\\sim p(h), \\\\& & \\forall k\\in [K], \\forall i\\in S. \\\\\\end{array}$ To solve the integer program, a common heuristic is to solve its linear program relaxation.", "Unfortunately, the constraint set in the above formulation has fractional extreme points, leading to possibly fractional solutions to the relaxed problem.", "It is not clear how to perform rounding to obtain good prunings.", "Consider the first tree in Figure REF .", "Feature 1 is used at the root node and feature 2 is used at node 3.", "There are 7 variables (assuming there is only one example and it goes to leaf 4): $z_1,z_2,z_3,z_4,z_5,w_{1,1},w_{2,1}$ .", "The LP relaxed constraints are: $& z_1+z_3+z_4=1 , z_1+z_3+z_5=1 , z_1+z_2=1, \\\\& w_{1,1}\\ge z_4, w_{1,1}\\ge z_3, w_{2,1}\\ge z_4, 0\\le z\\le 1.$ The following is a basic feasible solution: $z_1=0, z_2=1 , z_3=z_4=z_5=0.5, w_{1,1}=w_{2,1}=0.5,$ because the following set of 7 constraints are active: $& z_1+z_3+z_4=1, z_1+z_3+z_5=1, \\\\& w_{1,1}\\ge z_4, w_{1,1}\\ge z_3,w_{2,1}\\ge z_4, z_1=0,z_2=1.$ Even if we were to interpret the fractional solution of $z_h$ as probabilities of $h$ being a leaf node, we see an issue with this formulation: the example has $0.5$ probability of stopping at node 3 or 4 ($z_3=z_4=0.5$ ).", "In both cases, feature 1 at the root node has to be used, however $w_{1,1}=0.5$ indicates that it is only being used half of the times.", "This solution is not a feasible pruning and fails to capture the cost of the pruning.", "Attempting to use an LP relaxation of this formulation fails to capture the desired behavior of the integer program.", "In the main paper we propose a better integer program formulation and show that solving the LP relaxation yields the optimal solution to the integer program.", "Transformation to Network Matrices and Shortest Path Problems To illustrate the transformation to network matrix in Lemma 3.1, we provide the following illustration in Figure REF .", "Note in the main paper we have shown the example of the first tree.", "For simplicity we consider only one example being routed to nodes 4 and 11 respectively on the two trees.", "The equality constraints in (IP2) can be separated based on the trees and put in matrix form: ${& z_1 & z_2 & z_3 & z_4 & z_5 & w^{(1)}_{1,1} & w^{(1)}_{2,1}\\\\r_1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\r_2 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\\\r_3 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\\\r_4 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\\\r_5 & 1 & 0 & 0 & 0 & 0 & 1 & 0},$ for tree 1 and ${& z_6 & z_7 & z_8 & z_9 & z_{10} & z_{11} & z_{12} & w^{(2)}_{2,1} & w^{(2)}_{3,1}\\\\r_1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\r_2 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\r_3 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0\\\\r_4 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\\\r_5 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\\\r_6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0},$ for tree 2.", "Through row operations they can be turned into network matrices, where there is exactly two non-zeros in each column, a 1 and a $-1$ .", "${& z_1 & z_2 & z_3 & z_4 & z_5 & w^{(1)}_{1,1} & w^{(1)}_{2,1}\\\\-r_1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\\\r_1-r_2 & 0 & 1 & -1 & -1 & 0 & 0 & 0 \\\\r_2-r_3 & 0 & 0 & 0 & 1 & -1 & 0 & 0 \\\\r_3-r_4 & 0 & 0 & 0 & 0 & 1 & 0 & -1 \\\\r_4-r_5 & 0 & 0 & 1 & 0 & 0 & -1 & 1 \\\\r_5 & 1 & 0 & 0 & 0 & 0 & 1 & 0},$ for tree 1 and ${& z_6 & z_7 & z_8 & z_9 & z_{10} & z_{11} & z_{12} & w^{(2)}_{2,1} & w^{(2)}_{3,1}\\\\-r_1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\\\r_1-r_2 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\\\r_2-r_3 & 0 & 1 & 0 & 1 & -1 & -1 & 0 & 0 & 0\\\\r_3-r_4 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\\\r_4-r_5 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1\\\\r_5-r_6 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 1\\\\r_6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0}$ for tree 2.", "Note the above transformation to network matrices can always be done as long as the leaf nodes are arranged in a pre-order fashion.", "In the primal-dual algorithm, the inner minimization can be decomposed to shortest path problems corresponding to individual trees.", "Figure REF illustrates such a construction based on the network matrices shown above.", "The nodes in the graphs correspond to rows in the network matrices and the arcs correspond to the columns, which are the primal variables $z_h, w^{(t)}_{k,i}$ 's.", "There is a cost associated with each arc in the objective of the minimization problem.", "The task is to find a path from the first node (source) to the last node (sink) such that the sum of arc costs is minimized.", "Note each path from source to sink corresponds to a feasible pruning.", "For example, in (a) of Figure REF , consider the path of 1-2-5-6, the active arcs are $z_2, z_3$ and $w^{(1)}_{1,1}$ , Setting these variables to 1 and others to 0, we see that it corresponds to pruning Tree 1 at node 3 in Figure REF .", "(Note the nodes in Figure REF and Figure REF are not to be confused - they do not have a relation with each other. )", "Figure: Turning pruning to equivalent shortest path problems.", "Proof of Theorem 3.2 Denote the equality constraints of (IP) with index set $J_1$ .", "They can be divided into each tree.", "Each constraint matrix in $J_1$ associated with a tree can be turned into a network matrix according to Lemma 3.1.", "Stacking these matrices leads to a larger network matrix.", "Denote the $w^{(t)}_{k,i}\\le w_{k,i}$ constraints with index set $J_2$ .", "Consider the constraint matrix for $J_2$ .", "Each $w^{(t)}_{k,i}$ only appears once in $J_2$ , which means the column corresponding to $w^{(t)}_{k,i}$ has only one element equal to 1 and the rest equal to 0.", "If we arrange the constraints in $J_2$ such that for any given $k,i$ $w^{(t)}_{k,i}\\le w_{k,i}$ are put together for $t\\in [T]$ , the constraint matrix for $J_2$ has interval structure such that the non-zeros in each column appear consecutively.", "Finally, putting the network matrix from $J_1$ and the matrix from $J_2$ together.", "Assign $J_1$ and the odd rows of $J_2$ to the first partition $Q_1$ and assign the even rows of $J_2$ to the second partition $Q_2$ .", "Note the upper bound constraints on the variables can be ignored as this is an minimization problem.", "We conclude that the constraint matrix of (IP) is totally unimodular according to Theorem 2.7, Part 3 of [18] with partition $Q_1$ and $Q_2$ .", "By Proposition 2.1 and 2.2, Part 3 of [18] we can conclude the proof.", "every tree node/.style=minimum width=1em,draw,circle, blank/.style=draw=none, edge from parent/.style= draw, edge from parent path=() – (), level distance=1cm Figure: An ensemble of two decision trees with node numbers and associated feature in subscripts Additional Details of Experiments In this section we provide additional details of the experiment setup and explore how some parameter choices may affect BudgetPrune.", "Additional details of datasets The MiniBooNE data set is a binary classification task to distinguish electron neutrinos from muon neutrinos.", "There are $45523/19510/65031$ examples in training/validation/test sets.", "Each example has 50 features, each with unit cost.", "The Forest data set contains cartographic variables to predict 7 forest cover types.", "There are $36603/15688/58101$ examples in training/validation/test sets.", "Each example has 54 features, each with unit cost.", "We use 1000 trees for GreedyMiser and search over learning rates in $[10^{-5}, 10^2]$ for MiniBooNE and Forest.", "The Yahoo and Scene15 datasets have actual feature acquisition costs in terms of CPU time.", "We use 3000 trees for GreedyMiser and search over learning rates in $[10^{-5}, 1]$ .", "We use the multi-class logistic loss for Scene15 and the squared loss for other datasets in GreedyMiser.", "For the Scene15 dataset, we use a diverse set of visual discriptors varying in computation time: GIST, spatial HOG, Local Binary Pattern, self-similarity, texton histogram, geometric texton, geometric color and 177 object detectors from the Object Bank [14].", "We treat each individual detector as an independent descriptor so we have 184 different visual descriptors in total.", "The acquisition costs of these visual descriptors range from 0.0374 to 9.2820.", "For each descriptor we train 15 one-vs-rest kernel SVMs and use the output (margins) as features.", "The best classifier based on individual descriptors achieves an accuracy of 77.8%.", "Note the features are grouped based on the visual descriptors.", "Once any feature corresponding to a visual descriptor is used for a test example, an acquisition cost of the visual descriptor is incurred and subsequent usage of features from the same group is free for the test example.", "Next, we perform additional experiments to evaluate BudgetPrune with different costs, input RFs.", "Non-uniform cost on MiniBooNE We observe that CCP performs similarly to BudgetPrune on MiniBooNE when the costs are uniform in the case of entropy splitting criteria, indicating little gain from global optimization with respect to feature usage.", "We suspect that uniform feature costs work in favor of CCP because there's no loss in treating each feature equally.", "To confirm this intuition we assign the features non-uniform costs and re-run prunings on the same RF.", "We first normalize the data so that the data vectors corresponding to the features have the same $l$ -2 norm.", "We then train a linear SVM on it and obtain the weight vector corresponding to the learned hyperplane.", "We around the absolute values of the weights and make them the costs for the features.", "Intuitively the feature with higher weight tends to be more relevant for the classification task so we assign it a higher acquisition cost.", "The resulting costs lie in the range of $[1,40]$ and we normalize them so that the sum of all feature costs is 50 - the number of features.", "We plot BudgetPrune and CCP for uniform cost as well as the non-uniform cost described above in Figure REF .", "BudgetPrune still achieves similar performance as uniform cost while CCP performance drops significantly with non-uniform feature cost.", "This shows again the importance of taking into account feature costs in the pruning process.", "Figure: Comparing BudgetPrune and CCP with uniform and non-uniform feature cost on MiniBooNE dataset.", "BudgetPrune is robust when the feature cost is non-uniform.", "Entropy Vs Pairs How does BudgetPrune depend on the splitting criteria used in the underlying random forest?", "On two data sets we build RFs using the popular entropy splitting criteria and the mini-max Pairs criteria used in [17] and the results are shown in Figure REF .", "We observe that entropy splitting criteria lead to RFs with higher accuracy while the Pairs criteria lead to RFs with lower cost.", "This is expected as using Pairs biases to more balanced splits and thus provably low cost [17].", "In (a) of Figure REF we observe that as more of the RF is pruned away BudgetPrune and CCP results for entropy and Pairs coincide.", "This suggests that the two criteria actually lead to similar tree structures in the initial tree-building process.", "However, as the trees are built deeper their structures diverge.", "Plot (b) in Figure REF shows that pruning based on the RFs from the Pairs criteria can achieve higher accuracy in the low cost region.", "But if high accuracy in the high cost region is desirable then the entropy criteria should be used.", "Figure: Comparisons of various pruning methods based on entropy and Pairs splitting criteria on MiniBooNE and Forest datasetsFigure: Comparing various pruning approaches on RF built with k=20 and k=120 on Scene15 dataset.", "The initial RF has higher accuracy and higher cost for k=20.", "GreedyPrune performs very well in k=20 but very poorly in k=120.", "Size of random feature subset at each split At each split in RF building, it is possible to restrict the choice of splitting feature to be among a random subset of all features.", "Such restriction tends to further reduce correlation among trees and gain prediction accuracy.", "The drawback is that test examples tend to encounter a diverse set of features, increasing feature acquisition cost.", "For illustration purpose, we plot various pruning results on Scene15 dataset for feature subset sizes $k=20$ and $k=120$ in Figure REF .", "The initial RF has higher accuracy and higher cost for $k=20$ as expected.", "BudgetPrune achieves slightly better accuracy in $k=20$ than $k=120$ .", "Note also how GreedyPrune performance drops significantly for $k=120$ so it is not robust.", "In our main experiments $k$ is chosen on validation data to achieve highest accuracy for the initial RF." ], [ "A Primal-Dual Algorithm", "Even though we can solve (IP) via its LP relaxation, the resulting LP can be too large in practical applications for any general-purpose LP solver.", "In particular, the number of variables and constraints is roughly $O(T \\times |\\mathcal {T}_{\\text{max}}|+N \\times T \\times K_{\\text{max}})$ , where $T$ is the number of trees; $|\\mathcal {T}_{\\text{max}}|$ is the maximum number of nodes in a tree; $N$ is the number of examples; $K_{\\text{max}}$ is the maximum number of features an example uses in a tree.", "The runtime of the LP thus scales $O(T^{3})$ with the number of trees in the ensemble, limiting the application to only small ensembles.", "In this section we propose a primal-dual approach that effectively decomposes the optimization into many sub-problems.", "Each sub-problem corresponds to a tree in the ensemble and can be solved efficiently as a shortest path problem.", "The runtime per iteration is $O(\\frac{T}{p}(|\\mathcal {T}_{\\text{max}}|+N \\times K_{\\text{max}})\\log (|\\mathcal {T}_{\\text{max}}|+N \\times K_{\\text{max}}))$ , where $p$ is the number of processors.", "We can thus massively parallelize the optimization and scale to much larger ensembles as the runtime depends only linearly on $\\frac{T}{p}$ .", "To this end, we assign dual variables $\\beta _{k,i}^{(t)}$ for the inequality constraints $ w^{(t)}_{k,i} \\le w_{k,i}$ and derive the dual problem.", "$\\hspace{-8.5359pt}\\begin{array}{rlll}\\displaystyle \\max _{\\beta _{k,i}^{(t)}\\ge 0} \\min _{\\begin{array}{c}z^{(t)}_h\\in [0,1] \\\\ w^{(t)}_{k,i}\\in [0,1] \\\\w_{k,i}\\in [0,1]\\end{array}} & \\multicolumn{2}{l}{\\frac{1}{NT}\\displaystyle \\sum _{t=1}^{T}\\sum _{h\\in \\mathcal {T}_t} \\hat{e}^{(t)}_h z^{(t)}_h +\\lambda \\left( \\sum _{k=1}^{K}c_k(\\frac{1}{N}\\sum _{i=1}^{N}w_{k,i}) \\right) + \\sum _{t=1}^T \\sum _{i=1}^N \\sum _{k\\in K_{t,i}} \\beta _{k,i}^{(t)}(w_{k,i}^{(t)}-w_{k,i})}\\\\\\textrm {s.t.}", "& \\displaystyle \\sum _{u\\in p(h)} z^{(t)}_u=1, & \\forall h \\in \\tilde{\\mathcal {T}}_t, \\forall t \\in [T], \\\\& w^{(t)}_{k,i}+ \\displaystyle \\sum _{h\\in p(u_{t,k,i})} z^{(t)}_h=1 , & \\forall k\\in K_{t,i},\\forall i\\in S, \\forall t \\in [T], \\\\\\end{array}$ where for simplicity we have combined coefficients of $z_h^{(t)}$ in the objective of (IP) to $\\hat{e}_h^{(t)}$ .", "The primal-dual algorithm is summarized in Algorithm REF .", "It alternates between updating the primal and the dual variables.", "The key is to observe that given dual variables, the primal problem (inner minimization) can be decomposed for each tree in the ensemble and solved in parallel as shortest path problems due to Lemma REF .", "(See also Appendix).", "The primal variables $w_{k,i}$ can be solved in closed form: simply compute $\\mu _{k,i}=\\lambda c_k / N-\\sum _{t\\in T_{k,i}} \\beta _{k,i}^{(t)}$ , where $T_{k,i}$ is the set of trees in which example $i$ encounters feature $k$ .", "So $w_{k,i}$ should be set to 0 if $\\mu _{k,i}>0$ and $w_{k,i}=1$ if $\\mu _{k,i}<0$ .", "Note that our prediction rule aggregates the leaf distributions from all trees instead of just their predicted labels.", "In the case where the leaves are pure (each leaf contains only one class of examples), this prediction rule coincides with the majority vote rule commonly used in random forests.", "Whenever the leaves contain mixed classes, this rule takes into account the prediction confidence of each tree in contrast to majority voting.", "Empirically, this rule consistently gives lower prediction error than majority voting with pruned trees." ], [ "Experiments", "We test our pruning algorithm BudgetPrune on four benchmark datasets used for prediction-time budget algorithms.", "The first two datasets have unknown feature acquisition costs so we assign costs to be 1 for all features; the aim is to show that BudgetPrune successfully selects a sparse subset of features on average to classify each example with high accuracy.", "In contrast to traditional sparse feature selection, our algorithm allows adaptivity, meaning different examples use different subsets of features.", "The last two datasets have real feature acquisition costs measured in terms of CPU time.", "BudgetPrune achieves high prediction accuracy spending much less CPU time in feature acquisition.", "For each dataset we first train a RF and apply BudgetPrune on it using different $\\lambda $ 's to obtain various points on the accuracy-cost tradeoff curve.", "We use in-bag data to estimate error probability at each node and the validation data for the feature cost variables $w_{k,i}$ 's.", "We implement BudgetPrune using CPLEX [1] network flow solver for the primal update step.", "The running time is significantly reduced (from hours down to minutes) compared to directly solving the LP relaxation of (IP) using standard solvers such as Gurobi [10].", "Futhermore, the standard solvers simply break trying to solve the larger experiments whereas BudgetPrune handles them with ease.", "We run the experiments for 10 times and report the means and standard deviations.", "Competing methods: We compare against four other approaches.", "(i) BudgetRF[17]: the recursive node splitting process for each tree is stopped as soon as node impurity (entropy or Pairs) falls below a threshold.", "The threshold is a measure of impurity tolerated in the leaf nodes.", "This can be considered as a naive pruning method as it reduces feature acquisition cost while maintaining low impurity in the leaves.", "Figure: Comparison of BudgetPrune against CCP, BudgetRF with early stopping, GreedyPrune and GreedyMiser on 4 real world datasets.", "BudgetPrune (red) outperforms competing state-of-art methods.", "GreedyMiser dominates ASTC , CSTC  and DAG  significantly on all datasets.", "We omit them in the plots to clearly depict the differences between competing methods.", "(ii) Cost-Complexity Pruning (CCP) [4]: it iteratively prunes subtrees such that the resulting tree has low error and small size.", "We perform CCP on individual trees to different levels to obtain various points on the accuracy-cost tradeoff curve.", "CCP does not take into account feature costs.", "(iii) GreedyPrune: is a greedy global feature pruning strategy that we propose; at each iteration it attempts to remove all nodes corresponding to one feature from the RF such that the resulting pruned RF has the lowest training error and average feature cost.", "The process terminates in at most K iterations, where K is the number of features.", "The idea is to reduce feature costs by successively removing features that result in large cost reduction yet small accuracy loss.", "We also compare against the state-of-the-art methods in budgeted learning (iv) GreedyMiser [27]: it is a modification of gradient boosted regression tree [8] to incorporate feature cost.", "Specifically, each weak learner (a low-depth decision tree) is built to minimize squared loss with respect to current gradient at the training examples plus feature acquisition cost.", "To build each weak learner the feature costs are set to zero for those features already used in previous weak learners.", "Other prediction-time budget algorithms such as ASTC [12], CSTC [26] and cost-weighted $l$ -1 classifiers are shown to perform strictly worse than GreedyMiser by a significant amount [12], [17] so we omit them in our plots.", "Since only the feature acquisition costs are standardized, for fair comparison we do not include the computation cost term in the objective of (IP) and focus instead on feature acquisition costs.", "MiniBooNE Particle Identification and Forest Covertype Datasets:[7] Feature costs are uniform in both datasets.", "Our base RF consists of 40 trees using entropy split criteria and choosing from the full set of features at each split.", "As shown in (a) and (b) of Figure REF , BudgetPrune (in red) achieves the best accuracy-cost tradeoff.", "The advantage of BudgetPrune is particularly large in (b).", "GreedyMiser has lower accuracy in the high budget region compared to BudgetPrune in (a) and significantly lower accuracy in (b).", "The gap between BudgetPrune and other pruning methods is small in (a) but much larger in (b).", "This indicates large gains from globally encouraging feature sharing in the case of (b) compared to (a).", "In both datasets, BudgetPrune successfully prunes away large number of features while maintaining high accuracy.", "For example in (a), using only 18 unique features on average instead of 40, we can get essentially the same accuracy as the original RF.", "Yahoo!", "Learning to Rank:[6] This ranking dataset consists of 473134 web documents and 19944 queries.", "Each example in the dataset contains features of a query-document pair together with the relevance rank of the document to the query.", "There are $141397/146769/184968$ examples in the training/validation/test sets.", "There are 519 features for each example; each feature is associated with an acquisition cost in the set $\\lbrace 1,5,20,50,100,150,200\\rbrace $ , which represents the units of CPU time required to extract the feature and is provided by a Yahoo!", "employee.", "The labels are binarized so that the document is either relevant or not relevant to the query.", "The task is to learn a model that takes a new query and its associated set of documents to produce an accurate ranking using as little feature cost as possible.", "As in [17], we use the Average Precision@5 as the performance metric, which gives a high reward for ranking the relevant documents on top.", "Our base RF consists of 140 trees using cost weighted entropy split criteria as in [17] and choosing from a random subset of 400 features at each split.", "As shown in (c) of Figure REF , BudgetPrune achieves similar ranking accuracy as GreedyMiser using only 30% of its cost.", "Scene15 [13]: This scene recognition dataset contains 4485 images from 15 scene classes (labels).", "Following [27] we divide it into $1500/300/2685$ examples for training/validation/test sets.", "We use a diverse set of visual descriptors and object detectors from the Object Bank [14].", "We treat each individual detector as an independent descriptor so we have a total of 184 visual descriptors.", "The acquisition costs of these visual descriptors range from 0.0374 to 9.2820.", "For each descriptor we train 15 one-vs-rest kernel SVMs and use the output (margins) as features.", "Once any feature corresponding to a visual descriptor is used for a test example, an acquisition cost of the visual descriptor is incurred and subsequent usage of features from the same group is free for the test example.", "Our base RF consists of 500 trees using entropy split criteria and choosing from a random subset of 20 features at each split.", "As shown in (d) of Figure REF , BudgetPrune and GreedyPrune significantly outperform other competing methods.", "BudgetPrune has the same accuracy at the cost of 9 as at the full cost of 32.", "BudgetPrune and GreedyPrune perform similarly, indicating the greedy approach happen to solve the global optimization in this particular initial RF." ], [ "Discussion & Concluding Comments", "We have empirically evaluated several resource constrained learning algorithms including BudgetPrune and its variations on benchmarked datasets here and in the Appendix.", "We highlight key features of our approach below.", "(i) State-of-the-art Methods.", "Recent work has established that GreedyMiser and BudgetRF are among the state-of-the-art methods dominating a number of other methods [12], [26], [25] on these benchmarked datasets.", "GreedyMiser requires building class-specific ensembles and tends to perform poorly and is increasingly difficult to tune in multi-class settings.", "RF, by its nature, can handle multi-class settings efficiently.", "On the other hand, as we described earlier, [12], [25], [26] are fundamentally \"tree-growing\" approaches, namely they are top-down methods acquiring features sequentially based on a surrogate utility value.", "This is a fundamentally combinatorial problem that is known to be NP hard [5], [26] and thus requires a number of relaxations and heuristics with no guarantees on performance.", "In contrast our pruning strategy is initialized to realize good performance (RF initialization) and we are able to globally optimize cost-accuracy objective.", "(ii) Variations on Pruning.", "By explicitly modeling feature costs, BudgetPrune outperforms other pruning methods such as early stopping of BudgetRF and CCP that do not consider costs.", "GreedyPrune performs well validating our intuition (see Table.", "1) that pruning sparsely occurring feature nodes utilized by large fraction of examples can improve test-time cost-accuracy tradeoff.", "Nevertheless, the BudgetPrune outperforms GreedyPrune, which is indicative of the fact that apart from obvious high-budget regimes, node-pruning must account for how removal of one node may have an adverse impact on another downstream one.", "(iii) Sensitivity to Impurity, Feature Costs, & other inputs.", "We explore these issues in Appendix.", "We experiment BudgetPrune with different impurity functions such as entropy and Pairs [17] criteria.", "Pairs-impurity tends to build RFs with lower cost but also lower accuracy compared to entropy and so has poorer performance.", "We also explored how non-uniform costs can impact cost-accuracy tradeoff.", "An elegant approach has been suggested by [2], who propose an adversarial feature cost proportional to feature utility value.", "We find that BudgetPrune is robust with such costs.", "Other RF parameters including number of trees and feature subset size at each split do impact cost-accuracy tradeoff in obvious ways with more trees and moderate feature subset size improving prediction accuracy while incurring higher cost.", "To conclude, our proposed formulation possesses 1) elegant theoretical properties, 2) an algorithm scalable to large problems and 3) superior empirical performance." ], [ "Acknowledgment", "We thank Dr Kilian Weinberger for helpful discussions and Dr David Castanon for the insights on the primal dual algorithm." ], [ "A Naive Pruning Formulation", "The nice property of totally unimodular constraints in Theorem 3.2 is due to our specific formulation.", "Here we present an alternative integer program formulation and show its deficiency.", "Recall we defined the following node variables $z_h=\\left\\lbrace \\begin{array}{rl}1 & \\text{if node } h \\text{ is a leaf in the pruned tree} ,\\\\0 & \\text{otherwise}.\\end{array} \\right.$ and indicator variables of feature usage: $w_{k,i}=\\left\\lbrace \\begin{array}{rl}1 & \\text{ if feature } k \\text{ is used by }\\mathbf {x}^{(i)} \\text{ in any } \\mathcal {T}_t, t=1,\\dots ,T\\\\0 & \\text{ otherwise}.\\end{array} \\right.$ First, note that if $z_h=1$ for some node $h$ , then the examples that are routed to $h$ must have used all the features in the predecessor nodes $p(h)$ , excluding $h$ .", "We use $k\\sim p(h)$ to denote feature $k$ is used in any predecessor of $h$ , excluding $h$ .", "Then for each feature $k$ and example $i$ , we must have $w_{k,i}\\ge z_h$ for all nodes $h$ such that $i\\in S_h$ and $k\\sim p(h)$ .", "Combining these constraints with the pruning constraints we formulate pruning as a 0-1 integer program for an individual tree: $\\begin{array}{rlll}\\displaystyle \\min _{\\begin{array}{c}z_h \\in \\lbrace 0,1\\rbrace \\\\w_{k,i}\\in \\lbrace 0,1\\rbrace \\end{array}} & \\multicolumn{2}{l}{\\frac{1}{N}\\displaystyle \\sum _{h\\in \\mathcal {N}} e_h z_h +\\lambda \\sum _{k=1}^{K}c_k(\\frac{1}{N}\\sum _{i=1}^{N}w_{k,i})} \\\\\\textrm {s.t.}", "& z_h+ \\sum _{u\\in p(h)} z_u=1 & \\forall h \\in \\tilde{\\mathcal {T}}, \\\\& w_{k,i}\\ge z_h & \\forall h:i\\in S_h \\wedge k\\sim p(h), \\\\& & \\forall k\\in [K], \\forall i\\in S. \\\\\\end{array}$ To solve the integer program, a common heuristic is to solve its linear program relaxation.", "Unfortunately, the constraint set in the above formulation has fractional extreme points, leading to possibly fractional solutions to the relaxed problem.", "It is not clear how to perform rounding to obtain good prunings.", "Consider the first tree in Figure REF .", "Feature 1 is used at the root node and feature 2 is used at node 3.", "There are 7 variables (assuming there is only one example and it goes to leaf 4): $z_1,z_2,z_3,z_4,z_5,w_{1,1},w_{2,1}$ .", "The LP relaxed constraints are: $& z_1+z_3+z_4=1 , z_1+z_3+z_5=1 , z_1+z_2=1, \\\\& w_{1,1}\\ge z_4, w_{1,1}\\ge z_3, w_{2,1}\\ge z_4, 0\\le z\\le 1.$ The following is a basic feasible solution: $z_1=0, z_2=1 , z_3=z_4=z_5=0.5, w_{1,1}=w_{2,1}=0.5,$ because the following set of 7 constraints are active: $& z_1+z_3+z_4=1, z_1+z_3+z_5=1, \\\\& w_{1,1}\\ge z_4, w_{1,1}\\ge z_3,w_{2,1}\\ge z_4, z_1=0,z_2=1.$ Even if we were to interpret the fractional solution of $z_h$ as probabilities of $h$ being a leaf node, we see an issue with this formulation: the example has $0.5$ probability of stopping at node 3 or 4 ($z_3=z_4=0.5$ ).", "In both cases, feature 1 at the root node has to be used, however $w_{1,1}=0.5$ indicates that it is only being used half of the times.", "This solution is not a feasible pruning and fails to capture the cost of the pruning.", "Attempting to use an LP relaxation of this formulation fails to capture the desired behavior of the integer program.", "In the main paper we propose a better integer program formulation and show that solving the LP relaxation yields the optimal solution to the integer program." ], [ "Transformation to Network Matrices and Shortest Path Problems", "To illustrate the transformation to network matrix in Lemma 3.1, we provide the following illustration in Figure REF .", "Note in the main paper we have shown the example of the first tree.", "For simplicity we consider only one example being routed to nodes 4 and 11 respectively on the two trees.", "The equality constraints in (IP2) can be separated based on the trees and put in matrix form: ${& z_1 & z_2 & z_3 & z_4 & z_5 & w^{(1)}_{1,1} & w^{(1)}_{2,1}\\\\r_1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\r_2 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\\\r_3 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\\\r_4 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\\\r_5 & 1 & 0 & 0 & 0 & 0 & 1 & 0},$ for tree 1 and ${& z_6 & z_7 & z_8 & z_9 & z_{10} & z_{11} & z_{12} & w^{(2)}_{2,1} & w^{(2)}_{3,1}\\\\r_1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\r_2 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\r_3 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0\\\\r_4 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\\\r_5 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\\\r_6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0},$ for tree 2.", "Through row operations they can be turned into network matrices, where there is exactly two non-zeros in each column, a 1 and a $-1$ .", "${& z_1 & z_2 & z_3 & z_4 & z_5 & w^{(1)}_{1,1} & w^{(1)}_{2,1}\\\\-r_1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\\\r_1-r_2 & 0 & 1 & -1 & -1 & 0 & 0 & 0 \\\\r_2-r_3 & 0 & 0 & 0 & 1 & -1 & 0 & 0 \\\\r_3-r_4 & 0 & 0 & 0 & 0 & 1 & 0 & -1 \\\\r_4-r_5 & 0 & 0 & 1 & 0 & 0 & -1 & 1 \\\\r_5 & 1 & 0 & 0 & 0 & 0 & 1 & 0},$ for tree 1 and ${& z_6 & z_7 & z_8 & z_9 & z_{10} & z_{11} & z_{12} & w^{(2)}_{2,1} & w^{(2)}_{3,1}\\\\-r_1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\\\r_1-r_2 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\\\r_2-r_3 & 0 & 1 & 0 & 1 & -1 & -1 & 0 & 0 & 0\\\\r_3-r_4 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\\\r_4-r_5 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1\\\\r_5-r_6 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 1\\\\r_6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0}$ for tree 2.", "Note the above transformation to network matrices can always be done as long as the leaf nodes are arranged in a pre-order fashion.", "In the primal-dual algorithm, the inner minimization can be decomposed to shortest path problems corresponding to individual trees.", "Figure REF illustrates such a construction based on the network matrices shown above.", "The nodes in the graphs correspond to rows in the network matrices and the arcs correspond to the columns, which are the primal variables $z_h, w^{(t)}_{k,i}$ 's.", "There is a cost associated with each arc in the objective of the minimization problem.", "The task is to find a path from the first node (source) to the last node (sink) such that the sum of arc costs is minimized.", "Note each path from source to sink corresponds to a feasible pruning.", "For example, in (a) of Figure REF , consider the path of 1-2-5-6, the active arcs are $z_2, z_3$ and $w^{(1)}_{1,1}$ , Setting these variables to 1 and others to 0, we see that it corresponds to pruning Tree 1 at node 3 in Figure REF .", "(Note the nodes in Figure REF and Figure REF are not to be confused - they do not have a relation with each other. )", "Figure: Turning pruning to equivalent shortest path problems." ], [ "Proof of Theorem 3.2", "Denote the equality constraints of (IP) with index set $J_1$ .", "They can be divided into each tree.", "Each constraint matrix in $J_1$ associated with a tree can be turned into a network matrix according to Lemma 3.1.", "Stacking these matrices leads to a larger network matrix.", "Denote the $w^{(t)}_{k,i}\\le w_{k,i}$ constraints with index set $J_2$ .", "Consider the constraint matrix for $J_2$ .", "Each $w^{(t)}_{k,i}$ only appears once in $J_2$ , which means the column corresponding to $w^{(t)}_{k,i}$ has only one element equal to 1 and the rest equal to 0.", "If we arrange the constraints in $J_2$ such that for any given $k,i$ $w^{(t)}_{k,i}\\le w_{k,i}$ are put together for $t\\in [T]$ , the constraint matrix for $J_2$ has interval structure such that the non-zeros in each column appear consecutively.", "Finally, putting the network matrix from $J_1$ and the matrix from $J_2$ together.", "Assign $J_1$ and the odd rows of $J_2$ to the first partition $Q_1$ and assign the even rows of $J_2$ to the second partition $Q_2$ .", "Note the upper bound constraints on the variables can be ignored as this is an minimization problem.", "We conclude that the constraint matrix of (IP) is totally unimodular according to Theorem 2.7, Part 3 of [18] with partition $Q_1$ and $Q_2$ .", "By Proposition 2.1 and 2.2, Part 3 of [18] we can conclude the proof.", "every tree node/.style=minimum width=1em,draw,circle, blank/.style=draw=none, edge from parent/.style= draw, edge from parent path=() – (), level distance=1cm Figure: An ensemble of two decision trees with node numbers and associated feature in subscripts" ], [ "Additional Details of Experiments", "In this section we provide additional details of the experiment setup and explore how some parameter choices may affect BudgetPrune." ], [ "Additional details of datasets", "The MiniBooNE data set is a binary classification task to distinguish electron neutrinos from muon neutrinos.", "There are $45523/19510/65031$ examples in training/validation/test sets.", "Each example has 50 features, each with unit cost.", "The Forest data set contains cartographic variables to predict 7 forest cover types.", "There are $36603/15688/58101$ examples in training/validation/test sets.", "Each example has 54 features, each with unit cost.", "We use 1000 trees for GreedyMiser and search over learning rates in $[10^{-5}, 10^2]$ for MiniBooNE and Forest.", "The Yahoo and Scene15 datasets have actual feature acquisition costs in terms of CPU time.", "We use 3000 trees for GreedyMiser and search over learning rates in $[10^{-5}, 1]$ .", "We use the multi-class logistic loss for Scene15 and the squared loss for other datasets in GreedyMiser.", "For the Scene15 dataset, we use a diverse set of visual discriptors varying in computation time: GIST, spatial HOG, Local Binary Pattern, self-similarity, texton histogram, geometric texton, geometric color and 177 object detectors from the Object Bank [14].", "We treat each individual detector as an independent descriptor so we have 184 different visual descriptors in total.", "The acquisition costs of these visual descriptors range from 0.0374 to 9.2820.", "For each descriptor we train 15 one-vs-rest kernel SVMs and use the output (margins) as features.", "The best classifier based on individual descriptors achieves an accuracy of 77.8%.", "Note the features are grouped based on the visual descriptors.", "Once any feature corresponding to a visual descriptor is used for a test example, an acquisition cost of the visual descriptor is incurred and subsequent usage of features from the same group is free for the test example.", "Next, we perform additional experiments to evaluate BudgetPrune with different costs, input RFs." ], [ "Non-uniform cost on MiniBooNE", "We observe that CCP performs similarly to BudgetPrune on MiniBooNE when the costs are uniform in the case of entropy splitting criteria, indicating little gain from global optimization with respect to feature usage.", "We suspect that uniform feature costs work in favor of CCP because there's no loss in treating each feature equally.", "To confirm this intuition we assign the features non-uniform costs and re-run prunings on the same RF.", "We first normalize the data so that the data vectors corresponding to the features have the same $l$ -2 norm.", "We then train a linear SVM on it and obtain the weight vector corresponding to the learned hyperplane.", "We around the absolute values of the weights and make them the costs for the features.", "Intuitively the feature with higher weight tends to be more relevant for the classification task so we assign it a higher acquisition cost.", "The resulting costs lie in the range of $[1,40]$ and we normalize them so that the sum of all feature costs is 50 - the number of features.", "We plot BudgetPrune and CCP for uniform cost as well as the non-uniform cost described above in Figure REF .", "BudgetPrune still achieves similar performance as uniform cost while CCP performance drops significantly with non-uniform feature cost.", "This shows again the importance of taking into account feature costs in the pruning process.", "Figure: Comparing BudgetPrune and CCP with uniform and non-uniform feature cost on MiniBooNE dataset.", "BudgetPrune is robust when the feature cost is non-uniform." ], [ "Entropy Vs Pairs", "How does BudgetPrune depend on the splitting criteria used in the underlying random forest?", "On two data sets we build RFs using the popular entropy splitting criteria and the mini-max Pairs criteria used in [17] and the results are shown in Figure REF .", "We observe that entropy splitting criteria lead to RFs with higher accuracy while the Pairs criteria lead to RFs with lower cost.", "This is expected as using Pairs biases to more balanced splits and thus provably low cost [17].", "In (a) of Figure REF we observe that as more of the RF is pruned away BudgetPrune and CCP results for entropy and Pairs coincide.", "This suggests that the two criteria actually lead to similar tree structures in the initial tree-building process.", "However, as the trees are built deeper their structures diverge.", "Plot (b) in Figure REF shows that pruning based on the RFs from the Pairs criteria can achieve higher accuracy in the low cost region.", "But if high accuracy in the high cost region is desirable then the entropy criteria should be used.", "Figure: Comparisons of various pruning methods based on entropy and Pairs splitting criteria on MiniBooNE and Forest datasetsFigure: Comparing various pruning approaches on RF built with k=20 and k=120 on Scene15 dataset.", "The initial RF has higher accuracy and higher cost for k=20.", "GreedyPrune performs very well in k=20 but very poorly in k=120." ], [ "Size of random feature subset at each split", "At each split in RF building, it is possible to restrict the choice of splitting feature to be among a random subset of all features.", "Such restriction tends to further reduce correlation among trees and gain prediction accuracy.", "The drawback is that test examples tend to encounter a diverse set of features, increasing feature acquisition cost.", "For illustration purpose, we plot various pruning results on Scene15 dataset for feature subset sizes $k=20$ and $k=120$ in Figure REF .", "The initial RF has higher accuracy and higher cost for $k=20$ as expected.", "BudgetPrune achieves slightly better accuracy in $k=20$ than $k=120$ .", "Note also how GreedyPrune performance drops significantly for $k=120$ so it is not robust.", "In our main experiments $k$ is chosen on validation data to achieve highest accuracy for the initial RF." ] ]
1606.05060
[ [ "Universal criterion for designability of heteropolymers" ], [ "Abstract Proteins are an example of heteropolymers able to self-assemble in specific target structures.", "The self-assembly of designed artificial heteropolymers is still, to the best of our knowledge, not possible with control over the single chain self-assembling properties comparable to what natural proteins can achieve.", "What artificial heteropolymers lacks compared to bio-heteropolymers that grants the latter such a versatility?", "Is the geometry of the protein skeleton the only a particular choice to be designable?", "Here we introduce a general criteria to discriminate which polymer backbones can be designed to adopt a predetermined structure.", "With our approach we can explore different polymer backbones and different amino acids alphabets.", "By comparing the radial distribution functions of designable and not-designable scenarios we identify as designability criteria the presence of a particular peak in the radial distribution function that dominates over the random packing of the heteropolymer.", "We show that the peak is a universal feature of all designable heteropolymers, as it is dominating also the radial distribution function of natural proteins.", "Our finding can help in understanding the key features that make proteins a highly designable system.", "The criteria that we present can be applied to engineer new types of self-assembling modular polymers that will open new applications for polymer based material science." ], [ "Acknowledgment", "We would like to thank Achille Giacometti and Tatjana Škrbić for fruitful discussions.", "We acknowledge support from the Austrian Science Fund (FWF) project P23846-N16, the Mahlke-Obermann Stiftung and the European Union's Seventh Framework Programme for research, technological development and demonstration under grant agreement no 609431, VSC Research Center funded by the Austrian Federal Ministry of Science, Research and Economy (bmwfw).", "The computational results presented have been achieved using the Vienna Scientific Cluster (VSC)." ], [ "Methods", "A guiding theory for heteropolymer designability is given by the Random Energy Model (REM) [10], [11], [12].", "A clear review can be found in the seminal works of Pande et al.", "[12], where it is shown how the designability of a heteropolymer increases with the total number of possible bonds for each bead (valence) and decreases with the conformational entropy per bead.", "Hence, it is reasonable to assume that directionality (the patches) combined with isotropic interactions would increase designability, because the valence (i.e.", "the total number of possible bonds) remains constant, while the conformational entropy per bead decreases.", "In fact, the introduction of the patches decreases the entropy by favouring the system to populate more specific structures with the patches along particular directions.", "On the other hand, if the number of patches increases to much, the interactions become again close to isotropic and the designability decreases again.", "Hence, a model is needed that explicitly brings about the designability from a basic heteropolymer model by controlling the alphabet size and the conformational entropy per particle.", "Here we will use the patchy polymer model [17], [18] that has already been proven to be effective to refold artificial sequences into unique target structures for some specific cases of configurations of the patches.", "With patchy polymers we are able to change both the alphabet size of the isotropic interactions, the number of directional interactions incrementing them from the heteropolymer limit without patches, and finally change also the geometrical arrangement of the patches.", "The isotropic interaction energy $E_{AB}\\left(r\\right)$ between two different sub-units of types A and B is represented as a simple-square-well like shape (Fig.", "REF ) $E_{AB}\\left(r\\right)= {\\left\\lbrace \\begin{array}{ll}\\epsilon _{AB}\\left[1-\\dfrac{1}{1.0 + \\textrm {e}^{2.5 \\left(r_{max} - r\\right)}}\\right] &\\mbox{if } r > R_{bead} \\\\\\infty , =&\\mbox{if } r < R_{bead}\\end{array}\\right.", "}$ where $r$ is the distance of the centres of the beads and $R_{bead}$ is the hard core radius, which is the same for each bead.", "$\\epsilon _{AB}$ is a different pre-factor for every different pair of monomers.", "The cut-off distance $r_{max}=6 R_{bead}$ is the distance at which $E_{AB}=\\epsilon _{AB}/2$ and was derived with a trial and error approach on coarse-grained proteins in the Caterpillar protein model [14], [26].", "As directional interaction between the patches we employ the potential derived by Irbäck et al.", "[27], commonly used to model hydrogen bonds.", "It is represented by a 10-12 Lennard-Jones type potential multiplied by a factor containing the angles between the patches and the bead radius (Fig.", "REF ), so that the energy is minimum if the patches face each others (when they are opposite to each others the radial part of the potential is $\\sim 0$ ) $E_{p}=s\\,\\epsilon _{p}\\left(\\cos {\\theta _1}\\cos {\\theta _2}\\right)^\\nu \\left[5\\left(\\frac{\\sigma }{R}\\right)^{12}-6\\left(\\frac{\\sigma }{R}\\right)^{10}\\right].$ Here $R$ is the distance of the patches as in Fig.", "REF (right inset), $\\epsilon _{p}=3.1~k_BT$ and $\\nu =2$  [27] while we set $\\sigma =R_{bead}$ .", "The scaling factor $s$ is chosen to not over favour the patch contribution over the isotropic one.", "If its value is too large all sequences form regular structures that depend solely on the symmetries of the patch arrangements on the beads.", "On the other hand, if it is too small all sequences fail to self-assemble and collapse into random glassy three dimensional structures.", "A good number was found to be 4 in Ref. [18].", "The neighbour beads along the chain are bonded via a harmonic spring potential.", "In order to find if the system is designable or not, we identify for each configuration of the patches and an alphabet size at least one pattern (sequence) that has a global free energy minimum into a given structure.", "Hence, we have to find such a pattern (via DESIGN MC simulation [18]) and verify if it is capable of folding (via the FOLDING MC simulation [18]).", "In all Monte Carlo simulations we enhance the sampling with the Virtual Move Parallel Tempering algorithm [28], performing each simulation at 16 different temperatures in the set [3, 2.5, 2.0, 1.6, 1.4, 1.2, 1.0, 0.9, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.4].", "To increase the chances to find such a pattern, we first perform a SEEK MC simulation [18], in order to find potentially designable target structures (see Supplementary Figure REF ).", "The target structures are represented for each case in Fig.", "REF .", "Then we perform the DESIGN, where we explore different sequences while keeping the target structure frozen.", "Here we choose the optimized sequence in the global minimum of the free energy, which corresponds in our method to a low potential energy and a high heterogeneity of the sequence [18].", "Starting from a fully stretched structure we then perform a FOLDING Monte Carlo simulation, in order to study the self-assembling properties of this pattern.", "Here we explore the conformational space keeping the pattern fixed in the designed sequence.", "We project the FOLDING free energy onto an order parameter, namely the root mean square displacement of the inter-particle distance ($DRMSD$ ) between the target structure and each sampled structure: $DRMSD = \\frac{1}{N} \\sqrt{\\sum _{ij} (|\\Delta \\vec{r_{ij}}| - |\\Delta \\vec{r_{ij}}^T| )^2 }$ where $\\vec{r_{ij}}$ is the distance between the sphere $i$ and $j$ while $\\vec{r_{ij}}^T$ is the same distance calculated over the target structure, and $N$ is the chain length (50 in our case).", "The $DRMSD$ has already been shown to be a proper order parameter to study the folding process [14].", "$DRMSD=0$ corresponds uniquely to the target structure.", "The closer the global minimum is to $DRMSD=0$ , the smaller the corresponding ensemble of structures.", "Thus, if the free energy landscape has a clear global minimum close to $DRMSD=0$ , we can identify at least one pattern that drives the system to fold into a unique target structure: the configuration is designable.", "However, not all global minima correspond to folded conformations.", "Indeed, the different cases can be discriminated only by increasing the temperature and pushing the system to unfold.", "If the chain at low temperature is folded, then there will be an intermediate temperature at which the system explore conformations with higher values of $DRMSD$ corresponding to disordered globular structures (molten globule).", "The progressive unfolding results in either the appearance of a second minimum (Supplementary Figure REF ) or the spread of the width of the initial global minimum.", "On the other hand, for patterns that do not have a folded state the equilibrium configuration is always a molten globule and the temperature increase does not significantly change the position of the minimum, which corresponds to values of $DRMSD\\simeq 1:1.5$ in grey in Fig.", "REF .", "If the system does not reach the folded state, the SDF trials fail, then associated to the tested configuration of the patches there might be only a handful of structures that are designable, if they exists at all.", "A heteropolymer with few and hard-to-find designable structures is anyhow not a good candidate for potential applications and is categorized as not-designable.", "The radial distribution functions of the patchy polymers in Fig.", "REF a $g(r)$ has been averaged on the 20 most designable structures for every case.", "For proteins in Fig.", "REF b the $g(R)$ is calculated for the C$_{\\alpha }$ in three characteristics examples out of 20 analysed natural proteins, each averaged on multiple equilibrium PDB (Protein Data Bank).", "The other analysed proteins are not shown but give similar $g(R)$ .", "The normalization has been performed on the same ideal gas with average density, to make the $g(r)$ of proteins with different length comparable.", "All the $g(r)$ have been calculated by neglecting the contribution of the beads (or amino acids) directly connected along the chain, in order to ignore their trivial contribution to the first neighbours peak." ], [ "SUPPLEMENTARY MATERIAL", "We use Monte Carlo simulations to investigate patchy polymers composed by 50 patchy particles decorated with $n$ patches, where $n=1,2$ for freely jointed chains and $n=0,1,3,4,6,10$ for freely rotating chains.", "The patches are equispaced on the equator and placed on the vertices of a equilateral triangle and a tetrahedron for the $n=3$ and $n=4$ cases, respectively.", "For $n>4$ the patches are placed on the surface in the most symmetric way by using the following numerical procedure: $n$ patches are randomly placed on a sphere, their positions given by the set of vectors $\\lbrace \\vec{r_1},\\ldots ,\\vec{r_n}\\rbrace $ we assign a fictitious energy to the system, defined as $U = \\frac{1}{2} \\sum _{i\\ne j} |\\vec{r_i} - \\vec{r_j}|^{-1}$ we minimise $U$ by attempting to move a randomly chosen patch, accepting the move if the total energy of the system consequently decreases.", "Formally, this can be regarded as a Monte Carlo (MC) simulation performed at temperature zero.", "We iterate this procedure until convergence of $U$ .", "For completeness, we note that the above method produces a patch distribution which is independent of the definition of distance between two patches, being it the Euclidean distance or a spherical distance, for all the values of $n$ considered here.", "In addition, the patch distribution makes sure that two particles cannot be involved in more than one bond.", "Figure: Free energy landscape sampled by SEEK for one patch and alphabet 3 in the freely rotating chain model.", "The free energy is in function of the total number of contacts between the spheres (distance below 6R bead 6~R_{bead}) and the total number of contacts between the patches (distance below 1.25R bead 1.25~R_{bead} and angles θ 1 \\theta _1 and θ 2 \\theta _2 >0.8π> 0.8~\\pi ).", "The target structure is chosen in the global minimum of this landscape.", "Following the above definition of patches contacts, in the target structure the 80%80\\% of the patches are maximally oriented between each others.", "However, we observe by looking closer at the structure, that all the patches are interacting.", "Nevertheless, the close packing peak in the radial distribution function dominates on the directional interaction peak.", "Hence, the close packing is not suppressed even when all the directional interactions are fulfilled.", "Thus, increasing the relative strength of the directional interactions will not make the first peak disappear.Figure: Radial distribution functions (g(r)g(r)) of proteins.", "The g(r)g(r) are pair distribution functions between the C α _{\\alpha } in two characteristics examples out of 20 analysed natural proteins, each averaged on multiple equilibrium PDB (Protein Data Bank).", "The peaks correspond to the typical distances imposed by the secondary structure .", "In α\\alpha -helix they are 5.0Å(C 0-2 _{0-2} in figure), 5.4Å(C 0-3 _{0-3} in figure).", "The peak outside from the grey area corresponds to the fourth neighbour along the chain.", "In anti-parallel β\\beta -sheets the peaks a and b correspond to the C α _{\\alpha } facing each others.", "The peak outside from the grey area correspond to the second neighbour along the strand.Figure: FOLDING free energy landscape as a function of the distance root mean square displacement (DRMSDDRMSD) for an example of non-designable system (left) and designable system (right).", "By increasing the temperature, the position of the minimum of the non-designable system does not change significantly, while in the designable system we observe a significant shift of the minimum due to the disordered globular structures (molten globule).Figure: FOLDING free energy landscapes.", "The free energy is plotted as a function of the distance root mean square displacement (DRMSDDRMSD) for freely rotating chain (left) and freely jointed chain (right), for different patches numbers and alphabet sizes at temperature 0.40.4." ] ]
1606.05253
[ [ "Three-dimensional Critical Dirac semimetal in KMgBi" ], [ "Abstract We predicted that AMgBi (A=K,Rb Cs), which have the same lattice structures as the 111 family of iron-based superconductors (Na/LiFeAs), are symmetry-protected Dirac semimetals located near the boundary of type-I and type-II Dirac semimetal phases.", "Doping Rb or Cs into KMgBi can drive the transition between the two phases.", "The materials can also be turned into Weyl semimetals and topological insulators by explicitly or spontaneously breaking time-reversal symmetry and C$_4$ lattice symmetry respectively." ], [ "Three-dimensional Topological Critical Dirac semimetal in $A$ MgBi ($A$ =K, Rb, Cs) Congcong Lethese authors contributed equally to this paper Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Shengshan Qinthese authors contributed equally to this paper Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Xianxin Wu Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaInstitute for Theoretical Physics and Astrophysics, Julius-Maximilians University of W¡§urzburg, Am Hubland, D-97074 W¡§urzburg, Germany Xia Dai Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Peiyuan Fu Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Chen Fang Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Jiangping Hu [email protected] Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaCollaborative Innovation Center of Quantum Matter, Beijing, China We predicted, using first principles calculation, that $A$ MgBi ($A$ =K, Rb, Cs) are symmetry-protected topological semimetals near the boundary of type-I and type-II Dirac semimetal phases, dubbed topological critical Dirac semimetals.", "Doping Rb or Cs into KMgBi can drive the transition between the two phases.", "An effective theory is developed to describe the bands near the Fermi energy, by which we calculate the surface Fermi arcs and the Landau levels throughout the transition.", "We predict the key features of critical Dirac semimetals that can be observed in photoemission, quantum oscillation and transport measurements.", "73.43.-f, 73.20.-r, 71.20.-b Topological semimetals (TSs) are semimetals whose Fermi surfaces carry nontrivial topological numbers.", "These quantum numbers lead to a series of exotic effects such as the existence of Fermi arcs on the surface [1], [2] and the chiral anomaly [3], [4] in the bulk transport.", "Crystal symmetries afford us a large variety of topological semimetals including Weyl semimetals (WS) [5], [6], [1], [7], [8], [9], [10], [11], [12], Dirac semimetals (DS) [13], [14], [15] and nodal line semimetals [16], [17].", "For Weyl/Dirac semimetals, a further distinction has been made between the type-I and the type-II classes [18], where the Fermi surfaces (at ideal half-filling) are point-like and pocket-like, respectively.", "The physical consequences of both types have been studied [19], [20].", "For its unique feature and potential application, TSs have drawn great attention in the field of topological materials.", "In recent years, great progress has been made both theoretically and experimentally.", "Na$_3$ Bi and Cd$_3$ As$_2$ have been predicted to be three-dimensional (3D) linear DSs theoretically [14], [15] and have been verified by angle-resolved photoemission spectroscopy (ARPES) measurements [21], [22].", "In transport experiments, Cd$_3$ As$_2$ and several other DSs exhibit strong linear magnetoresistance[23], [24], which is also a strong evidence for DSs.", "In this Letter, we propose, by using first principles calculation, a family of materials, $A$ MgBi, as topological semimetals that lie in between the type-I and the type-II Dirac semimetals, where $A$ is an alkaline metal ($A$ =K, Rb, Cs).", "While KMgBi, which has been synthesized[25], is a type-I Dirac semimetal, both RbMgBi and CsMgBi, are both type-II Dirac semimetals.", "Doping Rb and Cs into KMgBi hence drives a transition from type-I to type-II Dirac semimetals.", "Hence we call these compounds topological critical Dirac semimetals, which can help us understand this topological phase transition.", "To further study the physical observables in these compounds, we develop an effective model that captures the key features near the Fermi energy throughout the topological transition.", "Using the model, we computed the surface states with Fermi arcs and also the Landau levels in the presence of magnetic field.", "These results can be directly observed in ARPES, Scanning Tunneling Microscope (STM) and transport measurements.", "Method and Crystal Structure Our calculations are performed using density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) code [26], [27], [28].", "The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional and the projector-augmented-wave (PAW) approach are used.", "Throughout the work, the cutoff energy is set to be 500 eV for expanding the wave functions into plane-wave basis.", "In the calculation, the BZ is sampled in the k space within Monkhorst-Pack scheme[29].", "On the basis of the equilibrium structure, the k mesh used is $10\\times 10\\times 6$ .", "We relax the lattice constants and internal atomic positions with GGA, where the plane wave cutoff energy is 600 eV.", "Forces are minimized to less than 0.01 eV/Åin the relaxation.", "The crystal structure of KMgBi [25]with the space group $P4/nmm$ is similar to the 111 family of iron-based superconductors[30], [31], [32].", "The Mg$_2$ Bi$_2$ layers possess an anti-PbO-type atom arrangement, consisting of a square lattice sheet of Mg coordinated by Bi above and below the plane to form face sharing MgBi$_4$ tetrahedra.", "Details about the optimized and experimental structural parameters are summarized in the supplementary materials.", "We find that the optimized structural parameters are within 2% from the corresponding experimental data.", "The bond angle of MgBi$_4$ tetrahedra is very close to that of the perfect tetrahedron.", "To further test the stability of optimized structural parameters, we calculate dispersions by using PHONOPY code[33], [34](see Supplementary materials).", "No imaginary frequencies are observed throughout the whole Brillouin zone in phonon dispersions, confirming its dynamically structural stability.", "Therefore, we adopt the optimized structural parameters in the following calculations.", "Electronic structure The band structure and density of states (DOS) of KMgBi with GGA optimized structural parameters are displayed in Fig.REF .", "The paramagnetic state is insulating with an energy gap of 362.8 meV, as shown in Fig.REF (a).", "As spin orbital coupling (SOC) is large in Bi atoms, when SOC is included, the band inversion occurs around $\\Gamma $ point as shown in Fig.REF (c).", "Near the Fermi level, the valence and conduction bands are mainly attributed to the Bi-6p orbitals.", "The band inversion happens between Bi-6p$_{x,y}$ and Bi-6p$_z$ orbitals, which is different from the known DS Na$_3$ Bi[14] where the inverted band structure is formed by the Na-3s and Bi-6p$_{x,y}$ orbitals near the Fermi level.", "To further confirm the topological property in KMgBi, due to the presence of inversion symmetry, we can adopt the parity check method proposed by Fu and Kane[35] to calculate the topological invariance.", "The parities of the eigenstates at $\\Gamma $ and Z points near the Fermi level are displayed in Fig.REF (c).", "The odd parity states are attributed to the Bi-6p$_{x,y}$ orbitals, while the even parity states are contributed by Bi-6p$_z$ orbital.", "Therefore, the band structure is topologically nontrivial according to the calculation of parity product of occupied states at the time reversal invariant momenta.", "Figure: (color online) (a) and (b) are band structure and DOS of KMgBi without SOC.", "(c) and (d) are band structure and DOS of KMgBi with SOC.", "The orbital characters of bands are represented by different colors.", "The parities of the eigenstates and the irreducible representations they belong to at Γ\\Gamma point near the Fermi level are represented by ±\\pm and Γ i \\Gamma _i respectively.", "The inset is Band structure of KMgBi in the k z =k z c k_z=k_z^{c} plane.When SOC is turned on, the inverted bands of KMgBi near the Fermi level do not reopen a gap.", "Thus, it is a Dirac semimetal with two Dirac nodes located at (0, 0, $\\pm $ 0.1345$\\times \\frac{\\pi }{c}$ ) along the $\\Gamma $ -Z line exactly at the Fermi level, and the Fermi surfaces of KMgBi are composed of two isolated Fermi points.", "To further confirm the band crossings, the irreducible representations of the bands along the $\\Gamma $ -Z line are calculated, shown in Fig.REF (c).", "The two crossing bands belong to $\\Gamma _6$ and $\\Gamma _7$ irreducible representations respectively, indicating that the crossing points cannot open a gap.", "Furthermore, due to the presence of both time reversal and inversion symmetries in KMgBi, each of the crossing points has a four-fold degeneracy and is robustly protected by these symmetries.", "The surface states of KMgBi can be obtained by calculating the surface Green function of the semi-infinite system using an iterative method[36], [37].", "Fig.REF (b) shows the edge states on the [100] surface, and the corresponding Fermi surface and in-plane spin texture are shown in Fig.REF (a).", "Similar to the case of Na$_3$ Bi[14], on the [100] surface, the surface states are topologically nontrivial in the area $-k_z^c<k_z<k_z^c$ while it is topologically trivial in the area $|k_z|>k_z^c$ .", "The corresponding Fermi surface on the [100] edge is a closed Fermi pocket composed of two Fermi arcs, and the two Fermi arcs connect each other at the projection points of the two Dirac points.", "Due to the spin-momentum lock, the spin texture on the the Fermi pocket is helical and it vanishes at the connecting points.", "Now we address the lattice parameter effect on KMgBi.", "In principle, the lattice parameters can be changed by external pressure or by internal pressure created by chemical substitution.", "We have summarized the influence of the structural parameters on the gap size between the conduction band bottom and valence band top at $\\Gamma $ point in the presence (absence) of SOC while the point group is kept unchanged in the supplementary materials.", "We find that, without SOC the gap at the $\\Gamma $ point decreases with the increasing of structural parameters.", "This can be easily understood because the energy gap between the bonding and anti-bonding states of Bi-6p$_{x,y,z}$ orbitals decreases with the increasing of lattice constants.", "In experiments, the lattice constants can also be tuned by doping Rb or Cs into the KMgBi, namely K$_{1-x}$ R$_x$ MgBi(R=Rb, Cs).", "In the supplementary materials, we have listed the optimized structural parameters and the energy gap between conduction band bottom and valence band top at $\\Gamma $ point for KMgBi, K$_{0.5}$ Rb$_{0.5}$ MgBi and RbMgBi.", "It shows that with the increasing of Rb concentration both the lattice constants and the inverted gap increase.", "Figure: (color online) (a) The spin texture for the [100] surface states.", "(b) The projected surface states of KMgBi for [100] surface.In addition, for KMgBi, the dispersion of the $\\Gamma _7$ band is very weak along the $\\Gamma $ -Z line.", "The energy of the $\\Gamma _7$ band at $\\Gamma $ point is only about $3.8$ meV higher than the one at Z point, as shown in Fig.REF (c).", "Thus, KMgBi is located at the edge of type-I and type-II DS phases and it is a intriguing type-I DS.", "A small perturbation may drive KMgBi to be a type-II DS.", "To study the transition between type-I DS phase and type-II DS phase of KMgBi, we replace K with Rb atom.", "The band structures of RbMgBi shown in Fig.REF (a) are very similar to those of KMgBi, except that the $\\Gamma _7$ band of RbMgBi has a different dispersion along the $\\Gamma $ -Z line.", "The $\\Gamma _7$ band for RbMgBi at $\\Gamma $ point is about 52 meV lower than the one at Z point and the inset in Fig.REF shows band structure of $\\Gamma $ -Z line.", "The highly tilted conical dispersion around Dirac point clearly indicates that RbMgBi is a type-II DS.", "Effective Hamiltonian In order to get the effective Hamiltonian near the $\\Gamma $ point where the band inversion occurs, we use the theory of invariants method in a similar way as for the Bi$_2$ Se$_3$ family of materials[38], [39].", "From the band structure, the states around the $\\Gamma $ point are mainly attributed to Bi-$p_{z}$ and Bi-$p_{x,y}$ orbitals, so we use these orbitals to construct the basis.", "Since Bi and Bi$^{\\prime }$ atoms are related by the inversion symmetry, it is convenient to combine these orbitals to form the eigenstates of the inversion symmetry, which are given by $|P^{\\mp }_{\\alpha }\\rangle =\\frac{1}{\\sqrt{2}}(|Bi_{\\alpha }\\rangle \\pm |Bi^{\\prime }_{\\alpha }\\rangle )$ where the superscript denotes the parity and $\\alpha =p_{x,y,z}$ .", "Figure: (color online) (a) Band structure of RbMgBi under the optimized structural parameters.", "Inset shows band structure of Γ\\Gamma -Z line.", "(b) Fermi surface of RbMgBi with fermi level tuned to -0.0375 eV.", "The color represents the k z k_z value.Taking SOC into account in the atomic picture, we can write the atomic Bi-6p states with SOC as $|P^{\\pm }_{\\frac{3}{2}}, \\pm \\frac{3}{2}\\rangle $ , $|P^{\\pm }_{\\frac{3}{2}}, \\pm \\frac{1}{2}\\rangle $ , $|P^{\\pm }_{\\frac{1}{2}}, \\pm \\frac{1}{2}\\rangle $ , where the subscript indicates the total angular momentum $J$ .", "Due to the crystal field effect, $|P^{\\pm }_{\\frac{3}{2}}, \\pm \\frac{1}{2}\\rangle $ and $|P^{\\pm }_{\\frac{1}{2}}, \\pm \\frac{1}{2}\\rangle $ are mixed to form new eigenstates $|P^{\\pm },\\pm \\frac{1}{2}\\rangle $ that define the energy spectra of the low energy model.", "Taking $|P^{+}, \\frac{1}{2}\\rangle $ , $|P^{-}, \\frac{3}{2}\\rangle $ , $|P^{+} , -\\frac{1}{2}\\rangle $ , $|P^{-}, -\\frac{3}{2}\\rangle $ as the basis in $k\\cdot {p}$ theory, we can construct the effective Hamiltonian around the $\\Gamma $ point as $H_{0}(\\mathbf {k}) = \\epsilon _0(\\mathbf {k})+ \\left(\\begin{array}{cccc}M(\\mathbf {k}) & ak_{-} & 0 & 0 \\\\ak_{+} & -M(\\mathbf {k}) & 0 & 0 \\\\0 & 0 & M(\\mathbf {k}) & -ak_{+} \\\\0& 0 & -ak_{-} & -M(\\mathbf {k}) \\\\\\end{array}\\right),$ where $\\epsilon _0(\\mathbf {k})=C_0+C_1k^2_{z}+C_{2}(k^2_{x}+k^2_{y})$ , $k_{\\pm }=k_x{\\pm }ik_y$ and $M(\\mathbf {k})=M_0-M_{1}k^2_{z}-M_{2}(k^2_{x}+k^2_{y})$ .", "$M_0M_2<0$ corresponds to the topologically trivial regime while $M_0M_2>0$ corresponds to the band inversion regime, namely, the topologically nontrivial regime.", "The energy spectrum of the $k\\cdot {p}$ Hamiltonian is $E(\\mathbf {k})=\\epsilon _0(\\mathbf {k})\\pm \\sqrt{M(\\mathbf {k})^2+a^2k_{+}k_{-}},$ where each band here is two fold degenerate.", "Obviously, the dispersion results in two band crossing points (0, 0, $\\pm k^{c}_{z}$ ) along $\\Gamma $ -Z line with $k^{c}_{z}=\\sqrt{\\frac{M_0}{M_1}}$ .", "By fitting the bands of the effective model with those of the DFT calculation around $\\Gamma $ point, the parameters in the effective model, for KMgBi, are given by $C_0$ =-0.0794 eV, $C_1$ =37.7590 eV Å$^2$ , $C_2$ =16.7234 eV Å$^2$ , $M_0$ =-0.0797 eV, $M_1$ =-39.7258 eV Å$^2$ , $M_2$ =-31.6298 eV Å$^2$ , and $a$ =4.0074 eV Å.", "For the sake of $|C_1|< |M_1|$ , KMgBi is a type-I Dirac semimetal[18].", "What's more, considering the fact $\\frac{|C_1|}{|M_1|}=0.9505$ , KMgBi is quite near the phase transition point $\\frac{|C_1|}{|M_1|}=1.0$ between the two Dirac semimetal phases.", "Therefore, KMgBi is located at the boundary between type-I and type-II DS phases.", "Similar to KMgBi, We also extract the $k\\cdot {p}$ model for RbMgBi around the $\\Gamma $ point.", "The fitting parameters are obtained as follows: $C_0$ =-0.1534 eV, $C_1$ =37.9605 eV Å$^2$ , $C_2$ =21.6337 eV Å$^2$ , $M_0$ =-0.1122 eV, $M_1$ =-36.9687 eV Å$^2$ , $M_2$ =-30.0223 eV Å$^2$ and $a$ =1.1008 eV Å.", "As $|C_1|> |M_1|$ , the model describes a type-II DS [18] (see Supplementary materials).", "As the type-II DS is characterized by intriguing Fermi surfaces, we also obtain the Fermi surfaces of RbMgBi,shown in Fig.REF (b).", "Fig.REF (b) shows the Fermi surfaces at the energy of Dirac point -0.0375 eV.", "From Fig.REF (b), we can find that the Fermi surfaces of RbMgBi consist of an electron pocket and a hole pocket, which connect each other just at the two Dirac points in the bulk Brillouin zone.", "This is the signature of the type-II DS.", "For the partially doped material K$_{0.5}$ Rb$_{0.5}$ MgBi, we find that it has already turned out to be a type-II DS.", "In this material, the $\\Gamma _7$ band at $\\Gamma $ point is about $35.3$ meV lower than that at Z point.", "Here, we also want to emphasize that: for $A$ MgBi ($A$ =K, Rb, Cs), no matter for the type-I or the type-II DS phase, there is no other Fermi surface but for that contributed by the Dirac points; the Dirac points here are rather near the Fermi energy.", "Therefore, we expect that the Dirac cone and the surface states can be well measured in ARPES experiments.", "Landau levels In the presence of magnetic field, TSs have many exotic transport properties[23], [24], [40], [41], [42], [43], which are usually closely related to the Landau levels contributed by the Weyl/Dirac points.", "Hence we investigate the Landau levels for the topological critical DS $A$ MgBi ($A$ =K,Rb, Cs).", "We first consider a magnetic field applied in the $xy$ -plane.", "In earlier studies[44], [45], [46], [47], [48], it was shown that the Landau levels in type-II Dirac/Weyl semimetals collapse due to open quasi-classical orbitals for field in the $xy$ -plane, while in type-I Dirac/Weyl semimetals the Landau levels are well-defined.", "Therefore, we conclude that the Landau levels of KMgBi are well-defined but close to collapse, and in RbMgBi they will collapse due to the type-II nature of the Dirac point.", "When the magnetic field is applied along $z$ -axis, however, the Landau levels always exist for both cases.Fig.REF (a) and (c) show the Landau levels contributed by the Dirac point at $(0,0,k_z^c)$ for KMgBi and RbMgBi, respectively.", "Obviously, each Dirac point will contribute to two chiral modes corresponding to the two degenerate Weyl points.", "For both cases, there is one nearly nondispersive chiral mode, which is a reflection of the criticality of the bulk Dirac point.", "The flat chiral mode leads to a large DOS near the Fermi level, shown in Fig.REF (b), which can be detected by STM measurement.", "However, there are key differences between the two cases.", "The two chiral modes of KMgBi have different slopes—one has a positive slope and the other a negative slope, while for RbMgBi, the slope of both chiral modes is positive, shown in Fig.REF (a) and (c).", "This leads to the result that at large $k_z$ , the $n\\ne 0$ Landau levels can always cross the Fermi level for RbMgBi, which can not happen for KMgBi.", "Hence, RbMgBi has a much larger DOS at the Fermi level than KMgBi.", "For the fact that the $n=0$ Landau levels are independent of the magnetic field, the DOS at the Fermi level of KMgBi will not vary with the magnetic field.", "Nevertheless, since the energy gap between different Landau levels will increase with the increasing of magnetic field, the DOS at the Fermi level of RbMgBi will decrease rapidly, shown in Fig.REF (d).", "These results can also be clearly verified by the STM measurement and the difference between KMgBi and RbMgBi can also be verified by the magneto-optical measurements[44].", "Figure: (color online) (a) and (c) are the Landau levels dispersion of KMgBi and RbMgBi with a magnetic field B=2 mT along the z axis, respectively.", "(b) The density of states of KMgBi(The density of states of RbMgBi similar to that of KMgBi).", "(d) The relation between density of states at the fermi level and magnetic field, the red (blue) line stand for KMgBi (RbMgBi).Discussion Above results suggest that AMgBi can be a very special family of materials to explore semimetal physics that are absent in the previously known DSs Na$_3$ Bi and Cd$_3$ As$_2$ .", "Because of their special location near the boundary of type-I and type-II Dirac semimetal phases, which is characterized by flat bands, one interesting physics is the effect of electron-electron correlation in these materials.", "The effect can be strongly enhanced due to the effective dimension reduction in the electronic structure near the boundary, which may result in intriguing many-body emergent physics.", "The other is the possibility of the development of superconductivity in these materials by carrier doping.", "As the materials have identical lattice structure of the 111 iron-based superconductors, it is very likely that the carrier doping can induce superconductivity.", "The specific studies on these issues will be reported in the future.", "In summary, we predict that the AMgBi are symmetry protected Dirac semimetals located near the boundary of type-I and type-II Dirac semimetal phases.", "The transition between the two types can be driven by doping Rb or Cs into KMgBi.", "One of the features of the topological critical Dirac semimetal is that there is always a nondispersive chiral mode.", "These results can be well verified by STM and ARPES measurement for its clean Fermi surfaces.", "The work is supported by \"973\" program (Grant No.", "2010CB922904 and No.", "2012CV821400), as well as national science foundation of China (Grant No.", "NSFC-1190024, 11175248 and 11104339).", "Note added: During the revision of the present work, we notice that an experimental work on the physical properties of KMgBi [49] has been carried out." ] ]
1606.05042
[ [ "Multiple imputation of missing covariate values in multilevel models\n with random slopes: A cautionary note" ], [ "Abstract Multiple imputation (MI) has become one of the main procedures used to treat missing data, but the guidelines from the methodological literature are not easily transferred to multilevel research.", "For models including random slopes, proper MI can be difficult, especially when the covariate values are partially missing.", "In the present article, we discuss applications of MI in multilevel random-coefficient models, theoretical challenges posed by slope variation, and the current limitations of standard MI software.", "Our findings from three simulation studies suggest that (a) MI is able to recover most parameters, but is currently not well suited to capture slope variation entirely when covariate values are missing; (b) MI offers reasonable estimates for most parameters, even in smaller samples or when its assumptions are not met; and (c) listwise deletion can be an alternative worth considering when preserving the slope variance is particularly important." ], [ "Study 1", "The first study compared the performance of LD, conditional MI, and multivariate MI when the only outcome had missing values.", "For conditional MI, both the analyst's model $f(Y|X,\\theta _X)$ and the imputation model $g(Y|X,\\omega _X)$ were RC models where $\\omega _X$ took on the same role as $\\theta _X$ but denoted a distinct set of model parameters.", "These models were equally complex and fit the clustered structure of the data.", "Multivariate MI was set up as described earlier, and LD was applied by restricting the analysis to complete cases only.", "Two standardized normal variables $X$ and $Y$ were simulated.", "Both varied at two levels as indicated by their intraclass correlations (ICCs) $\\rho _X$ and $\\rho _Y$ , respectively.", "The covariate $X$ was simulated from its within- and between-group portions $X^W \\sim N(0,1-\\rho _X)$ and $X^B \\sim N(0,\\rho _X)$ , respectively.", "Then $Y$ was simulated conditionally on $X$ according to Equation REF with fixed effects $\\beta = (\\beta _0, \\beta _1)$ , where $\\beta _0$ was zero due to standardization.", "The covariance matrix of random effects was $\\Psi = \\left( {\\begin{matrix} \\psi _{11}^2 & 0 \\\\ 0 & \\psi _{22}^2 \\end{matrix}} \\right)$ .", "Thus, the intercepts and slopes were uncorrelated.", "The Level 1 residual variance was $\\Sigma = \\sigma ^2$ .", "The variables in this study were parametrized by their ICC rather than their actual variance components.", "Given the ICC and a slope variance $\\psi _{22}^2$ , the other variance components followed [32] as $\\begin{aligned}\\sigma ^2 &= (1-\\rho _Y) - \\beta _1^2(1-\\rho _X) - \\psi _{22}^2(1-\\rho _X) \\\\\\psi _{11}^2 &= \\rho _Y - \\beta _1^2\\rho _X - \\psi _{22}^2\\rho _X \\text{.", "}\\end{aligned}$ Missing values on $Y$ were imposed using a linear model for the latent response variable $R^{*}$ .", "Values in $Y$ were set to be missing if their respective $R^{*}>0$ according to $\\begin{aligned}R^{*} &= \\alpha + \\lambda _1 X + \\lambda _2 Y + \\varepsilon _{R^{*}} \\text{,}\\end{aligned}$ where $\\alpha $ is a value of the standard normal distribution according to a missing data probability (e.g., $\\alpha = -0.67$ for 25% missing data), and $\\lambda _1$ and $\\lambda _2$ are used to control the missing data mechanism.", "The residuals were distributed normally with mean zero and variance $\\sigma _{R^{*}}^2 = 1 - \\lambda _1^2 - \\lambda _2^2 - 2\\;\\!", "\\lambda _1 \\lambda _2 \\text{Cov}(X,Y) \\text{.", "}$ Table REF provides an overview of the conditions included in all three studies.", "The two ICCs were set to be equal, that is, $\\rho _X=\\rho _Y=\\rho $ .", "In order for $Y$ to be MCAR, we set $\\lambda _1=\\lambda _2=0$ , and for MAR, we set $\\lambda _1=0.5$ and $\\lambda _2=0$ .", "For $Y$ to be MNAR, we chose equal values for $\\lambda _1$ and $\\lambda _2$ such that the error variance in $R^{*}$ was the same as in the MAR condition.", "Hence, with $\\text{Cov}(X,Y)=\\beta _1=0.5$ , we had $\\lambda _1=\\lambda _2=\\sqrt{0.25/3} \\approx 0.289$ .", "The conditions were chosen to mimic typical data in psychology and the behavioral sciences [1], [20], [22].", "Table: Simulation Designs of Study 1, Study 2, and Study 3In summary, each simulated setting was defined by the number of groups (G), the number of individuals within each group (N), the ICC of $X$ and $Y$ ($\\rho $ ), the fixed slope ($\\beta _1$ ), the slope variance ($\\psi _{22}^2$ ), the proportion of missing data, and the missing data mechanism (including the missing data effects $\\lambda _1$ and $\\lambda _2$ ).", "Each setting was replicated 1,000 times." ], [ "Imputation and data analysis", "The R package pan was used to impute missing values [30].", "We let pan perform 10,000 burn-in cycles before drawing one imputed dataset for every 200 cycles, leading to $M=50$ imputed datasets and 20,000 cycles in total [15].", "Diagnostic plots regarding the convergence behavior of pan's Gibbs sampler are presented in Supplement B in the online supplemental materials.", "Least-informative inverse-Wishart priors for $\\Sigma $ and $\\Psi $ were chosen with $\\Sigma \\sim W^{-1}(\\mathbf {I}_1,1)$ and $\\Psi \\sim W^{-1}(\\mathbf {I}_2,2)$ for conditional MI, and $\\Sigma \\sim W^{-1}(\\mathbf {I}_2,2)$ and $\\Psi \\sim W^{-1}(\\mathbf {I}_2,2)$ for multivariate MI, where $\\mathbf {I}_n$ denotes the identity matrix of size $n$ .", "We fit the analyst's model to each imputed dataset using the R package lme4 [6].", "The final parameter estimates were obtained according to Rubin's (1987) rules.", "We note that choosing least-informative priors implies a prior expectation of variances of $.50$ , which might induce bias into small variance components.", "However, because non-informative priors are often desirable for MI, the same priors were used throughout the three studies.", "Possible alternative specifications of the prior distribution will be reviewed in the General Discussion.", "The computer code for running conditional and multivariate MI, with least-informative or alternative priors, is provided in Supplement C of the supplemental online materials.", "Bias and the root-mean-square error (RMSE) were calculated for each condition and each parameter.", "The bias is the mean difference between a parameter estimate $\\hat{\\theta }$ and its true value $\\theta $ , and is crucial for statistical reasoning in general.", "The RMSE is the root of the mean squared difference between $\\hat{\\theta }$ and $\\theta $ and represents both accuracy and precision (i.e., the variability) of an estimator.", "Thus, it is an important measure of practical utility.", "Due to the large simulation design, only the most important findings will be reported.", "Furthermore, only results for 25% missing data will be reported as higher rates did not yield interesting results.", "The complete results for Study 1 are given in Supplement D in the online supplemental materials.", "Table REF shows the results of the first study for samples that featured small variance components (i.e., $\\text{ICC}=.05$ , $\\psi _{22}^2=.01$ ) for MCAR and MAR data in smaller (N = 10, G = 50) and larger samples (N = 30, G = 150).", "Notable values for bias and RMSE are presented in bold.", "Bias presented in bold is at least $\\pm 5\\%$ off the true value for fixed effects, and $\\pm 30\\%$ off for variance components.", "For parameters whose true value was zero, a threshold of $\\pm .05$ was used.", "For each simulated condition, the highest RMSE is printed in bold as long as it was significantly larger than that found for the complete datasets (at least twice as large).", "As can be seen in Table REF , neither LD nor MI produced strongly biased results, but bias emerged under specific conditions for both MI procedures.", "The multivariate imputation model underestimated the slope variance by as much as 50% unless it was essentially zero (i.e., $.01$ ), but overestimated the intercept variance.", "Conditional MI (using the RC model) overestimated both the intercept and slope variance (Table REF , top panel).", "A sufficient sample size reduced bias to acceptable proportions even for the smallest variance components (Table REF , bottom panel).", "For larger values of the ICC (i.e., $.15$ and $.25$ ) and the slope variance (i.e., $.05$ , $.10$ , and $.20$ ), this bias was reduced to essentially zero (see Supplement D).", "Using LD, the intercept and slope variance were sometimes biased when samples were not sufficiently large.", "When data were MNAR, all approaches yielded biased results (see Supplement D).", "Listwise deletion has previously been shown to provide essentially unbiased estimates when the outcome is ignorably missing [19].", "Surprisingly, the imputation models overestimated small random effects variances in small samples.", "We argue that this is a side effect of the least-informative prior which expects variances to be larger, and that bias may be reduced to zero when the prior is set on an appropriate scale (see general discussion).", "From the data at hand, both LD and conditional MI can be recommended for univariate missing data on $Y$ provided that the sample is sufficiently large or the prior is set on an appropriate scale.", "Care should be taken when small variance components are to be estimated, as overly non-informative priors may inflate them.", "The multivariate model is useful if the slope variance is close to zero.", "The second study examined the performance of MI and LD with missing values on the covariate $X$ .", "The analyst's model was again the RC model $f(Y|X,\\theta _X)$ , whereas conditional MI was carried out using the reversed RC model $g(X|Y,\\omega _Y)$ .", "The two models fit the clustered structure of the data but differed in the way the slope variability was attributed.", "Multivariate MI and LD were administered as before." ], [ "Simulation and Methods", "The same procedures as applied in Study 1 were used to simulate data and impose missing values on the covariate $X$ , whereas MAR was now dependent on the outcome $Y$ .", "Imputations were created by pan using the least-informative priors as chosen in Study 1.", "The analyst's model was fit using lme4, and the bias and RMSE were calculated for each parameter in each setting." ], [ "Results", "The results of Study 2 are reported in full in Supplement D. Here, we will report the most important findings.", "Table REF provides a brief overview of the results for samples that featured small variance components.", "Estimating the fixed effects of the RC model proved to be more accurate and efficient using MI.", "Specific difficulties emerged again for small variance components, that is, when samples featured small ICCs or little slope variation.", "In contrast to when data were missing on $Y$ , however, estimates of larger slope variances were not necessarily unbiased." ], [ "Fixed effects", "As shown in Table REF , LD led to biased estimates for the fixed effects unless the data were MCAR (see Supplement D).", "Bias for the fixed intercept varied between $-.098$ and $-.161$ with MAR data and between $-.055$ and $-.101$ with MNAR data.", "The fixed slope was underestimated by approximately 6-10% when the data were not MCAR.", "Results from MI were essentially unbiased, but the reversed model exhibited a small downward bias across conditions.", "The RMSE suggested that estimates obtained from MI were at least as efficient as those obtained by LD across conditions, and more efficient when data were not MCAR.", "Figure: Bias in estimating the fixed slope for univariate missing data on XX (Study 2) for different MD mechanisms, MD techniques, and different amounts of slope variance.", "ψ 22 2 \\psi _{22}^2 = true slope variance; MCAR = missing completely at random; MAR = missing at random; MNAR = missing not at random; CD = complete data; LD = listwise deletion; MV = multivariate imputation; RC = conditional imputation using the reversed RC model.Interestingly, bias from both LD and conditional MI was dependent on the amount of slope variation that was present in the dataset.", "As slope variation increased, bias became weaker with LD, and stronger with conditional MI.", "This result is illustrated in Figure REF for small samples (N = 10, G = 150), moderate ICCs (i.e., .15), and MAR data.", "Nonetheless, estimates obtained from MI were more accurate and efficient across all conditions." ], [ "Variance and covariance of random effects", "Conditional and multivariate MI underestimated the intercept variance when the ICCs were small but provided unbiased estimates otherwise.", "Listwise deletion followed the same pattern for MCAR data, but otherwise underestimated the intercept variance.", "This bias was strongest in the MAR condition, weaker with MNAR data, and increased as the ICCs grew larger.", "Figure REF (top row) illustrates this finding for different levels of the ICC.", "Results for the slope variance differed from Study 1.", "Although conditional MI again overestimated small amounts of slope variation, this bias was much weaker and practically disappeared in larger samples (see Table REF ).", "Moderate slope variation could be estimated almost without bias.", "In contrast to Study 1, however, large and very large slope variances were not estimated correctly by conditional MI but increasingly suffered from a downward bias.", "Listwise deletion provided practically unbiased estimates of the slope variance if the sample size was sufficiently large.", "The positive bias for conditional MI was also present with MNAR data, whereas the negative bias was smaller.", "Figure REF (bottom row) illustrates these findings for different levels of slope variation.", "Figure: Bias in estimating the intercept (top row) and slope variance (bottom row) for univariate missing data on XX (Study 2) for different values of the ICC or slope variance, respectively, and different MD mechanisms and MD techniques.", "ICC = true intraclass correlation; ψ 22 2 \\psi _{22}^2 = true slope variance; MCAR = missing completely at random; MAR = missing at random; MNAR = missing not at random; CD = complete data; LD = listwise deletion; MV = multivariate imputation; RC = conditional imputation using the reversed RC model.According to the RMSE, the intercept variance could occasionally be estimated more efficiently using MI, whereas the slope variance could be estimated more accurately using LD.", "However, these differences were usually very small.", "Supplement D even suggests that conditional MI occasionally estimated the slope variance more efficiently in small samples." ], [ "Other parameters", "The covariance between random intercepts and slopes was well recovered across all conditions.", "The Level 1 residual variance was overestimated using MI, where conditional MI was less biased, but it was underestimated by LD when data were not MCAR.", "For higher amounts of slope variation, the bias associated with LD became smaller, whereas the bias grew for MI.", "These patterns were observed with MAR and MNAR data, but the bias was relatively small.", "Regarding most parameters of the analyst's model, better estimates could be obtained using the reversed MI procedure, especially when the covariate $X$ was not MCAR.", "This was true for the fixed regression coefficients but also applied to the intercept variance and even transferred to MNAR data.", "However, reversed MI seemed to provide unstable estimates of the slope variance, which could be positively or negatively biased.", "The positive bias for small slope variances became essentially zero as the samples grew larger.", "For larger slope variances, the bias did not approach zero (as in Study 1) but turned negative regardless of sample size.", "The negative bias was, however, rather small and could be viewed as negligible considering that it only occurred for large slope variances, which are rarely found in empirical studies.", "Furthermore, the overall precision of the estimates, as indicated by the RMSE, was often comparable to LD because the data were handled more efficiently using MI.", "The reversed model seemed to share many but not all of the desirable properties of the regular RC model.", "The multivariate imputation model is applicable if little slope variation is present in the data, but it will suppress even moderate amounts of slope variation and inflate the Level 1 residual variance.", "Estimates of the fixed slope obtained from multivariate MI were even less biased and more efficient than those from the reversed MI procedure.", "Listwise deletion offered little benefit as most of its parameter estimates were biased unless the data were MCAR.", "However, LD provided surprisingly accurate results for the slope variance.", "Small variance components were again positively biased but less so than in the previous study.", "We will return to this point in the General Discussion.", "The final study examined the performance of MI and LD with multivariate missing data.", "The analyst's model was once again the RC model $f(Y|X,\\theta _X)$ , but only the multivariate imputation model $g(X,Y|\\omega _0)$ could be applied.", "This imputation model ignores slope variability, but may provide reasonable results for the remaining parameters of the analyst's model." ], [ "Simulation and Methods", "The same procedures that were used in the previous studies could be used for most tasks.", "Because the pattern of missing data was no longer univariate, the missing data model had to be adjusted.", "We excluded unit-nonresponse from our considerations; thus, every participant was expected to have at least one observation on either $X$ or $Y$ .", "This allowed us to implement the same mechanisms as described before (i.e., MCAR, MAR, MNAR) for both $X$ and $Y$ .", "For each case, a coin toss decided whether $X$ or $Y$ could be missing (i.e., each was equally likely).", "The actual missing values were then imposed on either $X$ or $Y$ with the probability that was given in the simulation design.", "Thus the amount of missing values in each dataset was the same in all three studies." ], [ "Results and Discussion", "The results of the third study provided little further insight into the performance of LD and multivariate MI because the bias and RMSE were usually halfway between those reported in Studies 1 and 2.", "Results for small variance components are presented in Table REF .", "The complete results are available in Supplement D. Multivariate MI provided approximately unbiased estimates of all parameters as long as the slope variance was close to zero and the values were either MCAR or MAR.", "The slope variance was underestimated by as much as 40%, especially in larger samples where more values were imputed under false assumptions.", "When the data were MNAR, multivariate MI underestimated the fixed regression coefficient, but the bias was relatively small compared with the true values.", "Estimates obtained from LD were approximately unbiased when data were MCAR.", "When data were MAR or MNAR, the fixed effects were biased downward and were estimated less efficiently than with multivariate MI, where higher values for the ICC and slope variance reduced bias with LD (see Supplement D).", "The results of the third study suggest that MI is necessary for proper estimation of the fixed regression coefficients.", "Unfortunately, pan's multivariate imputation model could not preserve the slope variance.", "If the slope variance was small and the number of missing values was not very high, then the bias was relatively small in absolute size.", "Limiting the analysis to complete cases only distorted the parameter estimates, but provided reasonable estimates of the slope variance." ], [ "General Discussion", "We investigated the performance of conditional and multivariate MI for univariate and multivariate patterns of missing data.", "Both conditional MI and LD provided unbiased estimates if only the outcome was missing.", "Care should be taken if covariates are partially unobserved.", "Imputing the covariate in a reversed manner accounted for, but also misspecified the slope variation.", "Only vague estimates could be obtained for the slope variance, but bias was not extreme, and the remaining estimates exhibited either no or less bias than what would have been obtained by deleting cases.", "The multivariate imputation model rarely induced any bias but strongly underestimated the slope variance.", "Thus, it is appropriate only if the true slope variance is close to zero and not too many values are unobserved.", "We recommend that LD be avoided when covariate data are missing unless the data are strictly MCAR.", "As is true for all computer simulations, our study was limited in several ways.", "The missing data mechanisms were based on linear models and may behave quite differently in nature.", "Other implementations are possible, and results may vary especially for MAR and MNAR data [2], [10].", "We focused on descriptive measures of approximate performance but ignored statistical inference.", "Testing for slope variation [18] as well as Type-I and Type-II error rates associated with LD and MI should be a subject of future research.", "Rather than estimating the slope variance, researchers often wish to explain it using predictor variables at Level 2 [20], [1].", "Cross-level interaction effects might be relatively easy to recover even if the slope variance is not.", "Interestingly, small variance components were positively biased across the three studies.", "We argue that this is due to the standard least-informative prior, which induces bias into small variance components.", "Ad hoc procedures might combine the specific advantages of LD and MI and lead to less biased and more stable estimates.", "For example, choosing $D^{-1} = 2 \\cdot \\hat{\\Psi }_{LD}$ as the scale matrix of the inverse Wishart prior for the covariance matrix of random effects, where $\\hat{\\Psi }_{LD}$ is an estimate of this covariance matrix obtained from LD, would loosely center the prior distribution around appropriate values.", "The computer code for this specification is provided in Supplement C of the supplemental online materials.", "We conducted a small simulation to examine whether the bias for the intercept and slope variance could be reduced by rescaling the prior distribution in this manner.", "The simulation featured small samples, univariate MAR data on either $X$ or $Y$ , small values for the ICCs, as well as small and very large values for the slope variance.", "Estimates of small variance components that utilized the adjusted prior did not exhibit any more bias than LD did and were often more efficient.", "The positive bias reported in Studies 1 and 2 could therefore be viewed as an artifact of specifying the least-informative prior.", "The negative bias for large slope variances in Study 2, however, could not be improved in this manner.", "Using least squares or maximum likelihood estimation might further strengthen this approach.", "The methodological literature offers alternatives to pan for multilevel MI.", "It has been suggested that multilevel data be imputed using dummy variables in random intercept models but that imputations should be conducted separately for each group if random slopes are involved [13], [14].", "However, [4] found that the first approach leads to biased results, and unreported simulation results indicate that very large samples are needed to treat even small amounts of missing data with the second approach.", "Alternative MI procedures include fully conditional specification using chained equations [37].", "These procedures might lead to better results, but may face similar problems with respect to the slope variance.", "However, recent developments in the context of substantive model compatible MI have offered promising results for interaction effects and nonlinear terms among covariates that have missing values [38], [5].", "Extending this approach to multilevel MI [11], [12] and applying it to random slope models should be the subject of future research.", "Adaptations of the pan model have been proposed by [31] and [39].", "The latter approach specifies a joint model that allows the within-group covariance matrix to vary across groups, and has been recently discussed by [7].", "However, it is currently not available in standard software and has yet to be evaluated in a systematic manner.", "In general, we believe that MI is a flexible and powerful tool that can be used to treat missing data in multilevel research.", "More research should be conducted to generalize the current formulations of MI and to evaluate recent developments as well as sensible ad hoc solutions to missing data in multilevel models with random slopes." ] ]
1606.05204
[ [ "Natural Inflation on the brane with a TeV-scale gravity: Parameter\n constraints after Planck 2015" ], [ "Abstract In the present work we have studied Natural Inflation in the framework of the Randall-Sundrum II brane model (RS-II) in the light of the latest Planck results.", "Adopting the Randall-Sundrum fine-tuning, the model is characterized by 3 parameters in total, namely the 5-dimensional Planck mass $M_5$ and the two mass scales of the inflaton potential $f$ and $\\Lambda$.", "We show in the $n_s-r$ plane the theoretical predictions of the model together with the allowed contour plots, and we conclude that the model is viable.", "By using the Planck results only it is possible to determine the two mass scales of the inflaton potential in terms of $M_5$, which remains undetermined.", "However, there are several good theoretical reasons to consider a higher-dimensional Planck mass of the order of $10 TeV$, which is compatible with primordial nucleosynthesis.", "If we insist on considering a $M_5$ of this order of magnitude all parameters are known and a sub-Planckian excursion of the inflaton scalar field is achieved." ], [ "Introduction", "The inflationary universe has become one of the central paradigms in modern cosmology.", "This is due to the fact that many long-standing problems of the Big Bang model, such as the horizon, flatness, homogeneity and monopole problems, find a natural explanation in the framework of the inflationary universe [1], [2], [3], [4], [5], [6].", "However, the essential feature of inflation is that it generates a mechanism to explain the Large-Scale Structure (LSS) of the universe [7], [8], [9], [10], [11] and provides a causal interpretation of the origin of the anisotropies observed in the Cosmic Microwave Background (CMB) radiation[12], [13], [14], [15], [16], [17], since primordial density perturbations may be produced from quantum fluctuations during the inflationary era.", "Despite its success, a theory of inflation is still missing, all we have is a large collection of inflationary models (see e.g.", "[18], and for a classification of inflationary models see[19]).", "The inflaton potential cannot be derived from a fundamental theory in a unique way.", "Moreover, the potential must be specially designed to be flat, and this is the fine-tuning problem of inflation.", "Natural Inflation (NI) with a pseudo-Nambu Goldstone boson (pNGB) as the inflaton[20] is provided in certain particle physics model[21].", "The scalar potential, which is flat due to shift symmetries, has the form $V(\\phi )=\\Lambda ^4\\left[1-\\cos \\left(\\frac{\\phi }{f}\\right)\\right].$ and it is characterized by two mass scales $f$ and $\\Lambda $ with $f \\gg \\Lambda $ .", "It is assumed that a global symmetry is spontaneously broken at some scale $f$ , with a soft explicit symmetry breaking at a lower scale $\\Lambda $ .", "Natural inflation has been already studied in standard cosmology based on Einstein's General Relativity[22].", "Upon comparison to the current cosmological and astronomical observations, specially those related with the CMB temperature anisotropies, it is possible to constrain several inflation models.", "Particularly, the constraints in the $n_s-r$ plane give us the predictions of a number of representative inflationary potentials.", "Recently, the Planck collaboration has published new data of enhanced precision of the CMB anisotropies [23].", "Here, the Planck full mission data has improved the upper bound on the tensor-to-scalar ratio $r_{0.002} < 0.11$ ($95\\%$ CL) which is similar to obtained from [17] , in which $r < 0.12$ ($95\\%$ CL).", "In particular, Natural Inflation is consistent with current data [17], [23] for trans-Planckian values of the symmetry breaking scale $f$ , for which it may be expect the low-energy effective theory, on which (REF ) is based, to break down [24].", "Regarding the last data of Planck, this model is consistent for $\\log _{10}( f /M_p) > 0.84$ , where $M_p$ is the reduced Planck mass $M_p\\equiv 1/\\sqrt{8\\pi G_N}=2\\times 10^8$ GeV.", "Nevertheless, several mechanisms which yield the potential (REF ) with super-Planckian values for the symmetry breaking scale $f$ consistent with a low energy description, have been proposed recently [25], [26], [27], [28], [29], [30], [31], [32], [33], [34].", "On the other hand, it would be challenging to study natural inflation in non-standard cosmologies.", "Considering non-standard cosmologies is motivated by at least two facts, namely i) deviations from the standard Friedmann equation arise in higher-dimensional theories of gravity, and ii) there is no observational test of the Friedmann equation before the primordial big-bang nucleosynthesis (BBN) epoch.", "A well-studied example of a novel higher-dimensional theory is brane-world models.", "Brane models are inspired from M/superstring theory and although they are not yet derivable from the fundamental theory, at least they contain the basic ingredients, such as extra dimensions, higher-dimensional objects (branes), higher-curvature corrections to gravity (Gauss-Bonnet) etc.", "Since superstring theory claims to give us a fundamental description of nature it is important to study what kind of cosmology it predicts.", "Regarding the realization of Natural Inflation in non-standard cosmologies, some works have been put forward in the literature so far [35], [36], [37], achieving sub-Planckian values for the symmetry breaking scale $f$ , being consistent with the data available at that time.", "The main goal of the present work is to study the realization of NI in the high-energy regime of the RS-II brane model, in the light of the recent Planck results.", "Later on we will show that our results are modified significantly compared to[35] using the Planck results.", "By comparing the theoretical predictions of the model together with the allowed contour plots, and we conclude that the model is viable.", "Using the latest Planck results only the inflaton potential mass scales $f, \\Lambda $ are given in terms of the five-dimensional Planck mass, which remains unconstraint though.", "However, we insist on considering a higher-dimensional Planck mass of the order of, say, $10 TeV$ , since there are several good theoretical reasons for that.", "We organize our work as follows: After this introduction, in the next section we summarize the basics of the brane model as well as the dynamics of inflation.", "In the third section we analyze natural inflation in the framework of RS-II model and present our results, and in the last section we finish with our conclusions.", "We choose units so that $c=\\hbar =1$ .", "In the brane-world scenario the main idea is that our four-dimensional world and the standard model of particle physics are confined to live on a 3-dimensional brane, while gravity lives in the higher-dimensional bulk.", "Since the higher-dimensional Plank mass $M_D$ is the fundamental mass scale instead of the usual four-dimensional Planck mass $M_4$ , the brane concept has been used to address the hierarchy problem of particle physics, initially in the simple framework of a flat (4+n) spacetime with 4 large dimensions and n small compact dimensions[38], and later refined by Randall and Sundrum[39], [40].", "For an introduction to brane cosmology see e.g.[41].", "In the RS-II model [40], the four-dimensional Einstein equations may be written as [42] $^{(4)}G_{\\mu \\nu }=-\\Lambda _4g_{\\mu \\nu }+\\frac{8\\pi }{M_4^2} \\tau _{\\mu \\nu }+\\left(\\frac{8\\pi }{M_5^3}\\right)^2\\pi _{\\mu \\nu }-E_{\\mu \\nu },$ where $\\Lambda _4$ is the four-dimensional cosmological constant, $\\tau _{\\mu \\nu }$ is the energy-momentum tensor of matter on the brane, $\\pi _{\\mu \\nu }=(1/12) \\tau \\tau _{\\mu \\nu }+(1/8) g_{\\mu \\nu } \\tau _{\\alpha \\beta } \\tau ^{\\alpha \\beta }-(1/4) \\tau _{\\mu \\alpha } \\tau _\\nu ^\\alpha -(1/24) \\tau ^2 g_{\\mu \\nu }$ , and $E_{\\mu \\nu }=C_{\\beta \\rho \\sigma }^\\alpha n_\\alpha n^\\rho g_\\mu ^\\beta g_\\nu ^\\sigma $ is the projection of the five-dimensional Weyl tensor $C_{\\alpha \\beta \\rho \\sigma }$ on the brane, where $n^\\alpha $ is the unit vector normal to the brane.", "$E_{\\mu \\nu }$ and $\\pi _{\\mu \\nu }$ encode the information about the bulk.", "The four-dimensional quantities can be computed in terms of the five-dimensional ones as follows[43] $M_4=\\sqrt{\\frac{3}{4\\pi }}\\left(\\frac{M_5^2}{\\sqrt{\\lambda }}\\right)M_5$ $\\Lambda _4 = \\frac{4 \\pi }{M_5^3} \\left( \\Lambda _5+\\frac{4 \\pi \\lambda ^2}{3M_5^3} \\right) $ The Friedmann equation for a flat FRW background is given by [44] $H^2=\\frac{\\Lambda _4}{3}+\\frac{8\\pi }{3M_4^2}\\rho \\left(1+\\frac{\\rho }{2\\lambda }\\right)+\\frac{\\mathcal {E}}{a^4}.$ where $\\mathcal {E}$ is an integration constant coming from $E_{\\mu \\nu }$ .", "The term $\\frac{\\mathcal {E}}{a^4}$ is known as the dark radiation, since it decays in the same way as radiation.", "However, during inflation this term will be rapidly diluted, and we can neglect it.", "The five-dimensional Planck mass is constraint by the standard bib-bang nucleosynthesis to be $M_5 \\ge 10 TeV$[45].", "In the following, we will take the four-dimensional cosmological constant $\\Lambda _4$ to be zero, or in other words we adopt the RS fine tuning $\\Lambda _5=-4 \\pi \\lambda ^2/(3 M_5^3)$ so that model can explain the cosmic acceleration without cosmological constant, and neglecting the term $\\frac{\\mathcal {E}}{a^4}$ the Friedmann equation (REF ) becomes $H^2=\\frac{8\\pi }{3M_4^2}\\rho \\left(1+\\frac{\\rho }{2\\lambda }\\right),$ which becomes the basis of our study on brane-world inflation." ], [ "Inflationary dynamics", "At low energies, i.e., when $\\rho \\ll \\lambda $ , inflation in the brane-world scenario behaves in exactly the same way as standard inflation.", "But at higher energies we would expect the dynamics of inflation to be changed.", "We consider slow-roll inflation driven by a scalar field $\\phi $ , for which the energy density $\\rho $ and the pressure $P$ are given by $\\rho =\\frac{\\dot{\\phi }^2}{2}+V(\\phi )$ and $P=\\frac{\\dot{\\phi }^2}{2}-V(\\phi )$ , respectively, where $V(\\phi )$ is the scalar potential.", "We assume that the scalar field is confined to the brane, so the four-dimensional Klein-Gordon equation still holds $\\ddot{\\phi }+3H\\dot{\\phi }+V^{\\prime }=0,$ where prime indicates derivative with respect to $\\phi $ , and dot a derivative with respect to cosmic time.", "We can use the slow-roll approximation to write (REF ) and (REF ) as $H^2\\simeq \\frac{8\\pi }{3M_4^2}V\\left(1+\\frac{V}{2\\lambda }\\right),$ and $3H\\dot{\\phi }\\simeq -V^{\\prime }.$ In this way, using these two equations, it is possible to write the slow-roll parameters on the brane as[43] $\\epsilon _V & \\equiv & \\frac{M_4^2}{16 \\pi }\\left(\\frac{V^{\\prime }}{V}\\right)^2\\frac{1+V/\\lambda }{\\left(1+V/2\\lambda \\right)^2},\\\\\\eta _V & \\equiv & \\frac{M_4^2}{8 \\pi }\\frac{V^{\\prime \\prime }}{V}\\frac{1}{1+V/2\\lambda }.$ Slow-roll inflation implies that $\\epsilon _V\\ll 1$ and $\\left|\\eta _V\\right|\\ll 1$ , as in standard cosmology.", "These reduce to standard slow-roll parameters at the the low-energy limit $V\\ll \\lambda $ .", "On the other hand, in the high-energy limit, i.e., $V\\gg \\lambda $ , these expressions become $\\epsilon _V &\\simeq & \\frac{ M_4^2\\lambda }{4 \\pi }\\frac{V^{\\prime \\,2}}{V^3},\\\\\\eta _V &\\simeq & \\frac{ M_4^2\\lambda }{4 \\pi }\\frac{V^{\\prime \\prime }}{V^2}.$ The deviations from standard slow-roll inflation can be seen in the high-energy as both the parameters are suppressed by a factor $V/\\lambda $ .", "The number of $e$ -folds in the slow-roll approximation, using (REF ) and (REF ), yields $N \\simeq -\\frac{8\\pi }{M_4^2}\\int _{\\phi _{*}}^{\\phi _{end}}\\frac{V}{V^{\\prime }}\\left(1+\\frac{V}{2\\lambda }\\right)\\, d\\phi ,$ where $\\phi _{*}$ and $\\phi _{end}$ are the values of the scalar field when the cosmological scales cross the Hubble-radius and at the end of inflation, respectively.", "As it can be seen, the number of $e$ -folds is increased due to an extra term of $V/\\lambda $ .", "This implies a more amount of inflation, between these two values of the field, compared to standard inflation." ], [ "Perturbations", "In the following, we will give a review of cosmological perturbations in brane-world inflation.", "We consider the gauge invariant quantity $\\zeta =-\\psi -H\\frac{\\delta \\rho }{\\dot{\\rho }}$ .", "Here, $\\zeta $ is defined on slices of uniform density and reduces to the curvature perturbation at super-horizon scales.", "A fundamental feature of $\\zeta $ is that it is nearly constant on super-horizon scales[46], and in fact this property does not depend on the gravitational field equations[47].", "Therefore, for the spatially flat gauge, we have $\\zeta =H\\frac{\\delta \\phi }{\\dot{\\phi }}$ , where $\\left|\\delta \\phi \\right|=H/2\\pi $ .", "In this way, using the slow-roll approximation, the spectra of scalar perturbations is given by[43] $\\mathcal {P}_{\\mathcal {R}}=\\frac{H^2}{\\dot{\\phi }^2}\\left(\\frac{H}{2\\pi }\\right)^2 \\simeq \\frac{128\\pi }{3 M_4^6}\\frac{V^3}{V^{\\prime 2}}\\left(1+\\frac{V}{2\\lambda }\\right)^3.$ On the other hand, the tensor perturbations are more involved since the gravitons can propagate in the bulk.", "The amplitude of tensor perturbations is given by[43] $\\mathcal {P}_{g}=\\frac{64\\pi }{M^2_4}\\left(\\frac{H}{2\\pi }\\right)^2F^2(x),$ where $F(x) &=& \\left[\\sqrt{1+x^2}-x^2\\ln \\left(\\frac{1}{x}+\\sqrt{1+\\frac{1}{x^2}}\\,\\right)\\,\\right]^2\\nonumber \\\\&=& \\left[\\sqrt{1+x^2}-x^2 \\sinh ^{-1}\\left(\\frac{1}{x}\\right)\\right]^{-1/2},$ and $x$ is given by $x=HM_4\\sqrt{\\frac{3}{4\\pi \\lambda }}.$ The expressions for the spectra are, as always, to be evaluated at Hubble radius crossing $k = aH$ .", "As expected, in the the low-energy limit the expressions for the scalar and tensor spectra become the same as those derived without considering the brane effects.", "However, in the high-energy limit, these expressions become $\\mathcal {P}_{\\mathcal {R}} &\\simeq & \\frac{16\\pi }{3 M_4^6 \\lambda ^3}\\frac{V^6}{V^{\\prime \\,2}},\\\\\\mathcal {P}_{g} &\\simeq & \\frac{32 V^3}{M_4^4 \\lambda ^2}.$ The scale dependence of the scalar power spectra is determined by the scalar spectral index, which under the slow-roll approximation, obeys the usual relation $n_s &=&1+\\frac{d \\ln \\mathcal {P}_{\\mathcal {R}}}{d \\ln k}\\nonumber \\\\n_s &\\simeq & 1-6\\epsilon _V+2\\eta _V.", "$ The amplitude of tensor perturbations can be parameterized by the tensor-to-scalar ratio, defined to be[48] $r\\equiv \\frac{\\mathcal {P}_{g}}{\\mathcal {P}_{\\mathcal {R}}},$ which implies that in the low-energy limit this expression becomes $r\\simeq 16 \\epsilon _V$ , where $\\epsilon _V$ is the standard slow-roll parameter, whereas in the high-energy limit we have that $r \\simeq 24 \\epsilon _V$ , with $\\epsilon _V$ corresponding to Eq.", "(REF ).", "As we have seen, at late times the brane-world cosmology is identical to the standard one.", "During the early universe, particularly during inflation, there may be changes to the perturbations predicted by the standard cosmology, if the energy density is sufficiently high compared with the brane tension.", "In the following, we will obtain the predictions for the natural inflationary model in the brane-world scenario in the high-energy limit, and to try to ascertain whether these predictions are compatible with current observational constraints." ], [ "Dynamics of inflation", "Natural inflation in the Randall-Sundrum brane-world scenario is characterized by 3 parameters in total, namely the 5-dimensional Planck mass and the mass scales of the inflaton potential (REF ), $\\Lambda $ and $f$ , respectively.", "In the high-energy limit, the slow-roll parameters $\\epsilon _V$ and $\\eta _V$ , using Eqs.", "(REF ) and () are given by $\\epsilon _V =\\alpha \\frac{(1+\\cos (y))}{(1-\\cos (y))^2},\\\\\\eta _V =\\alpha \\frac{\\cos (y)}{(1-\\cos (y))^2},$ where $y \\equiv \\phi /f$ and $\\alpha $ is a dimensionless parameter defined as $\\alpha \\equiv \\frac{M_4^2 \\lambda }{4\\pi f^2 \\Lambda ^4}.$ For this model, the condition for the end of inflation is found to be $\\epsilon (y_{end})=1$ , leading to $\\cos (y_{end})=\\cos \\left(\\frac{\\phi _{end}}{f}\\right)=\\frac{1}{2}\\left(2+\\alpha -\\sqrt{\\alpha }\\sqrt{8+\\alpha }\\right).$ The number of inflationary $e$ -folds that occur between the values of the scalar field when a given perturbation scale leaves the Hubble-radius and at the end of inflation, can be computed from (REF ) and the high-energy limit of (REF ), yielding $N=\\frac{1}{\\alpha }\\left[\\cos (y_{*})-\\cos {(y_{end})}-2\\ln \\left(\\frac{1+\\cos (y_{*})}{1+\\cos (y_{end})}\\right)\\right],$ with $\\cos (y_{end})$ given by (REF ).", "Solving Eq.", "(REF ) for $\\cos (y_{*})$ , after replacing Eq.", "(REF ), we may obtain the value of the scalar field at the time of Hubble-radius crossing, given by $\\cos (y_{*})=\\cos \\left( \\frac{\\phi _{*}}{f}\\right)=-1-2\\, W\\left[z(N,\\alpha )\\right],$ where $z(N,\\alpha )\\equiv -\\frac{\\sqrt{e^{-1-\\frac{\\sqrt{\\alpha }}{2}(2N\\sqrt{\\alpha }+\\sqrt{\\alpha }-\\sqrt{\\alpha +8})}\\left(\\alpha ^2+8 \\alpha +8-\\alpha ^{3/2}\\sqrt{\\alpha +8}-4\\sqrt{\\alpha }\\sqrt{\\alpha +8}\\right)}}{2\\sqrt{2 e}},$ and $W$ denotes the Lambert $W$ function [49]." ], [ "Cosmological perturbations", "Regarding the cosmological perturbations, the amplitude of scalar perturbations, using Eqs.", "(REF ) and (REF ), is found to be $\\mathcal {P}_{\\mathcal {R}}=\\frac{1}{12 \\pi ^2 \\alpha ^3}\\gamma ^4\\frac{(1-\\cos (y))^5}{(1+\\cos (y))},$ where $\\gamma \\equiv \\frac{\\Lambda }{f}$ is the ratio between both mass scales of the inflaton potential (REF ), $\\Lambda $ and $f$ .", "The scalar spectral index, using Eqs.", "(REF ), (REF ), and (), becomes $n_s=1-2\\alpha \\frac{(3+2\\cos (y))}{(1-\\cos (y))^2}.$ Finally, the tensor-to-scalar ratio can be obtained from the high-energy limit of Eq.", "(REF ), yielding $r=24\\alpha \\frac{(1+\\cos (y))}{(1-\\cos (y))^2}.$ After evaluating these inflationary observables at the value of the scalar field when a given perturbation scale leaves the Hubble-radius, given by (REF ), we may compare the theoretical predictions of our model with the observational data in order to obtain constraints on the parameters that characterize it.", "The amplitude of the scalar perturbations, the scalar spectral index, and the tensor-to-scalar ratio, evaluated at the Hubble-radius crossing $k=aH$ , become $\\mathcal {P}_{\\mathcal {R}} &=& \\frac{4}{3 \\pi \\alpha ^3}\\gamma ^4\\frac{(1+W\\left[z(N,\\alpha )\\right])^5}{\\left(-W\\left[z(N,\\alpha )\\right]\\right)},\\\\n_s &=& 1-\\frac{\\alpha }{2}\\frac{\\left(-W\\left[z(N,\\alpha )\\right]\\right)}{(1+W\\left[z(N,\\alpha )\\right])^2} ,\\\\r &=& 12\\alpha \\frac{\\left(-W\\left[z(N,\\alpha )\\right]\\right)}{(1+W\\left[z(N,\\alpha )\\right])^2}.$ Figure: Plot of the tensor-to-scalar ratio rr versus the scalar spectral index n s n_s for natural inflation in the high-energy limit of braneworld scenario.", "Here, we have considered thetwo-dimensional marginalized joint confidence contours for (n s ,r)(n_s,r), at the 68%68\\% and 95%95\\% CL, from the latest Planck data .", "In this plot wehave used 3 different values for the number of ee-folds NN: the solid, dashed, and dotted lines correspond to N=50,60N=50,\\,60, and 70, respectively.The trajectories in the $n_s$ - $r$ plane for the model studied here may be generated by plotting Eqs.", "() and () parametrically.", "In particular, we have obtained three different curves by fixing the number of $e$ -folds to $N=50,\\,60$ , and 70, and plotting with respect to the parameter $\\alpha $ in a wide range.", "The Fig.", "(REF ) shows the plot of the tensor-to-scalar ratio $r$ versus the scalar spectral index $n_s$ for natural inflation in the high-energy limit of brane-world scenario.", "Here, we have considered the two-dimensional marginalized joint confidence contours for $(n_s,r)$ , at the $68\\%$ and $95\\%$ CL, from the latest Planck data [23].", "The corresponding allowed range for the dimensionless parameter $\\alpha $ defined by Eq.", "(REF ), for each $r(n_s)$ curve, may be inferred by finding the points when the trajectory enters and exits the $95\\%$ CL region from Planck.", "We can determinate numerically from Eqs.", "() and () that, by fixing $N$ , the tensor-to-scalar ratio decreases as the parameter $\\alpha $ is increasing.", "On the other hand, after to reach a maximum value, the scalar spectral index begins to decrease as $\\alpha $ is increasing.", "In this way, we may obtain a lower limit as well an upper limit for $\\alpha $ .", "The trajectory for $N=50$ lies well outside of the joint $95\\%$ CL region in the $n_s$ - $r$ plane, so it is ruled out by the data.", "For $N=60$ , the trajectory lies inside the $95\\%$ CL region, obtaining the constraint on $\\alpha $ given by $3\\text{.", "}1\\times 10^{-2}\\lesssim \\alpha \\lesssim 7\\text{.", "}1\\times 10^{-2}$ .", "Finally, for $N=70$ the trajectory lies inside the the joint $95\\%$ CL region as well the $68\\%$ CL region, obtaining the constraint on $\\alpha $ given by $4\\text{.", "}8\\times 10^{-3}\\lesssim \\alpha \\lesssim 8\\text{.", "}5\\times 10^{-2}$ .", "On the other hand, by combining the scalar power spectrum (REF ), the constraints on $\\alpha $ already obtained, and the observational value for amplitude of the scalar power spectrum $\\mathcal {P}_{\\mathcal {R}}\\simeq 2 \\times 10^{-9}$ [23], me may obtain the allowed range for the ratio $\\gamma \\equiv \\Lambda /f$ for each value of $N$ .", "For $N=60$ , this constraint becomes $6\\text{.", "}6\\times 10^{-3}\\lesssim \\gamma \\lesssim 7\\text{.", "}4\\times 10^{-3}$ , and for $N=70$ we have that $4\\text{.", "}3\\times 10^{-3}\\lesssim \\gamma \\lesssim 6\\text{.", "}5\\times 10^{-3}$ .", "Table (REF ) summarizes the constraints obtained on $\\alpha $ and $\\gamma $ using the last data of Planck.", "Table: Results for the constraints on the parameters α\\alpha andγ\\gamma for natural inflation in the high-energy of Randall-Sundrum brane model, using the last data of Planck.As we can see, using the latest Planck results we find that the $\\gamma $ parameter must be one order of magnitude larger than that found in[35], while the ratio $f/M_5 \\sim 10$ (see equation (REF ) below) must be one order of magnitude lower compared to that obtained in[35].", "Clearly, we were not able to obtain the allowed range for all three parameters of the model, namely the 5-dimensional Planck mass $M_5$ and the mass scales of the inflaton potential $\\Lambda $ and $f$ , only by considering the Planck data.", "We have found the allowed range for certain combinations of these parameter, $\\alpha $ and $\\gamma $ , instead.", "However, using the definitions of $\\alpha , \\gamma $ as well as the formulas relating the four-dimensional quantities with the five-dimensional ones, we can express everything in terms of the fundamental Planck mass $M_5$ as follows $ \\ f = \\left( \\frac{3}{16 \\pi ^2 \\alpha \\gamma ^4} \\right)^{1/6} M_5 $ $\\Lambda = \\gamma f=\\gamma \\left( \\frac{3}{16 \\pi ^2 \\alpha \\gamma ^4} \\right)^{1/6} M_5 $ However, a string scale/higher-dimensional Planck mass of the order of a few TeV is very attractive from the theoretical point of view for several reasons, as it addresses the hierarchy problem, provides an alternative to gauge coupling unification in D-brane constructions of the Standard Model[50], and the evaporation a la Hawking of TeV mini-black holes can be seen at the colliders[51], and possibly explain anomalies related to cosmic ray observations[52].", "Therefore, we shall take it seriously and in the rest of this article we shall assume for $M_5$ a value of $10 TeV$ .", "Then, by combining the constraints found earlier on $\\alpha $ and $\\gamma $ , and this value for the five-dimensional Planck mass, Eqs.", "(REF ) and (REF ) give us the allowed range for the spontaneous symmetry breaking scale $f$ and the soft explicit symmetry breaking scale $\\Lambda $ , yielding $211.4 \\,\\text{TeV} \\lesssim f \\lesssim 261.9 \\, \\text{TeV}$ and $1.5 \\,\\text{TeV}\\lesssim \\Lambda \\lesssim 1.8 \\, \\text{TeV}$ for $N=60$ , whereas for $N=70$ , the allowed ranges become $223.7 \\,\\text{TeV} \\lesssim f \\lesssim 475.6 \\, \\text{TeV}$ and $1.3 \\,\\text{TeV} \\lesssim \\Lambda \\lesssim 2.3 \\, \\text{TeV}$ , respectively.", "These results imply a hierarchy between the mass scales consistent with $f\\gg \\Lambda $ .", "On the other hand, the constraints found on $\\alpha $ and Eqs.", "(REF ) and (REF ) imply that, during inflation $\\phi \\sim f$ , therefore natural inflation in the framework of the high-energy regime of the RS-II brane model takes place at sub-Planckian values of the scalar field." ], [ "Conclusions", "To summarize, in this article we have studied natural inflation in the framework of the Randall-Sundrum II brane model in the light of the recent Planck results.", "Adopting the Randall-Sundrum fine-tuning, the brane model is characterized only by the 5-dimensional Planck mass, while the inflationary model is characterized by the two mass scales $f, \\Lambda $ of the inflaton potential.", "We have used the COBE normalization as well as the allowed contour plots in the $n_s-r$ plane.", "First, in the $n_s-r$ plane we show the theoretical predictions of the model for three different values of e-folds $N=50, 60, 70$ .", "According to the plot the $N=50$ case is excluded, while for the $N=60, 70$ cases the model is viable for a certain range of the $\\alpha $ parameter (a combination of the three parameters of the model $f, M_5, \\Lambda $ ) defined in the text.", "After that, using the constraint for the amplitude of scalar perturbations we determined the ratio $\\Lambda /f$ .", "We have expressed the mass scales of the inflaton potential in terms of the five-dimensional Planck mass which remains unconstraint using the Planck results only.", "It is known, however, that there are good theoretical reasons, such us hierarchy problem, alternative to gauge coupling unification, mini black hole evaporation etc., to believe that $M_5$ is at the TeV scale.", "If we take it seriously and insist on a $M_5 ~ 10 TeV$ all parameters of the model are known.", "We would like to thank G. Barenboim for helping us with the figures.", "G.P.", "was supported by Comisión Nacional de Ciencias y Tecnología of Chile through Anillo project ACT1122.", "N.V. was supported by Comisión Nacional de Ciencias y Tecnología of Chile through FONDECYT Grant N$^{0}$ 3150490." ] ]
1606.04888
[ [ "On a geometric inequality related to fractional integration" ], [ "Abstract In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman's paper.", "Based on it we investigate its multilinear analogue inequalities.", "Combining with the Gressman's work on multilinear integral, we establish this new kind of geometric inequalities with bilinear form and multilinear form in more general settings.", "Moreover, in some cases we also find the best constants and optimisers for these geometric inequalities on Euclidean spaces with Lebesgue measure settings with $L^p$ bounds." ], [ "Introduction", "Several fractional integral inequalities have been studied.", "The Hardy-Littlewood-Sobolev inequality asserted $\\Vert \\int _{\\mathbb {R}} g(t) \\frac{1}{|s-t|^{2-\\frac{1}{p}-\\frac{1}{q}}} dt\\Vert _{p^{\\prime }} \\le C_{p,q} \\ \\Vert g\\Vert _{q},$ for $1< p$ , $q< \\infty $ , $\\frac{1}{p}+\\frac{1}{q} > 1$ and all functions $g$ in $L^{q}(\\mathbb {R})$ .", "Applying Hölder's inequality gives the forward Hardy-Littlewood-Sobolev inequality $(1.1)$ : for $1< p$ , $q< \\infty $ , $\\frac{1}{p}+\\frac{1}{q} > 1$ $| \\int _{\\mathbb {R}} \\int _{\\mathbb {R}} f(s) g(t) \\frac{1}{|s-t|^{2-\\frac{1}{p}-\\frac{1}{q}}} ds \\ dt| \\le C_{p,q} \\ \\Vert f\\Vert _{p}\\Vert g\\Vert _{q} \\qquad \\mathrm {(1.1)}$ holds for all functions $f \\in L^{p}(\\mathbb {R})$ and $g \\in L^{q}(\\mathbb {R})$ .", "Gressman [1] showed the equivalence between the forward Hardy-Littlewood-Sobolev inequality $(1.1)$ and the inverse Hardy-Littlewood-Sobolev inequality $(1.2)$ which follows.", "For $0< p$ , $q< 1$ , and all functions $f \\in L^{p}(\\mathbb {R})$ and $g \\in L^{q}(\\mathbb {R})$ , we have $\\Vert f\\Vert _{p} \\Vert g\\Vert _{q} \\le C_{p,q} \\int _{\\mathbb {R}} \\int _{\\mathbb {R}} | f(s) g(t)| |s-t|^{\\frac{1}{p}+\\frac{1}{q}-2} dsdt .", "\\qquad \\mathrm {(1.2)}$ As a result, for $0 < p$ , $q < r <\\infty $ and all measurable functions $f, g$ , $\\Vert f^{r}\\Vert _{\\frac{p}{r}} \\Vert g^{r}\\Vert _{\\frac{q}{r}} \\le C_{\\frac{p}{r},\\frac{q}{r}}\\int _{\\mathbb {R}} \\int _{\\mathbb {R}} | f^{r}(s) g^{r}(t)| |s-t|^{\\frac{r}{p}+\\frac{r}{q}-2} ds dt .$ Then, for $0 < p$ , $q < r <\\infty $ and all functions $f \\in L^{p}(\\mathbb {R})$ and $g \\in L^{q}(\\mathbb {R})$ , we have $\\Vert f\\Vert _{p}\\ \\Vert g\\Vert _{q} \\le C_{\\frac{p}{r},\\frac{q}{r}}^{\\frac{1}{r}} \\ \\Vert f(s) g(t) |s-t|^{\\frac{1}{p}+\\frac{1}{q}-\\frac{2}{r}} \\Vert _{L^{r}(dsdt)}.", "\\qquad \\mathrm {(1.3)}$ It is natural to ask what the inequality $(1.3)$ leads to if we let $r \\rightarrow \\infty $ , and what happens to the constant $C_{\\frac{p}{r},\\frac{q}{r}}^{\\frac{1}{r}}$ as $r \\rightarrow \\infty $ .", "One approach is to consider the behaviour of $C_{\\frac{p}{r},\\frac{q}{r}}^{\\frac{1}{r}}$ as $r \\rightarrow \\infty $ , but as we are interested in geometrical questions, we prefer a more direct approach.", "Thus, we will be asking the following questions mainly motivated by the multilinear fractional integrals in Gressman's paper [1].", "Question 1.", "Let $f, g$ be measurable functions defined on $\\mathbb {R}$ with Lebesgue measure.", "Does there exist $C_{p,q}$ such that for any $0 < p, q <\\infty $ , $\\gamma = \\frac{1}{p}+\\frac{1}{q}$ $\\Vert f\\Vert _{p}\\ \\Vert g\\Vert _{q} \\le C_{p,q} \\ \\displaystyle {\\sup _{s,t}} \\ | f(s)g(t)||s-t|^{\\gamma }?", "\\qquad \\mathrm {(1.4)}$ The reason why we take $\\gamma = \\frac{1}{p}+\\frac{1}{q}$ follows from homogeneity.", "We assume that $\\Vert f\\Vert _{p}\\ \\Vert g\\Vert _{q} \\le C_{p,q} \\ \\displaystyle {\\sup _{s,t}} \\ | f(s)g(t)||s-t|^{\\gamma }$ holds.", "Then consider functions $f(\\frac{\\cdot }{R})$ , $g(\\frac{\\cdot }{R})$ for all $R >0$ : $\\Vert f(\\frac{\\cdot }{R})\\Vert _{p} = R^{\\frac{1}{p}} \\Vert f\\Vert _{p}, \\ \\Vert g(\\frac{\\cdot }{R})\\Vert _{q} = R^{\\frac{1}{q}} \\Vert g\\Vert _{p},$ and $\\displaystyle {\\sup _{s,t}} \\ | f(\\frac{s}{R})g(\\frac{t}{R})||s-t|^{\\gamma }&= R^{\\gamma } \\displaystyle {\\sup _{s,t}} \\ | f(\\frac{s}{R})g(\\frac{t}{R})||\\frac{s}{R}-\\frac{t}{R}|^{\\gamma } \\\\&= R^{\\gamma } \\displaystyle {\\sup _{s,t}} \\ | f(s)g(t)||s-t|^{\\gamma } .$ So $R^{\\frac{1}{p}+\\frac{1}{q}} \\Vert f\\Vert _{p}\\ \\Vert g\\Vert _{p} \\le C_{p,q} \\ R^{\\gamma } \\displaystyle {\\sup _{s,t}} \\ | f(s)g(t)||s-t|^{\\gamma }.$ This indicates for all $R > 0$ , $R^{\\frac{1}{p}+\\frac{1}{q}} \\le C_{p,q} \\ R^{\\gamma }$ which implies $\\gamma = \\frac{1}{p}+\\frac{1}{q}$ .", "If we consider the simple case when $f$ , $g$ are supported in an interval $E \\subset \\mathbb {R}$ , we find $\\Vert f\\Vert _{p}\\ \\Vert g\\Vert _{q} \\le \\Vert f\\Vert _{\\infty } \\ \\Vert g\\Vert _{\\infty } |E|^{\\frac{1}{p} + \\frac{1}{q}}= \\displaystyle {\\sup _{s}} |f(s)| \\ \\displaystyle {\\sup _{t}} |g(t)| \\ \\displaystyle {\\sup _{s,t}} |s-t|^{\\frac{1}{p}+\\frac{1}{q}},$ where $|E|$ is the Lebesgue measure of $E$ .", "Clearly the right side of (1.4) is in principle smaller than $\\displaystyle {\\sup _{s}} |f(s)| \\ \\displaystyle {\\sup _{t}} |g(t)| \\ \\displaystyle {\\sup _{s,t}} |s-t|^{\\frac{1}{p}+\\frac{1}{q}}$ .", "We establish that the answer is positive as can be seen in Section 2.", "More precisely, we prove that if $f \\in L^{p}(\\mathbb {R}^{n}), g \\in L^{q}(\\mathbb {R}^{n})$ then $\\Vert f\\Vert _{ L^{p}(\\mathbb {R}^{n})}\\ \\Vert g\\Vert _{ L^{q}(\\mathbb {R}^{n})} \\le C_{p,q,n} \\ \\displaystyle {\\sup _{s,t}} \\ | f(s)g(t)||s-t|^{\\gamma },$ for any $0 < p,q <\\infty $ , $\\gamma = \\frac{n}{p}+\\frac{n}{q}$ .", "Question 2.", "What are the analogues of inequality $(1.4)$ in more general settings?", "We investigate what inequality (1.4) would be like in general metric space with a certain geometric measure as shown in Theorem 2.1.", "Question 3.", "Furthermore based on Question 1 and 2, what are the multilinear analogues of inequality (1.4)?", "Below we give two possible multilinear versions of inequality (1.4).", "Firstly, let $f_{j}$ be measurable functions defined on $\\mathbb {R}^{n}$ with Lebesgue measure.", "Does there exist a finite constant $C$ independent of functions $f_{j}$ such that the following multilinear geometric inequality (1.5) holds for any $0 < p_{j} <\\infty $ , $j=1, \\dots , n+1$ , $\\gamma =\\displaystyle { \\sum _{j=1}^{n+1} } \\frac{1}{p_{j}}$ , $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert f_{j}\\Vert _{p_{j}}\\le C \\ \\displaystyle {\\sup _{y_{j}}} \\displaystyle {\\prod _{j=1}^{n+1}} | f_{j}(y_{j})| \\det (y_{1}, \\dots , y_{n+1})^{\\gamma }?", "\\qquad \\mathrm {(1.5)}$ The condition $\\gamma =\\displaystyle { \\sum _{j=1}^{n+1} } \\frac{1}{p_{j}}$ follows from homogeneity.", "Here the notation $\\det (y_{1}, \\dots , y_{n+1})$ denotes $n!$ times the Euclidean $n$ -dimensional volume of the simplex with vertices $y_{1}, \\dots , y_{n+1}$ , so $\\det (y_{1}, \\dots , y_{n+1}) \\ge 0$ throughout the paper.", "Furthermore, combining with Gressman's work [1] we investigate what inequality (1.5) would be like in more general settings apart from in the Euclidean space cases , for instance, in a real finite-dimensional Hilbert space $H$ with a certain geometric measure as discussed in [1].", "The second possible multilinear form we study is to replace the determinant form by “product form\" as follows.", "Let $f_{j}$ be measurable functions defined on $\\mathbb {R}^{n}$ , $ r_{12}, r_{13}, r_{23}>0 $ .", "Does there exist a finite constant $C$ independent of the functions $f_{j}$ such that for any $0< p_{j}< \\infty $ satisfying $\\displaystyle { \\sum _{j=1}^{3} } \\frac{1}{p_{j}}= \\frac{1}{n} ( r_{12} +r_{13}+ r_{23})$ , $ \\Vert f_{1}\\Vert _{p_{1}} \\Vert f_{2}\\Vert _{p_{2}} \\Vert f_{3}\\Vert _{p_{3}}\\le C \\ \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{3} f_{j}(y_{j}) |y_{1}-y_{2}|^{r_{12}} |y_{1}-y_{3}|^{r_{13}} |y_{2}-y_{3}|^{r_{23}} ?", "\\qquad \\mathrm {(1.6)}$ Homogeneity requires $\\displaystyle { \\sum _{j=1}^{3} } \\frac{1}{p_{j}}= \\frac{1}{n} ( r_{12} +r_{13}+ r_{23})$ .", "Question 4.", "Do there exist sharp versions and optimisers for these geometric inequalities above?", "The main results under this heading are Theorem 4.1 and Theorem 4.6.", "The purpose of this paper is to study these geometric inequalities related to fractional integration in these questions above.", "The results can be discussed as follows.", "Section 2 is devoted to studying the bilinear geometric inequalities raised in Question 1 and 2, and the main results are established in Theorem 2.1 and Corollary 2.3.", "In Section 3, we give two analogues of multilinear form, such as the determinant form as shown in Theorem 3.1, Theorem 3.3 and the product form shown in Theorem 3.5, Theorem 3.7.", "In Section 4, in the Euclidean space setting we prove the existence of extremal functions for the geometric inequality bilinear form (1.4) when $p=q$ and for the multilinear form (1.5) when $p_{j}=p, j=1, \\dots , n+1$ .", "Meanwhile, we get the corresponding conformally equivalent formulations in unit sphere space $\\mathbb {S}^{n}$ and in hyperbolic space $\\mathbb {H}^{n}$ .", "Throughout this paper $\\sup $ is the essential supremum of function.", "$|\\cdot |$ denotes the Lebesgue measure on Euclidean space $\\mathbb {R}^{n}$ and the norm in a Hilbert space.", "$A \\lesssim B$ means there exists a positive constant $C$ independent of the essential variables such that $A \\le C B$ .", "$A \\sim B$ means there exist positive constants $C, C^{\\prime }$ independent of functions such that $C^{\\prime } B \\le A \\le C B$ .", "For the rest of this paper, all functions considered are nonnegative." ], [ "Bilinear forms of geometric inequalities", "Let $(M, \\ d)$ be a metric space and $\\mu $ a $\\sigma $ -finite nonnegative Borel measure on $M$ .", "Let $f, g$ be nonnegative measurable functions defined on $M$ .", "To answer Question 1, 2 we consider the two conditions: (i) For any $x \\in M, r>0$ , $\\mu $ satisfies $\\mu (B(x, r))\\le C_{\\alpha } r^{\\alpha } \\qquad \\mathrm {(2.1)}$ with a finite constant $C_{\\alpha }$ , $ \\alpha >0$ .", "(ii) $\\Vert f\\Vert _{L^{p}(d \\mu )}\\ \\Vert g\\Vert _{L^{q}(d \\mu )} \\le C_{p,q, \\gamma }\\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\gamma } \\qquad \\mathrm {(2.2)}$ holds for all nonnegative functions $f \\in L^{p}(d \\mu ), g \\in L^{q}(d \\mu )$ with a finite constant $C_{p,q,\\gamma }$ independent of the funtions $f, g$ .", "The main results are as follows.", "Theorem 2.1 Let $(M, \\ d)$ be a metric space and $\\mu $ a $\\sigma $ -finite, nonnegative Borel measure on $M$ .", "(a) If condition (i) holds, then (ii) holds for all nonnegative functions $f \\in L^{p}(d \\mu ), g \\in L^{q}(d \\mu )$ for all $0 < p, q <\\infty $ , $\\gamma $ such that $\\gamma = \\alpha (\\frac{1}{p}+\\frac{1}{q})$ .", "(b) If condition (ii) holds for all nonnegative functions $f \\in L^{p}(d \\mu ), g \\in L^{q}(d \\mu )$ for some $p, q>0, \\gamma >0$ , then condition (i) holds for all $\\alpha $ such that $\\alpha = \\gamma (\\frac{1}{p}+\\frac{1}{q})^{-1}$ .", "We begin by studying an endpoint case of (2.2) in the following Lemma 2.2, before studying Theorem 2.1 itself.", "Lemma 2.2 Let $f, g$ be nonnegative measurable functions defined on a metric space $(M, \\ d)$ with the $\\sigma $ -finite and nonnegative Borel measure $\\mu $ which satisfies $\\mu (B(x, r))\\le C_{\\alpha } r^{\\alpha }$ for any $x \\in M, r>0$ .", "Then for all $0 < p, q <\\infty $ we have $\\Vert f\\Vert _{L^{p, \\infty }(d \\mu ) } \\Vert g\\Vert _{L^{\\infty } (d \\mu ) } \\le C_{p, \\alpha } \\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\frac{\\alpha }{p}}.", "\\qquad \\mathrm {(2.3)}$ $\\Vert f\\Vert _{L^{\\infty } (d \\mu ) }\\ \\Vert g\\Vert _{L^{q, \\infty }(d \\mu )} \\le C_{q, \\alpha }\\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\frac{\\alpha }{q}}.", "\\qquad \\mathrm {(2.4)}$ If $\\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\frac{\\alpha }{p}} = \\infty $ , then the inequality (2.3) is trivial.", "If $\\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\frac{\\alpha }{p}} = A < \\infty $ , there exists a measure zero set $E \\subset M \\times M$ , $\\mu \\otimes \\mu (E)=0$ , such that for any $(s, t) \\in (M \\times M )\\setminus E$ $f(s)g(t)d(s, t)^{\\frac{\\alpha }{p}} \\le A.", "\\qquad \\mathrm {(2.5)}$ Note that for any $\\varepsilon > 0$ , there exists $F \\subset M$ , $\\mu (F) > 0$ , such that for all $t \\in F$ $ g(t) > \\Vert g\\Vert _{L^{\\infty }(d \\mu )}-\\varepsilon .", "\\qquad \\mathrm {(2.6)}$ It follows from (2.5) and (2.6) that for all $(s, t) \\in (M \\times F) \\setminus E$ $f(s) \\le \\frac{A}{d(s, t)^{\\frac{\\alpha }{p}} (\\Vert g\\Vert _{\\infty } - \\varepsilon ) }.$ So we can choose a $t \\in F$ such that for any $\\beta > 0$ , $ \\mu (\\lbrace s: f(s) > \\beta \\rbrace ) \\le \\mu (\\lbrace s: d(s, t)^{\\frac{\\alpha }{p}} < \\frac{A}{\\beta (\\Vert g\\Vert _{\\infty }-\\varepsilon )} \\rbrace ).$ This is because $\\mu \\otimes \\mu (E)=0$ implies that for almost every $t \\in M$ , $\\mu ( \\lbrace s \\in M: (s, t) \\in E\\rbrace )=0.$ And since $\\mu (F) >0$ , we can find $t \\in F$ such that $(s, t) \\in (M \\times F) \\setminus E$ for almost every $s \\in M$ .", "Calculate the weak $ L^{p}$ “norm\" of $f$ , $\\Vert f\\Vert _{L^{p, \\infty }(d \\mu )}&= \\displaystyle {\\sup _{\\beta > 0}} \\ \\beta \\ \\mu (\\lbrace s: f(s) > \\beta \\rbrace )^{\\frac{1}{p}} \\\\&\\le \\displaystyle {\\sup _{\\beta > 0}} \\ \\beta \\ \\mu ( \\lbrace s: d(s, t)^{\\frac{k \\alpha }{p}} < \\frac{A}{\\beta (\\Vert g\\Vert _{\\infty }-\\varepsilon )} \\rbrace )^{\\frac{1}{p}} \\\\&= \\displaystyle {\\sup _{\\beta > 0}} \\ \\beta \\ \\mu ( \\lbrace s: d(s, t) < (\\frac{A}{\\beta (\\Vert g\\Vert _{\\infty }-\\varepsilon )} )^{ \\frac{p}{\\alpha } } \\rbrace )^{\\frac{1}{p}}.$ Since $\\mu (B(x, r))\\le C_{\\alpha } r^{\\alpha }$ for any $x\\in M, r>0$ , $ \\mu (\\lbrace s: d(s, t) < r \\rbrace ) \\le C_{\\alpha } r^{\\alpha }$ .", "Hence $\\mu ( \\lbrace s: d(s, t) < (\\frac{A}{\\beta (\\Vert g\\Vert _{\\infty }-\\varepsilon )} )^{ \\frac{p}{\\alpha } } \\rbrace )\\le C_{\\alpha } (\\frac{A}{\\beta (\\Vert g\\Vert _{\\infty }-\\varepsilon )} )^{p}.$ Then we get $\\Vert f\\Vert _{L^{p, \\infty }(d \\mu )}&\\le \\displaystyle {\\sup _{\\beta > 0}} \\ \\beta \\ \\mu ( \\lbrace s: d(s, t) < (\\frac{A}{\\beta (\\Vert g\\Vert _{\\infty }-\\varepsilon )} )^{ \\frac{p}{\\alpha } } \\rbrace )^{\\frac{1}{p}} \\\\&\\le C_{\\alpha }^{\\frac{1}{p}}\\displaystyle {\\sup _{\\beta > 0}} \\ \\beta \\ \\frac{A}{\\beta (\\Vert g\\Vert _{\\infty }-\\varepsilon )} \\\\&= C_{\\alpha }^{\\frac{1}{p}} \\frac{A}{ \\Vert g\\Vert _{\\infty }-\\varepsilon },$ that is $\\Vert f\\Vert _{L^{p, \\infty }(d \\mu ) } ( \\Vert g\\Vert _{L^{\\infty }(d \\mu )} - \\varepsilon ) \\le C_{\\alpha }^{\\frac{1}{p}} A.", "$ Let $\\varepsilon \\rightarrow 0$ , we have $\\Vert f\\Vert _{L^{p, \\infty }(d \\mu ) } \\Vert g\\Vert _{L^{\\infty } (d \\mu ) } \\le C_{\\alpha }^{\\frac{1}{p}} A = C_{\\alpha }^{\\frac{1}{p}} \\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\frac{\\alpha }{p}}.$ Likewise, $\\Vert f\\Vert _{L^{\\infty } (d \\mu ) }\\ \\Vert g\\Vert _{L^{q, \\infty }(d \\mu )} \\le C_{\\alpha }^{\\frac{1}{q}} \\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\frac{\\alpha }{q}}.$ Proof of Theorem 2.1    (a) Suppose condition (i) holds, that is $\\mu (B(t, r))\\le C r^{\\alpha }$ holds for any $t \\in M, r>0$ .", "Let $m= \\frac{1}{\\frac{1}{p}+\\frac{1}{q}}$ , so $m < p, \\ q <\\infty $ .", "Then by the layer cake representation $\\Vert f\\Vert _{L^{p}(d \\mu )}^{p} &=p \\int _{0}^{\\infty } \\beta ^{p-1} \\mu (\\lbrace s: f(s) > \\beta \\rbrace ) \\ d\\beta \\\\&= p \\int _{0}^{\\Vert f\\Vert _{L^{\\infty }(d \\mu )}} \\beta ^{p-1} \\mu ( \\lbrace s: f(s) > \\beta \\rbrace ) \\ d\\beta +p \\int _{\\Vert f\\Vert _{L^{\\infty }(d \\mu )}}^{\\infty } \\beta ^{p-1} \\mu ( \\lbrace s: f(s) > \\beta \\rbrace ) \\ d\\beta \\\\&= p \\int _{0}^{\\Vert f\\Vert _{L^{\\infty }(d \\mu )}} \\beta ^{p-m-1} \\beta ^{m} \\mu ( \\lbrace s: |f(s)| > \\beta \\rbrace ) \\ d\\beta \\\\&\\le p \\ \\Vert f\\Vert _{L^{m, \\infty }(d \\mu )}^{m} \\int _{0}^{\\Vert f\\Vert _{L^{\\infty }(d \\mu )}} \\beta ^{p-m-1} \\ d\\beta \\\\&= \\frac{p}{p-m} \\ \\Vert f\\Vert _{L^{m, \\infty }(d \\mu )}^{m} \\Vert f\\Vert _{L^{\\infty }(d \\mu )}^{p-m},$ which means for $f$ in $L^{m, \\infty }(d \\mu )\\cap L^{\\infty }(d \\mu )$ , we have $f \\in L^{p}(d \\mu )$ , and $\\Vert f\\Vert _{L^{p}(d \\mu )} \\le (\\frac{p}{p-m})^{\\frac{1}{p}} \\ \\Vert f\\Vert _{L^{m, \\infty }(d \\mu )}^{\\frac{m}{p}} \\Vert f\\Vert _{L^{\\infty }(d \\mu )}^{1-\\frac{m}{p}}.", "\\qquad \\mathrm {(2.7)}$ Meanwhile if $g$ is in $L^{m, \\infty }(d \\mu )\\cap L^{\\infty }(d \\mu )$ , then $g \\in L^{q}(d \\mu )$ , and $\\Vert g\\Vert _{L^{q}(d \\mu )} \\le (\\frac{q}{q-m})^{\\frac{1}{q}} \\ \\Vert g\\Vert _{L^{m, \\infty }(d \\mu )}^{\\frac{m}{q}} \\Vert g\\Vert _{L^{\\infty }(d \\mu )}^{1-\\frac{m}{q}}.", "\\qquad \\mathrm {(2.8)}$ Since simple functions are in $L^{m, \\infty }(d \\mu ) \\cap L^{\\infty }(d \\mu )$ , we can apply Lemma 2.2 for simple functions $f, g$ .", "Inequalities (2.7) and (2.8) indicate $\\Vert f\\Vert _{L^{p}(d \\mu )}\\ \\Vert g\\Vert _{L^{q}(d \\mu )} &\\le (\\frac{p}{p-m})^{\\frac{1}{p}} \\ (\\frac{q}{q-m})^{\\frac{1}{q}}\\Vert f\\Vert _{m, \\infty }^{\\frac{m}{p}} \\Vert f\\Vert _{\\infty }^{1-\\frac{m}{p}} \\Vert g\\Vert _{m, \\infty }^{\\frac{m}{q}} \\Vert g\\Vert _{\\infty }^{1-\\frac{m}{q}} \\\\&= (\\frac{p}{p-m})^{\\frac{1}{p}} \\ (\\frac{q}{q-m})^{\\frac{1}{q}}( \\Vert f\\Vert _{m, \\infty }^{\\frac{m}{p}} \\Vert g\\Vert _{\\infty }^{1-\\frac{m}{q}} ) ( \\Vert g\\Vert _{m, \\infty }^{\\frac{m}{q}} \\Vert f\\Vert _{\\infty }^{1-\\frac{m}{p}} ).$ It follows from Lemma 2.2 that $\\Vert f\\Vert _{m, \\infty }^{\\frac{m}{p}} \\Vert g\\Vert _{\\infty }^{1-\\frac{m}{q}}= \\Vert f\\Vert _{m, \\infty }^{\\frac{m}{p}} \\Vert g\\Vert _{\\infty }^{\\frac{m}{p}}\\le C_{\\alpha }^{\\frac{1}{p}} \\displaystyle {\\sup _{s,t}} \\ ( f(s)g(t)d(s, t)^{\\frac{\\alpha }{m}} )^{\\frac{m}{p}},$ and $\\Vert g\\Vert _{m, \\infty }^{\\frac{m}{q}} \\Vert f\\Vert _{\\infty }^{1-\\frac{m}{p}} = \\Vert g\\Vert _{m, \\infty }^{\\frac{m}{q}} \\Vert f\\Vert _{\\infty }^{\\frac{m}{q}}\\le C_{\\alpha }^{\\frac{1}{q}} \\displaystyle {\\sup _{s,t}} \\ ( f(s)g(t)d(s, t)^{\\frac{\\alpha }{m}} )^{\\frac{m}{q}}.$ Therefore $\\Vert f\\Vert _{L^{p}(d \\mu )}\\ \\Vert g\\Vert _{L^{q}(d \\mu )}&\\le C_{\\alpha }^{\\frac{1}{p}+\\frac{1}{q}} (\\frac{p}{p-m})^{\\frac{1}{p}} \\ (\\frac{q}{q-m})^{\\frac{1}{q}}\\displaystyle {\\sup _{s,t}} ( f(s)g(t)d(s, t)^{\\frac{\\alpha }{m}} ) ^{\\frac{m}{p}}\\displaystyle {\\sup _{s, t}} ( f(s)g(t)d(s, t)^{\\frac{\\alpha }{m}} ) ^{\\frac{m}{q}} \\\\&= C_{\\alpha }^{\\frac{1}{p}+\\frac{1}{q}} (\\frac{p+q}{p})^{\\frac{1}{p}} (\\frac{p+q}{q})^{\\frac{1}{q}}\\displaystyle {\\sup _{s, t}} \\ f(s)g(t)d(s, t)^{\\frac{\\alpha }{p}+\\frac{\\alpha }{q}}.$ For general functions $f\\in L^{p}(d \\mu ), g\\in L^{q}(d \\mu )$ , there exist sequences of simple functions $\\lbrace f_{n}\\rbrace \\uparrow f$ , and $\\lbrace g_{n}\\rbrace \\uparrow g$ as $n \\rightarrow \\infty $ .", "Under the discussion above, we have already obtained that (2.2) holds for simple functions, $\\Vert f_{n}\\Vert _{L^{p}(d \\mu )}\\ \\Vert g_{n}\\Vert _{L^{q}(d \\mu )} \\le C_{p,q,\\alpha }\\displaystyle {\\sup _{s,t}} \\ f_{n}(s) g_{n}(t) d(s, t)^{ \\alpha (\\frac{1}{p}+\\frac{1}{q})}\\le C_{p,q,\\alpha } \\displaystyle {\\sup _{s, t}} \\ f(s)g(t)d(s, t)^{\\alpha (\\frac{1}{p}+\\frac{1}{q})}.$ Then let $n \\rightarrow \\infty $ , we have $\\Vert f\\Vert _{L^{p}(d \\mu )}\\ \\Vert g\\Vert _{L^{q}(d \\mu )} \\le C_{p,q,\\alpha } \\displaystyle {\\sup _{s, t}} \\ f(s)g(t)d(s, t)^{ \\alpha (\\frac{1}{p}+\\frac{1}{q})}.$ (b) Suppose $\\Vert f\\Vert _{L^{p}(d \\mu )}\\ \\Vert g\\Vert _{L^{q}(d \\mu )} \\le C_{p,q, \\gamma }\\displaystyle {\\sup _{s,t}} \\ f(s)g(t)d(s, t)^{\\gamma }$ holds for some $p, q >0$ , $\\gamma $ .", "For any $x \\in M, r >0$ , let $f=g=\\chi _{B(x, r)}$ , then we have $\\mu (B(x, r))^{\\frac{1}{p}+\\frac{1}{q}}\\le C_{p,q, \\gamma } \\displaystyle {\\sup _{s, t \\in B(x, r)}} d(s, t)^{ \\gamma }.$ Together with the fact $\\displaystyle {\\sup _{s, t \\in B(x, r)}} d(s, t) \\le 2r $ we deduce that $\\mu $ has the property $\\mu (B(x, r)) \\le C_{\\alpha } r^{\\alpha },$ where $\\alpha = \\gamma (\\frac{1}{p}+\\frac{1}{q})^{-1}$ .", "$\\hfill \\Box $ Corollary 2.3 Let $f, g$ be measurable functions defined on $\\mathbb {R}^{n}$ with Lebesgue measure, then for all $0 < p, q <\\infty $ , $\\gamma >0$ such that $\\gamma = n(\\frac{1}{p}+\\frac{1}{q})$ , $\\Vert f\\Vert _{L^{p}(\\mathbb {R}^{n}) } \\Vert g\\Vert _{L^{q} (\\mathbb {R}^{n})} \\le C_{p,q,n} \\displaystyle {\\sup _{s,t}} \\ f(s)g(t)|s-t|^{\\gamma } .", "\\qquad \\mathrm {(2.9)}$ On the other hand, it is not true that $\\Vert f\\Vert _{L^{p}(\\mathbb {R}^{n}) } \\Vert g\\Vert _{L^{\\infty } (\\mathbb {R}^{n}) } \\lesssim \\displaystyle {\\sup _{s,t}} \\ f(s)g(t)|s-t|^{\\frac{n}{p}}.", "\\qquad \\mathrm {(2.10)}$ holds for all $ f\\in L^{p}(\\mathbb {R}^{n}), g \\in L^{\\infty } (\\mathbb {R}^{n})$ .", "(1) Observe that $|B(x, r)| \\le C_{n} r^{n}$ for any $x\\in \\mathbb {R}^{n}, r>0$ , then we can apply Theorem 2.1 to give (2.9).", "More precisely, $\\gamma $ here must be $n(\\frac{1}{p}+\\frac{1}{q})$ which follows from the homogeneity mentioned in the introduction.", "(2) We use a counterexample to show that (2.10) fails.", "For any positive $N$ , let $f_{N}(s)= (1+|s|)^{-\\frac{n}{p}} \\chi _{(1 \\le |s| \\le N)}$ , $g(t)= \\chi _{ (|t| \\le 1)}(t)$ .", "Then $\\Vert g\\Vert _{L^{\\infty } (\\mathbb {R}^{n}) }=1$ and $\\displaystyle {\\sup _{s,t}} \\ \\ f_{N}(s)g(t) \\ |s-t|^{\\frac{n}{p}}&= \\displaystyle {\\sup _{s,t}} \\ \\frac{|s-t|^{\\frac{n}{p}}}{(1+|s|)^{\\frac{n}{p}}} \\ \\chi _{(1 \\le |s| \\le N)}(s) \\ \\chi _{ (|t| \\le 1)}(t) \\\\&\\le \\frac{(|s|+1)^{\\frac{n}{p}}}{(1+|s|)^{\\frac{n}{p}}} =1.$ While by polar coordinates $\\Vert f_{N}\\Vert _{L^{p}(\\mathbb {R}^{n}) }^{p}= \\int _{1 \\le |s| \\le N} \\ \\frac{ds}{(1+|s|)^{n}}=C \\int _{1}^{N} \\frac{r^{n-1}}{(1+r)^{n}} dr. $ Let $ u=1+r$ to make the change of variables $ \\int _{1}^{N} \\frac{r^{n-1}}{(1+r)^{n}} dr= \\int _{2}^{N+1} \\frac{(u-1)^{n-1}}{u^{n}} du\\ge \\int _{2}^{N+1} \\frac{1}{2^{n-1} } \\frac{1}{u} du $ and $\\int _{2}^{N+1} \\frac{1}{u} du = \\ln (N+1) -\\ln 2 \\rightarrow \\infty ,$ as $N \\rightarrow \\infty $ ." ], [ "Multilinear forms of geometric inequalities", "We first recall some terminology, notation and lemmas which are all given in [1].", "$(H, \\langle \\cdot , \\cdot \\rangle _{H})$ is a real finite-dimensional Hilbert space with inner product $\\langle \\cdot , \\cdot \\rangle _{H}$ .", "For any positive integer $k \\le \\mathrm {dim} \\ H$ , we use $\\det (y_{1}, \\dots , y_{k+1})$ to denote the square root of the determinant of the $k \\times k$ Gram matrix $(a_{i,j})_{k \\times k}$ , where $a_{i,j} =\\langle y_{i}-y_{k+1}, y_{j}-y_{k+1} \\rangle _{H}.$ Clearly, the Gram matrix $(a_{i,j})_{k \\times k}$ is positive semidefinite, since $(a_{i,j})_{k \\times k}$ can be written as $A^{\\prime } A$ , where $ A$ is the matrix whose $ j$ -th column is $ y_{j}-y_{k+1}$ , and $A^{\\prime }$ is the transpose of $A$ .", "So $\\det (y_{1}, \\dots , y_{k+1}) \\ge 0$ throughout the paper.", "Especially in Euclidean $\\mathbb {R}^{k}$ space, the determinant of the matrix $(a_{i,j})_{k \\times k}$ is the square of the volume of the parallelotope formed by the vectors $y_{1}, \\dots , y_{k+1}$ .", "Thus, $\\det (y_{1}, \\dots , y_{k+1})$ is also $k!$ times the Euclidean $k$ -dimensional volume of the simplex with vertices $y_{1}, \\dots , y_{k+1}$ .", "Definition 1.", "A subset $B \\subset H$ is called an ellipsoid when it may be written as $ B \\equiv \\lbrace x\\in H: \\displaystyle {\\sum _{i}} \\frac{ | \\langle x-x_{0}, \\omega _{i} \\rangle |^{2}}{l_{i}^{2}} \\le 1 \\rbrace $ for some $x_{0}\\in H$ , some orthonormal basis $\\lbrace \\omega _{i}\\rbrace $ of $H$ , and lengths $l_{i}\\in [0, \\infty ]$ .", "For example, $\\lbrace (t,0, \\dots , 0): t\\in \\mathbb {R}\\rbrace \\subset \\mathbb {R}^{n}$ is an ellipsoid in $\\mathbb {R}^{n}$ .", "It could be written as $\\frac{ | \\langle x, e_{1} \\rangle |^{2}}{\\infty }+ \\frac{ | \\langle x, e_{2} \\rangle |^{2}}{0}+\\cdots + \\frac{ | \\langle x, e_{n} \\rangle |^{2} }{0} \\le 1 ,$ where $ l_{1}= \\infty , l_{2}=0, \\dots , l_{n}=0$ , $x_{0}=0$ , and $\\lbrace e_{1}, \\dots , e_{n}\\rbrace $ are the standard orthonormal basis vectors for $ \\mathbb {R}^{n}$ .", "The ellipsoid will be called centred when $x_{0}=0$ .", "Given an ellipsoid $B \\subset H$ and an integer $k$ with $k \\le $ dim $H$ , denote $|B|_{k}=\\sup \\lbrace l_{i_{1}} \\dots l_{i_{k}}: i_{1}< i_{2}< \\dots <i_{k} \\rbrace ,$ which is called the $k$ -content of $B$ .", "Definition 2.", "A $\\sigma $ -finite and nonnegative Borel measure $\\mu $ is called $k$ -curved with exponent $\\alpha > 0$ , if there exists a finite constant $C_{\\alpha }$ such that $\\mu (B) \\le C_{\\alpha } |B|_{k}^{\\alpha } \\qquad \\mathrm {(3.1)}$ for all ellipsoids $B$ in $H$ .", "This kind of geometric measure describes the amount of mass of $\\mu $ supported on $k$ -dimensional subspaces of $H$ .", "For instance, the Lebesgue measure in $\\mathbb {R}^{n}$ is $n$ -curved with exponent 1.", "It is $k$ -curved with exponent $\\frac{n}{k}$ as well for $k <n$ .", "If we see the Lebesgue measure restricted on $x_{1}$ axis, it is 1-curved with exponent 1.", "It cannot be $k$ -curved for $k \\ge 2$ .", "Let $S$ be a hypersurface in $\\mathbb {R}^{n}$ with non-vanishing Gaussian curvature, then its surface area measure $\\mu _{S}$ is $n$ -curved with exponent $\\frac{n-1}{n+1}$ .", "We now recall some results of Gressman in [1].", "Lemma 3.", "[1] Let $\\mu $ be a $\\sigma $ -finite and nonnegative Borel measure such that (3.1) holds for all ellipsoids $B$ in $H$ .", "Then for any measurable sets $E_{1}, \\dots , E_{k}$ in $H$ we have $\\mu \\otimes \\cdots \\otimes \\mu (\\lbrace (y_{1}, \\dots , y_{k})\\in E_{1} \\times \\cdots \\times E_{k}:\\det (0, y_{1}, \\dots , y_{k}) < \\delta \\rbrace )\\le C_{k, \\alpha } \\delta ^{\\alpha } \\displaystyle {\\prod _{j=1}^{k}} \\mu (E_{j})^{1-\\frac{1}{k}}.$ Lemma 4.", "[1] Under the above assumptions, for any centred ellipsoid $B$ in $H$ , we have $\\displaystyle {\\sup _{x_{j}\\in B}} \\ \\det (0, x_{1}, \\dots , x_{k}) \\le C_{k} |B|_{k}, \\qquad \\mathrm {(3.2)}$ where $|B|_{k}$ is the $k$ -content of $B$ .", "Lemma 5.", "[1] Let $f_{j}$ be nonnegative measurable functions defined on a real finite-dimensional Hilbert space $H$ , and let $\\mu $ be a $\\sigma $ -finite nonnegative Borel measure on $H$ which satisfies inequality (3.1).", "Then for all $1 \\le p_{j} \\le \\infty $ satisfying $\\frac{1}{p_{j}} > 1- \\frac{\\gamma }{k \\alpha }$ , $j=1, \\dots , k+1$ , and $k+1 - \\displaystyle { \\sum _{j=1}^{k+1} } \\frac{1}{p_{j}} = \\frac{\\gamma }{\\alpha }$ , $ \\int _{H} \\cdots \\int _{H} \\displaystyle {\\prod _{j=1}^{k+1}} f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{-\\gamma } d\\mu (y_{1}) \\cdots d\\mu (y_{k+1}) \\le C \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{L^{p_{j}}(d \\mu )} \\qquad \\mathrm {(3.3)}$ holds with a finite constant $C$ independent of the functions $f_{j}$ .", "The first kind of multilinear analogue of the fractional integral inequality we start to study is the determinant form as given in the following theorem, mainly discussing the two conditions with $1 \\le k \\le \\mathrm {dim} \\ H$ fixed: (i) There exists a finite constant $C_{\\alpha }$ such that for all ellipsoids $B$ in $H$ , $\\mu (B) \\le C_{\\alpha } |B|_{k}^{\\alpha }.", "\\qquad \\mathrm {(3.4)}$ (ii) $ \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{L^{p_{j}}(d \\mu )}\\le C \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{k+1})^{\\gamma } \\qquad \\mathrm {(3.5)}$ for all nonnegative functions $f_{j} \\in L^{p_{j}}(d \\mu ), j=1, \\dots , k+1$ , where $C$ is a finite constant independent of functions $f_{j}$ which only depends on $p_{j}, k, \\gamma $ .", "Theorem 3.1 Let $(H, \\langle \\cdot , \\cdot \\rangle _{H})$ be a real finite-dimensional Hilbert space.", "Let $\\mu $ be a $\\sigma $ -finite nonnegative Borel measure .", "(a) If condition (i) holds, then (ii) holds for all nonnegative functions$f_{j} \\in L^{p_{j}}(d \\mu )$ , for all $0 <p_{j}< \\infty , \\gamma $ which satisfy $ \\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ and $\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}= \\frac{\\gamma }{\\alpha }$ , $j=1, \\dots , k+1$ .", "(b) If condition (ii) holds for all nonnegative functions$f_{j} \\in L^{p_{j}}(d \\mu )$ , $j=1, \\dots , k+1$ , for some $p_{j}>0$ , $\\gamma >0$ , then condition (i) holds for all $\\alpha $ such that $\\alpha = \\gamma (\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}})^{-1}$ .", "If we consider the special case when $k=1$ , the condition (3.4) is equivalent to the condition (2.1).", "It is clear that (3.4) implies (2.1).", "Conversely, suppose $\\mu (B(x,r)) \\le C_{\\alpha } r^{\\alpha } $ holds for any $ x\\in H, r>0$ .", "Given an ellipsoid $K$ centred at $x_{0}$ , clearly $K \\subset B(x_{0}, |K|_{1})$ .", "So $\\mu (K) \\le \\mu (B(x_{0}, |E|_{1})) \\le C_{\\alpha }|E|_{1}^{\\alpha } ,$ which gives that $\\mu $ is 1-curved with exponent $\\alpha $ .", "When $k=1$ , inequality (3.5) becomes the bilinear form (2.2).", "In Section 2 we stated that $\\Vert f_{1}\\Vert _{L^{p_{1}}(d \\mu )}\\ \\Vert f_{2}\\Vert _{L^{p_{2}}(d \\mu )} \\lesssim \\displaystyle {\\sup _{s, t} } \\ f_{1}(s) f_{2}(t)|s-t|^{\\gamma }$ holds for any $0 <p_{1}, p_{2}< \\infty $ , $ \\frac{1}{p_{1}}+ \\frac{1}{p_{2}} = \\frac{\\gamma }{ \\alpha }$ .", "Note that the condition $\\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }, j=1, 2$ , in Theorem 3.1 (a) is automatic in this case, since $0 <p_{1}, p_{2}< \\infty $ .", "We begin by studying why condition (ii) implies condition (i).", "Proof of Theorem 3.1 (b)    Let $f_{j}=\\chi _{B}$ , where $B \\subset H$ is an ellipsoid centred at $x_{0} \\in H$ , $ j=1, \\dots , k+1$ .", "Since condition (ii) holds for some $p_{j}, j=1, \\dots , k+1, \\gamma $ , then we have $ \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert \\chi _{B} \\Vert _{L^{p_{j}}(d \\mu )} \\lesssim \\displaystyle {\\sup _{y_{j}\\in B}} \\det (y_{1}, \\dots , y_{k+1})^{\\gamma },$ that is, $\\displaystyle {\\prod _{j=1}^{k+1}} \\mu (B)^{\\frac{1}{p_{j}}} = \\mu (B)^{\\sum \\limits _{j=1}^{k+1} \\frac{1}{p_{j}}}\\lesssim \\displaystyle {\\sup _{y_{j}\\in B}} \\det (y_{1}, \\dots , y_{k+1})^{\\gamma }.", "\\qquad \\mathrm {(3.6)}$ We use a fact that for any centred ellipsoid $E$ , $E-E \\subset 2E$ .", "Suppose $E= \\lbrace x\\in H: \\displaystyle {\\sum _{i}} \\frac{ | \\langle x, \\omega _{i} \\rangle |^{2}}{l_{i}^{2}} \\le 1 \\rbrace $ where $\\lbrace \\omega _{i}\\rbrace $ is the orthonormal basis of $H$ .", "Let $y, z \\in E$ , since for every $\\omega _{i}$ $| \\langle y-z, \\omega _{i} \\rangle |^{2}= | \\langle y, \\omega _{i} \\rangle - \\langle z, \\omega _{i} \\rangle |^{2}\\le 2(| \\langle y, \\omega _{i} \\rangle |^{2}+ | \\langle z, \\omega _{i} \\rangle |^{2}),$ it is easy to verify that $y-z \\in 2E = \\lbrace x\\in H: \\displaystyle {\\sum _{i}} \\frac{ | \\langle x, \\omega _{i} \\rangle |^{2}}{(2 l_{i})^{2}} \\le 1 \\rbrace .$ Thus we have $B-B= (B-x_{0})-(B-x_{0})\\subset 2(B-x_{0}).$ Therefore, it follows form Lemma 4 that $\\sup \\limits _{y_{j}\\in B} \\det (y_{1}, y_{2}, \\dots , y_{k}, y_{k+1})&= \\sup \\limits _{y_{j}\\in B} \\det (0, y_{1}-y_{k+1} , y_{2}-y_{k+1}, \\dots , y_{k}-y_{k+1}) \\\\&\\le \\sup \\limits _{x_{j}\\in 2(B-x_{0})} \\det (0, x_{1} , x_{2}, \\dots , x_{k}) \\\\&\\le 2^{k}C_{k} |B-x_{0}|_{k}=2^{k}C_{k} |B|_{k}.$ So $\\displaystyle {\\sup _{y_{j}\\in B}} \\det (y_{1}, \\dots , y_{k+1})^{\\gamma } \\lesssim |B|_{k}^{\\gamma }.", "$ Together with (3.6), we conclude that $\\mu (B)^{\\sum \\limits _{j=1}^{k+1} \\frac{1}{p_{j}}} \\lesssim \\displaystyle {\\sup _{y_{j}\\in B}} \\det (y_{1}, \\dots , y_{k+1})^{\\gamma }\\lesssim |B|_{k}^{\\gamma }.$ So $\\mu (B) \\lesssim |B|_{k}^{\\alpha }$ with $\\alpha = \\gamma (\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}})^{-1}$ .", "$\\hfill \\Box $ On the other hand, in order to see what inequality (3.5) will be like if $\\mu $ is $k$ -curved with exponent $\\alpha $ , we first investigate an endpoint case of (3.5) as follows.", "Lemma 3.2 Let $f_{j}$ be measurable functions defined on real finite-dimensional Hilbert space $H$ with the $\\sigma $ -finite and nonnegative Borel measure $\\mu $ which satisfies $\\mu (B) \\le C_{\\alpha } |B|_{k}^{\\alpha }$ for all ellipsoids $B \\subset H$ .", "Then for any positive $\\gamma $ we have $ \\displaystyle {\\prod _{j=1}^{k}} \\Vert f_{j}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu )} \\Vert f_{k+1}\\Vert _{L^{\\infty }(d \\mu )} \\le C_{k,\\alpha , \\gamma }\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{k+1})^{\\gamma }.", "\\qquad \\mathrm {(3.7)}$ Likewise for each $1 \\le l \\le k+1$ $ \\displaystyle {\\prod _{i\\ne l} } \\Vert f_{i}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu )} \\Vert f_{l}\\Vert _{L^{\\infty }(d \\mu )} \\le C_{k,\\alpha , \\gamma }\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{k+1})^{\\gamma } \\qquad \\mathrm {(3.8)}$ holds by symmetry.", "If $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{k+1})^{\\gamma } = \\infty $ , the inequality (3.7) is trivial.", "Suppose $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{k+1})^{\\gamma } = A < \\infty $ , then there exists measure zero set $E \\subset H \\times \\dots \\times H$ , $\\mu \\otimes \\cdots \\otimes \\mu (E)=0$ , such that $\\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{k+1})^{\\gamma } \\le A, \\qquad \\mathrm {(3.9)}$ for all $(y_{1}, \\dots , y_{k+1}) \\in (H \\times \\dots \\times H) \\setminus E$ .", "Note that for any $\\varepsilon > 0$ , there exists $F \\subset H$ such that $\\mu (F)>0$ , and for all $y_{k+1} \\in F$ $f_{k+1}(y_{k+1}) > \\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon .", "\\qquad \\mathrm {(3.10)}$ From (3.9) and (3.10) it follows that for $(y_{1}, \\dots ,y_{k+1}) \\in (H \\times \\dots \\times H \\times F) \\setminus E$ , $f_{1}(y_{1}) \\le \\frac{A}{\\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon } \\displaystyle {\\prod _{j=2}^{k}} f_{j}(y_{j})^{-1} \\det ( y_{1}, \\dots , y_{k+1})^{-\\gamma }.", "\\qquad \\mathrm {(3.11)}$ For any positive $\\alpha _{j}$ , denote $C_{j}= \\lbrace y_{j}: f_{j}(y_{j}) > \\alpha _{j} \\rbrace $ , $ j=1, \\dots ,k$ .", "Note that $\\mu \\otimes \\cdots \\otimes \\mu (E)=0$ , which implies that for almost every $y_{k+1} \\in H$ , $\\mu \\otimes \\cdots \\otimes \\mu ( \\lbrace (y_{1}, \\dots , y_{k}): (y_{1}, \\dots , y_{k}, y_{k+1}) \\in E \\rbrace ) =0.$ Denote $\\lbrace (y_{1}, \\dots , y_{k}): (y_{1}, \\dots , y_{k}, y_{k+1}) \\in E \\rbrace $ by $G_{y_{k+1}} \\subset H^{k}$ .", "Since $\\mu (F) > 0$ , we can choose a $y_{k+1} \\in F$ such that $\\mu \\otimes \\cdots \\otimes \\mu (G_{y_{k+1}})=0$ , and for all $(y_{1}, \\dots , y_{k}) \\in H^{k} \\setminus G_{y_{k+1}}$ $(y_{1}, \\dots , y_{k}, y_{k+1}) \\in (H \\times \\dots \\times H \\times F) \\setminus E.$ Since $\\mu \\otimes \\cdots \\otimes \\mu (G_{y_{k+1}})=0$ , for almost every $y_{1} \\in H$ $\\mu \\otimes \\cdots \\otimes \\mu ( \\lbrace (y_{2}, \\dots , y_{k}) \\in H^{k-1} : (y_{1},y_{2}, \\dots , y_{k}) \\in G_{y_{k+1}} \\rbrace ) =0.$ That is to say, for almost every $y_{1}$ , almost every $ (y_{2}, \\dots , y_{k}) \\in H^{k-1}$ $ (y_{1},y_{2}, \\dots , y_{k}) \\in (H \\times \\dots \\times H \\times F) \\setminus E.$ Therefore, together with (3.11) implies that for any $\\alpha _{1} >0$ $\\ \\ \\ &\\mu (\\lbrace y_{1}: f_{1}(y_{1}) > \\alpha _{1} \\rbrace ) \\\\\\le & \\mu ( \\lbrace y_{1}: \\det (y_{1}, \\dots , y_{k+1})^{\\gamma }< \\frac{A}{\\alpha _{1} (\\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon ) } \\displaystyle {\\prod _{j=2}^{k}} f_{j}(y_{j})^{-1} , \\ (y_{2}, \\dots , y_{k}) \\in H^{k-1} \\ a.e.", "\\ \\rbrace ).$ Due to the definition of $C_{j}$ , we get for any $\\alpha _{1} >0 $ , $&\\ \\ \\ \\mu (\\lbrace y_{1}: f_{1}(y_{1}) > \\alpha _{1} \\rbrace ) \\\\&\\le \\mu ( \\lbrace y_{1} \\in C_{1} : \\det (y_{1}, \\dots , y_{k+1})^{\\gamma }< \\frac{A}{\\alpha _{1} (\\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon )}\\displaystyle {\\prod _{j=2}^{k}} \\alpha _{j}^{-1} ,\\ (y_{2}, \\dots , y_{k}) \\in C_{2} \\times \\dots \\times C_{k} \\ a.e.", "\\ \\rbrace ) \\\\&\\le \\mu ( \\lbrace y_{1} \\in C_{1} : \\det (y_{1}, \\dots , y_{k+1}) < (\\frac{A}{\\alpha _{1}(\\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon )})^{\\frac{1}{\\gamma }}\\displaystyle {\\prod _{j=2}^{k}} \\alpha _{j}^{-\\frac{1}{\\gamma }} ,\\ (y_{2}, \\dots , y_{k}) \\in C_{2} \\times \\dots \\times C_{k} \\ a.e.", "\\ \\rbrace ) \\\\&= \\displaystyle {\\prod _{j=2}^{k}} \\mu (C_{j})^{-1} \\mu \\otimes \\dots \\otimes \\mu (\\lbrace (y_{1}, \\dots ,y_{k})\\in C_{1} \\times \\dots \\times C_{k}: \\det (y_{1}, \\dots , y_{k+1}) < (\\frac{A}{ \\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon })^{\\frac{1}{\\gamma }}\\displaystyle {\\prod _{j=1}^{k}} \\alpha _{j}^{-\\frac{1}{\\gamma }} \\rbrace ).$ Denote $ \\frac{A}{ \\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon }^{\\frac{1}{\\gamma }} \\displaystyle {\\prod _{j=1}^{k}} \\alpha _{j}^{-\\frac{1}{\\gamma }} $ by $M$ , then it follows from Lemma 3 that $& \\ \\ \\ \\mu (\\lbrace y_{1}: f_{1}(y_{1})> \\alpha _{1}\\rbrace ) \\\\&\\le \\displaystyle {\\prod _{j=2}^{k}} \\mu (C_{j})^{-1} \\mu \\otimes \\dots \\otimes \\mu (\\lbrace (y_{1}, \\dots ,y_{k})\\in C_{1} \\times \\dots \\times C_{k}:\\det (y_{1}, \\dots , y_{k+1}) < M \\rbrace ) \\\\&=\\displaystyle {\\prod _{j=2}^{k}} \\mu (C_{j})^{-1} \\mu \\otimes \\dots \\otimes \\mu (\\lbrace (y_{1}, \\dots ,y_{k})\\in C_{1} \\times \\dots \\times C_{k}: \\det (0,y_{1}-y_{k+1}, \\dots , y_{k}-y_{k+1}) <M \\rbrace ) \\\\&\\le C_{k, \\alpha } \\displaystyle {\\prod _{j=2}^{k}} \\mu (C_{j})^{-1} M^{\\alpha }\\displaystyle {\\prod _{j=1}^{k}} \\mu (C_{j})^{1- \\frac{1}{k}} \\\\&= C_{k, \\alpha } M^{\\alpha }\\mu (C_{1})^{1-\\frac{1}{k}} \\mu (C_{2})^{-\\frac{1}{k}} \\dots \\mu (C_{k})^{-\\frac{1}{k}}.$ Hence $\\mu (\\lbrace y_{1}: f_{1}(y_{1}) > \\alpha _{1}\\rbrace ) = \\mu (C_{1})\\le C_{k, \\alpha } (\\frac{A}{ \\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon })^{\\frac{\\alpha }{\\gamma }} \\prod _{j=1}^{k} \\alpha _{j}^{-\\frac{\\alpha }{\\gamma }}\\mu (C_{1})^{1-\\frac{1}{k}} \\mu (C_{2})^{-\\frac{1}{k}} \\dots \\mu (C_{k})^{-\\frac{1}{k}}.$ That is, we obtain for any $\\alpha _{j} > 0$ $\\mu (C_{1})^{\\frac{\\gamma }{\\alpha }}\\le C_{k, \\alpha }^{\\frac{\\gamma }{\\alpha }} \\frac{A}{ \\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon }\\displaystyle {\\prod _{j=1}^{k}} \\alpha _{j}^{-1} \\mu (C_{1})^{(1-\\frac{1}{k})\\frac{\\gamma }{\\alpha }} \\mu (C_{2})^{-\\frac{\\gamma }{k \\alpha }} \\dots \\mu (C_{k})^{-\\frac{\\gamma }{k \\alpha }}.$ Simplify it to give that $\\alpha _{1} \\mu (C_{1})^{\\frac{\\gamma }{k \\alpha }}\\le C_{k, \\alpha }^{\\frac{\\gamma }{\\alpha }} \\frac{A}{ \\Vert f_{k+1}\\Vert _{\\infty }- \\varepsilon }\\frac{1}{ \\displaystyle {\\prod _{j=2}^{k}} \\alpha _{j} \\mu (C_{j})^{\\frac{\\gamma }{k \\alpha }} } .", "$ Let $\\varepsilon \\rightarrow 0$ , we get for any $\\alpha _{j} > 0$ $\\displaystyle {\\prod _{j=1}^{k}} \\alpha _{j} \\mu (C_{j})^{\\frac{\\gamma }{k \\alpha }}\\le C_{k, \\alpha }^{\\frac{\\gamma }{\\alpha }}\\frac{A}{ \\Vert f_{k+1}\\Vert _{\\infty } } .", "\\qquad \\mathrm {(3.12)}$ Since $\\alpha _{j}$ are arbitrary, this allows us to take the infimum over all $\\alpha _{j} > 0$ on (3.12), $j=1, \\dots , k+1$ , which gives $\\displaystyle {\\prod _{j=1}^{k}} \\Vert f_{j}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu )} \\Vert f_{k+1}\\Vert _{L^{\\infty }(d \\mu )} \\le C_{k, \\alpha }^{\\frac{\\gamma }{\\alpha }} A=C_{k, \\alpha }^{\\frac{\\gamma }{\\alpha }} \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.$ This proves the endpoint case (3.7).", "Meanwhile by symmetry (3.8) holds.", "Proof of Theorem 3.1 (a)    For any general $f_{j} \\in L^{p_{j}}(d \\mu )$ , there exist sequences of simple functions $\\lbrace f_{jn}\\rbrace \\uparrow f_{j}$ as $n \\rightarrow \\infty $ .", "We apply Lemma 3.2 for simple functions $f_{jn}$ , this is because simple functions are in $L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu ) \\cap L^{\\infty }(d \\mu )$ .", "For each $1 \\le j \\le k+1$ , for every $n$ , we have $ \\prod _{i \\ne j} \\Vert f_{in}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu )} \\Vert f_{jn}\\Vert _{L^{\\infty }(d \\mu )} \\lesssim \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{jn}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.", "\\qquad \\mathrm {(3.13)}$ Based on this, by the layer cake representation it is easy to obtain that for $\\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ , $f_{jn} \\in L^{p_{j}}(d \\mu )$ and $\\Vert f_{jn}\\Vert _{L^{p_{j}}(d \\mu )} \\lesssim \\Vert f_{jn}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu )}^{\\frac{k \\alpha }{\\gamma p_{j}}}\\Vert f_{jn}\\Vert _{L^{\\infty }(d \\mu )}^{1-\\frac{k \\alpha }{\\gamma p_{j}}}.", "\\qquad \\mathrm {(3.14)}$ We assume that for every $n$ $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{jn}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }=A_{n}<\\infty ,$ then from (3.13), (3.14) and $\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}= \\frac{\\gamma }{\\alpha }$ it follows that $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{jn}\\Vert _{L^{p_{j}}(d \\mu )}&\\lesssim \\prod _{j=1}^{k+1} \\Vert f_{jn}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu )}^{\\frac{k \\alpha }{\\gamma p_{j}}} \\Vert f_{jn}\\Vert _{L^{\\infty }(d \\mu )}^{1-\\frac{k \\alpha }{\\gamma p_{j}}} \\\\&=\\prod _{j=1}^{k+1} ( \\prod _{i \\ne j} \\Vert f_{in}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }, \\infty }(d \\mu )} \\Vert f_{jn}\\Vert _{L^{\\infty }(d \\mu )} )^{1- \\frac{k \\alpha }{\\gamma p_{j} }} \\\\&\\lesssim \\prod _{j=1}^{k+1} (\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{jn}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma })^{1- \\frac{k \\alpha }{\\gamma p_{j}}} \\\\&= \\prod _{j=1}^{k+1} A_{n}^{1- \\frac{k \\alpha }{\\gamma p_{j} }}.$ Note that $\\sum \\limits _{j=1}^{k+1} (1- \\frac{k \\alpha }{\\gamma p_{j}})=1$ , since $\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}= \\frac{\\gamma }{\\alpha }$ .", "Hence, $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{jn}\\Vert _{L^{p_{j}}(d \\mu )} \\lesssim \\prod _{j=1}^{k+1} A_{n}^{1- \\frac{k \\alpha }{\\gamma p_{j} }}= A_{n} \\equiv \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{jn}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.$ Therefore, for every $n$ $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{jn}\\Vert _{L^{p_{j}}(d \\mu )}\\le \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{jn}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }\\le \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.$ Let $n \\rightarrow \\infty $ to deduce that $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{L^{p_{j}}(d \\mu )} \\lesssim \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.$ This completes the proof of this theorem.", "$\\hfill \\Box $ We shall now present an alternative method to show that condition (i) implies condition (ii), mainly applying Gressman's result Lemma 5 above.", "Alternative proof of Theorem 3.1 (a)    Suppose $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{\\gamma } = A < \\infty $ .", "We can write $& \\ \\ \\ \\Vert f_{1}\\Vert _{p_{1}}^{p_{1}} \\cdots \\Vert f_{k+1}\\Vert _{p_{k+1}}^{p_{k+1}} \\\\&= \\int _{H} \\cdots \\int _{H} \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j})^{p_{j}} d\\mu (y_{1}) \\dots d\\mu (y_{k+1}) \\\\&= \\int _{H} \\cdots \\int _{H} \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }\\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j})^{p_{j}-1} \\det (y_{1}, \\dots ,y_{k+1})^{-\\gamma }d\\mu (y_{1}) \\dots d\\mu (y_{k+1})$ Since $ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma } \\le A,$ and $p_{j}-1=\\frac{p_{j}}{p_{j}^{\\prime }}$ , so $\\Vert f_{1}\\Vert _{p_{1}}^{p_{1}} \\cdots \\Vert f_{k+1}\\Vert _{p_{k+1}}^{p_{k+1}}\\le A \\int _{H} \\cdots \\int _{H} \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j})^{\\frac{p_{j}}{p_{j}^{\\prime }}}\\det (y_{1}, \\dots ,y_{k+1})^{-\\gamma } d\\mu (y_{1}) \\dots d\\mu (y_{k+1}).", "$ From Lemma 5 it follows that for all $1 \\le p_{j}^{\\prime } \\le \\infty $ satisfying $\\frac{1}{p_{j}^{\\prime }} > 1- \\frac{\\gamma }{k \\alpha }$ , $j=1, \\dots , k+1$ , $\\int _{H}\\cdots \\int _{H} \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j})^{\\frac{p_{j}}{p_{j}^{\\prime }}} \\det (y_{1}, \\dots ,y_{k+1})^{-\\gamma } d\\mu (y_{1}) \\dots d\\mu (y_{k+1})\\le C \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}^{\\frac{p_{j}}{p_{j}^{\\prime }}}\\Vert _{p_{j}^{\\prime }}$ holds, where $k+1 - \\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}^{\\prime }}= \\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}= \\frac{\\gamma }{\\alpha }$ .", "Therefore, $\\Vert f_{1}\\Vert _{p_{1}}^{p_{1}} \\cdots \\Vert f_{k+1}\\Vert _{p_{k+1}}^{p_{k+1}}\\le C A \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}^{\\frac{p_{j}}{p_{j}^{\\prime }}}\\Vert _{p_{j}^{\\prime }}= C A \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{p_{j}}^{p_{j}-1},$ which indicates for all $1 \\le p_{j} < \\infty $ satisfying $\\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ and $\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}= \\frac{\\gamma }{\\alpha }$ , we have $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{p_{j}} \\le C \\ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.", "\\qquad \\mathrm {(3.15)}$ As for other $0 < p_{j} < 1$ , $1 \\le j \\le k+1$ , such that $\\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ and $\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}= \\frac{\\gamma }{\\alpha }$ , it is easy to see $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{p_{j}}= \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}^{p_{1} \\dots p_{k+1}}\\Vert _{\\frac{1}{q_{j}}}^{ \\frac{1}{p_{1} \\dots p_{k+1}}}$ , with $q_{j}= p_{1} \\dots p_{j-1} p_{j+1} \\dots p_{k+1}$ , and $\\displaystyle { \\sum _{j=1}^{k+1} } q_{j}= \\frac{\\gamma }{\\alpha }(p_{1} \\dots p_{k+1})$ .", "Since $\\frac{1}{q_{j}} >1$ , $q_{j} < \\frac{\\gamma }{k \\alpha }$ , we can apply (3.15) to give $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}^{p_{1} \\dots p_{k+1}}\\Vert _{\\frac{1}{q_{j}}}\\le C \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j})^{p_{1} \\dots p_{k+1}} \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma (p_{1} \\dots p_{k+1})}.$ Thus $\\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{p_{j}}&= \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}^{p_{1} \\dots p_{k+1}}\\Vert _{\\frac{1}{q_{j}}}^{\\frac{1}{p_{1} \\dots p_{k+1}}} \\\\&\\le C \\ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{k+1})^{\\gamma }.$ In conclusion, we obtain that for all $0 < p_{j} < \\infty $ satisfying $\\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ and $\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}= \\frac{\\gamma }{\\alpha }$ , $1 \\le j \\le k+1$ , $ \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{p_{j}} \\le C \\ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.", "$ This also completes the proof of part (a) of Theorem 3.1.", "$\\hfill \\Box $ It should be pointed out that we find condition (i) and (ii) are equivalent to the inequality (3.3) in Lemma 5 as well from the the alternative method of proof (a).", "Lemma 5 states condition (i) implies inequality (3.3), and we use inequality (3.3) to get the inequality (3.5) in the alternative method of proof (a).", "Besides, Theorem 3.1 shows that condition (i) and (ii) i.e.", "inequality (3.5) are equivalent.", "If we strengthen the condition (i) to $\\mu (B) \\sim |B|_{k}^{\\alpha }$ for all ellipsoids $B$ in $H$ , then $ \\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ for all $1 \\le j \\le k+1$ and $\\frac{\\gamma }{\\alpha } = \\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}} $ are necessary and sufficient conditions for inequality (3.5) to hold, which can be seen in the following theorem.", "Theorem 3.3 Let $f_{j}$ be nonnegative measurable functions defined on real finite-dimensional Hilbert space $H$ .", "Let $\\mu $ be a $\\sigma $ -finite, nonnegative Borel measure with satisfying $\\mu (B) \\sim |B|_{k}^{\\alpha }$ for all ellipsoids $B$ in $H$ .", "Then for all $0< p_{j}<\\infty $ $ \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{L^{p_{j}}(d \\mu )} \\le C_{k, \\alpha , p_{j}} \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\ (\\det (y_{1}, \\dots ,y_{k+1}))^{\\gamma } \\qquad \\mathrm {(3.16)}$ holds, if and only if $p_{j}$ satisfy $ \\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ for all $1 \\le j \\le k+1$ and $\\frac{\\gamma }{\\alpha } = \\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}} $ .", "$\\mu (B) \\sim |B|_{k}^{\\alpha }$ for all ellipsoids $B$ in $H$ , so the measure $\\mu $ is $k$ -curved with exponent $\\alpha $ .", "Theorem 3.1 (a) gave the sufficient conditions for inequality (3.16) to hold.", "To see the converse, we study the necessary conditions for inequality (3.16) to hold.", "Suppose (3.16) holds for all nonnegative functions $f_{j} \\in L^{p_{j}}(d \\mu )$ , then $\\frac{\\gamma }{\\alpha } = \\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}$ which follows from homogeneity.", "Let $f_{j}= \\chi _{B}$ where $B$ is a ball in $H$ , $j=1, \\dots ,k+1$ .", "We consider functions $\\chi _{B}(\\frac{\\cdot }{R})$ for all $R >0$ : for $j=1, \\dots ,k+1$ , we have $\\Vert \\chi _{B}(\\frac{\\cdot }{R})\\Vert _{L^{p_{j}}(d \\mu )} \\sim R^{\\frac{k \\alpha }{p_{j}}}( \\mu (B))^{\\frac{1}{p_{j}}},$ this is because for all $R >0$ $ \\mu (R B) \\sim |R B|_{k}^{\\alpha } = R^{k \\alpha } |B|_{k}^{\\alpha } \\sim R^{k \\alpha } \\mu (B).$ From the property of $\\det (y_{1}, \\dots ,y_{k+1})$ it follows that $&\\ \\ \\ \\ \\ \\ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ \\chi _{B}(\\frac{y_{j}}{R}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma } \\\\&= R^{k \\gamma } \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ \\chi _{B}(\\frac{y_{j}}{R}) \\det (\\frac{y_{1}}{R}, \\dots ,\\frac{y_{k+1}}{R})^{\\gamma } \\\\&= R^{k \\gamma } \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ \\chi _{B}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }.$ So if (3.16) holds, then $\\displaystyle { \\prod _{j=1}^{k+1} } \\ R^{\\frac{k \\alpha }{p_{j}}} \\lesssim R^{k \\gamma }$ for all $R >0$ , which implies $\\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{k \\alpha }{p_{j}} = k \\gamma .$ That is $\\frac{\\gamma }{\\alpha } = \\displaystyle { \\sum _{j=1}^{k+1} } \\ \\frac{1}{p_{j}}.", "\\qquad \\mathrm {(3.17)}$ We now claim that if (3.16) holds for all nonnegative functions $f_{j} \\in L^{p_{j}}(d \\mu )$ , $p_{j}$ must satisfy $\\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ for all $1 \\le j \\le k+1$ .", "Let $f_{1} \\in L^{p_{1}}(d \\mu )$ be supported on $\\lbrace y_{1}: |y_{1}| \\ge 10 \\rbrace $ .", "For $2 \\le j \\le k+1$ , let $f_{j}= \\chi _{B(0, \\frac{1}{2})}$ where $B(0, \\frac{1}{2})$ denotes the ball in $H$ centred at 0 with radius $\\frac{1}{2}$ .", "So $|y_{1}-y_{j}| \\sim |y_{1}|$ for all $2 \\le j \\le k+1$ .", "We consider the new functions $f_{1}, f_{j}(\\frac{\\cdot }{\\epsilon })$ with $0< \\epsilon < 1$ , $2 \\le j \\le k+1$ .", "Suppose that inequality $(3.16)$ holds for all nonnegative functions $f_{j} \\in L^{p_{j}}(d \\mu )$ , then $\\Vert f_{1}\\Vert _{L^{p_{1}}(d \\mu )}\\displaystyle {\\prod _{j=2}^{k+1}} \\Vert f_{j}(\\frac{\\cdot }{\\epsilon })\\Vert _{L^{p_{j}}(d \\mu )}\\lesssim \\ \\sup \\limits _{y_{j}} \\ f_{1}(y_{1}) \\prod _{j=2}^{k+1} f_{j}(\\frac{y_{j}}{\\epsilon }) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma } .$ By the Hadamard inequality $\\det (y_{1}, \\dots ,y_{k+1}) \\le |y_{1}-y_{k+1}| |y_{2}-y_{k+1}| \\cdots \\ |y_{k}-y_{k+1}|,$ we have $&\\ \\ \\ \\ \\sup \\limits _{y_{j}} \\ f_{1}(y_{1}) \\prod _{j=2}^{k+1} f_{j}(\\frac{y_{j}}{\\epsilon }) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma } \\\\& \\le \\ \\sup \\limits _{y_{j}} \\ f_{1}(y_{1}) \\prod _{j=2}^{k+1} f_{j}(\\frac{y_{j}}{\\epsilon }) \\ (|y_{1}-y_{k+1}| |y_{2}-y_{k+1}| \\cdots \\ |y_{k}-y_{k+1}|)^{\\gamma } \\\\& \\sim \\epsilon ^{(k-1) \\gamma } \\ \\sup \\limits _{y_{j}} \\ f_{1}(y_{1}) \\prod _{j=2}^{k+1} f_{j}(\\frac{y_{j}}{\\epsilon }) \\ (|y_{1}-\\frac{y_{k+1}}{\\epsilon }| |\\frac{y_{2}}{\\epsilon }-\\frac{y_{k+1}}{\\epsilon }| \\cdots \\ |\\frac{y_{k}}{\\epsilon }-\\frac{y_{k+1}}{\\epsilon }|)^{\\gamma } \\\\&\\sim \\epsilon ^{(k-1) \\gamma } \\ \\sup \\limits _{y_{j}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\ (|y_{1}-y_{k+1}| |y_{2}-y_{k+1}| \\cdots \\ |y_{k}-y_{k+1}|)^{\\gamma }.$ On the other hand, for $2 \\le j \\le k+1$ $\\Vert \\chi _{B(0, \\frac{1}{2})}(\\frac{\\cdot }{\\epsilon })\\Vert _{L^{p_{j}}(d \\mu )} \\sim \\epsilon ^{ \\frac{k \\alpha }{p_{j}}} \\mu (B(0, \\frac{1}{2}))^{\\frac{1}{p_{j}}},$ this is because for $ \\epsilon >0$ $ \\mu (\\epsilon B(0, \\frac{1}{2}))\\sim | \\epsilon B(0, \\frac{1}{2})|_{k}^{\\alpha }= \\epsilon ^{k \\alpha } |B(0, \\frac{1}{2})|_{k}^{\\alpha }\\sim \\epsilon ^{k \\alpha } \\mu (B(0, \\frac{1}{2})).$ Then $\\Vert f_{1}\\Vert _{L^{p_{1}}(d \\mu )}\\displaystyle {\\prod _{j=2}^{k+1}} \\Vert f_{j}(\\frac{\\cdot }{\\epsilon })\\Vert _{L^{p_{j}}(d \\mu )}= \\prod _{j=2}^{k+1} \\epsilon ^{ \\frac{k \\alpha }{p_{j}} } \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\Vert f_{j}\\Vert _{L^{p_{j}}(d \\mu )}.$ So if (3.16) holds, then for all $0< \\epsilon < 1$ , $\\displaystyle {\\prod _{j=2}^{k+1}} \\epsilon ^{ \\frac{k \\alpha }{p_{j}} } \\lesssim \\epsilon ^{(k-1) \\gamma },$ then we have $\\sum _{j=2}^{k+1} \\frac{k \\alpha }{p_{j}} \\ge (k-1) \\gamma ,$ which means $\\frac{1}{p_{1}}= \\frac{\\gamma }{\\alpha }- \\sum _{j=2}^{k+1} \\frac{1}{p_{j}}\\le \\frac{\\gamma }{\\alpha }- \\frac{k-1}{k \\alpha } \\gamma = \\frac{\\gamma }{k \\alpha }.", "\\qquad \\mathrm {(3.18)}$ By symmetry, for any $1 \\le j \\le k+1$ we have $\\frac{1}{p_{j}} \\le \\frac{\\gamma }{k \\alpha }$ provided (3.16) holds for all nonnegative functions $f_{j} \\in L^{p_{j}}(d \\mu )$ .", "As for the boundary case, the following counterexample shows that we must have $\\frac{1}{p_{j}} < \\frac{\\gamma }{k \\alpha }$ for all $1 \\le j \\le k+1$ .", "For any positive $N$ , let $f_{1}(y_{1})= \\frac{1}{|y_{1}|^{\\gamma }} \\chi _{2 \\le |y_{1}| \\le N }$ , $f_{j}(y_{j})= \\chi _{|y_{j}| \\le 1/4}, 2 \\le j \\le k+1$ .", "The Hadamard inequality tells us $\\det (y_{1}, \\dots ,y_{k+1}) \\le |y_{1}-y_{k+1}| \\cdots |y_{k}-y_{k+1}|,$ then $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{k+1})^{\\gamma }&\\le \\displaystyle {\\sup _{ y_{j}} } \\ \\displaystyle {\\prod _{j=1}^{k+1}} \\ f_{j}(y_{j}) \\ |y_{1}-y_{k+1}|^{\\gamma } \\cdots |y_{k}-y_{k+1}|^{\\gamma } \\\\&\\lesssim \\displaystyle {\\sup _{2 \\le |y_{1}| \\le N}} |y_{1}|^{-\\gamma } (|y_{1}|+\\frac{1}{4})^{\\gamma }\\lesssim 1.$ On the other hand, by polar coordinates we obtain $\\displaystyle { \\limsup _{N \\rightarrow \\infty }} \\ \\Vert f_{1}\\Vert _{L^{\\frac{k \\alpha }{\\gamma }(d \\mu )}}&= \\limsup _{N \\rightarrow \\infty } \\int _{2 \\le |y_{1}| \\le N} |y_{1}|^{-k \\alpha } d \\mu (y_{1}) \\\\&= \\sum \\limits _{j>0} \\int _{|y_{1}| \\sim 2^{j}} |y_{1}|^{-k \\alpha } d \\mu (y_{1}) \\\\&\\gtrsim \\sum \\limits _{j>0} 2^{-k \\alpha j} 2^{j k \\alpha } =\\infty ,$ which gives the contradiction to (3.16).", "The last inequality follows due to the fact that $\\mu (B) \\sim |B|_{k}^{\\alpha }$ for all ellipsoids $B$ in $H$ , which implies $\\mu (\\lbrace y_{1} \\in H: |y_{1}| \\sim 2^{j} \\rbrace ) \\sim 2^{j k \\alpha }.$ As is well known, the Lebesgue measure on $\\mathbb {R}^{n}$ is not only $n$ -curved with exponent 1, but also it satisfies $|B| \\sim |B|_{n}$ for all ellipsoids $B$ in $H$ .", "Hence we obtain the following corollary immediately.", "Corollary 3.4 Let $ f_{j} \\in L^{p_{j}}(\\mathbb {R}^{n}) $ with Lebesgue measure, then $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert f_{j}\\Vert _{p_{j}} \\le C_{n, p_{j}} \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{\\gamma } \\qquad \\mathrm {(3.19)}$ holds, if and only if $p_{j}$ satisfy $ \\frac{1}{p_{j}} < \\frac{\\gamma }{n}$ for all $1 \\le j \\le n+1$ and $\\gamma = \\displaystyle { \\sum _{j=1}^{n+1} } \\ \\frac{1}{p_{j}} $ .", "We now consider the second class of multilinear inequalities where we have a product form rather than a determinant.", "Theorem 3.5 Let $ r_{ij} >0 $ and $r_{i j} = r_{j i}$ .", "Let $f_{j}$ be nonnegative measurable functions defined on $\\mathbb {R}^{n}$ , then $\\prod _{j=1}^{3} \\Vert f_{j}\\Vert _{p_{j}} \\le C_{p_{j}, r_{ij}, n } \\ \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{3} f_{j}(y_{j}) \\prod _{1 \\le i < j \\le 3} |y_{i}-y_{j}|^{r_{ij}} \\qquad \\mathrm {(3.20)}$ holds, if and only if $p_{j}$ satisfy $\\displaystyle { \\sum _{j=1}^{3} } \\frac{1}{p_{j}}= \\frac{1}{n} ( r_{12} +r_{13}+ r_{23})$ , $\\frac{1}{p_{j}} < \\frac{1}{n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij} $ for every $j$ .", "$\\displaystyle { \\sum _{j=1}^{3} } \\frac{1}{p_{j}}= \\frac{1}{n} ( r_{12} +r_{13}+ r_{23})$ just follows from homogeneity.", "Besides, by applying the similar example in the proof of Theorem 3.3 we can get the necessary conditions for (3.20) to hold: for every $ j$ $\\frac{1}{p_{j}} \\le \\frac{1}{n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij}.", "$ The following counterexample shows that we must have $\\frac{1}{p_{j}} < \\frac{1}{n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij} $ for each $j$ .", "If we assume $\\frac{1}{p_{1}}= \\frac{r_{12}+r_{13}}{n} $ , for any positive $N$ , let $f_{1}(y_{1})= |y_{1}|^{-(r_{12}+r_{13})} \\chi _{2 \\le |y_{1}| \\le N }$ , $f_{2}(y_{2})= \\chi _{|y_{2}| \\le 1/4}$ , $f_{3}(y_{3})= \\chi _{|y_{3}| \\le 1/4}$ .", "Suppose $A= \\displaystyle {\\sup _{y_{j}}} \\ f_{1}(y_{1})f_{2}(y_{2})f_{3}(y_{3}) |y_{1}-y_{2}|^{r_{12}} |y_{1}-y_{3}|^{r_{13}} |y_{2}-y_{3}|^{r_{23}},$ then $A &\\lesssim \\displaystyle {\\sup _{2 \\le |y_{1}| \\le N}} |y_{1}|^{-(r_{12}+r_{13})} (|y_{1}|+\\frac{1}{4})^{r_{12}} (|y_{1}|+\\frac{1}{4})^{r_{13}} \\\\&\\lesssim 1.$ However, by polar coordinates we obtain $\\Vert f_{1}\\Vert _{\\frac{n}{r_{12}+r_{13}}}&=\\int _{2 \\le |y_{1}| \\le N} \\frac{1}{|y_{1}|^{n}} dy_{1}\\\\&=C \\int _{2}^{N} \\frac{r^{n-1}}{r^{n}} dr\\\\&= C(\\ln N-\\ln 2) \\rightarrow \\infty ,$ as $N \\rightarrow \\infty $ .", "For the converse, suppose $A= \\displaystyle {\\sup _{y_{j}}} \\ f_{1}(y_{1})f_{2}(y_{2})f_{3}(y_{3}) |y_{1}-y_{2}|^{r_{12}} |y_{1}-y_{3}|^{r_{13}} |y_{2}-y_{3}|^{r_{23}} < \\infty ,$ then there exists measure zero set $E \\subset \\mathbb {R}^{n} \\times \\mathbb {R}^{n} \\times \\mathbb {R}^{n}$ , such that $f_{1}(y_{1})f_{2}(y_{2})f_{3}(y_{3}) |y_{1}-y_{2}|^{r_{12}} |y_{1}-y_{3}|^{r_{13}} |y_{2}-y_{3}|^{r_{23}}\\le A,$ for all $(y_{1}, y_{2}, y_{3}) \\in (\\mathbb {R}^{n} \\times \\mathbb {R}^{n} \\times \\mathbb {R}^{n}) \\setminus E$ .", "By the definition of $\\Vert f_{3}\\Vert _{\\infty }$ , for any $\\varepsilon > 0$ there exists $F \\subset \\mathbb {R}^{n} $ such that $|F|>0$ , and for all $y_{3} \\in F$ $f_{3}(y_{3}) > \\Vert f_{3}\\Vert _{\\infty }- \\varepsilon .$ So for all $(y_{1},...,y_{3}) \\in (\\mathbb {R}^{n} \\times \\mathbb {R}^{n} \\times F) \\setminus E$ , $f_{2}(y_{2}) (\\Vert f_{3}\\Vert _{\\infty }- \\varepsilon ) \\le \\frac{1}{|y_{1}-y_{2}|^{r_{12}} |y_{2}-y_{3}|^{r_{23}}} \\frac{A}{|y_{1}-y_{3}|^{r_{13}} f_{1}(y_{1})}.$ Since $|E|=0$ , for almost every $y_{3} \\in \\mathbb {R}^{n}$ $| \\lbrace (y_{1},y_{2}) \\in \\mathbb {R}^{n} \\times \\mathbb {R}^{n}: (y_{1},y_{2},y_{3}) \\in E \\rbrace |=0.$ Denote $ \\lbrace (y_{1},y_{2}) \\in \\mathbb {R}^{n} \\times \\mathbb {R}^{n} : (y_{1},y_{2},y_{3}) \\in E \\rbrace $ by $G_{y_{3}}$ .", "Because $|F|>0$ , we can choose a $y_{3} \\in F$ such that $|G_{y_{3}}|=0 $ , which implies for almost every $y_{1} \\in \\mathbb {R}^{n}$ , $| \\lbrace y_{2} \\in \\mathbb {R}^{n} : (y_{1},y_{2}) \\in G_{y_{3}} \\rbrace |=0.$ That means for almost every $ y_{1}$ , almost every $y_{2}$ $ (y_{1},y_{2},y_{3}) \\in (\\mathbb {R}^{n} \\times \\mathbb {R}^{n} \\times \\mathbb {R}^{n}) \\setminus E.$ Thus for almost every $y_{1}$ , any small $\\theta >0$ , $\\Vert f_{2}\\Vert _{\\frac{n}{r_{12}+r_{23}-\\theta }} ( \\Vert f_{3}\\Vert _{\\infty }- \\varepsilon )&\\le ( \\int _{\\mathbb {R}^{n}} (|y_{1}-y_{2}|^{-r_{12}} |y_{2}-y_{3}|^{-r_{23}})^{\\frac{n}{r_{12}+r_{23}-\\theta }} dy_{2} )^{\\frac{r_{12}+r_{23}-\\theta }{n}} \\frac{A}{|y_{1}-y_{3}|^{r_{13}} f_{1}(y_{1})} \\\\&= C (|y_{1}-y_{3}|^{n- \\frac{r_{12}n + r_{23}n}{r_{12}+r_{23}-\\theta }} )^{\\frac{r_{12}+r_{23}-\\theta }{n}} \\frac{A}{|y_{1}-y_{3}|^{r_{13}} f_{1}(y_{1})} \\\\&= C |y_{1}-y_{3}|^{-\\theta } \\frac{A}{|y_{1}-y_{3}|^{r_{13}} f_{1}(y_{1})} \\\\&= C \\frac{A}{|y_{1}-y_{3}|^{r_{13}+\\theta } f_{1}(y_{1})}.$ Take the infimum over $y_{1}$ , then let $\\varepsilon \\rightarrow 0$ , $\\Vert f_{2}\\Vert _{\\frac{n}{r_{12}+r_{23}-\\theta }} \\Vert f_{3}\\Vert _{\\infty } \\le C \\ \\displaystyle {\\inf _{y_{1}} } \\frac{A}{|y_{1}-y_{3}|^{r_{13}+\\theta } f_{1}(y_{1})}= C \\ \\frac{A}{\\displaystyle {\\sup _{y_{1}} } \\ |y_{1}-y_{3}|^{ r_{13}+\\theta } f_{1}(y_{1})}.", "\\qquad \\mathrm {(3.21)}$ In the proof of Lemma 2.2, we have stated that for the bilinear form, $\\Vert f_{1}\\Vert _{\\frac{n}{r_{13}+\\theta }, \\infty } \\lesssim \\displaystyle {\\sup _{y_{1}} } \\ f_{1}(y_{1}) |y_{1}-y_{3}|^{r_{13}+\\theta }.$ Therefore, together with (3.21) we conclude that for any small $\\theta >0$ , $\\Vert f_{1}\\Vert _{\\frac{n}{r_{13}+\\theta }, \\infty } \\Vert f_{2}\\Vert _{\\frac{n}{r_{12}+r_{23}-\\theta }} \\Vert f_{3}\\Vert _{\\infty } \\lesssim A.", "\\qquad \\mathrm {(3.22)}$ Meanwhile applying the similar arguments we have $\\Vert f_{1}\\Vert _{\\infty } \\Vert f_{2}\\Vert _{\\frac{n}{r_{12}+r_{23}-\\theta }} \\Vert f_{3}\\Vert _{\\frac{n}{r_{13}+\\theta }, \\infty } \\lesssim A,$ $\\Vert f_{1}\\Vert _{\\frac{n}{r_{12}+r_{13}-\\theta }} \\Vert f_{2}\\Vert _{\\infty } \\Vert f_{3}\\Vert _{\\frac{n}{r_{23}+\\theta }, \\infty } \\lesssim A, \\ \\Vert f_{1}\\Vert _{\\frac{n}{r_{12}+r_{13}-\\theta }} \\Vert f_{2}\\Vert _{\\frac{n}{r_{23}+\\theta }, \\infty } \\Vert f_{3}\\Vert _{\\infty }\\lesssim A.", "\\qquad \\mathrm {(3.23)}$ and $\\Vert f_{1}\\Vert _{\\infty } \\Vert f_{2}\\Vert _{\\frac{n}{r_{12}+\\theta }, \\infty } \\Vert f_{3}\\Vert _{\\frac{n}{r_{13}+r_{23}-\\theta }} \\lesssim A, \\ \\Vert f_{1}\\Vert _{\\frac{n}{r_{12}+\\theta }, \\infty } \\Vert f_{2}\\Vert _{\\infty } \\Vert f_{3}\\Vert _{\\frac{n}{r_{13}+r_{23}-\\theta }} \\lesssim A.", "\\qquad \\mathrm {(3.24)}$ Since for all $ 0< p_{j} < \\infty $ , $1 \\le j \\le 3$ satisfying $\\displaystyle { \\sum _{j=1}^{3} } \\frac{1}{p_{j}}= \\frac{1}{n} ( r_{12} +r_{13}+ r_{23})$ and $\\frac{1}{p_{j}} < \\frac{1}{n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij} $ , we can always find a small $\\theta $ such that $(\\frac{1}{p_{1}}, \\frac{1}{p_{2}}, \\frac{1}{p_{3}})$ lies in the interior of the convex hull of $(\\frac{r_{13}+\\theta }{n}, \\frac{r_{12}+r_{23}-\\theta }{n}, 0 )$ , $(\\frac{r_{12}+r_{13}-\\theta }{n}, 0, \\frac{r_{23}+\\theta }{n} )$ , $(0, \\frac{r_{12}+\\theta }{n}, \\frac{r_{13}+r_{23}-\\theta }{n} ) $ , $(0, \\frac{r_{12}+r_{23}-\\theta }{n}, \\frac{r_{13}+\\theta }{n} )$ , $(\\frac{r_{12}+r_{13}-\\theta }{n}, \\frac{r_{23}+\\theta }{n}, 0 )$ , $( \\frac{r_{12}+\\theta }{n}, 0, \\frac{r_{13}+r_{23}-\\theta }{n} ) $ .", "Similar to the discussion in Section 2 and Section 3, inequality (3.20) follows immediately from (3.22)-(3.24) together with the following property $\\Vert f\\Vert _{q} \\le C_{p,q} \\ \\Vert f\\Vert _{p, \\infty }^{\\frac{p}{q}} \\Vert f\\Vert _{\\infty }^{1-\\frac{p}{q}} $ provided $0< p< q < \\infty $ .", "Remark 3.6 However, our method does not work for multilinear cases more than three functions.", "Beckner [2] gave a multilinear fractional integral inequality as follows, mainly applying the general rearrangement inequality (Theorem 3.8 [3]) and the conformally invariant property of (3.27) below.", "For nonnegative functions $f_{j}\\in L^{p_{j}}(\\mathbb {R}^{n})$ , $j=1, \\dots , N$ and $p_{j}>1$ , $\\sum \\limits _{j=1}^{N}\\frac{1}{p_{j}}>1$ .", "Let $0 \\le r_{ij} = r_{j i} <n $ be real numbers satisfying $\\displaystyle { \\sum _{j=1}^{N} } \\frac{1}{p_{j}^{\\prime }}= \\frac{1}{n} \\sum \\limits _{1 \\le i<j \\le N} r_{ij} \\qquad \\mathrm {(3.25),}$ and for every $j$ $\\frac{1}{p_{j}^{\\prime }} = \\frac{1}{2n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij} \\qquad \\mathrm {(3.26)}$ with $p_{j}$ and $p_{j}^{\\prime }$ dual exponents.", "Then $\\int _{(\\mathbb {R}^{n})^{N}} \\prod \\limits _{j=1}^{N} f_{j}(y_{j}) \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{-r_{ij}} dy_{1} \\dots dy_{N}\\le C_{p_{j}, r_{ij}, n , N} \\prod \\limits _{j=1}^{N} \\Vert f_{j}\\Vert _{p_{j}} \\qquad \\mathrm {(3.27)}$ Condition (3.25) follows from homogeneity.", "Condition (3.26) is to ensure conformal invariance of inequality (3.27).", "Similarly to the arguments in the alternative proof of part (a) of Theorem 3.1, we have the following theorem.", "Theorem 3.7 Let $ r_{ij} >0 $ and $r_{i j} = r_{j i}$ .", "Let $f_{j}$ be nonnegative measurable functions defined on $\\mathbb {R}^{n}$ , $1 \\le j \\le N$ .", "Then $\\displaystyle {\\prod _{j=1}^{N}} \\Vert f_{j}\\Vert _{p_{j}} \\le C_{p_{j}, r_{ij}, n , N} \\ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j}) \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{r_{ij}}, \\qquad \\mathrm {(3.28)}$ holds for any $0 < p_{j} < \\infty $ satisfying $\\frac{1}{p_{j}}= \\frac{1}{2n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij}, \\ \\ \\displaystyle {\\sum _{j=1}^{N}} \\frac{1}{p_{j}}= \\frac{1}{n} \\sum \\limits _{1 \\le i<j \\le N} r_{ij}.", "\\qquad \\mathrm {(3.29)}$ For any $ r_{ij} >0 $ , denote $\\alpha =\\displaystyle {\\sum _ {i \\ne j}} r_{ij}$ , then it is easy to see (3.28) is equivalent to the following inequality.", "$\\displaystyle {\\prod _{j=1}^{N}} \\Vert f_{j}^{1/\\alpha }\\Vert _{p_{j} \\alpha } \\le C_{p_{j}, r_{ij}, n , N}^{1/\\alpha } \\ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j})^{1/\\alpha } \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{\\frac{r_{ij}}{\\alpha }}.$ Below it is enough to show that $\\displaystyle {\\prod _{j=1}^{N}} \\Vert f_{j}\\Vert _{p_{j} \\alpha } \\lesssim \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j}) \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{\\frac{r_{ij}}{\\alpha }}.", "\\qquad \\mathrm {(3.30)}$ holds for any $f_{j} \\in L^{p_{j}\\alpha }(\\mathbb {R}^{n})$ with $0 < p_{j}\\alpha < \\infty $ satisfying $\\frac{1}{p_{j}\\alpha }= \\frac{1}{2n} \\displaystyle {\\sum _ {i \\ne j}} \\frac{r_{ij}}{\\alpha }, \\ \\ \\displaystyle {\\sum _{j=1}^{N}} \\frac{1}{p_{j}\\alpha }= \\frac{1}{n} \\sum \\limits _{1 \\le i<j \\le N} \\frac{r_{ij}}{\\alpha }.", "\\qquad \\mathrm {(3.31)}$ Suppose $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j}) \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{\\frac{r_{ij}}{\\alpha }} = A < \\infty $ .", "We can write $& \\ \\ \\ \\ \\ \\Vert f_{1}\\Vert _{p_{1}\\alpha }^{p_{1}\\alpha } \\cdots \\Vert f_{N}\\Vert _{p_{N}\\alpha }^{p_{N}\\alpha } \\\\&= \\int _{(\\mathbb {R}^{n})^{N}} \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j})^{p_{j}\\alpha } dy_{1} \\dots dy_{N} \\\\&= \\int _{(\\mathbb {R}^{n})^{N}} \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j}) \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{\\frac{r_{ij}}{\\alpha }}\\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j})^{p_{j}\\alpha -1} \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{-\\frac{r_{ij}}{\\alpha }}dy_{1} \\dots dy_{N} \\\\&\\le A \\int _{(\\mathbb {R}^{n})^{N}} \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j})^{p_{j}\\alpha -1}\\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{-\\frac{r_{ij}}{\\alpha }} dy_{1} \\dots dy_{N}.$ Since every $p_{j}\\alpha $ satisfies (3.31) and $\\sum \\limits _{1 \\le i<j \\le N} \\frac{r_{ij}}{\\alpha }=1$ , we have $(p_{j}\\alpha )^{\\prime }>1$ and $\\sum \\limits _{j=1}^{N}\\frac{1}{(p_{j}\\alpha )^{\\prime }}>1$ .", "This allows us to apply inequality (3.27) to get $&\\ \\ \\ \\ \\ \\int _{(\\mathbb {R}^{n})^{N}} \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j})^{p_{j}\\alpha -1} \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{-\\frac{r_{ij}}{\\alpha }} dy_{1} \\dots dy_{N} \\\\&=\\int _{(\\mathbb {R}^{n})^{N}} \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j})^{\\frac{p_{j}\\alpha }{(p_{j}\\alpha )^{\\prime }}}\\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{-\\frac{r_{ij}}{\\alpha }} dy_{1} \\dots dy_{N} \\\\&\\lesssim \\displaystyle {\\prod _{j=1}^{N}} \\Vert f_{j}^{\\frac{p_{j}\\alpha }{(p_{j}\\alpha )^{\\prime }}}\\Vert _{(p_{j}\\alpha )^{\\prime }}= \\displaystyle {\\prod _{j=1}^{N}} \\Vert f_{j}\\Vert _{p_{j}\\alpha }^{p_{j}\\alpha -1}.$ Combining them together gives $\\Vert f_{1}\\Vert _{p_{1}\\alpha }^{p_{1}\\alpha } \\cdots \\Vert f_{k+1}\\Vert _{p_{N}\\alpha }^{p_{N}\\alpha } \\lesssim A \\ \\displaystyle {\\prod _{j=1}^{N}} \\ \\Vert f_{j}\\Vert _{p_{j}\\alpha }^{p_{j}\\alpha -1}.$ This implies that $\\displaystyle {\\prod _{j=1}^{N}} \\Vert f_{j}\\Vert _{p_{j}\\alpha } \\lesssim A=\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{N}} f_{j}(y_{j}) \\prod \\limits _{1 \\le i < j \\le N} |y_{i}-y_{j}|^{\\frac{r_{ij}}{\\alpha }},$ which gives (3.30).", "Therefore by the equivalence as discussed above, this completes the proof of Theorem 3.7.", "Question.", "Similarly to the proof of Theorem 3.5, it is not hard to see the necessary conditions for inequality (3.28) to hold are homogeneity condition and for every $j$ , $1 \\le j \\le N$ , $\\frac{1}{p_{j}} < \\frac{1}{n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij}.", "$ We have already shown that it is sufficient for (3.28) to hold in the trilinear case together with the homogeneity condition.", "An interesting problem is whether inequality (3.28) holds for any $p_{j}$ satisfying $\\frac{1}{p_{j}} < \\frac{1}{n} \\displaystyle {\\sum _ {i \\ne j}} r_{ij}, \\ \\ \\ \\displaystyle { \\sum _{j=1}^{N} } \\frac{1}{p_{j}}= \\frac{1}{n} \\sum \\limits _{1 \\le i<j \\le N} r_{ij},$ where $N>3$ ." ], [ "Sharp versions for geometric inequalities", "1.", "Sharp constant for bilinear geometric inequality    Theorem 4.1 Let $0<p< \\infty $ and $f$ , $g$ be in $ L^{p}(\\mathbb {R}^{n})$ .", "For the geometric inequality $\\Vert f\\Vert _{L^{p}(\\mathbb {R}^{n})} \\Vert g\\Vert _{L^{p}(\\mathbb {R}^{n})} \\le C_{p,n} \\ \\displaystyle {\\sup _{x,y}} \\ f(x)g(y)|x-y|^{\\frac{2n}{p}}, \\qquad \\mathrm {(4.1)}$ the minimum constant $C_{p,n}$ is obtained for $ f= const \\cdot h$ , and $ g= const \\cdot h$ , where $h(x)=(1+|x|^{2})^{-\\frac{n}{p}}.$ Later we can see the sharp constant $C_{p,n}= 2^{-\\frac{2n}{p}} |\\mathbb {S}^{n}|^{\\frac{2}{p}}$ , where $|\\mathbb {S}^{n}|$ is the surface area of the unit sphere $\\mathbb {S}^{n}$ .", "Let $q \\in (0, \\infty )$ , $p \\in (1, \\infty )$ .", "Form $\\displaystyle {\\sup _{x,y}} \\ f(x)^{\\frac{p}{q}} g(y)^{\\frac{p}{q}} |x-y|^{\\frac{2n}{q}}= (\\displaystyle {\\sup _{x,y}} \\ f(x) g(y)|x-y|^{\\frac{2n}{p}})^{\\frac{p}{q}}, $ and $\\Vert f^{\\frac{p}{q}}\\Vert _{L^{q}(\\mathbb {R}^{n})}= ( \\Vert f\\Vert _{L^{p}(\\mathbb {R}^{n})})^{\\frac{p}{q}}, \\ \\Vert g^{\\frac{p}{q}}\\Vert _{L^{q}(\\mathbb {R}^{n})}= ( \\Vert g\\Vert _{L^{p}(\\mathbb {R}^{n})})^{\\frac{p}{q}},$ we observe that if $f, g$ is a pair of extremals for $p \\in (1, \\infty )$ , then $f^{\\frac{p}{q}}$ , $g^{\\frac{p}{q}}$ is a pair of extremals for any $q \\in (0, \\infty )$ .", "So it suffices to study the extremals for the case when $1< p <\\infty $ .", "In this section, we only consider such nonnegative measurable functions $f$ , $g$ that the right hand side of (4.1) is finite.", "For every nonnegative measurable function $f$ , its layer cake representation is $f(x)= \\int _{0}^{\\infty } \\chi _{\\lbrace f>t\\rbrace } (x) dt $ , where $ \\chi _{\\lbrace f>t\\rbrace }$ is the characteristic function of the level set $ \\lbrace x: f(x)>t \\rbrace $ .", "For $A\\subset \\mathbb {R}^{n} $ of finite Lebesgue measure, we define the symmetric rearrangement of $A$ as $A^{\\ast }:=\\lbrace x: |x|<r \\rbrace \\equiv B (0, r)$ with $|A^{\\ast }| =|A|$ .", "That is, $r^{n}=\\frac{|A|}{v_{n}}$ , and $v_{n}$ is the volume of unit ball in $\\mathbb {R}^{n}$ .", "We then define the symmetric decreasing rearrangement of nonnegative measurable function $f$ as $\\mathcal {R}f(x)=f^{\\ast } (x):= \\int _{0}^{\\infty } \\chi _{\\lbrace f>t\\rbrace ^{\\ast }} (x) dt, $ and define the Steiner symmetrisation of $f$ with respect to the $j$ -th coordinate as $\\mathcal {R}_{j}f(x_{1}, \\dots ,x_{n})=f^{\\ast j}(x_{1}, \\dots ,x_{n}) := \\int _{0}^{\\infty } \\chi _{\\lbrace f(x_{1}, \\dots ,x_{j-1}, \\cdot , x_{j+1}, \\dots , x_{n})>t\\rbrace ^{\\ast }}(x_{j}) dt.$ We observe that $f$ and $f^{\\ast }$ are equimeasurable which means $|\\lbrace x: f(x) >t\\rbrace | = | \\lbrace x: f^{\\ast }(x) > t\\rbrace |.$ Together with the layer cake representation of $f$ , hence $\\Vert f\\Vert _{p} = \\Vert \\mathcal {R}f\\Vert _{p}$ for any $f\\in L^{p}(\\mathbb {R}^{n})$ , $1\\le p \\le \\infty $ .", "Besides, $\\Vert f\\Vert _{p}= \\Vert \\mathcal {R}_{n} \\dots \\mathcal {R}_{1}f \\Vert _{p}$ follows from Fubini's theorem.", "We recall another related decreasing rearrangement of $f$ defined on $[0, \\infty )$ as $f_{\\ast }(t)= \\inf \\lbrace \\lambda > 0: m_{f}(\\lambda ) \\le t\\rbrace ,$ where $m_{f}$ is the distrution function of $f$ , $m_{f}(\\lambda ) := |\\lbrace x\\in \\mathbb {R}^{n}: f(x)> \\lambda \\rbrace |.$ Then it is easy to see for any $x \\in \\mathbb {R}^{n}$ , $f^{\\ast }(x)= f_{\\ast }(v_{n} |x|^{n} ).$ As is well known, for $0 \\le s, t< \\infty $ $ f_{\\ast }(s) > t$ if and only if $| \\lbrace x \\in \\mathbb {R}^{n}: f(x) >t \\rbrace | >s$ .", "By the relation of $f^{\\ast }$ and $f_{\\ast }$ , we have for any $s\\in \\mathbb {R}^{n}$ , $t \\ge 0$ $ f^{\\ast }(s) > t$ if and only if $| \\lbrace x \\in \\mathbb {R}^{n}: f(x) >t \\rbrace | >v_{n} |s|^{n}$ .", "Lemma 4.2 Let $f, g$ be defined on $\\mathbb {R}^{n}$ , then $\\displaystyle {\\sup _{s,t}} f^{\\ast }(s) g^{\\ast }(t) |s-t| \\le \\displaystyle {\\sup _{x,y}} f(x) g(y) |x-y|.", "\\qquad \\mathrm {(4.2)}$ Suppose $\\displaystyle {\\sup _{x,y}} f(x) g(y) |x-y|=A$ .", "We assume for a contradiction that $\\displaystyle {\\sup _{s,t}} f^{\\ast }(s) g^{\\ast }(t) |s-t| > A.$ Then there exist positive $\\varepsilon $ and a set $ G \\subset \\mathbb {R}^{n} \\times \\mathbb {R}^{n}$ such that $|G|>0$ and for all $(s_{0}, t_{0}) \\in G$ we have $f^{\\ast }(s_{0}) g^{\\ast }(t_{0}) |s_{0}-t_{0} | > A+\\varepsilon .$ It follows from $f^{\\ast }(s_{0}) > (A+ \\varepsilon ) (g^{\\ast }(t_{0}) | s_{0}-t_{0} |)^{-1}$ and the property of decreasing rearrangement discussed above that $ | \\lbrace x: f(x) > (A+\\varepsilon ) (g^{\\ast }(t_{0}) | s_{0}-t_{0} |)^{-1} \\rbrace |> v_{n} |s_{0}|^{n}.", "\\qquad \\mathrm {(4.3)}$ Denote the set $\\lbrace x: f(x) > (A+\\varepsilon ) (g^{\\ast }(t_{0}) | s_{0}-t_{0} |)^{-1} \\rbrace $ by $E$ , so $g^{\\ast }(t_{0}) >( A+\\frac{\\varepsilon }{2}) \\ (\\displaystyle {\\inf _{x\\in E }} f(x) | s_{0}-t_{0} | )^{-1}.$ Applying the property of decreasing rearrangement again, we have $| \\lbrace y: g(y) > ( A+\\frac{\\varepsilon }{2}) (\\displaystyle {\\inf _{x\\in E }} f(x) | s_{0}-t_{0} | )^{-1} \\rbrace |> v_{n} |t_{0}|^{n}.", "\\qquad \\mathrm {(4.4)}$ Denote the set $\\lbrace y: g(y) >( A+\\frac{\\varepsilon }{2}) (\\displaystyle {\\inf _{x\\in E }} f(x) | s_{0}-t_{0} | )^{-1} \\rbrace $ by $F$ .", "Then $s_{0}\\in E^{\\ast }, t_{0}\\in F^{\\ast }$ .", "It turns out that $ \\displaystyle {\\sup _{x\\in E, y\\in F}} |x-y| \\ge | s_{0}-t_{0} |.", "\\qquad \\mathrm {(4.5)}$ The reason is as follows.", "In the first place, it is easy to observe for any measurable set $C\\subset \\mathbb {R}^{n} $ $ \\displaystyle {\\sup _{x\\in C}} \\ |x| \\ge \\displaystyle {\\sup _{x\\in C^{\\ast }}} \\ |x|.$ If $\\displaystyle {\\sup _{x\\in C}} |x| < \\displaystyle {\\sup _{x\\in C^{\\ast }}} |x| \\equiv s$ , there exist positive $\\delta $ and a measure zero set $M \\subset \\mathbb {R}^{n}$ , such that $|x| < s-\\delta $ for any $x \\in C \\setminus M$ .", "So $C \\setminus M \\subset B(0, s-\\delta ),$ where $B(0, s-\\delta )$ is the ball centred at 0 with radius $s-\\delta $ .", "This shows $(C \\setminus M)^{\\ast } = C^{\\ast } \\subset B(0, s-\\delta )$ , which is a contradiction.", "Hence based on the property $\\displaystyle {\\sup _{x\\in C}} \\ |x| \\ge \\displaystyle {\\sup _{x\\in C^{\\ast }}} \\ |x|$ for any set $C\\subset \\mathbb {R}^{n}$ of finite Lebesgue measure, we have $\\displaystyle {\\sup _{x\\in E, y\\in F}} |x-y| = \\displaystyle {\\sup _{z\\in E- F}} |z| \\ge \\displaystyle {\\sup _{z\\in (E- F)^{\\ast }}} |z|.$ The Brunn-Minkowski inequality tells for measurable sets with finite volume $E$ and $F$ , $|E-F|^{1/n} \\ge |E|^{1/n}+|F|^{1/n}.", "$ By the definition of symmetric rearrangement of $E$ and $F$ , we have $ E^{\\ast } = B(0, r_{1} ), \\ \\ F^{\\ast } = B(0, r_{2}), $ where their radius are $r_{1}=(\\frac{|E|}{v_{n}})^{1/n}$ , $r_{2}= (\\frac{|F|}{v_{n}})^{1/n} $ respectively.", "Then $E^{\\ast }+F^{\\ast }$ is the ball centred at 0 with radius $r_{1}+r_{2}$ , and $E^{\\ast }- F^{\\ast } = E^{\\ast }+ F^{\\ast } = B(0, r_{1}+r_{2} ).$ Together with the Brunn-Minkowski inequality, we have $|(E-F)^{\\ast }|^{1/n} &=|E-F|^{1/n} \\ge |E|^{1/n}+|F|^{1/n} \\\\&=|E^{\\ast }|^{1/n}+|F^{\\ast }|^{1/n}\\\\&= v_{n}^{1/n} r_{1} + v_{n}^{1/n} r_{2},$ which means $|(E-F)^{\\ast }| \\ge v_{n} (r_{1} + r_{2})^{n}= | E^{\\ast }+F^{\\ast }|.$ Therefore $s_{0}- t_{0}\\in E^{\\ast }-F^{\\ast }= E^{\\ast }+F^{\\ast } \\subset (E-F)^{\\ast }.$ Moreover $ \\displaystyle {\\sup _{x\\in E, y\\in F}} |x-y| = \\displaystyle {\\sup _{z\\in E- F}} |z| \\ge \\displaystyle {\\sup _{z\\in (E- F)^{\\ast }}} |z| \\ge \\displaystyle {\\sup _{x\\in E^{\\ast }, y \\in F^{\\ast } }} |x-y| \\ge | s_{0}-t_{0} |, \\qquad \\mathrm {(4.6)}$ which completes the proof of (4.5).", "Now (4.4) implies that for any $x \\in E$ , $y\\in F$ $ f(x) g(y) |x-y| > (A + \\frac{\\varepsilon }{2} ) | s_{0}-t_{0} | ^{-1} |x-y|,$ thus $ \\displaystyle {\\sup _{x\\in E, y\\in F}} f(x) g(y) |x-y| \\ge (A + \\frac{\\varepsilon }{2} ) | s_{0}-t_{0} | ^{-1} \\displaystyle {\\sup _{x\\in E, y\\in F}} |x-y|.$ Consequently, together with (4.5) we get $\\displaystyle {\\sup _{x,y}} f(x) g(y) |x-y|& \\ge (A + \\frac{\\varepsilon }{2} ) | s_{0}-t_{0} | ^{-1} \\displaystyle {\\sup _{x\\in E, y\\in F}} |x-y| \\\\& \\ge (A + \\frac{\\varepsilon }{2} ) | s_{0}-t_{0} | ^{-1} | s_{0}-t_{0} | \\\\&>A.$ This is a contradiction.", "However, we do not know when there is equality in (4.2).", "One might guess that strict inequality (4.2) holds only if $f(x)=f^{\\ast }(x-y)$ and $g(y)=f^{\\ast }(x-y)$ for some $y$ in $\\mathbb {R}^{n}$ .", "By the following counterexample, we show that this is not true.", "In the one-dimensional case, let $f(x)=4 \\chi _{|x| \\le |E_{1}|}+ \\chi _{|E_{1}| < x \\le |E_{1}|+ 2|E_{2}|}$ with $|E_{1}|>|E_{2}|$ , and $f=g$ .", "Then $f^{\\ast }(x)=4 \\chi _{|x| \\le |E_{1}|}+ \\chi _{|E_{1}| <|x| \\le |E_{1}|+ |E_{2}|}.$ It is easy to check that $\\displaystyle {\\sup _{x,y}} f(x) g(y) |x-y|= \\max \\lbrace 32|E_{1}|, 8(|E_{1}|+|E_{2}|), 2|E_{2}| \\rbrace = 32|E_{1}|,$ and $\\displaystyle {\\sup _{x,y}} f^{\\ast }(x) g^{\\ast }(y) |x-y| = \\max \\lbrace 32|E_{1}|, 4(2|E_{1}|+|E_{2}|), 2(|E_{1}|+|E_{2}|) \\rbrace = 32|E_{1}|.$ So there are other classes of examples where equality holds.", "Due to Lemma 4.2, it suffices to seek optimisers amongst the class of all symmetric decreasing functions.", "Let $\\mathcal {S}$ be the stereographic projection from $\\mathbb {R}^{n}$ to the unit sphere $\\mathbb {S}^{n}$ with $\\mathcal {S}(x)=(\\frac{2x_{1}}{1+|x|^{2}}, \\dots , \\frac{2x_{n}}{1+|x|^{2}}, \\frac{1-|x|^{2}}{1+|x|^{2}}),$ where $ x=( x_{1}, \\dots , x_{n}) \\in \\mathbb {R}^{n}$ .", "So $\\mathcal {S}^{-1} (s) = (\\frac{s_{1}}{1+s_{n+1}}, \\dots , \\frac{s_{n}}{1+s_{n+1}} ).$ For $f\\in L^{p}(\\mathbb {R}^{n})$ , define $(\\mathcal {S}^{\\ast }f)(s):= |J_{\\mathcal {S}^{-1}}(s)|^{1/p}f(\\mathcal {S}^{-1}(s)), \\ (\\mathcal {S}^{\\ast }g)(t):=|J_{\\mathcal {S}^{-1}}(t)|^{1/p}g(\\mathcal {S}^{-1}(t)), \\qquad \\mathrm {(4.7)}$ where $J_{\\mathcal {S}^{-1}}$ is the Jacobian determinant of the map $\\mathcal {S}^{-1}$ $ |J_{\\mathcal {S}^{-1}}(s)| = (\\frac{1}{1+ s_{n+1}})^{n}= \\frac{1}{2^{n}} (1+|\\mathcal {S}^{-1} (s)|^{2})^{n}.", "\\qquad \\mathrm {(4.8)}$ Then we have the invariance of the geometric inequality under the stereographic projection shown as the following lemma.", "Lemma 4.3 For $f, g \\in L^{p}(\\mathbb {R}^{n})$ , denote $F(s)=(\\mathcal {S}^{\\ast }f)(s), G(t)= (\\mathcal {S}^{\\ast }g)(t)$ .", "Then $\\displaystyle {\\sup _{x,y\\in \\mathbb {R}^{n}}} f(x) g(y) |x-y|^{\\frac{2n}{p}} =\\displaystyle {\\sup _{s,t\\in \\mathbb {S}^{n}}} F(s) G(t) |s-t|^{\\frac{2n}{p}},$ and $\\Vert F\\Vert _{L^{p}(\\mathbb {S}^{n})}= \\Vert f\\Vert _{L^{p}(\\mathbb {R}^{n})}, \\ \\Vert G\\Vert _{L^{p}(\\mathbb {S}^{n})} = \\Vert g\\Vert _{L^{p}(\\mathbb {R}^{n})}.$ By the stereographic projection $\\mathcal {S}$ , we have (4.8) $| J_{\\mathcal {S}^{-1}}(s)|=( \\frac{1+|x|^{2}}{2})^{n},$ and let $x=\\mathcal {S}^{-1} (s), y=\\mathcal {S}^{-1}(t)$ , then $|x-y|=|s-t| (\\frac{1+|x|^{2}}{2})^{1/2} (\\frac{1+|y|^{2}}{2})^{1/2}= |J_{\\mathcal {S}^{-1}}(s)|^{\\frac{1}{2n}}|J_{\\mathcal {S}^{-1}}(t)|^{\\frac{1}{2n}} |s-t|.$ So $\\displaystyle {\\sup _{s,t\\in \\mathbb {S}^{n}}} F(s) G(t) |s-t|^{\\frac{2n}{p}}&= \\displaystyle {\\sup _{s,t\\in \\mathbb {S}^{n}}} |J_{\\mathcal {S}^{-1}}(s)|^{1/p}f(\\mathcal {S}^{-1}(s))|J_{\\mathcal {S}^{-1}}(t)|^{1/p}g(\\mathcal {S}^{-1}(t)) |s-t|^{\\frac{2n}{p}} \\\\&=\\displaystyle {\\sup _{x,y\\in \\mathbb {R}^{n}}} f(x) g(y) |x-y|^{\\frac{2n}{p}}.$ The invariance of $L^{p}$ norm can be obtained as follows, $\\Vert f\\Vert _{L^{p}(\\mathbb {R}^{n})} = (\\int _{\\mathbb {R}^{n}} |f(x)|^{p} dx)^{1/p}= (\\int _{\\mathbb {S}^{n}} |f(\\mathcal {S}^{-1}(s))|^{p} |J_{\\mathcal {S}^{-1}}(s)| ds)^{1/p}= (\\int _{\\mathbb {S}^{n}} |F(s)|^{p} ds)^{1/p}.$ Applying a similar argument implies $\\Vert g\\Vert _{L^{p}(\\mathbb {R}^{n})}=\\Vert G\\Vert _{L^{p}(\\mathbb {S}^{n})}$ .", "Now we turn to study the sharp case of inequality (4.1).", "Proof of Theorem 4.1    For $f\\in L^{p}(\\mathbb {R}^{n})$ , consider a rotation $D: \\mathbb {S}^{n} \\rightarrow \\mathbb {S}^{n}$ with $D(s)=(s_{1}, \\dots ,s_{n-1}, s_{n+1}, -s_{n}).$ Specifically, it is a rotation of the sphere by $90^{\\circ }$ which keeps the other basis vectors fixed except $n$ -th and $(n+1)$ -th vectors in the direction of mapping the $(n+1)$ -th vector $e_{n+1}= (0, \\dots ,0,1) $ to $n$ -th vector $e_{n}=(0, \\dots ,0,1,0)$ .", "Define $(D^{\\ast }F)(s)= |J_{D^{-1}}(s)|^{\\frac{1}{p}} F(D^{-1}(s)) = F(D^{-1}(s)) $ for any $F \\in L^{p}(\\mathbb {S}^{n})$ .", "Then $\\Vert D^{\\ast } F \\Vert _{p}= \\Vert F \\Vert _{p},$ which shows $D^{\\ast }$ is norm preserving.", "We consider the new function $(\\mathcal {S}^{\\ast })^{-1} D^{\\ast } \\mathcal {S}^{\\ast } f$ , where $(\\mathcal {S}^{\\ast }f)(s)$ is the same as $ (4.7)$ .", "Denote $ (\\mathcal {S}^{\\ast }f)(s)$ by $ F(s)$ , and let $x= \\mathcal {S}^{-1} (s)$ .", "From the discussion above, we have already shown $F(s)= (\\frac{1+|x|^{2}}{2})^{\\frac{n}{p}} f(x).$ The definition of $D$ and $ \\mathcal {S}$ implies $D^{-1} (s)=(s_{1}, \\dots ,s_{n-1}, -s_{n+1}, s_{n})=( \\frac{2x_{1}}{1+|x|^{2}}, \\dots , \\frac{2x_{n}}{1+|x|^{2}}, \\frac{|x|^{2}-1}{1+|x|^{2}}, \\frac{2x_{n}}{1+|x|^{2}} ).$ Then $ (D^{\\ast } \\mathcal {S}^{\\ast } f)(s)= (D^{\\ast }F)(s)= F(D^{-1}(s))= (\\frac{1+|x|^{2}}{|x+e_{n}|})^{\\frac{n}{p}} f( \\frac{2x_{1}}{|x+e_{n}| ^{2}}, \\dots , \\frac{2x_{n-1}}{|x+e_{n}|^{2}}, \\frac{|x|^{2}-1}{|x+e_{n}|^{2}} ), $ this is because $\\mathcal {S}^{-1} (D^{-1} (s)) = ( \\frac{2x_{1}}{|x+e_{n}| ^{2}}, \\dots , \\frac{2x_{n-1}}{|x+e_{n}|^{2}}, \\frac{|x|^{2}-1}{|x+e_{n}|^{2}} ).$ Finally we find $( \\mathcal {S}^{\\ast -1} D^{\\ast } \\mathcal {S}^{\\ast }f ) (x) &= ( \\frac{1+|x|^{2}}{2})^{-\\frac{n}{p}} F(D^{-1}(s)) \\\\&= (\\frac{2}{|x+e_{n}|})^{\\frac{n}{p}} f( \\frac{2x_{1}}{|x+e_{n}| ^{2}}, \\dots , \\frac{2x_{n-1}}{|x+e_{n}|^{2}}, \\frac{|x|^{2}-1}{|x+e_{n}|^{2}} ).$ Briefly speaking, we lift $f$ to the sphere by (4.7) first, then rotate it by $90^{\\circ }$ in a specific direction which maps the north pole $e_{n+1}$ to $ e_{n}$ , lastly push back to $\\mathbb {R}^{n}$ .", "For simplicity we denote $\\mathcal {S}^{\\ast -1} D^{\\ast } \\mathcal {S}^{\\ast }f$ by $\\mathcal {D}f$ .", "Let $f\\in L^{p}(\\mathbb {R}^{n})$ .", "Applying the transformation $\\mathcal {D}$ and the symmetric rearrangement to $f$ many times gives the sequence $\\lbrace f_{k}\\rbrace _{k \\in \\mathbb {N}}$ .", "Specifically, $f_{0}=f$ , $f_{k} =(\\mathcal {R}\\mathcal {D})^{k}f$ .", "Note that both $\\mathcal {D}$ and $\\mathcal {R}$ are norm-preserving.", "This is because Lemma 4.3 implies $ \\Vert \\mathcal {S}^{\\ast -1} D^{\\ast } \\mathcal {S}^{\\ast }f \\Vert _{p} = \\Vert D^{\\ast } \\mathcal {S}^{\\ast }f \\Vert _{p}.$ Due to the norm preserving property of $ D^{\\ast }$ , we have $ \\Vert D^{\\ast } \\mathcal {S}^{\\ast }f \\Vert _{p} = \\Vert \\mathcal {S}^{\\ast }f \\Vert _{p}$ So apply Lemma 4.3 again to get $ \\Vert \\mathcal {S}^{\\ast -1} D^{\\ast } \\mathcal {S}^{\\ast }f \\Vert _{p} = \\Vert \\mathcal {S}^{\\ast }f \\Vert _{p}= \\Vert f \\Vert _{p}.$ It follows from Theorem 4.6 in Lieb-Loss [3] that for all $ f \\in L^{p}(\\mathbb {R}^{n})$ , the sequence $f_{k}$ converges to $h_{f}$ in $L^{p}$ norm as $k \\rightarrow \\infty $ .", "Here $ h_{f}= c \\ h, \\ \\ h(x)=(1+|x|^{2})^{-\\frac{n}{p}}$ and $c$ is the constant such that $\\Vert f\\Vert _{p} = \\Vert h_{f}\\Vert _{p}$ , so the constant $c$ is $c=2^{\\frac{n}{p}} |\\mathbb {S}^{n}|^{-1/p} \\Vert f\\Vert _{p} ,$ where $|\\mathbb {S}^{n}|$ means the area of unit sphere in $\\mathbb {R}^{n+1}$ .", "Since $f_{k}$ converges to $ h_{f}$ in $L^{p}$ norm for all f $ \\in L^{p}(\\mathbb {R}^{n})$ , there exist subsequences $\\lbrace f_{k_{l}} \\rbrace , \\lbrace g_{k_{l}} \\rbrace $ such that $ f_{k_{l}} \\rightarrow h_{f}$ and $ g_{k_{l}} \\rightarrow h_{g}$ pointwise almost everywhere as $l \\rightarrow \\infty $ .", "Clearly, Lemma 4.2, Lemma 4.3 and the rearrangement property $(f^{p})^{\\ast }= (f^{\\ast })^{p}$ , for $0<p<\\infty $ indicate that $ \\displaystyle {\\sup _{x,y}} f_{k}(x) g_{k}(y) |x-y|^{\\frac{2n}{p}} $ decreases monotonically as $k$ grows.", "Hence for all $ x, y, k_{l}$ $ f_{k_{l}}(x) g_{k_{l}}(y) |x-y|^{\\frac{2n}{p}} \\le \\displaystyle {\\sup _{x,y}} f(x) g(y) |x-y|^{\\frac{2n}{p}} < \\infty .$ Together with the dominated convergence theorem it follows that $f_{k_{l}}(x) g_{k_{l}}(y) |x-y|^{\\frac{2n}{p}} \\xrightarrow{} h_{f}(x) h_{g}(y) |x-y|^{\\frac{2n}{p}} $ in $ L^{\\infty }(\\mathbb {R}^{n} \\times \\mathbb {R}^{n} )$ as $l \\rightarrow \\infty $ .", "Hence by the $\\mathrm {weak}^*$ lower semicontinuity of the $L^{\\infty }$ norm we have $\\displaystyle {\\sup _{x,y}} \\ h_{f}(x) h_{g}(y) |x-y|^{\\frac{2n}{p}}&\\le \\displaystyle {\\liminf _{l}} \\ (\\displaystyle {\\sup _{x,y}} \\ f_{k_{l}}(x) g_{k_{l}}(y) |x-y|^{\\frac{2n}{p}} ) \\\\&= \\displaystyle {\\inf _{l}} \\ (\\displaystyle {\\sup _{x,y}} \\ f_{k_{l}}(x) g_{k_{l}}(y) |x-y|^{\\frac{2n}{p}} ).$ Therefore for every $f, g \\in L^{p}(\\mathbb {R}^{n})$ and every $k_{l}$ $ \\frac{ \\displaystyle {\\sup _{x,y}} \\ f(x) g(y) |x-y|^{\\frac{2n}{p}} }{ \\Vert f \\Vert _{p} \\Vert g\\Vert _{p} } \\ge \\frac{ \\displaystyle {\\sup _{x,y}} \\ f_{k_{l}}(x) g_{k_{l}}(y) |x-y|^{\\frac{2n}{p}} }{ \\Vert f_{k_{l}} \\Vert _{p} \\Vert g_{k_{l}}\\Vert _{p}}\\ge \\frac{ \\displaystyle {\\sup _{x,y}} \\ h_{f}(x) h_{g}(y) |x-y|^{\\frac{2n}{p}} }{ \\Vert h_{f} \\Vert _{p} \\Vert h_{g}\\Vert _{p}},$ because of the norm-preserving of $\\mathcal {R}\\mathcal {D}$ .", "Obviously, $\\frac{ \\displaystyle {\\sup _{x,y}} \\ h_{f}(x) h_{g}(y) |x-y|^{\\frac{2n}{p}} }{ \\Vert h_{f} \\Vert _{p} \\Vert h_{g}\\Vert _{p}}= \\frac{ \\displaystyle {\\sup _{x,y}} \\ h(x) h(y) |x-y|^{\\frac{2n}{p}} }{ \\Vert h\\Vert _{p}^{2} }.$ $\\hfill \\Box $ Therefore, the conformally invariant property of (4.1) implies that if $f$ and $g$ are the same conformal transformation of $h$ , equality still holds.", "However, here we can not characterise the optimisers.", "From the sharp version for $\\mathbb {R}^{n}$ case in Theorem 4.1 together with (4.7), (4.8) and the conformally invariant property in Lemma 4.3, it follows that the geometric inequality (4.1) has conformally equivalent form on the unit sphere $\\mathbb {S}^{n}$ as follows.", "Theorem 4.4 For $0 < p < \\infty $ , let $F, G$ be nonnegative functions in $L^{p}(\\mathbb {S}^{n})$ .", "Then $\\Vert F\\Vert _{L^{p}(\\mathbb {S}^{n})} \\ \\Vert G\\Vert _{L^{p}(\\mathbb {S}^{n})} \\le B_{p,n} \\ \\displaystyle {\\sup _{s,t\\in \\mathbb {S}^{n}}} \\ F(s) G(t) \\ |s-t|^{\\frac{2n}{p}}.", "\\qquad \\mathrm {(4.9)}$ The best constant $B_{p,n}$ is obtained for $F$ , $G$ are constant functions, and the corresponding $B_{p,n} = 2^{-\\frac{2n}{p}} |\\mathbb {S}^{n}|^{\\frac{2}{p}}$ .", "Meanwhile, let $\\mathbb {H}^{n}$ be the hyperbolic space in $\\mathbb {R}^{n+1}$ : $\\mathbb {H}^{n}= \\lbrace q=(q_{1}, \\dots ,q_{n}, q_{n+1}) \\in \\mathbb {R}^{n}\\times \\mathbb {R}: q_{1}^{2}+ \\dots + q_{n}^{2}- q_{n+1}^{2}= -1 )\\rbrace ,$ with the Lorenz group $O(1,n)$ invariant measure $d \\nu (q)$ .", "We find the geometric inequality (4.1) also has the conformally equivalent form in $\\mathbb {H}^{n}$ space as shown in the following theorem.", "Theorem 4.5 For $0 < p < \\infty $ , let $F, G$ be nonnegative functions in $L^{p}(\\mathbb {H}^{n})$ .", "Then $\\Vert F\\Vert _{L^{p}(\\mathbb {H}^{n})} \\ \\Vert G\\Vert _{L^{p}(\\mathbb {H}^{n})}\\le E_{p, n} \\ \\displaystyle {\\sup _{q,t}} \\ F(q) G(t) \\ |qt-1|^{\\frac{n}{p}}, \\qquad \\mathrm {(4.10)}$ $qt= -q_{1}t_{1}- \\dots -q_{n}t_{n}+q_{n+1}t_{n+1}$ .", "The best constant $E_{p, n}$ is obtained when $F= const \\cdot H$ , $G= const \\cdot H$ , where $H(q)= |q_{n+1}|^{-\\frac{n}{p}}, \\ q=(q_{1}, \\dots ,q_{n}, q_{n+1}) .$ Consider the stereographic projection $\\mathcal {H}$ which is conformal transformation from $\\mathbb {R}^{n} \\backslash \\lbrace |x|=1\\rbrace $ to $\\mathbb {H}^{n}$ as $\\mathcal {H}(x)=(\\frac{2x_{1}}{1-|x|^{2}}, \\dots ,\\frac{2x_{n}}{1-|x|^{2}}, \\frac{1+|x|^{2}}{1-|x|^{2}}),$ so $\\mathcal {H}^{-1}(q)= (\\frac{q_{1}}{1+q_{n+1}}, \\dots ,\\frac{q_{n}}{1+q_{n+1}}).$ The Jacobian determinant of the map $\\mathcal {H}^{-1}$ is $|J_{\\mathcal {H}^{-1}}(q)|= (\\frac{1- |\\mathcal {H}^{-1}(q)|^{2}}{2})^{n}.$ Let $x=\\mathcal {H}^{-1}(q), y=\\mathcal {H}^{-1}(t)$ , then we have $|x-y|= (\\frac{|1-|x|^{2}|}{2})^{1/2} (\\frac{|1-|y|^{2}|}{2})^{1/2} |qt-1|^{1/2}=|J_{\\mathcal {H}^{-1}}(q)|^{\\frac{1}{2n}}|J_{\\mathcal {H}^{-1}}(t)|^{\\frac{1}{2n}} |qt-1|^{1/2},$ where $qt= -q_{1}t_{1}- \\dots -q_{n}t_{n}+q_{n+1}t_{n+1}$ .", "Define $F(q):= |J_{\\mathcal {H}^{-1}}(q)|^{1/p} f(\\mathcal {H}^{-1}(q)), \\ G(t):= |J_{\\mathcal {H}^{-1}}(t)|^{1/p} g(\\mathcal {H}^{-1}(t)).$ Thus from the above, we easily get the conformal invariance as follows.", "$\\displaystyle {\\sup _{q,t}} \\ F(q) G(t) |qt-1|^{\\frac{n}{p}}&=\\displaystyle {\\sup _{q,t}} \\ |J_{\\mathcal {H}^{-1}}(q)|^{1/p} f(\\mathcal {H}^{-1}(q)) |J_{\\mathcal {H}^{-1}}(t)|^{1/p} g(\\mathcal {H}^{-1}(t))|qt-1|^{\\frac{n}{p}} \\\\&=\\displaystyle {\\sup _{x,y}} \\ f(x) g(y) |x-y|^{\\frac{2n}{p}},$ and $\\Vert f\\Vert _{L^{p}(\\mathbb {R}^{n})}=(\\int _{\\mathbb {R}^{n}} |f(x)|^{p} dx)^{1/p}=(\\int _{\\mathbb {H}^{n}} |f(\\mathcal {H}^{-1}(q))|^{p} |J_{\\mathcal {H}^{-1}}(q)| dq)^{1/p}=(\\int _{\\mathbb {H}^{n}} |F(q)|^{p} dq)^{1/p} .$ Applying a similar argument gives $\\Vert g\\Vert _{L^{p}(\\mathbb {R}^{n})}=\\Vert G\\Vert _{L^{p}(\\mathbb {H}^{n})}$ .", "When $f(x)=c (1+|x|^{2})^{-\\frac{n}{p}}$ , $F(q)= c \\ (\\frac{1-|x|^{2}}{1+|x|^{2}})^{\\frac{n}{p}}= c \\ |q_{n+1}|^{-\\frac{n}{p}}$ .", "Hence the conformally equivalent form (4.10) follows from Theorem 4.1.", "2.", "Sharp constant for multilinear geometric inequality    Theorem 4.6 Let $0<p< \\infty $ and $f_{j}$ be in $ L^{p}(\\mathbb {R}^{n})$ .", "For multilinear geometric inequality $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert f_{j}\\Vert _{L^{p}(\\mathbb {R}^{n})} \\le C_{p,n} \\ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\gamma } \\qquad \\mathrm {(4.11)}$ with $ \\gamma =\\frac{n+1}{p}$ , the minimum constant is obtained when $ f_{j}= const \\cdot h$ , $1 \\le j \\le n+1$ , where $h (x)=({1+|x|^{2}})^{-\\frac{n+1}{2p}}.$ Later we can see the sharp constant $C_{p,n}=(\\frac{1}{2}|\\mathbb {S}^{n}| )^{\\frac{n+1}{p}}$ , where $|\\mathbb {S}^{n}|$ is the surface area of the unit sphere $\\mathbb {S}^{n}$ .", "As before, it suffices to study the extremals for the case when $1< p <\\infty $ .", "Because for any $q \\in (0, \\infty )$ , $p \\in (1, \\infty )$ , we have $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j})^{\\frac{p}{q}} \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{q}}=(\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}})^{\\frac{p}{q}}.$ and for each $j$ , $\\Vert f_{j}^{\\frac{p}{q}}\\Vert _{L^{q}(\\mathbb {R}^{n})}= ( \\Vert f_{j}\\Vert _{L^{p}(\\mathbb {R}^{n})})^{\\frac{p}{q}}$ Thus if $ \\lbrace f_{j}\\rbrace $ are the extremal functions for $p \\in (1, \\infty )$ , then $\\lbrace f_{j}^{\\frac{p}{q}}\\rbrace $ are the extremal functions for any $q \\in (0, \\infty )$ .", "Lemma 4.7 Let $a_{j} \\in \\mathbb {R}$ and $A_{j}$ be sets in $\\mathbb {R}$ with finite measure, $j=1, \\dots , l$ .", "Then $\\displaystyle {\\sup _{x_{j}\\in A_{j}^{\\ast } }} \\ |\\sum _{j=1}^{l} a_{j}x_{j}| \\le \\displaystyle {\\sup _{x_{j}\\in A_{j} }} \\ |\\sum _{j=1}^{l} a_{j}x_{j}|.", "\\qquad \\mathrm {(4.12)}$ For simplicity we just show $\\displaystyle {\\sup _{x\\in A^{\\ast }, y\\in B^{\\ast }, z\\in C^{\\ast }}}|a x+ b y + c z| \\le \\displaystyle {\\sup _{x\\in A, y\\in B, z\\in C}}|a x+ b y + c z|,\\qquad \\mathrm {(4.13)}$ where $a,b,c \\in \\mathbb {R}$ , $A, B, C\\subset \\mathbb {R}$ with finite measure.", "Then (4.12) can be obtained by induction on $l$ .", "The definition of symmetric rearrangement of set implies $ A^{\\ast } = (-\\frac{|A|}{2}, \\frac{|A|}{2}), \\ B^{\\ast } = (-\\frac{|B|}{2}, \\frac{|B|}{2}),$ so $A^{\\ast }+B^{\\ast }= (-\\frac{|A|+|B|}{2}, \\frac{|A|+|B|}{2}).$ The Brunn-Minkowski inequality tells $|A+B| \\ge |A|+|B|$ , then $ |(A+B)^{\\ast }| = |A+B| \\ge |A|+|B| = |A^{\\ast } +B^{\\ast }|$ which means $(A+B)^{\\ast } \\supset A^{\\ast }+B^{\\ast }.$ Together with the property $(aA)^{\\ast }=a A^{\\ast }$ we have $\\displaystyle {\\sup _{x\\in A^{\\ast }, y\\in B^{\\ast }, z\\in C^{\\ast }}}|a x+ b y + c z|&= \\displaystyle {\\sup _{\\overline{x}\\in a A^{\\ast }, \\overline{y}\\in b B^{\\ast }, \\overline{z}\\in c C^{\\ast } } } | \\overline{x}+ \\overline{y} + \\overline{z} | \\\\&= \\displaystyle {\\sup _{\\overline{m}\\in a A^{\\ast } + b B^{\\ast }, \\overline{z}\\in c C^{\\ast }} }| \\overline{m}+ \\overline{z} | \\\\&\\le \\displaystyle {\\sup _{\\overline{m}\\in (aA+bB)^{\\ast }, \\overline{z}\\in c C^{\\ast }} }| \\overline{m}+ \\overline{z} |.$ In the proof of Lemma 4.2 we stated that $\\displaystyle {\\sup _{x\\in A^{\\ast }, y\\in B^{\\ast }}} | x + y | \\le \\displaystyle {\\sup _{x\\in A, y\\in B}} | x + y |.$ Hence $\\displaystyle {\\sup _{\\overline{m}\\in (aA+bB)^{\\ast }, \\overline{z}\\in c C^{\\ast }}} | \\overline{m}+ \\overline{z} |&\\le \\displaystyle {\\sup _{m\\in aA+bB, z \\in c C }}| m + z| \\\\&= \\displaystyle {\\sup _{x\\in A, y\\in B, z\\in C}} |a x+ b y + c z|$ which gives (4.13).", "Lemma 4.8 (the general form of Lemma 4.2) Let $f_{j}$ be defined on $\\mathbb {R}$ and let $ a_{j}$ be real numbers, then $\\displaystyle {\\sup _{x_{j} }} \\prod _{j=1}^{l} f_{j}^{\\ast }(x_{j})|\\sum _{j}^{l} a_{j}x_{j}| \\le \\displaystyle {\\sup _{x_{j} }} \\prod _{j=1}^{l} f_{j}(x_{j})|\\sum _{j}^{l} a_{j}x_{j}|.", "\\qquad \\mathrm {(4.14)}$ For simplicity, we just see why $\\displaystyle {\\sup _{x, y, z}} \\ f^{\\ast }(x) g^{\\ast }(y) h^{\\ast }(z) |a x+ b y + c z| \\le \\displaystyle {\\sup _{x, y, z} } \\ f(x)g(y)h(z)|a x+ b y + c z|; \\qquad \\mathrm {(4.15)}$ $(4.14)$ can be obtained by induction and similar arguments as the proof of $(4.15)$ .", "Suppose for a contradiction that $\\displaystyle {\\sup _{x, y, z }} f^{\\ast }(x)g^{\\ast }(y)h^{\\ast }(z)|a x+ b y + c z| > \\displaystyle {\\sup _{x, y, z}} f(x)g(y)h(z)|a x+ b y + c z| = \\alpha ,$ which implies there exist positive $\\varepsilon $ and a set $G \\subset \\mathbb {R} \\times \\dots \\times \\mathbb {R}$ such that $|G| >0$ , and $f^{\\ast }(\\overline{x}) g^{\\ast }(\\overline{y}) h^{\\ast }(\\overline{z}) |a \\overline{x}+ b \\overline{y} + c \\overline{z}| > \\alpha +\\varepsilon $ for all $(\\overline{x}, \\overline{y}, \\overline{z}) \\in G$ .", "So $f^{\\ast }(\\overline{x}) > (\\alpha +\\varepsilon ) (g^{\\ast }(\\overline{y})h^{\\ast }(\\overline{z})|a \\overline{x}+ b \\overline{y} + c \\overline{z}|)^{-1}.", "$ By the property of decreasing rearrangement, we have $|E_{1}|\\equiv | \\lbrace x: f(x) > (\\alpha +\\varepsilon ) (g^{\\ast }(\\overline{y}) h^{\\ast }(\\overline{z})|a \\overline{x}+ b \\overline{y} + c \\overline{z}|)^{-1} \\rbrace | > 2 | \\overline{x}| .", "\\qquad \\mathrm {(4.16)}$ Based on $(4.16)$ , we find that $\\overline{x} \\in E_{1}^{\\ast }$ and $ f(x)g^{\\ast }(\\overline{y})h^{\\ast }(\\overline{z})|a \\overline{x}+ b \\overline{y} + c \\overline{z}| > \\alpha +\\varepsilon $ for all $x\\in E_{1}$ .", "Then $g^{\\ast }(\\overline{y}) > (\\alpha + \\frac{\\varepsilon }{2}) (\\displaystyle {\\inf _{x\\in E_{1}}}f(x) h^{\\ast }(\\overline{z}) |a \\overline{x}+ b \\overline{y} + c \\overline{z}|)^{-1}.$ Applying the property of symmetric rearrangement again we get $|E_{2}|\\equiv | \\lbrace y: g(y) >(\\alpha + \\frac{\\varepsilon }{2}) (\\displaystyle {\\inf _{x \\in E_{1}}}f(x) h^{\\ast }(\\overline{z}) |a \\overline{x}+ b \\overline{y} + c \\overline{z}| )^{-1} \\rbrace | > 2 | \\overline{y}| .", "\\qquad \\mathrm {(4.17)}$ Based on it we have $\\overline{y} \\in E_{2}^{\\ast }$ and for any $y \\in E_{2}$ , $\\displaystyle {\\inf _{x \\in E_{1}}}f(x) g(y) h^{\\ast }(\\overline{z}) |a \\overline{x}+ b \\overline{y} + c \\overline{z}| >\\alpha + \\frac{\\varepsilon }{2}.$ Obviously, $ \\displaystyle {\\inf _{x \\in E_{1}}}f(x) \\displaystyle {\\inf _{y \\in E_{2}}} g(y) h^{\\ast }(\\overline{z}) |a \\overline{x}+ b \\overline{y} + c \\overline{z}| > \\alpha + \\frac{\\varepsilon }{3}.", "$ It follows from $ h^{\\ast }(\\overline{z}) > (\\alpha + \\frac{\\varepsilon }{3}) (\\displaystyle {\\inf _{x \\in E_{1}}}f(x) \\displaystyle {\\inf _{y \\in E_{2}}}g(y) |a \\overline{x}+ b \\overline{y} + c \\overline{z}| )^{-1}$ and the property of decreasing rearrangement once more that $|E_{3}|\\equiv | \\lbrace y: h(z) > (\\alpha + \\frac{\\varepsilon }{3} ) (\\displaystyle {\\inf _{x \\in E_{1}}}f(x) \\displaystyle {\\inf _{y \\in E_{2}}}g(y) |a \\overline{x}+ b \\overline{y} + c \\overline{z}| )^{-1} \\rbrace | > 2 | \\overline{z}| , \\qquad \\mathrm {(4.18)}$ so $\\overline{z} \\in E_{3}^{\\ast }$ .", "From (4.18) we get for any $x \\in E_{1}$ , $y \\in E_{2}$ , $z \\in E_{3}$ , $f(x)g(y)h(z) > (\\alpha + \\frac{\\varepsilon }{3} ) (|a \\overline{x}+ b \\overline{y} + c \\overline{z}| )^{-1},$ which implies $\\displaystyle {\\sup _{x\\in E_{1}, y\\in E_{2}, z\\in E_{3}}} f(x)g(y)h(z) |a x+ b y + c z| \\ge (\\alpha + \\frac{\\varepsilon }{3} ) (|a \\overline{x}+ b \\overline{y} + c \\overline{z}| )^{-1} \\displaystyle {\\sup _{x\\in E_{1}, y\\in E_{2}, z\\in E_{3}}} |a x+ b y + c z|.$ Therefore, $\\displaystyle {\\sup _{x, y, z}} f(x)g(y)h(z)|a x+ b y + c z|&\\ge \\displaystyle {\\sup _{x\\in E_{1}, y\\in E_{2}, z\\in E_{3}}} f(x)g(y)h(z)|a x+ b y + c z| \\\\&\\ge (\\alpha + \\frac{\\varepsilon }{3} ) (|a \\overline{x}+ b \\overline{y} + c \\overline{z}|)^{-1} \\displaystyle {\\sup _{x\\in E_{1}, y\\in E_{2}, z\\in E_{3}}} |a x+ b y + c z|.$ Lemma 4.7 gives that $ \\displaystyle {\\sup _{x\\in E_{1}, y\\in E_{2}, z\\in E_{3}}} |a x+ b y + c z| \\ge \\displaystyle {\\sup _{x\\in E_{1}^{\\ast }, y\\in E_{2}^{\\ast }, z\\in E_{3}^{\\ast }} } |a x+ b y + c z|\\ge |a \\overline{x}+ b \\overline{y} + c \\overline{z}|,$ since $\\overline{x} \\in E_{1}^{\\ast }$ , $ \\overline{y} \\in E_{2}^{\\ast }$ , $\\overline{z} \\in E_{3}^{\\ast }$ .", "Hence $(\\alpha + \\frac{\\varepsilon }{3} ) (|a \\overline{x}+ b \\overline{y} + c \\overline{z}|)^{-1} \\displaystyle {\\sup _{x\\in E_{1}, y\\in E_{2}, z\\in E_{3}}}|a x+ b y + c z| \\ge \\alpha + \\frac{\\varepsilon }{3} > \\alpha , $ which is a contradiction.", "This completes the proof of Lemma 4.8.", "It follows from Lemma 4.8 and the rearrangement property $(f^{p})^{\\ast }= (f^{\\ast })^{p}$ , for $0<p<\\infty $ that that we have the following corollary.", "Corollary 4.9 Let $f_{j}$ be defined on $\\mathbb {R}$ , $j=1, \\dots , n+1$ .", "Then for any $1 \\le i \\le n$ , $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}^{\\ast i}(y_{j}) \\det ( y_{1}, \\dots , y_{n+1})^{\\frac{n+1}{p}}\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{\\frac{n+1}{p}}.", "\\qquad \\mathrm {(4.19)}$ Furthermore, $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} \\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} f_{j}(y_{j}) \\det ( y_{1}, \\dots , y_{n+1})^{\\frac{n+1}{p}}\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{\\frac{n+1}{p}}.", "$ This is because for each $i$ , $\\det (y_{1}, \\dots , y_{n+1})$ is the linear combination of $y_{1i}, \\dots ,y_{(n+1)i}$ , where $y_{ki}$ is the $ i$ -th coordinate of $y_{k}\\in \\mathbb {R}^{n}, 1 \\le k \\le n+1$ .", "We now turn to study the optimiser for multilinear form inequality (4.11).", "By the above, we need only look amongst the class of all functions which are decreasing and symmetric separately in each coordinate variable.", "Proof of Theorem 4.6   Let $\\mathcal {S}$ be the stereographic projection from $\\mathbb {R}^{n}$ to the nothern hemisphere $\\mathbb {S}_{+}^{n}$ with $\\mathcal {S}(x)=(\\frac{x_{1}}{\\sqrt{1+|x|^{2}}}, \\dots , \\frac{x_{n}}{\\sqrt{1+|x|^{2}}}, \\frac{1}{\\sqrt{1+|x|^{2}}}),$ where $ x=( x_{1}, \\dots , x_{n}) \\in \\mathbb {R}^{n}$ .", "So $\\mathcal {S}^{-1}(s)= (\\frac{s_{1}}{s_{n+1}}, \\dots , \\frac{s_{n}}{s_{n+1}} ).$ For $f\\in L^{p}(\\mathbb {R}^{n})$ , define $ (\\mathcal {S}^{\\ast }f)(s):= |J_{\\mathcal {S}^{-1}}(s)|^{1/p} f(\\mathcal {S}^{-1}(s)) \\qquad \\mathrm {(4.20)}$ where $J_{\\mathcal {S}^{-1}}$ is the Jacobian determinant of the map $\\mathcal {S}^{-1}$ , $ |J_{\\mathcal {S}^{-1}}(s)| = (\\frac{1}{s_{n+1}})^{n+1}= (1+|\\mathcal {S}^{-1}(s)|^{2})^{\\frac{n+1}{2}}.", "\\qquad \\mathrm {(4.21)}$ Denote $ \\mathcal {S}^{\\ast }f_{j}$ by $F_{j}$ , and let $y_{j} = \\mathcal {S}^{-1} (s_{j})$ , then we have $\\det (y_{1}, \\dots , y_{n+1}) &= \\prod _{j=1}^{n+1} ( 1+|y_{j}|^{2})^{1/2} \\det (s_{1}, \\dots , s_{n+1}) \\\\&= \\prod _{j=1}^{n+1} |J_{\\mathcal {S}^{-1}}(s_{j})| ^{ \\frac{1}{n+1} } \\det (s_{1}, \\dots , s_{n+1}),$ where $ \\det (s_{1}, \\dots , s_{n+1})$ is the absolute value of the determinant of the matrix $(s_{1}, \\dots , s_{n+1})_{(n+1) \\times (n+1)}$ .", "Together with (4.20) and (4.21), we also have the conformally invariant property of multilinear geometric inequality (4.11) as follows.", "$\\displaystyle {\\sup _{s_{j}\\in \\mathbb {S}_{+}^{n}}} \\prod _{j=1}^{n+1} F_{j}(s_{j}) \\det (s_{1}, \\dots , s_{n+1})^{\\frac{n+1}{p}}&= \\displaystyle {\\sup _{s_{j}\\in \\mathbb {S}_{+}^{n}}} |J_{\\mathcal {S}^{-1}}(s_{j})|^{1/p} f_{j}(\\mathcal {S}^{-1}(s_{j})) \\det (s_{1}, \\dots , s_{n+1})^{\\frac{n+1}{p}} \\\\&= \\displaystyle {\\sup _{y_{j}\\in \\mathbb {R}^{n}}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{\\frac{n+1}{p}},$ and the invariance of $L^{p}$ norm: for every $j$ $\\Vert f_{j}\\Vert _{L^{p}(\\mathbb {R}^{n})} = (\\int _{\\mathbb {R}^{n}} |f(x)|^{p} dx )^{1/p}= (\\int _{\\mathbb {S}_{+}^{n}} |f(\\mathcal {S}^{-1}(s_{j}))|^{p} |J_{\\mathcal {S}^{-1}}(s_{j})| ds_{j} )^{1/p}= (\\int _{\\mathbb {S}_{+}^{n}} |F_{j}(s_{j})|^{p} ds_{j})^{1/p}.", "$ For $f\\in L^{p}(\\mathbb {R}^{n})$ , pick $\\alpha $ which is not a rational multiple of $\\pi $ .", "For $1\\le i \\le n$ , we define $U_{\\alpha }^{i}: \\mathbb {S}_{+}^{n} \\rightarrow \\mathbb {S}_{+}^{n}$ be a rotation of the sphere $\\mathbb {S}^{n}$ by angle $\\alpha $ which keeps the other basis vectors fixed except the $i$ -th and $(n+1)$ -th vectors.", "If the point after rotation is in the southern hemisphere, we then send the point to its antipodal point in $\\mathbb {S}_{+}^{n}$ .", "For $ F \\in L^{p} (\\mathbb {S}_{+}^{n})$ , define $(( U_{\\alpha }^{i})^{\\ast } F)(s):= |J_{ (U_{\\alpha }^{i})^{-1}}(s)|^{\\frac{1}{p} } F ( (U_{\\alpha }^{i})^{-1}s ) = F((U_{\\alpha }^{i})^{-1}s).", "$ With the same $\\mathcal {S}^{\\ast }$ in (4.20), we consider the new function $(\\mathcal {S}^{\\ast })^{-1} (U_{\\alpha }^{i} )^{\\ast } \\mathcal {S}^{\\ast } f$ .", "In brief we denote this new function $(\\mathcal {S}^{\\ast })^{-1} (U_{\\alpha }^{i} )^{\\ast } \\mathcal {S}^{\\ast } f$ by $ \\mathcal {U}_{\\alpha }^{i} f$ .", "For any $f\\in L^{p} (\\mathbb {R}^{n})$ , we define a sequence $\\lbrace f^{k}\\rbrace $ as in [4] as follows, $f^{0}= f, \\ f^{1} =\\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1} f$ , $f^{2} = \\mathcal {R}_{1} \\mathcal {R}_{n} \\dots \\mathcal {R}_{2} \\mathcal {U}_{\\alpha }^{2} f^{1}$ , $f^{3} = \\mathcal {R}_{2} \\mathcal {R}_{1} \\mathcal {R}_{n} \\dots \\mathcal {R}_{3} \\mathcal {U}_{\\alpha }^{3} f^{2}, \\cdots , f^{n+1} = \\mathcal {R}_{n} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1} f^{n} \\cdots $ Note that $\\mathcal {U}_{\\alpha }^{i}$ and $\\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1}$ are norm-preserving.", "It follows from the proof of Theorem 8 in [4] that for any $f\\in L^{p} (\\mathbb {R}^{n}) $ , we have $\\lbrace f^{k}\\rbrace $ converges to $h_{f}$ in $ L^{p}$ norm, where $h_{f}= c h, \\ \\ \\ \\ h (x)=(\\frac{1}{1+|x|^{2}})^{\\frac{n+1}{2p}}$ and $c$ is the constant such that $\\Vert f\\Vert _{p}= \\Vert c h\\Vert _{p}$ .", "So $c= |\\mathbb {S}^{n}_{+}|^{-\\frac{1}{p}} \\Vert f\\Vert _{p} .$ Here we will only sketch the argument, mainly using the competing symmetries in one dimension.", "First it is enough to consider the bounded functions that vanish outside a bounded set which are dense in $L^{p}$ , so there exists a constant $C$ such that $f(x) \\le Ch_{f}(x)$ .", "Note that $\\mathcal {R}_{j}$ and $\\mathcal {U}_{\\alpha }^{j}$ are order-preserving, then we have $f^{k}(x) \\le Ch_{f}(x)$ for every $f^{k}$ and all $x$ .", "By Helly's selection principle we can find a subsequence $f^{k_{l}}$ such that $f^{k_{l}}$ converges to some $g$ almost everywhere as $l \\rightarrow \\infty $ .", "The dominated convergence theorem implies that $g \\in L^{p}$ .", "We define $A:= \\inf \\limits _{n} \\Vert h_{f}-f^{k}\\Vert _{p}= \\lim \\limits _{n} \\Vert h_{f}-f^{k}\\Vert _{p},$ this is because $\\Vert h_{f}-f^{k}\\Vert _{p}$ decreases monotonically which follows from the property $\\Vert \\mathcal {R}_{j}f-\\mathcal {R}_{j}g\\Vert _{p}\\le \\Vert f-g\\Vert _{p}, \\ \\ \\Vert \\mathcal {U}_{\\alpha }^{j}f-\\mathcal {U}_{\\alpha }^{j}g\\Vert _{p}= \\Vert f-g\\Vert _{p} \\qquad \\mathrm {(4.22)}$ and the invariance of $h_{f}$ under each $\\mathcal {R}_{j}$ and $\\mathcal {U}_{\\alpha }^{j}$ .", "Applying these properties again gives that $\\begin{array}{ll}A&= \\lim \\limits _{n} \\Vert h_{f}-f^{k_{l}+1}\\Vert _{p} \\\\&= \\Vert h_{f}- \\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g\\Vert _{p} \\\\&= \\Vert \\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}h_{f}-\\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g\\Vert _{p} \\\\&\\le \\Vert \\mathcal {U}_{\\alpha }^{1}h_{f}- \\mathcal {U}_{\\alpha }^{1}g\\Vert _{p} \\\\&= \\Vert h_{f}-g\\Vert _{p} =A,\\end{array}\\qquad \\mathrm {(4.23)}$ then we must have equality everywhere $\\begin{array}{ll}\\Vert h_{f}- \\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g\\Vert _{p}&= \\Vert h_{f}-\\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g\\Vert _{p} \\\\&= \\dots \\\\&=\\Vert h_{f}-\\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g\\Vert _{p} \\\\&=\\Vert h_{f}- \\mathcal {U}_{\\alpha }^{1}g\\Vert _{p}\\end{array}\\qquad \\mathrm {(4.24)}$ which implies (see Theorem 3.5 of [3]) $ \\mathcal {R}_{n} \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g= \\mathcal {R}_{n-1} \\dots \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g= \\dots = \\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g= \\mathcal {U}_{\\alpha }^{1}g \\qquad \\mathrm {(4.25)}$ It turns out that $\\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g= \\mathcal {U}_{\\alpha }^{1}g$ and $\\mathcal {R}_{1}g=g$ imply $ \\mathcal {U}_{2 \\alpha }^{1}g=g$ which shows $\\mathcal {S}^{\\ast } g$ is invariant under the rotation through an angle $2 \\alpha $ which keeps the other basis vectors fixed except the 1-th and $(n+1)$ -th ones.", "In particular, $2 \\alpha $ is an irrational multiple of $\\pi $ .", "Therefore, for any fixed $s_{2}, \\dots , s_{n}$ , $(\\mathcal {S}^{\\ast } g)(\\cdot , s_{2}, \\dots , s_{n}, \\cdot )$ is a constant.", "Also we have $\\mathcal {R}_{1} \\mathcal {U}_{\\alpha }^{1}g= \\mathcal {U}_{\\alpha }^{1}g=g.", "\\qquad \\mathrm {(4.26)}$ Similarly, if we replace $f^{k_{l}+1}$ in (4.23) by $f^{k_{l}+2}$ , together with (4.25)-(4.26) and Theorem 3.5 of [3] we have $ \\mathcal {R}_{1}\\mathcal {R}_{n} \\dots \\mathcal {R}_{2} \\mathcal {U}_{\\alpha }^{2} g= \\mathcal {R}_{n} \\dots \\mathcal {R}_{2} \\mathcal {U}_{\\alpha }^{2} g= \\dots = \\mathcal {R}_{2} \\mathcal {U}_{\\alpha }^{2} g= \\mathcal {U}_{\\alpha }^{2} g. \\qquad \\mathrm {(4.27)}$ From $\\mathcal {R}_{2} \\mathcal {U}_{\\alpha }^{2}g= \\mathcal {U}_{\\alpha }^{2}g$ and $\\mathcal {R}_{2}g=g$ we obtain that $ \\mathcal {U}_{2 \\alpha }^{2}g=g$ which shows $\\mathcal {S}^{\\ast } g$ is invariant under the rotation through an angle $2 \\alpha $ which keeps the other basis vectors fixed except the 2-th and $(n+1)$ -th ones.", "So for any fixed $s_{1}, s_{3}, \\dots , s_{n}$ , $(\\mathcal {S}^{\\ast } g)(s_{1}, \\cdot , s_{3}, \\dots , s_{n}, \\cdot )$ is a constant, since $2 \\alpha $ is an irrational multiple of $\\pi $ .", "Meanwhile we have $\\mathcal {R}_{2} \\mathcal {U}_{\\alpha }^{2}g= \\mathcal {U}_{\\alpha }^{2}g=g.", "\\qquad \\mathrm {(4.28)}$ So far based on the discussion above, we've got for any fixed $ s_{3}, \\dots , s_{n}$ , $(\\mathcal {S}^{\\ast } g)(\\cdot , \\cdot , s_{3}, \\dots , s_{n}, \\cdot )$ must be a constant.", "By induction we can obtain $ \\mathcal {S}^{\\ast }g$ is a constant function on $\\mathbb {S}_{+}^{n}$ , and thus the corresponding function $g$ on $\\mathbb {R}^{n}$ is $C h_{f}$ .", "Note that $\\mathcal {R}_{j}$ and $\\mathcal {U}_{\\alpha }^{j}$ are norm-preserving, so $\\Vert g\\Vert _{p}= \\lim \\limits _{n} \\Vert f^{k_{l}}\\Vert _{p}= \\Vert f\\Vert _{p}, $ which gives $C=1$ , $g=h_{f}$ .", "Therefore, the sequence $f^{k}$ converges to $h_{f}$ in $ L^{p}$ norm.", "It follows from Lemma 4.8, inequality (4.19) of Corollary 4.9 and the invariance of the multilinear geometric inequality under stereographic projection that $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}$ decreases monotonically as $k$ grows.", "That is for all $k \\in \\mathbb {N}$ , $ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}} \\ge \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k+1} (y_{j}) \\ \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}} .$ Since $\\lbrace f_{j}^{k}\\rbrace $ converges to $ h_{f_{j}}$ in $ L^{p}$ norm , $1 \\le j \\le n+1$ , then there exist subsequences $\\lbrace f_{j}^{k_{l}}\\rbrace $ such that $f_{j}^{k_{l}} \\rightarrow h_{f_{j}}$ pointwise almost everywhere as $ l \\rightarrow \\infty $ .", "From $ \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k_{l}} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}} \\le \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j} (y_{j}) \\ \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}} < \\infty $ for all $k_{l}$ together with the dominated convergence theorem it follows that $ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k_{l}} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}} \\xrightarrow{} \\displaystyle {\\prod _{j=1}^{n+1}} \\ h_{f_{j}}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}},$ in $ L^{\\infty }(\\mathbb {R}^{n}) \\times \\dots \\times L^{\\infty }(\\mathbb {R}^{n})$ as $l \\rightarrow \\infty $ .", "Hence by the $\\mathrm {weak}^*$ lower semicontinuity of the $L^{\\infty }$ norm , $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ h_{f_{j}}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}&\\le \\displaystyle {\\liminf _{l}} \\ ( \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k_{l}} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}} \\\\&= \\displaystyle {\\inf _{l}} \\ ( \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k_{l}} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}.$ Combining this with the norm-preserving property $\\Vert f_{j} \\Vert _{p} = \\Vert f_{j}^{k} \\Vert _{p}$ for every $k$ and the decreasing property of $\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k} (y_{j}) \\ \\det (y_{1},...,y_{n+1})^{\\frac{n+1}{p}}$ , we get for all $f_{j} \\in L^{p} (\\mathbb {R}^{n})$ and every $k_{l}$ , $\\frac{\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}}{ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert f_{j}\\Vert _{p} }&\\ge \\frac{\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}^{k_{l}} (y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}}{ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert f_{j}^{k_{l}}\\Vert _{p} } \\\\&\\ge \\frac{\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ h_{f_{j}}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}}{ \\prod \\limits _{j=1}^{n+1} \\Vert h_{f_{j}}\\Vert _{p} }$ Obviously, $\\frac{\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ h_{f_{j}}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}}{ \\prod \\limits _{j=1}^{n+1} \\Vert h_{f_{j}}\\Vert _{p} }= \\frac{\\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ h(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}}{ \\Vert h\\Vert _{p}^{n+1} }.$ $\\hfill \\Box $ Based on Theorem 4.6 and the conformal invariance under the stereographic projection from $\\mathbb {R}^{n}$ to $\\mathbb {S}_{+}^{n}$ , the geometric inequality (4.11) has the conformally equivalent form in $\\mathbb {S}^{n}$ space.", "Theorem 4.10 For $0 < p < \\infty $ , let $F_{j}$ be nonnegative functions in $ L^{p}(\\mathbb {S}^{n})$ .", "Then $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {S}^{n})} \\le B_{p,n} \\ \\displaystyle {\\sup _{s_{j}\\in \\mathbb {S}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j}(s_{j}) \\det (s_{1}, \\dots , s_{n+1})^{\\frac{n+1}{p}}, \\qquad \\mathrm {(4.29)}$ where $\\det (s_{1}, \\dots , s_{n+1})$ is the absolute value of the determinant of the matrix $(s_{1}, \\dots , s_{n+1})_{(n+1) \\times (n+1)}$ .", "The best constant $B_{p,n}$ is obtained when $F_{j}(s_{j})$ are constant, and the corresponding $B_{p,n}= |\\mathbb {S}^{n}|^{\\frac{n+1}{p}}$ .", "From Theorem 4.6 and and the conformal invariance of (4.11) under the stereographic projection from $\\mathbb {R}^{n}$ to $\\mathbb {S}_{+}^{n}$ , we obtain for nonnegative functions $F_{j} \\in L^{p}(\\mathbb {S}_{+}^{n})$ , $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {S}_{+}^{n})} \\le C_{p,n} \\ \\displaystyle {\\sup _{s_{j}\\in \\mathbb {S}_{+}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j}(s_{j}) \\det (s_{1},..., s_{n+1})^{\\frac{n+1}{p}} \\qquad \\mathrm {(4.30)}$ holds.", "The best constant $C_{p,n}$ is obtained when $F_{j}(s_{j})$ are constant, and the corresponding $C_{p,n}= |\\mathbb {S}_{+}^{n}|^{\\frac{n+1}{p}}=(\\frac{1}{2}|\\mathbb {S}^{n}| )^{\\frac{n+1}{p}}$ .", "Note that $\\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}_{+}^{n} }} \\ \\det (s_{1}, \\dots ,s_{n+1})=\\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}^{n} }} \\ \\det (s_{1}, \\dots ,s_{n+1})=1.$ Let $F_{j}$ be nonnegative functions in $ L^{p}(\\mathbb {S}^{n})$ .", "We define $\\overline{F}_{j}(s_{j})=\\max \\lbrace F_{j}(s_{j}), F_{j}(\\overline{s}_{j})\\rbrace ,$ where $\\overline{s}_{j}$ is the antipodal point of $s_{j}$ , $1 \\le j \\le n+1$ , $s_{j} \\in \\mathbb {S}_{+}^{n}$ .", "Then $\\overline{F}_{j} \\in L^{p}(\\mathbb {S}_{+}^{n})$ , and $\\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}_{+}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ \\overline{F}_{j}(s_{j}) \\det (s_{1}, \\dots ,s_{n+1})^{\\frac{n+1}{p}}=\\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j}(s_{j}) \\det (s_{1}, \\dots ,s_{n+1})^{\\frac{n+1}{p}}, \\qquad \\mathrm {(4.31)}$ this is because for any $s_{j} \\in \\mathbb {S}_{+}^{n}$ , $\\det (s_{1}, \\dots , \\overline{s}_{j}, \\dots ,s_{n+1})= \\det (s_{1}, \\dots ,s_{j}, \\dots ,s_{n+1}).$ Besides, $2 \\Vert \\overline{F}_{j}\\Vert _{L^{p}(\\mathbb {S}_{+}^{n})}^{p}\\ge \\int _{\\mathbb {S}_{+}^{n}} (F_{j}(s_{j}))^{p} d s_{j} + \\int _{\\mathbb {S}_{+}^{n}} (F_{j}(\\overline{s}_{j}))^{p} d s_{j}=\\Vert F_{j}\\Vert _{L^{p}(\\mathbb {S}^{n})}^{p}.", "$ Thus for each $j$ $\\Vert \\overline{F}_{j}\\Vert _{L^{p}(\\mathbb {S}_{+}^{n})} \\ge 2^{-\\frac{1}{p}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {S}^{n})}.", "\\qquad \\mathrm {(4.32)}$ It follows from (4.30)-(4.32) that for any nonnegative $F_{j} \\in L^{p}(\\mathbb {S}^{n})$ , $\\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {S}^{n})}&\\le 2^{\\frac{n+1}{p}} \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert \\overline{F}_{j}\\Vert _{L^{p}(\\mathbb {S}_{+}^{n}) } \\\\&\\le 2^{\\frac{n+1}{p}} |\\mathbb {S}_{+}^{n}|^{\\frac{n+1}{p}} \\ \\displaystyle {\\sup _{s_{j}\\in \\mathbb {S}_{+}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ \\overline{F}(s_{j}) \\det (s_{1}, \\dots , s_{n+1})^{\\frac{n+1}{p}} \\\\&= |\\mathbb {S}^{n}|^{\\frac{n+1}{p}} \\ \\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j} (s_{j}) \\det (s_{1}, \\dots ,s_{n+1})^{\\frac{n+1}{p}},$ which proves (4.29).", "To show that $|\\mathbb {S}^{n}|^{\\frac{n+1}{p}}$ is the best constant in (4.29), suppose for a contradiction that $F_{j}(s_{j}) \\in L^{p}(\\mathbb {S}^{n})$ is an optimiser for (4.29) that satisfies $\\frac{\\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j} (s_{j}) \\det (s_{1}, \\dots ,s_{n+1})^{\\frac{n+1}{p}}}{ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {S}^{n}) } }< \\frac{\\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}^{n} }} \\ \\det (s_{1}, \\dots ,s_{n+1})^{\\frac{n+1}{p}}}{ |\\mathbb {S}^{n}|^{\\frac{1}{p}} \\cdots |\\mathbb {S}^{n}|^{\\frac{1}{p}} }= |\\mathbb {S}^{n}|^{-\\frac{n+1}{p}} .$ Then by (4.31)-(4.32) we find $\\overline{F}_{j}(s_{j})$ defined as above satisfying $\\frac{\\displaystyle {\\sup _{s_{j}\\in \\mathbb {S}_{+}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ \\overline{F}_{j}(s_{j}) \\det (s_{1}, \\dots , s_{n+1})^{\\frac{n+1}{p}}}{\\displaystyle {\\prod _{j=1}^{n+1}} \\Vert \\overline{F}_{j}\\Vert _{L^{p}(\\mathbb {S}_{+}^{n}) }}&\\le \\frac{\\displaystyle {\\sup _{s_{j} \\in \\mathbb {S}^{n} }} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j} (s_{j}) \\det (s_{1}, \\dots ,s_{n+1})^{\\frac{n+1}{p}}}{2^{-\\frac{n+1}{p}} \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {S}^{n}) } } \\\\&< 2^{\\frac{n+1}{p}} |\\mathbb {S}^{n}|^{-\\frac{n+1}{p}} = |\\mathbb {S}_{+}^{n}|^{-\\frac{n+1}{p}}.$ This is in contradiction to the best constant in (4.30).", "Hence the best constant $B_{p,n}$ in (4.29) is $|\\mathbb {S}^{n}|^{\\frac{n+1}{p}}$ .", "Also the geometric inequality (4.11) has the conformally equivalent form in $\\mathbb {H}^{n}$ space as shown in the following theorem.", "Theorem 4.11 For $0 < p < \\infty $ , let $F_{j}$ be nonnegative functions in $ L^{p}(\\mathbb {H}^{n})$ .", "Then $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {H}^{n})} \\le E_{p,n} \\ \\displaystyle {\\sup _{q_{j}\\in \\mathbb {H}^{n}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j}(q_{j}) \\det (q_{1}, \\dots ,q_{n+1})^{\\frac{n+1}{p}}, \\qquad \\mathrm {(4.33)}$ where $\\det (q_{1}, \\dots ,q_{n+1})$ is the absolute value of the determinant of the matrix $(q_{1}, \\dots , q_{n+1})_{(n+1) \\times (n+1)}$ .", "Consider the stereographic projection $\\mathcal {H}$ which is a conformal transformation from the unit disk $D^{n}$ in $\\mathbb {R}^{n}$ to $\\mathbb {H}_{+}^{n}$ as $\\mathcal {H}(x)=(\\frac{x_{1}}{\\sqrt{1-|x|^{2}}}, \\dots ,\\frac{x_{n}}{\\sqrt{1-|x|^{2}}}, \\frac{1}{\\sqrt{1-|x|^{2}}}),$ where $x= (x_{1}, \\dots , x_{n}) \\in \\mathbb {R}^{n}$ .", "So $\\mathcal {H}^{-1}(q)= (\\frac{q_{1}}{q_{n+1}}, \\dots ,\\frac{q_{n}}{q_{n+1}}).$ Here the Jacobian determinant of the map $\\mathcal {H}^{-1}$ is $|J_{\\mathcal {H}^{-1}(q)}|= (1- |\\mathcal {H}^{-1}(q)|^{2})^{\\frac{n+1}{2}}.$ Let $y_{j}=\\mathcal {H}^{-1}(q_{j})$ , then we have $\\det (y_{1}, \\dots , y_{n+1}) &= \\prod _{j=1}^{n+1} ( 1-|y_{j}|^{2})^{1/2} \\det (q_{1}, \\dots , q_{n+1}) \\\\&= \\prod _{j=1}^{n+1} |J_{\\mathcal {H}^{-1}}(q_{j})| ^{ \\frac{1}{n+1} } \\det (q_{1}, \\dots , q_{n+1}).$ Define $F_{j}(q_{j}):= |J_{\\mathcal {H}^{-1}}(q_{j})|^{1/p} f_{j}(\\mathcal {H}^{-1}(q_{j}))$ , then from above we easily get the conformal invariance as follows.", "$\\displaystyle {\\sup _{q_{j}\\in \\mathbb {H}_{+}^{n}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j}(q_{j}) \\det (q_{1}, \\dots ,q_{n+1})^{\\frac{n+1}{p}}&= \\displaystyle {\\sup _{q_{j}\\in \\mathbb {H}_{+}^{n}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ |J_{\\mathcal {H}^{-1}}(q_{j})|^{1/p} f_{j}(\\mathcal {H}^{-1}(q_{j})) \\det (q_{1}, \\dots ,q_{n+1})^{\\frac{n+1}{p}} \\\\&= \\displaystyle {\\sup _{y_{j}\\in D^{n}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}},$ and for every $j$ $\\Vert f_{j}\\Vert _{L^{p}(D^{n})} &= (\\int _{D^{n}} |f_{j}(y_{j})|^{p} dy_{j})^{1/p} \\\\&= (\\int _{\\mathbb {H}_{+}^{n}} |f_{j}(\\mathcal {H}^{-1}(q_{j}))|^{p} |J_{\\mathcal {H}^{-1}(q)}| dq_{j})^{1/p} \\\\&= (\\int _{\\mathbb {H}_{+}^{n}} |F_{j}(q_{j})|^{p} dq_{j})^{1/p}= \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {H}_{+}^{n})}.$ Theorem 4.6 implies that $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert f_{j}\\Vert _{L^{p}(D^{n})} \\le C_{p,n}\\displaystyle {\\sup _{y_{j}\\in D^{n}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots ,y_{n+1})^{\\frac{n+1}{p}}, \\qquad \\mathrm {(4.34)}$ where $C_{p,n}= |\\mathbb {S}_{+}^{n}|^{\\frac{n+1}{p}}$ .", "Thus from the discussion above we have $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {H}_{+}^{n})} \\le |\\mathbb {S}_{+}^{n}|^{\\frac{n+1}{p}} \\ \\displaystyle {\\sup _{q_{j}\\in \\mathbb {H}_{+}^{n}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j}(q_{j}) \\det (q_{1}, \\dots ,q_{n+1})^{\\frac{n+1}{p}}.$ Similarly to the arguments in the proof of Theorem 4.10, we also have for any nonnegative $F_{j} \\in L^{p}(\\mathbb {H}^{n})$ , $0 < p < \\infty $ , there exists a finite constant $E_{p,n}= 2^{\\frac{n+1}{p}} |\\mathbb {S}_{+}^{n}|^{\\frac{n+1}{p}}$ such that $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert F_{j}\\Vert _{L^{p}(\\mathbb {H}^{n})} \\le E_{p,n} \\ \\displaystyle {\\sup _{q_{j}\\in \\mathbb {H}^{n}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ F_{j}(q_{j}) \\det (q_{1}, \\dots ,q_{n+1})^{\\frac{n+1}{p}}.", "$ However we do not know the extremal functions and the best constant $C_{p,n}$ for inequality (4.34).", "So a problem is to determine the optimisers and the best constant $E_{p,n}$ for the multilinear geometric inequality (4.33) in hyperbolic space $\\mathbb {H}^{n}$ .", "Acknowledgments.", "I am grateful to my supervisor Professor Carbery for his helpful advice and constructive suggestions.", "I would like to thank Professor Carbery for repeatedly making discussion and revision on this paper.", "This work was supported by the scholarship from China Scholarship Council." ] ]
1606.05206
[ [ "Modules over cluster-tilted algebras that do not lie on local slices" ], [ "Abstract We characterize the indecomposable transjective modules over an arbitrary cluster-tilted algebra that do not lie on a local slice, and we provide a sharp upper bound for the number of (isoclasses of) these modules." ], [ "Introduction", "Cluster-tilted algebras were introduced by Buan, Marsh and Reiten [6] and, independently in [7] for type $\\mathbb {A}$ .", "In [1] is given a procedure for constructing cluster-tilted algebras: let $C$ be a triangular algebra of global dimension two over an algebraically closed field $k$ , and consider the $C$ -$C$ -bimodule $\\textup {Ext}^2_C(DC,C)$ , where $D=\\textup {Hom}_k(-,k)$ is the standard duality, with its natural left and right $C$ -actions.", "The trivial extension of $C$ by this bimodule is called the relation-extension of $C$ .", "It is shown there that, if $C$ is tilted, then its relation-extension is cluster-tilted, and every cluster-tilted algebra occurs in this way.", "This relation between tilted and cluster-tilted algebras has been studied further in [2].", "Inspired by the complete slices in the module categories of tilted algebras, the authors introduced the concept of local slices as a generalization of complete slices, by relaxing a convexity condition.", "In [2] it is shown that every cluster-tilted algebra $B$ admits a local slice $\\Sigma $ and that, for every such local slice $\\Sigma $ , the quotient algebra $B/\\textup {Ann}\\,\\Sigma $ of $B$ by the annihilator of $\\Sigma $ is a tilted algebra with complete slice $\\Sigma $ .", "Furthermore, there is a unique component in the Auslander-Reiten quiver of $B$ , called the transjective component, that contains all local slices.", "Indecomposable modules in this transjective component are called transjective.", "In the module category of a tilted algebra, a complete slice should be thought of as a rather special configuration reproducing the quiver of a hereditary algebra to which our algebra tilts.", "It is well-known that an algebra is tilted if and only if it admits a complete slice, see, for instance [9].", "In contrast to the above situation, the existence of a local slice does not characterize cluster-tilted algebras.", "In [2], it is shown that if the cluster-tilted algebra is of tree type, then every indecomposable transjective module lies on a local slice.", "On the other hand, the authors also gave an example of an indecomposable transjective module over a cluster-tilted algebra of type $\\widetilde{\\mathbb {A}}_{2,1}$ that does not lie on a local slice.", "However, the questions `which indecomposable transjective modules do not lie on local slices\", and “how many of these modules do exist\", remained open.", "It is the purpose of the current paper to answer both questions for arbitrary cluster-tilted algebras.", "First, we characterize the indecomposable transjective modules that do not lie on a local slice in Theorem REF , using the completion of strong sinks defined in [3].", "Then we prove that the number of isoclasses (= isomorphism classes) of indecomposable transjective modules not lying on local slices is finite, and we actually give a sharp bound for this number in Corollary REF .", "Throughout this paper, algebras are basic and connected finite dimensional algebras over a fixed algebraically closed field $k$ .", "For an algebra $B$ , we denote by $\\text{mod}\\,B$ the category of finitely generated right $B$ -modules.", "All subcategories are full, and identified with their object classes.", "Given a category $\\mathcal {C}$ , we sometimes write $M\\in \\mathcal {C}$ to express that $M$ is an object in $\\mathcal {C}$ .", "For a point $x$ in the ordinary quiver of a given algebra $B$ , we denote by $P(x)$ , $I(x)$ , $S(x)$ respectively, the indecomposable projective, injective and simple $B$ -modules corresponding to $x$ .", "We denote by $\\Gamma (\\text{mod}\\,B)$ the Auslander-Reiten quiver of $B$ and by $\\tau , \\tau ^{-1} $ the Auslander-Reiten translations.", "For further definitions and facts, we refer the reader to [4]." ], [ "Tilting", "Let $Q$ be a finite connected and acyclic quiver.", "A module $T$ over the path algebra $kQ$ of $Q$ is called tilting if $\\text{Ext}^1_{kQ}(T,T)=0$ and the number of isoclasses of indecomposable summands of $T$ equals $|Q_0|$ , see [4].", "An algebra $C$ is called tilted of type $Q$ if there exists a tilting $kQ$ -module $T$ such that $C=\\text{End}_{kQ} T$ .", "An algebra $C$ is tilted if and only if it contains a complete slice $\\Sigma $ , see [9], that is, a finite set of indecomposable modules such that 1) $\\bigoplus _{U\\in \\Sigma } U$ is a sincere $C$ -module.", "2) If $U_0\\!\\rightarrow \\!", "U_1\\!", "\\rightarrow \\!", "\\dots \\!\\rightarrow \\!", "U_t$ is a sequence of nonzero morphisms between indecomposable modules with $U_0,U_t\\in \\Sigma $ then $U_i\\in \\Sigma $ for all $i$ (convexity).", "3) If $M$ is an indecomposable non-projective $C$ -module then at most one of $M$ , $\\tau M$ belongs to $\\Sigma $ .", "4) If $M,S$ are indecomposable $C$ -modules, $f\\colon M\\rightarrow S$ an irreducible morphism and $S\\in \\Sigma $ , then either $M\\in \\Sigma $ or $M$ is non-injective and $\\tau ^{-1} M\\in \\Sigma $ ." ], [ "Cluster-tilted algebras", "Let $Q$ be a finite, connected and acyclic quiver.", "The cluster category $\\mathcal {C}_Q$ of $Q$ is defined as follows, see [5].", "Let $F$ denote the composition $\\tau ^{-1}_{\\mathcal {D}}[1]$ , where $\\tau ^{-1}_{\\mathcal {D}}$ denotes the inverse Auslander-Reiten translation in the bounded derived category $\\mathcal {D} = \\mathcal {D}^b(\\text{mod}\\, kQ)$ , and [1] denotes the shift of $\\mathcal {D}$ .", "Then $\\mathcal {C}_Q$ is the orbit category $\\mathcal {D}/F$ : its objects are the $F$ -orbits $\\widetilde{X}=(F^i X)_{i\\in \\mathbb {Z}}$ of the objects $X\\in \\mathcal {D}$ , and the space of morphisms from $\\widetilde{X}=(F^i X)_{i\\in \\mathbb {Z}}$ to $\\widetilde{Y}=(F^i Y)_{i\\in \\mathbb {Z}}$ is $\\text{Hom}_{\\mathcal {C}_Q}(\\widetilde{X}, \\widetilde{Y}) = \\bigoplus _{i\\in \\mathbb {Z}} \\text{Hom}_{\\mathcal {D}}(X, F^i Y).$ Then $\\mathcal {C}_Q$ is a triangulated category with almost split triangles and, moreover, for $\\widetilde{X}, \\widetilde{Y}\\in \\mathcal {C}_Q$ we have a bifunctorial isomorphism $\\text{Ext}^1_{\\mathcal {C}_Q}(\\widetilde{X}, \\widetilde{Y})\\cong D\\text{Ext}^1_{\\mathcal {C}_Q}(\\widetilde{Y},\\widetilde{X})$ .", "This is expressed by saying that the category $\\mathcal {C}_Q$ is 2-Calabi-Yau.", "An object $\\widetilde{T}\\in \\mathcal {C}_Q$ is called tilting if $\\text{Ext}^1_{\\mathcal {C}_Q}(\\widetilde{T}, \\widetilde{T})=0$ and the number of isoclasses of indecomposable summands of $\\widetilde{T}$ equals $|Q_0|$ .", "The endomorphism algebra $B=\\text{End}_{\\mathcal {C}_Q} \\widetilde{T}$ is then called cluster-tilted of type $Q$ .", "Let now $T$ be a tilting $kQ$ -module, and $C=\\text{End}_{kQ} T$ the corresponding tilted algebra.", "Then it is shown in [1] that the trivial extension $\\widetilde{C}$ of $C$ by the $C$ -$C$ -bimodule $\\text{Ext}^2_C (DC,C)$ with the two natural actions of $C$ , the so-called relation-extension of $C$ , is cluster-tilted.", "Conversely, if $B$ is cluster-tilted, then there exists a tilted algebra $C$ such that $B=\\widetilde{C}$ ." ], [ "Local slices", "Let $B$ be a cluster-tilted algebra, then a full connected subquiver $\\Sigma $ of $\\Gamma (\\text{mod}\\,B)$ is a local slice, see [2], if: 1) $\\Sigma $ is a presection, that is, if $X\\rightarrow Y$ is an arrow then: (a) $X\\in \\Sigma $ implies that either $Y\\in \\Sigma $ or $\\tau Y \\in \\Sigma $ (b) $Y\\in \\Sigma $ implies that either $X\\in \\Sigma $ or $\\tau ^{-1} X\\in \\Sigma $ .", "2) $\\Sigma $ is sectionally convex, that is, if $X=X_0\\rightarrow X \\rightarrow \\dots \\rightarrow X_t = Y$ is a sectional path in $\\Gamma (\\text{mod}\\,B)$ then $X,Y\\in \\Sigma $ imply that $X_i\\in \\Sigma $ for all $i$ .", "3) $|\\Sigma _0| = \\text{rk}\\,K_0(B)$ .", "Let $C$ be tilted, then, under the standard embedding $\\text{mod}\\,C \\rightarrow \\text{mod}\\,\\widetilde{C}$ , any complete slice in the tilted algebra $C$ embeds as a local slice in $\\text{mod}\\,\\widetilde{C}$ , and any local slice in $\\textup {mod}\\,\\widetilde{C}$ occurs in this way.", "If $B$ is a cluster-tilted algebra, then a tilted algebra $C$ is such that $B=\\widetilde{C}$ if and only if there exists a local slice $\\Sigma $ in $\\Gamma (\\text{mod}\\,B)$ such that $C=B/\\text{Ann}_B \\Sigma $ , where $\\text{Ann}_B \\Sigma = \\bigcap _{X\\in \\Sigma } \\text{Ann}_B X$ , see [2]." ], [ "Completions and reflections", "We recall the definition of reflections from [3].", "Let $B$ be a cluster-tilted algebra.", "Let $\\Sigma $ be a local slice in the transjective component of $\\Gamma (\\text{mod}\\,B)$ having the property that all the sources in $\\Sigma $ are injective $B$ -modules.", "Then $\\Sigma $ is called a rightmost slice of $B$ .", "Let $x$ be a point in the quiver of $B$ such that $I(x)$ is an injective source of the rightmost slice $\\Sigma $ .", "The completion $H_x$ of $x$ is defined by the following three conditions.", "(a) $I(x)\\in H_x$ .", "(b) $H_x$ is closed under predecessors in $\\Sigma $ .", "(c) If $L\\rightarrow M$ is an arrow in $\\Sigma $ with $L\\in H_x$ having an injective successor in $H_x$ then $M\\in H_x$ .", "The completion $H_x$ can be constructed inductively in the following way.", "We let $H_1=I(x)$ , and $H_2^{\\prime }$ be the closure of $H_1$ with respect to (c).", "We then let $H_2$ be the closure of $H_2^{\\prime }$ with respect to predecessors in $\\Sigma $ .", "Then we repeat the procedure; given $H_i$ , we let $H_{i+1}^{\\prime }$ be the closure of $H_i$ with respect to (c) and $H_{i+1}$ be the closure of $H_{i+1}^{\\prime }$ with respect to predecessors.", "This procedure must stabilize, because the slice $\\Sigma $ is finite.", "If $H_j=H_k$ with $k>j$ , we let $H_x=H_j$ .", "We can decompose $H_x$ as the disjoint union of three sets as follows.", "Let $\\mathcal {J}$ denote the set of injectives in $H_x$ , let $\\mathcal {J}^-$ be the set of non-injectives in $H_x$ which have an injective successor in $H_x$ , and let $\\mathcal {E}=H_x\\setminus (\\mathcal {J}\\cup \\mathcal {J}^-)$ denote the complement of $(\\mathcal {J}\\cup \\mathcal {J}^-)$ in $H_x$ .", "Thus $H_x=\\mathcal {J}\\sqcup \\mathcal {J}^-\\sqcup \\mathcal {E}$ is a disjoint union.", "The reflection of the slice $\\Sigma $ in $x$ is defined as $\\sigma _x^+\\Sigma =\\tau ^{-2}(\\mathcal {J}\\cup \\mathcal {J}^-)\\cup \\tau ^{-1}\\mathcal {E}\\cup (\\Sigma \\setminus H_x),$ where $\\tau ^{-2}\\mathcal {J}$ stands for the set of all indecomposable projectives $P(y)$ such that the corresponding injective $I(y)$ is in the set $\\mathcal {J}$ .", "Theorem 2.1 [3] Let $\\Sigma $ be a rightmost local slice in $\\textup {mod}\\,B$ with injective source $I(x)$ .", "Then the reflection $\\sigma _x^+\\Sigma $ is a local slice as well." ], [ "Main results", "In this section, we prove our main results.", "We start with two preparatory lemmas.", "Definition 3.1 Let $B$ be a representation-infinite cluster-tilted algebra and let $\\Sigma ,\\Sigma ^{\\prime }$ be two local slices in $\\textup {mod}\\,B$ and $\\widetilde{\\Sigma },\\widetilde{\\Sigma ^{\\prime }}$ be their lifts in the cluster category $\\mathcal {C}$ .", "Then for every indecomposable module $X$ in $\\Sigma $ , we define $d_X(\\Sigma ,\\Sigma ^{\\prime })$ to be the unique integer $k$ such that $\\tau _\\mathcal {C}^{-k}\\widetilde{X} $ lies in $\\widetilde{\\Sigma ^{\\prime }}$ , where $\\widetilde{X} $ is the lift of $X$ in $\\mathcal {C}$ .", "Remark 3.2 In the above definition, the condition that $B$ is representation-infinite is necessary for the uniqueness of the integer $k$ .", "Lemma 3.3 Let $B$ be a representation-infinite cluster-tilted algebra.", "Let $\\Sigma $ be a rightmost local slice in $ \\textup {mod}\\,B$ with source $I(x)$ , and $H_x$ the completion in $\\Sigma $ .", "Suppose that $\\Sigma ^{\\prime }$ is another local slice such that $d_{I(x)}(\\Sigma ,\\Sigma ^{\\prime }) \\ge 2$ .", "Then for every indecomposable module $Y$ in $H_x$ we have $d_Y(\\Sigma ,\\Sigma ^{\\prime })\\ge 1.$ In particular, for every injective indecomposable $I(y)$ in $H_x$ we have $d_{I(y)}(\\Sigma ,\\Sigma ^{\\prime })\\ge 2.$ Let $\\lbrace I(x)\\rbrace =H_1\\subset H_2\\subset \\cdots \\subset H_r= H_x$ be the recursive construction of $H_x$ as in section REF above.", "Recall that given $H_{i-1}$ , the set $H_{i}^{\\prime }$ is the closure of $H_{i-1}$ with respect to condition (c) of the definition of $H_x$ , and $H_{i}$ is the closure of $H_{i}^{\\prime }$ under predecessors.", "Let $Y\\in H_i\\setminus H_{i-1}$ .", "We will prove the result by induction on $i$ .", "If $i=1$ then $Y=I(x)$ and we have $d_Y(\\Sigma ,\\Sigma ^{\\prime })\\ge 2$ by assumption.", "Now assume that $i>1$ .", "Then there are two possibilities a) Suppose first that $Y\\in H_i^{\\prime }$ .", "Then there exists an arrow $L\\rightarrow Y$ in $\\Sigma $ with $L\\in H_{i-1}$ having an injective successor $I$ in $H_{i-1}$ .", "So there is a path $\\ell :L=L_0\\rightarrow L_1 \\rightarrow \\cdots \\rightarrow L_{s-1}\\rightarrow L_s=I$ in $H_{i-1}$ and our induction hypothesis yields $d_{L_s}(\\Sigma ,\\Sigma ^{\\prime })=k\\ge 2.$ In the cluster category $\\mathcal {C}$ , denote by $\\widetilde{I}$ , $\\widetilde{L}_i$ and $\\widetilde{\\Sigma ^{\\prime }}$ the lifts of $I$ , $L_i$ and $\\Sigma ^{\\prime }$ , respectively.", "Then $\\tau ^{-k}_\\mathcal {C}\\widetilde{I} \\in \\widetilde{\\Sigma ^{\\prime }}$ .", "Moreover, since $\\widetilde{L}_{s-1}\\rightarrow \\widetilde{L}_s$ is an arrow in $\\widetilde{\\Sigma }$ , there is an arrow $\\tau ^{-k}_\\mathcal {C}\\widetilde{L}_{s-1}\\rightarrow \\tau ^{-k}_\\mathcal {C}\\widetilde{L}_s$ in the Auslander-Reiten quiver of $\\mathcal {C}$ , and because $\\widetilde{\\Sigma ^{\\prime }}$ is a local slice, this implies that either $\\tau ^{-k}_\\mathcal {C}\\widetilde{L}_{s-1}$ or $\\tau ^{-(k+1)}_\\mathcal {C}\\widetilde{L}_{s-1}$ is in $\\widetilde{\\Sigma ^{\\prime }}$ .", "In particular $d_{L_{s-1}}(\\Sigma ,\\Sigma ^{\\prime })\\ge d_{L_{s}}(\\Sigma ,\\Sigma ^{\\prime })\\ge 2.$ Repeating this argument for every arrow in the path $\\ell $ we see that $d_{L_{i}}(\\Sigma ,\\Sigma ^{\\prime })\\ge 2, \\textup { for all $ i$, } $ and thus $d_{L}(\\Sigma ,\\Sigma ^{\\prime })\\ge 2$ .", "This implies that $d_{Y}(\\Sigma ,\\Sigma ^{\\prime })\\ge 1$ , since there is an arrow $L\\rightarrow Y$ .", "b) Now suppose that $Y\\in H_i\\setminus H_i^{\\prime }$ .", "Thus $Y$ is obtained by closing under predecessors.", "Hence there is a path $\\ell ^{\\prime }: Y=L_0^{\\prime }\\rightarrow L_1^{\\prime }\\rightarrow \\cdots \\rightarrow L_t^{\\prime }$ with $L_t^{\\prime }\\in H_i^{\\prime }$ .", "In particular, $d_{L^{\\prime }_t}(\\Sigma ,\\Sigma ^{\\prime })\\ge 1$ , by part a).", "By the same argument as in case a), going back along the path $\\ell ^{\\prime }$ will not decrease the values of the function $d$ , so we see that $d_{Y}(\\Sigma ,\\Sigma ^{\\prime })\\ge d_{L^{\\prime }_{t}}(\\Sigma ,\\Sigma ^{\\prime })\\ge 1.$ This shows the first claim.", "Now, if $Y$ is injective then $d_{Y}(\\Sigma ,\\Sigma ^{\\prime })$ cannot be equal to 1, because $\\tau ^{-1} Y=0$ is not in $\\Sigma ^{\\prime }$ .", "This shows the second claim.", "For the proof of the next lemma, we need the following construction.", "Let $(\\Gamma , \\tau )$ be a translation quiver, and $X$ be a point in $\\Gamma $ .", "Then we define ${\\Sigma (\\rightarrow X) = \\left\\lbrace Y\\in \\Gamma \\left| \\begin{array}{c}\\textup { \\footnotesize there exists a sectional path from Y to X in \\Gamma }\\\\ \\textup {\\footnotesize and every path from Y to X in \\Gamma is sectional.", "}\\end{array}\\right.\\right\\rbrace },$ ${\\Sigma ( X\\rightarrow ) = \\left\\lbrace Y\\in \\Gamma \\left|\\footnotesize \\begin{array}{c}\\textup {there exists a sectional path from X to Y in \\Gamma }\\\\ \\textup {and every path from } X \\text{ to } Y\\text{ in } \\Gamma \\text{ is sectional.", "}\\end{array}\\right.\\right\\rbrace }.$ Proposition 3.4 [9] Let $Y$ be an indecomposable sincere module in a postprojective or preinjective component.", "Then both $\\Sigma (\\rightarrow Y)$ and $\\Sigma (Y\\rightarrow )$ are complete slices.", "Lemma 3.5 Let $M$ be an indecomposable transjective $B$ -module which does not lie on a local slice.", "Then there exist an indecomposable injective $B$ -module $I(j)$ and a local slice $\\Sigma $ containing a sectional path $v: \\tau M\\rightarrow \\cdots \\rightarrow I(j).$ Let $A$ be a hereditary algebra and $T\\in \\mathcal {C}_A$ a cluster-tilting object such that $B=\\text{End}_{\\mathcal {C}_A}(T)$ .", "Let $M$ be an indecomposable $B$ -module in the transjective component $\\mathcal {T}$ of $\\Gamma (\\text{mod}\\,B)$ , and let $\\widetilde{M}\\in \\mathcal {C}_A$ be an indecomposable object such that $\\text{Hom}_{\\mathcal {C}_A}(T, \\widetilde{M})=M$ .", "Finally, let $\\widetilde{\\Sigma }=\\Sigma (\\widetilde{M}\\rightarrow )$ in the cluster category $\\mathcal {C}_A$ .", "Since $B\\cong \\text{End}_{\\mathcal {C}_A}(\\tau _{\\mathcal {C}_A}^l T)$ for all $l\\in \\mathbb {Z}$ , we may assume without loss of generality that $\\widetilde{\\Sigma }$ lies in the postprojective component of $\\text{mod}\\,A$ .", "Furthermore, we may assume that every postprojective successor of $\\widetilde{\\Sigma }$ in $\\text{mod}\\,A$ in sincere.", "Indeed this follows from the fact that there are only finitely many isoclasses of indecomposable postprojective $A$ -modules that are not sincere.", "For tame algebras this holds, because non-sincere modules are supported on a Dynkin quiver, and for wild algebras see [8].", "Now since $\\widetilde{M}$ is a sincere $A$ -module, Proposition REF implies that $\\widetilde{\\Sigma }$ is a slice in $\\text{mod}\\,A$ , and therefore a local slice in $\\mathcal {C}_A$ .", "Let $\\Sigma _1=\\text{Hom}_{\\mathcal {C}_A}(T, \\Sigma (\\widetilde{M}\\rightarrow ))$ .", "Then $M\\in \\Sigma _1$ , and thus by assumption $\\Sigma _1$ is not a local slice in $\\text{mod}\\,B$ .", "Therefore, there exists an indecomposable direct summand $T_j$ of $T$ such that $\\tau T_j\\in \\Sigma (\\widetilde{M}\\rightarrow )$ .", "Moreover, by definition of $\\Sigma (\\widetilde{M}\\rightarrow )$ there is a sectional path $\\widetilde{M}\\rightarrow \\cdots \\rightarrow \\tau T_j $ and every path from $\\widetilde{M}$ to $\\tau T_j$ is sectional.", "Applying $\\tau $ we see that there exists a sectional path $\\tilde{v}: \\tau \\widetilde{M}\\rightarrow \\cdots \\rightarrow \\tau ^2 T_j$ and every path from $\\tau \\widetilde{M}$ to $\\tau ^2 T_j$ is sectional.", "Thus the local slice $\\Sigma (\\rightarrow \\tau ^2 T_j)$ in $\\mathcal {C}_A$ contains the path $\\tilde{v}$ .", "If there exists a summand $T_i$ of $T$ such that $\\tau T_i\\in \\Sigma (\\rightarrow \\tau ^2 T_j)$ then $ 0\\ne \\text{Hom}_{\\mathcal {C}_A}(\\tau T_i, \\tau ^2 T_j)\\cong D\\text{Ext}^1_{\\mathcal {C}_A}(T_j, T_i)$ which is impossible.", "Thus, the local slice $\\Sigma (\\rightarrow \\tau ^2 T_j)$ does not contain summands of $\\tau T$ .", "Therefore, $\\Sigma = \\text{Hom}_{\\mathcal {C}_A}(T, \\Sigma (\\rightarrow \\tau ^2 T_j))$ is a local slice in $\\text{mod}\\,B$ containing $\\tau M$ and containing a sectional path $v=\\text{Hom}_{\\mathcal {C}_A}(T, \\tilde{v}): \\tau M\\rightarrow \\cdots \\rightarrow I(j).$ We are now ready for our main result.", "Theorem 3.6 Let $B$ be a cluster-tilted algebra and $M$ an indecomposable transjective $B$ -module.", "Then the following are equivalent.", "(a) $M$ does not lie on a local slice.", "(b) There exist a rightmost slice $\\Sigma $ with source $I(x)$ such that the completion $H_x$ contains a sectional path $\\omega :I(x)\\rightarrow \\cdots \\rightarrow \\tau M\\rightarrow \\cdots \\rightarrow I(j)$ with $ I(j)$ injective.", "In particular $\\tau M\\in \\mathcal {J}^- (H_x)$ .", "(a)$ \\,\\Rightarrow \\,$ (b).", "By Lemma REF , there is an indecomposable injective $I(j)$ and a local slice $\\Sigma _1$ containing a sectional path $v: \\tau M\\rightarrow \\cdots \\rightarrow I(j).$ Without loss of generality we may assume that there is no other injective on the path $v$ and that $\\Sigma _1$ is a rightmost local slice.", "Let $u_1:I(x_1)\\rightarrow \\cdots \\rightarrow \\tau M$ be a maximal path in $\\Sigma _1$ ending in $\\tau M$ .", "Thus $I(x_1)$ is a source in the rightmost local slice $\\Sigma _1$ , hence $I(x_1)$ is injective.", "Moreover, since $\\tau M$ is not an injective module, $ I(x_1) \\ne \\tau M$ .", "We distinguish two cases.", "(1) If $I(j)\\in H_{x_1}$ , then the composition $\\omega = u_1v$ lies entirely inside ${ H_{x_1}}$ , because ${H_{x_1}}$ is closed under predecessors and we are done.", "(2) Now suppose that $I(j)\\notin H_{x_1}$ .", "(2.1) If $\\tau M\\in H_{x_1}$ then $\\tau M$ must lie in $\\mathcal {J}^-$ of $H_{x_1}$ , because otherwise the reflection $\\sigma _{x_1}^+\\Sigma _1$ would be a local slice containing $M$ which is impossible by (a).", "But $\\tau M\\in \\mathcal {J}^-$ implies the existence of a path $v^{\\prime }:\\tau M\\rightarrow \\cdots \\rightarrow I^{\\prime }$ in $H_{x_1}\\subset \\Sigma _1$ with $I^{\\prime }$ injective, and then the path $ \\omega =u_1v^{\\prime }:I(x_1)\\rightarrow \\cdots \\rightarrow \\tau M\\rightarrow \\cdots \\rightarrow I^{\\prime }$ lies entirely inside $H_{x_1}$ , and we are done.", "Note that $\\omega $ is sectional since it is a path in a local slice.", "(2.2) If $\\tau M\\notin H_{x_1}$ , then the path $v$ lies entirely in $\\Sigma _1\\setminus H_{x_1}$ and thus $v$ lies entirely in the local slice $\\Sigma _2=\\sigma _{x_1}^+\\Sigma _1$ .", "Repeating the argument, we either obtain a local slice with source $I(x)$ such that $I(j)\\in H_{x}$ and we conclude by the argument of case (1), or we obtain a local slice $\\Sigma _k=\\sigma _{x_{k-1}}^+\\cdots \\sigma _{x_2}^+\\sigma _{x_1}^+\\Sigma _1$ containing $v$ and a path $u_k:I(x_k)\\rightarrow \\cdots \\rightarrow \\tau M$ with $I(x_k)$ an injective source and $\\tau M\\in H_{x_k}$ , and we conclude by the argument of case (2.1).", "(b)$ \\,\\Rightarrow \\,$ (a).", "We want to show that $M$ does not lie on a local slice.", "Suppose to the contrary that there exists a local slice $\\Sigma _M$ containing $M$ .", "Let $\\omega , \\Sigma $ and $H_x$ be as in the statement of the theorem.", "By the argument of the first part of the proof, we may assume without loss of generality that $I(j)\\in H_x$ .", "We use the following notation for the path $\\omega $ $I(x)\\!\\rightarrow \\!\\cdots \\!\\rightarrow \\!", "I(i)\\!\\rightarrow \\!L_{\\text{-}s}\\!\\rightarrow \\!", "\\cdots \\!\\rightarrow \\!", "L_{\\text{-}2} \\!\\rightarrow \\!", "L_{\\text{-}1}\\!\\rightarrow \\!\\tau M \\!\\rightarrow \\!L_1\\!\\rightarrow \\!", "L_2\\!\\rightarrow \\!", "\\cdots \\!\\rightarrow \\!", "L_r\\!\\rightarrow \\!I(j),$ and we assume without loss of generality that none of the $L_i$ is injective.", "Let $\\gamma $ be the path obtained by applying $\\tau ^{-1}$ to a part of $\\omega $ , such that $\\gamma : \\tau ^{-1} L_{\\text{-}s}\\!\\rightarrow \\!\\cdots \\!\\rightarrow \\!\\tau ^{-1}L_{\\text{-}2}\\!\\rightarrow \\!\\tau ^{-1} L_{\\text{-}1}\\rightarrow M \\!\\rightarrow \\!", "\\tau ^{-1}L_1\\!\\rightarrow \\!\\tau ^{-1} L_2\\!\\rightarrow \\!", "\\cdots \\!\\rightarrow \\!", "\\tau ^{-1}L_r.$ Since $M$ lies in the local slice $\\Sigma _M$ and $M\\rightarrow \\tau ^{-1}L_1$ is an arrow in the Auslander-Reiten quiver, we have that either $\\tau ^{-1}L_1$ or $L_1$ is in $\\Sigma _M$ .", "If $\\tau ^{-1}L_1\\in \\Sigma _M$ then by the same argument, we have that either $\\tau ^{-1}L_2 $ or $L_2$ is in $\\Sigma _M$ .", "Repeating this reasoning, we see that either there is an $L_i\\in \\Sigma _M$ or $\\Sigma _M$ contains all the $\\tau ^{-1}L_i$ for $i=1,2,\\cdots ,r$ .", "In the latter case, we have an arrow $I(j)\\rightarrow \\tau ^{-1}L_r$ with $\\tau ^{-1} L_r\\in \\Sigma _M$ and thus $I(j)$ must be in $\\Sigma _M$ , since $\\tau ^{-1}I(j)=0$ .", "Thus in both cases $\\Sigma _M\\cap \\omega \\ne \\emptyset $ and $d_{I(j)}(\\Sigma ,\\Sigma _M)\\le 0.$ A similar argument along the part of the path $\\gamma $ from $\\tau ^{-1}L_{-s}$ to $M$ , we see that $\\Sigma _M\\cap \\tau ^{-1}\\gamma \\ne \\emptyset $ and $d_{I(i)}(\\Sigma ,\\Sigma _M)\\ge 2$ .", "Going back along the initial segment of the path $\\omega :I(x)\\rightarrow \\cdots \\rightarrow I(i)$ the values of the function $d$ cannot decrease, thus $d_{I(x)}(\\Sigma ,\\Sigma _M)\\ge 2$ as well.", "Now using Lemma REF , we see that $d_{I(j)}(\\Sigma ,\\Sigma _M)\\ge 2$ , which is a contradiction to the inequality (REF ).", "Remark 3.7 For cluster-tilted algebras of tree type, in particular for representation-finite cluster-tilted algebras, we know from [2] that every indecomposable module lies on a local slice.", "Thus condition (b) cannot hold in a cluster-tilted algebra of tree type.", "We now prove that the number of transjective modules over a cluster-tilted algebra which do not lie on a local slice is finite.", "Corollary 3.8 Let $B$ be a cluster-tilted algebra.", "Denote by $n$ the number of isoclasses of indecomposable projective $B$ -modules, and define $t$ as the maximum of the number 1 and the number of isoclasses of indecomposable transjective projective $B$ -modules.", "Then the number of isoclasses of indecomposable transjective $B$ -modules that do not lie on a local slice is at most $ (2^{t-1}-1)(n-2).$ First observe that if $B$ is representation-finite, then the result trivially holds by Remark REF .", "Assume therefore that $B$ is representation-infinite.", "By Theorem REF we have that the number of indecomposable transjective $B$ -modules that do not lie on a local slice is bounded above by the cardinality of the set $ \\cup _{\\Sigma } \\cup _x \\mathcal {J}^-(H_x),$ where $\\Sigma $ runs over all rightmost local slices and $x$ runs over all points such that $I(x) $ is a source in $\\Sigma $ .", "Since $\\mathcal {J}^-(H_x)\\subset \\Sigma $ , we have $ \\cup _x\\, \\mathcal {J}^-(H_x) \\subset \\lbrace L\\in \\Sigma \\mid L \\text{ is a noninjective indecomposable } B\\text{-module}\\rbrace $ and thus $ \\left| \\cup _x \\,\\mathcal {J}^-(H_x)\\right| \\le n-2$ because we need at least two injectives in $\\Sigma $ for $\\mathcal {J}^- (H_x)\\ne \\emptyset $ .", "Let $B=\\text{End}_{\\mathcal {C}} (T)$ where $T$ is a cluster-tilting object over a cluster category $\\mathcal {C}$ .", "Given a local slice $\\Sigma $ in $\\text{mod}\\,B$ let $\\widetilde{\\Sigma }$ be the lift of $\\Sigma $ to the cluster category, that is $\\Sigma = \\text{Hom}_{\\mathcal {C}}(T, \\widetilde{\\Sigma })$ .", "We claim that the number of rightmost local slices $\\Sigma $ in $\\text{mod}\\,B$ is at most $2^{t-1}-1$ .", "Observe that for every indecomposable transjective summand $T_i$ of $T$ we have that $\\tau T_i$ is a predecessor or a successor of the local slice $\\widetilde{\\Sigma }$ in $\\mathcal {C}$ .", "Moreover, since the slice $\\Sigma $ is rightmost it is determined by the predecessors and successors in $\\tau T$ of the corresponding $\\widetilde{\\Sigma }$ .", "We have to subtract 1 because if $\\widetilde{\\Sigma }$ has no transjective successors in $\\tau T$ then $\\Sigma $ is not rightmost.", "This shows that the number of local slices is at most $2^t-1$ .", "Finally, for $\\mathcal {J}^-(H_x)\\ne \\emptyset $ there must be at least two summands of $\\tau T$ which cannot be separated by a local slice, because $\\mathcal {J}^-(H_x)\\ne \\emptyset $ implies that there is a sectional path $\\omega : I(i)\\rightarrow \\cdots \\rightarrow \\tau M \\rightarrow \\cdots \\rightarrow I(j)$ and in the cluster category this yields a sectional path $\\tau ^{-1}\\tilde{\\omega }: \\tau T_i \\rightarrow \\cdots \\widetilde{M} \\rightarrow \\cdots \\rightarrow \\tau T_j$ and $M = \\text{Hom}_{\\mathcal {C}}(T, \\widetilde{M})$ does not lies on a local slice.", "This shows that the number of local slices is at most $2^{t-1}-1$ ." ], [ "Two examples", "We conclude with two examples.", "The first example shows that the bound in Corollary REF is sharp, and the second example illustrates the statement of the theorem.", "Example 4.1 Let $B$ be the cluster-tilted algebra of type $\\widetilde{\\mathbb {A}}_{2,1}$ given by the following quiver with relations $\\alpha \\beta =\\beta \\gamma =\\gamma \\alpha =0$ .", "$\\scalebox {0.8}{@R10pt@C20pt{1@<-1pt>[rr]_{\\alpha }@<1pt>[rr]&&3[dl]^{\\beta }\\\\&2[ul]^{\\gamma }}}$ The projective $B$ -modules $P(1)$ and $P(3)$ lie in the transjective component of $\\Gamma (\\text{mod}\\,B)$ while the projective $P(2)$ lies in a tube.", "The only transjective $B$ -module not lying on a local slice is $S(2)$ .", "On the other hand the formula in Corollary REF gives $ (2^{t-1}-1)(n-2) = (2^{2-1}-1)(3-2)=1.$ Example 4.2 We give an example to illustrate Theorem REF .", "Let $A$ be the path algebra of the quiver $\\scalebox {0.8}{@R8pt@C40pt{&1&2[l][dl]& 3@/_7pt/[ll][l]\\\\5&4[l]& 6@/^7pt/[ll][l]\\\\ }}$ Mutating at the vertices 2,4 and 6 yields the cluster-tilted algebra $B$ with quiver $\\scalebox {0.8}{@R10pt@C40pt{4[rrd]@/^10pt/[rrrr]&6[l]&2[l]&1[l]&3@<1pt>[l]@<-1pt>[l][dll]\\\\&&5@<1pt>[lu]@<-1pt>[lu]}}$ In the Auslander-Reiten quiver of $\\textup {mod}\\,B$ we have the following local configuration.", "$\\scalebox {0.8}{@!", "@R0pt@C1pt{&&& I(3) [rrrd]&&&& \\circ &&&& P(3)\\\\&& {{\\begin{matrix} 6\\\\4\\\\3 \\end{matrix}}} [ru][rrrddd]&&&& {{\\begin{matrix} 2\\\\6\\\\4 \\end{matrix}}} [rrrd][rrrddd]&&&&R[ru]\\\\&I(1)[ru]@/^15pt/[rruu]&&&&\\circ &&&&{{\\begin{matrix} 1\\\\2\\\\6\\\\4 \\end{matrix}}}[ru]@/^15pt/[rruu]\\\\&&{{\\begin{matrix} 4\\end{matrix}}}[drrr]&&&&{{\\begin{matrix} 6 \\end{matrix}}}[drrr][rrdd]&&&&\\cdot \\\\&{{\\begin{matrix} 4\\\\3 \\end{matrix}}}[ru][ruuu]&&&&{{\\begin{matrix} 6\\\\4 \\end{matrix}}}[ru][ruuu][rrrd]&&&&{{\\begin{matrix} 2\\ 5\\\\666\\\\4 \\end{matrix}}}[ru][ruuu]\\\\I(5)[ru]@/_15pt/[rruu]&&&&\\circ &&&&P(5)[ru]@/_15pt/[rruu]}} $ $\\small \\begin{array}{ccccccc} I(1)= {{\\begin{matrix} 2\\ \\ \\,\\\\ 6\\ 6\\\\4\\ 4\\\\3\\ 3\\\\1 \\end{matrix}}} &I(3)= {{\\begin{matrix} 2\\\\6\\\\4\\\\3 \\end{matrix}}}&I(5) ={{\\begin{matrix} 4\\ \\\\ \\ 3\\,4\\\\\\ 5 \\end{matrix}}}&P(5)={{\\begin{matrix} 5\\ \\\\ 6\\,6\\ \\\\ \\ 4 \\end{matrix}}}&R={{\\begin{matrix} 1\\quad \\\\2\\,\\ 5\\\\666\\\\4 \\end{matrix}}}&P(3)= {{\\begin{matrix}&3&\\\\1&& 1\\quad \\\\ 2&& 2\\,\\ 5\\\\ 6&& 666\\\\4&& 4 \\end{matrix}}}\\end{array}$ The 6 modules on the left form a rightmost local slice $\\Sigma =\\lbrace I(1), {{\\begin{matrix} 6\\\\4\\\\3 \\end{matrix}}},I(3),I(5),{{\\begin{matrix} 4\\\\3 \\end{matrix}}},{{\\begin{matrix} 4 \\end{matrix}}}\\rbrace $ in which both $I(1)$ and $I(5)$ are sources.", "Their completions are $H_1=\\Sigma $ and $H_5=\\lbrace I(5) ,{{\\begin{matrix} 4\\\\3 \\end{matrix}}},{{\\begin{matrix} 4 \\end{matrix}}}\\rbrace $ .", "The module ${{\\begin{matrix} 6\\\\4\\\\3 \\end{matrix}}}$ satisfies condition (b) of the theorem with respect to $H_1$ .", "Therefore the module $\\tau ^{-1}{{\\begin{matrix} 6\\\\4\\\\3 \\end{matrix}}}={{\\begin{matrix} 2\\\\6\\\\4 \\end{matrix}}}$ does not lie on a local slice.", "The module ${{\\begin{matrix} 4\\\\3 \\end{matrix}}}$ does not satisfy condition (b) of the theorem.", "Indeed, in $H_5$ it does not have an injective successor, and in $H_1$ it is not a successor of $I(1)$ .", "The theorem implies that the module $\\tau ^{-1}{{\\begin{matrix} 4\\\\3 \\end{matrix}}}={{\\begin{matrix} 6\\\\4 \\end{matrix}}}$ does lie on a local slice.", "This local slice is the reflection $\\sigma _5^+\\Sigma =\\lbrace I(1), {{\\begin{matrix} 6\\\\4\\\\3 \\end{matrix}}},I(3),{{\\begin{matrix} 6\\\\4 \\end{matrix}}},{{\\begin{matrix} 6 \\end{matrix}}},P(5)\\rbrace .$" ] ]
1606.05161
[ [ "Stationary Schr\\\"odinger equation in the semi-classical limit: numerical\n coupling of oscillatory and evanescent regions" ], [ "Abstract This paper is concerned with a 1D Schr\\\"odinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid \"turning points\".", "We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem.", "Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions.", "We provide a complete error analysis of this hybrid method and illustrate our convergence results by numerical tests." ], [ "Introduction", "This paper deals with the design, error analysis, and numerical study of an asymptotically correct scheme for the numerical solution of the stationary Schrödinger equation in one dimensional scattering situations: $\\displaystyle \\varepsilon ^2 \\psi ^{\\prime \\prime }(x) + a(x) \\psi (x) =0\\,, \\quad x \\in \\mathbb {R}\\,,$ where $0<\\varepsilon \\ll 1$ is a very small parameter and $a(x)$ a piecewise (sufficiently) smooth, real function.", "On the one hand, for $a(x)>0$ , the solution is highly oscillatory, with the small (local) wave length $\\lambda (x)={2\\pi \\varepsilon \\over \\sqrt{a(x)}}$ .", "On the other hand, for $a(x)<0$ , the wave function $\\psi $ is (essentially) exponentially de/increasing, typically exhibiting a thin boundary layer with thickness of the order $\\mathcal {O}\\big ({\\varepsilon \\over \\sqrt{|a(x)|}}\\big )$ .", "A key aspect of this paper is that $a(x)$ takes both signs.", "Hence, we have to cope with a classical multi-scale problem, combining different types of arduousnesses and multi-scale behaviours.", "Numerically, we aim at recovering these fine structures of the solution, however without using a fine spatial grid.", "To this end we shall develop a (non-overlapping) domain decomposition method (DDM) to separate the oscillatory and evanescent regions, as they require very different numerical approaches.", "This DDM allows to recover at the continuous level the exact analytical solution in a single sweep (ag!", "ainst the direction of the incoming plane wave) with appropriate interface conditions and a final scaling.", "The study of such multi-scale problems is very challenging from a theoretical as well as numerical point of view.", "In both situations (or regions) a classical discretization (like in [8], [9]) requires a very fine mesh in order to accurately resolve the oscillations and boundary layers.", "However, with a step size requirement of $h={\\cal O}(\\varepsilon )$ , standard numerical methods would be very costly and inefficient here.", "Concerning the oscillatory case, several WKB-based numerical schemes (named after the physicists Wentzel, Kramers, and Brillouin) have been developed and analysed in the last decade.", "Their goal is to use a coarse spatial grid with step size $h>\\lambda $ (see Figure REF ), reducing the limitation to at least $h={\\cal O}(\\sqrt{\\varepsilon })$ .", "For marching schemes we refer to [2], [10], [14], whereas a finite element method (FEM) using oscillatory WKB-basis functions was introduced in §2 of [16] and in [17].", "This FEM approach has the disadvantage that it requires a non-resonance condition between the mesh-size $h$ and the wave length $\\lambda $ of the solution.", "By contrast, this restriction is not necessary in the above mentioned marching schemes.", "Figure: In standard numerical methodshighly oscillating solutions require a very fine mesh to capture the oscillations.However, with the analytic pre-processing of our WKB-marching method an accurate solution can be obtained on a coarse grid(dots).", "Plotted is the solution ℜψ(x)\\Re \\psi (x) of () with ε=0.01,h=0.125,\\varepsilon =0.01,\\,h=0.125, and a=(x+1 2) 2 a=(x+\\frac{1}{2})^2.Numerical approaches for the evanescent regime (as $\\varepsilon \\rightarrow 0$ ) have been considered much less, so far.", "We refer to §3 in [16] for the formulation of a WKB-based multiscale FEM-scheme, including its numerical coupling to the oscillatory region (also based on a FEM).", "But a numerical analysis has, to our knowledge, not been carried out yet.", "This paper also aims at closing this gap.", "In this evanescent regime the problem (REF ) is elliptic, and for the example of $a=const.$ , a solution is given by a linear combination of the basis-functions $\\exp (\\pm \\frac{\\sqrt{|a|}}{\\varepsilon }x)$ .", "Hence this region must be considered as a boundary value problem (BVP) and solved e.g.", "by a finite difference or a FEM method.", "A reformulation as an initial value problem (IVP) and the use of a marching scheme would be inherently unstable, due to the unbounded growth of $\\exp (\\frac{\\sqrt{|a|}}{\\varepsilon }x)$ (in $\\varepsilon $ ).", "This contrasts with the oscillatory regime, where a basis of the solution would be given by the bounded functions $\\exp (\\pm i\\frac{\\sqrt{a}}{\\varepsilon }x)$ .", "Consequently, we are faced with coupling two different approaches for solving (REF ) in the semi-classical limit: an IVP for $a(x)>0$ and a BVP for $a(x)<0$ .", "The goal of this paper is to analyse the numerical coupling of oscillatory and evanescent regimes, using WKB-ansatz functions for both situations.", "In the oscillatory regime we shall use the marching scheme from [2], and in the evanescent regime a FEM like in §3 of [16].", "In the first case, the key idea is to eliminate analytically the dominant oscillations of the solution to (REF ).", "The transformed problem then has a much smoother solution, in the sense that the amplitude of the residual oscillations is much smaller than in the original problem – often by many orders of magnitude (in fact by the order $\\varepsilon ^2$ , cf.", "Propositions 2.1, 2.2 in [2]).", "Hence, the new problem can be solved numerically on a coarse grid, still yielding a very accurate approximation.", "In the evanescent regime, the key idea of the WKB-FEM method is to use WKB-ansatz functions (of exponential type), rather than the standard polynomials.", "Finally the strategy is the same as in the oscillatory region: to filter out the boundary layer via well-chosen basis functions.", "Since WKB-basis functions are asymptotic solutions to (REF ) (as $\\varepsilon \\rightarrow 0$ ), this method is again very accurate on a coarse grid.", "In this paper we shall provide first the numerical analysis of the WKB-FEM method for the evanescent regime (from scratch), and then the error analysis of the hybrid DDM – building on the convergence results in [2] for the WKB-marching method.", "Problems similar to (REF ) or in general that require the numerical integration of highly oscillatory equations play an essential role in a wide range of physical phenomena: e.g.", "electromagnetic and acoustic scattering (Maxwell and Helmholtz equations in the high frequency regime), wave evolution problems in quantum and plasma physics (Schrödinger equation in the semi-classical regime), and stiff mechanical systems.", "The application we are interested in here, stems from the electron transport in nanoscale semiconductor devices, like quantum wave-guides [1], resonant tunnelling diodes (RTDs) [4], MOSFETs [18], etc.", "In a 1D model setting, which is appropriate for RTDs or for the longitudinal dynamics in each transversal mode in MOSFETs, the governing equation is the stationary Schrödinger equation.", "In an idealized model we assume that the given electrostatic potential $V(x)$ is constant in the left lead $x\\in (-\\infty ,0]$ , with value $V(0)$ , and equally in the right lead $x\\in [1,\\infty )$ , with value $V(1)$ .", "Hence the Schrödinger equation can be complemented with open boundary conditions at both ends: $\\displaystyle \\left\\lbrace \\begin{array}{l}\\displaystyle - \\varepsilon ^2 \\psi ^{\\prime \\prime }(x) + V(x) \\psi (x)=E \\psi (x)\\,, \\quad x \\in (0,1)\\,, \\\\[3mm]\\displaystyle \\varepsilon \\psi ^{\\prime }(0)+{i} \\sqrt{a(0)} \\,\\psi (0) =0\\,, \\quad \\mbox{ if }a(0)>0\\,,\\\\[3mm]\\displaystyle \\varepsilon \\psi ^{\\prime }(0)- \\sqrt{|a(0)|} \\,\\psi (0) =0\\,, \\quad \\mbox{ if }a(0)<0\\,,\\\\[3mm]\\displaystyle \\varepsilon \\psi ^{\\prime }(1) -{i} \\sqrt{a(1)} \\,\\psi (1) = -2 {i} \\sqrt{a(1)}\\,.\\end{array}\\right.\\quad a(x):=E-V(x)\\,,$ This equation describes the state of an electron that is injected with the prescribed energy $E>0$ from the right boundary (or lead) into an electronic device (diode, e.g.", "), modelled on the interval $[0,1]$ .", "The corresponding (complex valued) wave function is denoted by $\\psi (x)$ , where $|\\psi (x)|^2$ is related to the spatial probability density of the electron.", "Due to the continuous injection of a plane wave function at $x=1$ , we cannot expect $|\\psi |^2$ to be normalised here (in $L^1(0,1)$ ).", "When considering the equivalent problem on $\\mathbb {R}$ , $\\psi $ rather describes a scattering state with $\\psi \\in L^\\infty (\\mathbb {R})$ .", "The small parameter $0<\\varepsilon <1$ is the re-scaled Planck constant.", "To make the link with (REF ), the coefficient function $a(x)$ is given by $a(x):=E-V(x)$ .", "To allow for an injection at $x=1$ , we have to require that $a(1)>0$ , cf.", "Figure REF .", "In fact, $E>V(x)$ characterises the oscillatory, classically allowed regime, whereas $E<V(x)$ characterises the evanescent, classically forbidden regime.", "Fig.", "REF sketches a tunnelling structure including both regimes, which are rather different.", "The boundary conditions in (REF ) are the so called open or transparent boundary conditions, permitting an electron wave to enter or leave the device without reflections [13].", "Due to the injected plane wave of electrons, the boundary condition at $x=1$ is inhomogeneous.", "But at $x=0$ it is homogeneous, due to the free outflow of the electron wave.", "In the present work we shall not discuss (in detail) situations with turning points, i.e.", "zeros of $a$ , but rather concentrate on devices with abrupt jumps at the interfaces between oscillatory and evanescent regions.", "This first step is simpler to treat and will be extended in a subsequent work.", "In Section we shall comment on situations incorporating turning points.", "Figure: Sketch of a tunnelling structure with two oscillatory and one evanescent regions.", "Electrons are injected from the right boundary with energy EE.", "The coefficient function in () is a(x):=E-V(x)a(x):=E-V(x).For the solvability of this model, the following simple result holds: Proposition 1.1 Let $a\\in L^\\infty (0,1)$ with $a(0)\\ne 0$ and $a(1)>0$ .Here and in the sequel, $a(0)$ and $a(1)$ are not meant as the point values of the function $a$ (which would not be defined for an $L^\\infty $ –function), but rather as the constant potential in the left and, resp., right leads.", "Then the boundary value problem (REF ) has a unique solution $\\psi \\in W^{2,\\infty }(0,1)\\hookrightarrow C^1[0,1]$ .", "Proof.", "For the case of an oscillatory outflow, i.e.", "$a(0)>0$ , the proof was provided in Proposition 2.3 of [3].", "For an evanescent outflow, i.e.", "$a(0)<0$ , the proof is analogous (multiplying the Schrödinger equation by ${\\overline{\\psi }}$ , integrating by parts, and taking the imaginary part).", "WKB-technique.", "Both parts of the hybrid numerical method studied in § will be based on WKB functions.", "Hence, let us first review the well-known WKB-approximation (cf.", "[11]) for the singularly perturbed ODE (REF ).", "In the standard approach, for the oscillatory regime (i.e.", "$a>0$ ), the WKB-ansatz $\\psi (x)\\sim \\exp \\left(\\frac{1}{\\varepsilon }\\sum _{p=0}^\\infty \\varepsilon ^p \\phi _p(x)\\right)\\,,$ is inserted in (REF ) and after comparison of the $\\varepsilon ^p$ -terms, leads to $\\phi _0(x)&=&\\pm {i}\\int _0^x\\sqrt{a(y)}\\,dy {+const.", "}\\,,\\nonumber \\\\\\phi _1(x)&=&\\ln a(x)^{-1/4}+const.\\,,\\\\\\phi _2(x)&=&\\mp {i}\\int _0^x\\beta (y)\\,dy {+const.", "}\\,,\\quad \\beta :=-\\frac{1}{2|a|^{1/4}}(|a|^{-1/4})^{\\prime \\prime }\\,.\\nonumber $ Truncating the ansatz (REF ) after $p=0,\\,1,$ or 2, yields the asymptotic approximation $\\psi (x)\\approx C\\varphi _p^{os}(x)$ , with the following oscillatory WKB-functions (of the three lowest orders in $\\varepsilon $ ): $\\varphi _0^{os}(x)&=&\\exp \\left(\\pm \\frac{{i}}{\\varepsilon }\\int _0^x\\sqrt{a(y)}\\,dy \\right)\\,,\\nonumber \\\\\\varphi _1^{os}(x)&=&\\frac{\\exp \\left(\\pm \\frac{{i}}{\\varepsilon }\\int _0^x\\sqrt{a(y)}\\,dy \\right)}{\\@root 4 \\of {a(x)}}\\,,\\\\\\varphi _2^{os}(x)&=&\\frac{\\exp \\left(\\pm \\frac{{i}}{\\varepsilon }\\int _0^x\\big [\\sqrt{a(y)}-\\varepsilon ^2\\beta (y)\\big ]\\,dy \\right)}{\\@root 4 \\of {a(x)}}\\,.\\nonumber $ Proceeding analogously for the evanescent regime (i.e.", "$a<0$ ) yields the following evanescent WKB-functions (of the three lowest orders in $\\varepsilon $ ): $\\varphi _0^{ev}(x)&=&\\exp \\left(\\pm \\frac{1}{\\varepsilon }\\int _0^x\\sqrt{|a(y)|}\\,dy \\right)\\,,\\nonumber \\\\\\varphi _1^{ev}(x)&=&\\frac{\\exp \\left(\\pm \\frac{1}{\\varepsilon }\\int _0^x\\sqrt{|a(y)|}\\,dy \\right)}{\\@root 4 \\of {|a(x)|}}\\,,\\\\\\varphi _2^{ev}(x)&=&\\frac{\\exp \\left(\\pm \\frac{1}{\\varepsilon }\\int _0^x\\big [\\sqrt{|a(y)|}+\\varepsilon ^2\\beta (y)\\big ]\\,dy \\right)}{\\@root 4 \\of {|a(x)|}}\\,.\\nonumber $ In the hybrid numerical method analysed in § we shall use the first order WKB ansatz functions $\\varphi _1^{ev}$ for the FEM in the evanescent region.", "And in the oscillatory region we shall use the second order WKB functions $\\varphi _2^{os}$ to transform (REF ) into a smoother problem that can be solved accurately and efficiently on a coarse grid.", "Since we shall use two different numerical approaches in the two regimes, also the corresponding error orders will be rather different (both with respect to $\\varepsilon $ and to the grid size $h$ ).", "Hence there is, a-priori, no obvious natural choice for the orders of the two WKB-methods.", "We choose a first order WKB-method in the evanescent region to keep the complexity of the numerical scheme and the technicalities of its convergence analysis low.", "Furthermore we choose a second order WKB-method in the oscillatory region such that we can use the results from [2] (without having to redo a first order WKB-analysis).", "Anyhow, our hybrid method is second order with respect to $h$ .", "This paper is organised as follows.", "In Section we present and analyse the (non-overlapping) domain decomposition method for the singularly perturbed ODE (REF ) on the continuous level.", "Propositions REF and REF establish that this DDM yields the analytical solution in one sweep for cases consisting of two and, resp., three distinct regions.", "In § we first review the two different numerical WKB-methods for the two distinct regions and establish convergence of the WKB-FEM.", "Then we prove convergence of the overall hybrid WKB-method (WKB-FEM in the evanescent regime coupled to a WKB-marching scheme for the oscillatory region), with Theorem REF as the main result.", "In § we illustrate our convergence results on some numerical examples treated with our scheme, including an example with a tunnelling structure.", "And finally, in § we briefly discuss extensions of our WKB method to coefficient functions with turning points." ], [ "Domain decomposition of the Schrödinger boundary value problem", "In this section we shall consider the Schrödinger BVP (REF ) with given coefficient functions $a(x)=E-V(x)$ corresponding to two different scenarios – first two coupled regions, then three regions.", "We confine ourselves here to these examples only for practical reasons: This setup already shows all the interesting features of the BVP, and it can be generalised easily to more regions." ], [ "Two coupled regions", "We start with the situation illustrated in Figure REF : It consists of two regimes, an evanescent region with $a:=E-V<0$ (adjacent to the left boundary) and an oscillatory region with $a>0$ (adjacent to the right boundary).", "Since we exclude turning points here, the function $a$ is assumed to have a jump discontinuity (and a sign change) at the interface $x=x_d$ .", "Moreover, for this section we shall assume: Hypothesis a2 Let $a\\in L^\\infty (0,1)$ with $a\\big |_{(0,x_d)}<0$ , $a\\big |_{(x_d,1)}>0$ , $a(0)<0$ , and $a(1)>0$ .", "Figure: Potential barrier: While electrons are injected from the right boundary with energy E>0E>0,the decomposed problem has to be solved from left to right (as a BVP–IVP).", "Following the basic idea from [2] we shall solve the BVP (REF ) as two consecutive sub-problems: We start with the evanescent region $(0,x_d)$ , where a BVP is solved (for stability reasons, as mentioned in §).", "This is followed by an IVP on the oscillatory region $(x_d,1)$ .", "So we shall proceed in the opposite direction of the injection direction (see Figure REF ).", "Due to Proposition REF , the solution to (REF ) is in $C^1[0,1]$ .", "Hence the solutions on these two sub-regions are matched by continuity of $\\psi $ and $\\psi ^{\\prime }$ at $x=x_d$ .", "For the BVP on $(0,x_d)$ , the original problem (REF ) provides only one homogeneous Robin boundary condition (BC) at $x=0$ .", "Hence, we supply the BVP with an auxiliary, artificial BC at $x=x_d$ .", "Here, both an inhomogeneous Dirichlet or Neumann BC would work from an algorithmic point of view.", "In order to simplify the numerical analysis in §REF below, we shall use at this point $\\varepsilon \\chi ^{\\prime }(x_d)=1$ for the auxiliary wave function $\\chi $ .", "While this auxiliary value has the correct $\\varepsilon $ –order (cf.", "Proposition REF and Lemma REF below), it will in general not be the correct derivative of the global solution $\\psi $ .", "Its correct value will finally be obtained by scaling the auxiliary functions using the remaining inhomogeneous Robin BC at $x=1$ (cf.", "(REF )).", "This leads to the following domain decomposition and problem coupling for the auxiliary wave functions $\\chi ,\\,\\varphi $ : Step 1 – BVP for $\\chi $ in region (1): $\\displaystyle \\,\\,\\, \\left\\lbrace \\begin{array}{l}\\displaystyle \\varepsilon ^2 \\chi ^{\\prime \\prime }(x) + a(x) \\chi (x)=0\\,, \\qquad x \\in (0,x_d)\\,, \\qquad \\quad a(x)= E -V(x) <0\\,,\\\\[3mm]\\displaystyle \\varepsilon \\, \\chi ^{\\prime }(0) - \\sqrt{|a(0)|}\\, \\chi (0)=0\\,, \\quad \\mbox{(Robin BC for $\\chi $ )}\\\\[3mm]\\displaystyle \\varepsilon \\chi ^{\\prime }(x_d) =1\\,.", "\\qquad \\qquad \\qquad \\quad \\: \\mbox{(auxiliary Neumann BC )}\\end{array}\\right.$ Step 2 – IVP for $\\varphi $ in region (2): $\\displaystyle \\,\\,\\, \\left\\lbrace \\begin{array}{l}\\displaystyle \\varepsilon ^2 \\varphi ^{\\prime \\prime }(x) + a(x) \\varphi (x)=0\\,, \\qquad x \\in (x_d,1)\\,, \\qquad \\quad a(x)= E -V(x)>0\\,, \\\\[3mm]\\displaystyle \\varphi (x_d) =\\chi (x_d)\\,, \\qquad \\qquad \\:\\: \\mbox{(implies continuity of $\\psi $ at $x_d$)}\\\\[3mm]\\displaystyle \\varepsilon \\varphi ^{\\prime }(x_d) =\\varepsilon \\chi ^{\\prime }(x_d)=1 \\,.\\quad \\: \\mbox{(implies continuity of $\\psi ^{\\prime }$ at $x_d$)}\\end{array}\\right.$ Step 3 – Scaling of the auxiliary wave functions: $\\displaystyle \\psi (x):=\\left\\lbrace \\begin{array}{l}\\displaystyle \\alpha \\,\\chi (x)\\,, \\quad x \\in (0,x_d)\\,, \\\\[3mm]\\displaystyle \\alpha \\,\\varphi (x)\\,, \\quad x \\in (x_d,1)\\,,\\end{array}\\right.$ with the scaling parameter $\\alpha \\in \\mathbb {C}$ defined via $\\displaystyle \\qquad \\alpha =\\alpha (\\varphi (1),\\,\\varphi ^{\\prime }(1))=\\frac{-2{i}\\sqrt{a(1)}}{\\varepsilon \\varphi ^{\\prime }(1)-i \\sqrt{a(1)}\\,\\varphi (1)} \\,, \\quad \\mbox{(due to the right BC in (\\ref {SchBVP}))}\\,.$ We note that the denominator in this expression for $\\alpha $ is non-zero: On the one hand $\\chi $ and $\\varphi $ are both real valued, and on the other hand $\\varphi (1)$ and $\\varphi ^{\\prime }(1)$ cannot vanish simultaneously (as otherwise $\\varphi \\equiv 0$ would contradict the Neumann BC in (REF )).", "In this whole section we shall only require that $a\\in L^\\infty (0,1)$ .", "As in Proposition REF , $a(0)$ and $a(1)$ are hence not meant as the point values of the function $a$ , but rather as the constant potential in the left and right leads.", "Next we address the solvability of the algorithm (REF )-(REF ).", "Lemma 2.1 Let Hypothesis a2 be satisfied.", "Then: The BVP (REF ) has $\\forall \\varepsilon >0$ a unique solution $\\chi \\in W^{2,\\infty }(0,x_d) \\subset C^1[0,x_d]$ .", "The IVP (REF ) has $\\forall \\varepsilon >0$ a unique solution $\\varphi \\in W^{2,\\infty }(x_d,1)\\subset C^1[x_d,1]$ .", "Both functions $\\chi $ and $\\varphi $ are real functions and the parameter $\\alpha \\in \\mathbb {C}$ given by (REF ) is well-defined.", "The above lemma, whose proof is very easy, shows that the domain decomposition algorithm (REF )-(REF ) yields a unique function $\\psi $ that is piecewise in $W^{2,\\infty }$ and piecewise (on the two regions) a solution to the Schrödinger equation (REF ).", "In fact, this DDM yields the unique solution of (REF ) as stated in the following proposition: Proposition 2.2 Let Hypothesis a2 be satisfied.", "Then the function $\\psi $ obtained from (REF )-(REF ) belongs to $W^{2,\\infty }(0,1)$ and is the unique solution of (REF ) (as guaranteed by Proposition REF ).", "Proof.", "The initial conditions in (REF ) imply $C^1$ –continuity of $\\psi $ at $x_d$ .", "Hence, $\\psi \\in C^1[0,1]$ , and this proves the claim.", "The following result provides the uniform-in-$\\varepsilon $ boundedness of this solution $\\psi $ .", "It generalizes Theorem 2.2 from [17], which holds only for one purely oscillatory region: Proposition 2.3 Let Hypothesis a2 hold.", "Moreover, let the potential in the oscillatory region satisfy $a\\in W^{1,\\infty }(x_d,1)$ and $0<\\tau _{os}\\le a(x)$ $\\forall x\\in (x_d,1)$ .", "Then, the solution of (REF ) satisfies $\\Vert \\psi \\Vert _{L^\\infty (0,1)} + \\varepsilon \\Vert \\psi ^{\\prime }\\Vert _{L^\\infty (0,1)} \\le C\\,,$ independently of $0<\\varepsilon \\le 1$ .", "The simple, but lengthy proof is deferred to the Appendix." ], [ "Three coupled regions", "In this subsection we consider the Schrödinger BVP (REF ) with a given coefficient function $a(x)$ as illustrated in Figure REF : It consists of three regimes, two oscillatory regions at the interval boundaries and an evanescent region in the middle.", "Since we exclude turning points here, $a$ is assumed to have jump discontinuities (and sign changes) at the interfaces $x=x_c$ and $x=x_d$ .", "The solution $\\psi $ to the BVP (REF ) for such an example is illustrated in Fig.", "REF below.", "Moreover, for this section we shall assume on $a(x)$ : Hypothesis a3 Let $a\\in L^\\infty (0,1)$ with $a\\big |_{(x_c,x_d)}<0$ , $a\\big |_{(0,x_c)\\cup (x_d,1)}>0$ , $a(0)>0$ , and $a(1)>0$ .", "Figure: Tunnelling structure: While electrons are injected from the right boundary with energy EE,the decomposed problem has to be solved from left to right (as an IVP–BVP–IVP).", "The coefficient function in () is a(x):=E-V(x)a(x):=E-V(x).Following the basic idea from [2] we shall solve the BVP (REF ) as an IVP-BVP-IVP problem in the opposite direction of the injection direction, i.e.", "starting at $x=0$ (see Figure REF ).", "In (REF ), the Robin boundary condition (BC) at $x=0$ only fixes the ratio $\\frac{\\psi ^{\\prime }(0)}{\\psi (0)}$ , hence an auxiliary Dirichlet (or Neumann) boundary value has to be invoked here.", "Its correct value will then be obtained by scaling the final equation using the inhomogeneous Robin BC at $x=1$ .", "In contrast to [2], (REF ) includes the evanescent region (2), cf.", "Fig.", "REF , which still has to be formulated as a BVP (for stability reasons).", "This leads to the following domain decomposition and problem coupling for the auxiliary wave functions $\\zeta ,\\,\\chi ,\\,\\varphi $ : Step 1 – IVP for $\\zeta $ in region (1): $\\displaystyle \\left\\lbrace \\begin{array}{l}\\displaystyle \\varepsilon ^2 \\zeta ^{\\prime \\prime }(x) + a(x) \\zeta (x)=0\\,, \\quad x \\in (0,x_c)\\,, \\\\[3mm]\\displaystyle \\zeta (0) =1\\,, \\qquad \\qquad \\qquad \\mbox{(auxiliary Dirichlet BC)}\\\\[3mm]\\displaystyle \\varepsilon \\zeta ^{\\prime }(0) =-{i} \\sqrt{a(0)} \\,.\\qquad \\mbox{(due to the left BC in (\\ref {SchBVP}))}\\end{array}\\right.$ Step 2 – BVP for $\\chi $ in region (2): $\\displaystyle \\left\\lbrace \\begin{array}{l}\\displaystyle \\varepsilon ^2 \\chi ^{\\prime \\prime }(x) + a(x) \\chi (x)=0\\,, \\quad x \\in (x_c,x_d)\\,, \\\\[3mm]\\displaystyle \\zeta ^{\\prime }(x_c)\\chi (x_c) - \\zeta (x_c)\\chi ^{\\prime }(x_c)=0\\,, \\quad \\mbox{(Robin BC for $\\chi $ implies continuity of $\\frac{\\psi ^{\\prime }}{\\psi }$ at $x_c$)}\\\\[3mm]\\displaystyle \\varepsilon \\chi ^{\\prime }(x_d) =1\\,.", "\\qquad \\qquad \\qquad \\qquad \\quad \\mbox{(auxiliary Neumann BC)}\\end{array}\\right.$ Step 3 – IVP for $\\varphi $ in region (3): $\\displaystyle \\left\\lbrace \\begin{array}{l}\\displaystyle \\varepsilon ^2 \\varphi ^{\\prime \\prime }(x) + a(x) \\varphi (x)=0\\,, \\quad x \\in (x_d,1)\\,, \\\\[3mm]\\displaystyle \\varphi (x_d) =\\chi (x_d)\\,, \\qquad \\:\\: \\mbox{(implies continuity of $\\psi $ at $x_d$)}\\\\[3mm]\\displaystyle \\varphi ^{\\prime }(x_d) =\\chi ^{\\prime }(x_d) \\,.\\qquad \\mbox{(implies continuity of $\\psi ^{\\prime }$ at $x_d$)}\\end{array}\\right.$ Step 4 – scaling of the auxiliary wave functions: $\\displaystyle \\psi (x):=\\left\\lbrace \\begin{array}{l}\\displaystyle \\beta \\,\\zeta (x)\\,, \\quad x \\in (0,x_c)\\,, \\\\[3mm]\\displaystyle \\alpha \\,\\chi (x)\\,, \\quad x \\in (x_c,x_d)\\,, \\\\[3mm]\\displaystyle \\alpha \\,\\varphi (x)\\,, \\quad x \\in (x_d,1)\\,,\\end{array}\\right.$ with the scaling parameters $\\alpha ,\\,\\beta \\in \\mathbb {C}$ still to be defined.", "This procedure can be explained as follows: First we note that the BCs of (REF ) imply $\\varepsilon \\zeta ^{\\prime }(0)+i\\sqrt{a(0)}\\zeta (0)=0$ , just as in the left BC of the BVP (REF ).", "Hence, the IVP (REF ) coincides on region (1) with the BVP (REF ), except for the auxiliary Dirichlet BC $\\zeta (0)=1$ .", "The true solution of (REF ) satisfies instead $\\psi (0)=\\beta $ with some a-priori unknown $\\beta \\in \\mathbb {C}$ .", "Hence, the auxiliary wave function $\\zeta $ is related to $\\psi $ by the scaling $\\psi \\big |_{[0,x_c]}=\\beta \\zeta $ , as postulated in the first line of (REF ).", "Clearly, this implies $\\frac{\\psi ^{\\prime }}{\\psi }= \\frac{\\zeta ^{\\prime }}{\\zeta }$ on $[0,x_c]$ .", "In the above Step 2, the Robin BC allows to carry over this relation to region (2): $\\frac{\\psi ^{\\prime }}{\\psi }= \\frac{\\chi ^{\\prime }}{\\chi }$ , and the auxiliary wave function $\\chi $ is related to $\\psi $ by the scaling $\\psi \\big |_{[x_c,x_d]}=\\alpha \\chi $ , with some $\\alpha \\in \\mathbb {C}$ to be determined.", "The initial conditions for the auxiliary wave function $\\varphi $ in Step 3 imply $C^1$ –continuity of $\\psi $ when using again the scaling $\\psi \\big |_{[x_d,1]}=\\alpha \\varphi $ .", "So far, the wave function $\\psi $ defined in (REF ) neither satisfies continuity at $x_c$ nor the right BC from (REF ).", "Therefore we define the scaling parameters $\\alpha ,\\,\\beta \\in \\mathbb {C}$ via $&&\\qquad \\alpha \\,[\\varepsilon \\varphi ^{\\prime }(1)-i \\sqrt{a(1)}\\,\\varphi (1)]=-2i \\sqrt{a(1)} \\,,\\qquad \\mbox{(due to the right BC in (\\ref {SchBVP}))}\\\\&&\\qquad \\beta \\,\\zeta (x_c)=\\alpha \\,\\chi (x_c) \\,.\\qquad \\qquad \\qquad \\qquad \\qquad \\:\\:\\: \\mbox{(implies continuity of $\\psi $ at $x_c$)}$ Remark 2.4 The key aspect of the above algorithm is to prescribe in the BVP (REF ) the continuity of $\\frac{\\zeta ^{\\prime }}{\\zeta }$ to $\\frac{\\chi ^{\\prime }}{\\chi }$ at $x_c$ .", "Note that this continuity is invariant under the scaling (REF ).", "Hence it is inherited by $\\frac{\\psi ^{\\prime }}{\\psi }$ , implying (with the continuity of $\\psi $ ) the required $C^1$ –continuity of $\\psi $ .", "The simpler alternative to prescribe in (REF ) continuity of $\\zeta $ to $\\chi $ would typically be paired with a discontinuity of $\\zeta ^{\\prime }$ to $\\chi ^{\\prime }$ at $x=x_c$ (as a result of solving the BVP).", "Then, the scaling of (REF ) would lead to an unwanted discontinuity of $\\psi ^{\\prime }$ at $x=x_c$ .", "Lemma 2.5 Let Hypothesis a3 be satisfied.", "Then: The IVPs (REF ), resp.", "(REF ) admit $\\forall \\varepsilon >0$ unique solutions $\\zeta \\in W^{2,\\infty }(0,x_c)\\subset C^1[0,x_c]$ , resp.", "$\\varphi \\in W^{2,\\infty }(x_d,1)\\subset C^1[x_d,1]$ .", "The BVP (REF ) has $\\forall \\varepsilon >0$ a unique solution $\\chi \\in W^{2,\\infty }(x_c,x_d)\\subset C^1[x_c,x_d]$ .", "The scaling parameters $\\alpha ,\\,\\beta \\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ are well-defined by (REF ), ().", "Proof.", "Part (i) is straightforward.", "For (ii), let us first consider the IVP (REF ).", "Its unique solution $\\zeta $ has the property: The values $\\zeta (x_c)$ and $\\zeta ^{\\prime }(x_c)$ are linearly independent over the field $\\mathbb {R}$ .", "Otherwise, the backward IVP (starting at $x_c$ ) would yield “final values” $\\zeta (0)$ and $\\zeta ^{\\prime }(0)$ that are linearly dependent over $\\mathbb {R}$ , which is in contradiction with the initial condition in (REF ).", "To solve the BVP (REF ), let $\\chi _1$ , $\\chi _2$ be a (real valued) basis of solutions for that Schrödinger equation on $(x_c,x_d)$ , with $&&\\chi _1(x_c)=1\\,,\\qquad \\chi _1^{\\prime }(x_c)=0\\,,\\\\&&\\chi _2(x_c)=0\\,,\\qquad \\chi _2^{\\prime }(x_c)=1\\,.$ Setting $\\chi =c_1\\chi _1+c_2\\chi _2$ with some $c_1,\\,c_2\\in \\mathbb {C}$ , the BCs of (REF ) give rise to the following linear equation: $\\left(\\begin{array}{cc}\\zeta ^{\\prime }(x_c)&-\\zeta (x_c)\\\\\\varepsilon \\chi _1^{\\prime }(x_d)&\\varepsilon \\chi _2^{\\prime }(x_d)\\end{array}\\right)\\:\\left(\\begin{array}{c}c_1\\\\c_2\\end{array}\\right)\\:=\\:\\left(\\begin{array}{c}0\\\\1\\end{array}\\right)\\,.$ The determinant of this system satisfies $\\varepsilon [\\zeta ^{\\prime }(x_c)\\chi _2^{\\prime }(x_d)+\\zeta (x_c)\\chi _1^{\\prime }(x_d)]\\ne 0$ , since $\\chi _{1,2}^{\\prime }(x_d)$ $\\in \\mathbb {R}$ , but $\\zeta (x_c)$ and $\\zeta ^{\\prime }(x_c)$ are linearly independent over $\\mathbb {R}$ .", "Hence, (REF ) is uniquely solvable for $c_1,\\,c_2$ , thus providing the unique solution to (REF ).", "For part (iii) we shall first argue that (REF ) yields a well-defined $\\alpha \\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ .", "To this end we shall prove that $\\varepsilon \\varphi ^{\\prime }(1)\\ne i \\sqrt{a(1)}\\,\\varphi (1)$ , using the quantum mechanical current of the model (REF ): $j(x):=\\varepsilon \\Im [{\\overline{\\psi }}(x)\\,\\psi ^{\\prime }(x)]\\,.$ Assume now that $\\varepsilon \\varphi ^{\\prime }(1)= i \\sqrt{a(1)}\\,\\varphi (1)$ .", "Then, (REF ) implies on the one hand $j(1)=\\varepsilon |\\alpha |^2 \\,\\Im [{\\overline{\\varphi }}(1)\\,\\varphi ^{\\prime }(1)]=|\\alpha |^2 \\,\\sqrt{a(1)}\\,|\\varphi (1)|^2\\ge 0\\,.$ But, on the other hand, (REF ) yields $j(0)=\\varepsilon |\\beta |^2 \\Im [{\\overline{\\zeta }}(0)\\,\\zeta ^{\\prime }(0)]=-|\\beta |^2 \\sqrt{a(0)}\\,|\\zeta (0)|^2\\le 0\\,.$ Since the current in a stationary quantum model is constant in $x$ , this implies $j\\equiv 0$ .", "Since $a(1)>0$ and $\\alpha \\ne 0$ (otherwise $\\psi (1)=\\psi ^{\\prime }(1)=0$ would contradict the BC at $x=1$ in (REF )), (REF ) shows that $\\varphi (1)=0$ , and hence –by our assumption– $\\varphi ^{\\prime }(1)=0$ .", "But this leads to a contradiction in the BC at $x=1$ in (REF ).", "Hence, (REF ) yields indeed a unique $\\alpha \\in \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ .", "Finally, () yields a well-defined $\\beta \\in \\mathbb {C}$ since $\\zeta (x_c)$ and $\\zeta ^{\\prime }(x_c)$ are linearly independent over $\\mathbb {R}$ (as shown in part (i) above).", "Moreover, $\\beta \\ne 0$ since $\\chi (x_c)=c_1\\ne 0$ (otherwise the first line of (REF ) would also yield $c_2=0$ ).", "The above lemma shows that the domain decomposition algorithm (REF )-() yields a unique function $\\psi $ that is piecewise in $W^{2,\\infty }$ and piecewise (on the three regions) a solution to the Schrödinger equation (REF ).", "Moreover, one has the proposition: Proposition 2.6 Let Hypothesis a3 be satisfied.", "Then the function $\\psi $ obtained from (REF )-() belongs to $W^{2,\\infty }(0,1)$ and is the unique solution of (REF ) (as guaranteed by Proposition REF ).", "Proof.", "The matching conditions in (REF ) and () imply $C^1$ –continuity of $\\psi $ at $x_c$ , and the initial conditions in (REF ) imply $C^1$ –continuity of $\\psi $ at $x_d$ .", "Hence, $\\psi \\in C^1[0,1]$ , and this proves the claim." ], [ "Numerical analysis of the hybrid WKB-method", "To keep the presentation simple we shall consider here only the two-zone model of §REF .", "It has a coefficient function $a(x)$ that corresponds to Figure REF .", "In the following subsections we shall thus study step by step the numerical errors obtained when solving the BVP (REF ) in the evanescent region with a multiscale WKB-FEM and the IVP (REF ) in the oscillatory region with the marching method introduced in [2].", "We shall always assume that the phase function $\\int ^x\\sqrt{|a(y)|}\\,dy$ in the WKB-functions (REF ), (REF ) can be computed exactly, e.g., this holds for piecewise linear $a(x)$ .", "Otherwise, an additional quadrature error of the phase would need to be included in our subsequent analysis." ], [ "Variational formulation for the evanescent region BVP (", "Let us introduce in this section the variational formulation of the evanescent region problem (REF ) and study the well-posedness of the problem.", "As pointed out previously, we consider in the current paper situations with an abrupt potential jump, avoiding turning points, such that we shall suppose: Hypothesis A Let $V \\in W^{2,\\infty }(0,x_d)$ and $E>0$ satisfy the bounds $0<\\tau _{ev} \\le -a(x):=V(x)-E\\le M_{ev}\\,, \\quad \\forall x \\in (0,x_d)\\,.$ Furthermore let in the following $0 < \\varepsilon <1$ be arbitrary.", "We are now searching for a weak solution of (REF ) in the Hilbert space ${\\mathcal {V}}:=H^1(0,x_d)\\,, \\quad (\\chi ,\\theta )_{\\mathcal {V}}:=(\\chi ,\\theta )_{L^2(0,x_d)}+\\varepsilon ^2(\\chi ^{\\prime },\\theta ^{\\prime })_{L^2(0,x_d)}\\,.$ This $\\varepsilon $ –dependent norm gives rise to the following weighted Sobolev embedding, where the Gagliardo-Nirenberg inequality for bounded domains is used in the first estimate: $\\displaystyle \\begin{array}{ccc}\\displaystyle \\varepsilon \\Vert \\chi \\Vert _{C[0,x_d]}^2 &\\le &\\displaystyle C \\Vert \\chi \\Vert _{L^2(0,x_d)}\\,\\big (\\varepsilon \\Vert \\chi ^{\\prime }\\Vert _{L^2(0,x_d)}\\big )+C \\varepsilon \\Vert \\chi \\Vert ^2_{L^2(0,x_d)}\\\\[3mm]& \\le &\\displaystyle C \\big (\\Vert \\chi \\Vert _{L^2(0,x_d)}^2 + \\varepsilon ^2\\Vert \\chi ^{\\prime }\\Vert _{L^2(0,x_d)}^2\\big ) =C \\Vert \\chi \\Vert ^2_{\\mathcal {V}}\\,.\\end{array}$ The variational formulation reads: Find $\\chi \\in {\\mathcal {V}}$ , solution of $\\displaystyle b(\\chi ,\\theta )=L(\\theta )\\,, \\quad \\forall \\theta \\in \\mathcal {V}\\,,$ with the sesquilinear form $b:\\mathcal {V} \\times \\mathcal {V} \\rightarrow \\mathbb {C}$ and the antilinear form $L:\\mathcal {V} \\rightarrow \\mathbb {C}$ defined as $\\displaystyle \\begin{array}{lll}\\displaystyle b(\\chi ,\\theta )\\!\\!\\!\\!&:=&\\!\\!\\!\\!", "\\displaystyle \\varepsilon ^2 \\int _0^{x_d}\\!\\!", "\\chi ^{\\prime }(x)\\, \\overline{\\theta }^{\\prime }(x)\\, dx -\\int _0^{x_d}\\!\\!", "a(x)\\, \\chi (x)\\, \\overline{\\theta }(x)\\, dx + \\varepsilon \\sqrt{|a(0)|}\\, \\chi (0)\\,\\overline{\\theta }(0)\\,, \\quad \\forall \\chi ,\\theta \\in \\mathcal {V}\\\\[3mm]\\displaystyle L(\\theta )\\!\\!\\!\\!&:=&\\!\\!\\!\\!", "\\displaystyle \\varepsilon \\, \\overline{\\theta }(x_d)\\,, \\quad \\forall \\theta \\in \\mathcal {V}\\,.\\end{array}$ The BVP (REF ) is a standard elliptic problem, meaning that the forms $b(\\cdot ,\\cdot )$ and $L(\\cdot )$ are continuous and $b(\\cdot ,\\cdot )$ is coercive, i.e.", "there exists a constant $C>0$ independent of $\\varepsilon $ , such that for all $\\chi ,\\theta \\in \\mathcal {V}$ one has $|b(\\chi ,\\theta )|\\le C\\, \\Vert \\chi \\Vert _\\mathcal {V}\\, \\Vert \\theta \\Vert _\\mathcal {V}\\,, \\quad |L(\\theta )| \\le C\\sqrt{\\varepsilon }\\, \\Vert \\theta \\Vert _\\mathcal {V}\\,, \\quad b(\\theta ,\\theta ) \\ge \\min \\lbrace 1, \\tau _{ev} \\rbrace \\, \\Vert \\theta \\Vert _\\mathcal {V}^2\\,.$ The Lax-Milgram theorem implies then for each $\\varepsilon >0$ the existence and uniqueness of a weak solution $\\chi \\in \\mathcal {V}$ of (REF ).", "We have moreover the following lemma: Let Hypothesis A be satisfied.", "Then the weak solution $\\chi \\in {\\mathcal {V}}$ of (REF ) or (REF ) belongs even to $\\in H^2(0,x_d) \\hookrightarrow C^1[0,x_d]$ and satisfies the following estimates, with a constant $C>0$ independent on $\\varepsilon $ $\\displaystyle \\Vert \\chi \\Vert _{L^2(0,x_d)}^2 \\le C \\varepsilon \\,, \\quad \\varepsilon ^2\\, \\Vert \\chi ^{\\prime }\\Vert _{L^2(0,x_d)}^2 \\le C \\varepsilon \\,,$ as well as $\\displaystyle \\Vert \\chi \\Vert _{C[0,x_d]} \\le C\\,, \\quad \\varepsilon \\, \\Vert \\chi ^{\\prime }\\Vert _{C[0,x_d]} \\le C\\,.$ Proof.", "The Lax-Milgram theorem yields immediately $\\Vert \\chi \\Vert _{\\mathcal {V}} \\le \\frac{C\\sqrt{\\varepsilon }}{\\min \\lbrace 1,\\tau _{ev}\\rbrace }\\,,$ which implies (REF ) and, with (REF ), the first inequality of (REF ).", "To show the second estimate of (REF ), we observe that $\\begin{array}{lll}\\varepsilon ^2 (\\chi ^{\\prime }(x))^2&=&\\displaystyle \\varepsilon ^2 (\\chi ^{\\prime }(0))^2 + 2 \\int _0^x \\chi ^{\\prime }(y)\\, [\\varepsilon ^2\\, \\chi ^{^{\\prime \\prime }}(y)]\\, dy\\\\[3mm]&=& \\displaystyle |a(0)|\\, |\\chi (0)|^2 - 2 \\int _0^x a(y) \\chi ^{\\prime }(y)\\, \\chi (y)\\, dy\\\\[3mm]&\\le & \\displaystyle C+C\\Vert \\chi ^{\\prime }\\Vert _{L^2}\\, \\Vert \\chi \\Vert _{L^2} \\le C+C\\varepsilon \\Vert \\chi ^{\\prime }\\Vert _{L^2}^2+{C \\over \\varepsilon }\\, \\Vert \\chi \\Vert ^2_{L^2}\\le C\\,,\\end{array}$ where we used the other, just proved estimates." ], [ "Convergence analysis for the WKB-FEM method in the evanescent region", "The multi-scale WKB-FEM method we shall use for an efficient resolution of the evanescent region problem (REF ) is based on a specific choice of WKB-basis functions from (REF ).", "In more detail, the Hilbert space $\\mathcal {V}$ will be approximated by an appropriate finite-dimensional Hilbert space $\\mathcal {V}_h \\subset {\\mathcal {V}}$ , spanned by well chosen basis functions, and the continuous problem (REF ) will be approximated by the following discrete problem: Find $\\chi _h \\in {\\mathcal {V}_h}$ , solution of $\\displaystyle b(\\chi _h,\\theta _h)=L(\\theta _h)\\,, \\quad \\forall \\theta _h \\in \\mathcal {V}_h\\,.$ To introduce the finite-dimensional space $\\mathcal {V}_h$ , let us partition the interval $[0,x_d]$ into $0= x_1 < x_2 < \\cdots < x_N = x_d$ and denote the mesh size by $h_n:=x_{n+1}-x_n$ as well as $h:= \\max _{n=1, \\cdots ,N-1} \\lbrace h_n\\rbrace $ .", "The appropriate Hilbert space $\\mathcal {V}_h$ is then defined as $\\mathcal {V}_h:= \\left\\lbrace \\theta _h \\in \\mathcal {V} \\,\\, \\Big | \\,\\, \\theta _h(x)= \\sum _{n=1}^N z_n\\, \\zeta _n(x)\\,, \\quad z_n \\in \\mathbb {C}\\right\\rbrace \\,,$ with the WKB-hat functions defined as $\\displaystyle \\zeta _n(x):=\\left\\lbrace \\begin{array}{ll}v_{n-1}(x)\\,,& x \\in [x_{n-1},x_n]\\,,\\\\[3mm]w_{n}(x)\\,,& x \\in [x_{n},x_{n+1}]\\,,\\\\[3mm]0\\,,& \\mbox{otherwise}\\,.\\end{array}\\right.$ Here we used the notation $\\displaystyle \\begin{array}{lll}\\displaystyle w_n(x):= \\alpha _n(x)\\, q_{n}(x)&;&\\displaystyle v_n(x):= \\beta _n(x)\\, q_{n+1}(x)\\,,\\\\[3mm]\\displaystyle \\alpha _n(x):=- {\\sinh \\sigma _{n+1}(x) \\over \\sinh \\gamma _n}&;& \\displaystyle \\beta _n(x):= {\\sinh \\sigma _{n}(x) \\over \\sinh \\gamma _n}\\,, \\\\[3mm]\\displaystyle \\sigma _n(x):= { 1 \\over \\varepsilon } \\int _{x_n}^x \\sqrt{|a(y)|}\\, dy&;&\\displaystyle \\gamma _n:= { 1 \\over \\varepsilon } \\int _{x_n}^{x_{n+1}} \\sqrt{|a(y)|}\\,dy\\,,\\\\[3mm]\\displaystyle q_n(x):= \\frac{(V(x_n)-E)^{1/4}}{(V(x)-E)^{1/4}}\\,.\\end{array}$ Assuming Hypothesis A, $\\zeta _n$ is piecewise in $W^{2,\\infty }(x_j,x_{j+1})$ $\\forall j\\in \\lbrace 1,...,N-1\\rbrace $ and globally in $W^{1,\\infty }(0,x_d)\\hookrightarrow C[0,x_d]$ .", "Note that both components $v_n$ and $w_n$ of these (non-standard) hat functions are linear combinations of the evanescent WKB-functions of first order, i.e.", "$\\varphi _1^{ev}$ given in (REF ).", "Hence these hat functions are solutions of our Schrödinger equation up to an error of order $\\mathcal {O}(\\varepsilon ^2)$ , i.e.", "$\\varepsilon ^2 \\zeta _n^{^{\\prime \\prime }}(x) + a(x)\\, \\zeta _n(x) = \\varepsilon ^2 \\left[ {5 \\over 16}\\, {(V^{\\prime }(x))^2 \\over (V(x)-E)^2 } + {1 \\over 4} {V^{^{\\prime \\prime }}(x) \\over E-V(x)}\\right]\\, \\zeta _n(x)\\,, &&\\\\\\forall x \\in (x_{n-1},x_n)\\cup (x_n,x_{n+1})\\,.&&$ This peculiarity signifies that the hat functions incorporate already some essential information about the solutions we are searching for, leading to a scheme which will be asymptotically correct in the limit $\\varepsilon \\rightarrow 0$ , as will be seen later on.", "For later purposes let us introduce here the differential operator ${\\mathcal {A}}_{\\varepsilon }(\\xi ) := -\\varepsilon ^2 \\xi ^{^{\\prime \\prime }}(x) - a(x)\\, \\xi (x) +\\varepsilon ^2 r(x)\\, \\xi (x)\\,,$ with the function $r(x) := \\displaystyle {5 \\over 16}\\, {(V^{\\prime }(x))^2 \\over (V(x)-E)^2 } + {1 \\over 4} {V^{^{\\prime \\prime }}(x) \\over E-V(x)}\\,.$ For each $\\xi \\in {\\mathcal {V}}_h$ one has ${\\mathcal {A}}_{\\varepsilon }(\\xi ) =0$ in every interval $I_n:=(x_n,x_{n+1})$ .", "But we note that ${\\mathcal {A}}_{\\varepsilon }$ cannot be applied globally on $(0,x_d)$ , as functions in ${\\mathcal {V}}_h$ typically have discontinuous derivatives at the grid points $x_n$ .", "We also remark that, due to Lax-Milgram's theorem, the discrete problem (REF ) admits $\\forall \\varepsilon >0$ also a unique solution $\\chi _h \\in {\\mathcal {V}}_h$ .", "The aim is now to investigate the error between the exact solution of (REF ), denoted by $\\chi _{ex}$ , and the solution of the discrete problem (REF ), denoted by $\\chi _h$ .", "We denote by $\\Pi _h^{\\varepsilon } \\chi _{ex} \\in \\mathcal {V}_h$ the interpolant of the exact solution in the finite dimensional Hilbert space $\\mathcal {V}_h$ , i.e.", "$\\displaystyle \\Pi _h^{\\varepsilon } \\chi _{ex} (x) := \\sum _{n=1}^N \\chi _{ex}(x_n)\\, \\zeta _n(x)\\,, \\quad \\forall x \\in [0,x_d]\\,.$ Then, the numerical error can be split as follows $e_h(x):= \\chi _{ex}(x)-\\chi _h(x) = \\left(\\chi _{ex}(x)- \\Pi _h^{\\varepsilon } \\chi _{ex}\\right) + \\left(\\Pi _h^{\\varepsilon } \\chi _{ex}-\\chi _h \\right)=:e_h^1(x)+e_h^2(x)\\,,$ where $e_h^1$ corresponds to the interpolation error (consistency) and $e_h^2$ is the stability error.", "These two error parts shall be now estimated separately." ], [ "Consistency error estimate", "The goal of this section is to estimate the interpolation error $e_h^1(x):=\\chi _{ex}(x)-\\Pi _h^{\\varepsilon } \\chi _{ex}(x)$ in the $\\mathcal {V}$ -norm.", "To this end, note that the equation satisfied by $e_h^1$ in $I_n$ is $\\left\\lbrace \\begin{array}{l}{\\mathcal {A}}_{\\varepsilon }(e_h^1) = \\varepsilon ^2\\, r(x)\\, \\chi _{ex}(x)\\,, \\qquad \\forall x \\in I_n\\,,\\\\[3mm]\\displaystyle e_h^1(x_n)=e_h^1(x_{n+1})=0\\,.\\end{array}\\right.$ The variation of constants method, i.e.", "writing $e_h^1(x)=c_1(x)\\, w_n(x)+c_2(x)\\, v_n (x)$ in $I_n$ leads after some lengthy but straightforward computations (see [17] for the oscillatory case) to the following explicit expressions for the error function $\\displaystyle e_h^1(x)=\\mathcal {E}_1(x)+\\mathcal {E}_2(x)\\, , \\qquad x \\in I_n\\,,$ with $\\begin{array}{lll}\\displaystyle \\mathcal {E}_1(x)&\\!\\!\\!", "=&\\!\\!\\!", "\\displaystyle -\\frac{\\varepsilon }{(V(x)-E)^{1/4}}\\, { \\sinh \\sigma _{n+1} (x)\\over \\sinh \\gamma _n }\\, \\int _{x_n}^x \\frac{r(y)\\, \\chi _{ex}(y)}{(V(y)-E)^{1/4}} \\sinh \\sigma _n(y)\\, dy\\, ,\\\\[5mm]\\displaystyle \\mathcal {E}_2(x)&\\!\\!\\!", "=&\\!\\!\\!", "\\displaystyle \\frac{\\varepsilon }{(V(x)-E)^{1/4}} \\, \\frac{\\sinh \\sigma _n(x)}{\\sinh \\gamma _n} \\, \\int _{x_{n+1}}^{x} \\!\\!", "\\frac{r(y)\\, \\chi _{ex}(y)}{(V(y)-E)^{1/4}} \\sinh \\sigma _{n+1}(y)\\, dy\\, .\\end{array}$ Differentiating the interpolation error function yields $\\displaystyle (e_{h}^1)^{\\prime }(x)=\\mathcal {D}_1(x)+\\mathcal {D}_2(x)+\\mathcal {D}_3(x)+\\mathcal {D}_4(x)= \\mathcal {D}_2(x)+\\mathcal {D}_4(x)\\, ,$ with $\\begin{array}{lll}\\displaystyle \\mathcal {D}_1(x)&\\!\\!\\!", "=&\\!\\!\\!", "-\\mathcal {D}_3(x) =\\displaystyle -\\varepsilon \\, { \\sinh \\sigma _{n+1} (x)\\over \\sinh \\gamma _n }\\, \\frac{\\sinh \\sigma _n(x)}{\\sqrt{V(x)-E}}\\, r(x)\\, \\chi _{ex}(x)\\,,\\\\[5mm]\\displaystyle \\mathcal {D}_2(x)&\\!\\!\\!=&\\!\\!\\!", "\\displaystyle - {\\varepsilon \\over \\sinh \\gamma _n} \\, \\left[ {(V(x)-E)^{1/4} \\over \\varepsilon } \\cosh \\sigma _{n+1}(x) - {V^{\\prime }(x)\\, \\sinh \\sigma _{n+1}(x) \\over 4\\, (V(y)-E)^{5/4} } \\right] \\\\[5mm]&& \\times \\,\\displaystyle \\int _{x_n}^x \\frac{r(y) \\, \\chi _{ex}(y)\\, \\sinh \\sigma _n(y)}{(V(y)-E)^{1/4}} \\, dy\\, , \\\\[5mm]\\displaystyle \\mathcal {D}_4(x)&\\!\\!\\!", "=&\\!\\!\\!", "\\displaystyle {\\varepsilon \\over \\sinh \\gamma _n} \\, \\left[ {(V(x)-E)^{1/4} \\over \\varepsilon } \\cosh \\sigma _{n}(x) - {V^{\\prime }(x)\\, \\sinh \\sigma _{n}(x) \\over 4\\, (V(y)-E)^{5/4} } \\right] \\\\[5mm]&& \\times \\,\\displaystyle \\int _{x_{n+1}}^x \\frac{r(y) \\, \\chi _{ex}(y)\\, \\sinh \\sigma _{n+1}(y)}{(V(y)-E)^{1/4}} \\, dy\\, .\\end{array}$ In order to estimate the interpolation error in the $\\mathcal {V}$ -norm, we shall investigate each of these terms separately.", "In this study, the behaviour of the following functions is very important: $\\Theta _{ss}(x):=\\frac{\\sinh (\\sigma _n(x))\\,\\sinh (-\\sigma _{n+1}(x)) }{\\sinh \\gamma _n}\\,, \\quad \\Theta _{sc}(x):=\\frac{\\sinh (\\sigma _n(x))\\,\\cosh (-\\sigma _{n+1}(x)) }{\\sinh \\gamma _n}\\,,$ $\\Theta _{cs}(x):=\\frac{\\cosh (\\sigma _n(x))\\,\\sinh (-\\sigma _{n+1}(x)) }{\\sinh \\gamma _n}\\,, \\qquad \\forall x \\in I_n\\,.$ Next we shall use $\\left|\\frac{r(y)}{(V(y)-E)^{1/4}}\\right|\\le C$ and the fact that $\\sinh \\sigma _n(x)$ and $\\sinh (- \\sigma _{n+1}(x))$ are both non-negative on $I_n$ and, respectively, increasing and decreasing.", "Then one can show for $x \\in I_n$ : $|\\mathcal {E}_1(x)| \\le C \\varepsilon \\, \\Theta _{ss}(x)\\, \\int _{x_n}^x |\\chi _{ex}(y)|\\, dy\\,, \\qquad |\\mathcal {E}_2(x)| \\le C \\varepsilon \\, \\Theta _{ss}(x)\\, \\int _{x}^{x_{n+1}} |\\chi _{ex}(y)|\\, dy \\,,$ and $|\\mathcal {D}_2(x)|\\le C\\, \\Theta _{sc}(x) \\int _{x_n}^x |\\chi _{ex}(y)|\\, dy + C \\varepsilon \\, \\Theta _{ss}(x) \\int _{x_n}^x |\\chi _{ex}(y)|\\, dy \\,,$ $|\\mathcal {D}_4(x)|\\le C\\, \\Theta _{cs}(x) \\int _{x}^{x_{n+1}} |\\chi _{ex}(y)|\\, dy + C \\varepsilon \\, \\Theta _{ss}(x) \\int _{x}^{x_{n+1}} |\\chi _{ex}(y)|\\, dy \\,.$ In the above estimates, the constant $C$ depends only on our data $a(x)$ and $E$ , but not on $\\varepsilon $ and $\\chi _{ex}$ .", "Using $\\sigma _n(x)-\\sigma _{n+1}(x)=\\gamma _n$ we easily find $0 \\le \\Theta _{ss}(x) \\le \\frac{\\cosh \\gamma _n -1 }{2\\, \\sinh \\gamma _n} \\le {1 \\over 2}\\,, \\quad \\forall x \\in I_n\\,,$ $0 \\le \\Theta _{sc}(x) = {1 \\over 2} +\\frac{\\sinh (\\sigma _n(x) + \\sigma _{n+1}(x)) }{2\\, \\sinh \\gamma _n} \\le 1\\,,\\quad \\forall x \\in I_n\\,,$ $0 \\le \\Theta _{cs}(x) = {1 \\over 2}-\\frac{\\sinh (\\sigma _n(x) +\\sigma _{n+1}(x)) }{2\\, \\sinh \\gamma _n}\\le 1\\,, \\quad \\forall x \\in I_n\\,.$ With the asymptotic behaviour $\\frac{\\cosh \\xi -1}{2\\, \\sinh \\xi } \\;\\stackrel{\\xi \\sim 0}{\\sim } \\;{\\xi \\over 4}\\,, \\quad \\frac{\\cosh \\xi -1}{2\\, \\sinh \\xi } \\stackrel{\\xi \\rightarrow \\pm \\infty }{\\longrightarrow } \\pm {1 \\over 2}$ we obtain (using $|\\gamma _n|\\le C\\frac{h}{\\varepsilon }$ ) $\\varepsilon | \\Theta _{ss}(x)| \\le C \\min \\lbrace \\varepsilon ,h\\rbrace \\,.$ With Lemma REF this permits to prove the following lemma: Let Hypothesis A be satisfied.", "Then the following estimates hold for the interpolation error $e_h^1\\in {\\mathcal {V}}_h \\subset C[0,x_d]$ of the exact solution $\\chi _{ex} \\in \\mathcal {V}$ of (REF ): $\\Vert e_h^1\\Vert _{L^2(0,x_d)} \\le C \\sqrt{\\varepsilon }\\, h\\, \\min \\lbrace \\varepsilon ,h\\rbrace \\,, \\quad \\varepsilon \\,\\Vert (e_h^1)^{\\prime }\\Vert _{L^2(0,x_d)} \\le C \\varepsilon ^{3/2}\\, h\\,,$ $\\Vert e_h^1\\Vert _{C[0,x_d]} \\le C \\,\\sqrt{h}\\, \\min \\lbrace \\varepsilon ^{3/2},h^{3/2}\\rbrace \\,, \\quad \\varepsilon \\Vert (e_h^1)^{\\prime }\\Vert _{L^{\\infty }(0,x_d)}\\le C \\varepsilon \\, \\sqrt{h}\\,\\min \\lbrace \\sqrt{\\varepsilon },\\sqrt{h}\\rbrace \\,.$ Let $\\varphi _{ex}\\in W^{2,\\infty }(x_d,1)$ be the exact solution of (REF ) as guaranteed by Lemma REF .", "In the above vector notation it will be denoted by $U_{ex}(x)$ or simply $U(x)$ , and is solution to the system $ \\left\\lbrace \\begin{array}{l}\\displaystyle U^{\\prime }(x)=\\left[ \\frac{1}{\\varepsilon } A_0(x)+\\varepsilon A_1(x)\\right] U(x)\\,,\\quad x_d<x<1\\,, \\\\[3mm]\\displaystyle U(x_d)=A(x_d+)\\, (\\chi _{ex}(x_d), 1)^\\top \\,,\\end{array}\\right.$ with the two matrices $A_0(x):=\\sqrt{a(x)}\\left(\\begin{array}{cc}0&1\\\\-1&0\\end{array}\\right)\\,; \\quad A_1(x):=\\left(\\begin{array}{cc}0&0\\\\2 \\beta (x)&0\\end{array}\\right)\\, .$ Here, $\\beta =-\\frac{1}{2a^{1/4}}(a^{-1/4})^{\\prime \\prime }$ which was already defined in (REF ), and the matrix element $a(x_d+)$ of $A(x_d+)$ denotes the right-sided limit of $a$ at the jump discontinuity $x_d$ .", "We also use the analogous notation for $a^{\\prime }(x_d+)$ .", "In the sequel we shall need an a-priori estimate on this solution.", "The upper bound was already given in §2.1 of [2].", "But for the scaling Step 3 we shall also need an $\\varepsilon $ -uniform lower bound on the solution: Let Hypothesis B hold.", "Then, the ODE (REF ) admits a unique solution $U \\in W^{1,\\infty }(x_d,1)$ , which satisfies $\\displaystyle \\Vert U(x_d)\\Vert \\exp \\Big [ -\\varepsilon \\int _{x_d}^x |\\beta (y)| dy\\Big ] \\le \\Vert U(x)\\Vert \\le \\Vert U(x_d)\\Vert \\exp \\Big [ \\varepsilon \\int _{x_d}^x |\\beta (y)| dy\\Big ]\\,,\\quad x_d\\le x\\le 1\\,.$ Thus, there exist constants $C_3,C_4>0$ independent on $\\varepsilon $ such that $\\displaystyle C_3\\le \\Vert U\\Vert _{C[x_d,1]} \\le C_4\\,,\\qquad \\forall 0<\\varepsilon \\le \\varepsilon _0\\,.$ Proof.", "For the norm $\\Vert U\\Vert ^2 := |u_1|^2 + |u_2|^2$ we compute for (REF ): $\\left|\\frac{d}{dx} \\Vert U(x)\\Vert ^2\\right| =\\left|2\\varepsilon \\beta (x)\\,\\big (u_1 {\\overline{u}}_2 + {\\overline{u}}_1 u_2\\big )\\right| \\le 2\\varepsilon |\\beta (x)| \\,\\Vert U(x)\\Vert ^2\\,.$ This implies (REF ).", "The estimate (REF ) is now a simple consequence of (REF ), presupposing that one proves some $\\varepsilon $ -independent bounds on the initial condition $\\Vert U_{ex}(x_d)\\Vert $ , or equivalently $\\Vert (\\chi _{ex}(x_d),\\,1)^\\top \\Vert $ .", "The latter norm is clearly bounded below by 1, and it is also bounded above due to the a-priori estimate on $\\Vert \\chi _{ex}\\Vert _{C[0,x_d]}$ from Lemma REF .", "Hence there exist constants $0<C_1,\\,C_2$ , independent of $0<\\varepsilon \\le \\varepsilon _0$ , such that $\\displaystyle C_1\\le \\Vert U_{ex}(x_d)\\Vert \\le C_2\\,,\\qquad \\forall 0<\\varepsilon \\le \\varepsilon _0\\,,$ leading to (REF )." ], [ "Review of the WKB-marching method for the oscillatory region", "In this subsection we shall first review the WKB-marching method for solving the IVP (REF ) (or, equivalently, (REF )).", "Then we recall its error estimates from [2].", "Following [2] this method consists of two parts, first an analytic transformation of (REF ) or (REF ) into a less oscillatory problem, and second the discretization of the smooth problem on a coarse grid in an $\\varepsilon $ -uniform manner.", "As shown in [2], the analytic WKB-transformation reviewed here is related to using oscillatory WKB-functions of second order, $\\varphi _2^{os}(x)$ .", "Part 1 – analytic transformation: The starting point is the vectorial IVP (REF ).", "The vector function $U\\in \\mathbb {C}^2$ is then transformed to the new unknown $Z\\in \\mathbb {C}^2$ by $Z(x) =\\left(\\begin{array}{c}z_1\\\\[2mm]\\displaystyle z_2\\end{array}\\right):= \\exp \\left(-{i\\over \\varepsilon } \\Phi ^\\varepsilon (x) \\right) P\\,U(x)\\,, \\quad \\forall x \\in [x_d,1]\\,,$ with the matrices $P := {1\\over \\sqrt{2}}\\left(\\begin{array}{cc}i&1\\\\1&i\\end{array}\\right)\\quad ; \\quad \\Phi ^\\varepsilon (x) :=\\left(\\begin{array}{cc}\\displaystyle \\phi ^{\\varepsilon }(x) &0\\\\0&-\\phi ^{\\varepsilon }(x) \\end{array}\\right)\\,,$ and the (real valued) phase function $ \\phi ^{\\varepsilon }(x):=\\int _{x_d}^x \\left( \\sqrt{a(y)} - \\varepsilon ^2 \\beta (y)\\right) \\,dy\\,.$ This change of unknown leads to the smooth ODE-system $ \\left\\lbrace \\begin{array}{l}\\displaystyle {dZ\\over dx} = \\varepsilon N^\\varepsilon Z\\,,\\quad x_d<x<1\\,,\\\\[3mm]\\displaystyle Z(x_d):=P\\,U_{ex}(x_d)\\,.\\end{array}\\right.$ Here, the $2\\times 2$ –matrix function $N^\\varepsilon (x) := \\beta (x)\\, \\exp ( -{i\\over \\varepsilon } \\Phi ^\\varepsilon )\\, \\left(\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right) \\, \\exp ( {i\\over \\varepsilon } \\Phi ^\\varepsilon )\\,,$ is bounded independently of $\\varepsilon $ .", "It is off-diagonal, with the entries $N^\\varepsilon _{1,2} (x)= {\\beta }(x)e^{-\\frac{2i}{ \\varepsilon } \\phi ^{\\varepsilon }(x)}\\,,\\quad N^\\varepsilon _{2,1} (x)= {\\beta }(x) e^{\\frac{2i}{ \\varepsilon } \\phi ^{\\varepsilon }(x)}\\,.$ This finishes the analytical transformation, and the goal of the second part is to provide an $\\varepsilon $ -uniform discretization of (REF ) that is second order w.r.t.", "the mesh size.", "Part 2 – numerical discretization: First we partition the interval $[x_d,1]$ into $x_d=x_N < x_{N+1} < \\cdots < x_M = 1$ .", "As in §REF we denote the mesh size by $h_n:=x_{n+1}-x_n$ as well as $h:= \\max _{n=1, \\cdots ,M-1} \\lbrace h_n\\rbrace $ .", "With the initial condition $Z_N:=P\\,U_N\\in \\mathbb {C}^2$ and $U_N:=U_{ex}(x_d)\\in \\mathbb {C}^2$ given, the marching scheme reads as follows (see [2]): $\\displaystyle Z_{n+1} = (I + A_{n}^1+ A_{n}^2) \\,Z_{n}\\,,\\qquad n=N,...,M-1\\,,$ with the $2\\times 2$ –matrices $\\begin{array}{lll}\\displaystyle A_{n}^1:=& \\\\[3mm]&&\\hspace{-56.9055pt} - i \\varepsilon ^2 \\!\\!\\left(\\!\\!\\begin{array}{cc}0&\\hspace{-22.76219pt} \\beta _0(x_{n})e^{-{2i \\over \\varepsilon } \\phi (x_{n})} -\\beta _0(x_{n+1})e^{-{2i \\over \\varepsilon } \\phi (x_{n+1})}\\\\[3mm]\\beta _0(x_{n+1}) e^{{2i \\over \\varepsilon } \\phi (x_{n+1})} -\\beta _0(x_{n})e^{{2i \\over \\varepsilon } \\phi (x_{n})}&\\hspace{-22.76219pt} 0\\end{array}\\!\\!\\!\\right)\\\\[6mm]&&\\hspace{-56.9055pt} \\displaystyle + \\varepsilon ^3 \\!\\!\\left(\\!\\!\\begin{array}{cc}0&\\hspace{-22.76219pt} \\beta _1(x_{n+1})e^{-{2i \\over \\varepsilon } \\phi (x_{n+1})} -\\beta _1(x_{n})e^{-{2i \\over \\varepsilon } \\phi (x_{n})}\\\\[3mm]\\beta _1(x_{n+1}) e^{{2i \\over \\varepsilon } \\phi (x_{n+1})} -\\beta _1(x_{n})e^{{2i \\over \\varepsilon } \\phi (x_{n})}&\\hspace{-22.76219pt} 0\\end{array}\\!\\!\\right)\\\\[6mm]&&\\hspace{-56.9055pt} \\displaystyle + i \\varepsilon ^4 \\beta _2(x_{n+1}) \\left(\\begin{array}{cc}0& -e^{-{2i \\over \\varepsilon } \\phi (x_{n})} H_1(-{2 \\over \\varepsilon } S_{n})\\\\[3mm]e^{{2i \\over \\varepsilon } \\phi (x_{n})} H_1({2 \\over \\varepsilon } S_{n})&0\\end{array}\\!", "\\right)\\\\[6mm]&&\\hspace{-56.9055pt} \\displaystyle - \\varepsilon ^5 \\beta _3(x_{n+1}) \\left(\\begin{array}{cc}0& e^{-{2i \\over \\varepsilon } \\phi (x_{n})} H_2(-{2 \\over \\varepsilon } S_{n})\\\\[3mm]e^{{2i \\over \\varepsilon } \\phi (x_{n})} H_2({2 \\over \\varepsilon } S_{n})&0\\end{array}\\!", "\\right)\\,,\\end{array}$ $\\begin{array}{lll}\\displaystyle A_{n}^2&:=& \\displaystyle - i \\varepsilon ^3 (x_{n+1} -x_n) { \\beta (x_{n+1}) \\beta _0(x_{n+1}) +\\beta (x_{n}) \\beta _0(x_{n}) \\over 2}\\left(\\begin{array}{cc}\\displaystyle 1&0\\\\\\displaystyle 0&\\displaystyle -1\\end{array}\\right)\\\\[6mm]&& \\displaystyle - \\varepsilon ^4 \\beta _0(x_{n}) \\beta _0(x_{n+1})\\left(\\begin{array}{cc}\\displaystyle H_1(-{2\\over \\varepsilon }S_{n})&0\\\\[5mm]\\displaystyle 0& \\displaystyle H_1({2\\over \\varepsilon }S_{n})\\end{array}\\right)\\\\[7mm]&& \\displaystyle +i \\varepsilon ^5\\beta _1(x_{n+1}) [\\beta _0(x_{n})-\\beta _0(x_{n+1})] \\left(\\begin{array}{cc}\\displaystyle H_2(-{2\\over \\varepsilon }S_{n})&\\displaystyle 0\\\\[4mm]\\displaystyle 0&\\displaystyle -H_2({2\\over \\varepsilon }S_{n})\\end{array}\\right)\\,.\\end{array}$ Here we used the notation $\\beta _0(y) := \\frac{\\beta }{2 (\\sqrt{a} -\\varepsilon ^2\\beta )} (y)\\,;\\qquad \\beta _{k+1}(y) := \\frac{1}{2 \\phi ^{\\prime }(y)} {d\\beta _k \\over dy}(y)\\,,\\quad k=0,\\,1,\\,2,$ $H_1(\\eta ) := e^{i\\eta } - 1\\,, \\qquad H_2(\\eta ) := e^{i\\eta } - 1-i\\eta \\,,$ and the discrete phase increments $S_{n}\\!", ":= \\phi (x_{n+1}) - \\phi (x_{n})=\\!", "\\int _{x_n}^{x_{n+1}} \\!\\!\\!", "\\left(\\sqrt{a(y)}-\\varepsilon ^2 \\beta (y)\\right) dy\\,.$ Remark that for notational reasons we omitted in the aforementioned description of the scheme the $\\varepsilon $ -index.", "Furthermore we assumed that the two functions $\\phi $ and $\\beta $ (the latter involving the derivatives $a^{\\prime }$ , $a^{\\prime \\prime }$ ) are explicitly “available”.", "Alternatively, $\\phi $ , $a^{\\prime }$ and $a^{\\prime \\prime }$ could be approximated numerically.", "But, for simplicity, we shall not include such errors in the subsequent error analysis.", "Finally we have to transform back to the $U$ -solution vector via $\\displaystyle U_n=P^{-1} \\,e^{{i \\over \\varepsilon } \\Phi ^{\\varepsilon } (x_n)} \\,Z_n\\,, \\qquad n=N+1,...,M\\,,$ which concludes the review of the WKB-marching algorithm.", "The following lemma is the discrete analogue of Lemma REF .", "Let Hypothesis B hold and let the initial condition $U_N\\in \\mathbb {R}^2$ .", "Then the iteration (REF )-(REF ) determines a well-defined sequence satisfying $U_{n}\\in \\mathbb {R}^2$ $\\forall n=N+1,...,M$ .", "Furthermore $\\exists \\, \\tilde{\\varepsilon }_0\\in (0,\\varepsilon _0]$ such that $\\displaystyle C_5\\le \\Vert U_n\\Vert \\le C_6\\,, \\qquad n=N,...,M\\,,$ with some positive constants $C_5,\\,C_6$ that are independent of $0<\\varepsilon \\le \\tilde{\\varepsilon }_0$ and the numerical grid on $[x_d,1]$ .", "Proof.", "Let us start by analysing the propagation matrix $B_n:=I+A_n^1+A_n^2\\in \\mathbb {C}^{2\\times 2}$ of the vector $Z_n$ as defined in (REF ).", "A straightforward computation reveals its symmetry (which was also used in the proof of Proposition 3.3, [2]): $\\overline{(B_n)_{11}} = (B_n)_{22}\\,,\\quad \\overline{(B_n)_{12}} = (B_n)_{21}\\,,\\qquad n=N,...,M-1\\,.$ This symmetry carries over to the matrix $\\tilde{B}_n = \\left(\\begin{array}{cc}b_1 &\\displaystyle b_2\\\\[4mm]{\\overline{b}}_2&{\\overline{b}}_1\\end{array}\\right) := e^{{i \\over \\varepsilon } \\Phi ^{\\varepsilon } (x_{n+1})}\\,B_n\\,e^{-{i \\over \\varepsilon } \\Phi ^{\\varepsilon } (x_n)} \\,.$ With this notation, the propagation matrix for the vector $U_n$ reads (cf.", "(REF ), (REF )): $P^{-1}\\, \\tilde{B}_n\\, P = \\left(\\begin{array}{cc}\\Re b_1 + \\Im b_2 & \\Im b_1 + \\Re b_2 \\\\[4mm]-\\Im b_1 + \\Re b_2& \\Re b_1 - \\Im b_2\\end{array}\\right)\\in \\mathbb {R}^{2\\times 2}\\,,$ where we used $P^{-1}= {1\\over \\sqrt{2}}\\left(\\begin{array}{cc}-i&1\\\\1&-i\\end{array}\\right)\\,.$ This shows that $U_n\\in \\mathbb {R}^2$ .", "Coming now to the bounds of $U_n$ , a simple Taylor expansion for the matrices in (REF ) yields $\\Vert A_n^1\\Vert \\le C\\varepsilon h_n$ , $\\Vert A_n^2\\Vert \\le C\\varepsilon ^3 h_n$ , and hence with some constant $C_7>0$ : $\\Vert Z_{n+1}\\Vert \\le \\Vert Z_{n}\\Vert (1+C_7 \\varepsilon h_n) \\le \\Vert Z_{n}\\Vert e^{C_7 \\varepsilon h_n} \\le \\Vert Z_N \\Vert e^{C_7 \\varepsilon (1-x_d)}\\le C_2 \\Vert P\\Vert e^{C_7 \\varepsilon (1-x_d)}\\,,$ where we used $Z_N:=P\\,U_{ex}(x_d)$ and the estimate (REF ) in the last step.", "Next we consider the transformation matrices in (REF ): $P^{-1}$ is independent of $\\varepsilon $ and $h$ , and $e^{{i \\over \\varepsilon } \\Phi ^{\\varepsilon } (x_n)}$ is unitary.", "This implies the upper bound in (REF ).", "For the lower bound we choose $\\tilde{\\varepsilon }_0:=\\min \\lbrace \\varepsilon _0,\\,\\frac{1}{2C_7(1-x_d)}\\rbrace >0$ such that $\\Vert A_n^1+A_n^2\\Vert \\le C_7\\varepsilon h_n\\le 0.5\\,;\\qquad \\forall 0<\\varepsilon \\le \\tilde{\\varepsilon }_0\\,,\\quad \\forall 0<h_n\\le 1-x_d\\,.$ With the elementary estimate $\\frac{1}{1-y}\\le 4^y$ for $y\\in [0,0.5]$ we then obtain $\\Vert (I+A_n^1+A_n^2)^{-1}\\Vert \\le \\frac{1}{1-\\Vert A_n^1+A_n^2\\Vert } \\le 4^{\\Vert A_n^1+A_n^2\\Vert }\\,.$ This allows to estimate the backwards propagation $Z_n=(I+A_n^1+A_n^2)^{-1}\\,Z_{n+1}$ as $\\Vert Z_{n+1}\\Vert \\ge \\Vert Z_{n}\\Vert \\, 4^{-C_7 \\varepsilon h_n} \\ge \\Vert Z_{N}\\Vert \\, 4^{-C_7 \\varepsilon (1-x_d)}\\,,$ and the lower bound on $\\Vert U_n\\Vert $ follows as before.", "Due to the above lemma we have to restrict the range of admissible $\\varepsilon $ –values: Hypothesis B' Let the assumptions of Hypothesis B hold, but with $\\varepsilon _0$ replaced by $\\tilde{\\varepsilon }_0$ from Lemma REF ." ], [ "Error and stability estimates for the WKB-marching method", "In this subsection we recall the main Theorem 3.1 of [2], providing error bounds for the marching method (REF )-(REF ), used for solving the IVP (REF ) or, equivalently, (REF ).", "(Convergence WKB-IVP) Let Hypothesis B be satisfied and let $U_{ex}(x)$ denote the exact solution to the IVP (REF ).", "Then the global error of the second order scheme (REF )-(REF ) satisfies $\\displaystyle \\qquad \\Vert U_{ex}(x_n)-U_n\\Vert _{} \\le C {h^{\\gamma }\\over \\varepsilon } +C \\varepsilon ^3 h^2\\,, \\; N\\le n\\le M\\,,$ with $C$ independent of $n$ , $h$ , and $\\varepsilon $ .", "Here, $\\gamma >0$ is the order of the chosen numerical integration method for computing the phase integral $\\displaystyle \\Phi ^{\\varepsilon }(x)= \\int _{x_d}^x \\left(\\sqrt{a(y)} -\\varepsilon ^2 \\beta (y) \\right) dy\\, \\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right)\\,$ for the back-transformation (REF ).", "We remark that the term ${h^{\\gamma } \\over \\varepsilon }$ of (REF ) may or may not be present in real computations, depending on the chosen coefficient function $a(x)$ .", "If $a(x)$ is piecewise linear or piecewise quadratic, e.g., the phase integral $\\phi ^\\varepsilon (x)$ can be computed analytically.", "Hence, this term would not appear in such cases.", "In the numerical tests performed in § below, we shall only consider such examples of exactly computable phase functions and shall hence not include this error term in the error analysis of §REF ." ], [ "Convergence results for the overall hybrid WKB method", "In this section we shall combine the error analysis of the previous two sections and adapt it to the algorithm for coupling two regions.", "To this end we have to fix the numerical analogues of the continuous coupling conditions in (REF ).", "First we shall (of course) use $\\varphi _h(x_d):=\\chi _h(x_d)$ .", "But for the initial condition of the derivative there are two options, namely $\\varepsilon \\varphi _h^{\\prime }(x_d):=\\varepsilon \\chi _h^{\\prime }(x_d)$ or $\\varepsilon \\varphi _h^{\\prime }(x_d):=1$ (taken from the exact value in (REF )).", "We shall use the second option for the following reasons: On the one hand it avoids the numerical error of $\\chi _h^{\\prime }(x_d)$ , where we recall that $\\chi _h^{\\prime }$ is discontinuous (and hence less accurate) at the grid points.", "And on the other hand, this choice will facilitate the a-priori estimate needed for the scaling in Step 3.", "Since the numerically used initial data $\\chi _h(x_d)$ deviates from its exact value $\\chi _{ex}(x_d)$ , this gives rise to an additional error component to be considered: Let thus $\\hat{U}_{ex}(x)$ denote the exact solution to the ODE (REF ), but with the following perturbed initial condition: $\\hat{U}_{ex}(x_d):=A(x_d+)\\left( \\begin{array}{c}\\chi _{h}(x_d) \\\\ 1\\end{array} \\right)\\,.$ Using the a-priori estimate (REF ) leads to the error $&& \\Vert U_{ex}-\\hat{U}_{ex}\\Vert _{C[x_d,1]} \\le \\left\\Vert A(x_d+)\\,\\big (e_h(x_d),\\,0\\big )^\\top \\right\\Vert \\,\\exp \\Big [ \\varepsilon \\int _{x_d}^1 |\\beta (y)| dy\\Big ] \\nonumber \\\\&&\\qquad \\qquad \\quad = \\Big [ a(x_d+)^{1/2}+\\frac{\\varepsilon ^2}{16}a^{-5/2}(x_d+) a^{\\prime }(x_d+)^2\\Big ]^{1/2}\\,|e_h(x_d)|\\exp \\Big [ \\varepsilon \\int _{x_d}^1 |\\beta (y)| dy\\Big ]\\,,$ where we used the notation $e_h=\\chi _{ex}-\\chi _h$ .", "The following convergence analysis of the hybrid method uses several different solution functions (exact, numerical, etc.).", "To keep the notation straight we summarize it in the following table, both for the evanescent and oscillatory regions.", "The superscript ${}{}^{(}{}^{\\prime }{}^{)}$ signifies that we refer to both the function and its first derivative.", "$\\begin{array}{|r|l|}\\hline \\hbox{\\rm evanescent} & \\\\ \\hline \\phantom{I^{I^{I^I}}} \\chi _{ex}\\!\\!\\!", "{}^{(}{}^{\\prime }{}^{)}(x) & \\hbox{\\rm exact solution of Step 1, eq.", "(\\ref {region1_ev}), (\\ref {VF_ev})} \\\\\\chi _{h}\\!", "{}^{(}{}^{\\prime }{}^{)}(x) & \\hbox{\\rm numerical solution of Step 1, eq.", "(\\ref {VF_ev_h})} \\\\ \\hline \\hbox{\\rm oscillatory} & \\\\ \\hline \\phantom{I^{I^{I^I}}} \\varphi _{ex}\\!\\!\\!", "{}^{(}{}^{\\prime }{}^{)}(x) & \\hbox{\\rm exact solution of Step 2, with exact IC $\\chi _{ex}(x_d)$, eq.", "(\\ref {region2_ev})} \\\\U_{ex}(x) & \\hbox{\\rm exact solution vector of Step 2, with exact IC $\\chi _{ex}(x_d)$, eq.", "(\\ref {EQU})} \\\\\\hat{U}_{ex}(x) & \\hbox{\\rm exact solution vector of Step 2, with numerical IC $\\chi _{h}(x_d)$, eq.", "(\\ref {EQU}), (\\ref {numIC})} \\\\U_n & \\hbox{\\rm numerical solution vector of Step 2, with numerical IC $\\chi _{h}(x_d)$, eq.", "(\\ref {marching-scheme})-(\\ref {Transfo_ZU})} \\\\\\hline \\hbox{\\rm hybrid} & \\\\ \\hline \\phantom{I^{I^{I^I}}} \\psi _{ex}\\!\\!\\!", "{}^{(}{}^{\\prime }{}^{)}(x) & \\hbox{\\rm exact solution after scaling in Step 3, eq.", "(\\ref {SchBVP}), (\\ref {psi-scaling_ev})} \\\\\\psi _{h}\\!", "{}^{(}{}^{\\prime }{}^{)}(x)\\,;\\; \\psi _{h,n} & \\hbox{\\rm numerical solution after scaling in Step 3 (on $[0,x_d]\\, ;\\; [x_d,1]$)} \\\\\\tilde{U}_{ex}(x) \\mbox{ on } [x_d,1] & \\hbox{\\rm exact solution vector after scaling in Step 3, with exact IC} \\\\\\tilde{U}_n \\mbox{ on } [x_d,1] & \\hbox{\\rm numerical solution vector after scaling in Step 3, with numerical IC} \\\\\\hline \\end{array}$ For clarity, we summarize here also the numerical analogue of the three steps in (REF )-(REF ), referring to the two regions in Fig.", "REF : Step 1 – WKB-FEM for $\\chi _h$ in region (1): Find $\\chi _h\\in \\mathcal {V}_h$ solving $\\displaystyle b(\\chi _h,\\theta _h) = L(\\theta _h)\\,, \\quad \\forall \\theta _h\\in \\mathcal {V}_h\\,.$ This yields $\\chi _h(x_d)\\in \\mathbb {R}$ with an error $|e_h(x_d)| \\le C\\sqrt{h}\\min \\lbrace \\varepsilon ^{3/2},h^{3/2}\\rbrace $ (see Thm.", ").", "Step 2 – WKB-marching method for $\\varphi _h$ in region (2): As initial condition for the marching scheme we use $U_N:=\\hat{U}_{ex}(x_d)\\in \\mathbb {R}^2$ given by (REF ).", "Applying the scheme (REF )–(REF ) iteratively we compute the vectors $U_n=(u_n^1,\\,u_n^2)^\\top \\in \\mathbb {R}^2; \\,n=N+1, \\cdots ,M$ .", "Step 3 – Scaling of the auxiliary wave functions $\\chi _h,\\,\\varphi _h$ : $\\displaystyle \\qquad \\psi _h(x):=\\left\\lbrace \\begin{array}{l}\\displaystyle \\tilde{\\alpha }\\,\\chi _h(x)\\,, \\qquad \\qquad \\qquad \\qquad x \\in [0,x_d)\\,, \\\\[3mm]\\displaystyle \\tilde{\\alpha }\\,\\varphi _{h,n}=\\tilde{\\alpha }\\,u_n^1\\,a(x_n)^{-1/4}\\,, \\quad x\\in \\lbrace x_N,...,x_M\\rbrace \\,,\\end{array}\\right.$ $\\psi _h^{\\prime }(x):=\\left\\lbrace \\begin{array}{l}\\displaystyle \\tilde{\\alpha }\\,\\chi _h^{\\prime }(x)\\,, \\qquad \\qquad \\qquad \\qquad x \\in [0,x_d)\\,, \\\\[3mm]\\displaystyle \\tilde{\\alpha }\\,\\varphi _{h,n}^{\\prime }=\\tilde{\\alpha }\\Big [\\frac{u_n^2}{\\varepsilon }\\,a(x_n)^{1/4}-\\frac{a^{\\prime }(x_n)}{4}\\, a(x_n)^{-5/4}\\,u_n^1\\Big ]\\,, \\quad x\\in \\lbrace x_N,...,x_M\\rbrace \\,,\\end{array}\\right.$ with the scaling parameter $\\tilde{\\alpha }\\in \\mathbb {C}$ defined in analogy to (REF ): $\\displaystyle \\tilde{\\alpha }= \\tilde{\\alpha }(u_M^1,\\,u_M^2) := \\frac{-2{i}a(1)^{1/4}}{u_M^2-\\big [i+\\frac{\\varepsilon }{4} a(1)^{-3/2}a^{\\prime }(1)\\big ] u_M^1} \\,.$ The statement (REF ) reveals that our final numerical solution $\\psi _h$ is continuous in the evanescent region, but discrete in the oscillatory region.", "Note also that the relation between $U=(u_1,\\,u_2)^\\top $ and $(\\varphi ,\\,\\varepsilon \\varphi ^{\\prime })^\\top $ , given by (), provides a connection between the two scaling functions $\\alpha $ and $\\tilde{\\alpha }$ , i.e.", "$\\tilde{\\alpha }(u_1,\\,u_2) = \\alpha (\\varphi ,\\,\\varphi ^{\\prime })\\,.$ Let us finally also recall that the solution $\\chi _h$ as well as the vector $U_n\\ne 0$ of Step 2 are real valued.", "The final (numerical) solution $\\psi _h$ only becomes complex valued due to the multiplication by $\\tilde{\\alpha }$ in Step 3.", "Note that the denominator of (REF ) cannot vanish for $U\\in \\mathbb {R}^2\\setminus \\lbrace 0\\rbrace $ , which makes the scaling well defined.", "This map $\\tilde{\\alpha }$ satisfies moreover the following simple properties: For each fixed $\\delta >0$ , the ($\\varepsilon $ -dependent) map $\\tilde{\\alpha }:\\,U\\in \\mathbb {R}^2\\setminus B_{\\delta }(0) \\rightarrow \\mathbb {C}$ is Lipschitz continuous with some constant $L>0$ and bounded by some constant $C_8$ .", "Both constants can be chosen uniformly w.r.t.", "$0<\\varepsilon \\le \\varepsilon _0$ and are dependent on $\\delta $ .", "The following error analysis of the hybrid scheme is the main result of this paper.", "(Convergence WKB-hybrid) Let Hypotheses A and B' be satisfied.", "Then $\\psi _h$ , the numerical solution to the hybrid scheme (REF )-(REF ), satisfies the following error estimates, compared to the exact solution $\\psi _{ex}$ of the algorithm (REF )-(REF ): In the evanescent region $[0,x_d)$ we have $\\Vert \\tilde{e}_h\\Vert _{L^2(0,x_d)} &\\le & C \\sqrt{\\varepsilon }\\, h\\, \\min \\lbrace \\varepsilon ,h\\rbrace \\,, \\qquad \\quad \\varepsilon \\,\\Vert \\tilde{e}_h^{\\prime }\\Vert _{L^2(0,x_d)} \\le C \\varepsilon ^{3/2}\\, h\\,,\\nonumber \\\\\\Vert \\tilde{e}_h\\Vert _{C[0,x_d]} &\\le & C \\,\\sqrt{h}\\, \\min \\lbrace \\varepsilon ^{3/2},h^{3/2}\\rbrace \\,,\\quad \\varepsilon \\Vert \\tilde{e}_h^{\\prime }\\Vert _{L^{\\infty }(0,x_d)}\\le C \\varepsilon \\, \\sqrt{h}\\,\\min \\lbrace \\sqrt{\\varepsilon },\\sqrt{h}\\rbrace \\,,\\nonumber \\\\&&$ with the notation $\\tilde{e}_h(x):=\\psi _{ex}(x)-\\psi _h(x)$ .", "In the oscillatory region $[x_d,1]$ we have $\\displaystyle |\\tilde{e}_{h,n}| +\\varepsilon |\\tilde{e}_{h,n}^{\\prime }| \\le C \\sqrt{h}\\, \\min \\lbrace \\varepsilon ^{3/2},h^{3/2}\\rbrace \\,;\\qquad n=N,...,M\\,,$ with the notation $\\tilde{e}_{h,n}:=\\psi _{ex}(x_n)-\\psi _{h,n}$ and $\\tilde{e}_{h,n}^{\\prime }:=\\psi _{ex}^{\\prime }(x_n)-\\psi _{h,n}^{\\prime }$ .", "For the overall hybrid method one has then, over $[0,1]$ , the estimates $\\Vert \\tilde{e}_h\\Vert _\\infty \\le C \\,\\sqrt{h}\\, \\min \\lbrace \\varepsilon ^{3/2},h^{3/2}\\rbrace \\,,\\qquad \\varepsilon \\Vert \\tilde{e}_h^{\\prime }\\Vert _\\infty \\le C \\varepsilon \\, \\sqrt{h}\\,\\min \\lbrace \\sqrt{\\varepsilon },\\sqrt{h}\\rbrace \\,,$ with the notation $\\Vert \\tilde{e}_h\\Vert _\\infty := \\max \\lbrace \\Vert \\tilde{e}_h\\Vert _{L^{\\infty }(0,x_d)}\\,;\\; \\displaystyle \\max _{n=N,...,M} |\\tilde{e}_{h,n}| \\rbrace $ .", "Note that the error of $\\psi _h$ is globally of the order $\\mathcal {O}(h^2)$ .", "The derivative $\\psi _h^{\\prime }$ is also correct to order $\\mathcal {O}(h^2)$ in the oscillatory region, but only $\\mathcal {O}(h)$ in the evanescent region.", "This can be explained as follows: The $\\mathcal {O}(h)$ –error of $\\chi _h^{\\prime }$ is not propagated to the oscillatory region, as we use the exact value of the derivative in the matching condition (REF ).", "Moreover, the scaling step does not change the error orders in the evanescent region.", "Proof.", "[of Theorem REF ] Statement a) is a consequence of Thm.", ".", "Let us then continue by estimating the error of the numerical solution $U_n$ compared to the exact solution $U_{ex}(x)$ , i.e.", "prior to the scaling Step 3: Using (REF ), (REF ) and Theorem we obtain $\\Vert U_{ex}(x_n)-U_n\\Vert &\\le & \\Vert U_{ex}(x_n)-\\hat{U}_{ex}(x_n)\\Vert + \\Vert \\hat{U}_{ex}(x_n)-U_n\\Vert \\nonumber \\\\&\\le & C\\,|e_h(x_d)| + C\\varepsilon ^3h^2\\\\&\\le & C\\sqrt{h} \\min \\lbrace \\varepsilon ^{3/2},\\,h^{3/2}\\rbrace \\,;\\quad n=N,...,M\\,.\\nonumber $ We continue with estimating the error propagation due to the non-linear scaling in Step 3.", "Due to Lemma REF , the map $\\tilde{\\alpha }:\\,U\\in \\mathbb {R}^2\\setminus B_{\\min \\lbrace C_3,C_5\\rbrace }(0) \\rightarrow \\mathbb {C}$ is Lipschitz continuous with some constant $L>0$ and bounded by some constant $C_8$ .", "Both of these constants can be chosen independent of $0<\\varepsilon \\le \\varepsilon _0$ , as the choice $\\delta :=\\min \\lbrace C_3,C_5\\rbrace $ for the domain of $\\tilde{\\alpha }$ uses the lower bounds on $U_{ex}$ from (REF ) and on $U_n$ from Lemma REF .", "Here it is crucial that both the exact solution vector $U_{ex}(1)$ and the numerical solution vector $U_M$ have real components.", "Then, Lemma REF shows that $\\tilde{\\alpha }$ is Lipschitz and bounded on $\\mathbb {R}^2\\setminus B_\\delta (0)$ .", "Part a) For the evanescent region $[0,x_d)$ we estimate the difference between the exact solution and the numerical solution (both after scaling) $\\psi _{ex}(x) =\\tilde{\\alpha }(U_{ex}(1))\\,\\chi _{ex}(x)\\,, \\quad \\psi _{h}(x) = \\tilde{\\alpha }(U_M)\\,\\chi _{h}(x)\\,.$ This yields $|\\psi _{ex}(x) - \\psi _{h}(x)|&\\le & |\\tilde{\\alpha }(U_{ex}(1))-\\tilde{\\alpha }(U_M)|\\;|\\chi _{ex}(x)|+|\\tilde{\\alpha }(U_M)|\\;|\\chi _{ex}(x)-\\chi _h(x)| \\\\&\\le & L\\;\\Vert U_{ex}(1)- U_M\\Vert \\;|\\chi _{ex}(x)| + C_8\\; |e_h(x)|\\\\&\\le & C\\sqrt{h} \\min \\lbrace \\varepsilon ^{3/2},\\,h^{3/2}\\rbrace \\;|\\chi _{ex}(x)| + C_8\\; |e_h(x)|\\,,$ where we used (REF ) in the last step.", "Using now Lemma REF to estimate $\\chi _{ex}\\!\\!\\!", "{}^{(}{}^{\\prime }{}^{)}(x)$ and Theorem for $e_h\\!", "{}^{(}{}^{\\prime }{}^{)}(x)$ yields the four error estimates of (REF ).", "Part b) For the oscillatory region $[x_d,1]$ we estimate the difference between the exact solution and the numerical solution (both after scaling) $\\psi _{ex}(x_n) =\\tilde{\\alpha }(U_{ex}(1))\\,\\varphi _{ex}(x_n)\\,, \\quad \\psi _{h,n} = \\tilde{\\alpha }(U_M)\\,\\varphi _{h,n}\\,,$ at the grid points $x_n;\\,n=N,...,M$ .", "Here it is again more convenient to use the vector notation from (), where we introduce the notations $\\tilde{U}_{ex}(x) := \\tilde{\\alpha }U_{ex}(x)$ and $\\tilde{U}_{n} := \\tilde{\\alpha }U_{n}$ for the exact and, respectively, numerical solution after scaling in Step 3: $\\Vert \\tilde{U}_{ex}(x_n) - \\tilde{U}_{n}\\Vert &=& \\Vert \\tilde{\\alpha }(U_{ex}(1)) \\,U_{ex}(x_n) - \\tilde{\\alpha }(U_M)\\,U_{n}\\Vert \\\\&\\le & |\\tilde{\\alpha }(U_{ex}(1))-\\tilde{\\alpha }(U_M)|\\;\\Vert U_{ex}(x_n)\\Vert +|\\tilde{\\alpha }(U_M)|\\;\\Vert U_{ex}(x_n)- U_n\\Vert \\\\&\\le & L\\;\\Vert U_{ex}(1)- U_M\\Vert \\;C_4 + C_8\\; \\Vert U_{ex}(x_n)- U_n\\Vert \\\\&\\le & C\\sqrt{h} \\min \\lbrace \\varepsilon ^{3/2},\\,h^{3/2}\\rbrace \\,,$ where we used (REF ) in the penultimate line, and (REF ) twice in the last line.", "Using the norm equivalence then yields the error estimate (REF ).", "Part c) is just a combination of the previous two parts." ], [ "Numerical tests of the hybrid WKB method", "The aim of this section is to present numerical results obtained with the WKB-coupling scheme introduced in Section and to compare these results with the error analysis established in Section REF .", "In particular, we present the results for 3 zones (oscillating-evanescent-oscillating, cf.", "§REF ) corresponding to the passage or flow of electrons through a tunnelling structure (see Fig.", "REF ), with a piecewise linear and, respectively, piecewise quadratic potential $V(x)$ , chosen such that the phase $\\phi ^\\varepsilon (x)$ is explicitly calculable.", "The reason for such a choice is to avoid having to care about the ${h^\\gamma \\over \\varepsilon }$ –error term in (REF ), yielding hence an asymptotically correct scheme for fixed $h>0$ and $\\varepsilon \\rightarrow 0$ .", "Figure: Tunnelling structure with injection of a plane wave from the right boundary x=1x=1.Red curve: piecewise linear potential V(x)V(x) with applied bias (since V(1)>V(0)V(1)>V(0)).", "It is discontinuous at x c =0.5x_c=0.5 and x d =0.53125x_d=0.53125.Blue curve: ℜ(ψ(x))\\Re (\\psi (x)), real part of the wave function that is partly transmitted, but mainly reflected in this example; ε=0.01\\varepsilon =0.01.Green line: energy of the injected plane wave with E<maxV(x)E<\\max V(x).", "(colours only online)Example 1: We start with the piecewise linear potential graphed in Fig.", "REF .", "Note the small applied bias with $V(0)=0$ , $V(1)=0.2$ .", "In Fig.", "REF we plotted the numerical errors of the coupling method associated to the wave function $\\psi $ (left figure) and to its derivative $\\varepsilon \\psi ^{\\prime }$ (right figure), as functions of the mesh size $h$ (in $\\log -\\log $ scale) and for three different $\\varepsilon $ -values.", "In the oscillating regions we chose the second order method (REF )-(REF ) and in the evanescent region the FEM (REF ).", "The plotted errors are the $L^\\infty $ -errors between the numerical solution on the whole interval $[0,1]$ and a reference solution, computed with the same scheme but on a finer grid of $2^{18}$ points.", "It can be observed that the slopes in these two plots are approximatively one (for $h\\gtrsim 3\\cdot 10^{-5}$ ) and improving to $1.5$ for smaller values of $h$ .", "For $\\varepsilon =0.1$ the slope of the $\\psi $ –error even improves up to 2 for the smallest values of $h$ .", "This behaviour is in accordance with our numerical analysis in Theorem REF (c!", ").", "The $\\varepsilon $ -dependence seems to be like $\\mathcal {O} (\\sqrt{\\varepsilon })$ (for large values of $\\varepsilon $ ), improving to $\\mathcal {O} (\\varepsilon )$ (for small values of $\\varepsilon $ ), and even $\\mathcal {O} (\\varepsilon ^{3/2})$ (for small values of $\\varepsilon $ and large $h$ ).", "Summarizing, the error of $\\psi $ shows to be of order $\\mathcal {O} (\\min \\lbrace h^2,\\,\\sqrt{\\varepsilon }h^{3/2},\\,\\varepsilon h,\\,\\varepsilon ^{3/2}\\sqrt{h} \\rbrace )$ , which corresponds exactly to the estimates given in Theorem REF (c).", "The error of $\\varepsilon \\psi ^{\\prime }$ shows to be of order $\\mathcal {O} (\\min \\lbrace \\sqrt{\\varepsilon }h^{3/2},\\,\\varepsilon h,\\,\\varepsilon ^{3/2}\\sqrt{h} \\rbrace )$ , which is even slightly better than the estimate from Theorem REF (c) (in the sense of including also an $\\mathcal {O} (\\sqrt{\\varepsilon }h^{3/2})$ –behaviour).", "Figure: Absolute error (in the L ∞ L^\\infty -norm and log-log\\log -\\log scale) between the computed solution and a reference solution (obtained with h=2 -18 ≈4·10 -6 h=2^{-18}\\approx 4\\cdot 10^{-6}), for the piecewise linear potential from Fig.", ".", "Left: ||ψ ref -ψ num || ∞ ||\\psi _{ref}-\\psi _{num}||_\\infty .", "Right: ||εψ ref ' -εψ num ' || ∞ ||\\varepsilon \\psi ^{\\prime }_{ref}-\\varepsilon \\psi ^{\\prime }_{num}||_\\infty .We mention that the obtained numerical errors are mainly those introduced by the WKB-FEM of the evanescent region.", "Indeed, in this evanescent region, the numerical error of the WKB-FEM is larger than the one obtained from the second order WKB marching method of the oscillating region (compare the estimates in the Theorems and REF ).", "Example 2: Next we consider a piecewise quadratic potential given by $a(x):=c_1 (x + c_2)^2\\,, \\quad \\forall x \\in [0,x_c] \\cup [x_d,1]\\,; \\quad a(x):=-c_1 (x + c_2)^2\\,,\\quad \\forall x \\in (x_c,x_d)\\,,$ with $x_c=0.5$ , $x_d=0.5+2^{-5}=0.53125$ and $E=1.5\\, , \\quad V_1=V(1)=0.2\\, , \\quad c_2= -{ E + \\sqrt{E^2 - V_1\\, E} \\over V_1}\\,, \\quad c_1= { E \\over c_2^2}\\,.$ Before turning to the error plots we consider the condition number associated to solving the discrete variational problem (REF ) in the (intermediate) evanescent region.", "In Fig.", "REF we plot this condition number as a function of $h$ , for three different values of $\\varepsilon $ .", "For $\\varepsilon =10^{-1},\\,10^{-2}$ it grows like $\\mathcal {O}(h^{-2})$ when $h\\rightarrow 0$ , and for $\\varepsilon =10^{-3}$ it grows like $\\mathcal {O}(h^{-1})$ (on the shown interval of $h$ -values).", "We remark that this behaviour is not a problem in practice: For large $\\varepsilon $ , the solution $\\psi _{ex}$ is not highly oscillatory and hence does not need a high spatial resolution.", "For small values of $\\varepsilon $ , even a fine resolution would only lead to moderate condition numbers.", "Indeed, one observes a decrease of the condition number when $\\varepsilon $ gets smaller.", "This important feature is somehow related to the asymptotic-preserving property of the scheme.", "Figure: Condition number for the discrete BVP in Example 2, as a function of hh, for three values of ε\\varepsilon .Large condition numbers signify that the errors of numerical experiments also include significant contributions stemming from round-of errors and their accumulation.", "While the method-error (as estimated in Theorem REF (c)) decreases with decreasing $h$ , the round-of errors could then increase in some situations, due to the increasing condition number.", "These arguments may lead to the idea that one cannot trust too much the reference solution in Fig.", "REF , computed with $2^{18}$ points.", "In order to verify this suspicion, we decided to plot in the case of a piecewise quadratic potential two types of error curves.", "To be more precise, in Fig.", "REF we show the numerical errors of the wave function $\\psi $ (left figure) and its derivative $\\varepsilon \\psi ^{\\prime }$ (right figure), as functions of the mesh size $h$ (in $\\log -\\log $ scale) and for four different $\\varepsilon $ -values.", "The dashed lines correspond (as in Example 1) to the $L^\\infty $ -error between the numerical solution on the whole interval $[0,1]$ and a reference solution, computed with the same scheme but on a finer grid (here $h=2^{-19}$ ) whereas the solid lines correspond to the incremental error when iteratively doubling the grid size, i.e.", "$||\\psi _{h_j}-\\psi _{h_{j-1}}||_\\infty $ with $h_j=2h_{j-1}$ .", "For a first order method, the former error is about twice as large as the latter (incremental) error.", "This can be understood from the geometric series of the incremental errors, since the summands then differ by a factor of about 2.", "In Fig.", "REF this difference is clearly visible for the solid red curves, pertainin!", "g to $\\varepsilon =10^{-2}$ , and the corresponding dashed error curves (for large $h$ ).", "The minimum of the incremental error (as a function of $h$ ) indicates the onset of significant round-of errors when reducing $h$ .", "In Fig.", "REF this is best visible for the solid blue and red curves, pertaining to $\\varepsilon =10^{-1},\\,10^{-2}$ .", "Furthermore remark that for $\\varepsilon =10^{-1},\\,10^{-2},\\,10^{-3}$ and $h\\gtrsim 3\\cdot 10^{-5}$ , the error slopes are approximately one – just like in Example 1.", "For smaller values of $h$ the error then gets polluted by round-of errors.", "For $\\varepsilon =10^{-4}$ the shown errors seem to be mostly due to round-of errors.", "They again increase for $h\\lesssim 3\\cdot 10^{-5}$ .", "Figure: Absolute error (in the L ∞ L^\\infty -norm and log-log\\log -\\log scale) between the computed solution and a reference solution, for a piecewise quadratic potential.", "Left: ||ψ ref -ψ num || ∞ ||\\psi _{ref}-\\psi _{num}||_\\infty .", "Right: ||εψ ref ' -εψ num ' || ∞ ||\\varepsilon \\psi ^{\\prime }_{ref}-\\varepsilon \\psi ^{\\prime }_{num}||_\\infty .", "For the dashed curves, ψ ref \\psi _{ref} is computed with h=2 -19 h=2^{-19}.", "The solid curves show the incremental error when refining the mesh by a factor two.", "Left: ||ψ h j -ψ h j-1 || ∞ ||\\psi _{h_j}-\\psi _{h_{j-1}}||_\\infty .", "Right: ||εψ h j ' -εψ h j-1 ' || ∞ ||\\varepsilon \\psi ^{\\prime }_{h_j}-\\varepsilon \\psi ^{\\prime }_{h_{j-1}}||_\\infty ." ], [ "Turning points", "A turning point of the Schrödinger equation (REF ) is defined as a zero of the given coefficient function $a(x)$ .", "Accordingly one also speaks about the order of a turning point.", "We first remark that both error analyses, in §REF for the WKB-FEM and in [2] for the WKB-marching method are not valid for turning points.", "Therefore, we assumed in Hypothesis A and B that $a(x)$ is bounded away from zero.", "Furthermore, in the convergence Theorems and REF we did not keep track how the leading constant $C$ grows with $\\tau _{ev},\\,\\tau _{os}\\rightarrow 0$ .", "However, the failure of both WKB methods when approaching turning points also appeared in our numerical experiments (not included here).", "The paper [14] considers a matrix generalization of our equation (REF ), but only for the oscillatory case.", "Their coefficient matrix $A(x)$ (generalizing our $a(x)$ ) is there assumed to be symmetric positive definite, satisfying the uniform lower bound $A(x) \\ge \\delta ^2>0$ .", "The proof of their Theorem 6.1 shows that their $L^\\infty $ –error bounds would grow like $\\mathcal {O}(\\delta ^{-2}$ ).", "This numerical failure can be understood easily: The WKB-ansatz (REF )-(REF ) is not valid at turning points.", "In fact, close to a turning point of first order, solutions to (REF ) are neither exponential nor oscillatory, but in a transition layer of thickness $\\mathcal {O}(\\varepsilon ^{2/3})$ they behave rather like Airy functions: Indeed, for $a(x):=-x$ , a solution basis for (REF ) is given by Ai$(\\varepsilon ^{-2/3}x)$ , Bi$(\\varepsilon ^{-2/3}x)$ , where Ai and Bi denote the Airy functions of first and second kinds.", "At a turning point the solution to (REF ) clearly satisfies $\\psi ^{\\prime \\prime }=0$ .", "This motivated to use linear FEM-ansatz functions in the numerical cell containing a turning point (cf.", "§3.2.2 in [16]).", "The quest for an appropriate replacement of the WKB-ansatz close to turning points has a long history in asymptotic analysis, starting with Langer [12]: For general coefficients $a(x)$ with a zero, he found an asymptotic approximation for the solution of (REF ) that is valid uniformly in $x$ , including the turning point.", "For a first order turning point his approximation is a composite function involving Airy functions and the phase function (like $\\sigma (x)$ defined in (REF )).", "For details on first order turning points we refer to §4.3 in [7], and to §7.3 in [15] for higher order turning points.", "The above mentioned approximation formulas of Langer have also been used for numerical computations, mostly for Schrödinger eigenvalue problems [5], [6], [19].", "In the physics literature, this strategy is frequently called Modified Airy function (MAF) technique.", "It relies on evaluating the explicit formulas of approximate solutions, but it has not been the starting point of constructing a (convergent) numerical method.", "In a follow-up paper we shall use Langer's approximation functions as ansatz-functions for an $\\varepsilon $ –uniform numerical method that should also cover turning points." ], [ "Conclusion", "This paper is concerned with a 1D Schrödinger scattering problem in the semi-classical limit, with the inflow given by plane waves.", "The injection energy and potential are given such that the problem involves both oscillatory and evanescent regions.", "For the continuous boundary value problem we presented a new, non-overlapping domain decomposition method that separates the oscillatory and evanescent subproblems.", "The former are treated as IVPs, and the latter as BVPs.", "Key issues of this approach are the appropriate interface conditions and the final scaling of the solution function.", "We proved that the domain decomposition method yields the exact solution in a single sweep, performed in the opposite direction of the wave injection.", "The hybrid numerical discretization is based on WKB-methods in both types of regions: a WKB-FEM for evanescent regions [17], and a WKB-marching method for oscillatory regions [2].", "The objective of these WKB-methods is to provide an accurate solution – even on coarse grids and independently of $\\epsilon $ .", "Hence, they are asymptotic preserving.", "For the first time we present an error analysis for the WKB-FEM method.", "Together with the analysis of the WKB-marching method from [2], this constitutes the key ingredient for our complete convergence analysis of the hybrid WKB-method.", "Finally, these error bounds are illustrated and verified in numerical experiments." ], [ "Appendix: proof of Proposition ", "Proof.", "Step 1: The BVP (REF ) is equivalent to the Schrödinger equation (REF ) on the real line with constant potentials in the leads, and with an incoming plane wave at $x=1$ .", "Using the right BC from (REF ), its solution in the right lead $x\\ge 1$ hence reads $\\psi (x)=r\\,e^{i\\frac{\\sqrt{a(1)}}{\\varepsilon }(x-1)} + e^{-i\\frac{\\sqrt{a(1)}}{\\varepsilon }(x-1)}\\,.$ To estimate the reflection coefficient $r$ , we consider the current defined in (REF ).", "At $x=1$ it reads $j(1)=\\varepsilon \\,\\Im [{\\overline{\\psi }}(1)\\,\\psi ^{\\prime }(1)]=\\varepsilon \\sqrt{\\alpha (1)} \\,(|r|^2-1) \\,.$ But using the left BC from (REF ) yields $j(0)=\\varepsilon \\sqrt{\\alpha (0)} \\,\\Im [{\\overline{\\psi }}(0)\\,\\psi (0)] = 0\\,.$ Since the current in a stationary quantum model is constant in $x$ , this implies $|r|=1$ .", "Then (REF ) implies $|\\psi (1)|\\le 2\\,,\\qquad \\varepsilon \\Vert \\psi ^{\\prime }(1)\\Vert \\le 2\\sqrt{a(1)}\\,.$ With this bound for the initial condition at $x=1$ we now consider the IVP (REF ) on $[x_d,1]$ .", "Then, Theorem 2.2 from [17] yields the asserted estimate (REF ) on the oscillatory region.", "Step 2: For the evanescent region $[0,x_d]$ we consider the scaled, real valued solution $\\chi $ of the BVP (REF ).", "With an elementary argument we first show that $\\chi $ has no zero in $[0,x_d]$ : Assuming the opposite, let $\\chi (x_0)=0$ , which implies $\\chi ^{\\prime }(x_0)\\ne 0$ (as otherwise $\\chi \\equiv 0$ ).", "Then $\\chi $ is convex on one side of $x_0$ and concave on the other side, with $\\mbox{sgn} (\\chi ^{\\prime \\prime })=\\mbox{sgn} (\\chi )$ due to $a\\big |_{[0,x_d)}<0$ .", "But then $\\chi (0)$ and $\\chi ^{\\prime }(0)$ would have opposite signs, contradicting the left BC in (REF ).", "So we conclude that $\\chi $ does not change signs in $[0,x_d]$ .", "Assume now that $\\chi (0)<0$ which implies $\\chi ^{\\prime }(0)<0$ by the left BC in (REF ).", "Since then $\\chi ^{\\prime \\prime }<0$ on $[0,x_d)$ we conclude $\\chi ^{\\prime }(x_d)<0$ , contradicting the right BC in (REF ).", "This implies that $\\chi (0)>0$ , and we finally obtain $\\chi >0\\,,\\quad \\chi ^{\\prime }>0\\,,\\quad \\chi ^{\\prime \\prime }>0\\,,\\quad \\mbox{ on }(0,x_d)\\,.$ After scaling this auxiliary function, we find that also $|\\psi |$ and $|\\varepsilon \\psi ^{\\prime }|$ are increasing on $[0,x_d]$ .", "Therefore the uniform bound (REF ) carries over from $x=x_d$ to the evanescent region $[0,x_d]$ ." ], [ "Acknowledgements", "   The first author was supported by the FWF-doctoral school “Dissipation and dispersion in non-linear partial differential equations”, the ÖAD-Amadeus project “Quantum transport models for semiconductors and Bose-Einstein condensates”, Université de Toulouse, and by Clear Sky Ventures.", "He also thanks Kirian Döpfner for illustrating simulations for §5.", "The second author would like to acknowledge support from the ANR project MOONRISE (MOdels, Oscillations and NumeRIcal SchEmes, 2015-2019)." ] ]
1606.05207
[ [ "Ricci iteration on homogeneous spaces" ], [ "Abstract The Ricci iteration is a discrete analogue of the Ricci flow.", "We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not K\\\"ahler.", "The Ricci iteration in the non-K\\\"ahler setting exhibits new phenomena.", "Among them is the existence of so-called ancient Ricci iterations.", "As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times.", "In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov--Hausdorff topology) from a collapsed Einstein metric and converge smoothly to a second Einstein metric.", "In the case of compact homogeneous spaces with maximal isotropy, we prove a relative compactness result that excludes collapsing." ], [ "Introduction", "Let $(M,g_1)$ be a smooth Riemannian manifold.", "A Ricci iteration is a sequence of metrics $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ on $M$ satisfying $\\hbox{\\rm Ric}\\,g_{i+1}=g_i,\\quad i\\in \\mathbb {N},$ where $\\hbox{\\rm Ric}\\,g_{i+1}$ denotes the Ricci curvature of $g_{i+1}$ .", "One may think of (REF ) as a dynamical system on the space of Riemannian metrics on $M$ .", "We restrict our attention to the case of positive Ricci curvature in the present article; different Ricci iterations can be defined in the context of non-positive curvature.", "Part of the interest in the Ricci iteration is that, clearly, Einstein metrics are fixed points, and so (REF ) aims to provide a natural new approach to uniformization.", "In essence, the Ricci iteration aims to reduce the Einstein equation to a sequence of prescribed Ricci curvature equations.", "Introduced by the second-named author [31], [32] as a discretization of the Ricci flow, the Ricci iteration has been since studied by a number of authors, see the survey [34] and references therein.", "In all previous works, the underlying manifold $(M,g_1)$ is assumed to be Kähler; essentially nothing is known about the Ricci iteration in the general Riemannian setting.", "Given the central rôle of uniformization and of the Ricci flow in geometry, it seems of interest to investigate whether the Ricci iteration could be understood for general Riemannian manifolds which may not be Kähler.", "In this article we take a first step in this direction.", "Namely, we show that, indeed, the Ricci iteration can be defined on some non-Kähler manifolds, and that under some natural assumptions it converges to an Einstein metric with positive Ricci curvature.", "We also prove a relative compactness result.", "The manifolds we investigate here are compact homogeneous spaces.", "The study of Einstein metrics and the Ricci flow on such spaces is an active field, see, e.g., [5], [8], [7], [25], [3], [6], [1].", "As we find in this article, in the non-Kähler setting the Ricci iteration exhibits new types of behavior as compared to the Kähler setting.", "For instance, solutions to (REF ) may not exist for all $i\\in {\\mathbb {N}}$ .", "In addition, existence (and hence convergence) may fail even when $M$ admits a homogeneous Einstein metric.", "Finally, and perhaps surprisingly, we construct the first non-trivial ancient Ricci iterations.", "These are sequences of Riemannian metrics $\\lbrace g_1,g_0,g_{-1},g_{-2},\\ldots \\rbrace $ such that $\\begin{aligned}g_{i-1}=\\hbox{\\rm Ric}\\,g_{i}, \\qquad i=1,0,-1,-2,\\ldots .\\end{aligned}$ Thus, on the Riemannian manifold $(M,g_1)$ , the “time-reversed\" Ricci iteration $g_1,\\; \\hbox{\\rm Ric}\\,g_1,\\; \\hbox{\\rm Ric}\\,\\hbox{\\rm Ric}\\,g_1,\\; \\ldots $ exists.", "As far as we are aware, these are the first nontrivial examples in the literature of Riemannian metrics to which the Ricci operator can be applied infinitely many times, answering questions posed in [32].", "We find such metrics in dimensions as low as 6.", "To put ancient Ricci iterations in perspective, as explained in [32], the Ricci iteration (REF ) can be thought of as the backward Euler discretization of the Ricci flow, and hence one may expect good analytical properties for it.", "On the other hand, in doing the reverse procedure, i.e., iterating the Ricci operator, one typically loses two derivatives in each step.", "What is more, the Ricci operator can only be iterated as long as positivity is preserved.", "Therefore, at least from the analytical point of view, it seems unlikely that such ancient iterations should exist.", "We show they are closely tied to the geometry, exist in tandem with ancient solutions of the Ricci flow constructed by Buzano [9], and exhibit a relative compactness property similar to that of the Ricci iteration.", "In some cases, they emerge from one Einstein metric as $i\\rightarrow -\\infty $ and converge to a second Einstein metric as $i\\rightarrow \\infty $ .", "In other cases, they emerge from a collapsed Einstein limit $G/K$ where $H\\subsetneq K \\subsetneq G$ as $i\\rightarrow -\\infty $ and converge to an Einstein metric as $i\\rightarrow \\infty $ .", "We find examples of the latter phenomenon starting in dimension 7.", "One of the reasons the Ricci iteration is relatively well-understood in the Kähler setting is that the prescribed Ricci curvature equation reduces in that setting to a complex Monge–Ampère equation.", "The existence and uniqueness are then given by the Calabi–Yau theorem [10], [37].", "More precisely, there always exist unique Kähler metrics solving (REF ) for $i\\in {\\mathbb {N}}$ , provided that $g_1$ , and hence each $g_i$ , represents ($2\\pi $ times) the first Chern class.", "The first result on the Ricci iteration established smooth convergence to a Kähler–Einstein metric under some symmetry assumptions [32] that turn out to hold in a number of interesting cases [12], [13], [14], [11].", "A conjecture stipulates that the Ricci iteration should in fact always converge in an appropriate sense to a Kähler–Einstein metric when one exists [32].", "Recent progress in this direction is due to [4], [15].", "Clearly, the first obstacle in the general Riemannian setting is to understand the prescribed Ricci curvature equation.", "While this equation has been the subject of active research (e.g., [5], [23], [18], [19], [17], [16], [29], [28], [35])—when compared to the Kähler case—it is understood rather poorly on general Riemannian manifolds.", "Recent results of the first-named author [30] provide a replacement for the Calabi–Yau theorem on certain classes of homogeneous spaces and are crucial in the present article.", "The second obstacle is to show convergence.", "In order to do so, one may seek and analyze various monotonic quantities associated with the iteration.", "One key ingredient in the proofs of our Theorems REF and REF is the monotonicity of the quantity $\\alpha _i$ given by (REF ).", "Analogous monotonicity was exploited by Buzano in the analysis of the Ricci flow [9].", "While the iteration turns out to exhibit similar limiting behavior to the flow in many situations, there are cases where they differ; see, e.g., Remark REF .", "The third, and perhaps new, obstacle in the general Riemannian setting is that—as we find in this article—the Ricci iteration can exhibit a range of behaviors depending on the homogeneous space and the starting point, quite in contrast to the Kähler setting: there the Ricci iteration always exists and either converges or not depending only on whether a Kähler–Einstein metric exists or not (regardless of the initial condition).", "We will see that neither of these behaviors persists in the setting of homogeneous spaces.", "This is, perhaps, indicative of the relation between the Ricci iteration and the Ricci flow.", "Indeed, the behavior of the Ricci flow on homogeneous spaces is rather intricate as compared to the Kähler–Ricci flow on Fano manifolds (that always converges to a Kähler–Einstein metric when such a metric exists).", "Finally, we remark that the Ricci iteration seems harder to understand than the Ricci flow.", "This is certainly the case in the Kähler setting; perhaps, one reason is the wide variety of tools available to analyze parabolic flows.", "Currently, a complete theory of the Ricci iteration for all compact homogeneous manifolds seems out of reach, as even Einstein metrics and the Ricci flow are not completely understood in this setting.", "The study we carry out in the present article—while restricted to a certain class of highly symmetric manifolds in which the Ricci iteration reduces to a sequence of systems of algebraic equations—at least gives a setting in which the Ricci iteration can be fully understood and compared to the Ricci flow in detail.", "Among other things, it suggests what kind of phenomena to expect in general." ], [ "Main results", "Let $H $ be a closed connected subgroup of a compact connected Lie group $G$ .", "In this article, we focus on the compact homogeneous space $M:=G/H$ and $G$ -invariant Riemannian metrics on it." ], [ "Preliminaries", "The group $G$ acts on $M$ by associating to $\\lambda \\in G$ the diffeomorphism $L_\\lambda :M\\rightarrow M$ defined by $\\nu H\\mapsto \\lambda \\nu H$ for $\\nu \\in G$ .", "The action is called effective if the equality $L_\\lambda =\\hbox{id}$ implies that $\\lambda $ is the identity of $G$ .", "Henceforth, $G$ will be assumed to act effectively.", "This entails no loss of generality [5].", "Suppose $\\begin{aligned}n:=\\dim M\\ge 3.\\end{aligned}$ Let $\\mathfrak {g}$ denote the Lie algebra of $G$ , and let $\\hbox{\\rm Ad}_G$ be the adjoint representation of $G$ on $\\mathfrak {g}$ .", "We fix a bi-invariant Riemannian metric on $G$ .", "It induces an $\\hbox{\\rm Ad}_G(G)$ -invariant inner product $Q$ on $\\mathfrak {g}$ and a $G$ -invariant Riemannian metric $\\hat{g}$ on $M$ .", "Choose a point $\\mu \\in M$ .", "Then $H$ can be identified with $G\\cap \\hbox{Iso}_\\mu (M,\\hat{g})$ , where $\\hbox{Iso}_\\mu (M,\\hat{g})$ is the isotropy group of $(M,\\hat{g})$ at $\\mu $ .", "Let $\\mathfrak {h}$ denote the Lie algebra of $H$ .", "We know $H$ is compact (as $G$ is), and so is $\\hbox{\\rm Ad}_G(H)$ .", "The $Q$ -orthogonal complement of $\\mathfrak {h}$ in $\\mathfrak {g}$ is an $\\hbox{\\rm Ad}_G(H)$ -invariant subspace of $\\mathfrak {g}$ .", "We denote this subspace by $\\mathfrak {m}$ .", "Thus, $\\begin{aligned}\\mathfrak {g}=\\mathfrak {m}\\oplus \\mathfrak {h}.\\end{aligned}$ We naturally identify $\\mathfrak {m}\\cong \\mathfrak {g}/\\mathfrak {h} \\cong T_\\mu M.$ Every $G$ -invariant Riemannian metric on $M$ induces an $\\hbox{\\rm Ad}_G(H)$ -invariant inner product on $\\mathfrak {m}$ .", "The converse also holds [5], so $\\mathcal {M}&:=\\lbrace \\hbox{$G$-invariant Riemannian metrics on $M$}\\rbrace \\cr &\\cong \\lbrace \\hbox{$\\hbox{Ad}_G(H)$-invariant inner products on $\\mathfrak {m}$}\\rbrace .$ Unlike in the case of Kähler metrics on a Kähler manifold, the space of $G$ -invariant metrics on $M$ is finite-dimensional.", "Nevertheless, we will see that the Ricci iteration exhibits more complicated phenomena than in the Kähler setting, where the space of Kähler metrics is infinite-dimensional.", "Consider a $Q$ -orthogonal $\\hbox{\\rm Ad}_G(H)$ -invariant decomposition $\\mathfrak {m}=\\mathfrak {m}_1\\oplus \\cdots \\oplus \\mathfrak {m}_s$ such that $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_i}$ is irreducible for each $i=1,\\ldots ,s$ .", "Let $\\begin{aligned}d_i:=\\dim {\\mathfrak {m}_i}\\in {\\mathbb {N}}.\\end{aligned}$ While the space $\\mathfrak {m}$ may admit more than one decomposition of the form (REF ), the number of summands, $s$ , is determined by $G$ and $H$ , i.e., it is the same for all such decompositions.", "On the other hand, the summands $\\mathfrak {m}_1,\\ldots ,\\mathfrak {m}_s$ are determined uniquely up to order if $\\begin{aligned}\\hbox{$\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_i}$ is inequivalent to $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_k}$ whenever $i\\ne k$.", "}\\end{aligned}$ Indeed, suppose $\\mathfrak {m}_1^{\\prime }\\oplus \\cdots \\oplus \\mathfrak {m}_{s^{\\prime }}^{\\prime }$ is another decomposition.", "The $Q$ -orthogonal projection from $\\mathfrak {m}_i$ to $\\mathfrak {m}_j^{\\prime }$ is an equivariant map.", "Therefore, it is either an isomorphism or zero by Schur's lemma.", "We will assume (REF ) holds in order to state our first two results.", "For our other results, however, we will not assume it.", "Given $T\\in \\mathcal {M}$ , it is always possible to choose the decomposition (REF ) so that $T$ has a simple “diagonal\" form.", "More precisely, let $\\pi _i:\\mathfrak {m}\\rightarrow \\mathfrak {m}_i$ denote the natural projections induced by (REF ).", "Recall that $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_i}$ is irreducible for each $i$ .", "There exists a choice of (REF ) such that $T$ has the form $T=\\sum _{i=1}^s z_i\\pi _i^* Q,\\quad z_i>0,$ see [36].", "Equality (REF ) can also be written as $T(X,Y)=\\sum _{i=1}^s z_iQ(\\pi _i(X),\\pi _i(Y)),\\qquad X,Y\\in \\mathfrak {m}.$ As (REF ) shows, we can identify $\\mathcal {M}$ with a subset of $\\mathfrak {m}^*\\otimes \\mathfrak {m}^*$ .", "With this identification, the topology of $\\mathfrak {m}^*\\otimes \\mathfrak {m}^*$ induces a topology on $\\mathcal {M}$ .", "Convergence in $\\mathcal {M}$ is equivalent to the smooth convergence of Riemannian metrics on $M$ .", "In what follows, we say $H$ is maximal in $G$ if $H$ is a maximal connected Lie subgroup of $G$ or, equivalently, if $\\mathfrak {h}$ is a maximal Lie subalgebra of $\\mathfrak {g}$ .", "When $s=1$ , the space $\\cal M$ is one-dimensional, and so we will always assume $s>1$ .", "(We remark that our earlier assumption (REF ) is reasonable since the only compact two-dimensional manifolds admitting positively-curved metrics, $S^2=SO(3)/SO(2)$ and $\\mathbb {R}P^2=O(3)/(O(2)\\times O(1))$ , satisfy $s=1$ .)", "Theorems REF and REF below concern the case when $s=2$ and (REF ) holds.", "To formulate them, denote $\\begin{aligned}\\mathcal {E}:=\\lbrace \\hbox{Einstein metrics in $\\mathcal {M}$}\\rbrace .\\end{aligned}$ If $\\mathcal {E}$ is nonempty, define $\\alpha _-&:=\\inf \\big \\lbrace {z_1}/{z_2}\\,:\\,z_1,z_2~\\mbox{satisfy~(\\ref {T_def}) for some}~T\\in \\mathcal {E}\\big \\rbrace ,\\\\\\alpha _+&:=\\sup \\big \\lbrace {z_1}/{z_2}\\,:\\,z_1,z_2~\\mbox{satisfy~(\\ref {T_def}) for some}~T\\in \\mathcal {E}\\big \\rbrace .$ Lemma REF below implies that, when $s=2$ and (REF ) holds, there are only finitely many Einstein metrics in $\\mathcal {M}$ up to scaling; therefore, $\\alpha _-$ and $\\alpha _+$ lie in $(0,\\infty )$ .", "Theorems REF and REF distinguish between the maximal and non-maximal cases.", "When $H$ is maximal in $G$ , at least one Einstein metric exists in $\\mathcal {M}$  [36].", "When $H$ is not maximal, there is a connected Lie subgroup $K$ of $G$ such that $\\begin{aligned}H\\subsetneq K \\subsetneq G,\\end{aligned}$ and we write $\\mathfrak {k}$ for the Lie algebra of $K$ .", "If $s=2$ and (REF ) is satisfied, we assume without loss of generality that $\\mathfrak {k}=\\mathfrak {h}\\oplus \\mathfrak {m}_1.$ Theorems REF and REF will require the hypothesis $\\begin{aligned}Q([X,Y],Z)\\ne 0 \\hbox{\\ for some $X\\in \\mathfrak {m}_1$ and $Y,Z\\in \\mathfrak {m}_2$}.\\end{aligned}$ This hypothesis automatically holds when $H$ is maximal in $G$ (see Lemma REF ).", "If an intermediate subgroup $K$ satisfying (REF ) and (REF ) exists, formula (REF ) may fail to hold.", "However, when (REF ) is false, all the $G$ -invariant metrics on $M$ have the same Ricci curvature (see (REF )), and the study of the Ricci iteration is of no interest." ], [ "Two irreducible isotropy summands", "Our first result completely describes the Ricci iteration on compact homogeneous spaces with two inequivalent isotropy summands.", "We refer to [21], [24] for a classification of such spaces; see also Examples REF and REF below.", "The Ricci flow on them was studied in [2], [22], [9].", "Theorem 2.1 (Ricci iteration for $s=2$ ) Let $M=G/H$ .", "Assume that $s = 2$ in (REF ) and that (REF ) holds.", "Let $T\\in \\mathcal {M}$ (recall (REF )) be an arbitrary $G$ -invariant metric given by (REF ).", "Suppose $H$ is maximal in $G$ .", "There exists a unique sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {M}$ such that (REF ) holds for all $i\\in \\mathbb {N}$ and $g_1=cT$ for some $c>0$ .", "The metrics $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ converge smoothly to an Einstein metric on $M$ as $i$ tends to $\\infty $ .", "Suppose $G$ has a connected Lie subgroup $K$ satisfying (REF ).", "Let (REF ) and (REF ) hold.", "Assume $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial.", "Then $\\mathcal {E}\\ne \\emptyset $ .", "There exists a unique sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {M}$ such that (REF ) holds for all $i\\in \\mathbb {N}$ and $g_1=cT$ for some $c>0$ .", "This sequence converges smoothly to the unique $G$ -invariant Einstein metric on $M$ with Einstein constant 1.", "Assume $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is nontrivial and $\\mathcal {E}\\ne \\emptyset $ .", "A sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {M}$ satisfying (REF ) for all $i\\in \\mathbb {N}$ and $g_1=cT$ for some $c>0$ exists if and only if the inequality $z_1/z_2\\ge \\alpha _-$ holds.", "When it exists, the sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ is unique and converges smoothly to an Einstein metric on $M$ .", "If $\\mathcal {E}=\\emptyset $ , there is no sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {M}$ such that (REF ) holds for all $i\\in \\mathbb {N}$ .", "Remark 2.2 The Ricci iteration and the Ricci flow exhibit essentially the same limiting behavior except in two cases: in the case (ii-b) with initial condition satisfying $z_1/z_2<\\alpha _-$ and in the case (ii-c), the flow converges to a collapsed Einstein metric on $G/K$ [9], while the iteration stops after finitely many steps.", "One motivation for introducing the Ricci iteration in [32] was a question of DeTurck and Nadel [33], [26]: does the Ricci operator possess nontrivial (i.e., non-Einstein) fixed points (i.e., solutions to $\\hbox{\\rm Ric}\\,\\hbox{\\rm Ric}\\,\\cdots \\hbox{\\rm Ric}\\,g=g$ )?", "The proof of Theorem REF answers this question in the negative in the setting of the theorem since we exhibit a strictly monotone quantity along the iteration.", "Another natural question, posed in [32], is, roughly, whether there exist discrete analogs of ancient Ricci flows.", "More precisely, given a Riemannian metric $g$ , one may associate to it a Riemannian invariant $r(g):=\\sup \\big \\lbrace k\\in {\\mathbb {N}}\\,:\\,\\underbrace{\\hbox{\\rm Ric}\\,\\hbox{\\rm Ric}\\,\\ldots \\hbox{\\rm Ric}\\,}_{k-1~\\rm {times}} g~\\mbox{is a Riemannian metric}\\big \\rbrace $ called the Ricci index of $g$ [32].", "Following [32] one can then define a filtration of $\\mathcal {M}$ $\\mathcal {M}=\\mathcal {M}^{(1)}\\supset \\mathcal {M}^{(2)}\\supset \\cdots \\supset \\mathcal {M}^{(l)}\\supset \\cdots ,$ by setting $\\mathcal {M}^{(l)}=\\lbrace g\\in \\mathcal {M}\\,:\\, r(g)\\ge l\\rbrace .$ Question 2.3 ([32]) What is $\\mathcal {M}^{(\\infty )}:=\\cap _{l=1}^\\infty \\mathcal {M}^{(l)}$ ?", "What is the relation between $\\mathcal {M}^{(\\infty )}$ and the Ricci flow?", "Clearly, $\\mathcal {E}\\subset \\mathcal {M}^{(\\infty )}=\\lbrace g\\in \\mathcal {M}\\,:\\,r(g)=\\infty \\rbrace $ .", "But, do there exist $g$ with $r(g)=\\infty $ that are neither Einstein nor direct sums of Einstein metrics?", "We answer this last question in the affirmative.", "We also answer Question REF on compact homogeneous spaces with $s=2$ satisfying (REF ).", "The interest in metrics with infinite Ricci index is that one may use them as starting points for the ancient Ricci iteration (REF ).", "These seem to be the first nontrivial examples in the literature of ancient Ricci iterations, and we describe precisely their limits.", "Comparing our result to the study of the Ricci flow by Buzano [9] shows that the ancient Ricci iterations exist in tandem with ancient Ricci flows and have similar limits.", "Theorem 2.4 (ancient Ricci iterations for $s=2$ ) Let $M=G/H$ .", "Assume that $s = 2$ in (REF ) and that (REF ) holds.", "In the statements below, we always assume $T\\in \\mathcal {M}$ (recall (REF )) is given by (REF ).", "Suppose $H$ is maximal in $G$ .", "Then (recall (REF )) $\\begin{aligned}\\mathcal {M}^{(\\infty )}=\\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\in [\\alpha _-,\\alpha _+]\\rbrace .\\end{aligned}$ Whenever $r(g_1)=\\infty $ , the metrics $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ given by (REF ) converge smoothly to an Einstein metric on $M$ .", "Suppose $G$ has a connected Lie subgroup $K$ satisfying (REF ).", "Let (REF ) and (REF ) hold.", "Assume $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial.", "Then $\\mathcal {M}^{(\\infty )}=\\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\le \\alpha _-\\rbrace .$ Whenever $r(g_1)=\\infty $ , the metrics $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ given by (REF ) converge smoothly to a degenerate tensor that is the pullback of an Einstein metric $g_E$ on $G/K$ under the inclusion map $G/K\\hookrightarrow G/H$ .", "The manifolds $(M,g_{-i})$ converge in the Gromov–Hausdorff topology to $(G/K,g_E)$ as $i\\rightarrow \\infty $ .", "Assume $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is nontrivial.", "The set $\\mathcal {M}^{(\\infty )}$ is empty if $\\mathcal {E}$ is, and $\\begin{aligned}\\mathcal {M}^{(\\infty )}=\\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\le \\alpha _+\\rbrace .\\end{aligned}$ otherwise.", "Whenever $r(g_1)=\\infty $ , the metrics $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ given by (REF ) converge smoothly to an Einstein metric on $M$ .", "In particular, $\\mathcal {M}^{(\\infty )}\\ne \\mathcal {E}$ always in (REF ) and (REF ), as well as when $\\alpha _-<\\alpha _+$ (i.e., when there exist at least two distinct Einstein metrics of volume 1) in (REF ).", "We also observe that in the maximal case (in fact, even when $s\\ge 2$ and (REF ) does not necessarily hold) $\\begin{aligned}\\mathcal {M}^{(l)}\\setminus \\mathcal {M}^{(l+1)}=\\lbrace g\\in \\mathcal {M}\\,:\\, r(g)=l\\rbrace \\ne \\emptyset , \\quad l\\in {\\mathbb {N}},\\end{aligned}$ as can be seen by applying Theorem REF $l$ times starting with a degenerate positive-semidefinite nonzero $T$ .", "To compare with the Kähler setting, note that there also the filtration of the space of Kähler metrics similar to (REF ) is strictly decreasing in the sense of (REF ) (by applying the Calabi–Yau theorem $l$ times starting with a non-positive form); however, a characterization of the analogue of $\\mathcal {M}^{(\\infty )}$ is still missing.", "In general, we make the following conjecture.", "Conjecture 2.5 A manifold admits an ancient Ricci iteration only if it admits an ancient Ricci flow.", "Theorem REF and the results of Buzano [9] show that this conjecture holds in the case of compact homogeneous spaces satisfying $s=2$ and (REF ).", "In a forthcoming paper we study this conjecture on certain Lie groups and spheres, including the case of $S^3$ , equipped with left-invariant metrics (note that it is easy to see, as was also pointed out to us by W. Ziller after the completion of the present article, that some Berger metrics on $S^3$ have infinite Ricci index)." ], [ "Relative compactness for maximal isotropy", "Our next result concerns the case when $s\\ge 2$ and the isotropy summands are allowed to be equivalent (i.e., when (REF ) does not necessarily hold) but when $H$ is assumed to be maximal.", "It is generally unknown whether, in this situation, the prescribed Ricci curvature problem has unique solutions in $\\mathcal {M}$ .", "Therefore, it is not clear whether to expect convergence of the Ricci iteration or not.", "If indeed there is no uniqueness, then relative compactness is a reasonable replacement (cf.", "[4], [15] in the Kähler setting).", "We prove it in Theorem REF  (i) and Corollary REF  (i).", "It turns out that a similar result holds for ancient Ricci iterations.", "This is the content of Theorem REF  (ii) and Corollary REF  (ii).", "If a sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ satisfying (REF ) or a sequence $\\lbrace g_{-i}\\rbrace ^\\infty _{i=-1}$ satisfying (REF ) converges to some tensor field, this limit may fail to be positive-definite, i.e., a Riemannian metric.", "When $H$ is maximal, this is excluded by Corollary REF .", "Theorem 2.6 (existence and relative compactness for $s\\ge 2$ ) Let $M=G/H$ .", "Suppose that (REF ) holds with $s \\ge 2 $ and that $H$ is maximal in $G$ .", "Given $T\\in \\mathcal {M}$ , there exist a sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {M}$ satisfying (REF ) for all $i\\in {\\mathbb {N}}$ and $g_1=cT$ for some $c>0$ .", "Any such sequence is relatively compact in $\\mathcal {M}$ .", "Any sequence $\\lbrace g_{-i}\\rbrace ^\\infty _{i=-1}\\subset \\mathcal {M}$ satisfying (REF ) is relatively compact in $\\mathcal {M}$ .", "Remark 2.7 The existence part of Theorem REF  (i) becomes false if the maximality condition on $H$ is removed.", "Indeed, Theorem REF  (ii-b)–(ii-c) demonstrates that it may be impossible to find a sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ satisfying (REF ).", "If $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ is known to exist and $H$ is not maximal, whether or not $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ must be relatively compact in $\\mathcal {M}$ is an open question.", "Remark 2.8 The collapsing construction of Theorem REF  (ii-a) shows that none of the results for ancient Ricci iterations appearing in this subsection holds if the maximality assumption is dropped.", "We state two consequences of Theorem REF .", "Corollary 2.9 Let $M=G/H$ .", "Suppose that (REF ) holds with $s \\ge 2 $ and that $H$ is maximal in $G$ .", "Any sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}\\subset \\mathcal {M}$ satisfying (REF ) (or $\\lbrace g_{-i}\\rbrace ^\\infty _{i=-1}\\subset \\mathcal {M}$ satisfying (REF )) has a subsequence converging smoothly to some Riemannian metric in $\\mathcal {M}$ .", "Theorem REF implies that the closures of $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ and $\\lbrace g_{-i}\\rbrace ^\\infty _{i=-1}$ in $\\mathcal {M}$ are compact subsets of $\\mathcal {M}$ .", "Therefore, these sequences contain subsequences whose limits lie in $\\mathcal {M}$ .", "Similarly to (REF ), there exists an isomorphism $\\mathcal {T}&:=\\lbrace \\hbox{$G$-invariant symmetric (0,2)-tensor fields on $M$}\\rbrace \\\\&\\cong \\lbrace \\hbox{$\\hbox{Ad}_G(H)$-invariant symmetric bilinear forms on $\\mathfrak {m}$}\\rbrace .$ Thus, we can identify $\\mathcal {T}$ with a subset of $\\mathfrak {m}^*\\otimes \\mathfrak {m}^*$ .", "The topology of $\\mathfrak {m}^*\\otimes \\mathfrak {m}^*$ induces a topology on $\\mathcal {T}$ , as it did on $\\mathcal {M}$ .", "Corollary 2.10 Suppose that $M=G/H$ , that (REF ) holds with $s \\ge 2 $ , and that $H$ is maximal in $G$ .", "Assume a sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}\\subset \\mathcal {M}$ satisfying (REF ) (or $\\lbrace g_{-i}\\rbrace ^\\infty _{i=-1}\\subset \\mathcal {M}$ satisfying (REF )) converges smoothly to some tensor field $g_\\infty \\in \\mathcal {T}$ .", "Then $g_\\infty $ is an Einstein metric.", "Assume $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ converges to $g_\\infty $ .", "The other case is treated analogously.", "According to Corollary REF , $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ contains a subsequence that converges to some Riemannian metric in $\\mathcal {M}$ .", "This metric must coincide with $g_\\infty $ by uniqueness of limits.", "Thus, $g_\\infty $ lies in $\\mathcal {M}$ .", "To see that $g_\\infty $ satisfies the Einstein equation, one simply needs to pass to the limit in equality (REF ) as $i$ tends to $\\infty $ .", "Motivated by Theorems REF  (i) and REF  (i), we propose the following conjecture.", "Conjecture 2.11 Suppose $H$ is maximal in $G$ .", "Let $g_1$ be a Riemannian metric in $\\mathcal {M}$ .", "If a sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {M}$ starting with $g_1$ and satisfying (REF ) exists and is unique, then this sequence converges to an Einstein metric.", "If $r(g_1)=\\infty $ , then the sequence $\\lbrace g_{-i}\\rbrace ^\\infty _{i=-1}\\subset \\mathcal {M}$ given by (REF ) converges to an Einstein metric." ], [ "Organization", "This article is organized as follows.", "Section  recalls facts on homogeneous metrics and the prescribed Ricci curvature equation from [36], [27], [30] that play a central rôle in our arguments.", "Section  contains the proofs of our results for $s=2$ : Theorem REF  (i) is established in §REF , Theorem REF  (ii) in §REF , and Theorem REF in §REF .", "We conclude Section  by giving two explicit examples in §REF .", "In Section  we treat the case $s\\ge 2$ with $H$ maximal, proving Theorem REF ." ], [ "The prescribed Ricci curvature equation", "In this section we recall results on homogeneous metrics and the prescribed Ricci curvature equation, following [36], [27], [30].", "Our arguments will involve three arrays of numbers $\\begin{aligned}\\lbrace b_i\\rbrace _{i=1}^s,\\quad \\lbrace \\gamma _{ik}^l\\rbrace _{i,k,l=1}^s, \\quad \\lbrace \\zeta _i\\rbrace _{i=1}^s,\\end{aligned}$ associated with the inner product $Q$ and the decomposition (REF ).", "To define the first one, denote by $B$ the Killing form on the Lie algebra $\\mathfrak {g}$ .", "As $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_i}$ is irreducible, there exists $b_i\\ge 0$ such that $B|_{\\mathfrak {m}_i} = -b_iQ|_{\\mathfrak {m}_i}.$ Next, let $\\Gamma _{ik}^l\\in \\mathfrak {m}_i^*\\otimes \\mathfrak {m}_k^*\\otimes \\mathfrak {m}_l$ be the tensor $\\Gamma _{ik}^l(X,Y):=\\pi _l([X,Y]), \\quad X\\in \\mathfrak {m}_i, Y\\in \\mathfrak {m}_k,$ and denote by $\\gamma _{ik}^l$ the squared norm of $\\Gamma _{ik}^l$ with respect to $Q$ .", "Thus, $\\gamma _{ik}^l:=\\sum Q([e_{\\iota _i},e_{\\iota _k}], e_{\\iota _l})^2,$ where $\\lbrace e_j\\rbrace _{j=1}^n$ is a $Q$ -orthonormal basis of $\\mathfrak {m}$ adapted to the decomposition (REF ), and where the sum is taken over all $\\iota _i, \\iota _k$ and $\\iota _l$ such that $e_{\\iota _i} \\in \\mathfrak {m}_i$ , $e_{\\iota _k} \\in \\mathfrak {m}_k$ and $e_{\\iota _l} \\in \\mathfrak {m}_l$ .", "The $\\gamma _{ik}^l$ are often called the structure constants of $M$ .", "Since $Q$ is $\\hbox{\\rm Ad}_G(G)$ -invariant, it follows that $\\begin{aligned}Q([X,Y],Z)=Q(Y,[X,Z]),\\quad X,Y,Z\\in \\mathfrak {g}.\\end{aligned}$ It is, therefore, evident that $\\gamma _{ik}^l$ is symmetric in all three indices.", "For the third piece of notation, fix a $Q$ -orthonormal basis $\\lbrace w_j\\rbrace _{j=1}^q$ of the Lie algebra $\\mathfrak {h}$ .", "The irreducibility of $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_i}$ implies the existence of $\\zeta _i\\ge 0$ such that the Casimir operator $C_{\\mathfrak {m}_i,Q|_{\\mathfrak {h}}}:=-\\sum _{j=1}^q\\hbox{\\rm ad}w_j\\circ \\hbox{\\rm ad}w_j$ acting on $\\mathfrak {m}_i$ satisfies $C_{\\mathfrak {m}_i,Q|_{\\mathfrak {h}}}=\\zeta _i\\, \\hbox{id}.$ Remark 3.1 If $\\zeta _i$ vanishes, then $\\hbox{\\rm Ad}_G(H)$ is trivial on $\\mathfrak {m}_i$ and the dimension of $\\mathfrak {m}_i$ equals 1.", "To see this, simply observe that $Q(C_{\\mathfrak {m}_i,Q|_{\\mathfrak {h}}}e_k,e_k)=-\\sum _{j=1}^qQ([w_j,[w_j,e_k]],e_k)=-\\sum _{j=1}^qQ([w_j,e_k],[w_j,e_k]).$ If (REF ) holds with $\\zeta _i=0$ , the above formula implies $[w_j,e_k]=0$ for all $k$ , which means $\\hbox{\\rm ad}w_j=0$ on $\\mathfrak {m}_i$ .", "It follows that $\\hbox{\\rm Ad}_G(H)$ is trivial and $\\mathfrak {m}_i$ is 1-dimensional by irreducibility.", "Note also that $\\zeta _1$ and $\\zeta _2$ cannot vanish together when $s=2$ in (REF ).", "This would imply $\\dim M=2$ , contradicting (REF ).", "The constants (REF ) are related by the formula [36] $b_i=2\\zeta _i+\\frac{1}{d_i}\\sum _{k,l=1}^s\\gamma _{ik}^l,$ where $d_i$ is given by (REF ).", "The following well-known result is adapted from [27].", "We include the proof since our notation and assumptions are somewhat different (e.g., we do not assume $G$ is semisimple, and we work with $(0,2)$ -type and not $(1,1)$ -type tensors).", "Lemma 3.2 Suppose (REF ) holds.", "Let $g\\in \\mathcal {M}$ be given by $g=\\sum _{i=1}^s x_i\\pi _i^* Q, \\quad x_i>0.$ Then $\\begin{aligned}\\hbox{\\rm Ric}\\,g=\\sum _{i=1}^s r_i\\pi _i^* Q,\\end{aligned}$ where $r_i=\\frac{b_i}{2}+\\sum _{j,k=1}^s\\frac{\\gamma _{jk}^i}{4d_i}\\Big (\\frac{x_i^2}{x_jx_k}-2\\frac{x_j}{x_k}\\Big ).$ Since $g$ is $G$ -invariant, so is $\\hbox{\\rm Ric}\\,g$ , which means (REF ) holds.", "The $i$ -th component of $\\hbox{\\rm Ric}\\,g$ equals [5] $r_i=-\\frac{1}{2}B(X,X)-\\frac{1}{2}\\sum _{j=1}^ng([X,e_j]_\\mathfrak {m},[X,e_j]_\\mathfrak {m})+\\frac{1}{4}\\sum _{j,k=1}^ng([e_j,e_k]_\\mathfrak {m},X)^2,$ where $X\\in \\mathfrak {m}_i$ is any vector such that $Q(X,X)=1$ , the subscript $\\mathfrak {m}$ denotes the $g$ -orthogonal projection onto $\\mathfrak {m}$ , and $\\lbrace e_j\\rbrace _{j=1}^n$ is a $g$ -orthonormal basis of $\\mathfrak {m}$ adapted to the decomposition (REF ).", "Note that the vector $Z$ in [5] is identically zero for $G$ admitting a bi-invariant Haar measure [5] (in particular for $G$ compact).", "Define $I^{(i)}:=\\lbrace \\iota \\,:\\,e_{\\iota }\\in \\mathfrak {m}_i\\rbrace .$ In formula (REF ), set $X=\\sqrt{x_i}e_{\\iota _i}$ for some $\\iota _i\\in I^{(i)}$ .", "By (REF ), $r_i&=\\frac{b_i}{2}g(e_{\\iota _i},e_{\\iota _i})-\\frac{1}{2}\\sum _{j=1}^nx_ig([e_{\\iota _i},e_j]_\\mathfrak {m},[e_{\\iota _i},e_j]_\\mathfrak {m})+\\frac{1}{4}\\sum _{j,k=1}^nx_ig([e_j,e_k]_\\mathfrak {m},e_{\\iota _i})^2 \\\\&=\\frac{b_i}{2}-\\frac{1}{2}\\sum _{j,k=1}^nx_ig( g([e_{\\iota _i},e_j],e_k)e_k, g([e_{\\iota _i},e_j],e_k)e_k)+\\frac{1}{4}\\sum _{j,k=1}^nx_ig([e_j,e_k],e_{\\iota _i})^2\\\\&=\\frac{b_i}{2}-\\frac{1}{2}\\sum _{j,k=1}^nx_ig([e_{\\iota _i},e_j], e_k)^2+\\frac{1}{4}\\sum _{j,k=1}^nx_i^3Q([e_j,e_k],e_{\\iota _i})^2.$ Hence, $d_ir_i&=\\sum _{\\iota _i\\in I^{(i)}}r_i=\\frac{d_ib_i}{2}-\\frac{1}{2}\\sum _{\\iota _i\\in I^{(i)}}\\sum _{j,k=1}^nx_ig([e_{\\iota _i},e_j], e_k)^2+\\frac{1}{4}\\sum _{\\iota _i\\in I^{(i)}}\\sum _{j,k=1}^nx_i^3Q([e_j,e_k],e_{\\iota _i})^2.$ Observe that the vectors $\\lbrace \\tilde{e}_j\\rbrace _{j=1}^n$ defined by $\\tilde{e}_{\\iota _l}=\\sqrt{x_l}e_{\\iota _l},\\qquad \\iota _l\\in I^{(l)},~l\\in \\lbrace 1,\\ldots ,s\\rbrace ,$ form a $Q$ -orthonormal basis of $\\mathfrak {m}$ .", "Thus, $r_i&=\\frac{b_i}{2}-\\frac{1}{2d_i}\\sum _{j,k=1}^s\\frac{x_k}{x_j} \\sum Q([\\tilde{e}_{\\iota _i},\\tilde{e}_{\\iota _j}],\\tilde{e}_{\\iota _k})^2+\\frac{1}{4d_i}\\sum _{j,k=1}^s\\frac{x_i^2}{x_jx_k}\\sum Q([\\tilde{e}_{\\iota _j},\\tilde{e}_{\\iota _k}],\\tilde{e}_{\\iota _i})^2\\\\&=\\frac{b_i}{2}-\\frac{1}{2d_i}\\sum _{j,k=1}^s\\frac{x_k}{x_j}\\gamma _{ij}^k+\\frac{1}{4d_i}\\sum _{j,k=1}^s\\frac{x_i^2}{x_jx_k}\\gamma _{ij}^k,$ where the sums without limits are taken over all $\\iota _i\\in I^{(i)}$ , $\\iota _j\\in I^{(j)}$ and $\\iota _k\\in I^{(k)}$ .", "The next lemma is a special case of a claim from the proof of [36].", "Recall that we assume $s>1$ in the decomposition (REF ).", "Lemma 3.3 Suppose that $H$ is maximal in $G$ .", "Then, for any $i\\in \\lbrace 1,\\ldots ,s\\rbrace $ , there exists some $k\\in \\lbrace 1,\\ldots ,s\\rbrace \\setminus \\lbrace i\\rbrace $ such that $\\begin{aligned}\\gamma _{ii}^k>0.\\end{aligned}$ We give the proof for the case $s=2$ which generalizes in an obvious way.", "The assumption is equivalent to $\\mathfrak {h}$ being a maximal Lie subalgebra of $\\mathfrak {g}$ .", "In particular, $\\mathfrak {h}\\oplus \\mathfrak {m}_1$ cannot be a Lie subalgebra of $\\mathfrak {g}$ , so $\\pi _2([\\mathfrak {m}_1, \\mathfrak {m}_1])\\ne \\lbrace 0\\rbrace $ , i.e., $\\gamma _{11}^2\\ne 0$ .", "Similarly, $\\gamma _{22}^1\\ne 0$ since $\\mathfrak {h}\\oplus \\mathfrak {m}_2$ cannot be a Lie subalgebra of $\\mathfrak {g}$ .", "The next result implies that any $G$ -invariant metric with definite Ricci curvature must actually have positive Ricci curvature if $H$ is maximal.", "Lemma 3.4 Suppose that $g\\in \\cal M$ satisfies $\\hbox{\\rm Ric}\\,g=cT$ for some $c\\in \\mathbb {R}$ and $T\\in \\cal M$ .", "Then $c\\ge 0$ .", "If $H$ is maximal in $G$ , then $c>0$ .", "If $c<0$ , Bochner's theorem [5] implies that there are no Killing fields, so $G$ is trivial and $M$ is a point.", "If $c=0$ , then the connected component of the identity in the isometry group of $(M,g)$ is a torus (op.", "cit.).", "Since every Lie subgroup of a torus is abelian, $G$ must be abelian.", "Thus, $\\gamma _{ij}^k=0$ for all $i,j,k$ , contradicting Lemma REF .", "The next result states that, whenever $H$ is maximal in $G$ , there always exists a solution to the prescribed (positive-semidefinite) Ricci curvature equation [30].", "The sign of the constant $c$ below is explained by Lemma REF .", "Theorem 3.5 Let $H$ be maximal in $G$ .", "Given a positive-semidefinite nonzero $T\\in \\mathcal {T}$ , there exists a metric $g\\in \\mathcal {M}$ whose Ricci curvature coincides with $cT$ for some $c>0$ .", "Whenever $s=2$ and $H$ is not maximal in $G$ , the prescribed (positive-semidefinite) Ricci curvature equation is completely understood [30].", "We only need to state the result for prescribed positive-definite Ricci curvature.", "Recall that $\\zeta _1,\\zeta _2\\ge 0$ are given by (REF ).", "Proposition 3.6 Suppose $s=2$ in (REF ).", "Consider a metric $T\\in \\mathcal {M}$ given by (REF ).", "Let $G$ have a connected Lie subgroup $K$ satisfying (REF ).", "Assume (REF ) and (REF ) hold.", "A metric $g\\in \\mathcal {M}$ such that $\\hbox{\\rm Ric}\\,g=cT$ for some $c>0$ exists if and only if $\\bigg (\\zeta _2+\\frac{\\gamma _{22}^2}{4d_2}+\\frac{\\gamma _{22}^1}{d_2}\\bigg )z_1>\\bigg (\\zeta _1+\\frac{\\gamma _{11}^1}{4d_1}\\bigg )z_2.$ When $g$ exists, it is unique up to scaling." ], [ "Two irreducible isotropy summands", "This section contains the proofs of Theorems REF and REF , as well as two examples.", "We assume the number $s$ in (REF ) equals 2.", "Thus, $\\mathfrak {m}$ admits a $Q$ -orthogonal $\\hbox{\\rm Ad}_G(H)$ -invariant decomposition $\\mathfrak {m}=\\mathfrak {m}_1\\oplus \\mathfrak {m}_2$ such that $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ and $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_2}$ are irreducible.", "We further impose the assumption (REF ).", "The formula for the Ricci curvature in this case simplifies.", "Lemma 4.1 Suppose that $s=2$ in (REF ) and that (REF ) holds.", "If $g\\in \\mathcal {M}$ is given by (REF ), then $\\hbox{\\rm Ric}\\,g$ satisfies (REF ) with $r_1 &=\\frac{b_1}{2}-\\frac{\\gamma _{11}^1}{4d_1}-\\frac{\\gamma _{22}^1}{2d_1}+\\frac{\\gamma _{22}^1}{4d_1}\\alpha ^2-\\frac{\\gamma _{11}^2}{2d_1}\\frac{1}{\\alpha }, \\\\r_2 &=\\frac{b_2}{2}-\\frac{\\gamma _{22}^2}{4d_2}-\\frac{\\gamma _{11}^2}{2d_2}+\\frac{\\gamma _{11}^2}{4d_2}\\frac{1}{\\alpha ^2}-\\frac{\\gamma _{22}^1}{2d_2}\\alpha ,$ where $\\alpha =\\frac{x_1}{x_2}$ .", "The assumption (REF ) implies that all $G$ -invariant tensors are completely determined by two real numbers, as in (REF ).", "Thus, (REF ) holds.", "Using Lemma REF , we compute $r_1&=\\frac{b_1}{2}+\\frac{1}{4d_1}\\sum _{j,k=1}^2\\frac{x_1^2}{x_jx_k}\\gamma _{jk}^1-\\frac{1}{2d_1}\\sum _{j,k=1}^2\\frac{x_j}{x_k}\\gamma _{jk}^1\\\\ &=\\frac{b_1}{2}+\\frac{1}{4d_1}\\left(\\gamma _{11}^1+\\frac{x_1^2}{x_2^2}\\gamma _{22}^1+2\\frac{x_1}{x_2}\\gamma _{12}^1\\right)-\\frac{1}{2d_1}\\left(\\gamma _{11}^1+\\gamma _{22}^1+\\frac{x_1}{x_2}\\gamma _{12}^1+\\frac{x_2}{x_1}\\gamma _{21}^1\\right)\\\\ &=\\frac{b_1}{2}-\\frac{\\gamma _{11}^1}{4d_1}-\\frac{\\gamma _{22}^1}{2d_1}+\\frac{\\gamma _{22}^1}{4d_1}\\alpha ^2-\\frac{\\gamma _{11}^2}{2d_1}\\frac{1}{\\alpha },$ as desired.", "The formula for $r_2$ is proved analogously.", "The next result reduces the prescribed Ricci curvature equation to a single polynomial equation.", "Lemma 4.2 Suppose that $s=2$ in (REF ) and that (REF ) holds.", "Let $g\\in \\mathcal {M}$ and $T\\in \\cal M$ be given by (REF ) and (REF ).", "The equality $\\hbox{\\rm Ric}\\,g=cT$ holds for some $c\\ge 0$ if and only if $P(x_1/x_2,z_1/z_2)=0$ , where $P(x,y):=d_2\\gamma _{22}^1x^4+2d_1\\gamma _{22}^1yx^3+(\\theta _1-y\\theta _2)x^2-2d_2\\gamma _{11}^2x-d_1\\gamma _{11}^2y$ and $\\theta _1:=2d_1d_2b_1-d_2\\gamma _{11}^1-2d_2\\gamma _{22}^1, \\qquad \\theta _2:=2d_1d_2b_2-d_1\\gamma _{22}^2-2d_1\\gamma _{11}^2.$ Suppose that $\\hbox{\\rm Ric}\\,g=cT$ for some $c\\ge 0$ .", "Lemma REF implies that $\\frac{b_1}{2}-\\frac{\\gamma _{11}^1}{4d_1}-\\frac{\\gamma _{22}^1}{2d_1}+\\frac{\\gamma _{22}^1}{4d_1}\\alpha ^2-\\frac{\\gamma _{11}^2}{2d_1}\\frac{1}{\\alpha }&=cz_1, \\\\\\frac{b_2}{2}-\\frac{\\gamma _{22}^2}{4d_2}-\\frac{\\gamma _{11}^2}{2d_2}+\\frac{\\gamma _{11}^2}{4d_2}\\frac{1}{\\alpha ^2}-\\frac{\\gamma _{22}^1}{2d_2}\\alpha &=cz_2,$ where $\\alpha =\\frac{x_1}{x_2}$ .", "Since $T\\in \\cal M$ , the numbers $z_1,z_2$ are positive, and so $\\begin{aligned}\\frac{b_1}{2}-\\frac{\\gamma _{11}^1}{4d_1}-\\frac{\\gamma _{22}^1}{2d_1}+\\frac{\\gamma _{22}^1}{4d_1}\\alpha ^2-\\frac{\\gamma _{11}^2}{2d_1}\\frac{1}{\\alpha }=\\frac{z_1}{z_2}\\Big (\\frac{b_2}{2}-\\frac{\\gamma _{22}^2}{4d_2}-\\frac{\\gamma _{11}^2}{2d_2}+\\frac{\\gamma _{11}^2}{4d_2}\\frac{1}{\\alpha ^2}-\\frac{\\gamma _{22}^1}{2d_2}\\alpha \\Big ).\\end{aligned}$ Multiplying by $4d_1d_2\\alpha ^2$ and rearranging yields $P(x_1/x_2,z_1/z_2)=0$ .", "Conversely, suppose that $P(x_1/x_2,z_1/z_2)=0$ , and hence (REF ) holds.", "Define $c$ as the left-hand side of (REF ) divided by $z_1$ .", "Then, it follows from (REF ) that (REF ) holds.", "By Lemma REF , $c\\ge 0$ .", "Denote, for $x>0$ , $\\begin{aligned}g^x:=x\\pi _1^* Q+\\pi _2^* Q\\in \\mathcal {M}.\\end{aligned}$ The following result characterizes Einstein metrics in $\\cal M$ .", "Lemma 4.3 Suppose that $s=2$ in (REF ) and that (REF ) holds.", "Then $P(x,x)=0$ if and only if $g^x$ is Einstein with $\\hbox{\\rm Ric}\\,g^x=cg^x$ for some $c\\ge 0$ .", "This follows from Lemma REF .", "We state two identities that follow directly from (REF ).", "In particular, they show that $\\theta _1,\\theta _2$ are non-negative.", "Lemma 4.4 One has (recall (REF )) $\\theta _1=4d_1d_2\\zeta _1+d_2\\gamma _{11}^1+4d_2\\gamma _{11}^2, \\qquad \\theta _2=4d_1d_2\\zeta _2+d_1\\gamma _{22}^2+4d_1\\gamma _{22}^1.$" ], [ "Ricci iteration with maximal isotropy", "Let us prove Theorem REF  (i).", "We assume $H$ is maximal in $G$ .", "According to [36], in this case, a $G$ -invariant Einstein metric on $M$ is known to exist.", "Since $P(x,x)$ factors as $x$ times a cubic polynomial, Lemma REF implies that there are at most three such metrics, up to scaling; cf. [21].", "Remark 4.5 Suppose $g_\\infty $ is the limit of the sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ in Theorem REF  (i).", "When there exist more than one $G$ -invariant Einstein metric (up to scaling) on $M$ , the proof we are about to present identifies $g_\\infty $ .", "More precisely, suppose $\\epsilon _1\\le \\epsilon _2\\le \\epsilon _3$ are positive numbers such that $\\hbox{\\rm Ric}\\,g^{\\epsilon _i}= c_ig^{\\epsilon _i}$ with $c_i>0$ for all $i\\in \\lbrace 1,2,3\\rbrace $ .", "Assume every $x>0$ satisfying the equation $\\hbox{\\rm Ric}\\,g^x= cg^x$ for some $c>0$ coincides with $\\epsilon _1$ , $\\epsilon _2$ or $\\epsilon _3$ .", "In the notation of (REF ) and (REF ), if $\\alpha _{T}\\in (0,\\epsilon _1)$ , then $g_\\infty $ is proportional to $g^{\\epsilon _1}$ ; if $\\alpha _{T}\\in (\\epsilon _3,\\infty )$ , then $g_\\infty $ is proportional to $g^{\\epsilon _3}$ ; if $\\alpha _{T}\\in (\\epsilon _i,\\epsilon _{i+1})$ ($i\\in \\lbrace 1,2\\rbrace $ ), then $g_\\infty $ is proportional to $g^{\\epsilon _i}$ or $g^{\\epsilon _{i+1}}$ depending on whether $\\alpha _2<\\alpha _T$ or $\\alpha _2>\\alpha _T$ , respectively.", "[Proof of Theorem REF (i).]", "The existence of the sequence follows from Theorem REF , while the uniqueness is the content of Lemma REF established below.", "To prove the convergence, suppose $\\begin{aligned}g_i=x_1^{(i)}\\pi _1^*Q+x_2^{(i)}\\pi _2^*Q,\\end{aligned}$ and let $\\begin{aligned}\\alpha _i:=x_1^{(i)}/x_2^{(i)}.\\end{aligned}$ Lemma REF below demonstrates that $\\lbrace \\alpha _i\\rbrace _{i\\in {\\mathbb {N}}}$ is monotone, bounded below by $\\min \\lbrace \\alpha _1,\\alpha _-\\rbrace $ and bounded above by ${\\operatorname{max}}\\lbrace \\alpha _1,\\alpha _+\\rbrace $ (recall (REF )).", "Consequently, this sequence has a positive limit $\\alpha _\\infty $ .", "Thus, if the limits $x_1^{(\\infty )}:=\\lim _{i\\rightarrow \\infty } x_1^{(i)}\\hbox{\\ \\ and \\ \\ }x_2^{(\\infty )}:=\\lim _{i\\rightarrow \\infty } x_2^{(i)}$ both exist and are finite, then they must be simultaneously positive or zero.", "Note that $\\hbox{\\rm Ric}\\,(g_{i+1}/x_2^{(i+1)})=\\hbox{\\rm Ric}\\,g_{i+1}=g_i,$ i.e., $\\begin{aligned}\\hbox{\\rm Ric}\\,(\\alpha _{i+1}\\pi _1^*Q+\\pi _2^*Q)=x_1^{(i)}\\pi _1^*Q+x_2^{(i)}\\pi _2^*Q.\\end{aligned}$ This shows that the limits indeed both exist and are finite, as the left-hand side has a well-defined limit by Lemma REF (as $\\alpha _\\infty >0$ ).", "We thus have $\\hbox{\\rm Ric}\\,(\\alpha _\\infty \\pi _1^*Q+\\pi _2^*Q)=x_1^{(\\infty )}\\pi _1^*Q+x_2^{(\\infty )}\\pi _2^*Q.$ By Lemma REF the right-hand side cannot vanish; we conclude that $x_1^{(\\infty )}, x_2^{(\\infty )}>0$ .", "Finally, let $g_\\infty =x_1^{(\\infty )}\\pi _1^*Q+x_2^{(\\infty )}\\pi _2^*Q.$ We have shown that $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ converges to $g_\\infty $ , and as explained in §REF , this convergence is smooth.", "By passing to the limit in formula (REF ), we see that $\\hbox{\\rm Ric}\\,g_\\infty =g_\\infty $ , concluding the proof.", "Lemma 4.6 Suppose that $s=2$ in (REF ) and that (REF ) holds.", "The metric $g$ given by Theorem REF is unique up to scaling.", "Fix $b\\ge 0$ and set $\\begin{aligned}P_b(x):=P(x,b).\\end{aligned}$ By Lemma REF , it suffices to show that $P_b(x)=0$ for at most one $x\\in (0,\\infty )$ .", "The leading coefficient of $P_b$ (i.e., the coefficient of $x^4$ ) is positive by Lemma REF , so the leading coefficient of the quadratic polynomial $R_2:=\\frac{d^2}{dx^2}P_b(x)$ is positive.", "The $x$ coefficient of $R_2$ is nonnegative.", "Thus, $R_2$ has no more than one positive root, equivalently, $R_1:=\\frac{d}{dx}P_b(x)$ has at most one critical point, in $(0,\\infty )$ .", "Since $R_1 (0) < 0$ and $ \\lim _{x\\rightarrow \\infty }R_1 (x) =\\infty $ it follows that $R_1$ has at most one positive root.", "Consequently, the function $P_b$ has at most one critical point in $(0,\\infty )$ .", "Again, since $P_b(0) \\le 0$ and $ \\lim _{x\\rightarrow \\infty }P_b (x) =\\infty $ , it follows that $P_b$ has no more than one positive root.", "Remark 4.7 In regards to the previous lemma, we remark that in the context of the Ricci iteration we have a naturally imposed scaling normalization given by (REF ).", "Namely, assume that the hypotheses of Theorem REF  (i) hold.", "Let $g$ be a metric such that $\\hbox{\\rm Ric}\\,g=g_{i-1}$ .", "Then, $\\hbox{\\rm Ric}\\,cg=g_{i-1}$ for any $c>0$ .", "However, according to Lemma REF , there is at most one $c>0$ such that $\\hbox{\\rm Ric}\\,g_{i+1}=cg$ is solvable for some $g_{i+1}$ , and we set $g_i=cg$ .", "Let $T,g,h\\in \\mathcal {M}$ be given by (REF ), (REF ) and $h:=y_1\\pi _1^*Q+y_2\\pi _2^*Q.$ Denote $\\alpha _T:=\\frac{z_1}{z_2},\\quad \\alpha _g:=\\frac{x_1}{x_2},\\quad \\alpha _h:=\\frac{y_1}{y_2}.$ The next result gives the monotonicity needed in the proof of Theorem REF .", "Before stating it, we need to point out that, as Lemma REF shows, there are at most three numbers $x>0$ such that $g^x\\in \\mathcal {E}$ (recall (REF ) and (REF )).", "Lemma 4.8 Suppose that $H$ is maximal in $G$ , that $s = 2 $ in (REF ), and that (REF ) holds.", "Let the metrics $T,g,h\\in \\mathcal {M}$ given by (REF ), (REF ) and (REF ) satisfy $\\hbox{\\rm Ric}\\,g=cT$ for some $c>0$ and $\\hbox{\\rm Ric}\\,h=g$ .", "Suppose $\\epsilon ,\\epsilon _1,\\epsilon _2>0$ are such that $g^\\epsilon ,g^{\\epsilon _1},g^{\\epsilon _2}\\in \\cal E$ and $g^x\\notin \\cal E$ whenever $x\\in (\\epsilon _1,\\epsilon _2)$ .", "Then: If $\\alpha _T\\ge \\epsilon $ , then $\\alpha _g\\ge \\epsilon $ .", "If $\\alpha _T\\le \\epsilon $ , then $\\alpha _g\\le \\epsilon $ .", "If $\\alpha _T\\ge \\alpha _+$ (recall (REF )), then $\\alpha _T\\ge \\alpha _g\\ge \\alpha _+$ .", "If $\\epsilon _1\\le \\alpha _g\\le \\alpha _T\\le \\epsilon _2$ , then $\\epsilon _1\\le \\alpha _h\\le \\alpha _g$ .", "If $\\epsilon _1\\le \\alpha _T\\le \\alpha _g\\le \\epsilon _2$ , then $\\alpha _g\\le \\alpha _h\\le \\epsilon _2$ .", "If $\\alpha _T\\le \\alpha _-$ (recall (REF )), then $\\alpha _T\\le \\alpha _g\\le \\alpha _-$ .", "(i) The function $P(\\epsilon ,y)$ is decreasing in $y$ .", "Indeed, since $\\hbox{\\rm Ric}\\,g^\\epsilon = cg^\\epsilon $ for some $c>0$ , the second line in (REF ) implies $\\begin{aligned}c&=\\frac{b_2}{2}-\\frac{\\gamma _{22}^2}{4d_2}-\\frac{\\gamma _{11}^2}{2d_2}+\\frac{\\gamma _{11}^2}{4d_2}\\frac{1}{\\epsilon ^2}-\\frac{\\gamma _{22}^1}{2d_2}\\epsilon \\cr &=(\\theta _2+d_1\\gamma _{11}^2\\frac{1}{\\epsilon ^2}-2d_1\\gamma _{22}^1\\epsilon )/4d_1d_2.\\end{aligned}$ Thus, $\\frac{\\partial }{\\partial y}P(\\epsilon ,y)=2d_1\\gamma _{22}^1\\epsilon ^3-\\theta _2\\epsilon ^2-d_1\\gamma _{11}^2=-4d_1d_2\\epsilon ^2c<0,$ as claimed.", "Note that $P(\\epsilon ,\\epsilon )=0$ by Lemma REF .", "If $\\alpha _T\\ge \\epsilon $ , then $P(\\epsilon ,\\alpha _T)\\le P(\\epsilon ,\\epsilon )=0.$ As in the proof of Lemma REF , the leading coefficient of the polynomial $P_{\\alpha _T}(x)=P(x,\\alpha _T)$ is positive, so $\\lim _{x\\rightarrow \\infty }P_{\\alpha _T}(x)=\\infty $ .", "From this and (REF ), it follows that there exists $x\\ge \\epsilon $ such that $P_{\\alpha _T}(x)=0$ .", "However, Lemma REF implies that $P(\\alpha _g,\\alpha _T)=0$ , and Lemma REF then implies that $x=\\alpha _g$ , so $\\alpha _g\\ge \\epsilon $ as desired.", "(ii) The argument of (i) gives that $P(\\epsilon ,\\alpha _T)\\ge 0$ .", "Note that $P(0,\\alpha _T)=-d_1\\gamma _{11}^2\\alpha _T<0$ by Lemma REF .", "Thus, $P(x,\\alpha _T)=0$ for some $x\\in (0,\\alpha _T]$ , and the rest of the proof follows that of (i).", "(iii) Suppose $\\alpha _T\\ge \\alpha _+$ .", "Setting $\\epsilon =\\alpha _+$ in (i), we obtain $\\begin{aligned}\\alpha _g\\ge \\alpha _+.\\end{aligned}$ It remains to show that $\\alpha _g\\le \\alpha _T$ .", "Formulæ (REF ) yield $\\alpha _g-\\alpha _T&=\\alpha _g-\\frac{\\theta _1\\alpha _g^2+d_2\\gamma _{22}^1\\alpha _g^4-2d_2\\gamma _{11}^2\\alpha _g}{\\theta _2\\alpha _g^2+d_1\\gamma _{11}^2-2d_1\\gamma _{22}^1\\alpha _g^3} \\\\&=\\frac{\\alpha _g\\big (-(2d_1+d_2)\\gamma _{22}^1\\alpha _g^3+\\theta _2\\alpha _g^2-\\theta _1\\alpha _g+(d_1+2d_2)\\gamma _{11}^2\\big )}{\\theta _2\\alpha _g^2+d_1\\gamma _{11}^2-2d_1\\gamma _{22}^1\\alpha _g^3}\\,.$ Since $\\hbox{\\rm Ric}\\,g=cT$ , the denominator is a positive multiple of the second component of $T$ , i.e., is itself positive.", "Thus, the sign of $\\alpha _g-\\alpha _T$ coincides with the sign of $\\tilde{P}(\\alpha _g)$ , where $\\tilde{P}(x):=-(2d_1+d_2)\\gamma _{22}^1x^3+\\theta _2x^2-\\theta _1x+(d_1+2d_2)\\gamma _{11}^2,\\qquad x\\in \\mathbb {R}.$ Note that $\\tilde{P}(x)$ is a polynomial whose leading coefficient is negative.", "To prove that $\\alpha _g\\le \\alpha _T$ , it suffices to show that $\\alpha _g$ is greater than or equal to the largest root of this polynomial.", "Since $\\tilde{P}(x)=-{P(x,x)}/x$ for all $x>0$ , the positive roots of $\\tilde{P}(x)$ are precisely all $\\epsilon $ such that $g^\\epsilon \\in \\mathcal {E}$ , and we are done by (REF ).", "(iv) Assume $\\epsilon _1\\le \\alpha _g\\le \\alpha _T\\le \\epsilon _2$ .", "By assumption, $\\begin{aligned}P(x,x)\\ne 0,\\qquad x\\in (\\epsilon _1,\\epsilon _2).\\end{aligned}$ By (i) and (ii), $\\epsilon _1\\le \\alpha _h\\le \\epsilon _2$ .", "It suffices to show that $\\alpha _h-\\alpha _g\\le 0$ or, equivalently, that $\\tilde{P}(\\alpha _h)\\le 0$ .", "Since $\\tilde{P}(\\epsilon _1)=\\tilde{P}(\\epsilon _2)=0$ , assume that $\\alpha _h\\in (\\epsilon _1,\\epsilon _2)$ .", "We may also assume $\\epsilon _1<\\alpha _g<\\alpha _T<\\epsilon _2$ (otherwise, Lemma REF implies $g$ and $h$ are Einstein, which means $\\alpha _g=\\alpha _h$ , and we are done).", "By (REF ), $\\tilde{P}(x)$ does not change sign in the interval $(\\epsilon _1,\\epsilon _2)$ .", "Therefore, it suffices to prove that $\\tilde{P}(\\alpha _g)<0$ .", "But the sign of $\\tilde{P}(\\alpha _g)$ coincides with the sign of $\\alpha _g-\\alpha _T$ .", "Since $\\alpha _g<\\alpha _T$ by assumption, we are done.", "(v) The proof is essentially identical to that of (iv).", "(vi) The proof is essentially identical to that of (iii)." ], [ "Ricci iteration with non-maximal isotropy", "Let us prove Theorem REF  (ii).", "We assume $G$ has a connected Lie subgroup $K$ such that (REF ) holds.", "The number $s$ in (REF ) is still 2, and (REF ) is satisfied.", "Since $\\mathfrak {k}$ is a Lie algebra, $[\\mathfrak {k},\\mathfrak {k}]\\subset \\mathfrak {k}$ , and as $\\mathfrak {k}$ is $Q$ -orthogonal to $\\mathfrak {m}_2$ , $\\begin{aligned}\\gamma _{11}^2=0.\\end{aligned}$ The requirement (REF ) is equivalent to the formula $\\begin{aligned}\\gamma _{22}^1>0.\\end{aligned}$ As we mentioned in §REF , if $\\gamma _{22}^1=0$ , then Lemma REF implies that all the metrics in $\\mathcal {M}$ have the same Ricci curvature, namely, $\\begin{aligned}\\Big (\\frac{b_1}{2}-\\frac{\\gamma _{11}^1}{4d_1}\\Big )\\pi _1^* Q+\\Big (\\frac{b_2}{2}-\\frac{\\gamma _{22}^2}{4d_2}\\Big )\\pi _2^* Q.\\end{aligned}$ In this case, the Ricci iteration exists if and only if the initial metric is equal to (REF ), and hence is Einstein.", "Because (REF ) holds, the expression $P(x,x)$ is now a quadratic polynomial in $x$ times $x^2$ .", "Hence Lemma REF implies that there are at most two Einstein metrics in $\\mathcal {E}$ , up to scaling; cf. [21].", "Remark 4.9 By analogy with Remark REF , our proof Theorem REF  (ii) identifies the limit, $g_\\infty $ , of the sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ , provided this sequence exists.", "More precisely, suppose there are two distinct Einstein metrics of volume 1 in $\\mathcal {M}$ .", "In the notation of (REF ) and (REF ), if $\\alpha _{T}\\in (0,\\alpha _-)$ , then $g_\\infty $ is proportional to $g^{\\alpha _-}$ ; if $\\alpha _{T}\\in (\\alpha _-,\\infty )$ , then $g_\\infty $ is proportional to $g^{\\alpha _+}$ .", "[Proof of Theorem REF (ii).]", "(b) Suppose $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is nontrivial, $\\mathcal {E}\\ne \\emptyset $ and $\\alpha _T\\ge \\alpha _-$ .", "Then by Remark REF one has $\\begin{aligned}\\zeta _1>0.\\end{aligned}$ Lemma REF  (i) below implies that a sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ satisfying (REF ) exists and is unique.", "Let $\\alpha _i$ be as in (REF ) and (REF ).", "Lemma REF  (i) also implies that the sequence $\\lbrace \\alpha _i\\rbrace _{i\\in {\\mathbb {N}}}$ is monotone and $\\alpha _-\\le \\alpha _i\\le {\\operatorname{max}}\\lbrace \\alpha _1,\\alpha _+\\rbrace ,\\qquad i\\in \\mathbb {N}.$ Therefore, this sequence must converge to some number $\\alpha _\\infty $ .", "It is clear that $\\alpha _\\infty \\ge \\alpha _->0$ .", "An alternative way to see that $\\alpha _\\infty >0$ is to observe that, by Proposition REF , $\\begin{aligned}\\alpha _i>\\frac{\\zeta _1+\\frac{\\gamma _{11}^1}{4d_1}}{\\zeta _2+\\frac{\\gamma _{22}^2}{4d_2}+\\frac{\\gamma _{22}^1}{d_2}}=\\frac{\\tilde{\\eta }_1}{\\tilde{\\eta }_2},\\quad i\\in \\mathbb {N},\\end{aligned}$ where $\\tilde{\\eta }_1$ and $\\tilde{\\eta }_2$ denote the numerator and the denominator of the middle expression.", "The positivity of $\\alpha _\\infty $ then follows from (REF ) and (REF ).", "As in the proof of Theorem REF  (i), the Ricci iteration equation (REF ) implies that $x_1^{(\\infty )}=\\lim _{i\\rightarrow \\infty } x_1^{(i)}\\hbox{\\ \\ and \\ \\ }x_2^{(\\infty )}=\\lim _{i\\rightarrow \\infty } x_2^{(i)}$ exist and are finite.", "Since $\\alpha _\\infty >0$ , these limits are simultaneously positive or zero.", "Plugging (REF ) in Lemma REF gives $x_1^{(i)}&:=\\frac{b_1}{2}-\\frac{\\gamma _{11}^1}{4d_1}-\\frac{\\gamma _{22}^1}{2d_1}+\\frac{\\gamma _{22}^1}{4d_1}\\alpha _{i+1}^2, \\\\x_2^{(i)}&:=\\frac{b_2}{2}-\\frac{\\gamma _{22}^2}{4d_2}-\\frac{\\gamma _{22}^1}{2d_2}\\alpha _{i+1}.$ Combining (REF ), (REF ) and (REF ) yields $x_1^{(\\infty )}&=\\zeta _1+\\frac{\\gamma _{11}^1}{4d_1}+\\frac{\\gamma _{22}^1}{4d_1}\\alpha _\\infty ^2>0,$ and so $x_2^{(\\infty )}$ is positive.", "As in the proof of Theorem REF  (i), it follows that the Ricci iteration converges smoothly to some $g_\\infty \\in \\mathcal {M}$ .", "Passing to the limit shows that $g_\\infty \\in \\mathcal {E}$ .", "Assume now that $\\alpha _T<\\alpha _-$ .", "Suppose the sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ exists.", "Then it must be unique by Proposition REF .", "The corresponding sequence $\\lbrace \\alpha _i\\rbrace _{i\\in {\\mathbb {N}}}$ is monotone non-increasing by Lemma REF  (i) and converges to some $\\alpha _\\infty $ .", "Because (REF ) holds, the reasoning above shows that $\\alpha _\\infty \\in [{\\tilde{\\eta }_1}/{\\tilde{\\eta }_2},\\alpha _-)\\subset (0,\\alpha _-)$ and $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ converges to an Einstein metric $g_\\infty $ proportional to $g^{\\alpha _\\infty }$ (recall (REF )).", "But then $g^{\\alpha _\\infty }$ is Einstein, which is impossible since $\\alpha _\\infty <\\alpha _-$ .", "(a) Suppose $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial.", "Lemma REF  (ii) implies that a sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ satisfying (REF ) exists and is unique.", "As before, let $\\alpha _i$ be as in (REF ) and (REF ).", "Lemma REF  (ii) also implies that the sequence $\\lbrace \\alpha _i\\rbrace _{i\\in {\\mathbb {N}}}$ is monotone and $\\min \\lbrace \\alpha _1,\\alpha _-\\rbrace =\\min \\lbrace \\alpha _1,\\alpha _+\\rbrace \\le \\alpha _i\\le {\\operatorname{max}}\\lbrace \\alpha _1,\\alpha _+\\rbrace ={\\operatorname{max}}\\lbrace \\alpha _1,\\alpha _-\\rbrace ,\\qquad i\\in \\mathbb {N}.$ Therefore, $\\lbrace \\alpha _i\\rbrace _{i\\in {\\mathbb {N}}}$ must converge to some positive number $\\alpha _\\infty $ .", "As in the proof of (b), we use this fact to show that $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ converges to a $G$ -invariant Einstein metric on $M$ .", "The uniqueness of such metrics up to scaling follows from the equality $\\alpha _-=\\alpha _+$ .", "(c) Suppose $\\mathcal {E}=\\emptyset $ and a sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ satisfying (REF ) exists.", "Lemma REF  (iii) shows that $\\lbrace \\alpha _i\\rbrace _{i\\in {\\mathbb {N}}}$ is monotone decreasing.", "Arguing as in the proof of the “if\" portion of (b), we arrive at a contradiction.", "Lemma 4.10 Suppose that $s=2$ in (REF ) and that (REF ) holds.", "Consider a metric $T\\in \\mathcal {M}$ given by (REF ).", "Let $G$ have a connected Lie subgroup $K$ satisfying (REF ).", "Assume (REF ) and (REF ) hold.", "Suppose $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is nontrivial and $\\mathcal {E}\\ne \\emptyset $ .", "Then: If $\\alpha _T\\ge \\alpha _+$ , then there exists a metric $g\\in \\mathcal {M}$ , unique up to scaling, such that $\\hbox{\\rm Ric}\\,g=cT$ for some $c>0$ .", "Moreover, $\\alpha _T\\ge \\alpha _g\\ge \\alpha _+$ .", "If $\\alpha _T\\in (\\alpha _-,\\alpha _+)$ , then there exists $g\\in \\mathcal {M}$ , unique up to scaling, such that $\\hbox{\\rm Ric}\\,g=cT$ for some $c>0$ .", "Moreover, $\\alpha _T\\le \\alpha _g\\le \\alpha _+$ .", "If $\\alpha _T\\le \\alpha _-$ and $\\hbox{\\rm Ric}\\,g=cT$ for some $g\\in \\mathcal {M}$ and $c>0$ , then $\\alpha _g\\le \\alpha _T$ .", "Suppose $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial.", "Then: The set $\\mathcal {E}$ is nonempty, and the equality $\\alpha _-=\\alpha _+$ holds true.", "There exists a metric $g\\in \\mathcal {M}$ , unique up to scaling, such that $\\hbox{\\rm Ric}\\,g=cT$ for some $c>0$ .", "If $\\alpha _T\\ge \\alpha _-=\\alpha _+$ , then $\\alpha _T\\ge \\alpha _g\\ge \\alpha _-=\\alpha _+$ .", "If $\\alpha _T\\le \\alpha _-=\\alpha _+$ , then $\\alpha _T\\le \\alpha _g\\le \\alpha _-=\\alpha _+$ .", "If $\\mathcal {E}=\\emptyset $ and $\\hbox{\\rm Ric}\\,g=T$ for some $g\\in \\mathcal {M}$ , then $\\alpha _T\\ge \\alpha _g$ .", "Remark 4.11 Lemma REF can be proven in essentially the same way as Lemma REF .", "The alternative proof below is simpler.", "It will be convenient to use the slightly different normalization than (REF ): $\\eta _i:=\\frac{2d_2^2}{d_1\\gamma _{22}^1}\\tilde{\\eta }_i,\\quad i\\in \\lbrace 1,2\\rbrace .$ Observe that $\\eta _2\\ne 0$ .", "Condition (REF ) means $z_1/z_2>\\eta _1/\\eta _2$ .", "(i-a) The metric $g^{\\alpha _+}$ is Einstein.", "This fact and Proposition REF imply $\\alpha _+>\\frac{\\eta _1}{\\eta _2}$ , so our assumption gives $\\alpha _T> \\frac{\\eta _1}{\\eta _2}$ .", "One more application of Proposition REF yields the existence and the uniqueness of $g\\in \\mathcal {M}$ such that $\\hbox{\\rm Ric}\\,g=cT$ .", "Next, we show that $\\alpha _+\\le \\alpha _g$ .", "Lemmas REF and REF , together with (REF ), give $\\alpha _T=\\frac{\\tilde{\\eta }_1+\\gamma ^1_{22}\\alpha _g^2/4d_1}{\\tilde{\\eta }_2-\\gamma ^1_{22}\\alpha _g/2d_2}=\\frac{\\eta _1+\\frac{1}{2}(d_2\\alpha _g/d_1)^2}{\\eta _2-d_2\\alpha _g/d_1},$ so letting $y=d_2\\alpha _g/d_1$ , we obtain $y^2+2\\alpha _Ty+2\\eta _1-2\\eta _2\\alpha _T=0.$ Consequently, $\\alpha _g=F(\\alpha _T)$ with $F(x)=\\frac{d_1}{d_2}\\big (\\sqrt{x^2+2\\eta _2x-2\\eta _1}-x\\big ),\\quad x\\in ({\\eta _1}/{\\eta _2},\\infty ).$ Because $g^{\\alpha _-}$ and $g^{\\alpha _+}$ are Einstein, we have $\\alpha _-=F(\\alpha _-)$ and $\\alpha _+=F(\\alpha _+)$ .", "This implies $\\alpha _g-\\alpha _+=F(\\alpha _T)-F(\\alpha _+).$ Since $\\alpha _T\\ge \\alpha _+$ , it suffices to demonstrate that $F$ is an increasing function, to wit, $F^{\\prime }(x)=\\frac{d_1}{d_2}\\bigg (\\frac{x+\\eta _2-\\sqrt{(x+\\eta _2)^2-\\eta _2^2-2\\eta _1}}{\\sqrt{x^2+2\\eta _2x-2\\eta _1}}\\bigg )>0,\\quad x\\in ({\\eta _1}/{\\eta _2},\\infty ).$ Finally, let us prove that $\\alpha _g\\le \\alpha _T$ , i.e., $F(\\alpha _T)\\le \\alpha _T$ , or $\\frac{d_1}{d_2}\\sqrt{\\alpha _T^2+2\\eta _2\\alpha _T-2\\eta _1}\\le \\bigg (1+\\frac{d_1}{d_2}\\bigg )\\alpha _T.$ Squaring the nonnegative expressions on both sides shows that $\\alpha _g\\le \\alpha _T$ if and only if $\\tilde{F}(\\alpha _T)\\ge 0$ with $\\tilde{F}(x)=\\bigg (1+2\\frac{d_1}{d_2}\\bigg )x^2-2\\frac{d_1^2}{d_2^2}\\eta _2x+2\\frac{d_1^2}{d_2^2}\\eta _1,\\qquad x\\in \\mathbb {R}.$ When $x>\\eta _1/\\eta _2$ , the formula $\\tilde{F}(x)=0$ implies $F(x)=x$ .", "For such $x$ , the equality $F(x)=x$ holds if and only if $x=\\alpha _-$ or $x=\\alpha _+$ ; cf.", "Lemma REF .", "To prove that $\\tilde{F}(\\alpha _T)\\ge 0$ , it suffices to show that $\\alpha _T$ is greater than or equal to the largest root of the polynomial $\\tilde{F}(x)$ , i.e., $\\alpha _T\\ge \\alpha _+$ .", "But this inequality holds by assumption.", "(i-b)–(i-c) It suffices to repeat the arguments from the proof of (i-a) with very minor changes.", "(ii-a) Suppose $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial.", "Then $\\zeta _1=0$ .", "Because $\\mathfrak {m}_1$ is irreducible, it must be 1-dimensional.", "Ergo, the constants $\\gamma _{11}^1$ and $\\eta _1$ equal 0.", "Lemma REF shows that $\\theta _1$ equals 0 as well.", "The expression $P(x,x)$ (recall (REF )) factors into $x^3$ and a linear function of $x$ with negative free term.", "Therefore, it vanishes for exactly one positive $x$ .", "Since $P(\\alpha _-,\\alpha _-)=P(\\alpha _+,\\alpha _+)=0$ by Lemma REF , we conclude that $\\alpha _-=\\alpha _+$ .", "(ii-b) This claim is a direct consequence of Proposition REF and the equality $\\eta _1=0$ .", "(ii-c)–(ii-d)–(iii) It suffices to repeat the arguments from the proof of (i-a) with minor modifications." ], [ "Ancient Ricci iterations", "In this subsection, we prove Theorem REF .", "First, we observe that a metric has positive Ricci curvature as soon as it is “sandwiched\" between two metrics with positive Ricci curvature.", "Lemma 4.12 Suppose that $s=2$ in (REF ) and that (REF ) holds.", "Let $g_i\\in \\mathcal {M}$ and $\\alpha _i>0$ be given by (REF ) and (REF ) for $i=1,2,3$ .", "Assume that $\\alpha _1\\le \\alpha _2\\le \\alpha _3$ .", "If $\\hbox{\\rm Ric}\\,g_1$ and $\\hbox{\\rm Ric}\\,g_3$ are positive-definite, then so is $\\hbox{\\rm Ric}\\,g_2$ .", "According to Lemma REF , $\\hbox{\\rm Ric}\\,g_i = r^{(i)}_1\\pi _1^*Q+r^{(i)}_2\\pi _2^*Q$ with $r^{(i)}_1 &=A_1+B_1\\alpha _i^2-\\frac{C_1}{\\alpha _i}, \\\\r^{(i)}_2 &=A_2+\\frac{B_2}{\\alpha _i^2}-{C_2}\\alpha _i,$ where $A_1,A_2,B_1,B_2,C_1,C_2$ are constants independent of $\\alpha _i$ , and where $B_1,B_2,C_1,C_2\\ge 0$ in view of (REF ).", "By assumption, we have $r^{(1)}_1>0$ .", "Since $\\alpha _2\\ge \\alpha _1$ , it follows that $r^{(2)}_1\\ge r^{(1)}_1>0$ .", "Similarly, by assumption, $r^{(3)}_2>0$ .", "Since $\\alpha _2\\le \\alpha _3$ , it follows that $r^{(2)}_2\\ge r^{(3)}_2>0$ .", "Thus, $\\hbox{\\rm Ric}\\,g_2$ is positive-definite.", "In the non-maximal setting we will use also the following result to guarantee positivity of the Ricci tensor.", "Lemma 4.13 Assume that $s = 2$ in (REF ) and that (REF ) holds.", "Suppose $G$ has a connected Lie subgroup $K$ satisfying (REF ).", "Let (REF ) and (REF ) hold.", "If $T\\in \\mathcal {M}$ satisfies (REF ) with $z_1/z_2\\le \\alpha _-$ , then $\\hbox{\\rm Ric}\\,T\\in \\mathcal {M}$ .", "When $z_1/z_2=\\alpha _-$ , the result is obvious.", "Suppose that $z_1/z_2< \\alpha _-$ .", "Since $\\gamma _{11}^2=0$ by (REF ), Lemma REF gives that $\\hbox{\\rm Ric}\\,T = r_1\\pi _1^*Q+r_2\\pi _2^*Q$ with $r_1 &=A_1+B_1(z_1/z_2)^2,\\\\r_2 &=A_2-{C_2}z_1/z_2,$ where $A_1,A_2,B_1,C_2$ are constants independent of $T$ .", "Moreover, $B_1,C_2>0$ by (REF ), and $A_1= \\theta _1/4d_1d_2$ (recall (REF )).", "By Lemma REF , $\\theta _1=4d_1d_2\\zeta _1+d_2\\gamma _{11}^1+4d_2\\gamma _{11}^2\\ge 0$ , so $A_1\\ge 0$ .", "Thus, $r_1\\ge B_1(z_1/z_2)^2>0$ , while $r_2>A_2-{C_2}\\alpha _->0$ since $\\hbox{\\rm Ric}\\,g^{\\alpha _-}= g^{\\alpha _-}$ is positive-definite.", "Thus, $\\hbox{\\rm Ric}\\,T\\in \\mathcal {M}$ .", "As in the formulation of Theorem REF , we always assume $T\\in \\mathcal {M}$ is given by (REF ) in the proof below.", "[Proof of Theorem REF ] (i) Suppose that the assumptions of Theorem REF  (i) hold.", "Let $T$ satisfy $\\alpha _-< z_1/z_2<\\alpha _+$ .", "Applying Lemma REF to the triple $g^{\\alpha _-},T,g^{\\alpha _+}$ shows that $\\hbox{\\rm Ric}\\,T\\in \\cal M$ since $\\hbox{\\rm Ric}\\,g^{\\alpha _-}=g^{\\alpha _-}$ and $\\hbox{\\rm Ric}\\,g^{\\alpha _+}=g^{\\alpha _+}$ are positive-definite.", "Thus, $\\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\in [\\alpha _-,\\alpha _+]\\rbrace \\subset \\mathcal {M}^{(2)}.$ Next, let $T$ satisfy $\\alpha _-<z_1/z_2<\\alpha _+$ , and set $g_0:=\\hbox{\\rm Ric}\\,T\\in \\cal M$ .", "According to Lemma REF  (ii), if we had $\\alpha _0\\le \\alpha _-$ (recall (REF )), we would also have $\\alpha _T\\le \\alpha _-$ .", "It follows that $\\alpha _0>\\alpha _-$ .", "Similarly, according to Lemma REF  (i), if we had $\\alpha _0\\ge \\alpha _+$ , then we would have $\\alpha _g\\ge \\alpha _+$ .", "It follows that $\\alpha _0<\\alpha _+$ .", "Thus, we may apply Lemma REF to the triple $g^{\\alpha _-},g_0,g^{\\alpha _+}$ to conclude that $g_{-1}:=\\hbox{\\rm Ric}\\,g_0\\in \\mathcal {M}$ , i.e., $\\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\in [\\alpha _-,\\alpha _+]\\rbrace \\subset \\mathcal {M}^{(3)}.$ By induction, it follows that $\\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\in [\\alpha _-,\\alpha _+]\\rbrace \\subset \\mathcal {M}^{(\\infty )}.$ Moreover, when $g_1\\in \\mathcal {M}$ is such that $\\alpha _1\\in (\\alpha _-,\\alpha _+)$ , the ancient Ricci iteration (REF ) exists and $\\alpha _{-i}\\in (\\alpha _-,\\alpha _+)$ for all $i\\in {\\mathbb {N}}\\cup \\lbrace 0\\rbrace $ .", "Lemma REF (iv)–(v) implies that the sequence $\\lbrace \\alpha _{-i}\\rbrace _{i=-1}^\\infty $ is monotone.", "The arguments in the proof of Theorem REF (i) now apply verbatim to show that the limit of $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ exists in the smooth topology and is an Einstein metric.", "Finally, suppose that $T\\in \\mathcal {M}^{(\\infty )}$ and $z_1/z_2>\\alpha _+$ .", "Then (REF ) with $g_1=T$ is well-defined.", "The arguments of the previous paragraph show that $\\lim _{i\\rightarrow \\infty }\\alpha _{-i}=\\infty $ , or else $\\alpha _{-i}$ must converge to some $\\alpha _{-\\infty }>\\alpha _+$ such that $g^{\\alpha _{-\\infty }}$ an Einstein metric, a contradiction.", "However, there exists $C>0$ depending only on $d_1,d_2,\\lbrace \\gamma ^{l}_{jk}\\rbrace _{j,k,l=1}^2,b_1,b_2$ such that $r^{(i)}_2<0$ if $\\alpha _{i}>C$ , by Lemma REF (observing that $\\gamma _{22}^1>0$ by Lemma REF ).", "Thus, $r(T)<\\infty $ .", "Similar reasoning shows that $z_1/z_2<\\alpha _-$ implies $r(T)<\\infty $ (otherwise $r^{(i)}_1<0$ for some $i$ ).", "Thus, we obtain (REF ).", "(ii) Suppose that the hypoteses of Theorem REF  (ii) hold.", "First, assume $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is nontrivial and $\\mathcal {E}\\ne \\emptyset $ .", "Suppose $r(g_1)\\ge 2$ , i.e., $g_0:=\\hbox{\\rm Ric}\\,g_1\\in \\mathcal {M}$ .", "As above, Lemma REF (i)–(ii) implies that $\\alpha _0\\in (\\alpha _-,\\alpha _+)$ when $\\alpha _1\\in (\\alpha _-,\\alpha _+)$ .", "Similarly, $\\alpha _0<\\alpha _-$ when $\\alpha _1<\\alpha _-$ .", "If $\\alpha _1<\\alpha _-$ , then Lemma REF  (i-c) yields $\\alpha _1<\\alpha _0$ .", "Applying Lemma REF to the triple $g_1,g_0,g^{\\alpha _-}$ shows that $g_{-1}:=\\hbox{\\rm Ric}\\,g_0\\in \\cal M$ since, by assumption, $\\hbox{\\rm Ric}\\,g^{\\alpha _-}=g^{\\alpha _-}$ and $\\hbox{\\rm Ric}\\,g_1$ are positive-definite.", "Thus, $\\lbrace T\\in \\mathcal {M}^{(2)}\\,:\\, z_1/z_2<\\alpha _-\\rbrace \\subset \\mathcal {M}^{(3)}.$ By induction, it follows that $\\lbrace T\\in \\mathcal {M}^{(2)}\\,:\\, z_1/z_2<\\alpha _-\\rbrace \\subset \\mathcal {M}^{(\\infty )}.$ In addition, the ancient Ricci iteration (REF ) exists, and $\\alpha _{-i}\\in (0,\\alpha _-)$ for all $i\\in {\\mathbb {N}}\\cup \\lbrace 0\\rbrace $ .", "We also have $\\alpha _{-i}<\\alpha _{-i-1}$ .", "If $\\alpha _1\\in (\\alpha _-,\\alpha _+)$ holds, then applying Lemma REF to the triple $g^{\\alpha _-},g_1,g^{\\alpha _+}$ shows that $g_{0}:=\\hbox{\\rm Ric}\\,g_1\\in \\cal M$ since $\\hbox{\\rm Ric}\\,g^{\\alpha _-}=g^{\\alpha _-}$ and $\\hbox{\\rm Ric}\\,g^{\\alpha _-}=g^{\\alpha _-}$ are positive-definite, so $\\lbrace T\\in \\mathcal {M}\\,:\\, \\alpha _-\\le z_1/z_2\\le \\alpha _+\\rbrace \\subset \\mathcal {M}^{(2)}.$ (Thus, there is no need to make the assumption $\\hbox{\\rm Ric}\\,g_1\\in \\mathcal {M}$ .)", "Lemma REF  (i-b) implies that $\\alpha _1>\\alpha _0$ .", "As noted earlier, $\\alpha _0\\in (\\alpha _-,\\alpha _+)$ .", "Therefore, applying Lemma REF to the triple $g^{\\alpha _-},g_0,g^{\\alpha _+}$ shows that $g_{-1}:=\\hbox{\\rm Ric}\\,g_0\\in \\cal M$ , i.e., $\\lbrace T\\in \\mathcal {M}\\,:\\, \\alpha _-\\le z_1/z_2\\le \\alpha _+\\rbrace \\subset \\mathcal {M}^{(3)}.$ By induction, it follows that $\\lbrace T\\in \\mathcal {M}\\,:\\, \\alpha _-\\le z_1/z_2\\le \\alpha _+\\rbrace \\subset \\mathcal {M}^{(\\infty )}.$ Also, the ancient Ricci iteration (REF ) exists and $\\alpha _{-i}\\in (\\alpha _-,\\alpha _+)$ for all $i\\in {\\mathbb {N}}\\cup \\lbrace 0\\rbrace $ .", "Also, $\\alpha _{-i}>\\alpha _{-i-1}$ .", "The arguments in the proof of Theorem REF  (ii) now apply verbatim to show that, in both cases we just considered, i.e., when either $g_1\\in \\lbrace T\\in \\mathcal {M}^{(2)}\\,:\\, z_1/z_2<\\alpha _-\\rbrace $ or $g_1\\in \\lbrace T\\in \\mathcal {M}\\,:\\, \\alpha _-\\le z_1/z_2\\le \\alpha _+\\rbrace $ , the limit of $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ exists in the smooth topology and is an Einstein metric.", "Moreover, similar arguments to those employed in part (i) above show that actually $\\begin{aligned}\\lbrace T\\in \\mathcal {M}^{(2)}\\,:\\, z_1/z_2<\\alpha _-\\rbrace \\cup \\lbrace T\\in \\mathcal {M}\\,:\\, \\alpha _-\\le z_1/z_2\\le \\alpha _+\\rbrace =\\mathcal {M}^{(\\infty )}.\\end{aligned}$ Indeed, $\\gamma _{22}^1>0$ by (REF ), and so the argument employed in part (i) goes through in the same way to show that any $T$ with $z_1/z_2> \\alpha _+$ has finite Ricci index, while if $z_1/z_2<\\alpha _-$ and $T\\notin \\mathcal {M}^{(2)}$ , then, of course, $r(T)=1<\\infty $ .", "Finally, Lemma REF allows us to replace $\\mathcal {M}^{(2)}$ in (REF ) by $\\mathcal {M}$ since it shows that $T\\in \\mathcal {M}^{(2)}$ when $z_1/z_2<\\alpha _-$ .", "This proves (REF ).", "Suppose that $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is non-trivial and $\\mathcal {E}$ is empty.", "Assume $r(g_1)\\ge 2$ .", "Then, Lemma REF  (iii) shows that $\\alpha _0>\\alpha _1$ .", "Arguing as in the previous paragraphs, we show that $r(g_1)<\\infty $ (since $\\gamma _{22}^1>0$ by (REF )).", "Thus, $\\mathcal {M}^{(\\infty )}=\\emptyset $ .", "Next, if $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial, then $\\alpha _-=\\alpha _+>0$ (recall Lemma REF (ii-a)).", "Once again, since $\\gamma _{22}^1>0$ by (REF ), we see that any $T$ with $z_1/z_2> \\alpha _+$ has finite Ricci index.", "Suppose now $T$ is such that $z_1/z_2< \\alpha _+$ .", "Lemma REF implies that $r(T)\\ge 2$ .", "Denote $g_1:=T$ and $g_0:=\\hbox{\\rm Ric}\\,T$ .", "Lemma REF  (ii-d) shows that $\\alpha _0<\\alpha _1$ , and applying Lemma REF again, we obtain $r(T)\\ge 3$ .", "By induction, (REF ) must hold.", "Also, the ancient Ricci iteration (REF ) exists, and $\\alpha _{-i}\\in (0,\\alpha _-)$ for all $i\\in {\\mathbb {N}}\\cup \\lbrace 0\\rbrace $ .", "We have $\\alpha _{-i}>\\alpha _{-i-1}$ and, therefore, $\\lim _{i\\rightarrow \\infty }\\alpha _{-i}=0$ .", "To understand the limit of the sequence $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ , we analyze more closely formulæ (REF ).", "Because $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial, $\\zeta _1=0=\\gamma _{11}^1$ .", "By (REF ), $\\gamma _{11}^2=0$ .", "Lemma REF , together with (REF ) and (REF ) thus imply $A_1&=B_2=C_1=0, \\\\4d_1d_2A_2&=\\theta _2=4d_1d_2\\zeta _2+d_1\\gamma _{22}^2+4d_1\\gamma _{22}^1\\ge 4d_1\\gamma _{22}^1>0.$ Consequently, $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ converges smoothly to the degenerate tensor $A_2\\pi _2^*Q$ .", "This collapsed limit is the pull-back of a metric $g_E$ on $G/K$ under the inclusion map $G/K\\hookrightarrow G/H$ .", "Since $G/K$ is isotropy irreducible (i.e., $s=1$ for it), all $G$ -invariant metrics on it, and hence also $g_E$ , are Einstein.", "Finally, $(G/H,g_{-i})$ must converge in the Gromov–Hausdorff topology to $(G/K,g_E)$ [9].", "This concludes the proof of Theorem REF ." ], [ "Examples", "In this subsection, we assume that the group $G$ is simple and that the inner product $Q$ coincides with $-B$ , the negative of the Killing form.", "Let (REF ) and (REF ) hold for some connected Lie subgroup $K<G$ with Lie algebra $\\mathfrak {k}$ .", "We first consider a situation where $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial.", "Example 4.14 (collapsing) Suppose $ G=SO(2m),\\qquad K=U(m),\\qquad H=SU(m),\\qquad m\\in \\mathbb {N}\\cap [3,\\infty ).$ We identify $K$ and $H$ with subgroups of $G$ in the natural way.", "The reader will find a description of the isotropy representation of $G/H$ in [21].", "In particular, $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is trivial, and we have $\\zeta _1=0,\\quad d_1=1,\\quad d_2=m^2-m.$ Because $\\mathfrak {m}_1$ is 1-dimensional, the constant $\\gamma _{11}^1$ equals 0.", "Formulæ (REF ) and (REF ) imply $\\gamma _{22}^1=d_1=1.$ Since $G/K$ is symmetric, the inclusion $[\\mathfrak {m}_2,\\mathfrak {m}_2]\\subset \\mathfrak {k}$ holds, and $\\gamma _{22}^2$ vanishes.", "The expression $P(x,x)$ (recall (REF ) and (REF )) is now given by the formula $P(x,x)&=d_2\\gamma _{22}^1x^4+2d_1\\gamma _{22}^1x^4+(2d_1d_2-2d_2\\gamma _{22}^1-2d_1d_2x)x^2 \\\\ &=(m^2-m+2)x^4-2(m^2-m)x^3.$ Lemma REF implies (recall (REF )) $\\alpha _-=\\alpha _+=\\frac{2(m^2-m)}{m^2-m+2}\\,.$ Theorem REF  (ii-a) describes the Ricci iteration on $M$ .", "It shows that, given $T\\in \\mathcal {M}$ , there exists a unique sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ satisfying (REF ) for all $i\\in \\mathbb {N}$ and $g_1=cT$ for some $c>0$ .", "This sequence converges to an Einstein metric proportional to $g^{\\alpha _-}=g^{\\alpha _+}$ (recall (REF )).", "Theorem REF  (ii-a) shows that $\\mathcal {M}^{(\\infty )}=\\Big \\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\le \\frac{2(m^2-m)}{m^2-m+2}\\Big \\rbrace ,$ where $z_1,z_2$ are from (REF ).", "By formula (REF ), $\\zeta _2=\\frac{1}{2d_2}(-2\\gamma _{22}^1+d_2)=\\frac{m^2-m-2}{2(m^2-m)}.$ When $g_1\\in \\mathcal {M}^{(\\infty )}$ , the ancient Ricci iteration $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ given by (REF ) converges smoothly to the degenerate metric $\\bigg (\\zeta _2+\\frac{\\gamma _{22}^1}{d_2}\\bigg )\\pi _2^*Q=- \\frac{1}{2}\\pi _2^*B.$ Then, $SO(2m)/SU(m)$ collapses to $SO(2m)/U(m)$ .", "Note that $\\dim SO(6)/SU(3)=7$ .", "For the next example, we introduce some additional notation largely following [36].", "The space $K/H$ may not be effective, and we denote by $K^{\\prime }$ the quotient of $K$ acting effectively on $K/H$ .", "Assume $K^{\\prime }$ is semisimple and $B_{\\mathfrak {k}^{\\prime }}=\\alpha B|_{\\mathfrak {k}^{\\prime }}$ for some $\\alpha >0$ , where $\\mathfrak {k}^{\\prime }$ is the Lie algebra of $K^{\\prime }$ and $B_{\\mathfrak {k}^{\\prime }}$ is the Killing form of $\\mathfrak {k}^{\\prime }$ .", "Consider two Casimir operators $C_{\\mathfrak {m}_1,-B_{\\mathfrak {k}^{\\prime }}|_{\\mathfrak {h}}}:=-\\sum _{j=1}^{q_H}\\hbox{\\rm ad}u_j\\circ \\hbox{\\rm ad}u_j,\\qquad C_{\\mathfrak {m}_2,-B|_{\\mathfrak {k}}}:=-\\sum _{j=1}^{q_K}\\hbox{\\rm ad}v_j\\circ \\hbox{\\rm ad}v_j,$ where $\\lbrace u_j\\rbrace _{j=1}^{q_H}$ is an orthonormal basis of $\\mathfrak {h}$ with respect to $-B_{\\mathfrak {k}^{\\prime }}|_{\\mathfrak {h}}$ , and $\\lbrace v_j\\rbrace _{j=1}^{q_K}$ is an orthonormal basis of $\\mathfrak {k}$ with respect to $-B|_{\\mathfrak {k}}$ .", "These operators have domains $\\mathfrak {m}_1$ and $\\mathfrak {m}_2$ , respectively.", "Let $\\zeta _1^*$ and $\\zeta _2^*$ be the numbers such that $C_{\\mathfrak {m}_1,-B_{\\mathfrak {k}^{\\prime }}|_{\\mathfrak {h}}}=\\zeta _1^*\\, \\hbox{id},\\qquad C_{\\mathfrak {m}_2,-B|_{\\mathfrak {k}}}=\\zeta _2^*\\, \\hbox{id}.$ As shown in [36], $\\zeta _1&=\\alpha \\zeta _1^*, \\\\\\gamma _{22}^1&=d_1(1-\\alpha ), \\\\\\gamma _{11}^1&=d_1-\\gamma _{22}^1-2d_1\\zeta _1=d_1\\alpha (1-2\\zeta _1^*), \\\\\\gamma _{22}^2&=d_2-2d_2\\zeta _2^*.$ Example 4.15 (non-collapsing) Assume $ G&=SO(2m-1),\\qquad K=SO(2m-2), \\qquad H=U(m-1),\\qquad m\\in \\mathbb {N}\\cap [3,\\infty ).$ We identify $K$ and $H$ with subgroups of $G$ in the natural way.", "For descriptions of the isotropy representation of $G/H$ , see [36] and [21].", "Let us first find the relevant constants.", "We have $\\alpha =\\frac{2m-4}{2m-3},\\qquad \\zeta _1^*=\\zeta _2^*=\\frac{1}{2}, \\quad d_1=(m-1)(m-2),\\qquad d_2=2(m-1),$ see [21].", "In light of (REF ), this yields $\\gamma _{22}^1=\\frac{(m-1)(m-2)}{2m-3}, \\qquad \\gamma _{11}^1=\\gamma _{22}^2=0.$ The expression $P(x,x)$ (recall (REF ) and (REF )) is now given by the formula $P(x,x)&=d_2\\gamma _{22}^1x^4+2d_1\\gamma _{22}^1x^4+(2d_1d_2-2d_2\\gamma _{22}^1-2d_1d_2x)x^2\\\\&=(d_2+2d_1)\\gamma _{22}^1x^4-2d_1d_2x^3+2d_2(d_1-\\gamma _{22}^1)x^2 \\\\&=\\frac{2(m-1)^3(m-2)}{2m-3}x^4-4(m-1)^2(m-2)x^3+\\frac{8(m-1)^2(m-2)^2}{2m-3}x^2.$ The equality $P(x,x)=0$ holds for some $x>0$ if and only if $\\frac{m-1}{2m-3}x^2-2x+\\frac{4(m-2)}{2m-3}=0.$ This is a quadratic equation in $x$ with discriminant $\\mathcal {D}=\\frac{4}{(2m-3)^2}.$ Its solutions are the positive numbers $\\alpha _\\pm =\\frac{(2m-3)\\big (1\\pm \\sqrt{\\mathcal {D}/4}\\big )}{m-1}=\\frac{2m-3\\pm 1}{m-1}.$ These numbers satisfy (REF ).", "The representation $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ is nontrivial.", "Indeed, otherwise, the irreducibility of $\\hbox{\\rm Ad}_G(H)|_{\\mathfrak {m}_1}$ would imply that $d_1=1$ , which is impossible for any choice of $m$ .", "Theorem REF  (ii-b) describes the Ricci iteration on $M$ .", "In particular, it shows that a sequence $\\lbrace g_i\\rbrace _{i\\in \\mathbb {N}}$ satisfying (REF ) for all $i\\in \\mathbb {N}$ and $g_1=cT$ for some $c>0$ exists if and only if (recall (REF )) $\\alpha _T\\ge \\alpha _-=\\frac{2m-4}{m-1}\\in [1,2).$ Theorem REF  (ii-b) describes the ancient Ricci iterations on $M$ .", "It demonstrates that $\\mathcal {M}^{(\\infty )}=\\big \\lbrace T\\in \\mathcal {M}\\,:\\, z_1/z_2\\le \\alpha _+=2\\big \\rbrace ,$ where $z_1,z_2$ are given by (REF ).", "Note that $\\dim SO(5)/U(2)=6$ ." ], [ "Relative compactness for maximal isotropy", "This section contains the proof of Theorem REF .", "We no longer impose an upper bound on the number of summands in the isotropy representation.", "Nor do we require the pairwise inequivalence of the summands.", "Remark 5.1 The arguments in this section actually yield a statement that is stronger than the relative compactness of the sequences $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ and $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ satisfying (REF ) and (REF ).", "Namely, consider the set $\\Xi =\\lbrace h\\in \\mathcal {M}\\,:\\,h=\\hbox{\\rm Ric}\\,\\tilde{h}~\\mbox{for some}~\\tilde{h}\\in \\mathcal {M}~\\mbox{and}~S(h)\\ge 0\\rbrace ,$ where $S(h)$ denotes the scalar curvature of $h$ .", "We show that this set is compact in the topology it inherits from $\\mathcal {M}$ .", "Of course, if $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}$ satisfies (REF ) (or, if $\\lbrace g_{-i}\\rbrace _{i=-1}^\\infty $ satisfies (REF )), then $\\lbrace g_i\\rbrace _{i=2}^\\infty $ (or $\\lbrace g_{-i}\\rbrace _{i=0}^\\infty $ ) lies in $\\Xi $ .", "The proof of Theorem REF will be based on the following three lemmas.", "Denote $\\Theta :=\\lbrace X\\in T_\\mu M\\,:\\,Q(X,X)=1\\rbrace ,$ the $Q$ -unit sphere in $T_\\mu M$ .", "Lemma 5.2 Suppose $H$ is maximal in $G$ .", "If $g\\in \\mathcal {M}$ has nonnegative scalar curvature, then $\\frac{\\inf _{X\\in \\Theta }g(X,X)}{\\sup _{X\\in \\Theta }g(X,X)}\\ge a,$ for some constant $a\\in (0,1]$ depending only on $G$ , $H$ and $Q$ .", "Choose the decomposition (REF ) so that (REF ) is satisfied with $x_1\\le \\cdots \\le x_s$ (see [36]).", "Consider the $G$ -invariant metric $\\hat{g}:=\\frac{1}{x_s}g$ on $M$ .", "By definition, $\\inf _{X\\in \\Theta }\\hat{g}(X,X)\\le 1$ and $\\sup _{X\\in \\Theta }\\hat{g}(X,X)=1.$ According to [30], these formulæ imply the inequality $S(\\hat{g})&\\le a_1\\Big (\\frac{x_1}{x_s}\\Big )^{-1}-a_2\\bigg (\\Big (\\frac{x_1}{x_s}\\Big )^{-\\frac{2^{s-1}}{2^{s-1}-1}}+1\\bigg ) \\\\&\\le a_1\\Big (\\frac{x_1}{x_s}\\Big )^{-1}-a_2\\Big (\\frac{x_1}{x_s}\\Big )^{-\\frac{2^{s-1}}{2^{s-1}-1}}.$ Here, $S(\\hat{g})$ denotes the scalar curvature of $\\hat{g}$ , and $a_1,a_2>0$ are constants depending only on $G$ , $H$ and $Q$ .", "As $S(g)\\ge 0$ , also $S(\\hat{g})\\ge 0$ , and $a_2\\Big (\\frac{x_1}{x_s}\\Big )^{-\\frac{2^{s-1}}{2^{s-1}-1}}\\le a_1\\Big (\\frac{x_1}{x_s}\\Big )^{-1},$ so $\\frac{\\inf _{X\\in \\Theta }g(X,X)}{\\sup _{X\\in \\Theta }g(X,X)}=\\frac{x_1}{x_s}\\ge \\Big (\\frac{a_2}{a_1}\\Big )^{2^{s-1}-1},$ as desired.", "Lemma 5.3 Let $H$ be maximal in $G$ .", "Suppose $\\lbrace h_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {T}$ is a sequence of positive-semidefinite (0,2)-tensor fields on $M$ converging smoothly to some (0,2)-tensor field $h\\in \\mathcal {T}$ .", "If $h_i=\\hbox{\\rm Ric}\\,\\tilde{h}_i$ with $\\tilde{h}_i\\in \\mathcal {M}$ for every $i\\in \\mathbb {N}$ , then there exists $\\tilde{h}\\in \\mathcal {M}$ satisfying the equality $h=\\hbox{\\rm Ric}\\,\\tilde{h}$ .", "The inner product $Q$ induces an inner product on $\\mathfrak {m}^*\\otimes \\mathfrak {m}^*$ , which yields, via (REF ), an inner product on $\\mathcal {T}$ .", "We denote by $|\\cdot |_Q$ the corresponding norm on $\\mathcal {T}$ .", "By renormalizing each $\\tilde{h}_i$ (leaving $h_i=\\hbox{\\rm Ric}\\,\\tilde{h}_i$ unchanged), we assume that $|\\tilde{h}_i|_Q=1$ .", "Because the space $\\mathcal {T}$ is finite-dimensional, the unit sphere in it with respect to $|\\cdot |_Q$ is compact.", "Consequently, there exists a subsequence $\\lbrace \\tilde{h}_{i_m}\\rbrace _{m\\in {\\mathbb {N}}}$ converging to some $\\tilde{h}\\in \\mathcal {T}$ .", "We claim that $\\tilde{h}\\in \\mathcal {M}$ , or, equivalently, that $\\inf _{X\\in \\Theta }\\tilde{h}(X,X)>0.$ By Lemma REF , $\\inf _{X\\in \\Theta }\\tilde{h}_{i_m}(X,X)\\ge a\\sup _{X\\in \\Theta }\\tilde{h}_{i_m}(X,X)\\ge a\\frac{|\\tilde{h}_{i_m}|_Q}{\\sqrt{n}}=\\frac{a}{\\sqrt{n}}>0,$ and so the same holds for $\\tilde{h}$ by passing to the limit.", "Since $|\\hbox{\\rm Ric}\\,\\tilde{h}-h|_Q=\\lim _{m\\rightarrow \\infty }|\\hbox{\\rm Ric}\\,\\tilde{h}_{i_m}-h|_Q=\\lim _{m\\rightarrow \\infty }|h_{i_m}-h|_Q=0,$ one has $h=\\hbox{\\rm Ric}\\,\\tilde{h}$ .", "Lemma 5.4 Let $H$ be maximal in $G$ .", "Assume $g$ lies in the set $\\Xi $ given by (REF ).", "Then $a_-\\le \\inf _{X\\in \\Theta }g(X,X)\\le \\sup _{X\\in \\Theta }g(X,X)\\le a_+$ for some constants $a_-,a_+>0$ depending only on $G$ , $H$ and $Q$ .", "First, we prove that $\\inf _{X\\in \\Theta }g(X,X)\\ge a_-.$ Assume, for the sake of contradiction, that there exists a sequence $\\lbrace h_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\Xi $ such that $\\inf _{X\\in \\Theta }h_i(X,X)\\le \\frac{1}{i},\\quad i\\in \\mathbb {N}.$ According to Lemma REF , $\\sup _{X\\in \\Theta }h_i(X,X)\\le \\frac{1}{ai},\\quad i\\in \\mathbb {N}.$ Therefore, the formula $\\lim _{i\\rightarrow \\infty }|h_i|_Q=0$ holds, and the sequence $\\lbrace h_i\\rbrace _{i\\in {\\mathbb {N}}}$ converges to $0\\in \\mathcal {T}$ .", "In view of Lemma REF , $\\hbox{\\rm Ric}\\,\\tilde{h}=0$ for some $\\tilde{h}\\in \\mathcal {M}$ .", "However, by Lemma REF , there are no Ricci-flat metrics in $\\mathcal {M}$ .", "This contradiction proves (REF ).", "Next, we claim that $\\sup _{X\\in \\Theta }g(X,X)\\le a_+.$ Consider the set $\\Omega :=\\big \\lbrace h\\in \\mathcal {M}\\,:\\,\\sup _{X\\in \\Theta }h(X,X)=1,~S(h)\\ge 0\\big \\rbrace .$ Lemma REF shows that $\\Omega \\subset \\hat{\\Omega }:=\\lbrace h\\in \\mathcal {M}\\,:\\,a\\le h(X,X)\\le 1~\\mbox{for all}~X\\in \\Theta \\rbrace ,$ where $a>0$ .", "The topology of $\\mathcal {M}$ induces topologies on $\\Omega $ and $\\hat{\\Omega }$ .", "Fix a $Q$ -orthonormal basis $\\lbrace e_l\\rbrace _{l=1}^n$ in $T_\\mu M$ .", "Clearly, $h\\in \\mathcal {M}$ lies in $\\hat{\\Omega }$ if and only if the eigenvalues of the matrix of $h$ at $\\mu \\in M$ with respect to the basis $\\lbrace e_l\\rbrace _{l=1}^n$ belong to the interval $[a,1]$ .", "This observation implies that $\\hat{\\Omega }$ is compact.", "It is easy to see that $\\Omega $ is closed in $\\hat{\\Omega }$ .", "Therefore, $\\Omega $ must be compact.", "Define a function $f:\\Omega \\rightarrow \\mathbb {R}$ so that $f(h)$ is the largest of the sectional curvatures of $h$ at $\\mu \\in M$ .", "This function is continuous and, therefore, bounded.", "Given $h\\in \\Omega $ , it is easy to see that $bh$ has sectional curvatures less than $\\frac{1}{n-1}$ as long as $b>(n-1)f(h)$ .", "For such $b$ , the tensor field $bh$ cannot be the Ricci curvature of any metric in $\\mathcal {M}$  [20], [23].", "Define $\\tilde{g}:=g/{\\sup _{X\\in \\Theta }g(X,X)}.$ It is clear that $\\tilde{g}$ lies in $\\Omega $ .", "According to the assumptions of the lemma, $b\\tilde{g}$ is the Ricci curvature of a metric in $\\mathcal {M}$ for $b=\\sup _{X\\in \\Theta }g(X,X)$ .", "Consequently, $\\sup _{X\\in \\Theta }g(X,X)\\le (n-1)f(\\tilde{g}).$ Thus, (REF ) holds with $a_+:=(n-1){\\operatorname{max}}_{h\\in \\Omega }f(h).$ [Proof of Theorem REF .]", "(i) The existence of a sequence $\\lbrace g_i\\rbrace _{i\\in {\\mathbb {N}}}\\subset \\mathcal {M}$ satisfying (REF ) for all $i\\in {\\mathbb {N}}$ and $g_1=cT$ for some $c>0$ is a consequence of Theorem REF and the maximality assumption on $H$ .", "Lemma REF implies that any such sequence lies in the set $\\lbrace cT\\rbrace \\cup \\lbrace h\\in \\mathcal {M}\\,:\\,a_-\\le h(X,X)\\le a_+~\\mbox{for all}~X\\in \\Theta \\rbrace .$ This set, with the topology inherited from $\\mathcal {M}$ , is compact.", "Any of its subsets must, therefore, be relatively compact in $\\mathcal {M}$ .", "(ii) The proof is completely analogous to that of (i).", "The following remark may prove useful in the future analysis of the Ricci iteration.", "Given $i\\in \\mathbb {Z}$ , the Riemannian metric $g_i$ from Theorem REF induces an inner product on $\\mathfrak {m}^*\\otimes \\mathfrak {m}^*$ , which yields, via (REF ), an inner product on $\\mathcal {T}$ .", "Denote the corresponding norm on $\\mathcal {T}$ by $|\\cdot |_{g_i}$ .", "Remark 5.5 Lemma REF provides an estimate for the difference between $g_i$ and $g_{i-1}$ .", "Namely, we have $|g_i-g_{i-1}|_Q^2\\le a_+^2\\big (n+|\\hbox{\\rm Ric}\\,g_i|_{g_i}^2-2S(g_i)\\big ),\\qquad i\\in \\mathbb {Z}.$ To see this, fix a $Q$ -orthonormal basis $\\lbrace e_l\\rbrace _{l=1}^n$ of $T_\\mu M$ diagonalizing $g_i$ .", "The left-hand side of (REF ) equals $\\sum _{l,m=1}^n(g_i-g_{i-1})(e_l,e_m)^2=\\sum _{l,m=1}^ng_i(e_l,e_l)g_i(e_m,e_m)\\bigg ((g_i-g_{i-1})\\bigg (\\frac{e_l}{\\sqrt{g_i(e_l,e_l)}},\\frac{e_m}{\\sqrt{g_i(e_m,e_m)}}\\bigg )\\bigg )^2.$ Lemma REF implies that it is bounded from above by $a_+^2\\sum _{l,m=1}^n(g_i-g_{i-1})\\bigg (\\frac{e_l}{\\sqrt{g_i(e_l,e_l)}},\\frac{e_m}{\\sqrt{g_i(e_m,e_m)}}\\bigg )^2.$ The vectors $\\Big \\lbrace \\frac{e_l}{\\sqrt{g_i(e_l,e_l)}}\\Big \\rbrace _{l=1}^n$ form a $g_i$ -orthonormal basis of $T_\\mu M$ .", "Therefore, our last displayed expression equals $a_+^2|g_i-g_{i-1}|_{g_i}^2&= a_+^2\\langle g_i-g_{i-1},g_i-g_{i-1}\\rangle _{g_i}\\\\&=a_+^2\\big (|g_i|_{g_i}^2+|g_{i-1}|_{g_i}^2-2\\langle g_i,g_{i-1}\\rangle _{g_i}\\big )=a_+^2\\big (n+|\\hbox{\\rm Ric}\\,g_i|_{g_i}^2-2S(g_i)\\big ),$ where $\\langle \\cdot ,\\cdot \\rangle _{g_i}$ is the inner product on $\\mathcal {T}$ induced by $g_i$ .", "Formula (REF ) follows immediately." ], [ "Acknowledgments", "Research supported by the Australian Research Council Discovery Early-Career Researcher Award DE150101548 (A.P.)", "and by NSF grants DMS-1206284,1515703 and a Sloan Research Fellowship (Y.A.R.).", "Part of this work took place while Y.A.R.", "visited MSRI (supported by NSF grant DMS-1440140) during the Spring 2016 semester.", "The University of Queensland [email protected] University of Maryland [email protected]" ] ]
1606.05064
[ [ "Non-equilibrium quantum transport coefficients and the transient\n dynamics of full counting statistics in the strong coupling and non-Markovian\n regimes" ], [ "Abstract Non-equilibrium transport properties of quantum systems have recently become experimentally accessible in a number of platforms in so-called full-counting experiments that measure transient and steady state non-equilibrium transport dynamics.", "We show that the effect of the measurement back-action can be exploited to gain access to relevant transport coefficients.", "This relationship is general, but becomes most conspicuous in the transient dynamics of open quantum systems understrong coupling to non-Markovian environments.", "In order to explore this regime, a new simulation method for the generation of full counting statistics of non-Markovian, strong-coupling transport settings has been developed that is expressed in terms of a hierarchy of equations of motion.", "With this tool we gain access to the relevant regime and instantiate our proposal with the study of energetic conductance between two baths connected via a few level system." ], [ "Introduction", "The experimental ability to probe the statistical properties of quantum transport in mesoscopic systems, such as electrons in nanojunctions [1] or cold atoms [2], has provided new insights into the non-equilibrium behavior of quantum systems.", "Theoretical tools for their description have been developed in the form of so-called full counting statistics (FCS) methods [3], [4], which constitute a generalization of the theory of moment and cumulant generating functions and is conceptually based on a two-point measurement scheme [4], so that state projection of the bath after the first measurement is automatically incorporated.", "The effects of this state collapse quickly fade in Markovian baths, allowing for the derivation of steady-state fluctuation theorems that relate several transport coefficients governing the dynamics.", "In the linear response regime, these are none other than the celebrated fluctuation-dissipation theorems, such as the Kubo formula, or other properties such as the Onsager-Casimir relations.", "Far from equilibrium, these theorems can be generalized for non-linear coefficients [5].", "Femtosecond laser pulses and other ultrafast control techniques provide access to the statistical response of quantum systems in the transient regime.", "The Jarzynski equality [6], the Crooks theorem [7], [8] and related relationships confirm that steady-state fluctuation-dissipation theorems need not automatically carry over to the transient regime.", "An interpretation of these effects as the consequence of measurement back-action in the bath supports the intuition that the failure of steady-state relations must be particularly evident in situations where the system-environment coupling is very large or the environmental evolution is so slow that non-Markovian effects become relevant.", "In this situation, only an explicit computation of the full dynamics is so far known to provide the correct insight into transport properties.", "The behaviour of non-Markovian, strong coupling transport settings is captured in the Levitov-Lesovik formula [9] in the case of non-interacting particles.", "FCS for non-Markovian settings was studied from a general perspective in [10], [11], whereas specific treatments include harmonic chains [12], spin-boson or fermionic models in the perturbative regime [13], [14], [15], [16], [17], [18], [19], [20] and general bosonic or fermionic systems for the first and/or second moments of the dynamics [21], [22], [23], [24], [25].", "Bath statistics of open quantum sysems has provided access to universal oscillations in high order cumulants [26] and the Kondo signature in the spin-boson model [27] and fermionic models [28].", "FCS measurement strategies are attracting renewed attention [29] together with optimized cumulant evaluation methods [30].", "Additionally, discussion of classical and quantum initial correlation effects in shot noise has been addressed in [31] and thermodynamic consistency of FCS simulation methods has been studied in [32].", "Here we provide an alternative approach, which consists in quantitatively computing the deviation of the Saito-Utsumi coefficient relations [5] when applied in transient situations.", "We show that this deviation is directly associated to a physical picture where no part of the system-environment compound is initially subject to measurement, and is expressed in terms of a natural symmetry that affects the cumulant generating function.", "In order to investigate these effects, we have developed a simulation method that incorporates FCS into the formalism of hierarchy of equations of motion (HEOM) [33], an established method for the simulation of general, multilevel open quantum systems (OQS) that is non-perturbative in the coupling strength and which faithfully represents non-Markovian effects of the environment.", "With this method we gain access to cumulants of any desired order of environmental energetic and particle observables and arbitrary time-dependence of the Hamiltonian may be treated.", "As a first example, we consider an open quantum system that couples to a bosonic heat bath, although this procedure is equally valid for fermionic baths.", "This method generalizes previous attempts that involved first moments [34], [35] and to our knowledge it is the first time that high order cumulants are simulated with this formalism.", "We first introduce the FCS formalism and discuss a generalization thereof that allows for the isolation of the transient measurement back-action.", "A relationship between the back-action effect and transport coefficients is presented that holds both close to equilibrium and far from it.", "We further introduce the simulation method developed for the investigation of these transient effects, which is based on the HEOM formalism.", "We finally present simulation results on a bosonic transport setting and show that the energetic conductance can be reliably accessed through investigation of measurement back-action effects." ], [ "Steady state coefficient relations", "The formalism for the analysis of full counting experiments constitutes a well-established theoretical framework [4] involving a two point measurement prescription: the value of a specific observable of interest $\\hat{\\operatorname{O}}$ is measured at an initial time $t=0$ , where a result $o\\left(0\\right)$ is obtained, and at a final time $t>0$ , with an outcome $o\\left(t\\right)$ [see Fig.", "(REF .1)].", "Repetition of the experiment generates statistics of the measurement difference $\\Delta o\\left(t\\right)=o\\left(t\\right)-o\\left(0\\right)$ , which can be treated as a stochastic variable.", "Although more general cases can be considered, let us regard for simplicity a single measured operator and an initial state of the total system $\\pi (A)=\\rho \\otimes \\frac{\\exp (-A\\hat{\\operatorname{O}})}{\\operatorname{\\text{Tr}}\\left\\lbrace \\exp (-A\\hat{\\operatorname{O}})\\right\\rbrace }$ , where $\\rho $ is an arbitrary state of the subsystem where the measurement has no effect and $A$ is a thermodynamic constraint fixing the initial expected value of $\\hat{\\operatorname{O}}$ .", "An instance of such a setting is a system in contact with several baths, where $\\hat{\\operatorname{O}}$ is the Hamiltonian of one of the baths, $A$ is its initial inverse temperature and $\\rho $ is an arbitrary state of the system and all non-measured baths.", "Under these conditions, the cumulant generating function (CGF) takes the form [4] $G(\\chi ,A,t)=\\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}(t)}e^{-i\\chi \\hat{\\operatorname{O}}(0)}\\right\\rangle _{A},$ with $\\left\\langle \\bullet \\right\\rangle _{A}\\equiv \\operatorname{\\text{Tr}}\\left\\lbrace \\bullet \\pi (A)\\right\\rbrace $ .", "Its Taylor-expansion coefficients in the counting field $\\chi $ correspond to the cumulants of the measurement difference $\\Delta o\\left(t\\right)$ .", "In the steady state limit, the cumulant generating function for the currents $\\mathcal {F} (\\chi ,A)\\equiv \\lim _{\\tau \\rightarrow \\infty } \\frac{1}{\\tau } G(\\chi ,A,\\tau )$ often fulfills the symmetry $\\mathcal {F} (\\chi ,A)=\\mathcal {F} (-\\chi +iA,A),$ also known as fluctuation theorem, so that transport coefficients $L_m^n (A)\\equiv \\left.\\frac{\\partial ^{m+n}}{\\partial ( i \\chi )^n\\partial A ^m}\\mathcal {F}(\\chi ,A)\\right|_{\\chi =0}$ obey the Saito-Utsumi relations [5] $L_m^n(A)=\\sum _{j=0}^{m}\\left(\\begin{array}{c}m\\\\j\\end{array}\\right) (-1)^{n+j} L_{m-j}^{n+j}(A).$ Relations such as the Kubo formula and the Onsager-Casimir relations can be recast as specific cases of this equation, in particular when several counting fields $\\chi _k$ are involved and the associated thermodynamic constraints are close to an equilibrium state $A_k\\simeq A$ .", "Nevertheless, Eq.", "(REF ) is generally not valid in the transient regime.", "Figure: 1) Depiction of measurement schemes associated to Eq.", "() (1.a) and Eq.", "() (1.b) respectively.In both measurement schemes a separable initial state of the form π(A)\\pi (A) is assumed.2) Steady state energetic conductance as a function of the coupling strengthλ\\lambda and for different spectral density cutoff frequencies ω c \\omega _{c}.", "(2.a) Lines show results computed via numerical derivationfor (T R -T L )/T R =0.01(T_{R}-T_{L})/T_{R}=0.01.", "Dots are computed following Eq.", "()with no temperature bias.", "Other parameters are T L =T R =10ω 0 T_{L}=T_{R}=10\\omega _{0},J=0J=0.", "All values have been numerically converged by varying thehierarchical depth of the simulation.", "(2.b) Closeup of the weak couplinglimit, where red solid lines reproduce the analytical prediction ofthe weak coupling theory." ], [ "Relations for transient transport coefficients", "With the aim of quantifying the error of the fluctuation theorem Eq.", "(REF ) in the transient dynamics, one may define the difference between two CGFs $G_{S}(\\chi ,A,t)\\equiv \\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}(t)}\\right\\rangle _{A}-\\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}(0)}\\right\\rangle _{A},$ each associated to a single (S) measurement of the operator $\\hat{\\operatorname{O}}$ given an initial state of the form $\\pi (A)$ [see Fig.", "(REF .1)].", "Its $\\chi -$ derivatives provide the difference of the cumulants of the measurement outcome between two times $t$ and $t=0$ .", "Note the subtle difference in the statistical interpretation of functions Eq.", "(REF ) and Eq.", "(REF ).", "As shown in Appendix , both functions are related by the expression $G_S(\\chi ,A,t)=G(\\chi ,A-i\\chi ,t).$ Although this equation relates two physically distinct situations (two different measurement schemes), it bears a resemblance with Eq.", "(REF ) that can be exploited to obtain relations similar to Eq.", "(REF ) for the objects ${J_{(S)}}_m^n(A,t)\\equiv \\left.\\frac{\\partial ^{m+n}}{\\partial ( i \\chi )^n\\partial A ^m}G_{(S)}(\\chi ,A,t)\\right|_{\\chi =0}$ ${J_{S}}_m^n(A,t)=\\sum _{j=0}^{n}\\left(\\begin{array}{c}n\\\\j\\end{array}\\right) (-1)^{j} J_{m+j}^{n-j}(A,t).$ Note that $L_m^n(A)=\\lim _{\\tau \\rightarrow \\infty } \\frac{1}{\\tau } J_m^n(A,\\tau )$ , and this can be used to recover Eq.", "(REF ) in the steady state.", "This is a powerful relationship that establishes analogies of fluctuation-dissipation theorems on the transient dynamics by quantifying the deviation from the steady state expressions in terms of the values ${J_{S}}_m^n(A,t)$ , which correspond to measurable quantities in a well defined physical setting.", "The amplitude of these deviations grows with the strength of the coupling between the measured partition and the rest of the system, and also with the duration of the transient dynamics, which is associated to non-Markovian effects in the language of open quantum systems.", "We explore below these effects at hand of specific examples.", "Figure: (a) Steady state energetic transport coefficient lim t→∞ κ R t;T R ,T L \\lim _{t\\rightarrow \\infty }\\kappa _{R}\\left(t;T_{R},T_{L}\\right)as a function of the temperature of the left bath T L T_{L} and fordifferent spectral density cutoff frequencies ω c \\omega _{c}.", "Otherparameters are T R =10ω 0 T_{R}=10\\omega _{0}, J=ω 0 J=\\omega _{0}, λ=ω 0 \\lambda =\\omega _{0}.Lines show results computed vianumerical derivation of steady-state currents.", "Dots are computed followingEq.().", "(b) Transient dynamics of the energetictransport coefficient as a function of time and for different reorganizationenergies λ\\lambda .", "Other parameters are T R =T L =10ω 0 T_{R}=T_{L}=10\\omega _{0},J=ω 0 J=\\omega _{0}, ω c =3ω 0 \\omega _{c}=3\\omega _{0}.", "Results shown in blueare computed via numerical derivationof transient currents.", "Red lines are computed following Eq.", "()." ], [ "Applications", "In order to instantiate the relations in Eq.", "(REF ), let us consider an open quantum system of Hamiltonian $\\hat{\\operatorname{H}}_{\\mathrm {S}}$ interacting via operators $\\hat{\\operatorname{V}}_{\\nu }$ with one or several baths, so that the total Hamiltonian has the form $\\hat{\\operatorname{H}} & =\\hat{\\operatorname{H}}_{\\mathrm {S}}+\\sum _{\\nu }\\hat{\\operatorname{V}}_{\\nu }\\otimes \\hat{\\operatorname{B}}_{\\nu }+\\hat{\\operatorname{H}}_{\\mathrm {B}},\\\\\\hat{\\operatorname{H}}_{\\mathrm {B}} & =\\sum _{\\nu }\\hat{\\operatorname{H}}_{\\nu }=\\sum _{\\nu ,k}\\omega _{\\nu ,k}\\hat{\\operatorname{a}}_{\\nu k}^{\\dagger }\\hat{\\operatorname{a}}_{\\nu k}.$ where $\\hat{\\operatorname{B}}_{\\nu }$ is an arbitrary operator of bath $\\nu $ and $\\hat{\\operatorname{a}}_{\\nu ,k}$ and $\\hat{\\operatorname{a}}_{\\nu ,k}^{\\dagger }$ are the usual bosonic or fermionic annihilation and creation operators for the mode $k$ of frequency $\\omega _{\\nu ,k}$ .", "This encompasses a broad range of dissipative and transport settings.", "The energy of one of the baths $\\hat{\\operatorname{H}}_{\\nu }$ provides relevant information about the heat flows in the system.", "The thermodynamic constraint $\\beta _{\\nu }$ associated to it is the inverse temperature of the bath, so that Eq.", "(REF ) for $m=0$ takes the form ${J_{S}}_0^n(\\beta _\\nu ,t)=\\sum _{j=0}^{n}\\left(\\begin{array}{c}n\\\\j\\end{array}\\right) (-1)^{j} J_{n-j}^{j}(\\beta _\\nu ,t),$ which relates the cumulants of the bath energies with high order instantaneous energetic conductances.", "It has the same form as the case $n=0$ of Eq.", "(REF ) except for the deviation term ${J_{S}}_0^n$ and an extra minus sign for odd $n$ .", "For a two-bath setting ($\\nu \\in \\left\\lbrace R,L\\right\\rbrace $ ), we define a first order energetic transport coefficient as $\\kappa _{R}\\left(t;T_{R},T_{L}\\right)\\equiv \\beta ^{2}_R \\frac{\\partial ^{}}{\\partial {t}^{}} J_1^1(\\beta _R,t),$Please note the special choice of sign, such that subsequent numerical results remain positive.", "whose steady state, equilibrium limit energetic conductance $\\kappa \\equiv \\lim _{t\\rightarrow \\infty }\\lim _{T_R\\rightarrow T_L}\\kappa _{R}\\left(t;T_R,T_L\\right)$ can be understood in turn as a generalization of the concept of thermal conductance for the case of strong-coupling and non-Markovian regimes.", "Equation (REF ) for $n=2$ establishes a relationship between the first order energetic transport coefficient and the second order cumulants of the form $\\kappa _{R}\\left(t;T_{R},T_{L}\\right)=\\frac{\\beta ^2_R}{2} \\frac{\\partial ^{}}{\\partial {t}^{}}\\left[J_0^2\\left(\\beta _R,t\\right)-{J_{S}}_0^2\\left(\\beta _R,t\\right)\\right],$ which is analogous to the Kubo formula except for the deviation term ${J_{S}}_0^2$ .", "As in the case of the fluctuation-dissipation theorem, it puts forward the possibility to derive a conductance value from the difference of the fluctuations as obtained from a two measurement scheme and a single measurement scheme.", "This is a way to circumvent the necessity of a numerical derivative and avoids the accumulation of numerical error implicit in the choice of any small but finite temperature bias." ], [ "Numerical method", "In order to address the strong-coupling and non-Markovian regimes, we developed a hierarchy of equations of motion for the simulation of full counting statistics and high-order moments of relevant bath observables.", "The technique enables statistical analysis of any bath observable that commutes with its free Hamiltonian term, and we provide examples for the case of the energy of the bath $\\hat{\\operatorname{H}}_{\\mathrm {B}}$ .", "In particular, any linear combination of the operators $\\hat{\\operatorname{a}}_{\\nu k}^{\\dagger }\\hat{\\operatorname{a}}_{\\nu k}$ such as the particle number $\\hat{\\operatorname{N}}$ can be simulated with this approach.", "Furthermore, it is flexible enough to generate both the moments corresponding to the two-measurement picture Eq.", "(REF ) and the single measurement picture Eq.", "(REF ), and we will use it in both modes and apply the relationship Eq.", "(REF ) to obtain non-equilibrium transport coefficients.", "A central element in the derivation of the method is the counting-field-resolved bath correlation function $C_{\\nu }^{jk}(\\chi ,t)=(-)^{j+k}\\left\\langle \\tilde{\\hat{\\operatorname{B}}}_{\\nu }^{j}\\left[(-)^{j}\\frac{\\chi }{2}\\right](t)\\tilde{\\hat{\\operatorname{B}}}_{\\nu }^{k}\\left[(-)^{k}\\frac{\\chi }{2}\\right](0)\\right\\rangle ,$ where superindices $j$ and $k$ take two values 0 or 1 and indicate the side an operator acts from: $\\hat{\\operatorname{A}}^{0}\\rho \\equiv \\hat{\\operatorname{A}}\\rho $ or $\\hat{\\operatorname{A}}^{1}\\rho \\equiv \\rho \\hat{\\operatorname{A}}$ .", "The tilde indicates the interaction picture with respect to $\\hat{\\operatorname{H}}_{\\nu }$ , the dressing of an operator by the counting operator $\\hat{\\operatorname{O}}$ is denoted by $\\hat{\\operatorname{A}}[\\chi ](t)=e^{i\\chi \\hat{\\operatorname{O}}}\\hat{\\operatorname{A}}(t)e^{-i\\chi \\hat{\\operatorname{O}}}$ and the average is over the initial state of the bath.", "The hierarchy is based on a decomposition of the correlation function $C_{\\nu }^{jk}(\\chi ,t)=\\sum _{r}c_{\\nu r}^{jk}(\\chi )\\phi _{r}(t)$ by means of a set of functions $\\lbrace \\phi _{r}(t)\\rbrace $ whose derivatives are well defined within the set by $\\frac{\\mathrm {d}^{}}{\\mathrm {d}{t}^{}}\\phi _{r}(t)=\\sum _{r}\\eta _{rs}\\phi _{s}(t)$ [36].", "The set size determines the growth of the simulation requirements, traditionally making this method indicated for not too low temperature regimes.", "Nevertheless, a judicious choice of the basis can circumvent this limitation [36].", "Figure: Transient dynamics of the time derivatives of the first 5 cumulantsof the bath energy H ^ R \\hat{\\operatorname{H}}_{R} in the single measurement scheme (S)for different values of the cutoff frequency ω c \\omega _{c} and the reorganization energy λ\\lambda : λ=0.05ω c \\lambda =0.05\\omega _c for the dotted lines, λ=0.1ω c \\lambda =0.1\\omega _c for the dashed lines and λ=0.2ω c \\lambda =0.2\\omega _c for the solid lines.", "Other parameter values are: T L =T R =10ω 0 T_{L}=T_{R}=10\\omega _{0},J=0J=0.We refer the reader to the Appendices (sections B & C) for the technical aspects of the method.", "It involves the propagation of so-called auxiliary fields $\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace n \\right\\rbrace }\\left(t\\right)$ , labeled by a rank-three tensor $\\left\\lbrace n \\right\\rbrace $ and a vector $\\left\\lbrace m \\right\\rbrace $ of non-negative integer entries whose elements respectively sum up to the hierarchic level $n$ and represent a partition of the moment order $m$ .", "The zeroth hierarchic level contains the auxiliary field $\\sigma _{\\left\\lbrace 0 \\right\\rbrace }^{\\left\\lbrace 0 \\right\\rbrace }\\left(t\\right)$ , corresponding to the system density matrix $\\rho \\left(t\\right)$ , and $\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace 0 \\right\\rbrace }\\left(t\\right)$ , a linear combination of which (defined in the Appendix, Eq.REF ) can be traced to obtain the $m$ th moment.", "The auxiliary fields satisfy the equation $\\frac{\\mathrm {d}^{}}{\\mathrm {d}{t}^{}}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace n \\right\\rbrace } & (t)=-i\\hat{\\operatorname{H}}_{S}^{\\times }\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace n \\right\\rbrace }(t)\\nonumber \\\\+ & \\sum _{\\nu ,r;k=0,1}\\Bigg (\\hat{\\operatorname{\\bar{V}}}_{\\nu r}^{k}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}+1,\\dots \\right\\rbrace }(t)\\nonumber \\\\+ & \\sum _{s}n_{\\nu r}^{k}\\eta _{rs}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}-1,\\dots ,n_{\\nu s}^{k}+1,\\dots \\right\\rbrace }(t)\\nonumber \\\\+ & n_{\\nu r}^{k}\\phi _{r}(0)\\hat{\\operatorname{V}}_{\\nu }^{k}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}-1,\\dots \\right\\rbrace }(t)\\nonumber \\\\+ & \\sum _{q}m_{q}\\hat{\\operatorname{\\bar{V}}}_{\\nu rq}^{k}\\sigma _{\\left\\lbrace \\dots ,m_{q}-1,\\dots \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}+1,\\dots \\right\\rbrace }(t)\\Bigg ),$ where $\\hat{\\operatorname{\\bar{V}}}_{\\nu r}^{k}\\equiv \\sum _{j=0,1}c_{\\nu r}^{jk}(0)\\hat{\\operatorname{V}}_{\\nu }^{j}$ , $\\hat{\\operatorname{\\bar{V}}}_{\\nu rq}^{k}\\equiv \\sum _{j=0,1}c_{\\nu rq}^{jk}\\hat{\\operatorname{V}}_{\\nu }^{j}$ and $c_{\\nu rq}^{jk}\\equiv \\left.\\frac{\\mathrm {d}^{q}}{\\mathrm {d}{(i\\chi )}^{q}}c_{\\nu r}^{jk}(\\chi )\\right|_{\\chi =0}$ .", "The structure is identical to the usual hierarchy [33] but for the last term, which connects it to the hierarchy associated to the previous moment, which is compatible with results in the Markovian, weak-coupling case [30].", "A cutoff at a maximum hierarchic level $n_{max}$ is usually justified in terms of numerical convergence and is roughly proportional to the system-bath coupling strength.", "In order to demonstrate the numerical validity of Eq.", "(REF ), let us consider a two level system with Hamiltonian $\\hat{\\operatorname{H}}_{\\mathrm {S}}=\\omega _{0}\\sigma _{x}+J\\sigma _{z}$ , where $\\sigma _{i}$ with $i\\in \\left\\lbrace x,y,z\\right\\rbrace $ are the Pauli matrices, coupled to two bosonic baths $\\nu \\in \\lbrace R,L\\rbrace $ via $\\hat{\\operatorname{V}}_{\\nu }=\\sigma _z$ and $\\hat{\\operatorname{B}}_{\\nu }=\\sum _{k}\\gamma _{\\nu k}\\left(\\hat{\\operatorname{a}}_{\\nu k}+\\hat{\\operatorname{a}}_{\\nu k}^{\\dagger }\\right)$ , characterized by a spectral density $J_{\\nu }(\\omega )=\\sum _{k}\\gamma ^{2}_{\\nu k}\\delta (\\omega -\\omega _{\\nu k})$ and an inverse temperature $\\beta _{\\nu }$ .", "The following results are derived for $\\rho (0)=\\frac{\\sigma _x+1}{2}$ and the choice of an Ohmic spectral density with exponential cutoff $J(\\omega )=\\frac{\\lambda }{\\omega _{c}}\\omega e^{-\\frac{\\omega }{\\omega _{c}}}$ , where $\\lambda =\\int _{0}^{\\infty }\\frac{J\\left(\\omega \\right)}{\\omega }d\\omega $ is the reorganization energy and $\\omega _{c}$ is the scale of the cutoff.", "Energetic conductance values are shown in Fig.", "(REF .b).", "Following the discussion presented above, two methods are used for their computation.", "On the one hand, simulations under small temperature bias $(T_{R}-T_{L})/T_{R}=0.01$ are run in order to obtain steady state energy currents, which are related to the energetic conductance through the definition of $J_1^1(\\beta _R,t)$ .", "On the other hand, second order moments computed in the two-measurement and the single measurement schemes for equilibrium conditions ($T_{L}=T_{R}$ ) are further used to derive conductance values as per Eq.", "(REF ).", "Results show excellent agreement between both pictures and also approach predictions from the weak coupling theory and the non-interacting blip approximation (NIBA) [13] in their respective regimes of validity.", "Whereas a linear increase of conductance with the coupling strength $\\lambda $ is expected in the weak coupling limit, a turnover is reproduced for higher coupling strengths.", "The relationship Eq.", "(REF ) is valid also in the transient regime and far from equilibrium as shown in Fig.", "(REF ).", "In this case, the steady state energetic conductance of a biased two level system ($J\\ne 0$ ) is studied in situations where $T_{R}-T_{L}\\simeq T_{L}$ .", "The transient dynamics are also accurately reproduced by Eq.", "(REF ), where the oscillating effect is introduced by the tunneling, which acts as an effective driving.", "Finally, the transient deviation of high order transport coefficient relations Eq.", "(REF ) is shown in Fig.", "(REF ) for a range of coupling constants $\\lambda $ and spectral density cutoffs $\\omega _{c}$ .", "It constitutes a quantitative confirmation that the failure of steady-state fluctuation-dissipation theorems is proportional to the coupling strength to the bath and lasts longer for higher degrees of non-Markovianity.", "Additionally, the sign of the deviation changes for odd orders, as predicted from the comparison of Eq.", "(REF ) and Eq.", "(REF )." ], [ "Conclusions", "When considering the transient transport dynamics of non-Markovian systems, it is possible to quantify deviations from fluctuation-dissipation theorems and the nonlinear Saito-Utsumi relations, which are only valid in the steady state.", "These deviations have a physical interpretation and are associated to equilibration dynamics of the same system under a different measurement scheme.", "We demonstrate this relation by developing a tool that allows for the simulation of the full counting statistics of a broad range of bath observables under dissipative and non-equilibrium settings, which is a generalization of the celebrated hierarchy of equations of motion for non-Markovian and strong-coupling settings.", "By accessing high order cumulants of the bath energy, it is possible to derive energetic conductances and higher order derivatives thereof while, at the same time, avoiding finite bias simulations.", "This approach is immediately applicable to the study of observables such as the particle number, environments of fermionic nature, more complex and higher dimensional systems, and time-dependent driving.", "We are grateful to Dr. Gernot Schaller and Dr. Philipp Strasberg for helpful discussions.", "The authors gratefully acknowledge financial support from the DFG Grants No.", "BR 1528/7, No.", "BR 1528/8, No.", "BR 1528/9, No.", "SFB 910 and No.", "GRK 1558." ], [ "Relevant limits of the cumulant generating function", "Let us reproduce the definition of the cumulant generating function $G(\\chi ,A,t)&\\equiv \\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}}(t)e^{-i\\chi \\hat{\\operatorname{O}}}(0)\\right\\rangle _{A}\\nonumber \\\\&=\\ln \\operatorname{\\text{Tr}}\\left\\lbrace e^{i\\chi \\hat{\\operatorname{O}}}(t)e^{-i\\chi \\hat{\\operatorname{O}}}(0)\\pi \\left(A\\right)\\right\\rbrace ,$ with $\\pi \\left(A\\right)=\\rho \\otimes \\frac{e^{-A\\hat{\\operatorname{O}}}}{\\operatorname{\\text{Tr}}\\left\\lbrace e^{-A\\hat{\\operatorname{O}}}\\right\\rbrace }$ a separable initial state of the total system, where $\\rho $ is an arbitrary state of the subsystem where the measurement has no effect and $A$ is a thermodynamic constraint fixing the initial expected value of $\\hat{\\operatorname{O}}$ .", "Evaluating the second variable of the cumulant generating function at $A-i\\chi $ one obtains $G(\\chi ,&A-i\\chi ,t)=\\nonumber \\\\&=\\ln \\operatorname{\\text{Tr}}\\left\\lbrace e^{i\\chi \\hat{\\operatorname{O}}}(t)\\pi \\left(A\\right)\\right\\rbrace -\\ln \\frac{\\operatorname{\\text{Tr}}\\left\\lbrace e^{(-A+i\\chi )\\hat{\\operatorname{O}}}\\right\\rbrace }{\\operatorname{\\text{Tr}}\\left\\lbrace e^{-A\\hat{\\operatorname{O}}}\\right\\rbrace }\\nonumber \\\\&=\\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}}(t)\\right\\rangle _{A}-\\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}}(0)\\right\\rangle _{A}.$ One may interpret this function as the difference between two single-measurement cumulant generating functions: one where the measurement takes place at time $t$ and another one where the measurement takes place at time $t=0$ .", "We denote it by $G_S(\\chi ,A,t)\\equiv \\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}}(t)\\right\\rangle _{A}-\\ln \\left\\langle e^{i\\chi \\hat{\\operatorname{O}}}(0)\\right\\rangle _{A}.$" ], [ "Counting-field-resolved hierarchy of equations of motion", "As indicated in the main text Eq.", "(7-8), let us consider a general total Hamiltonian consisting of a system, interaction and bath parts $\\hat{\\operatorname{H}} & =\\hat{\\operatorname{H}}_{\\mathrm {S}}+\\sum _{\\nu }\\hat{\\operatorname{V}}_{\\nu }\\otimes \\hat{\\operatorname{B}}_{\\nu }+\\hat{\\operatorname{H}}_{\\mathrm {B}},\\\\\\hat{\\operatorname{H}}_{\\mathrm {B}} & =\\sum _{\\nu }\\hat{\\operatorname{H}}_{\\nu }=\\sum _{\\nu ,k}\\omega _{\\nu ,k}\\hat{\\operatorname{a}}_{\\nu k}^{\\dagger }\\hat{\\operatorname{a}}_{\\nu k}.$ The index $\\nu $ labels the baths, $\\hat{\\operatorname{a}}_{\\nu ,k}$ and $\\hat{\\operatorname{a}}_{\\nu ,k}^{\\dagger }$ are the usual bosonic or fermionic annihilation and creation operators for the mode $k$ in bath $\\nu $ .", "The goal is to derive the full counting cumulant generating function Eq.", "(REF ) of an observable $\\hat{\\operatorname{O}}$ of one of the baths which commutes with its free Hamiltonian $\\hat{\\operatorname{H}}_{\\nu }$ and with the initial state of the system and baths $\\pi \\left(0\\right)=\\rho (0)\\bigotimes _{\\nu }{\\frac{e^{-\\beta _\\nu \\hat{\\operatorname{H}}_\\nu }}{\\operatorname{\\text{Tr}}\\left\\lbrace e^{-\\beta _\\nu \\hat{\\operatorname{H}}_\\nu }\\right\\rbrace }}$ .", "For simplicity, we will omit the explicit dependence on the thermodynamic constraint in the derivation, so that the two-measurement cumulant generating function takes the form $G(\\chi ,t)=\\ln \\operatorname{\\text{Tr}}\\left\\lbrace e^{-i\\hat{\\operatorname{H}}\\left[\\frac{\\chi }{2}\\right]t}\\pi (0)e^{i\\hat{\\operatorname{H}}\\left[-\\frac{\\chi }{2}\\right]t}\\right\\rbrace ,$ where $\\hat{\\operatorname{A}}[\\chi ](t)=e^{i\\chi \\hat{\\operatorname{O}}}\\hat{\\operatorname{A}}(t)e^{-i\\chi \\hat{\\operatorname{O}}}$ .", "This problem can be formulated in terms of the solution to a hierarchy of equations of motion for the counting field resolved density matrix $\\rho (\\chi ,t)=\\operatorname{\\text{Tr}_{B}}\\left\\lbrace e^{-i\\hat{\\operatorname{H}}\\left[\\frac{\\chi }{2}\\right]t}\\pi (0)e^{i\\hat{\\operatorname{H}}\\left[-\\frac{\\chi }{2}\\right]t}\\right\\rbrace ,$ and the equation $G(\\chi ,t)=\\ln \\left[\\operatorname{\\text{Tr}}\\left\\lbrace \\rho (\\chi ,t)\\right\\rbrace \\right]$ directly relates both quantities.", "For the sake of clarity, we will derive the equations of motion for Eq.", "(REF ).", "The case associated to Eq.", "(REF ) immediately follows under modification of the initial state of the bath with the imaginary inverse temperature $\\beta _\\nu -i\\chi $ .", "The matrix $\\rho (\\chi ,t)$ satisfies the differential equation $\\frac{d}{dt}\\rho (\\chi ,t)=-i\\operatorname{\\text{Tr}_{B}}\\left\\lbrace \\hat{\\operatorname{H}}\\left[\\frac{\\chi }{2}\\right]\\pi (\\chi ,t)-\\pi (\\chi ,t)\\hat{\\operatorname{H}}\\left[-\\frac{\\chi }{2}\\right]\\right\\rbrace ,$ with $\\pi (\\chi ,t)\\equiv e^{-i\\hat{\\operatorname{H}}\\left[\\frac{\\chi }{2}\\right]t}\\pi (0)e^{i\\hat{\\operatorname{H}}\\left[-\\frac{\\chi }{2}\\right]t}$ .", "The formal solution of Eq.", "(REF ) in the interaction picture with respect to $\\hat{\\operatorname{H}}_{\\mathrm {S}}+\\hat{\\operatorname{H}}_{\\mathrm {B}}$ (denoted by an overhead tilde) can be obtained by means of Wick's theorem.", "For simplicity, we will focus on the derivation for the bosonic case, but all steps can be trivially generalized for the fermionic case.", "Wick's theorem simplifies the calculation of the partial trace of the bath by reducing products of 2n operators to n products of pairwise traces $\\left\\langle \\hat{\\mathrm {T}}\\tilde{\\hat{\\mathrm {B}}}_{\\nu }\\left(t_{2n}\\right)\\tilde{\\hat{\\mathrm {B}}}_{\\nu }\\left(t_{2n-1}\\right)\\cdots \\tilde{\\hat{\\mathrm {B}}}_{\\nu }\\left(t_{2}\\right)\\tilde{\\hat{\\mathrm {B}}}_{\\nu }\\left(t_{1}\\right)\\right\\rangle =\\nonumber \\\\\\sum _{app}\\prod _{ij}\\left\\langle \\hat{\\mathrm {T}}\\tilde{\\hat{\\mathrm {B}}}_{\\nu }\\left(t_{i}\\right)\\tilde{\\hat{\\mathrm {B}}}_{\\nu }\\left(t_{j}\\right)\\right\\rangle ,$ where the sum is over all possible pairs (app) of indices up to $2n$ , $\\hat{\\operatorname{T}}$ is the time ordering operator and the average may be performed over any Gaussian state.", "Therefore, the solution of Eq.", "(REF ) in the interaction picture is $\\tilde{\\rho }(\\chi ,t)=\\tilde{\\mathcal {U}}(\\chi ,t)\\rho (0,0)$ , with $\\tilde{\\mathcal {U}}(\\chi ,t)=\\prod _{\\nu }\\prod _{jk=0,1}\\exp _{+}\\left(\\int \\limits _{0}^{t}\\operatorname{\\text{d}s}\\tilde{\\hat{\\operatorname{W}}}_{\\nu }^{jk}(\\chi ,s)\\right),$ where $\\exp _{+}$ stands for the time ordered exponential and $\\tilde{\\hat{\\operatorname{W}}}_{\\nu }^{jk}(\\chi ,t)=-\\int \\limits _{0}^{t}\\operatorname{\\text{d}s}\\tilde{\\hat{\\operatorname{V}}}_{\\nu }^{j}(t)C_{\\nu }^{jk}(\\chi ,t-s)\\tilde{\\hat{\\operatorname{V}}}_{\\nu }^{k}(s).$ Here we introduce the superoperator notation $\\hat{\\operatorname{A}}^{0}\\rho \\equiv \\hat{\\operatorname{A}}\\rho $ and $\\hat{\\operatorname{A}}^{1}\\rho \\equiv \\rho \\hat{\\operatorname{A}}$ and the bath correlation functions $C_{\\nu }^{jk}(\\chi ,t)=(-)^{j+k}\\left\\langle \\tilde{\\hat{\\operatorname{B}}}_{\\nu }^{j}\\left[(-)^{j}\\frac{\\chi }{2}\\right](t)\\tilde{\\hat{\\operatorname{B}}}_{\\nu }^{k}\\left[(-)^{k}\\frac{\\chi }{2}\\right](0)\\right\\rangle ,$ where $\\left\\langle A\\right\\rangle =\\operatorname{\\text{Tr}}\\left\\lbrace A\\pi (0)\\right\\rbrace $ .", "Note that this definition should be replaced by $\\left\\langle A\\right\\rangle =\\operatorname{\\text{Tr}}\\left\\lbrace A\\pi (0)e^{i\\chi \\hat{\\operatorname{O}}}\\operatorname{\\text{Tr}}\\left\\lbrace e^{i\\chi \\hat{\\operatorname{O}}}\\pi (0)\\right\\rbrace ^{-1}\\right\\rbrace $ for Eq.", "(REF ).", "The HEOM formalism uses an approximate representation of the correlation functions by means of linear combinations of decaying exponential functions.", "The extended version generalizes the framework to more involved functional bases [36].", "In our case, the coefficients of the linear combination are functions dependent on the counting field $\\chi $ , so that we approximate $C_{\\nu }^{jk}(\\chi ,t)=\\sum _{r}c_{\\nu r}^{jk}(\\chi )\\phi _{r}(t)$ by means of a set of functions $\\lbrace \\phi _{r}(t)\\rbrace $ whose derivatives are well defined within the set by $\\frac{\\mathrm {d}^{}}{\\mathrm {d}{t}^{}}\\phi _{r}(t)=\\sum _{n}\\eta _{rs}\\phi _{s}(t)$ , where $\\eta $ is a matrix with complex entries.", "The form of $c_{\\nu r}^{jk}\\left(\\chi \\right)$ is general and depends on the specific observable of interest $\\hat{\\mathrm {O}}$ .", "For instance, in the case $\\hat{\\mathrm {O}}=\\mathrm {H}_{\\nu }$ and $\\hat{\\operatorname{B}}_{\\nu }=\\sum _{k}\\gamma _{\\nu k}\\left(\\hat{\\operatorname{a}}_{\\nu k}+\\hat{\\operatorname{a}}_{\\nu k}^{\\dagger }\\right)$ , $C_{\\nu }^{jk}\\left(\\chi ,t\\right)=C_{\\nu }^{jk}\\left(\\chi \\pm t\\right)$ and the dependence is expected to be similar to that of $\\phi _{r}\\left(t\\right)$ .", "With this representation, it is possible to define the auxiliary objects $&\\tilde{\\rho }^{\\left\\lbrace n \\right\\rbrace }(\\chi ,t)=\\nonumber \\\\&\\hat{\\operatorname{\\text{T}}}\\prod \\limits _{\\nu ,r;k=0,1}\\left(\\int \\limits _{0}^{t}\\operatorname{\\text{d}s}\\phi _{r}\\left(t-s\\right)\\tilde{\\hat{\\operatorname{V}}}_{\\nu }^{k}(s)\\right)^{n_{\\nu r}^{k}}\\tilde{\\mathcal {U}}(\\chi ,t)\\rho (0,0),$ where $\\left\\lbrace n \\right\\rbrace \\equiv \\left\\lbrace \\dots ,n_{\\nu r}^{k},\\dots \\right\\rbrace $ is a rank-three tensor of non-negative integer entries which sum up to $n$ and $n$ is the so-called hierarchic level.", "It is clear that $\\tilde{\\rho }^{\\left\\lbrace 0 \\right\\rbrace }(\\chi ,t)=\\tilde{\\rho }(\\chi ,t)$ and the auxiliary fields satisfy the equation $\\frac{\\mathrm {d}^{}}{\\mathrm {d}{t}^{}}\\rho ^{\\left\\lbrace n \\right\\rbrace }(\\chi ,t)= & -i\\hat{\\operatorname{H}}_{S}^{\\times }\\rho ^{\\left\\lbrace n \\right\\rbrace }(\\chi ,t)\\\\&+\\sum _{\\nu ,r;k=0,1}\\Bigg (\\bar{\\hat{\\operatorname{V}}}_{\\nu r}^{k}\\rho ^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}+1,\\dots \\right\\rbrace }(\\chi ,t)\\nonumber \\\\&+\\sum _{s}n_{\\nu r}^{k}\\eta _{rs}\\rho ^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}-1,\\dots ,n_{\\nu s}^{k}+1,\\dots \\right\\rbrace }(\\chi ,t)\\nonumber \\\\& +n_{\\nu r}^{k}\\phi _{r}(0)\\hat{\\operatorname{V}}_{\\nu }^{k}\\rho ^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}-1,\\dots \\right\\rbrace }(\\chi ,t)\\Bigg ),\\nonumber $ where we have used the notation $\\hat{\\operatorname{A}}^{\\times }\\rho \\equiv \\hat{\\operatorname{A}}\\rho -\\rho \\hat{\\operatorname{A}}$ and $\\bar{\\hat{\\operatorname{V}}}_{\\nu r}^{k}\\equiv \\sum _{j=0,1}c_{\\nu r}^{jk}(\\chi )\\hat{\\operatorname{V}}_{\\nu }^{j}$ .", "This is an extension of the usual HEOM formulation [33]." ], [ "Hierarchy of equations of motion for high order statistical moments", "Although Eq.", "(REF ) can be used on its own to obtain the generating function, in the case where one is interested in specific statistical moments, a specialized hierarchy can be derived.", "Let us define the object $\\tilde{\\sigma }_{m}(t)&\\equiv \\left.\\frac{\\partial ^{m}}{\\partial {(i\\chi )}^{m}}\\tilde{\\rho }^{\\left\\lbrace 0 \\right\\rbrace }(\\chi ,t) \\right|_{\\chi =0}\\nonumber \\\\&=\\hat{\\operatorname{\\text{T}}}\\left.\\frac{\\partial ^{m}}{\\partial {(i\\chi )}^{m}}\\tilde{\\mathcal {U}}(\\chi ,t)\\right|_{\\chi =0}\\rho (0,0),$ so that the moment $m$ of the full counting distribution may be obtained by tracing: $\\operatorname{\\text{Tr}}\\left\\lbrace \\tilde{\\sigma }_{m}(t)\\right\\rbrace $ .", "It contains correlation functions of the form $\\left.\\frac{\\partial ^{q}}{\\partial {(i\\chi )}^{q}}C_{\\nu }^{jk}(\\chi ,t)\\right|_{\\chi =0}\\equiv C_{\\nu q}^{jk}(t)$ , which are well defined in terms of the approximate representation as $C_{\\nu q}^{jk}(t)=\\sum _{r}c_{\\nu rq}^{jk}\\phi _{r}(t)$ with $c_{\\nu rq}^{jk}\\equiv \\left.\\frac{\\mathrm {d}^{q}}{\\mathrm {d}{(i\\chi )}^{q}}c_{\\nu r}^{jk}(\\chi )\\right|_{\\chi =0}$ .", "In a procedure analogous to the one followed for the obtention of Eq.", "(REF ), we define $\\tilde{\\sigma }_{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace n \\right\\rbrace }(t)&=\\hat{\\operatorname{\\text{T}}}\\prod \\limits _{\\nu ,k;j=0,1}\\left(\\int \\limits _{0}^{t}\\operatorname{\\text{d}s}\\phi _{k}\\left(t-s\\right)\\tilde{\\hat{\\operatorname{V}}}_{\\nu }^{j}(s)\\right)^{n_{\\nu k}^{j}}\\nonumber \\\\&\\prod _{q}\\left(\\int \\limits _{0}^{t}\\operatorname{\\text{d}s}\\tilde{\\hat{\\operatorname{W}}}_{\\nu q}(s)\\right)^{m_{q}}\\tilde{\\mathcal {U}}(0,t)\\rho (0,0),$ where $\\left\\lbrace m \\right\\rbrace \\equiv \\left\\lbrace \\dots ,m_{q},\\dots \\right\\rbrace $ is a vector of non-negative integer entries such that $\\sum _{q}m_{q}q=m$ and $\\tilde{\\hat{\\operatorname{W}}}_{\\nu q}(t)\\equiv \\sum _{j,k=0,1}-\\int \\limits _{0}^{t}\\operatorname{\\text{d}s}\\tilde{\\hat{\\operatorname{V}}}_{\\nu }^{j}(t)C_{\\nu q}^{jk}(t-s)\\tilde{\\hat{\\operatorname{V}}}_{\\nu }^{k}(s)$ .", "This object satisfies the equation $\\frac{\\mathrm {d}^{}}{\\mathrm {d}{t}^{}}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace n \\right\\rbrace }(t)= & -i\\hat{\\operatorname{H}}_{S}^{\\times }\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace n \\right\\rbrace }(t)\\\\&+\\sum _{\\nu ,r;k=0,1}\\Bigg (\\bar{\\hat{\\operatorname{V}}}_{\\nu r}^{k}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}+1,\\dots \\right\\rbrace }(t)\\\\&+\\sum _{s}n_{\\nu r}^{k}\\eta _{rs}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}-1,\\dots ,n_{\\nu s}^{k}+1,\\dots \\right\\rbrace }(t)\\\\& +n_{\\nu r}^{k}\\phi _{r}(0)\\hat{\\operatorname{V}}_{\\nu }^{k}\\sigma _{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}-1,\\dots \\right\\rbrace }(t)\\\\&+\\sum _{q}m_{q}\\bar{\\hat{\\operatorname{V}}}_{\\nu rq}^{k}\\sigma _{\\left\\lbrace \\dots ,m_{q}-1,\\dots \\right\\rbrace }^{\\left\\lbrace \\dots ,n_{\\nu r}^{k}+1,\\dots \\right\\rbrace }(t)\\Bigg ),$ where $\\bar{\\hat{\\operatorname{V}}}_{\\nu rq}^{k}\\equiv \\sum _{j=0,1}c_{\\nu rq}^{jk}\\hat{\\operatorname{V}}_{\\nu }^{j}$ .", "The structure is identical to the usual hierarchy (Eq.", "REF ) but for the last term, which connects it to the next tier elements of the hierarchy associated to the previous moment.", "This can be interpreted as an additional driving that each moment exerts on the next one.", "This naturally defines a cascade of hierarchies that can be exploited for parallel simulation of several moments with reduced overhead.", "The relationship between $\\tilde{\\sigma }_{m}(t)$ and the set of $\\tilde{\\sigma }_{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace n \\right\\rbrace }(t)$ follows $\\tilde{\\sigma }_{m}(t)=\\sum _{\\left\\lbrace m \\right\\rbrace }a_{\\left\\lbrace m \\right\\rbrace }\\tilde{\\sigma }_{\\left\\lbrace m \\right\\rbrace }^{\\left\\lbrace 0 \\right\\rbrace }(t).$ where the sum is over all partitions $\\left\\lbrace m \\right\\rbrace $ of $m$ (all vectors $m_{q}$ with the property $\\sum _{q=1}^{m}m_{q}q=m$ ) and $a_{\\left\\lbrace m \\right\\rbrace }\\equiv \\prod _{q=1}^{m}\\prod _{j=1}^{m_{q}}\\frac{1}{j}\\left(\\begin{array}{c}m-\\sum _{r=q}^{m}rm_{r}+jq\\\\k\\end{array}\\right)$ is the number of permutations associated to that partition.", "Finally, cumulants recursively relate to moments by means of the formula $\\left\\langle \\left\\langle A^{n}\\right\\rangle \\right\\rangle =\\left\\langle A^{n}\\right\\rangle -\\sum _{m=1}^{n-1}\\left(\\begin{array}{c}n-1\\\\m-1\\end{array}\\right)\\left\\langle \\left\\langle A^{m}\\right\\rangle \\right\\rangle \\left\\langle A^{n-m}\\right\\rangle .$ Furthermore factorial cumulants are obtained by $\\left\\langle \\left\\langle A^{n}\\right\\rangle \\right\\rangle _{F}=\\left\\langle \\left\\langle A^{n}\\right\\rangle \\right\\rangle -\\sum _{m=1}^{n-1}\\left\\lbrace \\begin{array}{c}n\\\\m\\end{array}\\right\\rbrace \\left\\langle \\left\\langle A^{m}\\right\\rangle \\right\\rangle _{F},$ where $\\left\\lbrace \\begin{array}{c}n\\\\m\\end{array}\\right\\rbrace $ are the Stirling numbers of the second kind." ] ]
1606.05074
[ [ "Energy Conservation and Coupling Error Reduction in Non-Iterative\n Co-Simulations" ], [ "Abstract When simulators are energetically coupled in a co-simulation, residual energies alter the total energy of the full coupled system.", "This distorts the system dynamics, lowers the quality of the results, and can lead to instability.", "By using power bonds to realize simulator coupling, the Energy-Conservation-based Co-Simulation method (ECCO) [Sadjina et al.", "2016] exploits these concepts to define non-iterative global error estimation and adaptive step size control relying on coupling variable data alone.", "Following similar argumentation, the Nearly Energy Preserving Coupling Element (NEPCE) [Benedikt et al.", "2013] uses corrections to the simulator inputs to approximately ensure energy conservation.", "Here, we discuss a modification to NEPCE for when direct feed-through is present in one of the coupled simulators.", "We further demonstrate how accuracy and efficiency in non-iterative co-simulations are substantially enhanced when combining NEPCE with ECCO's adaptive step size controller.", "A quarter car model with linear and nonlinear damping characteristics serves as a co-simulation benchmark, and we observe reductions of the coupling errors of up to 98% utilizing the concepts discussed here." ], [ "Introduction", "Co-simulation allows for the independent and parallel modeling and simulation of complex systems including multiple physical and engineering domains, the use of tailored software tools and expert knowledge, the efficient use of suited solvers, and the protection of intellectual property within models.", "All these properties make this kind of simulator coupling an attractive choice, especially from an industrial perspective.", "But the fact that coupled subsystems are solved independently of each other between discrete communication time points also emphasizes accuracy and stability issues.", "The flow and the conservation of energy between simulators in a co-simulation can be conveniently studied when using power bonds to realize the couplings.", "A power bond is a direct energetic bond between subsystems defined by inputs and outputs whose product gives a physical power: force and velocity, electric current and voltage, pressure and flow rate, to name a few.", "Because subsystems in a co-simulation advance in time independently of each other, energy transactions between them are inherently inaccurate.", "Energy residuals emerge as a consequence and directly affect the total energy of the overall coupled system.", "Consequently, system dynamics are distorted and co-simulation accuracy and stability are challenged.", "These concepts are exploited in the Energy-Conservation-based Co-Simulation method[1] (ECCO).", "Because energy residuals are a direct expression of coupling errors, they are a versatile tool to assess the quality of co-simulations.", "Based on such error estimators, ECCO defines an adaptive control of the co-simulation step size, and displays significant improvements in the accuracy and efficiency of non-iterative co-simulations.", "Similar arguments are used in the Nearly Energy Preserving Coupling Element[2] (NEPCE) to introduce corrections to the flow of (generalized) power between simulators in order to minimize coupling errors.", "Here, we have a closer look at NEPCE and its energy-conserving properties.", "We further propose a modification to include the presence of direct feed-through, enhancing its performance.", "NEPCE's efficiency is based on the assumption that the coupling variables are slowly varying functions of time.", "This assumption is challenged, however, by finding a suitable choice of the co-simulation (macro) time step.", "We demonstrate how this issue is efficiently handled by ECCO's energy-conservation-based adaptive step size control in order to substantially improve accuracy and efficiency.", "Because the resulting framework is non-iterative, it is computationally inexpensive and well suited for industrial applications.", "This paper is organized as follows: In Section , we start with a brief recapitulation of the flow and conservation of energy in co-simulations using power bonds.", "Next, we study NEPCE's non-iterative corrections to the simulator inputs in Section  and show how they should be modified in the presence of direct feed-through.", "Section  discusses how these corrections can be combined with ECCO's adaptive step size control, and a quarter car model is then used in Section  to demonstrate the performance of the proposed method and its influence on co-simulation accuracy and efficiency.", "Finally, we give a conclusion in Section ." ], [ "Energy Conservation in Co-Simulations", "Most commonly, co-simulations are realized by letting the simulators advance in time in parallel and independently of each other, and then synchronizing coupling data at discrete communication time points.", "This weak coupling approach is easily implemented and relatively efficient on paper: It is universally applicable for industrial applications (which usually prohibit iterative schemes) and the parallelization potential holds the promise of substantial simulation speed-ups.", "Its major weaknesses, however, are accuracy and stability.", "Input quantities are generally unknown during the time integrations inside the simulators.", "They must therefore be approximated, and are often simply held constant.", "A sufficiently small macro time step has to be chosen in order to keep the coupling errors which result from this scheme contained.", "Figure: Two coupled simulators exchange energy through a power bond in a co-simulation" ], [ "Power and Energy Residuals", "The use of power bonds from bond graph theory[3], [4] allows to reframe these issues in terms of energy conservation considerations.", "[1] A power bond $k$ is defined by a pair of power variables—a flow $f_k$ and an effort $e_k$ —whose product $P_k = e_k f_k$ gives a physical power.", "Powers and energies, the universal currencies of physical systems, are directly accessible in co-simulations when using power bonds.", "As an example, consider the flow of energy between two simulators S$_1$ and S$_2$ that are coupled via a power bond $k$ , see Fig.", "REF .", "From the point of view of S$_1$ , energy is transferred to S$_2$ at a rate $P_{k_1}(t)=\\mbox{$\\tilde{u}$}_{k_1}(t)y_{k_1}(t),$ where $y_{k_1}(t)$ is the output and $\\mbox{$\\tilde{u}$}_{k_1}(t) \\approx u_{k_1}(t)$ is an approximation of the generally unknown value $u_{k_1}(t)$ .", "If, instead, we consider the energy transfer from the other simulator's perspective, we conclude that $P_{k_2}(t)=\\mbox{$\\tilde{u}$}_{k_2}(t)y_{k_2}(t).$ This is problematic because it fundamentally violates the conservation of energy, $-(P_{k_1}+P_{k_2})\\ne 0,$ because, generally, $\\mbox{$\\tilde{u}$}_{k_1}(t) \\ne u_{k_1}(t)$ and $\\mbox{$\\tilde{u}$}_{k_2}(t) \\ne u_{k_2}(t)$ .", "Hence, a residual energy is incorrectly created due to the independent time integrations of the simulators during the macro time step $t_i \\rightarrow t_{i+1} = t_i + \\Delta t_i$ ,[1] $\\delta E_k(t_{i+1})&\\equiv \\int _{t_i}^{t_{i+1}}\\delta P_k(t)\\mathop {}\\!\\mathrm {d}t,\\multicolumn{2}{l}{\\text{where}}\\\\\\delta P_k&\\equiv -(P_{k_1}+P_{k_2})$ is the residual power for the power bond $k$ , see Fig.", "REF for an illustration.", "At each macro time step, the residual energy $\\delta E_k$ is directly added to the total energy of the overall coupled system.", "[1] As a consequence, system dynamics are distorted and the quality of the co-simulation reduced.", "Note that the power transmitted from S$_1$ to S$_2$ can be obtained from the simulator outputs as $P_{k_{12}}(t)=\\sigma _{k_{12}}\\big (y_{k_1}(t)y_{k_2}(t)\\big ),$ where the sign $\\sigma _{k_{12}} \\equiv ({\\mathbf {L}_k}_{12}-{\\mathbf {L}_k}_{21})/2$ is determined by the corresponding elements of the connection graph matrix $\\mathbf {L}$ .", "Figure: Total system dynamics are distorted by a residual power δP k \\delta P_k between two energetically coupled simulators due to the independent time integrationsLuckily, inaccurate energy transactions provide us with a versatile error estimator because the corresponding residual energies are a direct expression of the co-simulation coupling errors and the violation of energy conservation.", "[1] This is exploited by the ECCO algorithm to define an adaptive macro step size controller: For input extrapolation of order $m$ , the residual energy scales quadratic with the step size[1], $\\delta {E_k} = \\mathcal {O}({\\Delta t}^{m+2})$ .", "Consequently, the conservation of energy can be approximately satisfied by controlling the macro step size, optimizing the quality and efficiency of co-simulations." ], [ "Local Errors in the Coupling Variables", "Considering the time evolution of the internal states $\\mathbf {x} = \\lbrace x_1, x_2\\rbrace $ of the coupled simulators between the discrete communication time points $t_i$ and $t_{i+1}$ , $\\dot{\\mathbf {x}}(t)=\\mathbf {f}\\big (\\mathbf {x}(t),\\tilde{\\mathbf {u}}(t)\\big ),\\quad t \\in (t_i,t_{i+1}],$ simulator coupling can be expressed as $\\mathbf {y}(t_{i+1})&=\\mathbf {g}\\big (\\mathbf {x}(t_{i+1}),\\tilde{\\mathbf {u}}(t_{i+1})\\big ),\\\\\\mathbf {u}(t_{i+1})&=\\mathbf {L}\\mathbf {y}(t_{i+1}),$ where $\\mathbf {L}$ is a connection graph matrix that relates outputs $\\mathbf {y}$ and inputs $\\mathbf {u}$ at communication time points.", "In the non-iterative co-simulation, the inputs are generally unknown and have to be approximated during the time integrations inside the simulators, $\\tilde{\\mathbf {u}}(t) \\approx \\mathbf {u}(t)$ .", "Most commonly, they are simply held constant such that $\\tilde{\\mathbf {u}}(t) = \\mathbf {u}(t_i)$ for $t \\in (t_i,t_{i+1}]$ .", "Let us in the following have a closer look at the local coupling errors which stem from these approximations and the independent time integrations in the subsystems between communication time points.", "For the case of coupling via power bonds, these errors are conveniently represented as power and energy errors and directly related to the conservation of energy throughout the entire coupled system.", "Using energies and powers as error metrics instead of non-energetic quantities has two major advantages: i.)", "They offer a more holistic and intuitive approach by considering the flow of energy between subsystems directly ii.)", "They avoid that some simulator's contributions to the global error are given too much weight.", "If, for example, one simulator outputs a force and another a position, the numerical values of the force output will typically be much larger than those of the position output.", "The same will then generally be true for the numerical values of the corresponding errors, skewing the actual simulators' contributions to the global co-simulation error.", "The use of energy and power errors solves this issue in an elegant fashion.", "In the next section, we will discuss how we can minimize local coupling errors.", "The subsystem states are inaccessible in a typical co-simulation setting and can not be directly altered.", "Instead, corrections to the inputs can be derived such that the residual energies between simulators are minimized and energy conservation is approximately satisfied.", "These corrections ideally cancel the local errors in the inputs which are given by $\\begin{split}\\Delta \\mathbf {u}(t)&=\\tilde{\\mathbf {u}}(t)-\\mathbf {u}_0(t)\\\\&=\\tilde{\\mathbf {u}}(t)-\\mathbf {L}\\mathbf {y}_0(t)\\\\&=\\tilde{\\mathbf {u}}(t)-\\mathbf {L}\\big (\\mathbf {y}(t)-\\Delta \\mathbf {y}(t)\\big ),\\end{split}$ where $\\mathbf {u}_0(t)$ is the exact solution and we used that $\\mathbf {u}_0(t)=\\mathbf {L}\\mathbf {y}_0(t)$ for any time $t$ .", "The errors in the outputs evaluate to $\\begin{split}\\Delta \\mathbf {y}(t)&=\\mathbf {y}(t)-\\mathbf {y}_0(t)\\\\&=\\mathbf {g}\\big (\\mathbf {x}(t),\\tilde{\\mathbf {u}}(t)\\big )-\\mathbf {g}\\big (\\mathbf {x}_0(t),\\mathbf {u}_0(t)\\big )\\\\&=\\mathbf {J}_{\\mathbf {g}}(\\mathbf {u})\\Delta \\mathbf {u}(t)+\\mathbf {J}_{\\mathbf {g}}(\\mathbf {x})\\Delta \\mathbf {x}(t)\\\\&+\\mathcal {O}({\\Delta t}^{m+2}),\\end{split}$ where ${J_{\\mathbf {g}}}_{i j}(\\mathbf {u}) = \\mathop {}\\!\\mathrm {\\partial }g_i /\\mathop {}\\!\\mathrm {\\partial }u_j$ is the interface Jacobian and ${J_{\\mathbf {g}}}_{i j}(\\mathbf {x}) = \\mathop {}\\!\\mathrm {\\partial }g_i /\\mathop {}\\!\\mathrm {\\partial }x_j$ .", "While the error contributions from the state vector are $\\Delta \\mathbf {x}(t)=\\mathbf {x}(t)-\\mathbf {x}_0(t)=\\mathcal {O}({\\Delta t}^{m+2}),$ the input errors appear to order $\\Delta \\mathbf {u} = \\mathcal {O}({\\Delta t}^{m+1})$ .", "Consequently, if one of the simulators S$_k$ has direct feed-through, the output errors are also of order $\\mathcal {O}({\\Delta t}^{m+1})$ because then ${J_{\\mathbf {g}}}_{k k}(\\mathbf {u}) \\ne 0$ .", "Using Eq.", "(REF ) in Eq.", "(REF ) and rewriting thus gives $\\begin{split}\\Delta \\mathbf {u}(t)&=\\big (1-\\mathbf {L}\\mathbf {J}\\big )^{-1}\\big (\\tilde{\\mathbf {u}}(t)-\\mathbf {L}\\mathbf {y}(t)\\big )\\\\&+\\mathcal {O}({\\Delta t}^{m+2}),\\end{split}$ where we set $\\mathbf {J} \\equiv \\mathbf {J}_{\\mathbf {g}}(\\mathbf {u})$ for brevity." ], [ "Non-Iterative Energy-Conservation-Based Corrections", "Let us now take the idea of energy conservation in co-simulations a step further by directly modifying the coupling variables such that energy transactions between simulators are described more accurately.", "In this section, we will explore this concept which is used by NEPCE [2] and generalize it to include the presence of direct feed-through.", "In Section  we then discuss how the energy-conservation-based corrections studied here can be combined with ECCO's non-iterative adaptive step size controller, and Section  demonstrates the substantial improvements in accuracy and efficiency thus obtained using a quarter car co-simulation benchmark model.", "As can be seen from Eqs.", "(REF ) and (REF ), a residual energy $\\delta E_k(t_{i+1})=-\\int _{t_i}^{t_{i+1}}\\tilde{\\mathbf {u}}_k(t)\\cdot \\mathbf {y}_k(t)\\mathop {}\\!\\mathrm {d}t$ is accumulated during the time step $t_i\\rightarrow t_{i+1}$ for a power bond $k$ connecting the inputs $\\tilde{\\mathbf {u}}_k = \\lbrace \\mbox{$\\tilde{u}$}_{k_1},\\mbox{$\\tilde{u}$}_{k_2}\\rbrace $ and outputs $\\mathbf {y}_k = \\lbrace y_{k_1},y_{k_2}\\rbrace $ .", "The concept behind NEPCE is to find corrections $\\delta \\mathbf {u}_k = \\lbrace \\delta u_{k_1}, \\delta u_{k_2}\\rbrace $ to the inputs at communication time instant $t=t_i$ with the aim of reducing the residual energy by a factor of $(1-\\alpha )$ , such that $(\\alpha -1)\\delta E_k(t_{i+1})=\\int _{t_i}^{t_{i+1}}\\big (\\tilde{\\mathbf {u}}_k(t)+\\delta \\mathbf {u}_k(t)\\big )\\cdot \\mathbf {y}_k(t)\\mathop {}\\!\\mathrm {d}t$ with the tuning factor $\\alpha \\in [0,1]$ .", "Ideally, $\\alpha = 1$ if the corrections accurately track the errors in the inputs, $\\delta \\mathbf {u}_k(t) = - \\Delta \\mathbf {u}_k(t)$ .", "While this can not be realized in general for non-iterative co-simulations, however, corrections should be of the same order as the errors in the input (REF ), $\\delta \\mathbf {u}_k = \\mathcal {O}({\\Delta t}^{m+1})$ , to mitigate their effects.", "Moreover, a correction to the input will generally elicit a modification of the output $\\delta \\mathbf {y}_k$ , such that we generally need to consider $\\begin{split}&(\\alpha -1)\\delta E_k(t_{i+1})\\\\=&\\int _{t_i}^{t_{i+1}}\\big (\\tilde{\\mathbf {u}}_k(t)+\\delta \\mathbf {u}_k(t)\\big )\\cdot \\big (\\mathbf {y}_k(t)+\\delta \\mathbf {y}_k(t)\\big ) \\mathop {}\\!\\mathrm {d}t.\\end{split}$ If direct feed-through is present, this modification to the output is of the same order as the input corrections, $\\delta \\mathbf {y}_k = \\mathcal {O}({\\Delta t}^{m+1})$ , and should be included." ], [ "NEPCE", "But first, let us discuss the case where none of the simulators have direct feed-through.", "Then, the errors in the inputs (REF ) are simply $\\Delta \\mathbf {u}(t)=\\tilde{\\mathbf {u}}(t)-\\mathbf {L}\\mathbf {y}(t)+\\mathcal {O}({\\Delta t}^{m+2}),$ and Eq.", "(REF ) suffices.", "Choosing $\\delta \\mathbf {u}(t) = -\\Delta \\mathbf {u}(t) \\approx \\mathbf {L} \\mathbf {y}(t) - \\tilde{\\mathbf {u}}(t)$ would make the residual energy vanish and the coupling quantities exact to order $\\mathcal {O}({\\Delta t}^{m+1})$ .", "As already mentioned, this is not possible for non-iterative co-simulations because $\\mathbf {y}(t)$ is unknown a priori for $t = (t_i, t_{i+1}]$ .", "Instead, we realize the correction in terms of previous coupling data,[2] $\\delta \\mathbf {u}(t)\\approx \\frac{\\alpha }{\\Delta t_i}\\int _{t_{i-1}}^{t_i}\\big (\\mathbf {L}\\mathbf {y}(\\tau )-\\tilde{\\mathbf {u}}(\\tau )\\big ) \\mathop {}\\!\\mathrm {d}\\tau ,$ for $t \\in (t_i,t_{i+1}]$ , assuming that the coupling variables and the errors are slowly varying on the scale of the time step $\\Delta t$ .", "Note that this is a reasonable assumption in theory: In a co-simulation the macro time step should be chosen such that the dynamics of the system can be sufficiently well resolved in time.", "A violation of this assumption is equivalent to the macro time step simply being too large for the problem at hand.", "In section , we will take a big step towards ensuring that this crucial assumption holds by combining the energy-conservation-based input corrections discussed in the present section with the energy-conservation-based adaptive step size controller ECCO." ], [ "Corrections with Direct Feed-Through", "As discussed previously, corrections to the inputs cause modifications to the outputs which are of the same order $\\mathcal {O}({\\Delta t}^{m+1})$ in the presence of direct feed-through.", "The errors in the inputs are then given by Eq.", "(REF ), and Eq.", "(REF ) should be modified to $\\delta \\mathbf {u}(t)\\approx \\frac{\\alpha }{\\Delta t_i}\\big (1-\\mathbf {L}\\mathbf {J}\\big )^{-1}\\int _{t_{i-1}}^{t_i}\\big (\\mathbf {L}\\mathbf {y}(\\tau )-\\tilde{\\mathbf {u}}(\\tau )\\big ) \\mathop {}\\!\\mathrm {d}\\tau $ to include all coupling errors of order $\\mathcal {O}({\\Delta t}^{m+1})$ .", "It is important to point out that Eq.", "(REF ) requires the knowledge of the generally time-dependent interface Jacobian $J_{i j} = \\mathop {}\\!\\mathrm {\\partial }g_i /\\mathop {}\\!\\mathrm {\\partial }u_j$ .", "In practical applications, it will likely not be available and the unmodified NEPCE form (REF ) should be chosen.", "While disregarding the output error contribution in Eqs.", "(REF ), it is still an improvement over the uncorrected co-simulation in the presence of direct feed-through.", "Finally, note that we can safely disregard the case where both simulators have direct feed-through, because it amounts to an algebraic loop which indicates that the particular system reticulation is not suitable for non-iterative co-simulation and ill-chosen." ], [ "Energy-Conserving Adaptive Step Size Control", "The previous section discussed NEPCE and how it should be modified in the presence of direct feed-through.", "The approach to energy-conservation-based corrections to the inputs in non-iterative co-simulations relies on the assumption that the coupling variables are slowly varying functions of time on the scale of the macro time step.", "When this assumption does not hold the corrections become increasingly ineffective and can even lead to instability by exciting relatively fast dynamics in the subsystems[2].", "In other words, the smaller the chosen macro time step the more efficient and beneficial the input corrections become.", "The Energy-Conservation-based Co-Simulation method (ECCO) provides a framework that allows us to adaptively choose a macro step size which (given some tolerances) approximately ensures energy conservation in non-iterative co-simulations.", "This concept and its performance have recently been studied[1], and we shall in the following combine it with the energy-conservation-based input corrections from the previous section to define a non-iterative co-simulation framework yielding high accuracy and efficiency without the use of any simulator-internal data.", "An I-controller is used to determine a new optimal step size ${\\Delta t}_{i+1}=\\alpha _\\text{s}\\epsilon (t_i)^{-k_\\text{I}}\\Delta t_i$ as a function of an error indicator $\\epsilon $ .", "Here, $k_\\text{I} = 0.3/(m+2)$ is the integral gainThe denominator $m+2$ represents the order of the error, here $\\delta E_k = \\mathcal {O}({\\Delta t}^{m+2})$ .", "The corrections discussed in Sec.", "have the aim of canceling the energy error $\\delta E_k$ to leading order in $\\Delta t$ , and thus one should choose $k_\\text{I} = 0.3/(m+3)$ .", "Here, we decide against this alteration for two reasons: i.)", "As mentioned previously, the leading terms in the error can in general not be canceled exactly for the non-iterative case, as expressed by the tuning factor $\\alpha $ .", "The ii.)", "The actual benchmark results discussed in Sec.", "show little to no dependence on this chance., $m$ is the extrapolation order ($m=0$ for constant extrapolation), and $\\alpha _\\text{s} \\in [0.8,0.9]$ is a safety factor.", "The scalar error indicator can be defined as[1] $\\epsilon (t)\\equiv \\sqrt{\\frac{1}{N}\\sum _{k = 1}^N\\bigg (\\frac{\\delta E_k(t)}{r_k\\big ({E_0}_k+| E_k(t) |\\big )}\\bigg )^2},$ using the residual energies $\\delta E_k$ and energies $E_k(t_{i+1}) \\approx P_{k_{12}}(t_{i+1}) \\Delta t_i$ transmitted per time step for all $N$ power bonds.", "Here, the typical energy scale ${E_0}_k$ and the relative tolerance $r_k$ are freely configurable parameters which determine the energy resolution for the power bond $k$ .", "The I-controller (REF ) aims to find and maintain a balance between accuracy and efficiency by choosing a step size for which $\\epsilon \\approx 1$ : Efficiency can be improved if $\\epsilon < 1$ by increasing the step size, while accuracy needs to be increased by choosing smaller time steps of $\\epsilon > 1$ .", "In order to avoid rapid oscillations in the step size on one hand, and inefficiently small step sizes on the other, the step size itself and its rate of change are restricted by the parameters ${\\Delta t}_\\text{min}$ and ${\\Delta t}_\\text{max}$ , and $\\Theta _\\text{min}$ and $\\Theta _\\text{max}$ , respectively.", "Table REF lists the full configuration used for the benchmark tests of Section .", "Table: Configuration of the adaptive step size controller for the benchmark model in Sec." ], [ "Co-Simulation Benchmark Tests", "In order to assess the performance of the methods discussed in sections  and , we employ a quarter car model as described in Ref.", "Arnold2013 and split it into two subsystems connected via a power bond, see Fig.", "REF .", "This model can be considered two coupled Dahlquist test equations[7] and is thus well suited as a co-simulation benchmark test case.", "[8], [9], [6], [10], [1] We further study two different reticulations for the co-simulation and also investigate nonlinear damping characteristics.", "The corresponding model and the underlying equations are adapted directly from Ref.", "Sadjina2016, the parameters are summarized in Table REF for the linear test case and in Table REF for the nonlinear case.", "Figure: The quarter car benchmark model is split into the subsystems S 1 _1 and S 2 _2 for co-simulation using the two distinct reticulations 1 and 2Table: Parameters for the linear quarter car benchmark model according to Ref.", "Arnold2013Table: Parameter changes to include nonlinear damping forces in the benchmark model according to Ref.", "Busshardt1992We generally carry out the time integrations in the subsimulators using micro step sizes of ${\\Delta t}_{\\text{S}_1} = {\\Delta t}_{\\text{S}_2} = {\\Delta t}/256$ with the forward Euler method to focus on the co-simulation coupling errors.Even smaller micro step sizes affect the benchmark results only marginally.", "As mentioned previously, we use energies and powers as error metrics to assess the quality of the co-simulation results: On one hand, we consider the average error in the power (REF ) transmitted over the power bond from simulator S$_1$ to simulator S$_2$ , $\\Delta P(t_{i+1})\\equiv \\frac{1}{T}\\sum _{j=0}^i|P_{12}(t_{j+1}) - P^0_{12}(t_{j+1})|\\Delta t_j,$ where $P^0_{12}(t)$ is the exact solution and $T$ is the total (virtual) duration of the simulation run.", "On the other hand, the total accumulated residual energy $\\Delta E(t_{i+1})\\equiv \\sum _{j=0}^i\\delta P(t_{j+1})\\Delta t_j$ gives the amount of energy wrongfully added to the full system during the entire simulation time interval $t \\in [t_0, t_{i+1}]$ and is thus used as another indicator of co-simulation accuracy." ], [ "NEPCE", "Let us first use the quarter car model to benchmark NEPCE's performance.", "The tuning factor $\\alpha $ is chosen such that the errors are minimized while avoiding the excitation of fast oscillations and risking instability.", "The energy errors can be reduced throughout by 49 to 86 when using NEPCE compared to the uncorrected results.", "The results are summarized in Tables REF and REF , where the tuning factor, the total number of macro time steps, and the power transmitted over the power bond $P_{12}$ averaged over the entire simulation duration $T$ are shown.", "Furthermore, the error in the power $\\Delta P(T)$ and the total accumulated residual energy $\\Delta E(T)$ are given according to Eqs.", "() with respect to the simulation duration $T$ .", "Table: Linear quarter car benchmark results for reticulation 1 with NEPCE and with NEPCE with direct feed-through modificationTable: Linear quarter car benchmark results for reticulation 2 with NEPCE and with NEPCE with direct feed-through modificationThe quarter car benchmark model does exhibit direct feed-through (in simulator S$_2$ for system reticulation 1 and in S$_1$ in system reticulation 2).", "We thus expect improved performance when including the modification to NEPCE discussed in Section REF .", "Indeed, the average error in the power $\\Delta P(T)$ is reduced by about another 17 to 33 with the modification.", "Fig.", "REF exemplifies this enhancement by showing the average error in the transmitted power for system reticulation 2.", "Note, however, that the direct feed-through modification to NEPCE does not significantly influence the overall accumulated residual energy $\\Delta E(T)$ .", "Figure: Average error in the power for the linear quarter car benchmark with reticulation 2 and constant step size:NEPCE with direct feed-through modification (solid), NEPCE alone (dashed), and the uncorrected result (dotted)" ], [ "NEPCE with ECCO", "Let us now demonstrate how the corrections to the inputs are made more efficient by combining them with ECCO's energy-conservation-based adaptive step size control, as proposed in Sec. .", "For this purpose, the I controller (REF ) and the scalar error indicator (REF ) are configured according to the parameters listed in Table REF , and the starting step size is set to ${\\Delta t}_0 = {\\Delta t}_\\text{min}$ .", "The quarter car system is initially excited with an energy of 750 which thus determines the characteristic energy scale for the system, $E_0 = {750}{}$ .", "The tolerance $r$ is set such that the total number of macro time steps remains around a constant 4000 steps in order to keep the computational cost at the same level.", "Substantial improvements are observed when using NEPCE with ECCO's adaptive step size control: The energy errors in the benchmarks are reduced by 87 to 92 for system reticulation 1, see Table REF , and by 97 to 98 for system reticulation 2, see Table REF .", "This considerable enhancement of the quality of the co-simulation results is also exemplified in Fig.", "REF .", "Table: Linear quarter car benchmark results for reticulation 1 using residual-energy-based adaptive step size control with NEPCE and with NEPCE with direct feed-through modificationTable: Linear quarter car benchmark results for reticulation 2 using residual-energy-based adaptive step size control with NEPCE and with NEPCE with direct feed-through modificationFigure: Average error in the power for the linear quarter car benchmark with reticulation 2:ECCO with NEPCE (solid) against the constant step size results with NEPCE (dashed) and without any corrections (dotted)The situation is further improved by also including the direct feed-through modifications for NEPCE, as shown in Fig.", "REF .", "Then, an additional reduction of the average error in the power of 26 to 36 is achieved compared to the results without the modification.", "Again, however, the accumulated residual energy is almost unaffected.", "Figure: Average error in the power for the linear quarter car benchmark with reticulation 2:ECCO with NEPCE with direct feed-through modification (solid), ECCO with NEPCE alone (dashed), and uncorrected result with constant step sizes (dotted)In conclusion, the non-iterative energy-conservation-based co-simulation framework presented here (NEPCE with direct feed-through modification and ECCO) manages to reduce the energy errors by between 87 and 98 in the linear quarter car benchmark at no additional computational cost." ], [ "Nonlinear Damping", "Finally, let us study the effects of nonlinear damping as given in Table REF .", "Note that the total simulation duration is now set to $T = {2}{}$ (2000 macro time steps in total) because the excitations in the system are subdued faster with the more efficient nonlinear damper.", "In addition, system reticulation 2 is relatively unstable for nonlinear damping, and the macro step size is thus restricted to $t_\\text{max} = {2.5}{}$ for this setup.", "The energy-conservation-based corrections to the inputs (as expressed by the tuning factor $\\alpha $ ) have to be applied less aggressively to avoid rapid oscillations.", "Yet, using NEPCE alone without modifications yields a reduction in the energy errors of 32 to 60 when compared to the uncorrected results, as shown in Tables REF and REF .", "As was the case for the linear benchmark, significant improvements are obtained by combining NEPCE with ECCO: The energy errors are reduced by 79 to 91 compared to uncorrected results obtained with a constant step size.", "Also including the direct feed-through modifications with NEPCE leads to small additional reductions of 0 to 19.", "Table: Nonlinear quarter car benchmark results for reticulation 1Table: Nonlinear quarter car benchmark results for reticulation 2" ], [ "Conclusion", "The Energy-Conservation-based Co-Simulation method[1] (ECCO) provides a generic framework for error estimation and adaptive step size control in non-iterative co-simulations.", "Using power bonds to realize the simulator coupling, it directly monitors power flows between the subsystems and gives the exact amount of energy wrongfully added to the total energy of the full coupled system during co-simulation (macro) time steps.", "The resulting so-called residual energies are obtain from the coupling variable values alone, and ECCO uses them to propose an optimal macro time step to minimize energy errors throughout the co-simulation.", "The Nearly Energy Preserving Coupling Element[2] (NEPCE), on the other hand, corrects for coupling errors in non-iterative co-simulations directly to make the flow of (generalized) energy between subsimulators more accurate.", "In the present paper, we combine both methods to optimize the efficiency and accuracy of non-iterative co-simulations.", "NEPCE is based on the assumption that the coupling variables are slowly varying on the scale of the macro time step.", "ECCO, on the other hand, provides a systematic approach to fulfill this requirement by adaptively controlling the macro step size in order to minimize the violation of energy conservation.", "We also extend NEPCE to the case where direct feed-through is present.", "Then, the output errors give contributions to the residual energy which are of the same order as the ones stemming from the input errors.", "Put differently, additional contributions to the violation of energy conservation should be taken into account when constructing energy-conserving corrections to the coupling variables.", "This is, however, only possible if the interface Jacobian is known.", "The performance of the concepts discussed here is demonstrated by use of a quarter car co-simulation benchmark model.", "We study two distinct system reticulations, as well as the effects of including nonlinear damping characteristics.", "In these benchmarks, NEPCE alone generally yields a reduction in the energy errors of 32 to 86, depending on how aggressively it can be used before unwanted oscillations are induced.", "The proposed direct feed-through modification to NEPCE reduces the energy errors by another 0 to 36.", "Also employing ECCO's adaptive step size control leads to substantially higher accuracies in the co-simulation results: Energy errors are then reduced by up to 98 when compared to the uncorrected results with constant macro step sizes.", "This work was funded by the Research Council of Norway (project no.", "225322 MAROFF) and the industrial partners in the ViProMa project consortium (VARD, Rolls-Royce Marine and DNV GL).", "We are grateful for their financial support.", "The authors would further like to thank Stian Skjong for fruitful discussions." ] ]
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